Physics Wiki - User contributions [en]

Main Page/PHYS 4210 & 4211

2022-05-26T15:47:05Z

Gloria:

<h1>PHYS 4210 & 4211 3.0 </h1>

<p>Selected advanced experiments in physics related to
topics in solid state physics, atomic spectroscopy, microwaves, low-noise measurements, superconductivity, and nuclear and particle physics.

Prerequisites: SC/PHYS 3220 3.00; registration in a Bachelor or Honours Program in physics and astronomy or in biophysics. Co-requisite: SC/PHYS 3040 6.00.</p>

<p> See the course [http://moodle.yorku.ca eClass] for all administrative details. </p>

<h1>Laboratory Manual</h1>

IMPORTANT NOTE: Winter 2021 PHYS 4211 Lab Scheduling procedures will be announced via the Moodle eClass course website http://moodle.yorku.ca/ prior to the beginning of term.



<ul>
<li> [[Main Page/PHYS 4210/How to Choose Experiments|How to Choose Experiments]] </li>
<li> [[Main Page/PHYS 4210/How to Write Reports|How to Write Reports]] </li>
<li> [[Main Page/PHYS 3220/Lab Safety|Lab Safety]] </li>
</ul>

<h2> List of Available Experiments </h2>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 150 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]]</td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Muon Lifetime|Muon Lifetime]]</td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Waveguides|Waveguides*]] </td><td> 111 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Mass Spectrometer|Mass Spectrometer]]</td><td> 108 PSE</td><td>Currently Not Available</td></tr>
<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 150 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/He-Ne Lasers|He-Ne Lasers*]]</td><td> 150 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Zeeman Effect|Zeeman Effect*]] </td><td> 108 PSE</td><td>Currently Not Available</td></tr>

<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Electron Spin Resonance|Electron Spin Resonance]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Coaxial Cable|Coaxial Cable]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/High-TC Superconductivity|High-TC Superconductivity*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Fourier Optics|Fourier Optics*]] </td><td> 126 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Rutherford II|Rutherford Scattering II]]</td><td> 111 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Gamma Ray Spectroscopy|Gamma Ray Spectroscopy*]] </td><td> 111 PSE</td><td></td></tr>








<p> Experiments with an asterisk (*) are better suited for PHYS 4211 students. </p>





















<p> </p>

Main Page/PHYS 3220

2022-05-26T15:46:04Z

Gloria:

<h1>PHYS 3220 3.0 Experiments in Modern Physics </h1>
<p>A selection of experiments in fluid mechanics,
electromagnetism, optics, and atomic, nuclear, and
particle physics. Analysis of the data and detailed
write-ups are required. One lecture hour which is
devoted to techniques of data analysis and three
laboratory hours per week.</p>

<h1>Laboratory Manual</h1>
<p>'''IMPORTANT NOTE: Fall 2020 PHYS 3220 Lab Scheduling procedures will be announced via the Moodle eClass course website http://eclass.yorku.ca/ prior to the beginning of term.'''</p>

<ul>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Cavendish|Measurement of the Gravitation Constant G: The Cavendish Experiment]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 150 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Thermionic|Thermionic Emission]]</td><td> 150 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Speed of Light|A Measurement of the Velocity of Light: The Foucault-Michelson Experiment]] </td><td> 209 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Millikan|Determination of the Electric Charge Unit ''e'' : The Millikan Oil Drop Experiment]] </td><td> 111 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 150 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Semiconductors I|Semiconductors I]] </td><td> 150 PSE </td><td></td></tr>

</table>
</ul>

Main Page/PHYS 3220

2022-05-26T15:20:48Z

Gloria:

<h1>PHYS 3220 3.0 Experiments in Modern Physics </h1>
<p>A selection of experiments in fluid mechanics,
electromagnetism, optics, and atomic, nuclear, and
particle physics. Analysis of the data and detailed
write-ups are required. One lecture hour which is
devoted to techniques of data analysis and three
laboratory hours per week.</p>

<h1>Laboratory Manual</h1>
<p>'''IMPORTANT NOTE: Fall 2020 PHYS 3220 Lab Scheduling procedures will be announced via the Moodle eClass course website http://eclass.yorku.ca/ prior to the beginning of term.'''</p>

<ul>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Cavendish|Measurement of the Gravitation Constant G: The Cavendish Experiment]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 150 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Thermionic|Thermionic Emission]]</td><td> 150 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 150 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Speed of Light|A Measurement of the Velocity of Light: The Foucault-Michelson Experiment]] </td><td> 209 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Millikan|Determination of the Electric Charge Unit ''e'' : The Millikan Oil Drop Experiment]] </td><td> 111 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 150 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Semiconductors I|Semiconductors I]] </td><td> 150 PSE </td><td></td></tr>

</table>
</ul>

Main Page/PHYS 4210 & 4211

2022-05-26T15:19:38Z

Gloria:

<h1>PHYS 4210 & 4211 3.0 </h1>

<p>Selected advanced experiments in physics related to
topics in solid state physics, atomic spectroscopy, microwaves, low-noise measurements, superconductivity, and nuclear and particle physics.

Prerequisites: SC/PHYS 3220 3.00; registration in a Bachelor or Honours Program in physics and astronomy or in biophysics. Co-requisite: SC/PHYS 3040 6.00.</p>

<p> See the course [http://moodle.yorku.ca eClass] for all administrative details. </p>

<h1>Laboratory Manual</h1>

IMPORTANT NOTE: Winter 2021 PHYS 4211 Lab Scheduling procedures will be announced via the Moodle eClass course website http://moodle.yorku.ca/ prior to the beginning of term.



<ul>
<li> [[Main Page/PHYS 4210/How to Choose Experiments|How to Choose Experiments]] </li>
<li> [[Main Page/PHYS 4210/How to Write Reports|How to Write Reports]] </li>
<li> [[Main Page/PHYS 3220/Lab Safety|Lab Safety]] </li>
</ul>

<h2> List of Available Experiments </h2>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 150 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]]</td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Muon Lifetime|Muon Lifetime]]</td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Waveguides|Waveguides*]] </td><td> 111 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Mass Spectrometer|Mass Spectrometer]]</td><td> 108 PSE</td><td>Currently Not Available</td></tr>
<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 150 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/He-Ne Lasers|He-Ne Lasers*]]</td><td> 150 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Zeeman Effect|Zeeman Effect*]] </td><td> 108 PSE</td><td>Currently Not Available</td></tr>

<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 150 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Electron Spin Resonance|Electron Spin Resonance]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Coaxial Cable|Coaxial Cable]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/High-TC Superconductivity|High-TC Superconductivity*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Fourier Optics|Fourier Optics*]] </td><td> 126 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Rutherford II|Rutherford Scattering II]]</td><td> 111 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Gamma Ray Spectroscopy|Gamma Ray Spectroscopy*]] </td><td> 111 PSE</td><td></td></tr>








<p> Experiments with an asterisk (*) are better suited for PHYS 4211 students. </p>





















<p> </p>

Main Page/PHYS 4210 & 4211

2022-05-26T14:51:03Z

Gloria:

<h1>PHYS 4210 & 4211 3.0 </h1>

<p>Selected advanced experiments in physics related to
topics in solid state physics, atomic spectroscopy, microwaves, low-noise measurements, superconductivity, and nuclear and particle physics.

Prerequisites: SC/PHYS 3220 3.00; registration in a Bachelor or Honours Program in physics and astronomy or in biophysics. Co-requisite: SC/PHYS 3040 6.00.</p>

<p> See the course [http://moodle.yorku.ca eClass] for all administrative details. </p>

<h1>Laboratory Manual</h1>

IMPORTANT NOTE: Winter 2021 PHYS 4211 Lab Scheduling procedures will be announced via the Moodle eClass course website http://moodle.yorku.ca/ prior to the beginning of term.



<ul>
<li> [[Main Page/PHYS 4210/How to Choose Experiments|How to Choose Experiments]] </li>
<li> [[Main Page/PHYS 4210/How to Write Reports|How to Write Reports]] </li>
<li> [[Main Page/PHYS 3220/Lab Safety|Lab Safety]] </li>
</ul>

<h2> List of Available Experiments </h2>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 126 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]]</td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Muon Lifetime|Muon Lifetime]]</td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Waveguides|Waveguides*]] </td><td> 111 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Mass Spectrometer|Mass Spectrometer]]</td><td> 108 PSE</td><td>Currently Not Available</td></tr>
<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 150 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/He-Ne Lasers|He-Ne Lasers*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Zeeman Effect|Zeeman Effect*]] </td><td> 108 PSE</td><td>Currently Not Available</td></tr>

<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Electron Spin Resonance|Electron Spin Resonance]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Coaxial Cable|Coaxial Cable]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/High-TC Superconductivity|High-TC Superconductivity*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Fourier Optics|Fourier Optics*]] </td><td> 126 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Rutherford II|Rutherford Scattering II]]</td><td> 111 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Gamma Ray Spectroscopy|Gamma Ray Spectroscopy*]] </td><td> 111 PSE</td><td></td></tr>








<p> Experiments with an asterisk (*) are better suited for PHYS 4211 students. </p>





















<p> </p>

Main Page/PHYS 4210 & 4211

2022-05-26T14:50:35Z

Gloria:

<h1>PHYS 4210 & 4211 3.0 </h1>

<p>Selected advanced experiments in physics related to
topics in solid state physics, atomic spectroscopy, microwaves, low-noise measurements, superconductivity, and nuclear and particle physics.

Prerequisites: SC/PHYS 3220 3.00; registration in a Bachelor or Honours Program in physics and astronomy or in biophysics. Co-requisite: SC/PHYS 3040 6.00.</p>

<p> See the course [http://moodle.yorku.ca eClass] for all administrative details. </p>

<h1>Laboratory Manual</h1>

IMPORTANT NOTE: Winter 2021 PHYS 4211 Lab Scheduling procedures will be announced via the Moodle eClass course website http://moodle.yorku.ca/ prior to the beginning of term.



<ul>
<li> [[Main Page/PHYS 4210/How to Choose Experiments|How to Choose Experiments]] </li>
<li> [[Main Page/PHYS 4210/How to Write Reports|How to Write Reports]] </li>
<li> [[Main Page/PHYS 3220/Lab Safety|Lab Safety]] </li>
</ul>

<h2> List of Available Experiments </h2>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 126 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]]</td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Muon Lifetime|Muon Lifetime]]</td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Waveguides|Waveguides*]] </td><td> 111 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Mass Spectrometer|Mass Spectrometer]]</td><td> 150 PSE</td><td>Currently Not Available</td></tr>
<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 150 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/He-Ne Lasers|He-Ne Lasers*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Zeeman Effect|Zeeman Effect*]] </td><td> 108 PSE</td><td>Currently Not Available</td></tr>

<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Electron Spin Resonance|Electron Spin Resonance]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Coaxial Cable|Coaxial Cable]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/High-TC Superconductivity|High-TC Superconductivity*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Fourier Optics|Fourier Optics*]] </td><td> 126 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Rutherford II|Rutherford Scattering II]]</td><td> 111 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Gamma Ray Spectroscopy|Gamma Ray Spectroscopy*]] </td><td> 111 PSE</td><td></td></tr>








<p> Experiments with an asterisk (*) are better suited for PHYS 4211 students. </p>





















<p> </p>

Main Page/PHYS 3220

2022-05-26T14:10:34Z

Gloria:

<h1>PHYS 3220 3.0 Experiments in Modern Physics </h1>
<p>A selection of experiments in fluid mechanics,
electromagnetism, optics, and atomic, nuclear, and
particle physics. Analysis of the data and detailed
write-ups are required. One lecture hour which is
devoted to techniques of data analysis and three
laboratory hours per week.</p>

<h1>Laboratory Manual</h1>
<p>'''IMPORTANT NOTE: Fall 2020 PHYS 3220 Lab Scheduling procedures will be announced via the Moodle eClass course website http://eclass.yorku.ca/ prior to the beginning of term.'''</p>

<ul>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Cavendish|Measurement of the Gravitation Constant G: The Cavendish Experiment]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 126 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Thermionic|Thermionic Emission]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Speed of Light|A Measurement of the Velocity of Light: The Foucault-Michelson Experiment]] </td><td> 209 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Millikan|Determination of the Electric Charge Unit ''e'' : The Millikan Oil Drop Experiment]] </td><td> 111 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 150 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Semiconductors I|Semiconductors I]] </td><td> 150 PSE </td><td></td></tr>

</table>
</ul>

Main Page/PHYS 3220

2022-05-26T14:07:12Z

Gloria:

<h1>PHYS 3220 3.0 Experiments in Modern Physics </h1>
<p>A selection of experiments in fluid mechanics,
electromagnetism, optics, and atomic, nuclear, and
particle physics. Analysis of the data and detailed
write-ups are required. One lecture hour which is
devoted to techniques of data analysis and three
laboratory hours per week.</p>

<h1>Laboratory Manual</h1>
<p>'''IMPORTANT NOTE: Fall 2020 PHYS 3220 Lab Scheduling procedures will be announced via the Moodle eClass course website http://eclass.yorku.ca/ prior to the beginning of term.'''</p>

<ul>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Cavendish|Measurement of the Gravitation Constant G: The Cavendish Experiment]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 126 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Thermionic|Thermionic Emission]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Speed of Light|A Measurement of the Velocity of Light: The Foucault-Michelson Experiment]] </td><td> 209 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Millikan|Determination of the Electric Charge Unit ''e'' : The Millikan Oil Drop Experiment]] </td><td> 111 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 150 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Semiconductors I|Semiconductors I]] </td><td> 150 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Coupled Motion|Coupled Oscillatory and Rotational Motion]] </td><td> 150 PSE </td><td></td></tr>

</table>
</ul>

Main Page/PHYS 3220

2022-05-26T14:02:28Z

Gloria:

<h1>PHYS 3220 3.0 Experiments in Modern Physics </h1>
<p>A selection of experiments in fluid mechanics,
electromagnetism, optics, and atomic, nuclear, and
particle physics. Analysis of the data and detailed
write-ups are required. One lecture hour which is
devoted to techniques of data analysis and three
laboratory hours per week.</p>

<h1>Laboratory Manual</h1>
<p>'''IMPORTANT NOTE: Fall 2020 PHYS 3220 Lab Scheduling procedures will be announced via the Moodle eClass course website http://eclass.yorku.ca/ prior to the beginning of term.'''</p>

<ul>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Cavendish|Measurement of the Gravitation Constant G: The Cavendish Experiment]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 126 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Thermionic|Thermionic Emission]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Speed of Light|A Measurement of the Velocity of Light: The Foucault-Michelson Experiment]] </td><td> 209 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Millikan|Determination of the Electric Charge Unit ''e'' : The Millikan Oil Drop Experiment]] </td><td> 111 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 150 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Semiconductors I|Semiconductors I]] </td><td> 150 PSE </td><td></td></tr>

</table>
</ul>

Main Page/PHYS 4210 & 4211

2021-12-23T17:00:18Z

Gloria:

<h1>PHYS 4210 & 4211 3.0 </h1>

<p>Selected advanced experiments in physics related to
topics in solid state physics, atomic spectroscopy, microwaves, low-noise measurements, superconductivity, and nuclear and particle physics.

Prerequisites: SC/PHYS 3220 3.00; registration in a Bachelor or Honours Program in physics and astronomy or in biophysics. Co-requisite: SC/PHYS 3040 6.00.</p>

<p> See the course [http://moodle.yorku.ca eClass] for all administrative details. </p>

<h1>Laboratory Manual</h1>

IMPORTANT NOTE: Winter 2021 PHYS 4211 Lab Scheduling procedures will be announced via the Moodle eClass course website http://moodle.yorku.ca/ prior to the beginning of term.



<ul>
<li> [[Main Page/PHYS 4210/How to Choose Experiments|How to Choose Experiments]] </li>
<li> [[Main Page/PHYS 4210/How to Write Reports|How to Write Reports]] </li>
<li> [[Main Page/PHYS 3220/Lab Safety|Lab Safety]] </li>
</ul>

<h2> List of Available Experiments </h2>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 126 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]]</td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Muon Lifetime|Muon Lifetime]]</td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Waveguides|Waveguides*]] </td><td> 111 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Mass Spectrometer|Mass Spectrometer]]</td><td> 150 PSE</td><td>Currently Not Available</td></tr>
<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 150 PSE</td><td>Currently Not Available</td></tr>
<tr><td>[[Main Page/PHYS 4210/He-Ne Lasers|He-Ne Lasers*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Zeeman Effect|Zeeman Effect*]] </td><td> 108 PSE</td><td>Currently Not Available</td></tr>

<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Electron Spin Resonance|Electron Spin Resonance]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Coaxial Cable|Coaxial Cable]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/High-TC Superconductivity|High-TC Superconductivity*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Fourier Optics|Fourier Optics*]] </td><td> 126 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Rutherford II|Rutherford Scattering II]]</td><td> 111 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Gamma Ray Spectroscopy|Gamma Ray Spectroscopy*]] </td><td> 111 PSE</td><td></td></tr>








<p> Experiments with an asterisk (*) are better suited for PHYS 4211 students. </p>





















<p> </p>

Main Page/PHYS 4210/Gamma Ray Spectroscopy

2021-12-23T16:58:03Z

Gloria:

<h1>Gamma Ray Spectroscopy</h1>
<p>In this experiment we study the gamma ray spectra of several radioactive elements to learn about the interaction properties of gamma rays with matter. The gamma rays are detected through ionization of the material in a scintillation counter, and the output pulse, which is generated with a photomultiplier tube, is then recorded with the aid of a Multi Channel Analyser (MCA) connected to a computer interface. The different types of interaction of gamma rays with matter are understood from a detailed analysis of the observed spectra. Thus, this experiment not only illustrates the physics of the interaction properties of photons, but also introduces scintillation detectors and relevant electronics.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Radioactive Decay</li>
<li>Gamma Rays</li>
<li>Multi-Channel Analyser (MCA)</li>
<li>Photo-Electric Effect</li>
<li>Compton Scattering</li>
<li>Pair Production</li>

</ul>
</td>
<td width=250>
<ul>
<li>Crystal Detector</li>
<li>Photomultiplier Tube</li>
<li>Back Scatter Peak</li>
<li>Compton Edge</li>
<li>Compton Plateau</li>
<li>Photopeak</li>
</ul>
</td>
</table>

<h1>Introduction</h1>
<p>Gamma rays are photons of very short wavelength (~10<sup>-12</sup> cm) or very high frequency (10<sup>22</sup> Hz) that are emitted during nuclear transitions. The decay schemes of the three radionuclides that we study (Na22, Cs137, Co60) are shown below. Read ref. 1-5 to get a clear idea of the significance of the gamma energies.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Decay schemes of Cs-137, Co-60, and Na-22.
<br clear=right>
</p>
</td></table>

<h2>Method</h2>
<p>Photons are electrically neutral, and unlike charged particles, do not experience the Coulomb force. They are, however, carriers of the electromagnetic force, and are able to ionize atoms through their interaction with matter, and this leads to the deposition of energy in the medium as the ionized particle slows down in traversing the medium. This energy can then be detected. The three modes of interaction are: photoelectric effect, Compton scattering, and pair production where the photon interacts with an atom, an electron, and a nucleus respectively. You should read the details of each type of interaction in the references and include this in your write-up. We summarise it briefly below.</p>

<h3>Photoelectric effect</h3>
<p>The photon of energy ''E<sub>g</sub>'' is absorbed by an atom and an electron from one of the shells is emitted. If ''B<sub>e</sub>'' is the binding energy of the electron, then the energy of the emitted electron will have an energy ''E<sub>e</sub>'' = ''E<sub>g</sub>'' - ''B<sub>e</sub>''. Since ''B<sub>e</sub>'' is small (of the order 40 KeV) compared to ''E<sub>g</sub>'' (of the order 1MeV), the electron carries most of the energy of the photon.</p>

<h3>Compton scattering</h3>
<p>The photon scatters off an electron that is either free or is loosely bound in the atom, thereby scattering the electron. It can be shown (do so in your writeup) that the energy of the scattered electron is related to that of the incident photon, the angle of the scattered photon (theta), and the mass of the electron me through the relationship:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Gam-eqn1.png|190px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig2.png|500px|border|center]]
<b>Figure 2 -</b> Compton scattering.
<br clear=right>
</p>
</td></table>
<p>Hence the scattered photon (of energy hv’) has the freedom to move at any angle with respect to the incident photon (or gamma ray, as we call it here), whereas the scattered electron is bound by the laws of conservation of momentum to only go in the forward direction. The kinetic energy of the Compton electron is E<sub>C</sub> = hv- hv’. This, combined with the above formula for the energy of the electron shows that the maximum kinetic energy of the electron is when theta = 180° ,i.e., when the photon is scattered backwards. This maximum is known as the Compton edge. </p>

<h3>Pair production</h3>
<p>If the photon has more than twice the rest mass of an electron (0.511 MeV), the photon can produce an electron-positron pair. This must be done in the Coulomb field of the nucleus to balance linear momentum, as the photon cannot produce a pair in free space (its center of mass would have zero momentum). The energy in excess of the rest masses of the products is imparted as kinetic energy. </p>
<p>The energy dependencies of the three processes are quite different. At energies below a few KeV, the photoelectric effect dominates and the Coulomb effect is small, while pair production is energetically impossible. For the region of 0.1 to about 10 MeV, Compton scattering dominates. Above this, pair production is the predominant method for the interaction of the photons. It is important to realize that in photoelectric effect and pair production, the photon is eliminated in the process of the interaction, whereas in Compton scattering, the energy of the photon is only degraded. As explained further below the various processes involved lead to a complete or incomplete energy deposition of the gamma energy in the scintillator. We have to be concerned only with the first two methods; the photoelectric effect leads to complete energy deposition, while the Compton effect can lead to processes where the scattered gamma ray leaves the detector without depositing its energy.</p>

<h2>The Radioactive Sources</h2>
<p>You will be working with five radioactive sources: Mn-54, Na-22, Cs-137, and Co-60. You should note the radiation dosage and date marked on the source, and read about radiation safety and how to handle radiation sources from ref. 1, pg 326-328. Calculate the number of disintegrations per second for each of the sources based on the quoted dosage.</p>

<h2>The Detector</h2>
<p>The gamma rays that are emitted from the source are detected with a scintillator detector. In our laboratory, we use inorganic crystals of sodium iodide doped with impurity centres of thallium. This combination is denoted by NaI(Tl). A photon entering the detector ionizes the material through the processes described above. The positive ions and electrons created by the incoming photon diffuse through the lattice and are captured by the impurity centres i.e. the Tl atoms, which act as luminescence centres. Recombination produces an excited centre,which emits visible light upon its return to the ground state. The efficiency of these inorganic crystals is high, but the light output is spread over a time interval of the order of microseconds. </p>
<p>In general, for gamma rays less than 1 MeV, photons undergoing the photoelectric process will be completely absorbed and will give rise to a determined number of light quanta. These light quanta are then seen by the photomultiplier tube, and the output pulse will be proportional to the incoming energy of the gamma ray. The energy of the electrons produced by the Compton effect will depend on the angle at which they are scattered and hence there will be a spectrum of detected pulses. At energies above 1 MeV, pair production can occur and the electron and positron lose their kinetic energy by ionizing the medium. The electron is absorbed, while the positron annihilates within a few nanoseconds into two photons, each of 0.511 MeV. These may then interact through the photoelectric or Compton effect. There are three possibilities for the observed energy (show this in the writeup) depending on whether none, one or both of these photons from the positron annihilation are detected.</p>
<p>As mentioned, the produced light is proportional to the energy of the incident gamma ray. This light is then made incident on the photocathode face of a photomultiplier tube (PMT), which converts the photons to photoelectrons through the photoelectric effect. The electrons are amplified and then converted to a voltage pulse. The PMT used here has 10 stages and the multiplication for such tubes is about 10<sup>6</sup> for an operational voltage between the cathode and the anode of 1000V. The cascade of electrons produced at each of the multiplication stages of the dynodes is collected at the anode, converted to a voltage pulse, which is then amplified and analysed. The PMT is enclosed in a high permeability material to shield against magnetic fields, as such effects would affect the efficiency of the tube.</p>
<p>Note that the metal shield that encloses the scintillator crystal does not allow β (or other charged) particles to permeate and deposit their usually well-defined energies. Thus, you should not expect to observe structures associated with, e.g., the 0.514 MeV and 1.17 MeV electrons coming from the Cs137 source with branching ratios of 93.5% and 6.5% respectively (cf.. Fig. 1). Charged particles are created inside the crystal by γ ray impact. There will be, however, backscattered γ rays from Compton scattering outside the crystal, e.g., off the backing of the source and off the lead shield.</p>
<p>Read the specifications for the model of the scintillator detector and photomultiplier that you have. Note that the size of the crystal is small. How does this affect your results? The typical quantum efficiency of scintillators of this type is one photon of light produced per 100 eV of energy deposited in the scintillator. The width of the full-energy peak (i.e. when the gamma energy is fully absorbed) depends on the number of light quanta produced by the incident gamma ray. The energy resolution, ''dE/E'' is an important quantity to consider as this factor will determine if we can separate gamma rays very close in energy.</p>
<p>The chain of events may help illustrate this point. The incident gamma ray of energy ''E<sub>g</sub>'' produces a photoelectron with energy ''E<sub>e</sub>'' ~ ''E<sub>g</sub>''. The photoelectron produces N light quanta in the scintillator material via ionization and excitation, each with an energy ''E<sub>lq</sub>''. Since the light is visible (~400 nm), ''E<sub>lq</sub>'' is about 3 eV. This light falls on the photocathode, which has a quantum efficiency or sensitivity to the wavelength of the incident light. For the tube supplied, the efficiency peaks near 400 nm, making it quite suitable for use with a scintillator. Let
<ul>
<li>ε(light) be the efficiency for the conversion of the excitation energy into light quanta</li>
<li>ε(coll) be the efficiency for collecting the n lq light quanta at the photocathode of the PMT</li>
<li>ε(cathode) be the efficiency for the cathode to eject a photoelectron</li>
</ul>
Then, the number of photoelectrons produced at the photocathode is
</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Gam-eqn1b.png|330px|center]]
</p>
</td></table>

<p>Typical values for the efficiencies for NaI detectors are:
<ul>
<li>ε(light) = 0.1</li>
<li>ε(coll) = 0.4</li>
<li>ε(cathode) = 0.2</li>
</ul>
</p>
<p>For a 1MeV gamma ray, this yields about 3x10<sup>3</sup> photoelectrons at the photocathode. All the processes mentioned above are subject to statistical fluctuations, and contribute to the broadening of the line width. In addition, there is a contribution from the statistical process due to the multiplication of the photoelectrons in the stages of the PMT.</p>

<h2>The Multichannel Analyser (MCA)</h2>
<p> The electrons collected at the anode of the PMT pass directly into the Spectrum Techniques MCA (UCS 30). This pulse of electrons enters the internal pre-amplifier, followed by the internal amplifier. The voltage across the internal amplifier resistance is digitized into one of 1024 bins according to its maximum value (peak height)- this process of converting an analog voltage into a digital value is called analog-to-digital conversion (ADC). The MCA performs this task for all pulses and creates a histogram of counts in each ADC bin.</p>
<p>There are several important parameters which affect the performance of the MCA. For a detailed explanation of the pulse processing details refer to the book by Knoll, "Radiation Detection and Measurements" listed in the references section below.
<ul>

<li><b>Amplifier Gain: </b> This is an amplification factor applied to the detector pulse using the adjustable coarse gain and fine gain controls.
<li><b>Lower Level Discriminator (LLD) :</b> The amount above the average background which is required for the MCA to accept a particular pulse and record its statistics in the histogram.</li>
<li><b>Run Time: </b> The actual length of time for which to acquire data.</li>

<li><b>Live Time:</b> The time the detector is actually able to detect pulses.
<li><b>Dead Time:</b> The time during collection when the detector is unable to process additional events. Look up live time and dead time and discuss in your report.</li>
<li><b>High Voltage: </b> The parameter "high voltage" in the MCA client software does nothing. Bridgeport makes an HV supply for PMTs which connect directly to the eMorpho. We are using a separate power supply, hence adjusting this parameter in the software does nothing. However, changing the 1000V supplied to the PMT from the power supply will mean that more electrons are collected per pulse. If this dc voltage is set too high and and too many photons are present, there could be a buildup of charge inside the PMT and catastrophic damage could occur.<i>Please do not change the setting of the voltage to the PMT from 1000V.</i></li>

</ul>
</p>

<p> Turn the power on the spectrum analyzer (UCS 30) and run the program "USX" located on the desktop. (The UCS 30 manual is located on the desktop and should be referred to for additional information.) Hover over the toolbar icons and note the descriptions. Click on each icon and note the options available. For example, you can toggle between a linear and logarithmic y-axis scale by clicking on the "y-log" icon on the toolbar. Additionally, during data collection you can adjust the scale of the y-axis by scrolling up and down when the mouse is in the graph region. Review the operation section (pg 16-25) of the UCS30 lab manual. </p>


<p>The signal from the PMT is directly input to a multichannel analyser which is a device that sorts incoming pulses according to pulse height and keeps count of the number at each height in a multichannel memory. The contents of each channel is displayed on a screen to give a pulse height spectrum, which is then analysed. The amplitude of the incoming pulse is digitized with an Analogue to Digital Converter (ADC), and sorting is done based on how many pulses had a particular value of the digitized amplitude. The total number of channels into which the voltage range is digitized determines the resolution of the MCA. Refer ref. 1, 2 and 4 to understand the characteristics of the electronics you are supplied with, and to learn more about the ADC range and resolution that can be achieved. Also, use ref. 1, 2, 4 to understand the functions of a MCA. Review what is meant by ‘dead time’ and ‘live time’. Do not operate with a dead time of more than 30%, since too high count rates can cause the electronics to misbehave.</p>

<p>The natural linewidth of the gamma rays is extremely narrow (a few eV compared to the MeV range of the energies themselves!). The broadening observed in the recorded spectra is a result of the detection method (a cooled Germanium detector would show these lines being much narrower, but again at a resolution that depends on the detector itself). It is important to realize on the example of the Cs137 spectrum, for which a single energy at 0.662 MeV is expected that several effects occur in ‘real life’:</p>
<ol>
<li>Broadening of the photopeak at 0.662 MeV.</li>
<li>A Compton background (plateau) with two edges: at the higher end, below the
photopeak a maximum electron(!) energy of
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Gam-eqn2.png|170px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
Understand this result using eq. (1). At lower energies in the Compton plateau a distinct photopeak corresponding to backscattered gamma rays whose energy follows from eq. (1) for θ = π. Note how nearly backscattered gammas give about the same energy due to the slow variation of the cos function for θ = π.</li>
<li>A fluorescent K-shell Xray peak caused by soft gamma rays hitting the shielding material (typically lead), knocking out K-shell electrons. The vacancies are typically filled from the L-III shell (2p±1). Use the CRC Handbook (see Lab Technologist or Library) to verify that the difference in energy results in characteristic X rays of about 75 keV for Pb.</li>
</ol>
<p>A typical spectrum is shown in Fig. 2 on a linear scale. If we go to a logarithmic scale (the MCA software allows to do this), we find an additional weak peak at about twice the gamma energy. This is the so-called sum peak, which arises when two gamma rays from uncorrelated decay events are depositing their energy in the scintillator within a fraction of a microsecond, i.e., the timescale over which the scintillation in the crystal and photomultiplication in the tube occurs. You should be able to observe sum peaks in this experiment when accumulating enough statistics. The peaks appear more strongly when the source is brought closer to the PMT. </p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig3.png|500px|border|center]]
<b>Figure 3 -</b> Typical Cs-137 spectrum using an NaI(T1) scintillator.
<br clear=right>
</p>
</td></table>
<p>Besides sum peaks another complication can occur for high-energy gamma rays. When pair creation is an important energy deposition mechanism, so-called escape peaks are observed. These correspond to events where one or more of the created electrons/positrons escape the crystal without giving up their energy. Such peaks occur at ''E<sub>g</sub>'' - ''j'' (0.511 MeV), where ''j'' is the number of escaped electrons/positrons. Since we do not use sources with ''E<sub>g</sub>'' > 2 MeV in this experiment, we do not find this complication in our spectra.</p>

<h2>Experimental Procedure</h2>
<p>While conducting the experiment, make sure that only the source whose spectrum you are observing is near the apparatus otherwise your calibration results will be skewed. Take only one source out at a time, and keep the others in the box away from the detector. Be careful while handling sources. Become familiar with the documentation and all the apparatus before starting. </p>

<p>Ensure the following connections: the scintillator and PMT are connected, the HV output from the PMT is connected to the USC 30, the signal output from the PMT is connected to the MCA, and the MCA is connected to the computer.</p>
<p>In our setup, we have mounted the PMT vertically, allowing you to place the source on a tray at several different distances from the source. Once you have found an optimal distance for all three sources (i.e. you do not get ‘pile up’ effects (see Leo)), it is best to use this distance for calibration and determination of the peak energies. However, in the last part of the experiment you will be asked to draw qualitative conclusions by placing a source at varying distances. </p>

<h1>Settings</h1>
<p>Please ensure the following running parameters</p>
<ul>
<li>High-Voltage to PMT: +1000V (Click on the "Amp/HV/ADC" icon on the toolbar to set and turn on the voltage.)</li>
<li>Amplifier Gain: Use the course gain and fine gain controls to set the gain in the range of 16 - 18. </li>
<li>Conversion Gain: 1024</li>
<li>Peak Time: 1 micro sec</li>
</ul>


<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig4.png|700px|border|center]]
<b>Figure 4 -</b> Gamma source, NaI(T1) detector, PMT, lead shield showing relevant processes.
<br clear=right>
</p>
</td></table>
<ol>
<h3>Calibration Curve</h3>
<li><p>Use Experiment 2.1 as a guide for gamma ray calibration from Duggan in the red binder in the lab. The UCS 30 software will only be used for data collection and observing the spectra. All data analysis, including energy calibration, can be conducted using software like Excel or Mathematica. Use the 4 known sources Cs-137 (0.662 MeV), Co-60 (1.173 MeV, 1.332 MeV), Mn-54 (0.835 MeV) and Na-22 (0.511 MeV) provided in the RSS 8 radiation source kit (Spectrum Techniques). Centre each source (label down) about 5 cm from the face of the detector and collect a spectrum for at least 5 minutes with each source. Longer collection times correspond to smaller statistical errors, why? <b>Stop collection prior to saving and record the dead time, live time and real time for each spectrum.</b> Save the spectrum. (The .csv file generated contains a single column with the UCS 30 parameters in the first 18 rows followed by the spectrum data (number of events) in each channel from 0 to 1023. Create a calibration curve using all five sources i.e. for each spectrum locate the channel number corresponding to the gamma ray energy peaks and create a plot of the energy of each peak and the peak channel number. What is the error of a straight line fit to this curve? If the fit is poor, you have made a mistake in assigning peak-channel values, and may have to repeat the analysis. Discuss sources of error. </p>

<p>Using your calibration fit plot the calibrated Cs-137 spectrum, Co-60 spectrum and Na-22 spectrum separately(i.e. events as a function of energy). Explain all the peaks on the spectra. Do you see any 'sum peaks’? (Plot your data on a logarithmic scale for a better view!) Explain why this peak occurs. Does it occur at the value you expect? What is the source of the 511 keV peak in the Na-22 spectrum? The Na-22 has a gamma ray energy at 1275 keV. What value of the maximum position do you find based on your calibration curve? Is this consistent with the resolution ''dE/E''? What is the present activity of these three sources (note the date marked on the source)? Present the calibrated spectra in your report and explain all the interesting details.</p> </li>

<h3>Gamma-ray Experiments</h3>
<li><p> Put a sheet of lead on the shelf and then place the Co-60 source above. How does the spectrum change? Now place the sheet of lead on top of the source. What, if anything, has changed in the spectrum? You may have to move the source closer or farther from the detector to observe the backscattering signals. Collect a spectrum for each case for a sufficient collection time and save. <b> Caution: Always wear gloves when handling the lead sheets.</b></p> </li>

<li> Obtain the energy resolution for the photopeak for Cs-137 and for one of the Co-60 peaks (See Knoll, chapter 4). Does this agree with what you have learned about scintillator devices? </li>
<li> Place the Cs-137 source at varying distances from the detector and explain qualitatively the rates that you observe. Put the source at slot 2 below the detector, and place first one, then two, and then three sheets of lead on top of the source. Comment on the rates, and the reason for the observed behaviour. Repeat for sheets of aluminum.</li>
<li> A spectrum of the unknown source (the background in the room) will be collected by the Lab Technologist. The data will be saved and the file can be found in the "data" folder located on the desktop. Using this data, create a calibrated spectrum using your calibration fit. Use your now-calibrated gamma-ray spectrum to identify the radioisotope(s) found in the background.</li>
<li> After all data has been collected, turn the HV off from the software controls and then turn off the UCS 30 power switch.
</ol>

<h1>References</h1>
<ol>
<li>Preston and Dietz,[https://www.library.yorku.ca/find/Record/1038893 ''The Art of Experimental Physics''], Wiley.</li>
<li>G.F. Knoll, [https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/mc13rm/alma991008455689705164 ''Radiation Detection and Measurement''], Wiley. </li>
<li>J.L. Duggan, ''Laboratory Investigations in Nuclear Science'', Tennelec.</li>
<li>A.C. Melissinos, [https://www.library.yorku.ca/find/Record/1641963 ''Experiments in Modern Physics''], Academic Press.</li>
<li>Leo, [https://www.library.yorku.ca/find/Record/1178503 ''Techniques for Nuclear and Particle Physics Experiments''], Springer-Verlag.</li>
<li>H. Frauenfelder and Henley,[https://www.library.yorku.ca/find/Record/51764 ''Subatomic Physics''] ,Prentice-Hall.</li>
<li>K. Siegbahn, [https://www.library.yorku.ca/find/Record/51739 ''Alpha, Beta, and Gamma-Ray Spectroscopy''], vol. I, chpts 5,8a.</li>
</ol>

Main Page/PHYS 4210/Gamma Ray Spectroscopy

2021-12-23T16:54:28Z

Gloria:

<h1>Gamma Ray Spectroscopy</h1>
<p>In this experiment we study the gamma ray spectra of several radioactive elements to learn about the interaction properties of gamma rays with matter. The gamma rays are detected through ionization of the material in a scintillation counter, and the output pulse, which is generated with a photomultiplier tube, is then recorded with the aid of a Multi Channel Analyser (MCA) connected to a computer interface. The different types of interaction of gamma rays with matter are understood from a detailed analysis of the observed spectra. Thus, this experiment not only illustrates the physics of the interaction properties of photons, but also introduces scintillation detectors and relevant electronics.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Radioactive Decay</li>
<li>Gamma Rays</li>
<li>Multi-Channel Analyser (MCA)</li>
<li>Photo-Electric Effect</li>
<li>Compton Scattering</li>
<li>Pair Production</li>

</ul>
</td>
<td width=250>
<ul>
<li>Crystal Detector</li>
<li>Photomultiplier Tube</li>
<li>Back Scatter Peak</li>
<li>Compton Edge</li>
<li>Compton Plateau</li>
<li>Photopeak</li>
</ul>
</td>
</table>

<h1>Introduction</h1>
<p>Gamma rays are photons of very short wavelength (~10<sup>-12</sup> cm) or very high frequency (10<sup>22</sup> Hz) that are emitted during nuclear transitions. The decay schemes of the three radionuclides that we study (Na22, Cs137, Co60) are shown below. Read ref. 1-5 to get a clear idea of the significance of the gamma energies.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Decay schemes of Cs-137, Co-60, and Na-22.
<br clear=right>
</p>
</td></table>

<h2>Method</h2>
<p>Photons are electrically neutral, and unlike charged particles, do not experience the Coulomb force. They are, however, carriers of the electromagnetic force, and are able to ionize atoms through their interaction with matter, and this leads to the deposition of energy in the medium as the ionized particle slows down in traversing the medium. This energy can then be detected. The three modes of interaction are: photoelectric effect, Compton scattering, and pair production where the photon interacts with an atom, an electron, and a nucleus respectively. You should read the details of each type of interaction in the references and include this in your write-up. We summarise it briefly below.</p>

<h3>Photoelectric effect</h3>
<p>The photon of energy ''E<sub>g</sub>'' is absorbed by an atom and an electron from one of the shells is emitted. If ''B<sub>e</sub>'' is the binding energy of the electron, then the energy of the emitted electron will have an energy ''E<sub>e</sub>'' = ''E<sub>g</sub>'' - ''B<sub>e</sub>''. Since ''B<sub>e</sub>'' is small (of the order 40 KeV) compared to ''E<sub>g</sub>'' (of the order 1MeV), the electron carries most of the energy of the photon.</p>

<h3>Compton scattering</h3>
<p>The photon scatters off an electron that is either free or is loosely bound in the atom, thereby scattering the electron. It can be shown (do so in your writeup) that the energy of the scattered electron is related to that of the incident photon, the angle of the scattered photon (theta), and the mass of the electron me through the relationship:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Gam-eqn1.png|190px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig2.png|500px|border|center]]
<b>Figure 2 -</b> Compton scattering.
<br clear=right>
</p>
</td></table>
<p>Hence the scattered photon (of energy hv’) has the freedom to move at any angle with respect to the incident photon (or gamma ray, as we call it here), whereas the scattered electron is bound by the laws of conservation of momentum to only go in the forward direction. The kinetic energy of the Compton electron is E<sub>C</sub> = hv- hv’. This, combined with the above formula for the energy of the electron shows that the maximum kinetic energy of the electron is when theta = 180° ,i.e., when the photon is scattered backwards. This maximum is known as the Compton edge. </p>

<h3>Pair production</h3>
<p>If the photon has more than twice the rest mass of an electron (0.511 MeV), the photon can produce an electron-positron pair. This must be done in the Coulomb field of the nucleus to balance linear momentum, as the photon cannot produce a pair in free space (its center of mass would have zero momentum). The energy in excess of the rest masses of the products is imparted as kinetic energy. </p>
<p>The energy dependencies of the three processes are quite different. At energies below a few KeV, the photoelectric effect dominates and the Coulomb effect is small, while pair production is energetically impossible. For the region of 0.1 to about 10 MeV, Compton scattering dominates. Above this, pair production is the predominant method for the interaction of the photons. It is important to realize that in photoelectric effect and pair production, the photon is eliminated in the process of the interaction, whereas in Compton scattering, the energy of the photon is only degraded. As explained further below the various processes involved lead to a complete or incomplete energy deposition of the gamma energy in the scintillator. We have to be concerned only with the first two methods; the photoelectric effect leads to complete energy deposition, while the Compton effect can lead to processes where the scattered gamma ray leaves the detector without depositing its energy.</p>

<h2>The Radioactive Sources</h2>
<p>You will be working with five radioactive sources: Mn-54, Na-22, Cs-137, and Co-60. You should note the radiation dosage and date marked on the source, and read about radiation safety and how to handle radiation sources from ref. 1, pg 326-328. Calculate the number of disintegrations per second for each of the sources based on the quoted dosage.</p>

<h2>The Detector</h2>
<p>The gamma rays that are emitted from the source are detected with a scintillator detector. In our laboratory, we use inorganic crystals of sodium iodide doped with impurity centres of thallium. This combination is denoted by NaI(Tl). A photon entering the detector ionizes the material through the processes described above. The positive ions and electrons created by the incoming photon diffuse through the lattice and are captured by the impurity centres i.e. the Tl atoms, which act as luminescence centres. Recombination produces an excited centre,which emits visible light upon its return to the ground state. The efficiency of these inorganic crystals is high, but the light output is spread over a time interval of the order of microseconds. </p>
<p>In general, for gamma rays less than 1 MeV, photons undergoing the photoelectric process will be completely absorbed and will give rise to a determined number of light quanta. These light quanta are then seen by the photomultiplier tube, and the output pulse will be proportional to the incoming energy of the gamma ray. The energy of the electrons produced by the Compton effect will depend on the angle at which they are scattered and hence there will be a spectrum of detected pulses. At energies above 1 MeV, pair production can occur and the electron and positron lose their kinetic energy by ionizing the medium. The electron is absorbed, while the positron annihilates within a few nanoseconds into two photons, each of 0.511 MeV. These may then interact through the photoelectric or Compton effect. There are three possibilities for the observed energy (show this in the writeup) depending on whether none, one or both of these photons from the positron annihilation are detected.</p>
<p>As mentioned, the produced light is proportional to the energy of the incident gamma ray. This light is then made incident on the photocathode face of a photomultiplier tube (PMT), which converts the photons to photoelectrons through the photoelectric effect. The electrons are amplified and then converted to a voltage pulse. The PMT used here has 10 stages and the multiplication for such tubes is about 10<sup>6</sup> for an operational voltage between the cathode and the anode of 1000V. The cascade of electrons produced at each of the multiplication stages of the dynodes is collected at the anode, converted to a voltage pulse, which is then amplified and analysed. The PMT is enclosed in a high permeability material to shield against magnetic fields, as such effects would affect the efficiency of the tube.</p>
<p>Note that the metal shield that encloses the scintillator crystal does not allow β (or other charged) particles to permeate and deposit their usually well-defined energies. Thus, you should not expect to observe structures associated with, e.g., the 0.514 MeV and 1.17 MeV electrons coming from the Cs137 source with branching ratios of 93.5% and 6.5% respectively (cf.. Fig. 1). Charged particles are created inside the crystal by γ ray impact. There will be, however, backscattered γ rays from Compton scattering outside the crystal, e.g., off the backing of the source and off the lead shield.</p>
<p>Read the specifications for the model of the scintillator detector and photomultiplier that you have. Note that the size of the crystal is small. How does this affect your results? The typical quantum efficiency of scintillators of this type is one photon of light produced per 100 eV of energy deposited in the scintillator. The width of the full-energy peak (i.e. when the gamma energy is fully absorbed) depends on the number of light quanta produced by the incident gamma ray. The energy resolution, ''dE/E'' is an important quantity to consider as this factor will determine if we can separate gamma rays very close in energy.</p>
<p>The chain of events may help illustrate this point. The incident gamma ray of energy ''E<sub>g</sub>'' produces a photoelectron with energy ''E<sub>e</sub>'' ~ ''E<sub>g</sub>''. The photoelectron produces N light quanta in the scintillator material via ionization and excitation, each with an energy ''E<sub>lq</sub>''. Since the light is visible (~400 nm), ''E<sub>lq</sub>'' is about 3 eV. This light falls on the photocathode, which has a quantum efficiency or sensitivity to the wavelength of the incident light. For the tube supplied, the efficiency peaks near 400 nm, making it quite suitable for use with a scintillator. Let
<ul>
<li>ε(light) be the efficiency for the conversion of the excitation energy into light quanta</li>
<li>ε(coll) be the efficiency for collecting the n lq light quanta at the photocathode of the PMT</li>
<li>ε(cathode) be the efficiency for the cathode to eject a photoelectron</li>
</ul>
Then, the number of photoelectrons produced at the photocathode is
</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Gam-eqn1b.png|330px|center]]
</p>
</td></table>

<p>Typical values for the efficiencies for NaI detectors are:
<ul>
<li>ε(light) = 0.1</li>
<li>ε(coll) = 0.4</li>
<li>ε(cathode) = 0.2</li>
</ul>
</p>
<p>For a 1MeV gamma ray, this yields about 3x10<sup>3</sup> photoelectrons at the photocathode. All the processes mentioned above are subject to statistical fluctuations, and contribute to the broadening of the line width. In addition, there is a contribution from the statistical process due to the multiplication of the photoelectrons in the stages of the PMT.</p>

<h2>The Multichannel Analyser (MCA)</h2>
<p> The electrons collected at the anode of the PMT pass directly into the Spectrum Techniques MCA (UCS 30). This pulse of electrons enters the internal pre-amplifier, followed by the internal amplifier. The voltage across the internal amplifier resistance is digitized into one of 1024 bins according to its maximum value (peak height)- this process of converting an analog voltage into a digital value is called analog-to-digital conversion (ADC). The MCA performs this task for all pulses and creates a histogram of counts in each ADC bin.</p>
<p>There are several important parameters which affect the performance of the MCA. For a detailed explanation of the pulse processing details refer to the book by Knoll, "Radiation Detection and Measurements" listed in the references section below.
<ul>

<li><b>Amplifier Gain: </b> This is an amplification factor applied to the detector pulse using the adjustable coarse gain and fine gain controls.
<li><b>Lower Level Discriminator (LLD) :</b> The amount above the average background which is required for the MCA to accept a particular pulse and record its statistics in the histogram.</li>
<li><b>Run Time: </b> The actual length of time for which to acquire data.</li>

<li><b>Live Time:</b> The time the detector is actually able to detect pulses.
<li><b>Dead Time:</b> The time during collection when the detector is unable to process additional events. Look up live time and dead time and discuss in your report.</li>
<li><b>High Voltage: </b> The parameter "high voltage" in the MCA client software does nothing. Bridgeport makes an HV supply for PMTs which connect directly to the eMorpho. We are using a separate power supply, hence adjusting this parameter in the software does nothing. However, changing the 1000V supplied to the PMT from the power supply will mean that more electrons are collected per pulse. If this dc voltage is set too high and and too many photons are present, there could be a buildup of charge inside the PMT and catastrophic damage could occur.<i>Please do not change the setting of the voltage to the PMT from 1000V.</i></li>

</ul>
</p>

<p> Turn the power on the spectrum analyzer (UCS 30) and run the program "USX" located on the desktop. (The UCS 30 manual is located on the desktop and should be referred to for additional information.) Hover over the toolbar icons and note the descriptions. Click on each icon and note the options available. For example, you can toggle between a linear and logarithmic y-axis scale by clicking on the "y-log" icon on the toolbar. Additionally, during data collection you can adjust the scale of the y-axis by scrolling up and down when the mouse is in the graph region. Review the operation section (pg 16-25) of the UCS30 lab manual. </p>


<p>The signal from the PMT is directly input to a multichannel analyser which is a device that sorts incoming pulses according to pulse height and keeps count of the number at each height in a multichannel memory. The contents of each channel is displayed on a screen to give a pulse height spectrum, which is then analysed. The amplitude of the incoming pulse is digitized with an Analogue to Digital Converter (ADC), and sorting is done based on how many pulses had a particular value of the digitized amplitude. The total number of channels into which the voltage range is digitized determines the resolution of the MCA. Refer ref. 1, 2 and 4 to understand the characteristics of the electronics you are supplied with, and to learn more about the ADC range and resolution that can be achieved. Also, use ref. 1, 2, 4 to understand the functions of a MCA. Review what is meant by ‘dead time’ and ‘live time’. Do not operate with a dead time of more than 30%, since too high count rates can cause the electronics to misbehave.</p>

<p>The natural linewidth of the gamma rays is extremely narrow (a few eV compared to the MeV range of the energies themselves!). The broadening observed in the recorded spectra is a result of the detection method (a cooled Germanium detector would show these lines being much narrower, but again at a resolution that depends on the detector itself). It is important to realize on the example of the Cs137 spectrum, for which a single energy at 0.662 MeV is expected that several effects occur in ‘real life’:</p>
<ol>
<li>Broadening of the photopeak at 0.662 MeV.</li>
<li>A Compton background (plateau) with two edges: at the higher end, below the
photopeak a maximum electron(!) energy of
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Gam-eqn2.png|170px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
Understand this result using eq. (1). At lower energies in the Compton plateau a distinct photopeak corresponding to backscattered gamma rays whose energy follows from eq. (1) for θ = π. Note how nearly backscattered gammas give about the same energy due to the slow variation of the cos function for θ = π.</li>
<li>A fluorescent K-shell Xray peak caused by soft gamma rays hitting the shielding material (typically lead), knocking out K-shell electrons. The vacancies are typically filled from the L-III shell (2p±1). Use the CRC Handbook (see Lab Technologist or Library) to verify that the difference in energy results in characteristic X rays of about 75 keV for Pb.</li>
</ol>
<p>A typical spectrum is shown in Fig. 2 on a linear scale. If we go to a logarithmic scale (the MCA software allows to do this), we find an additional weak peak at about twice the gamma energy. This is the so-called sum peak, which arises when two gamma rays from uncorrelated decay events are depositing their energy in the scintillator within a fraction of a microsecond, i.e., the timescale over which the scintillation in the crystal and photomultiplication in the tube occurs. You should be able to observe sum peaks in this experiment when accumulating enough statistics. The peaks appear more strongly when the source is brought closer to the PMT. </p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig3.png|500px|border|center]]
<b>Figure 3 -</b> Typical Cs-137 spectrum using an NaI(T1) scintillator.
<br clear=right>
</p>
</td></table>
<p>Besides sum peaks another complication can occur for high-energy gamma rays. When pair creation is an important energy deposition mechanism, so-called escape peaks are observed. These correspond to events where one or more of the created electrons/positrons escape the crystal without giving up their energy. Such peaks occur at ''E<sub>g</sub>'' - ''j'' (0.511 MeV), where ''j'' is the number of escaped electrons/positrons. Since we do not use sources with ''E<sub>g</sub>'' > 2 MeV in this experiment, we do not find this complication in our spectra.</p>

<h2>Experimental Procedure</h2>
<p>While conducting the experiment, make sure that only the source whose spectrum you are observing is near the apparatus otherwise your calibration results will be skewed. Take only one source out at a time, and keep the others in the box away from the detector. Be careful while handling sources. Become familiar with the documentation and all the apparatus before starting. </p>

<p>Ensure the following connections: the scintillator and PMT are connected, the HV output from the PMT is connected to the USC 30, the signal output from the PMT is connected to the MCA, and the MCA is connected to the computer.</p>
<p>In our setup, we have mounted the PMT vertically, allowing you to place the source on a tray at several different distances from the source. Once you have found an optimal distance for all three sources (i.e. you do not get ‘pile up’ effects (see Leo)), it is best to use this distance for calibration and determination of the peak energies. However, in the last part of the experiment you will be asked to draw qualitative conclusions by placing a source at varying distances. </p>

<h1>Settings</h1>
<p>Please ensure the following running parameters</p>
<ul>
<li>High-Voltage to PMT: +1000V (Click on the "Amp/HV/ADC" icon on the toolbar to set and turn on the voltage.)</li>
<li>Amplifier Gain: Use the course gain and fine gain controls to set the gain in the range of 16 - 18. </li>
<li>Conversion Gain: 1024</li>
<li>Peak Time: 1 micro sec</li>
</ul>


<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig4.png|700px|border|center]]
<b>Figure 4 -</b> Gamma source, NaI(T1) detector, PMT, lead shield showing relevant processes.
<br clear=right>
</p>
</td></table>
<ol>
<h3>Calibration Curve</h3>
<li><p>Use Experiment 2.1 as a guide for gamma ray calibration from Duggan in the red binder in the lab. The UCS 30 software will only be used for data collection and observing the spectra. All data analysis, including energy calibration, can be conducted using software like Excel or Mathematica. Use the 4 known sources Cs-137 (0.662 MeV), Co-60 (1.173 MeV, 1.332 MeV), Mn-54 (0.835 MeV) and Na-22 (0.511 MeV) provided in the RSS 8 radiation source kit (Spectrum Techniques). Centre each source (label down) about 5 cm from the face of the detector and collect a spectrum for at least 5 minutes with each source. Longer collection times correspond to smaller statistical errors, why? <b>Stop collection prior to saving and record the dead time, live time and real time for each spectrum.</b> Save the spectrum. (The .csv file generated contains a single column with the UCS 30 parameters in the first 18 rows followed by the spectrum data (number of events) in each channel from 0 to 1023. Create a calibration curve using all five sources i.e. for each spectrum locate the channel number corresponding to the gamma ray energy peaks and create a plot of the energy of each peak and the peak channel number. What is the error of a straight line fit to this curve? If the fit is poor, you have made a mistake in assigning peak-channel values, and may have to repeat the analysis. Discuss sources of error. </p>

<p>Using your calibration fit plot the calibrated Cs-137 spectrum, Co-60 spectrum and Na-22 spectrum separately(i.e. events as a function of energy). Explain all the peaks on the spectra. Do you see any 'sum peaks’? (Plot your data on a logarithmic scale for a better view!) Explain why this peak occurs. Does it occur at the value you expect? What is the source of the 511 keV peak in the Na-22 spectrum? The Na-22 has a gamma ray energy at 1275 keV. What value of the maximum position do you find based on your calibration curve? Is this consistent with the resolution ''dE/E''? What is the present activity of these three sources (note the date marked on the source)? Present the calibrated spectra in your report and explain all the interesting details.</p> </li>

<h3>Gamma-ray Experiments</h3>
<li><p> Put a sheet of lead on the shelf and then place the Co-60 source above. How does the spectrum change? Now place the sheet of lead on top of the source. What, if anything, has changed in the spectrum? You may have to move the source closer or farther from the detector to observe the backscattering signals. Collect a spectrum for each case for a sufficient collection time and save. <b> Caution: Always wear gloves when handling the lead sheets.</b></p> </li>

<li> Obtain the energy resolution for the photopeak for Cs-137 and for one of the Co-60 peaks (See Knoll, chapter 4). Does this agree with what you have learned about scintillator devices? </li>
<li> Place the Cs-137 source at varying distances from the detector and explain qualitatively the rates that you observe. Put the source at slot 2 below the detector, and place first one, then two, and then three sheets of lead on top of the source. Comment on the rates, and the reason for the observed behaviour. Repeat for sheets of aluminum.</li>
<li> A spectrum of the unknown source (the background in the room) will be collected by the Lab Technologist. The data will be saved and the file can be found in the "data" folder located on the desktop. Using this data, create a calibrated spectrum using your calibration fit. Use your now-calibrated gamma-ray spectrum to identify the radioisotope(s) found in the background.</li>
<li> After all data has been collected, turn the HV off from the software controls and then turn off the UCS 30 power switch.
</ol>

<h1>References</h1>
<ol>
<li>Preston and Dietz,[https://www.library.yorku.ca/find/Record/1038893 ''The Art of Experimental Physics''], Wiley.</li>
<li>G.F. Knoll, ''Radiation Detection and Measurement, 3rd or 4th ed.'', Wiley. </li>
<li>J.L. Duggan, ''Laboratory Investigations in Nuclear Science'', Tennelec.</li>
<li>A.C. Melissinos, [https://www.library.yorku.ca/find/Record/1641963 ''Experiments in Modern Physics''], Academic Press.</li>
<li>Leo, [https://www.library.yorku.ca/find/Record/1178503 ''Techniques for Nuclear and Particle Physics Experiments''], Springer-Verlag.</li>
<li>H. Frauenfelder and Henley,[https://www.library.yorku.ca/find/Record/51764 ''Subatomic Physics''] ,Prentice-Hall.</li>
<li>K. Siegbahn, [https://www.library.yorku.ca/find/Record/51739 ''Alpha, Beta, and Gamma-Ray Spectroscopy''], vol. I, chpts 5,8a.</li>
</ol>

Main Page/PHYS 4210/Gamma Ray Spectroscopy

2021-12-23T16:46:55Z

Gloria:

<h1>Gamma Ray Spectroscopy</h1>
<p>In this experiment we study the gamma ray spectra of several radioactive elements to learn about the interaction properties of gamma rays with matter. The gamma rays are detected through ionization of the material in a scintillation counter, and the output pulse, which is generated with a photomultiplier tube, is then recorded with the aid of a Multi Channel Analyser (MCA) connected to a computer interface. The different types of interaction of gamma rays with matter are understood from a detailed analysis of the observed spectra. Thus, this experiment not only illustrates the physics of the interaction properties of photons, but also introduces scintillation detectors and relevant electronics.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Radioactive Decay</li>
<li>Gamma Rays</li>
<li>Multi-Channel Analyser (MCA)</li>
<li>Photo-Electric Effect</li>
<li>Compton Scattering</li>
<li>Pair Production</li>

</ul>
</td>
<td width=250>
<ul>
<li>Crystal Detector</li>
<li>Photomultiplier Tube</li>
<li>Back Scatter Peak</li>
<li>Compton Edge</li>
<li>Compton Plateau</li>
<li>Photopeak</li>
</ul>
</td>
</table>

<h1>Introduction</h1>
<p>Gamma rays are photons of very short wavelength (~10<sup>-12</sup> cm) or very high frequency (10<sup>22</sup> Hz) that are emitted during nuclear transitions. The decay schemes of the three radionuclides that we study (Na22, Cs137, Co60) are shown below. Read ref. 1-5 to get a clear idea of the significance of the gamma energies.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Decay schemes of Cs-137, Co-60, and Na-22.
<br clear=right>
</p>
</td></table>

<h2>Method</h2>
<p>Photons are electrically neutral, and unlike charged particles, do not experience the Coulomb force. They are, however, carriers of the electromagnetic force, and are able to ionize atoms through their interaction with matter, and this leads to the deposition of energy in the medium as the ionized particle slows down in traversing the medium. This energy can then be detected. The three modes of interaction are: photoelectric effect, Compton scattering, and pair production where the photon interacts with an atom, an electron, and a nucleus respectively. You should read the details of each type of interaction in the references and include this in your write-up. We summarise it briefly below.</p>

<h3>Photoelectric effect</h3>
<p>The photon of energy ''E<sub>g</sub>'' is absorbed by an atom and an electron from one of the shells is emitted. If ''B<sub>e</sub>'' is the binding energy of the electron, then the energy of the emitted electron will have an energy ''E<sub>e</sub>'' = ''E<sub>g</sub>'' - ''B<sub>e</sub>''. Since ''B<sub>e</sub>'' is small (of the order 40 KeV) compared to ''E<sub>g</sub>'' (of the order 1MeV), the electron carries most of the energy of the photon.</p>

<h3>Compton scattering</h3>
<p>The photon scatters off an electron that is either free or is loosely bound in the atom, thereby scattering the electron. It can be shown (do so in your writeup) that the energy of the scattered electron is related to that of the incident photon, the angle of the scattered photon (theta), and the mass of the electron me through the relationship:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Gam-eqn1.png|190px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig2.png|500px|border|center]]
<b>Figure 2 -</b> Compton scattering.
<br clear=right>
</p>
</td></table>
<p>Hence the scattered photon (of energy hv’) has the freedom to move at any angle with respect to the incident photon (or gamma ray, as we call it here), whereas the scattered electron is bound by the laws of conservation of momentum to only go in the forward direction. The kinetic energy of the Compton electron is E<sub>C</sub> = hv- hv’. This, combined with the above formula for the energy of the electron shows that the maximum kinetic energy of the electron is when theta = 180° ,i.e., when the photon is scattered backwards. This maximum is known as the Compton edge. </p>

<h3>Pair production</h3>
<p>If the photon has more than twice the rest mass of an electron (0.511 MeV), the photon can produce an electron-positron pair. This must be done in the Coulomb field of the nucleus to balance linear momentum, as the photon cannot produce a pair in free space (its center of mass would have zero momentum). The energy in excess of the rest masses of the products is imparted as kinetic energy. </p>
<p>The energy dependencies of the three processes are quite different. At energies below a few KeV, the photoelectric effect dominates and the Coulomb effect is small, while pair production is energetically impossible. For the region of 0.1 to about 10 MeV, Compton scattering dominates. Above this, pair production is the predominant method for the interaction of the photons. It is important to realize that in photoelectric effect and pair production, the photon is eliminated in the process of the interaction, whereas in Compton scattering, the energy of the photon is only degraded. As explained further below the various processes involved lead to a complete or incomplete energy deposition of the gamma energy in the scintillator. We have to be concerned only with the first two methods; the photoelectric effect leads to complete energy deposition, while the Compton effect can lead to processes where the scattered gamma ray leaves the detector without depositing its energy.</p>

<h2>The Radioactive Sources</h2>
<p>You will be working with five radioactive sources: Mn-54, Na-22, Cs-137, and Co-60. You should note the radiation dosage and date marked on the source, and read about radiation safety and how to handle radiation sources from ref. 1, pg 326-328. Calculate the number of disintegrations per second for each of the sources based on the quoted dosage.</p>

<h2>The Detector</h2>
<p>The gamma rays that are emitted from the source are detected with a scintillator detector. In our laboratory, we use inorganic crystals of sodium iodide doped with impurity centres of thallium. This combination is denoted by NaI(Tl). A photon entering the detector ionizes the material through the processes described above. The positive ions and electrons created by the incoming photon diffuse through the lattice and are captured by the impurity centres i.e. the Tl atoms, which act as luminescence centres. Recombination produces an excited centre,which emits visible light upon its return to the ground state. The efficiency of these inorganic crystals is high, but the light output is spread over a time interval of the order of microseconds. </p>
<p>In general, for gamma rays less than 1 MeV, photons undergoing the photoelectric process will be completely absorbed and will give rise to a determined number of light quanta. These light quanta are then seen by the photomultiplier tube, and the output pulse will be proportional to the incoming energy of the gamma ray. The energy of the electrons produced by the Compton effect will depend on the angle at which they are scattered and hence there will be a spectrum of detected pulses. At energies above 1 MeV, pair production can occur and the electron and positron lose their kinetic energy by ionizing the medium. The electron is absorbed, while the positron annihilates within a few nanoseconds into two photons, each of 0.511 MeV. These may then interact through the photoelectric or Compton effect. There are three possibilities for the observed energy (show this in the writeup) depending on whether none, one or both of these photons from the positron annihilation are detected.</p>
<p>As mentioned, the produced light is proportional to the energy of the incident gamma ray. This light is then made incident on the photocathode face of a photomultiplier tube (PMT), which converts the photons to photoelectrons through the photoelectric effect. The electrons are amplified and then converted to a voltage pulse. The PMT used here has 10 stages and the multiplication for such tubes is about 10<sup>6</sup> for an operational voltage between the cathode and the anode of 1000V. The cascade of electrons produced at each of the multiplication stages of the dynodes is collected at the anode, converted to a voltage pulse, which is then amplified and analysed. The PMT is enclosed in a high permeability material to shield against magnetic fields, as such effects would affect the efficiency of the tube.</p>
<p>Note that the metal shield that encloses the scintillator crystal does not allow β (or other charged) particles to permeate and deposit their usually well-defined energies. Thus, you should not expect to observe structures associated with, e.g., the 0.514 MeV and 1.17 MeV electrons coming from the Cs137 source with branching ratios of 93.5% and 6.5% respectively (cf.. Fig. 1). Charged particles are created inside the crystal by γ ray impact. There will be, however, backscattered γ rays from Compton scattering outside the crystal, e.g., off the backing of the source and off the lead shield.</p>
<p>Read the specifications for the model of the scintillator detector and photomultiplier that you have. Note that the size of the crystal is small. How does this affect your results? The typical quantum efficiency of scintillators of this type is one photon of light produced per 100 eV of energy deposited in the scintillator. The width of the full-energy peak (i.e. when the gamma energy is fully absorbed) depends on the number of light quanta produced by the incident gamma ray. The energy resolution, ''dE/E'' is an important quantity to consider as this factor will determine if we can separate gamma rays very close in energy.</p>
<p>The chain of events may help illustrate this point. The incident gamma ray of energy ''E<sub>g</sub>'' produces a photoelectron with energy ''E<sub>e</sub>'' ~ ''E<sub>g</sub>''. The photoelectron produces N light quanta in the scintillator material via ionization and excitation, each with an energy ''E<sub>lq</sub>''. Since the light is visible (~400 nm), ''E<sub>lq</sub>'' is about 3 eV. This light falls on the photocathode, which has a quantum efficiency or sensitivity to the wavelength of the incident light. For the tube supplied, the efficiency peaks near 400 nm, making it quite suitable for use with a scintillator. Let
<ul>
<li>ε(light) be the efficiency for the conversion of the excitation energy into light quanta</li>
<li>ε(coll) be the efficiency for collecting the n lq light quanta at the photocathode of the PMT</li>
<li>ε(cathode) be the efficiency for the cathode to eject a photoelectron</li>
</ul>
Then, the number of photoelectrons produced at the photocathode is
</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Gam-eqn1b.png|330px|center]]
</p>
</td></table>

<p>Typical values for the efficiencies for NaI detectors are:
<ul>
<li>ε(light) = 0.1</li>
<li>ε(coll) = 0.4</li>
<li>ε(cathode) = 0.2</li>
</ul>
</p>
<p>For a 1MeV gamma ray, this yields about 3x10<sup>3</sup> photoelectrons at the photocathode. All the processes mentioned above are subject to statistical fluctuations, and contribute to the broadening of the line width. In addition, there is a contribution from the statistical process due to the multiplication of the photoelectrons in the stages of the PMT.</p>

<h2>The Multichannel Analyser (MCA)</h2>
<p> The electrons collected at the anode of the PMT pass directly into the Spectrum Techniques MCA (UCS 30). This pulse of electrons enters the internal pre-amplifier, followed by the internal amplifier. The voltage across the internal amplifier resistance is digitized into one of 1024 bins according to its maximum value (peak height)- this process of converting an analog voltage into a digital value is called analog-to-digital conversion (ADC). The MCA performs this task for all pulses and creates a histogram of counts in each ADC bin.</p>
<p>There are several important parameters which affect the performance of the MCA. For a detailed explanation of the pulse processing details refer to the book by Knoll, "Radiation Detection and Measurements" listed in the references section below.
<ul>

<li><b>Amplifier Gain: </b> This is an amplification factor applied to the detector pulse using the adjustable coarse gain and fine gain controls.
<li><b>Lower Level Discriminator (LLD) :</b> The amount above the average background which is required for the MCA to accept a particular pulse and record its statistics in the histogram.</li>
<li><b>Run Time: </b> The actual length of time for which to acquire data.</li>

<li><b>Live Time:</b> The time the detector is actually able to detect pulses.
<li><b>Dead Time:</b> The time during collection when the detector is unable to process additional events. Look up live time and dead time and discuss in your report.</li>
<li><b>High Voltage: </b> The parameter "high voltage" in the MCA client software does nothing. Bridgeport makes an HV supply for PMTs which connect directly to the eMorpho. We are using a separate power supply, hence adjusting this parameter in the software does nothing. However, changing the 1000V supplied to the PMT from the power supply will mean that more electrons are collected per pulse. If this dc voltage is set too high and and too many photons are present, there could be a buildup of charge inside the PMT and catastrophic damage could occur.<i>Please do not change the setting of the voltage to the PMT from 1000V.</i></li>

</ul>
</p>

<p> Turn the power on the spectrum analyzer (UCS 30) and run the program "USX" located on the desktop. (The UCS 30 manual is located on the desktop and should be referred to for additional information.) Hover over the toolbar icons and note the descriptions. Click on each icon and note the options available. For example, you can toggle between a linear and logarithmic y-axis scale by clicking on the "y-log" icon on the toolbar. Additionally, during data collection you can adjust the scale of the y-axis by scrolling up and down when the mouse is in the graph region. Review the operation section (pg 16-25) of the UCS30 lab manual. </p>


<p>The signal from the PMT is directly input to a multichannel analyser which is a device that sorts incoming pulses according to pulse height and keeps count of the number at each height in a multichannel memory. The contents of each channel is displayed on a screen to give a pulse height spectrum, which is then analysed. The amplitude of the incoming pulse is digitized with an Analogue to Digital Converter (ADC), and sorting is done based on how many pulses had a particular value of the digitized amplitude. The total number of channels into which the voltage range is digitized determines the resolution of the MCA. Refer ref. 1, 2 and 4 to understand the characteristics of the electronics you are supplied with, and to learn more about the ADC range and resolution that can be achieved. Also, use ref. 1, 2, 4 to understand the functions of a MCA. Review what is meant by ‘dead time’ and ‘live time’. Do not operate with a dead time of more than 30%, since too high count rates can cause the electronics to misbehave.</p>

<p>The natural linewidth of the gamma rays is extremely narrow (a few eV compared to the MeV range of the energies themselves!). The broadening observed in the recorded spectra is a result of the detection method (a cooled Germanium detector would show these lines being much narrower, but again at a resolution that depends on the detector itself). It is important to realize on the example of the Cs137 spectrum, for which a single energy at 0.662 MeV is expected that several effects occur in ‘real life’:</p>
<ol>
<li>Broadening of the photopeak at 0.662 MeV.</li>
<li>A Compton background (plateau) with two edges: at the higher end, below the
photopeak a maximum electron(!) energy of
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Gam-eqn2.png|170px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
Understand this result using eq. (1). At lower energies in the Compton plateau a distinct photopeak corresponding to backscattered gamma rays whose energy follows from eq. (1) for θ = π. Note how nearly backscattered gammas give about the same energy due to the slow variation of the cos function for θ = π.</li>
<li>A fluorescent K-shell Xray peak caused by soft gamma rays hitting the shielding material (typically lead), knocking out K-shell electrons. The vacancies are typically filled from the L-III shell (2p±1). Use the CRC Handbook (see Lab Technologist or Library) to verify that the difference in energy results in characteristic X rays of about 75 keV for Pb.</li>
</ol>
<p>A typical spectrum is shown in Fig. 2 on a linear scale. If we go to a logarithmic scale (the MCA software allows to do this), we find an additional weak peak at about twice the gamma energy. This is the so-called sum peak, which arises when two gamma rays from uncorrelated decay events are depositing their energy in the scintillator within a fraction of a microsecond, i.e., the timescale over which the scintillation in the crystal and photomultiplication in the tube occurs. You should be able to observe sum peaks in this experiment when accumulating enough statistics. The peaks appear more strongly when the source is brought closer to the PMT. </p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig3.png|500px|border|center]]
<b>Figure 3 -</b> Typical Cs-137 spectrum using an NaI(T1) scintillator.
<br clear=right>
</p>
</td></table>
<p>Besides sum peaks another complication can occur for high-energy gamma rays. When pair creation is an important energy deposition mechanism, so-called escape peaks are observed. These correspond to events where one or more of the created electrons/positrons escape the crystal without giving up their energy. Such peaks occur at ''E<sub>g</sub>'' - ''j'' (0.511 MeV), where ''j'' is the number of escaped electrons/positrons. Since we do not use sources with ''E<sub>g</sub>'' > 2 MeV in this experiment, we do not find this complication in our spectra.</p>

<h1>Experimental Procedure</h1>
<p>While conducting the experiment, make sure that only the source whose spectrum you are observing is near the apparatus otherwise your calibration results will be skewed. Take only one source out at a time, and keep the others in the box away from the detector. Be careful while handling sources. Become familiar with the documentation and all the apparatus before starting. </p>

<p>Ensure the following connections: the scintillator and PMT are connected, the HV output from the PMT is connected to the USC 30, the signal output from the PMT is connected to the MCA, and the MCA is connected to the computer.</p>
<p>In our setup, we have mounted the PMT vertically, allowing you to place the source on a tray at several different distances from the source. Once you have found an optimal distance for all three sources (i.e. you do not get ‘pile up’ effects (see Leo)), it is best to use this distance for calibration and determination of the peak energies. However, in the last part of the experiment you will be asked to draw qualitative conclusions by placing a source at varying distances. </p>

<h2>Settings</h2>
<p>Please ensure the following running parameters</p>
<ul>
<li>High-Voltage to PMT: +1000V (Click on the "Amp/HV/ADC" icon on the toolbar to set and turn on the voltage.)</li>
<li>Amplifier Gain: Use the course gain and fine gain controls to set the gain in the range of 16 - 18. </li>
<li>Conversion Gain: 1024</li>
<li>Peak Time: 1 micro sec</li>
</ul>


<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig4.png|700px|border|center]]
<b>Figure 4 -</b> Gamma source, NaI(T1) detector, PMT, lead shield showing relevant processes.
<br clear=right>
</p>
</td></table>
<ol>
<h3>Calibration Curve</h3>
<li><p>Use Experiment 2.1 as a guide for gamma ray calibration from Duggan in the red binder in the lab. The eMorpho software will only be used for data collection and observing the spectra. All data analysis, including energy calibration, can be conducted using software like Excel or Mathematica. Use the 5 known sources Cs-137 (0.662 MeV), Co-60 (1.173 MeV, 1.332 MeV), Mn-54 (0.835 MeV), Na-22 (0.511 MeV) and Zn-65 (1.115 MeV) provided in the RSS 8 radiation source kit (Spectrum Techniques). Centre each source (label down) about 5 cm from the face of the detector and collect a spectrum for at least 5 minutes with each source. Longer collection times correspond to smaller statistical errors, why? <b>Stop collection prior to saving and record the dead time, live time and real time for each spectrum.</b> Save the spectrum. (The .csv file generated contains a single column with MCA parameters in the first 74 entries followed by the spectrum data (number of events) in each channel from 0 to 4096. Prior to data analysis delete the first 74 entries. Create a calibration curve using all five sources i.e. for each spectrum locate the channel number corresponding to the gamma ray energy peaks and create a plot of the energy of each peak and the peak channel number. What is the error of a straight line fit to this curve? If the fit is poor, you have made a mistake in assigning peak-channel values, and may have to repeat the analysis. Discuss sources of error. </p>

<p>Using your calibration fit plot the calibrated Cs-137 spectrum, Co-60 spectrum and Na-22 spectrum separately(i.e. events as a function of energy). Explain all the peaks on the spectra. Do you see any 'sum peaks’? (Plot your data on a logarithmic scale for a better view!) Explain why this peak occurs. Does it occur at the value you expect? What is the source of the 511 keV peak in the Na-22 spectrum? The Na-22 has a gamma ray energy at 1275 keV. What value of the maximum position do you find based on your calibration curve? Is this consistent with the resolution ''dE/E''? What is the present activity of these three sources (note the date marked on the source)? Present the calibrated spectra in your report and explain all the interesting details.</p> </li>

<li><p> Put a sheet of lead on the shelf and then place the Co-60 source above. How does the spectrum change? Now place the sheet of lead on top of the source. What, if anything, has changed in the spectrum? You may have to move the source closer or farther from the detector to observe the backscattering signals. Collect a spectrum for each case for a sufficient collection time and save. <b> Caution: Always wear gloves when handling the lead sheets.</b></p> </li>

<li> Obtain the energy resolution for the photopeak for Cs-137 and for one of the Co-60 peaks (See Knoll, chapter 4). Does this agree with what you have learned about scintillator devices? </li>
<li> Place the Cs-137 source at varying distances from the detector and explain qualitatively the rates that you observe. Put the source at slot 2 below the detector, and place first one, then two, and then three sheets of lead on top of the source. Comment on the rates, and the reason for the observed behaviour. Repeat for sheets of aluminum.</li>
<li> A spectrum of the unknown source (the background in the room) will be collected by the Lab Technologist. The data will be saved and the file can be found in the "data" folder located on the desktop. Using this data, create a calibrated spectrum using your calibration fit. Use your now-calibrated gamma-ray spectrum to identify the radioisotope(s) found in the background.</li>
</ol>

<h1>References</h1>
<ol>
<li>Preston and Dietz,[https://www.library.yorku.ca/find/Record/1038893 ''The Art of Experimental Physics''], Wiley.</li>
<li>G.F. Knoll, ''Radiation Detection and Measurement, 3rd or 4th ed.'', Wiley. </li>
<li>J.L. Duggan, ''Laboratory Investigations in Nuclear Science'', Tennelec.</li>
<li>A.C. Melissinos, [https://www.library.yorku.ca/find/Record/1641963 ''Experiments in Modern Physics''], Academic Press.</li>
<li>Leo, [https://www.library.yorku.ca/find/Record/1178503 ''Techniques for Nuclear and Particle Physics Experiments''], Springer-Verlag.</li>
<li>H. Frauenfelder and Henley,[https://www.library.yorku.ca/find/Record/51764 ''Subatomic Physics''] ,Prentice-Hall.</li>
<li>K. Siegbahn, [https://www.library.yorku.ca/find/Record/51739 ''Alpha, Beta, and Gamma-Ray Spectroscopy''], vol. I, chpts 5,8a.</li>
</ol>

Main Page/PHYS 4210/Gamma Ray Spectroscopy

2021-12-23T16:16:30Z

Gloria:

<h1>Gamma Ray Spectroscopy</h1>
<p>In this experiment we study the gamma ray spectra of several radioactive elements to learn about the interaction properties of gamma rays with matter. The gamma rays are detected through ionization of the material in a scintillation counter, and the output pulse, which is generated with a photomultiplier tube, is then recorded with the aid of a Multi Channel Analyser (MCA) connected to a computer interface. The different types of interaction of gamma rays with matter are understood from a detailed analysis of the observed spectra. Thus, this experiment not only illustrates the physics of the interaction properties of photons, but also introduces scintillation detectors and relevant electronics.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Radioactive Decay</li>
<li>Gamma Rays</li>
<li>Multi-Channel Analyser (MCA)</li>
<li>Photo-Electric Effect</li>
<li>Compton Scattering</li>
<li>Pair Production</li>

</ul>
</td>
<td width=250>
<ul>
<li>Crystal Detector</li>
<li>Photomultiplier Tube</li>
<li>Back Scatter Peak</li>
<li>Compton Edge</li>
<li>Compton Plateau</li>
<li>Photopeak</li>
</ul>
</td>
</table>

<h1>Introduction</h1>
<p>Gamma rays are photons of very short wavelength (~10<sup>-12</sup> cm) or very high frequency (10<sup>22</sup> Hz) that are emitted during nuclear transitions. The decay schemes of the three radionuclides that we study (Na22, Cs137, Co60) are shown below. Read ref. 1-5 to get a clear idea of the significance of the gamma energies.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Decay schemes of Cs-137, Co-60, and Na-22.
<br clear=right>
</p>
</td></table>

<h2>Method</h2>
<p>Photons are electrically neutral, and unlike charged particles, do not experience the Coulomb force. They are, however, carriers of the electromagnetic force, and are able to ionize atoms through their interaction with matter, and this leads to the deposition of energy in the medium as the ionized particle slows down in traversing the medium. This energy can then be detected. The three modes of interaction are: photoelectric effect, Compton scattering, and pair production where the photon interacts with an atom, an electron, and a nucleus respectively. You should read the details of each type of interaction in the references and include this in your write-up. We summarise it briefly below.</p>

<h3>Photoelectric effect</h3>
<p>The photon of energy ''E<sub>g</sub>'' is absorbed by an atom and an electron from one of the shells is emitted. If ''B<sub>e</sub>'' is the binding energy of the electron, then the energy of the emitted electron will have an energy ''E<sub>e</sub>'' = ''E<sub>g</sub>'' - ''B<sub>e</sub>''. Since ''B<sub>e</sub>'' is small (of the order 40 KeV) compared to ''E<sub>g</sub>'' (of the order 1MeV), the electron carries most of the energy of the photon.</p>

<h3>Compton scattering</h3>
<p>The photon scatters off an electron that is either free or is loosely bound in the atom, thereby scattering the electron. It can be shown (do so in your writeup) that the energy of the scattered electron is related to that of the incident photon, the angle of the scattered photon (theta), and the mass of the electron me through the relationship:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Gam-eqn1.png|190px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig2.png|500px|border|center]]
<b>Figure 2 -</b> Compton scattering.
<br clear=right>
</p>
</td></table>
<p>Hence the scattered photon (of energy hv’) has the freedom to move at any angle with respect to the incident photon (or gamma ray, as we call it here), whereas the scattered electron is bound by the laws of conservation of momentum to only go in the forward direction. The kinetic energy of the Compton electron is E<sub>C</sub> = hv- hv’. This, combined with the above formula for the energy of the electron shows that the maximum kinetic energy of the electron is when theta = 180° ,i.e., when the photon is scattered backwards. This maximum is known as the Compton edge. </p>

<h3>Pair production</h3>
<p>If the photon has more than twice the rest mass of an electron (0.511 MeV), the photon can produce an electron-positron pair. This must be done in the Coulomb field of the nucleus to balance linear momentum, as the photon cannot produce a pair in free space (its center of mass would have zero momentum). The energy in excess of the rest masses of the products is imparted as kinetic energy. </p>
<p>The energy dependencies of the three processes are quite different. At energies below a few KeV, the photoelectric effect dominates and the Coulomb effect is small, while pair production is energetically impossible. For the region of 0.1 to about 10 MeV, Compton scattering dominates. Above this, pair production is the predominant method for the interaction of the photons. It is important to realize that in photoelectric effect and pair production, the photon is eliminated in the process of the interaction, whereas in Compton scattering, the energy of the photon is only degraded. As explained further below the various processes involved lead to a complete or incomplete energy deposition of the gamma energy in the scintillator. We have to be concerned only with the first two methods; the photoelectric effect leads to complete energy deposition, while the Compton effect can lead to processes where the scattered gamma ray leaves the detector without depositing its energy.</p>

<h2>The Radioactive Sources</h2>
<p>You will be working with five radioactive sources: Mn-54, Na-22, Cs-137, and Co-60. You should note the radiation dosage and date marked on the source, and read about radiation safety and how to handle radiation sources from ref. 1, pg 326-328. Calculate the number of disintegrations per second for each of the sources based on the quoted dosage.</p>

<h2>The Detector</h2>
<p>The gamma rays that are emitted from the source are detected with a scintillator detector. In our laboratory, we use inorganic crystals of sodium iodide doped with impurity centres of thallium. This combination is denoted by NaI(Tl). A photon entering the detector ionizes the material through the processes described above. The positive ions and electrons created by the incoming photon diffuse through the lattice and are captured by the impurity centres i.e. the Tl atoms, which act as luminescence centres. Recombination produces an excited centre,which emits visible light upon its return to the ground state. The efficiency of these inorganic crystals is high, but the light output is spread over a time interval of the order of microseconds. </p>
<p>In general, for gamma rays less than 1 MeV, photons undergoing the photoelectric process will be completely absorbed and will give rise to a determined number of light quanta. These light quanta are then seen by the photomultiplier tube, and the output pulse will be proportional to the incoming energy of the gamma ray. The energy of the electrons produced by the Compton effect will depend on the angle at which they are scattered and hence there will be a spectrum of detected pulses. At energies above 1 MeV, pair production can occur and the electron and positron lose their kinetic energy by ionizing the medium. The electron is absorbed, while the positron annihilates within a few nanoseconds into two photons, each of 0.511 MeV. These may then interact through the photoelectric or Compton effect. There are three possibilities for the observed energy (show this in the writeup) depending on whether none, one or both of these photons from the positron annihilation are detected.</p>
<p>As mentioned, the produced light is proportional to the energy of the incident gamma ray. This light is then made incident on the photocathode face of a photomultiplier tube (PMT), which converts the photons to photoelectrons through the photoelectric effect. The electrons are amplified and then converted to a voltage pulse. The PMT used here has 10 stages and the multiplication for such tubes is about 10<sup>6</sup> for an operational voltage between the cathode and the anode of 1000V. The cascade of electrons produced at each of the multiplication stages of the dynodes is collected at the anode, converted to a voltage pulse, which is then amplified and analysed. The PMT is enclosed in a high permeability material to shield against magnetic fields, as such effects would affect the efficiency of the tube.</p>
<p>Note that the metal shield that encloses the scintillator crystal does not allow β (or other charged) particles to permeate and deposit their usually well-defined energies. Thus, you should not expect to observe structures associated with, e.g., the 0.514 MeV and 1.17 MeV electrons coming from the Cs137 source with branching ratios of 93.5% and 6.5% respectively (cf.. Fig. 1). Charged particles are created inside the crystal by γ ray impact. There will be, however, backscattered γ rays from Compton scattering outside the crystal, e.g., off the backing of the source and off the lead shield.</p>
<p>Read the specifications for the model of the scintillator detector and photomultiplier that you have. Note that the size of the crystal is small. How does this affect your results? The typical quantum efficiency of scintillators of this type is one photon of light produced per 100 eV of energy deposited in the scintillator. The width of the full-energy peak (i.e. when the gamma energy is fully absorbed) depends on the number of light quanta produced by the incident gamma ray. The energy resolution, ''dE/E'' is an important quantity to consider as this factor will determine if we can separate gamma rays very close in energy.</p>
<p>The chain of events may help illustrate this point. The incident gamma ray of energy ''E<sub>g</sub>'' produces a photoelectron with energy ''E<sub>e</sub>'' ~ ''E<sub>g</sub>''. The photoelectron produces N light quanta in the scintillator material via ionization and excitation, each with an energy ''E<sub>lq</sub>''. Since the light is visible (~400 nm), ''E<sub>lq</sub>'' is about 3 eV. This light falls on the photocathode, which has a quantum efficiency or sensitivity to the wavelength of the incident light. For the tube supplied, the efficiency peaks near 400 nm, making it quite suitable for use with a scintillator. Let
<ul>
<li>ε(light) be the efficiency for the conversion of the excitation energy into light quanta</li>
<li>ε(coll) be the efficiency for collecting the n lq light quanta at the photocathode of the PMT</li>
<li>ε(cathode) be the efficiency for the cathode to eject a photoelectron</li>
</ul>
Then, the number of photoelectrons produced at the photocathode is
</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Gam-eqn1b.png|330px|center]]
</p>
</td></table>

<p>Typical values for the efficiencies for NaI detectors are:
<ul>
<li>ε(light) = 0.1</li>
<li>ε(coll) = 0.4</li>
<li>ε(cathode) = 0.2</li>
</ul>
</p>
<p>For a 1MeV gamma ray, this yields about 3x10<sup>3</sup> photoelectrons at the photocathode. All the processes mentioned above are subject to statistical fluctuations, and contribute to the broadening of the line width. In addition, there is a contribution from the statistical process due to the multiplication of the photoelectrons in the stages of the PMT.</p>

<h2>The Multichannel Analyser (MCA)</h2>
<p> The electrons collected at the anode of the PMT pass directly into the Spectrum Techniques MCA (UCS 30). This pulse of electrons enters the internal pre-amplifier, followed by the internal amplifier. The voltage across the internal amplifier resistance is digitized into one of 1024 bins according to its maximum value (peak height)- this process of converting an analog voltage into a digital value is called analog-to-digital conversion (ADC). The MCA performs this task for all pulses and creates a histogram of counts in each ADC bin.</p>
<p>There are several important parameters which affect the performance of the MCA. For a detailed explanation of the pulse processing details refer to the book by Knoll, "Radiation Detection and Measurements" listed in the references section below.
<ul>

<li><b>Amplifier Gain: </b> This is an amplification factor applied to the detector pulse using the adjustable coarse gain and fine gain controls.
<li><b>Lower Level Discriminator (LLD) :</b> The amount above the average background which is required for the MCA to accept a particular pulse and record its statistics in the histogram.</li>
<li><b>Run Time: </b> The actual length of time for which to acquire data.</li>

<li><b>Live Time:</b> The time the detector is actually able to detect pulses.
<li><b>Dead Time:</b> The time during collection when the detector is unable to process additional events. Look up live time and dead time and discuss in your report.</li>
<li><b>High Voltage: </b> The parameter "high voltage" in the MCA client software does nothing. Bridgeport makes an HV supply for PMTs which connect directly to the eMorpho. We are using a separate power supply, hence adjusting this parameter in the software does nothing. However, changing the 1000V supplied to the PMT from the power supply will mean that more electrons are collected per pulse. If this dc voltage is set too high and and too many photons are present, there could be a buildup of charge inside the PMT and catastrophic damage could occur.<i>Please do not change the setting of the voltage to the PMT from 1000V.</i></li>

</ul>
</p>

<p> Turn the power on the spectrum analyzer (UCS 30) and run the program "USX" located on the desktop. </p>



<p>The signal from the PMT is directly input to a multichannel analyser which is a device that sorts incoming pulses according to pulse height and keeps count of the number at each height in a multichannel memory. The contents of each channel is displayed on a screen to give a pulse height spectrum, which is then analysed. The amplitude of the incoming pulse is digitized with an Analogue to Digital Converter (ADC), and sorting is done based on how many pulses had a particular value of the digitized amplitude. The total number of channels into which the voltage range is digitized determines the resolution of the MCA. Read the Bridgeport eMorpho manual to understand the characteristics of the electronics you are supplied with, and to learn more about the ADC range and resolution that can be achieved. Read ref. 1, 2, 4 to understand the functions of a MCA. Review what is meant by ‘dead time’ and ‘live time’. Do not operate with a dead time of more than 30%, since too high count rates can cause the electronics to misbehave.</p>
<p>To understand this experiment it is crucial to observe the pulses fed from the scintillator/PMT to the MCA. This can be done using the Bridgeport eMorpho Client v1.0 on the desktop. This is the software which controls the parameters of the MCA and displays the acquired data. By selecting the "Pulse" icon along the top, you can observe the shape of pulses coming from the MCA. You will observe high-amplitude pulses (high energy was recorded since many visible photons were produced), as well as lower-amplitude pulses occurring randomly in rapid succession. The MCA records the pulse heights and the software assembles them into a histogram according to channel number. One can then calibrate the channels using known gamma sources.</p>
<p>The natural linewidth of the gamma rays is extremely narrow (a few eV compared to the MeV range of the energies themselves!). The broadening observed in the recorded spectra is a result of the detection method (a cooled Germanium detector would show these lines being much narrower, but again at a resolution that depends on the detector itself). It is important to realize on the example of the Cs137 spectrum, for which a single energy at 0.662 MeV is expected that several effects occur in ‘real life’:</p>
<ol>
<li>Broadening of the photopeak at 0.662 MeV.</li>
<li>A Compton background (plateau) with two edges: at the higher end, below the
photopeak a maximum electron(!) energy of
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Gam-eqn2.png|170px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
Understand this result using eq. (1). At lower energies in the Compton plateau a distinct photopeak corresponding to backscattered gamma rays whose energy follows from eq. (1) for θ = π. Note how nearly backscattered gammas give about the same energy due to the slow variation of the cos function for θ = π.</li>
<li>A fluorescent K-shell Xray peak caused by soft gamma rays hitting the shielding material (typically lead), knocking out K-shell electrons. The vacancies are typically filled from the L-III shell (2p±1). Use the CRC Handbook (see Lab Technologist or Library) to verify that the difference in energy results in characteristic X rays of about 75 keV for Pb.</li>
</ol>
<p>A typical spectrum is shown in Fig. 2 on a linear scale. If we go to a logarithmic scale (the MCA software allows to do this), we find an additional weak peak at about twice the gamma energy. This is the so-called sum peak, which arises when two gamma rays from uncorrelated decay events are depositing their energy in the scintillator within a fraction of a microsecond, i.e., the timescale over which the scintillation in the crystal and photomultiplication in the tube occurs. You should be able to observe sum peaks in this experiment when accumulating enough statistics. The peaks appear more strongly when the source is brought closer to the PMT. </p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig3.png|500px|border|center]]
<b>Figure 3 -</b> Typical Cs-137 spectrum using an NaI(T1) scintillator.
<br clear=right>
</p>
</td></table>
<p>Besides sum peaks another complication can occur for high-energy gamma rays. When pair creation is an important energy deposition mechanism, so-called escape peaks are observed. These correspond to events where one or more of the created electrons/positrons escape the crystal without giving up their energy. Such peaks occur at ''E<sub>g</sub>'' - ''j'' (0.511 MeV), where ''j'' is the number of escaped electrons/positrons. Since we do not use sources with ''E<sub>g</sub>'' > 2 MeV in this experiment, we do not find this complication in our spectra.</p>

<h1>Experimental Procedure</h1>
<p>While conducting the experiment, make sure that only the source whose spectrum you are observing is near the apparatus otherwise your calibration results will be skewed. Take only one source out at a time, and keep the others in the box away from the detector. Be careful while handling sources. Become familiar with the documentation and all the apparatus before starting. </p>
<h2>Settings</h2>
<p>Please ensure the following running parameters</p>
<ul>
<li>High-Voltage to PMT: +1000V (set on the actual power supply)</li>
<li>Electronic Gain: 2</li>
<li>Digital Gain: 4096</li>
<li>Integration Time: 0.625</li>
</ul>
<p> <b>Note:</b> The <b>trigger threshold</b> located under the Settings tab should be set to 14. </p>

<p>Ensure the following connections: the scintillator and PMT is connected, the HV output from the power uspply is connected to the PMT, the signal output from the PMT is connected to the MCA, and the MCA is connected to the computer.</p>
<p>In our setup, we have mounted the PMT vertically, allowing you to place the source on a tray at several different distances from the source. Once you have found an optimal distance for all three sources (i.e. you do not get ‘pile up’ effects (see Leo)), it is best to use this distance for calibration and determination of the peak energies. However, in the last part of the experiment you will be asked to draw qualitative conclusions by placing a source at varying distances. </p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig4.png|700px|border|center]]
<b>Figure 4 -</b> Gamma source, NaI(T1) detector, PMT, lead shield showing relevant processes.
<br clear=right>
</p>
</td></table>
<ol>
<h3>Calibration Curve</h3>
<li><p>Use Experiment 2.1 as a guide for gamma ray calibration from Duggan in the red binder in the lab. The eMorpho software will only be used for data collection and observing the spectra. All data analysis, including energy calibration, can be conducted using software like Excel or Mathematica. Use the 5 known sources Cs-137 (0.662 MeV), Co-60 (1.173 MeV, 1.332 MeV), Mn-54 (0.835 MeV), Na-22 (0.511 MeV) and Zn-65 (1.115 MeV) provided in the RSS 8 radiation source kit (Spectrum Techniques). Centre each source (label down) about 5 cm from the face of the detector and collect a spectrum for at least 5 minutes with each source. Longer collection times correspond to smaller statistical errors, why? <b>Stop collection prior to saving and record the dead time, live time and real time for each spectrum.</b> Save the spectrum. (The .csv file generated contains a single column with MCA parameters in the first 74 entries followed by the spectrum data (number of events) in each channel from 0 to 4096. Prior to data analysis delete the first 74 entries. Create a calibration curve using all five sources i.e. for each spectrum locate the channel number corresponding to the gamma ray energy peaks and create a plot of the energy of each peak and the peak channel number. What is the error of a straight line fit to this curve? If the fit is poor, you have made a mistake in assigning peak-channel values, and may have to repeat the analysis. Discuss sources of error. </p>

<p>Using your calibration fit plot the calibrated Cs-137 spectrum, Co-60 spectrum and Na-22 spectrum separately(i.e. events as a function of energy). Explain all the peaks on the spectra. Do you see any 'sum peaks’? (Plot your data on a logarithmic scale for a better view!) Explain why this peak occurs. Does it occur at the value you expect? What is the source of the 511 keV peak in the Na-22 spectrum? The Na-22 has a gamma ray energy at 1275 keV. What value of the maximum position do you find based on your calibration curve? Is this consistent with the resolution ''dE/E''? What is the present activity of these three sources (note the date marked on the source)? Present the calibrated spectra in your report and explain all the interesting details.</p> </li>

<li><p> Put a sheet of lead on the shelf and then place the Co-60 source above. How does the spectrum change? Now place the sheet of lead on top of the source. What, if anything, has changed in the spectrum? You may have to move the source closer or farther from the detector to observe the backscattering signals. Collect a spectrum for each case for a sufficient collection time and save. <b> Caution: Always wear gloves when handling the lead sheets.</b></p> </li>

<li> Obtain the energy resolution for the photopeak for Cs-137 and for one of the Co-60 peaks (See Knoll, chapter 4). Does this agree with what you have learned about scintillator devices? </li>
<li> Place the Cs-137 source at varying distances from the detector and explain qualitatively the rates that you observe. Put the source at slot 2 below the detector, and place first one, then two, and then three sheets of lead on top of the source. Comment on the rates, and the reason for the observed behaviour. Repeat for sheets of aluminum.</li>
<li> A spectrum of the unknown source (the background in the room) will be collected by the Lab Technologist. The data will be saved and the file can be found in the "data" folder located on the desktop. Using this data, create a calibrated spectrum using your calibration fit. Use your now-calibrated gamma-ray spectrum to identify the radioisotope(s) found in the background.</li>
</ol>

<h1>References</h1>
<ol>
<li>Preston and Dietz,[https://www.library.yorku.ca/find/Record/1038893 ''The Art of Experimental Physics''], Wiley.</li>
<li>G.F. Knoll, ''Radiation Detection and Measurement, 3rd or 4th ed.'', Wiley. </li>
<li>J.L. Duggan, ''Laboratory Investigations in Nuclear Science'', Tennelec.</li>
<li>A.C. Melissinos, [https://www.library.yorku.ca/find/Record/1641963 ''Experiments in Modern Physics''], Academic Press.</li>
<li>Leo, [https://www.library.yorku.ca/find/Record/1178503 ''Techniques for Nuclear and Particle Physics Experiments''], Springer-Verlag.</li>
<li>H. Frauenfelder and Henley,[https://www.library.yorku.ca/find/Record/51764 ''Subatomic Physics''] ,Prentice-Hall.</li>
<li>K. Siegbahn, [https://www.library.yorku.ca/find/Record/51739 ''Alpha, Beta, and Gamma-Ray Spectroscopy''], vol. I, chpts 5,8a.</li>
</ol>

Main Page/PHYS 4210/Gamma Ray Spectroscopy

2021-12-23T16:06:13Z

Gloria:

<h1>Gamma Ray Spectroscopy</h1>
<p>In this experiment we study the gamma ray spectra of several radioactive elements to learn about the interaction properties of gamma rays with matter. The gamma rays are detected through ionization of the material in a scintillation counter, and the output pulse, which is generated with a photomultiplier tube, is then recorded with the aid of a Multi Channel Analyser (MCA) connected to a computer interface. The different types of interaction of gamma rays with matter are understood from a detailed analysis of the observed spectra. Thus, this experiment not only illustrates the physics of the interaction properties of photons, but also introduces scintillation detectors and relevant electronics.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Radioactive Decay</li>
<li>Gamma Rays</li>
<li>Multi-Channel Analyser (MCA)</li>
<li>Photo-Electric Effect</li>
<li>Compton Scattering</li>
<li>Pair Production</li>

</ul>
</td>
<td width=250>
<ul>
<li>Crystal Detector</li>
<li>Photomultiplier Tube</li>
<li>Back Scatter Peak</li>
<li>Compton Edge</li>
<li>Compton Plateau</li>
<li>Photopeak</li>
</ul>
</td>
</table>

<h1>Introduction</h1>
<p>Gamma rays are photons of very short wavelength (~10<sup>-12</sup> cm) or very high frequency (10<sup>22</sup> Hz) that are emitted during nuclear transitions. The decay schemes of the three radionuclides that we study (Na22, Cs137, Co60) are shown below. Read ref. 1-5 to get a clear idea of the significance of the gamma energies.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Decay schemes of Cs-137, Co-60, and Na-22.
<br clear=right>
</p>
</td></table>

<h2>Method</h2>
<p>Photons are electrically neutral, and unlike charged particles, do not experience the Coulomb force. They are, however, carriers of the electromagnetic force, and are able to ionize atoms through their interaction with matter, and this leads to the deposition of energy in the medium as the ionized particle slows down in traversing the medium. This energy can then be detected. The three modes of interaction are: photoelectric effect, Compton scattering, and pair production where the photon interacts with an atom, an electron, and a nucleus respectively. You should read the details of each type of interaction in the references and include this in your write-up. We summarise it briefly below.</p>

<h3>Photoelectric effect</h3>
<p>The photon of energy ''E<sub>g</sub>'' is absorbed by an atom and an electron from one of the shells is emitted. If ''B<sub>e</sub>'' is the binding energy of the electron, then the energy of the emitted electron will have an energy ''E<sub>e</sub>'' = ''E<sub>g</sub>'' - ''B<sub>e</sub>''. Since ''B<sub>e</sub>'' is small (of the order 40 KeV) compared to ''E<sub>g</sub>'' (of the order 1MeV), the electron carries most of the energy of the photon.</p>

<h3>Compton scattering</h3>
<p>The photon scatters off an electron that is either free or is loosely bound in the atom, thereby scattering the electron. It can be shown (do so in your writeup) that the energy of the scattered electron is related to that of the incident photon, the angle of the scattered photon (theta), and the mass of the electron me through the relationship:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Gam-eqn1.png|190px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig2.png|500px|border|center]]
<b>Figure 2 -</b> Compton scattering.
<br clear=right>
</p>
</td></table>
<p>Hence the scattered photon (of energy hv’) has the freedom to move at any angle with respect to the incident photon (or gamma ray, as we call it here), whereas the scattered electron is bound by the laws of conservation of momentum to only go in the forward direction. The kinetic energy of the Compton electron is E<sub>C</sub> = hv- hv’. This, combined with the above formula for the energy of the electron shows that the maximum kinetic energy of the electron is when theta = 180° ,i.e., when the photon is scattered backwards. This maximum is known as the Compton edge. </p>

<h3>Pair production</h3>
<p>If the photon has more than twice the rest mass of an electron (0.511 MeV), the photon can produce an electron-positron pair. This must be done in the Coulomb field of the nucleus to balance linear momentum, as the photon cannot produce a pair in free space (its center of mass would have zero momentum). The energy in excess of the rest masses of the products is imparted as kinetic energy. </p>
<p>The energy dependencies of the three processes are quite different. At energies below a few KeV, the photoelectric effect dominates and the Coulomb effect is small, while pair production is energetically impossible. For the region of 0.1 to about 10 MeV, Compton scattering dominates. Above this, pair production is the predominant method for the interaction of the photons. It is important to realize that in photoelectric effect and pair production, the photon is eliminated in the process of the interaction, whereas in Compton scattering, the energy of the photon is only degraded. As explained further below the various processes involved lead to a complete or incomplete energy deposition of the gamma energy in the scintillator. We have to be concerned only with the first two methods; the photoelectric effect leads to complete energy deposition, while the Compton effect can lead to processes where the scattered gamma ray leaves the detector without depositing its energy.</p>

<h2>The Radioactive Sources</h2>
<p>You will be working with five radioactive sources: Mn-54, Na-22, Cs-137, and Co-60. You should note the radiation dosage and date marked on the source, and read about radiation safety and how to handle radiation sources from ref. 1, pg 326-328. Calculate the number of disintegrations per second for each of the sources based on the quoted dosage.</p>

<h2>The Detector</h2>
<p>The gamma rays that are emitted from the source are detected with a scintillator detector. In our laboratory, we use inorganic crystals of sodium iodide doped with impurity centres of thallium. This combination is denoted by NaI(Tl). A photon entering the detector ionizes the material through the processes described above. The positive ions and electrons created by the incoming photon diffuse through the lattice and are captured by the impurity centres i.e. the Tl atoms, which act as luminescence centres. Recombination produces an excited centre,which emits visible light upon its return to the ground state. The efficiency of these inorganic crystals is high, but the light output is spread over a time interval of the order of microseconds. </p>
<p>In general, for gamma rays less than 1 MeV, photons undergoing the photoelectric process will be completely absorbed and will give rise to a determined number of light quanta. These light quanta are then seen by the photomultiplier tube, and the output pulse will be proportional to the incoming energy of the gamma ray. The energy of the electrons produced by the Compton effect will depend on the angle at which they are scattered and hence there will be a spectrum of detected pulses. At energies above 1 MeV, pair production can occur and the electron and positron lose their kinetic energy by ionizing the medium. The electron is absorbed, while the positron annihilates within a few nanoseconds into two photons, each of 0.511 MeV. These may then interact through the photoelectric or Compton effect. There are three possibilities for the observed energy (show this in the writeup) depending on whether none, one or both of these photons from the positron annihilation are detected.</p>
<p>As mentioned, the produced light is proportional to the energy of the incident gamma ray. This light is then made incident on the photocathode face of a photomultiplier tube (PMT), which converts the photons to photoelectrons through the photoelectric effect. The electrons are amplified and then converted to a voltage pulse. The PMT used here has 10 stages and the multiplication for such tubes is about 10<sup>6</sup> for an operational voltage between the cathode and the anode of 1000V. The cascade of electrons produced at each of the multiplication stages of the dynodes is collected at the anode, converted to a voltage pulse, which is then amplified and analysed. The PMT is enclosed in a high permeability material to shield against magnetic fields, as such effects would affect the efficiency of the tube.</p>
<p>Note that the metal shield that encloses the scintillator crystal does not allow β (or other charged) particles to permeate and deposit their usually well-defined energies. Thus, you should not expect to observe structures associated with, e.g., the 0.514 MeV and 1.17 MeV electrons coming from the Cs137 source with branching ratios of 93.5% and 6.5% respectively (cf.. Fig. 1). Charged particles are created inside the crystal by γ ray impact. There will be, however, backscattered γ rays from Compton scattering outside the crystal, e.g., off the backing of the source and off the lead shield.</p>
<p>Read the specifications for the model of the scintillator detector and photomultiplier that you have. Note that the size of the crystal is small. How does this affect your results? The typical quantum efficiency of scintillators of this type is one photon of light produced per 100 eV of energy deposited in the scintillator. The width of the full-energy peak (i.e. when the gamma energy is fully absorbed) depends on the number of light quanta produced by the incident gamma ray. The energy resolution, ''dE/E'' is an important quantity to consider as this factor will determine if we can separate gamma rays very close in energy.</p>
<p>The chain of events may help illustrate this point. The incident gamma ray of energy ''E<sub>g</sub>'' produces a photoelectron with energy ''E<sub>e</sub>'' ~ ''E<sub>g</sub>''. The photoelectron produces N light quanta in the scintillator material via ionization and excitation, each with an energy ''E<sub>lq</sub>''. Since the light is visible (~400 nm), ''E<sub>lq</sub>'' is about 3 eV. This light falls on the photocathode, which has a quantum efficiency or sensitivity to the wavelength of the incident light. For the tube supplied, the efficiency peaks near 400 nm, making it quite suitable for use with a scintillator. Let
<ul>
<li>ε(light) be the efficiency for the conversion of the excitation energy into light quanta</li>
<li>ε(coll) be the efficiency for collecting the n lq light quanta at the photocathode of the PMT</li>
<li>ε(cathode) be the efficiency for the cathode to eject a photoelectron</li>
</ul>
Then, the number of photoelectrons produced at the photocathode is
</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Gam-eqn1b.png|330px|center]]
</p>
</td></table>

<p>Typical values for the efficiencies for NaI detectors are:
<ul>
<li>ε(light) = 0.1</li>
<li>ε(coll) = 0.4</li>
<li>ε(cathode) = 0.2</li>
</ul>
</p>
<p>For a 1MeV gamma ray, this yields about 3x10<sup>3</sup> photoelectrons at the photocathode. All the processes mentioned above are subject to statistical fluctuations, and contribute to the broadening of the line width. In addition, there is a contribution from the statistical process due to the multiplication of the photoelectrons in the stages of the PMT.</p>

<h2>The Multichannel Analyser (MCA)</h2>
<p> The electrons collected at the anode of the PMT pass directly into the Spectrum Techniques MCA (UCS 30). This pulse of electrons enters the internal pre-amplifier, followed by the internal amplifier. The voltage across the internal amplifier resistance is digitized into one of 1024 bins according to its maximum value (peak height)- this process of converting an analog voltage into a digital value is called analog-to-digital conversion (ADC). The MCA performs this task for all pulses and creates a histogram of counts in each ADC bin.</p>
<p>There are several important parameters which affect the performance of the MCA.
<ul>

<li><b>Lower Level Discriminator (LLD) :</b> The amount above the average background which is required for the MCA to accept a particular pulse and record its statistics in the histogram.</li>
<li><b>Run Time: </b> The actual length of time for which to acquire data.</li>

<li><b>Live Time:</b> The time the detector is actually able to detect pulses.
<li><b>Dead Time:</b> The time during collection when the detector is unable to process additional events. Look up live time and dead time and discuss in your report.</li>
<li><b>High Voltage: </b> The parameter "high voltage" in the MCA client software does nothing. Bridgeport makes an HV supply for PMTs which connect directly to the eMorpho. We are using a separate power supply, hence adjusting this parameter in the software does nothing. However, changing the 1000V supplied to the PMT from the power supply will mean that more electrons are collected per pulse. If this dc voltage is set too high and and too many photons are present, there could be a buildup of charge inside the PMT and catastrophic damage could occur.<i>Please do not change the setting of the voltage to the PMT from 1000V.</i></li>
<li><b>Hold off time: </b>This is the time after detection of a pulse for which the MCA will not register the detection of another pulse. In practice, this should be set to a few times the 1/e decay time of a pulse to avoid another trigger event occurring from the decay of the pulse which was just recorded.</li>
</ul>
</p>

<p> Turn the power on the spectrum analyzer (UCS 30) and run the program "USX" located on the desktop. </p>



<p>The signal from the PMT is directly input to a multichannel analyser which is a device that sorts incoming pulses according to pulse height and keeps count of the number at each height in a multichannel memory. The contents of each channel is displayed on a screen to give a pulse height spectrum, which is then analysed. The amplitude of the incoming pulse is digitized with an Analogue to Digital Converter (ADC), and sorting is done based on how many pulses had a particular value of the digitized amplitude. The total number of channels into which the voltage range is digitized determines the resolution of the MCA. Read the Bridgeport eMorpho manual to understand the characteristics of the electronics you are supplied with, and to learn more about the ADC range and resolution that can be achieved. Read ref. 1, 2, 4 to understand the functions of a MCA. Review what is meant by ‘dead time’ and ‘live time’. Do not operate with a dead time of more than 30%, since too high count rates can cause the electronics to misbehave.</p>
<p>To understand this experiment it is crucial to observe the pulses fed from the scintillator/PMT to the MCA. This can be done using the Bridgeport eMorpho Client v1.0 on the desktop. This is the software which controls the parameters of the MCA and displays the acquired data. By selecting the "Pulse" icon along the top, you can observe the shape of pulses coming from the MCA. You will observe high-amplitude pulses (high energy was recorded since many visible photons were produced), as well as lower-amplitude pulses occurring randomly in rapid succession. The MCA records the pulse heights and the software assembles them into a histogram according to channel number. One can then calibrate the channels using known gamma sources.</p>
<p>The natural linewidth of the gamma rays is extremely narrow (a few eV compared to the MeV range of the energies themselves!). The broadening observed in the recorded spectra is a result of the detection method (a cooled Germanium detector would show these lines being much narrower, but again at a resolution that depends on the detector itself). It is important to realize on the example of the Cs137 spectrum, for which a single energy at 0.662 MeV is expected that several effects occur in ‘real life’:</p>
<ol>
<li>Broadening of the photopeak at 0.662 MeV.</li>
<li>A Compton background (plateau) with two edges: at the higher end, below the
photopeak a maximum electron(!) energy of
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Gam-eqn2.png|170px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
Understand this result using eq. (1). At lower energies in the Compton plateau a distinct photopeak corresponding to backscattered gamma rays whose energy follows from eq. (1) for θ = π. Note how nearly backscattered gammas give about the same energy due to the slow variation of the cos function for θ = π.</li>
<li>A fluorescent K-shell Xray peak caused by soft gamma rays hitting the shielding material (typically lead), knocking out K-shell electrons. The vacancies are typically filled from the L-III shell (2p±1). Use the CRC Handbook (see Lab Technologist or Library) to verify that the difference in energy results in characteristic X rays of about 75 keV for Pb.</li>
</ol>
<p>A typical spectrum is shown in Fig. 2 on a linear scale. If we go to a logarithmic scale (the MCA software allows to do this), we find an additional weak peak at about twice the gamma energy. This is the so-called sum peak, which arises when two gamma rays from uncorrelated decay events are depositing their energy in the scintillator within a fraction of a microsecond, i.e., the timescale over which the scintillation in the crystal and photomultiplication in the tube occurs. You should be able to observe sum peaks in this experiment when accumulating enough statistics. The peaks appear more strongly when the source is brought closer to the PMT. </p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig3.png|500px|border|center]]
<b>Figure 3 -</b> Typical Cs-137 spectrum using an NaI(T1) scintillator.
<br clear=right>
</p>
</td></table>
<p>Besides sum peaks another complication can occur for high-energy gamma rays. When pair creation is an important energy deposition mechanism, so-called escape peaks are observed. These correspond to events where one or more of the created electrons/positrons escape the crystal without giving up their energy. Such peaks occur at ''E<sub>g</sub>'' - ''j'' (0.511 MeV), where ''j'' is the number of escaped electrons/positrons. Since we do not use sources with ''E<sub>g</sub>'' > 2 MeV in this experiment, we do not find this complication in our spectra.</p>

<h1>Experimental Procedure</h1>
<p>While conducting the experiment, make sure that only the source whose spectrum you are observing is near the apparatus otherwise your calibration results will be skewed. Take only one source out at a time, and keep the others in the box away from the detector. Be careful while handling sources. Become familiar with the documentation and all the apparatus before starting. </p>
<h2>Settings</h2>
<p>Please ensure the following running parameters</p>
<ul>
<li>High-Voltage to PMT: +1000V (set on the actual power supply)</li>
<li>Electronic Gain: 2</li>
<li>Digital Gain: 4096</li>
<li>Integration Time: 0.625</li>
</ul>
<p> <b>Note:</b> The <b>trigger threshold</b> located under the Settings tab should be set to 14. </p>

<p>Ensure the following connections: the scintillator and PMT is connected, the HV output from the power uspply is connected to the PMT, the signal output from the PMT is connected to the MCA, and the MCA is connected to the computer.</p>
<p>In our setup, we have mounted the PMT vertically, allowing you to place the source on a tray at several different distances from the source. Once you have found an optimal distance for all three sources (i.e. you do not get ‘pile up’ effects (see Leo)), it is best to use this distance for calibration and determination of the peak energies. However, in the last part of the experiment you will be asked to draw qualitative conclusions by placing a source at varying distances. </p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig4.png|700px|border|center]]
<b>Figure 4 -</b> Gamma source, NaI(T1) detector, PMT, lead shield showing relevant processes.
<br clear=right>
</p>
</td></table>
<ol>
<h3>Calibration Curve</h3>
<li><p>Use Experiment 2.1 as a guide for gamma ray calibration from Duggan in the red binder in the lab. The eMorpho software will only be used for data collection and observing the spectra. All data analysis, including energy calibration, can be conducted using software like Excel or Mathematica. Use the 5 known sources Cs-137 (0.662 MeV), Co-60 (1.173 MeV, 1.332 MeV), Mn-54 (0.835 MeV), Na-22 (0.511 MeV) and Zn-65 (1.115 MeV) provided in the RSS 8 radiation source kit (Spectrum Techniques). Centre each source (label down) about 5 cm from the face of the detector and collect a spectrum for at least 5 minutes with each source. Longer collection times correspond to smaller statistical errors, why? <b>Stop collection prior to saving and record the dead time, live time and real time for each spectrum.</b> Save the spectrum. (The .csv file generated contains a single column with MCA parameters in the first 74 entries followed by the spectrum data (number of events) in each channel from 0 to 4096. Prior to data analysis delete the first 74 entries. Create a calibration curve using all five sources i.e. for each spectrum locate the channel number corresponding to the gamma ray energy peaks and create a plot of the energy of each peak and the peak channel number. What is the error of a straight line fit to this curve? If the fit is poor, you have made a mistake in assigning peak-channel values, and may have to repeat the analysis. Discuss sources of error. </p>

<p>Using your calibration fit plot the calibrated Cs-137 spectrum, Co-60 spectrum and Na-22 spectrum separately(i.e. events as a function of energy). Explain all the peaks on the spectra. Do you see any 'sum peaks’? (Plot your data on a logarithmic scale for a better view!) Explain why this peak occurs. Does it occur at the value you expect? What is the source of the 511 keV peak in the Na-22 spectrum? The Na-22 has a gamma ray energy at 1275 keV. What value of the maximum position do you find based on your calibration curve? Is this consistent with the resolution ''dE/E''? What is the present activity of these three sources (note the date marked on the source)? Present the calibrated spectra in your report and explain all the interesting details.</p> </li>

<li><p> Put a sheet of lead on the shelf and then place the Co-60 source above. How does the spectrum change? Now place the sheet of lead on top of the source. What, if anything, has changed in the spectrum? You may have to move the source closer or farther from the detector to observe the backscattering signals. Collect a spectrum for each case for a sufficient collection time and save. <b> Caution: Always wear gloves when handling the lead sheets.</b></p> </li>

<li> Obtain the energy resolution for the photopeak for Cs-137 and for one of the Co-60 peaks (See Knoll, chapter 4). Does this agree with what you have learned about scintillator devices? </li>
<li> Place the Cs-137 source at varying distances from the detector and explain qualitatively the rates that you observe. Put the source at slot 2 below the detector, and place first one, then two, and then three sheets of lead on top of the source. Comment on the rates, and the reason for the observed behaviour. Repeat for sheets of aluminum.</li>
<li> A spectrum of the unknown source (the background in the room) will be collected by the Lab Technologist. The data will be saved and the file can be found in the "data" folder located on the desktop. Using this data, create a calibrated spectrum using your calibration fit. Use your now-calibrated gamma-ray spectrum to identify the radioisotope(s) found in the background.</li>
</ol>

<h1>References</h1>
<ol>
<li>Preston and Dietz,[https://www.library.yorku.ca/find/Record/1038893 ''The Art of Experimental Physics''], Wiley.</li>
<li>G.F. Knoll, ''Radiation Detection and Measurement, 3rd or 4th ed.'', Wiley. </li>
<li>J.L. Duggan, ''Laboratory Investigations in Nuclear Science'', Tennelec.</li>
<li>A.C. Melissinos, [https://www.library.yorku.ca/find/Record/1641963 ''Experiments in Modern Physics''], Academic Press.</li>
<li>Leo, [https://www.library.yorku.ca/find/Record/1178503 ''Techniques for Nuclear and Particle Physics Experiments''], Springer-Verlag.</li>
<li>H. Frauenfelder and Henley,[https://www.library.yorku.ca/find/Record/51764 ''Subatomic Physics''] ,Prentice-Hall.</li>
<li>K. Siegbahn, [https://www.library.yorku.ca/find/Record/51739 ''Alpha, Beta, and Gamma-Ray Spectroscopy''], vol. I, chpts 5,8a.</li>
</ol>

Main Page/PHYS 4210 & 4211

2021-12-17T16:23:47Z

Gloria:

<h1>PHYS 4210 & 4211 3.0 </h1>

<p>Selected advanced experiments in physics related to
topics in solid state physics, atomic spectroscopy, microwaves, low-noise measurements, superconductivity, and nuclear and particle physics.

Prerequisites: SC/PHYS 3220 3.00; registration in a Bachelor or Honours Program in physics and astronomy or in biophysics. Co-requisite: SC/PHYS 3040 6.00.</p>

<p> See the course [http://moodle.yorku.ca eClass] for all administrative details. </p>

<h1>Laboratory Manual</h1>

IMPORTANT NOTE: Winter 2021 PHYS 4211 Lab Scheduling procedures will be announced via the Moodle eClass course website http://moodle.yorku.ca/ prior to the beginning of term.



<ul>
<li> [[Main Page/PHYS 4210/How to Choose Experiments|How to Choose Experiments]] </li>
<li> [[Main Page/PHYS 4210/How to Write Reports|How to Write Reports]] </li>
<li> [[Main Page/PHYS 3220/Lab Safety|Lab Safety]] </li>
</ul>

<h2> List of Available Experiments </h2>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 126 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]]</td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Muon Lifetime|Muon Lifetime]]</td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Waveguides|Waveguides*]] </td><td> 111 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Mass Spectrometer|Mass Spectrometer]]</td><td> 150 PSE</td><td>Currently Not Available</td></tr>
<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 150 PSE</td><td>Currently Not Available</td></tr>
<tr><td>[[Main Page/PHYS 4210/He-Ne Lasers|He-Ne Lasers*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Zeeman Effect|Zeeman Effect*]] </td><td> 108 PSE</td><td>Currently Not Available</td></tr>

<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Electron Spin Resonance|Electron Spin Resonance]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Coaxial Cable|Coaxial Cable]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/High-TC Superconductivity|High-TC Superconductivity*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Fourier Optics|Fourier Optics*]] </td><td> 126 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Rutherford II|Rutherford Scattering II]]</td><td> 111 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Gamma Ray Spectroscopy|Gamma Ray Spectroscopy*]] </td><td> 111 PSE</td><td>Currently Not Available</td></tr>








<p> Experiments with an asterisk (*) are better suited for PHYS 4211 students. </p>





















<p> </p>

Main Page/PHYS 4210/Gamma Ray Spectroscopy

2021-12-10T20:08:28Z

Gloria:

<h1>Gamma Ray Spectroscopy</h1>
<p>In this experiment we study the gamma ray spectra of several radioactive elements to learn about the interaction properties of gamma rays with matter. The gamma rays are detected through ionization of the material in a scintillation counter, and the output pulse, which is generated with a photomultiplier tube, is then recorded with the aid of a Multi Channel Analyser (MCA) connected to a computer interface. The different types of interaction of gamma rays with matter are understood from a detailed analysis of the observed spectra. Thus, this experiment not only illustrates the physics of the interaction properties of photons, but also introduces scintillation detectors and relevant electronics.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Radioactive Decay</li>
<li>Gamma Rays</li>
<li>Multi-Channel Analyser (MCA)</li>
<li>Photo-Electric Effect</li>
<li>Compton Scattering</li>
<li>Pair Production</li>

</ul>
</td>
<td width=250>
<ul>
<li>Crystal Detector</li>
<li>Photomultiplier Tube</li>
<li>Back Scatter Peak</li>
<li>Compton Edge</li>
<li>Compton Plateau</li>
<li>Photopeak</li>
</ul>
</td>
</table>

<h1>Introduction</h1>
<p>Gamma rays are photons of very short wavelength (~10<sup>-12</sup> cm) or very high frequency (10<sup>22</sup> Hz) that are emitted during nuclear transitions. The decay schemes of the three radionuclides that we study (Na22, Cs137, Co60) are shown below. Read ref. 1-5 to get a clear idea of the significance of the gamma energies.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Decay schemes of Cs-137, Co-60, and Na-22.
<br clear=right>
</p>
</td></table>

<h2>Method</h2>
<p>Photons are electrically neutral, and unlike charged particles, do not experience the Coulomb force. They are, however, carriers of the electromagnetic force, and are able to ionize atoms through their interaction with matter, and this leads to the deposition of energy in the medium as the ionized particle slows down in traversing the medium. This energy can then be detected. The three modes of interaction are: photoelectric effect, Compton scattering, and pair production where the photon interacts with an atom, an electron, and a nucleus respectively. You should read the details of each type of interaction in the references and include this in your write-up. We summarise it briefly below.</p>

<h3>Photoelectric effect</h3>
<p>The photon of energy ''E<sub>g</sub>'' is absorbed by an atom and an electron from one of the shells is emitted. If ''B<sub>e</sub>'' is the binding energy of the electron, then the energy of the emitted electron will have an energy ''E<sub>e</sub>'' = ''E<sub>g</sub>'' - ''B<sub>e</sub>''. Since ''B<sub>e</sub>'' is small (of the order 40 KeV) compared to ''E<sub>g</sub>'' (of the order 1MeV), the electron carries most of the energy of the photon.</p>

<h3>Compton scattering</h3>
<p>The photon scatters off an electron that is either free or is loosely bound in the atom, thereby scattering the electron. It can be shown (do so in your writeup) that the energy of the scattered electron is related to that of the incident photon, the angle of the scattered photon (theta), and the mass of the electron me through the relationship:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Gam-eqn1.png|190px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig2.png|500px|border|center]]
<b>Figure 2 -</b> Compton scattering.
<br clear=right>
</p>
</td></table>
<p>Hence the scattered photon (of energy hv’) has the freedom to move at any angle with respect to the incident photon (or gamma ray, as we call it here), whereas the scattered electron is bound by the laws of conservation of momentum to only go in the forward direction. The kinetic energy of the Compton electron is E<sub>C</sub> = hv- hv’. This, combined with the above formula for the energy of the electron shows that the maximum kinetic energy of the electron is when theta = 180° ,i.e., when the photon is scattered backwards. This maximum is known as the Compton edge. </p>

<h3>Pair production</h3>
<p>If the photon has more than twice the rest mass of an electron (0.511 MeV), the photon can produce an electron-positron pair. This must be done in the Coulomb field of the nucleus to balance linear momentum, as the photon cannot produce a pair in free space (its center of mass would have zero momentum). The energy in excess of the rest masses of the products is imparted as kinetic energy. </p>
<p>The energy dependencies of the three processes are quite different. At energies below a few KeV, the photoelectric effect dominates and the Coulomb effect is small, while pair production is energetically impossible. For the region of 0.1 to about 10 MeV, Compton scattering dominates. Above this, pair production is the predominant method for the interaction of the photons. It is important to realize that in photoelectric effect and pair production, the photon is eliminated in the process of the interaction, whereas in Compton scattering, the energy of the photon is only degraded. As explained further below the various processes involved lead to a complete or incomplete energy deposition of the gamma energy in the scintillator. We have to be concerned only with the first two methods; the photoelectric effect leads to complete energy deposition, while the Compton effect can lead to processes where the scattered gamma ray leaves the detector without depositing its energy.</p>

<h2>The Radioactive Sources</h2>
<p>You will be working with five radioactive sources: Mn-54, Zn-65, Na-22, Cs-137, and Co-60. You should note the radiation dosage and date marked on the source, and read about radiation safety and how to handle radiation sources from ref. 1, pg 326-328. Calculate the number of disintegrations per second for each of the sources based on the quoted dosage.</p>

<h2>The Detector</h2>
<p>The gamma rays that are emitted from the source are detected with a scintillator detector. In our laboratory, we use inorganic crystals of sodium iodide doped with impurity centres of thallium. This combination is denoted by NaI(Tl). A photon entering the detector ionizes the material through the processes described above. The positive ions and electrons created by the incoming photon diffuse through the lattice and are captured by the impurity centres i.e. the Tl atoms, which act as luminescence centres. Recombination produces an excited centre,which emits visible light upon its return to the ground state. The efficiency of these inorganic crystals is high, but the light output is spread over a time interval of the order of microseconds. </p>
<p>In general, for gamma rays less than 1 MeV, photons undergoing the photoelectric process will be completely absorbed and will give rise to a determined number of light quanta. These light quanta are then seen by the photomultiplier tube, and the output pulse will be proportional to the incoming energy of the gamma ray. The energy of the electrons produced by the Compton effect will depend on the angle at which they are scattered and hence there will be a spectrum of detected pulses. At energies above 1 MeV, pair production can occur and the electron and positron lose their kinetic energy by ionizing the medium. The electron is absorbed, while the positron annihilates within a few nanoseconds into two photons, each of 0.511 MeV. These may then interact through the photoelectric or Compton effect. There are three possibilities for the observed energy (show this in the writeup) depending on whether none, one or both of these photons from the positron annihilation are detected.</p>
<p>As mentioned, the produced light is proportional to the energy of the incident gamma ray. This light is then made incident on the photocathode face of a photomultiplier tube (PMT), which converts the photons to photoelectrons through the photoelectric effect. The electrons are amplified and then converted to a voltage pulse. The PMT used here has 10 stages and the multiplication for such tubes is about 10<sup>6</sup> for an operational voltage between the cathode and the anode of 1000V. The cascade of electrons produced at each of the multiplication stages of the dynodes is collected at the anode, converted to a voltage pulse, which is then amplified and analysed. The PMT is enclosed in a high permeability material to shield against magnetic fields, as such effects would affect the efficiency of the tube.</p>
<p>Note that the metal shield that encloses the scintillator crystal does not allow β (or other charged) particles to permeate and deposit their usually well-defined energies. Thus, you should not expect to observe structures associated with, e.g., the 0.514 MeV and 1.17 MeV electrons coming from the Cs137 source with branching ratios of 93.5% and 6.5% respectively (cf.. Fig. 1). Charged particles are created inside the crystal by γ ray impact. There will be, however, backscattered γ rays from Compton scattering outside the crystal, e.g., off the backing of the source and off the lead shield.</p>
<p>Read the specifications for the model of the scintillator detector and photomultiplier that you have. Note that the size of the crystal is small. How does this affect your results? The typical quantum efficiency of scintillators of this type is one photon of light produced per 100 eV of energy deposited in the scintillator. The width of the full-energy peak (i.e. when the gamma energy is fully absorbed) depends on the number of light quanta produced by the incident gamma ray. The energy resolution, ''dE/E'' is an important quantity to consider as this factor will determine if we can separate gamma rays very close in energy.</p>
<p>The chain of events may help illustrate this point. The incident gamma ray of energy ''E<sub>g</sub>'' produces a photoelectron with energy ''E<sub>e</sub>'' ~ ''E<sub>g</sub>''. The photoelectron produces N light quanta in the scintillator material via ionization and excitation, each with an energy ''E<sub>lq</sub>''. Since the light is visible (~400 nm), ''E<sub>lq</sub>'' is about 3 eV. This light falls on the photocathode, which has a quantum efficiency or sensitivity to the wavelength of the incident light. For the tube supplied, the efficiency peaks near 400 nm, making it quite suitable for use with a scintillator. Let
<ul>
<li>ε(light) be the efficiency for the conversion of the excitation energy into light quanta</li>
<li>ε(coll) be the efficiency for collecting the n lq light quanta at the photocathode of the PMT</li>
<li>ε(cathode) be the efficiency for the cathode to eject a photoelectron</li>
</ul>
Then, the number of photoelectrons produced at the photocathode is
</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Gam-eqn1b.png|330px|center]]
</p>
</td></table>

<p>Typical values for the efficiencies for NaI detectors are:
<ul>
<li>ε(light) = 0.1</li>
<li>ε(coll) = 0.4</li>
<li>ε(cathode) = 0.2</li>
</ul>
</p>
<p>For a 1MeV gamma ray, this yields about 3x10<sup>3</sup> photoelectrons at the photocathode. All the processes mentioned above are subject to statistical fluctuations, and contribute to the broadening of the line width. In addition, there is a contribution from the statistical process due to the multiplication of the photoelectrons in the stages of the PMT.</p>

<h2>The Multichannel Analyser (MCA)</h2>
<p> The electrons collected at the anode of the PMT pass directly into the Spectrum Techniques MCA (UCS 30). This pulse of electrons is passed across a resistor whose value is set by the user (using the "electronic gain" parameter). The voltage across this resistor is digitized into one of 1024 bins according to its maximum value (peak height)- this process of converting an analog voltage into a digital value is called analog-to-digital conversion (ADC). The MCA performs this task for all pulses and creates a histogram of counts in each ADC bin.</p>
<p>There are several important parameters which affect the performance of the MCA.
<ul>
<li><b>Electronic Gain: </b> As mentioned above, setting this parameter changes the value of the resistance that the PMT current is being passed through. </li>
<li><b>Digital Gain: </b> Once passing through the ADC, the resulting value can then be further scaled by this amount. This is useful if one is trying to match up bin number to a calibrated energy in eV.</li>
<li><b>Lower Level Discriminator (LLD) :</b> The amount above the average background which is required for the MCA to accept a particular pulse and record its statistics in the histogram.</li>
<li><b>Run Time: </b> The actual length of time for which to acquire data.</li>

<li><b>Live Time:</b> The time the detector is actually able to detect pulses.
<li><b>Dead Time:</b> The time during collection when the detector is unable to process additional events. Look up live time and dead time and discuss in your report.</li>
<li><b>High Voltage: </b> The parameter "high voltage" in the MCA client software does nothing. Bridgeport makes an HV supply for PMTs which connect directly to the eMorpho. We are using a separate power supply, hence adjusting this parameter in the software does nothing. However, changing the 1000V supplied to the PMT from the power supply will mean that more electrons are collected per pulse. If this dc voltage is set too high and and too many photons are present, there could be a buildup of charge inside the PMT and catastrophic damage could occur.<i>Please do not change the setting of the voltage to the PMT from 1000V.</i></li>
<li><b>Hold off time: </b>This is the time after detection of a pulse for which the MCA will not register the detection of another pulse. In practice, this should be set to a few times the 1/e decay time of a pulse to avoid another trigger event occurring from the decay of the pulse which was just recorded.</li>
</ul>
</p>

<p> Turn the power on the spectrum analyzer (UCS 30) and run the program "USX" located on the desktop. </p>



<p>The signal from the PMT is directly input to a multichannel analyser which is a device that sorts incoming pulses according to pulse height and keeps count of the number at each height in a multichannel memory. The contents of each channel is displayed on a screen to give a pulse height spectrum, which is then analysed. The amplitude of the incoming pulse is digitized with an Analogue to Digital Converter (ADC), and sorting is done based on how many pulses had a particular value of the digitized amplitude. The total number of channels into which the voltage range is digitized determines the resolution of the MCA. Read the Bridgeport eMorpho manual to understand the characteristics of the electronics you are supplied with, and to learn more about the ADC range and resolution that can be achieved. Read ref. 1, 2, 4 to understand the functions of a MCA. Review what is meant by ‘dead time’ and ‘live time’. Do not operate with a dead time of more than 30%, since too high count rates can cause the electronics to misbehave.</p>
<p>To understand this experiment it is crucial to observe the pulses fed from the scintillator/PMT to the MCA. This can be done using the Bridgeport eMorpho Client v1.0 on the desktop. This is the software which controls the parameters of the MCA and displays the acquired data. By selecting the "Pulse" icon along the top, you can observe the shape of pulses coming from the MCA. You will observe high-amplitude pulses (high energy was recorded since many visible photons were produced), as well as lower-amplitude pulses occurring randomly in rapid succession. The MCA records the pulse heights and the software assembles them into a histogram according to channel number. One can then calibrate the channels using known gamma sources.</p>
<p>The natural linewidth of the gamma rays is extremely narrow (a few eV compared to the MeV range of the energies themselves!). The broadening observed in the recorded spectra is a result of the detection method (a cooled Germanium detector would show these lines being much narrower, but again at a resolution that depends on the detector itself). It is important to realize on the example of the Cs137 spectrum, for which a single energy at 0.662 MeV is expected that several effects occur in ‘real life’:</p>
<ol>
<li>Broadening of the photopeak at 0.662 MeV.</li>
<li>A Compton background (plateau) with two edges: at the higher end, below the
photopeak a maximum electron(!) energy of
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Gam-eqn2.png|170px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
Understand this result using eq. (1). At lower energies in the Compton plateau a distinct photopeak corresponding to backscattered gamma rays whose energy follows from eq. (1) for θ = π. Note how nearly backscattered gammas give about the same energy due to the slow variation of the cos function for θ = π.</li>
<li>A fluorescent K-shell Xray peak caused by soft gamma rays hitting the shielding material (typically lead), knocking out K-shell electrons. The vacancies are typically filled from the L-III shell (2p±1). Use the CRC Handbook (see Lab Technologist or Library) to verify that the difference in energy results in characteristic X rays of about 75 keV for Pb.</li>
</ol>
<p>A typical spectrum is shown in Fig. 2 on a linear scale. If we go to a logarithmic scale (the MCA software allows to do this), we find an additional weak peak at about twice the gamma energy. This is the so-called sum peak, which arises when two gamma rays from uncorrelated decay events are depositing their energy in the scintillator within a fraction of a microsecond, i.e., the timescale over which the scintillation in the crystal and photomultiplication in the tube occurs. You should be able to observe sum peaks in this experiment when accumulating enough statistics. The peaks appear more strongly when the source is brought closer to the PMT. </p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig3.png|500px|border|center]]
<b>Figure 3 -</b> Typical Cs-137 spectrum using an NaI(T1) scintillator.
<br clear=right>
</p>
</td></table>
<p>Besides sum peaks another complication can occur for high-energy gamma rays. When pair creation is an important energy deposition mechanism, so-called escape peaks are observed. These correspond to events where one or more of the created electrons/positrons escape the crystal without giving up their energy. Such peaks occur at ''E<sub>g</sub>'' - ''j'' (0.511 MeV), where ''j'' is the number of escaped electrons/positrons. Since we do not use sources with ''E<sub>g</sub>'' > 2 MeV in this experiment, we do not find this complication in our spectra.</p>

<h1>Experimental Procedure</h1>
<p>While conducting the experiment, make sure that only the source whose spectrum you are observing is near the apparatus otherwise your calibration results will be skewed. Take only one source out at a time, and keep the others in the box away from the detector. Be careful while handling sources. Become familiar with the documentation and all the apparatus before starting. </p>
<h2>Settings</h2>
<p>Please ensure the following running parameters</p>
<ul>
<li>High-Voltage to PMT: +1000V (set on the actual power supply)</li>
<li>Electronic Gain: 2</li>
<li>Digital Gain: 4096</li>
<li>Integration Time: 0.625</li>
</ul>
<p> <b>Note:</b> The <b>trigger threshold</b> located under the Settings tab should be set to 14. </p>

<p>Ensure the following connections: the scintillator and PMT is connected, the HV output from the power uspply is connected to the PMT, the signal output from the PMT is connected to the MCA, and the MCA is connected to the computer.</p>
<p>In our setup, we have mounted the PMT vertically, allowing you to place the source on a tray at several different distances from the source. Once you have found an optimal distance for all three sources (i.e. you do not get ‘pile up’ effects (see Leo)), it is best to use this distance for calibration and determination of the peak energies. However, in the last part of the experiment you will be asked to draw qualitative conclusions by placing a source at varying distances. </p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig4.png|700px|border|center]]
<b>Figure 4 -</b> Gamma source, NaI(T1) detector, PMT, lead shield showing relevant processes.
<br clear=right>
</p>
</td></table>
<ol>
<h3>Calibration Curve</h3>
<li><p>Use Experiment 2.1 as a guide for gamma ray calibration from Duggan in the red binder in the lab. The eMorpho software will only be used for data collection and observing the spectra. All data analysis, including energy calibration, can be conducted using software like Excel or Mathematica. Use the 5 known sources Cs-137 (0.662 MeV), Co-60 (1.173 MeV, 1.332 MeV), Mn-54 (0.835 MeV), Na-22 (0.511 MeV) and Zn-65 (1.115 MeV) provided in the RSS 8 radiation source kit (Spectrum Techniques). Centre each source (label down) about 5 cm from the face of the detector and collect a spectrum for at least 5 minutes with each source. Longer collection times correspond to smaller statistical errors, why? <b>Stop collection prior to saving and record the dead time, live time and real time for each spectrum.</b> Save the spectrum. (The .csv file generated contains a single column with MCA parameters in the first 74 entries followed by the spectrum data (number of events) in each channel from 0 to 4096. Prior to data analysis delete the first 74 entries. Create a calibration curve using all five sources i.e. for each spectrum locate the channel number corresponding to the gamma ray energy peaks and create a plot of the energy of each peak and the peak channel number. What is the error of a straight line fit to this curve? If the fit is poor, you have made a mistake in assigning peak-channel values, and may have to repeat the analysis. Discuss sources of error. </p>

<p>Using your calibration fit plot the calibrated Cs-137 spectrum, Co-60 spectrum and Na-22 spectrum separately(i.e. events as a function of energy). Explain all the peaks on the spectra. Do you see any 'sum peaks’? (Plot your data on a logarithmic scale for a better view!) Explain why this peak occurs. Does it occur at the value you expect? What is the source of the 511 keV peak in the Na-22 spectrum? The Na-22 has a gamma ray energy at 1275 keV. What value of the maximum position do you find based on your calibration curve? Is this consistent with the resolution ''dE/E''? What is the present activity of these three sources (note the date marked on the source)? Present the calibrated spectra in your report and explain all the interesting details.</p> </li>

<li><p> Put a sheet of lead on the shelf and then place the Co-60 source above. How does the spectrum change? Now place the sheet of lead on top of the source. What, if anything, has changed in the spectrum? You may have to move the source closer or farther from the detector to observe the backscattering signals. Collect a spectrum for each case for a sufficient collection time and save. <b> Caution: Always wear gloves when handling the lead sheets.</b></p> </li>

<li> Obtain the energy resolution for the photopeak for Cs-137 and for one of the Co-60 peaks (See Knoll, chapter 4). Does this agree with what you have learned about scintillator devices? </li>
<li> Place the Cs-137 source at varying distances from the detector and explain qualitatively the rates that you observe. Put the source at slot 2 below the detector, and place first one, then two, and then three sheets of lead on top of the source. Comment on the rates, and the reason for the observed behaviour. Repeat for sheets of aluminum.</li>
<li> A spectrum of the unknown source (the background in the room) will be collected by the Lab Technologist. The data will be saved and the file can be found in the "data" folder located on the desktop. Using this data, create a calibrated spectrum using your calibration fit. Use your now-calibrated gamma-ray spectrum to identify the radioisotope(s) found in the background.</li>
</ol>

<h1>References</h1>
<ol>
<li>Preston and Dietz,[https://www.library.yorku.ca/find/Record/1038893 ''The Art of Experimental Physics''], Wiley.</li>
<li>G.F. Knoll, ''Radiation Detection and Measurement, 3rd or 4th ed.'', Wiley. </li>
<li>J.L. Duggan, ''Laboratory Investigations in Nuclear Science'', Tennelec.</li>
<li>A.C. Melissinos, [https://www.library.yorku.ca/find/Record/1641963 ''Experiments in Modern Physics''], Academic Press.</li>
<li>Leo, [https://www.library.yorku.ca/find/Record/1178503 ''Techniques for Nuclear and Particle Physics Experiments''], Springer-Verlag.</li>
<li>H. Frauenfelder and Henley,[https://www.library.yorku.ca/find/Record/51764 ''Subatomic Physics''] ,Prentice-Hall.</li>
<li>K. Siegbahn, [https://www.library.yorku.ca/find/Record/51739 ''Alpha, Beta, and Gamma-Ray Spectroscopy''], vol. I, chpts 5,8a.</li>
</ol>

Main Page/PHYS 4210/Gamma Ray Spectroscopy

2021-12-10T19:06:46Z

Gloria:

<h1>Gamma Ray Spectroscopy</h1>
<p>In this experiment we study the gamma ray spectra of several radioactive elements to learn about the interaction properties of gamma rays with matter. The gamma rays are detected through ionization of the material in a scintillation counter, and the output pulse, which is generated with a photomultiplier tube, is then recorded with the aid of a Multi Channel Analyser (MCA) connected to a computer interface. The different types of interaction of gamma rays with matter are understood from a detailed analysis of the observed spectra. Thus, this experiment not only illustrates the physics of the interaction properties of photons, but also introduces scintillation detectors and relevant electronics.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Radioactive Decay</li>
<li>Gamma Rays</li>
<li>Multi-Channel Analyser (MCA)</li>
<li>Photo-Electric Effect</li>
<li>Compton Scattering</li>
<li>Pair Production</li>

</ul>
</td>
<td width=250>
<ul>
<li>Crystal Detector</li>
<li>Photomultiplier Tube</li>
<li>Back Scatter Peak</li>
<li>Compton Edge</li>
<li>Compton Plateau</li>
<li>Photopeak</li>
</ul>
</td>
</table>

<h1>Introduction</h1>
<p>Gamma rays are photons of very short wavelength (~10<sup>-12</sup> cm) or very high frequency (10<sup>22</sup> Hz) that are emitted during nuclear transitions. The decay schemes of the three radionuclides that we study (Na22, Cs137, Co60) are shown below. Read ref. 1-5 to get a clear idea of the significance of the gamma energies.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Decay schemes of Cs-137, Co-60, and Na-22.
<br clear=right>
</p>
</td></table>

<h2>Method</h2>
<p>Photons are electrically neutral, and unlike charged particles, do not experience the Coulomb force. They are, however, carriers of the electromagnetic force, and are able to ionize atoms through their interaction with matter, and this leads to the deposition of energy in the medium as the ionized particle slows down in traversing the medium. This energy can then be detected. The three modes of interaction are: photoelectric effect, Compton scattering, and pair production where the photon interacts with an atom, an electron, and a nucleus respectively. You should read the details of each type of interaction in the references and include this in your write-up. We summarise it briefly below.</p>

<h3>Photoelectric effect</h3>
<p>The photon of energy ''E<sub>g</sub>'' is absorbed by an atom and an electron from one of the shells is emitted. If ''B<sub>e</sub>'' is the binding energy of the electron, then the energy of the emitted electron will have an energy ''E<sub>e</sub>'' = ''E<sub>g</sub>'' - ''B<sub>e</sub>''. Since ''B<sub>e</sub>'' is small (of the order 40 KeV) compared to ''E<sub>g</sub>'' (of the order 1MeV), the electron carries most of the energy of the photon.</p>

<h3>Compton scattering</h3>
<p>The photon scatters off an electron that is either free or is loosely bound in the atom, thereby scattering the electron. It can be shown (do so in your writeup) that the energy of the scattered electron is related to that of the incident photon, the angle of the scattered photon (theta), and the mass of the electron me through the relationship:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Gam-eqn1.png|190px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig2.png|500px|border|center]]
<b>Figure 2 -</b> Compton scattering.
<br clear=right>
</p>
</td></table>
<p>Hence the scattered photon (of energy hv’) has the freedom to move at any angle with respect to the incident photon (or gamma ray, as we call it here), whereas the scattered electron is bound by the laws of conservation of momentum to only go in the forward direction. The kinetic energy of the Compton electron is E<sub>C</sub> = hv- hv’. This, combined with the above formula for the energy of the electron shows that the maximum kinetic energy of the electron is when theta = 180° ,i.e., when the photon is scattered backwards. This maximum is known as the Compton edge. </p>

<h3>Pair production</h3>
<p>If the photon has more than twice the rest mass of an electron (0.511 MeV), the photon can produce an electron-positron pair. This must be done in the Coulomb field of the nucleus to balance linear momentum, as the photon cannot produce a pair in free space (its center of mass would have zero momentum). The energy in excess of the rest masses of the products is imparted as kinetic energy. </p>
<p>The energy dependencies of the three processes are quite different. At energies below a few KeV, the photoelectric effect dominates and the Coulomb effect is small, while pair production is energetically impossible. For the region of 0.1 to about 10 MeV, Compton scattering dominates. Above this, pair production is the predominant method for the interaction of the photons. It is important to realize that in photoelectric effect and pair production, the photon is eliminated in the process of the interaction, whereas in Compton scattering, the energy of the photon is only degraded. As explained further below the various processes involved lead to a complete or incomplete energy deposition of the gamma energy in the scintillator. We have to be concerned only with the first two methods; the photoelectric effect leads to complete energy deposition, while the Compton effect can lead to processes where the scattered gamma ray leaves the detector without depositing its energy.</p>

<h2>The Radioactive Sources</h2>
<p>You will be working with five radioactive sources: Mn-54, Zn-65, Na-22, Cs-137, and Co-60. You should note the radiation dosage and date marked on the source, and read about radiation safety and how to handle radiation sources from ref. 1, pg 326-328. Calculate the number of disintegrations per second for each of the sources based on the quoted dosage.</p>

<h2>The Detector</h2>
<p>The gamma rays that are emitted from the source are detected with a scintillator detector. In our laboratory, we use inorganic crystals of sodium iodide doped with impurity centres of thallium. This combination is denoted by NaI(Tl). A photon entering the detector ionizes the material through the processes described above. The positive ions and electrons created by the incoming photon diffuse through the lattice and are captured by the impurity centres i.e. the Tl atoms, which act as luminescence centres. Recombination produces an excited centre,which emits visible light upon its return to the ground state. The efficiency of these inorganic crystals is high, but the light output is spread over a time interval of the order of microseconds. </p>
<p>In general, for gamma rays less than 1 MeV, photons undergoing the photoelectric process will be completely absorbed and will give rise to a determined number of light quanta. These light quanta are then seen by the photomultiplier tube, and the output pulse will be proportional to the incoming energy of the gamma ray. The energy of the electrons produced by the Compton effect will depend on the angle at which they are scattered and hence there will be a spectrum of detected pulses. At energies above 1 MeV, pair production can occur and the electron and positron lose their kinetic energy by ionizing the medium. The electron is absorbed, while the positron annihilates within a few nanoseconds into two photons, each of 0.511 MeV. These may then interact through the photoelectric or Compton effect. There are three possibilities for the observed energy (show this in the writeup) depending on whether none, one or both of these photons from the positron annihilation are detected.</p>
<p>As mentioned, the produced light is proportional to the energy of the incident gamma ray. This light is then made incident on the photocathode face of a photomultiplier tube (PMT), which converts the photons to photoelectrons through the photoelectric effect. The electrons are amplified and then converted to a voltage pulse. The PMT used here has 10 stages and the multiplication for such tubes is about 10<sup>6</sup> for an operational voltage between the cathode and the anode of 1000V. The cascade of electrons produced at each of the multiplication stages of the dynodes is collected at the anode, converted to a voltage pulse, which is then amplified and analysed. The PMT is enclosed in a high permeability material to shield against magnetic fields, as such effects would affect the efficiency of the tube.</p>
<p>Note that the metal shield that encloses the scintillator crystal does not allow β (or other charged) particles to permeate and deposit their usually well-defined energies. Thus, you should not expect to observe structures associated with, e.g., the 0.514 MeV and 1.17 MeV electrons coming from the Cs137 source with branching ratios of 93.5% and 6.5% respectively (cf.. Fig. 1). Charged particles are created inside the crystal by γ ray impact. There will be, however, backscattered γ rays from Compton scattering outside the crystal, e.g., off the backing of the source and off the lead shield.</p>
<p>Read the specifications for the model of the scintillator detector and photomultiplier that you have. Note that the size of the crystal is small. How does this affect your results? The typical quantum efficiency of scintillators of this type is one photon of light produced per 100 eV of energy deposited in the scintillator. The width of the full-energy peak (i.e. when the gamma energy is fully absorbed) depends on the number of light quanta produced by the incident gamma ray. The energy resolution, ''dE/E'' is an important quantity to consider as this factor will determine if we can separate gamma rays very close in energy.</p>
<p>The chain of events may help illustrate this point. The incident gamma ray of energy ''E<sub>g</sub>'' produces a photoelectron with energy ''E<sub>e</sub>'' ~ ''E<sub>g</sub>''. The photoelectron produces N light quanta in the scintillator material via ionization and excitation, each with an energy ''E<sub>lq</sub>''. Since the light is visible (~400 nm), ''E<sub>lq</sub>'' is about 3 eV. This light falls on the photocathode, which has a quantum efficiency or sensitivity to the wavelength of the incident light. For the tube supplied, the efficiency peaks near 400 nm, making it quite suitable for use with a scintillator. Let
<ul>
<li>ε(light) be the efficiency for the conversion of the excitation energy into light quanta</li>
<li>ε(coll) be the efficiency for collecting the n lq light quanta at the photocathode of the PMT</li>
<li>ε(cathode) be the efficiency for the cathode to eject a photoelectron</li>
</ul>
Then, the number of photoelectrons produced at the photocathode is
</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Gam-eqn1b.png|330px|center]]
</p>
</td></table>

<p>Typical values for the efficiencies for NaI detectors are:
<ul>
<li>ε(light) = 0.1</li>
<li>ε(coll) = 0.4</li>
<li>ε(cathode) = 0.2</li>
</ul>
</p>
<p>For a 1MeV gamma ray, this yields about 3x10<sup>3</sup> photoelectrons at the photocathode. All the processes mentioned above are subject to statistical fluctuations, and contribute to the broadening of the line width. In addition, there is a contribution from the statistical process due to the multiplication of the photoelectrons in the stages of the PMT.</p>

<h2>The Multichannel Analyser</h2>
<p> The electrons collected at the anode of the PMT pass directly into the Spectrum Techniques MCA (UCS 30). This pulse of electrons is passed across a resistor whose value is set by the user (using the "electronic gain" parameter). The voltage across this resistor is digitized into one of 1024 bins according to its maximum value (peak height)- this process of converting an analog voltage into a digital value is called analog-to-digital conversion (ADC). The MCA performs this task for all pulses and creates a histogram of counts in each ADC bin.</p>
<p>There are several important parameters which affect the performance of the MCA.
<ul>
<li><b>Electronic Gain: </b> As mentioned above, setting this parameter changes the value of the resistance that the PMT current is being passed through. The allowed values are 0-100 Ohm, 1-430 Ohm, 2- 1100 Ohm, 4- 3400 Ohm.</li>
<li><b>Digital Gain: </b> Once passing through the ADC, the resulting value can then be further scaled by this amount. This is useful if one is trying to match up bin number to a calibrated energy in eV.</li>
<li><b>Trigger Threshhold :</b> The amount above the average background which is required for the MCA to accept a particular pulse and record its statistics in the histogram.</li>
<li><b>Run Time: </b> The length of time for which to acquire data.</li>
<li><b>Integration Time: </b> Each combination of scintillator/PMT will have a characteristic decay time for the pulses. This parameter sets the time for which the MCA averages the data, and should be set to roughly the 1/e time of the pulse decay.</li>
<li><b>High Voltage: </b> The parameter "high voltage" in the MCA client software does nothing. Bridgeport makes an HV supply for PMTs which connect directly to the eMorpho. We are using a separate power supply, hence adjusting this parameter in the software does nothing. However, changing the 1000V supplied to the PMT from the power supply will mean that more electrons are collected per pulse. If this dc voltage is set too high and and too many photons are present, there could be a buildup of charge inside the PMT and catastrophic damage could occur.<i>Please do not change the setting of the voltage to the PMT from 1000V.</i></li>
<li><b>Hold off time: </b>This is the time after detection of a pulse for which the MCA will not register the detection of another pulse. In practice, this should be set to a few times the 1/e decay time of a pulse to avoid another trigger event occurring from the decay of the pulse which was just recorded.</li>
</ul>
</p>

<p> Turn the power on the spectrum analyzer (UCS 30) and run the program "USX" located on the desktop. </p>



<p>The signal from the PMT is directly input to a multichannel analyser which is a device that sorts incoming pulses according to pulse height and keeps count of the number at each height in a multichannel memory. The contents of each channel is displayed on a screen to give a pulse height spectrum, which is then analysed. The amplitude of the incoming pulse is digitized with an Analogue to Digital Converter (ADC), and sorting is done based on how many pulses had a particular value of the digitized amplitude. The total number of channels into which the voltage range is digitized determines the resolution of the MCA. Read the Bridgeport eMorpho manual to understand the characteristics of the electronics you are supplied with, and to learn more about the ADC range and resolution that can be achieved. Read ref. 1, 2, 4 to understand the functions of a MCA. Review what is meant by ‘dead time’ and ‘live time’. Do not operate with a dead time of more than 30%, since too high count rates can cause the electronics to misbehave.</p>
<p>To understand this experiment it is crucial to observe the pulses fed from the scintillator/PMT to the MCA. This can be done using the Bridgeport eMorpho Client v1.0 on the desktop. This is the software which controls the parameters of the MCA and displays the acquired data. By selecting the "Pulse" icon along the top, you can observe the shape of pulses coming from the MCA. You will observe high-amplitude pulses (high energy was recorded since many visible photons were produced), as well as lower-amplitude pulses occurring randomly in rapid succession. The MCA records the pulse heights and the software assembles them into a histogram according to channel number. One can then calibrate the channels using known gamma sources.</p>
<p>The natural linewidth of the gamma rays is extremely narrow (a few eV compared to the MeV range of the energies themselves!). The broadening observed in the recorded spectra is a result of the detection method (a cooled Germanium detector would show these lines being much narrower, but again at a resolution that depends on the detector itself). It is important to realize on the example of the Cs137 spectrum, for which a single energy at 0.662 MeV is expected that several effects occur in ‘real life’:</p>
<ol>
<li>Broadening of the photopeak at 0.662 MeV.</li>
<li>A Compton background (plateau) with two edges: at the higher end, below the
photopeak a maximum electron(!) energy of
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Gam-eqn2.png|170px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
Understand this result using eq. (1). At lower energies in the Compton plateau a distinct photopeak corresponding to backscattered gamma rays whose energy follows from eq. (1) for θ = π. Note how nearly backscattered gammas give about the same energy due to the slow variation of the cos function for θ = π.</li>
<li>A fluorescent K-shell Xray peak caused by soft gamma rays hitting the shielding material (typically lead), knocking out K-shell electrons. The vacancies are typically filled from the L-III shell (2p±1). Use the CRC Handbook (see Lab Technologist or Library) to verify that the difference in energy results in characteristic X rays of about 75 keV for Pb.</li>
</ol>
<p>A typical spectrum is shown in Fig. 2 on a linear scale. If we go to a logarithmic scale (the MCA software allows to do this), we find an additional weak peak at about twice the gamma energy. This is the so-called sum peak, which arises when two gamma rays from uncorrelated decay events are depositing their energy in the scintillator within a fraction of a microsecond, i.e., the timescale over which the scintillation in the crystal and photomultiplication in the tube occurs. You should be able to observe sum peaks in this experiment when accumulating enough statistics. The peaks appear more strongly when the source is brought closer to the PMT. </p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig3.png|500px|border|center]]
<b>Figure 3 -</b> Typical Cs-137 spectrum using an NaI(T1) scintillator.
<br clear=right>
</p>
</td></table>
<p>Besides sum peaks another complication can occur for high-energy gamma rays. When pair creation is an important energy deposition mechanism, so-called escape peaks are observed. These correspond to events where one or more of the created electrons/positrons escape the crystal without giving up their energy. Such peaks occur at ''E<sub>g</sub>'' - ''j'' (0.511 MeV), where ''j'' is the number of escaped electrons/positrons. Since we do not use sources with ''E<sub>g</sub>'' > 2 MeV in this experiment, we do not find this complication in our spectra.</p>

<h1>Experimental Procedure</h1>
<p>While conducting the experiment, make sure that only the source whose spectrum you are observing is near the apparatus otherwise your calibration results will be skewed. Take only one source out at a time, and keep the others in the box away from the detector. Be careful while handling sources. Become familiar with the documentation and all the apparatus before starting. </p>
<h2>Settings</h2>
<p>Please ensure the following running parameters</p>
<ul>
<li>High-Voltage to PMT: +1000V (set on the actual power supply)</li>
<li>Electronic Gain: 2</li>
<li>Digital Gain: 4096</li>
<li>Integration Time: 0.625</li>
</ul>
<p> <b>Note:</b> The <b>trigger threshold</b> located under the Settings tab should be set to 14. </p>

<p>Ensure the following connections: the scintillator and PMT is connected, the HV output from the power uspply is connected to the PMT, the signal output from the PMT is connected to the MCA, and the MCA is connected to the computer.</p>
<p>In our setup, we have mounted the PMT vertically, allowing you to place the source on a tray at several different distances from the source. Once you have found an optimal distance for all three sources (i.e. you do not get ‘pile up’ effects (see Leo)), it is best to use this distance for calibration and determination of the peak energies. However, in the last part of the experiment you will be asked to draw qualitative conclusions by placing a source at varying distances. </p>
<table width=500 align=center><td>
<p align=justify>[[File:Gam-fig4.png|700px|border|center]]
<b>Figure 4 -</b> Gamma source, NaI(T1) detector, PMT, lead shield showing relevant processes.
<br clear=right>
</p>
</td></table>
<ol>
<h3>Calibration Curve</h3>
<li><p>Use Experiment 2.1 as a guide for gamma ray calibration from Duggan in the red binder in the lab. The eMorpho software will only be used for data collection and observing the spectra. All data analysis, including energy calibration, can be conducted using software like Excel or Mathematica. Use the 5 known sources Cs-137 (0.662 MeV), Co-60 (1.173 MeV, 1.332 MeV), Mn-54 (0.835 MeV), Na-22 (0.511 MeV) and Zn-65 (1.115 MeV) provided in the RSS 8 radiation source kit (Spectrum Techniques). Centre each source (label down) about 5 cm from the face of the detector and collect a spectrum for at least 5 minutes with each source. Longer collection times correspond to smaller statistical errors, why? <b>Stop collection prior to saving and record the dead time, live time and real time for each spectrum.</b> Save the spectrum. (The .csv file generated contains a single column with MCA parameters in the first 74 entries followed by the spectrum data (number of events) in each channel from 0 to 4096. Prior to data analysis delete the first 74 entries. Create a calibration curve using all five sources i.e. for each spectrum locate the channel number corresponding to the gamma ray energy peaks and create a plot of the energy of each peak and the peak channel number. What is the error of a straight line fit to this curve? If the fit is poor, you have made a mistake in assigning peak-channel values, and may have to repeat the analysis. Discuss sources of error. </p>

<p>Using your calibration fit plot the calibrated Cs-137 spectrum, Co-60 spectrum and Na-22 spectrum separately(i.e. events as a function of energy). Explain all the peaks on the spectra. Do you see any 'sum peaks’? (Plot your data on a logarithmic scale for a better view!) Explain why this peak occurs. Does it occur at the value you expect? What is the source of the 511 keV peak in the Na-22 spectrum? The Na-22 has a gamma ray energy at 1275 keV. What value of the maximum position do you find based on your calibration curve? Is this consistent with the resolution ''dE/E''? What is the present activity of these three sources (note the date marked on the source)? Present the calibrated spectra in your report and explain all the interesting details.</p> </li>

<li><p> Put a sheet of lead on the shelf and then place the Co-60 source above. How does the spectrum change? Now place the sheet of lead on top of the source. What, if anything, has changed in the spectrum? You may have to move the source closer or farther from the detector to observe the backscattering signals. Collect a spectrum for each case for a sufficient collection time and save. <b> Caution: Always wear gloves when handling the lead sheets.</b></p> </li>

<li> Obtain the energy resolution for the photopeak for Cs-137 and for one of the Co-60 peaks (See Knoll, chapter 4). Does this agree with what you have learned about scintillator devices? </li>
<li> Place the Cs-137 source at varying distances from the detector and explain qualitatively the rates that you observe. Put the source at slot 2 below the detector, and place first one, then two, and then three sheets of lead on top of the source. Comment on the rates, and the reason for the observed behaviour. Repeat for sheets of aluminum.</li>
<li> A spectrum of the unknown source (the background in the room) will be collected by the Lab Technologist. The data will be saved and the file can be found in the "data" folder located on the desktop. Using this data, create a calibrated spectrum using your calibration fit. Use your now-calibrated gamma-ray spectrum to identify the radioisotope(s) found in the background.</li>
</ol>

<h1>References</h1>
<ol>
<li>Preston and Dietz,[https://www.library.yorku.ca/find/Record/1038893 ''The Art of Experimental Physics''], Wiley.</li>
<li>G.F. Knoll, ''Radiation Detection and Measurement, 3rd or 4th ed.'', Wiley. </li>
<li>J.L. Duggan, ''Laboratory Investigations in Nuclear Science'', Tennelec.</li>
<li>A.C. Melissinos, [https://www.library.yorku.ca/find/Record/1641963 ''Experiments in Modern Physics''], Academic Press.</li>
<li>Leo, [https://www.library.yorku.ca/find/Record/1178503 ''Techniques for Nuclear and Particle Physics Experiments''], Springer-Verlag.</li>
<li>H. Frauenfelder and Henley,[https://www.library.yorku.ca/find/Record/51764 ''Subatomic Physics''] ,Prentice-Hall.</li>
<li>K. Siegbahn, [https://www.library.yorku.ca/find/Record/51739 ''Alpha, Beta, and Gamma-Ray Spectroscopy''], vol. I, chpts 5,8a.</li>
</ol>

Main Page/PHYS 4210 & 4211

2021-11-25T19:04:55Z

Gloria:

<h1>PHYS 4210 & 4211 3.0 </h1>

<p>Selected advanced experiments in physics related to
topics in solid state physics, atomic spectroscopy, microwaves, low-noise measurements, superconductivity, and nuclear and particle physics.

Prerequisites: SC/PHYS 3220 3.00; registration in a Bachelor or Honours Program in physics and astronomy or in biophysics. Co-requisite: SC/PHYS 3040 6.00.</p>

<p> See the course [http://moodle.yorku.ca eClass] for all administrative details. </p>

<h1>Laboratory Manual</h1>

IMPORTANT NOTE: Winter 2021 PHYS 4211 Lab Scheduling procedures will be announced via the Moodle eClass course website http://moodle.yorku.ca/ prior to the beginning of term.



<ul>
<li> [[Main Page/PHYS 4210/How to Choose Experiments|How to Choose Experiments]] </li>
<li> [[Main Page/PHYS 4210/How to Write Reports|How to Write Reports]] </li>
<li> [[Main Page/PHYS 3220/Lab Safety|Lab Safety]] </li>
</ul>

<h2> List of Available Experiments </h2>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 126 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]]</td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Muon Lifetime|Muon Lifetime]]</td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Waveguides|Waveguides*]] </td><td> 111 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Mass Spectrometer|Mass Spectrometer]]</td><td> 123 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 123 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/He-Ne Lasers|He-Ne Lasers*]]</td><td> 123 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Zeeman Effect|Zeeman Effect*]] </td><td> 123 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Electron Spin Resonance|Electron Spin Resonance]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Coaxial Cable|Coaxial Cable]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/High-TC Superconductivity|High-TC Superconductivity*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Fourier Optics|Fourier Optics*]] </td><td> 126 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Rutherford II|Rutherford Scattering II]]</td><td> 111 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>









<p> Experiments with an asterisk (*) are better suited for PHYS 4211 students. </p>





















<p> </p>

Main Page/PHYS 4210 & 4211

2021-11-19T16:00:16Z

Gloria:

<h1>PHYS 4210 & 4211 3.0 </h1>

<p>Selected advanced experiments in physics related to
topics in solid state physics, atomic spectroscopy, microwaves, low-noise measurements, superconductivity, and nuclear and particle physics.

Prerequisites: SC/PHYS 3220 3.00; registration in a Bachelor or Honours Program in physics and astronomy or in biophysics. Co-requisite: SC/PHYS 3040 6.00.</p>

<p> See the course [http://moodle.yorku.ca eClass] for all administrative details. </p>

<h1>Laboratory Manual</h1>

IMPORTANT NOTE: Winter 2021 PHYS 4211 Lab Scheduling procedures will be announced via the Moodle eClass course website http://moodle.yorku.ca/ prior to the beginning of term.



<ul>
<li> [[Main Page/PHYS 4210/How to Choose Experiments|How to Choose Experiments]] </li>
<li> [[Main Page/PHYS 4210/How to Write Reports|How to Write Reports]] </li>
<li> [[Main Page/PHYS 3220/Lab Safety|Lab Safety]] </li>
</ul>

<h2> List of Available Experiments </h2>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 126 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]]</td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Muon Lifetime|Muon Lifetime]]</td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Waveguides|Waveguides*]] </td><td> 111 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Mass Spectrometer|Mass Spectrometer]]</td><td> 123 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 123 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/He-Ne Lasers|He-Ne Lasers*]]</td><td> 123 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Zeeman Effect|Zeeman Effect*]] </td><td> 123 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Electron Spin Resonance|Electron Spin Resonance]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Coaxial Cable|Coaxial Cable]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/High-TC Superconductivity|High-TC Superconductivity*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Fourier Optics|Fourier Optics*]] </td><td> 126 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Rutherford II|Rutherford Scattering II]]</td><td> 111 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Gamma Ray Spectroscopy|Gamma Ray Spectroscopy*]] </td><td> 111 PSE</td><td></td></tr>








<p> Experiments with an asterisk (*) are better suited for PHYS 4211 students. </p>





















<p> </p>

Main Page/PHYS 4210 & 4211

2021-11-19T15:59:16Z

Gloria:

<h1>PHYS 4210 & 4211 3.0 </h1>

<p>Selected advanced experiments in physics related to
topics in solid state physics, atomic spectroscopy, microwaves, low-noise measurements, superconductivity, and nuclear and particle physics.

Prerequisites: SC/PHYS 3220 3.00; registration in a Bachelor or Honours Program in physics and astronomy or in biophysics. Co-requisite: SC/PHYS 3040 6.00.</p>

<p> See the course [http://moodle.yorku.ca eClass] for all administrative details. </p>

<h1>Laboratory Manual</h1>

IMPORTANT NOTE: Winter 2021 PHYS 4211 Lab Scheduling procedures will be announced via the Moodle eClass course website http://moodle.yorku.ca/ prior to the beginning of term.



<ul>
<li> [[Main Page/PHYS 4210/How to Choose Experiments|How to Choose Experiments]] </li>
<li> [[Main Page/PHYS 4210/How to Write Reports|How to Write Reports]] </li>
<li> [[Main Page/PHYS 3220/Lab Safety|Lab Safety]] </li>
</ul>

<h2> List of Available Experiments </h2>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 209 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]]</td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Muon Lifetime|Muon Lifetime]]</td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Waveguides|Waveguides*]] </td><td> 111 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Mass Spectrometer|Mass Spectrometer]]</td><td> 123 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 123 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/He-Ne Lasers|He-Ne Lasers*]]</td><td> 123 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Zeeman Effect|Zeeman Effect*]] </td><td> 123 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Electron Spin Resonance|Electron Spin Resonance]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Coaxial Cable|Coaxial Cable]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/High-TC Superconductivity|High-TC Superconductivity*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Fourier Optics|Fourier Optics*]] </td><td> 126 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Rutherford II|Rutherford Scattering II]]</td><td> 111 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Gamma Ray Spectroscopy|Gamma Ray Spectroscopy*]] </td><td> 111 PSE</td><td></td></tr>








<p> Experiments with an asterisk (*) are better suited for PHYS 4211 students. </p>





















<p> </p>

Main Page/PHYS 4210 & 4211

2021-11-19T15:57:25Z

Gloria:

<h1>PHYS 4210 & 4211 3.0 </h1>

<p>Selected advanced experiments in physics related to
topics in solid state physics, atomic spectroscopy, microwaves, low-noise measurements, superconductivity, and nuclear and particle physics.

Prerequisites: SC/PHYS 3220 3.00; registration in a Bachelor or Honours Program in physics and astronomy or in biophysics. Co-requisite: SC/PHYS 3040 6.00.</p>

<p> See the course [http://moodle.yorku.ca eClass] for all administrative details. </p>

<h1>Laboratory Manual</h1>

IMPORTANT NOTE: Winter 2021 PHYS 4211 Lab Scheduling procedures will be announced via the Moodle eClass course website http://moodle.yorku.ca/ prior to the beginning of term.



<ul>
<li> [[Main Page/PHYS 4210/How to Choose Experiments|How to Choose Experiments]] </li>
<li> [[Main Page/PHYS 4210/How to Write Reports|How to Write Reports]] </li>
<li> [[Main Page/PHYS 3220/Lab Safety|Lab Safety]] </li>
</ul>

<h2> List of Available Experiments </h2>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 209 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]]</td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Muon Lifetime|Muon Lifetime]]</td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Waveguides|Waveguides*]] </td><td> 111 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Mass Spectrometer|Mass Spectrometer]]</td><td> 123 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 123 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/He-Ne Lasers|He-Ne Lasers*]]</td><td> 123 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Zeeman Effect|Zeeman Effect*]] </td><td> 123 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Electron Spin Resonance|Electron Spin Resonance]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Coaxial Cable|Coaxial Cable]]</td><td> 126 PSE</td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/High-TC Superconductivity|High-TC Superconductivity*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Fourier Optics|Fourier Optics*]] </td><td> 126 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Rutherford II|Rutherford Scattering II]]</td><td> 111 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>








<p> Experiments with an asterisk (*) are better suited for PHYS 4211 students. </p>





















<p> </p>

Main Page/PHYS 3220

2021-08-25T18:47:01Z

Gloria:

<h1>PHYS 3220 3.0 Experiments in Modern Physics </h1>
<p>A selection of experiments in fluid mechanics,
electromagnetism, optics, and atomic, nuclear, and
particle physics. Analysis of the data and detailed
write-ups are required. One lecture hour which is
devoted to techniques of data analysis and three
laboratory hours per week.</p>

<h1>Laboratory Manual</h1>
<p>'''IMPORTANT NOTE: Fall 2020 PHYS 3220 Lab Scheduling procedures will be announced via the Moodle eClass course website http://eclass.yorku.ca/ prior to the beginning of term.'''</p>

<ul>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Cavendish|Measurement of the Gravitation Constant G: The Cavendish Experiment]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 126 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Thermionic|Thermionic Emission]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Speed of Light|A Measurement of the Velocity of Light: The Foucault-Michelson Experiment]] </td><td> 209 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Millikan|Determination of the Electric Charge Unit ''e'' : The Millikan Oil Drop Experiment]] </td><td> 111 PSE </td><td></td></tr>



</table>
</ul>

Main Page/PHYS 3220

2021-08-25T18:37:25Z

Gloria:

<h1>PHYS 3220 3.0 Experiments in Modern Physics </h1>
<p>A selection of experiments in fluid mechanics,
electromagnetism, optics, and atomic, nuclear, and
particle physics. Analysis of the data and detailed
write-ups are required. One lecture hour which is
devoted to techniques of data analysis and three
laboratory hours per week.</p>

<h1>Laboratory Manual</h1>
<p>'''IMPORTANT NOTE: Fall 2020 PHYS 3220 Lab Scheduling procedures will be announced via the Moodle eClass course website http://eclass.yorku.ca/ prior to the beginning of term.'''</p>

<ul>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Cavendish|Measurement of the Gravitation Constant G: The Cavendish Experiment]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Speed of Light|A Measurement of the Velocity of Light: The Foucault-Michelson Experiment]] </td><td> 209 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]] </td><td> 111 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Millikan|Determination of the Electric Charge Unit ''e'' : The Millikan Oil Drop Experiment]] </td><td> 111 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Thermionic|Thermionic Emission]]</td><td> 126 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 126 PSE </td><td></td></tr>
</table>
</ul>

Main Page/PHYS 3220

2021-08-25T18:36:11Z

Gloria:

<h1>PHYS 3220 3.0 Experiments in Modern Physics </h1>
<p>A selection of experiments in fluid mechanics,
electromagnetism, optics, and atomic, nuclear, and
particle physics. Analysis of the data and detailed
write-ups are required. One lecture hour which is
devoted to techniques of data analysis and three
laboratory hours per week.</p>

<h1>Laboratory Manual</h1>
<p>'''IMPORTANT NOTE: Fall 2020 PHYS 3220 Lab Scheduling procedures will be announced via the Moodle eClass course website http://moodle.yorku.ca/ prior to the beginning of term.'''</p>

<ul>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Cavendish|Measurement of the Gravitation Constant G: The Cavendish Experiment]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Speed of Light|A Measurement of the Velocity of Light: The Foucault-Michelson Experiment]] </td><td> 209 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]] </td><td> 111 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Millikan|Determination of the Electric Charge Unit ''e'' : The Millikan Oil Drop Experiment]] </td><td> 111 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Thermionic|Thermionic Emission]]</td><td> 126 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 126 PSE </td><td></td></tr>
</table>
</ul>

Main Page/PHYS 3220/new Excitation Potentials

2021-08-13T18:20:35Z

Gloria:

<h1>The Franck-Hertz Experiment: Excitation Potentials of Mercury and Neon</h1>

<h1>Introduction</h1>

<p>One of the most direct proofs of the existence of discrete energy states within the atom was first demonstrated in experiments on critical potentials, performed initially by Franck and Hertz in the early 1900's. Studying the way electrons lose energy in collisions with mercury vapour, they laid the basis for the quantum theory of atoms by observing that the electrons give energy to internal motion of mercury atoms in discrete units only.</p>
<p>The collision of a neutral atom with a fast particle (e.g., an electron) may result in the excitation or ionization of the atom. A slow electron in an elastic collision can give very little of its kinetic energy to the translational motion of a mercury atom (without changing the energy state of the atom) - just as a ping-pong ball cannot effectively move a billiard ball. If a moderately slow electron has enough kinetic energy to overcome an atomic excitation threshold (several eV) the collision may be inelastic and much of the energy of the electron can go into exciting a higher state of the atom. The energy in electron volts (eV) necessary to raise an atom from its normal ("ground") state to a given excited state is called the excitation potential for that state. For sufficiently high scattering energy of the impinging electron even ionization may occur.</p>
<p>The energy levels of mercury (Hg) are shown in Fig. 1; it is easy to see that the internal structure is complicated - a consequence of the many electrons in the atom. The diagram gives considerable information you need to know for this experiment. The numbers associated with the lines drawn between the energy levels are wavelengths (in Angstroms Å). In the present experiment we explore only the energy levels 6<sup>3</sup>P on the diagram, the first group of excited states. The electrons do not acquire enough energy to excite many of the other levels.</p>
<p>The Franck-Hertz apparatus consists of an evacuated glass envelope containing a cathode, screen, plate and a small drop of mercury, which can be vaporized by heating. The plate is always kept slightly negative with respect to the grid (that acts as an anode, i.e. accelerates the electrons) and both are set at various positive voltages with respect to the cathode. As the grid potential is raised, the plate current increases accordingly. For accelerating voltages below 5V all collisions with mercury atoms will be elastic (kinetic energy below about 5 eV). Hence, these electrons are energetic enough to overcome the negative plate-grid potential and are collected by the plate. The current flowing in the tube depends upon both the number of charged carriers (electrons) and their velocities (j = nev). Thus a significant change in the particle velocity can affect the size of the current. Once electrons with more than about 5eV energy excite a mercury atom, they slow down and the current flowing in the tube drops. If there is a larger voltage across the tube so that the electron can be re-accelerated to ~ 5 eV after giving it up once in the first collision, then we can see decreases in the current at higher voltages corresponding to a repeated inelastic collision. This process can yield a cyclic rise and fall of the current with the voltage.</p>

<table width=400 align=center><td>
<p align=justify>[[File:Fh-fig1.png|800px|border|center]]
<b>Figure 1 -</b> Energy Levels of Mercury.
<br clear=right>
</p>
</td></table>

<h1>Procedure</h1>

<h2>Required Components</h2>

<p> For this experiment you will be using equipment provided by Leybold®. Go to https://www.leybold-shop.com/physics/physics-equipment/atomic-and-nuclear-physics/franck-hertz-experiments/mercury/franck-hertz-supply-unit-555880.html and click on <b>Related Documents</b>. Read through the instruction sheet for the Franck-Hertz Supply Unit (pay attention to sections 1-4, 5.1, 5.2 and 5.6) and the Experiment Descriptions for Hg (P6.2.4.1) and Ne (P6.2.4.3). These leaflets provide useful information on how to use the equipment and optimize the Franck-Hertz signal. THE EXPERIMENTAL SETUP OF THE TEMPERATURE PROBE IS CRITICAL - THE PROBE MUST BE INSERTED INTO THE BLIND HOLE OF THE COPPER TUBE of the oven. Ensure the temperature sensor is properly connected and IS NOT TOUCHING THE Hg Franck-Hertz tube. </p>

<h3>Observing the Signal</h3>

<p>Electrons liberated from the filament and accelerated to the detector plate which do not collide with an Hg atom will register as a current. This current is amplified by the supply unit and be viewed on the oscilloscope. Note that evidence of collisions with Hg atoms will result in a deficit of current at specific accelerating voltages. This will be observed as dips on the oscilloscope trace. It is the origin and properties of these dips which is the focus of this experiment.</p>

<p>When the temperature is stable you can record the current-voltage characteristic of the Franck-Hertz tube. The current-voltage trace can be observed using the oscilloscope in XY mode. Make sure the signal you observe does not have horizontal clipping (the peaks cut off); see the leaflets for guidance on how to optimize your signal. (What is the meaning of the vertical cut-off?)</p>

<p>Record an I-V curve for an initial temperature of about 180ºC. Set the oscilloscope display mode to XY and the persist to 2 seconds to best visually observe the oscilloscope signal on the screen. </p>

<h3>Saving the Scope Traces</h3>
<p>Use a USB to save your optimized Franck-Hertz signals using the following procedure. (For more information on saving in XY mode refer to the user manual for the oscilloscope.) </p>
<ol>
<li>Connect your USB device to the oscilloscope. </li>
<li>Switch the oscilloscope display mode from XY to YT.</li>
<li>Push the Save/Recall button on the oscilloscope to activate the save menu. </li>
<li>Push the "Print" button to save all files to your USB drive. (The "Print" button is set to "Save All Files". This will save waveforms on Ch.1 and Ch.2 and a picture of the waveforms. </li>
</ol>

<h2>Measurements</h2>

<p>Find as many values as you can of the excitation energy ("excitation potential") for Hg from your record. Repeat these measurements for 5 different temperature values ranging from 140ºC to 195ºC. <b>DO NOT EXCEED a setpoint temperature of 195ºC on the supply unit.</b> Comment on the effect of the Hg pressure in the tube. Perform a full error analysis and compare your results with the expected values.</p>
<p> The neon tube is operated at room temperature. Optimize and record the Ne Franck-Hertz curve and find the excitation energy values for Ne. Compare with the expected values. Can you see the luminous layers in the neon tube? (Hint: Use the MAN operating mode to manually adjust the accelerating voltage and turn off the lights in the room.)

<h1>Questions</h1>
<ol>
<li>Explain the effect of changing the grid-to-plate voltage (V<sub>1</sub>)?</li>
<li>Find out what is meant by "contact potential" in the Franck-Hertz tube and explain how it could be determined. Can you estimate it from your record?</li>
<li>Determine from simple classical mechanics (using a head-on collision with recoil at 180 degrees) what fraction of an electron's kinetic energy can be transferred to a mercury atom in an '''elastic''' collision. Derive an approximate value of the fraction. Repeat for a neon atom. </li>
<li>Why are the other levels not observed? (e.g. 6<sup>3</sup>P<sub>2</sub>, 6<sup>3</sup>P<sub>o</sub>, 6<sup>1</sup>P<sub>1</sub>.)</li>
</ol>

<h1>References</h1>
<ol>
<li>Brehm, J., Mullin W, ''Modern Physics'', p. 168</li>
<li>Halliday, D., Resnick, R., ''Physics I'', pp. 522-24.</li>

<li>Hanne, G. F. “What Really Happens in the Franck–Hertz Experiment with Mercury?” American journal of physics 56.8 (1988): 696–700. Web. https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/sqt9v/cdi_scitation_primary_10_1119_1_15503 </li>
<li>Huebner, J. S. “Comment on the Franck–Hertz Experiment.” American journal of physics 44.3 (1976): 302–303. Web.
https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/sqt9v/cdi_crossref_primary_10_1119_1_10596</li>
<li>Liu, F. H. “Franck–Hertz Experiment with Higher Excitation Level Measurements.” American journal of physics 55.4 (1987): 366–369. Web. https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/sqt9v/cdi_crossref_primary_10_1119_1_15174</li>
<li>Preston, D., Dietz, E.,'' The Art of Experimental Physics'', pp. 197ff.</li>
</ol>

Main Page/PHYS 3220/new Excitation Potentials

2021-08-13T18:13:21Z

Gloria:

<h1>The Franck-Hertz Experiment: Excitation Potentials of Mercury and Neon</h1>

<h1>Introduction</h1>

<p>One of the most direct proofs of the existence of discrete energy states within the atom was first demonstrated in experiments on critical potentials, performed initially by Franck and Hertz in the early 1900's. Studying the way electrons lose energy in collisions with mercury vapour, they laid the basis for the quantum theory of atoms by observing that the electrons give energy to internal motion of mercury atoms in discrete units only.</p>
<p>The collision of a neutral atom with a fast particle (e.g., an electron) may result in the excitation or ionization of the atom. A slow electron in an elastic collision can give very little of its kinetic energy to the translational motion of a mercury atom (without changing the energy state of the atom) - just as a ping-pong ball cannot effectively move a billiard ball. If a moderately slow electron has enough kinetic energy to overcome an atomic excitation threshold (several eV) the collision may be inelastic and much of the energy of the electron can go into exciting a higher state of the atom. The energy in electron volts (eV) necessary to raise an atom from its normal ("ground") state to a given excited state is called the excitation potential for that state. For sufficiently high scattering energy of the impinging electron even ionization may occur.</p>
<p>The energy levels of mercury (Hg) are shown in Fig. 1; it is easy to see that the internal structure is complicated - a consequence of the many electrons in the atom. The diagram gives considerable information you need to know for this experiment. The numbers associated with the lines drawn between the energy levels are wavelengths (in Angstroms Å). In the present experiment we explore only the energy levels 6<sup>3</sup>P on the diagram, the first group of excited states. The electrons do not acquire enough energy to excite many of the other levels.</p>
<p>The Franck-Hertz apparatus consists of an evacuated glass envelope containing a cathode, screen, plate and a small drop of mercury, which can be vaporized by heating. The plate is always kept slightly negative with respect to the grid (that acts as an anode, i.e. accelerates the electrons) and both are set at various positive voltages with respect to the cathode. As the grid potential is raised, the plate current increases accordingly. For accelerating voltages below 5V all collisions with mercury atoms will be elastic (kinetic energy below about 5 eV). Hence, these electrons are energetic enough to overcome the negative plate-grid potential and are collected by the plate. The current flowing in the tube depends upon both the number of charged carriers (electrons) and their velocities (j = nev). Thus a significant change in the particle velocity can affect the size of the current. Once electrons with more than about 5eV energy excite a mercury atom, they slow down and the current flowing in the tube drops. If there is a larger voltage across the tube so that the electron can be re-accelerated to ~ 5 eV after giving it up once in the first collision, then we can see decreases in the current at higher voltages corresponding to a repeated inelastic collision. This process can yield a cyclic rise and fall of the current with the voltage.</p>

<table width=400 align=center><td>
<p align=justify>[[File:Fh-fig1.png|800px|border|center]]
<b>Figure 1 -</b> Energy Levels of Mercury.
<br clear=right>
</p>
</td></table>

<h1>Procedure</h1>

<h2>Required Components</h2>

<p> For this experiment you will be using equipment provided by Leybold®. Go to https://www.leybold-shop.com/physics/physics-equipment/atomic-and-nuclear-physics/franck-hertz-experiments/mercury/franck-hertz-supply-unit-555880.html and click on <b>Related Documents</b>. Read through the instruction sheet for the Franck-Hertz Supply Unit (pay attention to sections 1-4, 5.1, 5.2 and 5.6) and the Experiment Descriptions for Hg (P6.2.4.1) and Ne (P6.2.4.3). These leaflets provide useful information on how to use the equipment and optimize the Franck-Hertz signal. THE EXPERIMENTAL SETUP OF THE TEMPERATURE PROBE IS CRITICAL - THE PROBE MUST BE INSERTED INTO THE BLIND HOLE OF THE COPPER TUBE of the oven. Ensure the temperature sensor is properly connected and IS NOT TOUCHING THE Hg Franck-Hertz tube. </p>

<h3>Observing the Signal</h3>

<p>Electrons liberated from the filament and accelerated to the detector plate which do not collide with an Hg atom will register as a current. This current is amplified by the supply unit and be viewed on the oscilloscope. Note that evidence of collisions with Hg atoms will result in a deficit of current at specific accelerating voltages. This will be observed as dips on the oscilloscope trace. It is the origin and properties of these dips which is the focus of this experiment.</p>

<p>When the temperature is stable you can record the current-voltage characteristic of the Franck-Hertz tube. The current-voltage trace can be observed using the oscilloscope in XY mode. Make sure the signal you observe does not have horizontal clipping (the peaks cut off); see the leaflets for guidance on how to optimize your signal. (What is the meaning of the vertical cut-off?)</p>

<p>Record an I-V curve for an initial temperature of about 180ºC. Set the oscilloscope display mode to XY and the persist to 2 seconds to best visually observe the oscilloscope signal on the screen. </p>

<h3>Saving the Scope Traces</h3>
<p>Use a USB to save your optimized Franck-Hertz signals using the following procedure. (For more information on saving in XY mode refer to the user manual for the oscilloscope.) </p>
<ol>
<li>Connect your USB device to the oscilloscope. </li>
<li>Switch the oscilloscope display mode from XY to YT.</li>
<li>Push the Save/Recall button on the oscilloscope to activate the save menu. </li>
<li>Push the "Print" button to save all files to your USB drive. (The "Print" button is set to "Save All Files". This will save waveforms on Ch.1 and Ch.2 and a picture of the waveforms. </li>
</ol>

<h2>Measurements</h2>

<p>Find as many values as you can of the excitation energy ("excitation potential") for Hg from your record. Repeat these measurements for 5 different temperature values ranging from 140ºC to 195ºC. <b>DO NOT EXCEED a setpoint temperature of 195ºC on the supply unit.</b> Comment on the effect of the Hg pressure in the tube. Perform a full error analysis and compare your results with the expected values.</p>
<p> The neon tube is operated at room temperature. Optimize and record the Ne Franck-Hertz curve and find the excitation energy values for Ne. Compare with the expected values. Can you see the luminous layers in the neon tube? (Hint: Use the MAN operating mode to manually adjust the accelerating voltage and turn off the lights in the room.)

<h1>Questions</h1>
<ol>
<li>Explain the effect of changing the grid-to-plate voltage (V<sub>1</sub>)?</li>
<li>Find out what is meant by "contact potential" in the Franck-Hertz tube and explain how it could be determined. Can you estimate it from your record?</li>
<li>Determine from simple classical mechanics (using a head-on collision with recoil at 180 degrees) what fraction of an electron's kinetic energy can be transferred to a mercury atom in an '''elastic''' collision. Derive an approximate value of the fraction. Repeat for a neon atom. </li>
<li>Why are the other levels not observed? (e.g. 6<sup>3</sup>P<sub>2</sub>, 6<sup>3</sup>P<sub>o</sub>, 6<sup>1</sup>P<sub>1</sub>.)</li>
</ol>

<h1>References</h1>
<ol>
<li>Brehm, J., Mullin W, ''Modern Physics'', p. 168</li>
<li>Halliday, D., Resnick, R., ''Physics I'', pp. 522-24.</li>
<li>Carpenter, K.H., [http://ajp.aapt.org/resource/1/ajpias/v43/i2/p190_s1| Amer. J. Phys. '''43''' (1975) 190].</li>
<li>Hanne, G. F. “What Really Happens in the Franck–Hertz Experiment with Mercury?” American journal of physics 56.8 (1988): 696–700. Web. https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/sqt9v/cdi_scitation_primary_10_1119_1_15503 </li>
<li>Huebner, J. S. “Comment on the Franck–Hertz Experiment.” American journal of physics 44.3 (1976): 302–303. Web.
https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/sqt9v/cdi_crossref_primary_10_1119_1_10596</li>
<li>Liu, F. H. “Franck–Hertz Experiment with Higher Excitation Level Measurements.” American journal of physics 55.4 (1987): 366–369. Web. https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/sqt9v/cdi_crossref_primary_10_1119_1_15174</li>
<li>Preston, D., Dietz, E.,'' The Art of Experimental Physics'', pp. 197ff.</li>
</ol>

Main Page/PHYS 3220/new Excitation Potentials

2021-08-13T18:11:31Z

Gloria:

<h1>The Franck-Hertz Experiment: Excitation Potentials of Mercury and Neon</h1>

<h1>Introduction</h1>

<p>One of the most direct proofs of the existence of discrete energy states within the atom was first demonstrated in experiments on critical potentials, performed initially by Franck and Hertz in the early 1900's. Studying the way electrons lose energy in collisions with mercury vapour, they laid the basis for the quantum theory of atoms by observing that the electrons give energy to internal motion of mercury atoms in discrete units only.</p>
<p>The collision of a neutral atom with a fast particle (e.g., an electron) may result in the excitation or ionization of the atom. A slow electron in an elastic collision can give very little of its kinetic energy to the translational motion of a mercury atom (without changing the energy state of the atom) - just as a ping-pong ball cannot effectively move a billiard ball. If a moderately slow electron has enough kinetic energy to overcome an atomic excitation threshold (several eV) the collision may be inelastic and much of the energy of the electron can go into exciting a higher state of the atom. The energy in electron volts (eV) necessary to raise an atom from its normal ("ground") state to a given excited state is called the excitation potential for that state. For sufficiently high scattering energy of the impinging electron even ionization may occur.</p>
<p>The energy levels of mercury (Hg) are shown in Fig. 1; it is easy to see that the internal structure is complicated - a consequence of the many electrons in the atom. The diagram gives considerable information you need to know for this experiment. The numbers associated with the lines drawn between the energy levels are wavelengths (in Angstroms Å). In the present experiment we explore only the energy levels 6<sup>3</sup>P on the diagram, the first group of excited states. The electrons do not acquire enough energy to excite many of the other levels.</p>
<p>The Franck-Hertz apparatus consists of an evacuated glass envelope containing a cathode, screen, plate and a small drop of mercury, which can be vaporized by heating. The plate is always kept slightly negative with respect to the grid (that acts as an anode, i.e. accelerates the electrons) and both are set at various positive voltages with respect to the cathode. As the grid potential is raised, the plate current increases accordingly. For accelerating voltages below 5V all collisions with mercury atoms will be elastic (kinetic energy below about 5 eV). Hence, these electrons are energetic enough to overcome the negative plate-grid potential and are collected by the plate. The current flowing in the tube depends upon both the number of charged carriers (electrons) and their velocities (j = nev). Thus a significant change in the particle velocity can affect the size of the current. Once electrons with more than about 5eV energy excite a mercury atom, they slow down and the current flowing in the tube drops. If there is a larger voltage across the tube so that the electron can be re-accelerated to ~ 5 eV after giving it up once in the first collision, then we can see decreases in the current at higher voltages corresponding to a repeated inelastic collision. This process can yield a cyclic rise and fall of the current with the voltage.</p>

<table width=400 align=center><td>
<p align=justify>[[File:Fh-fig1.png|800px|border|center]]
<b>Figure 1 -</b> Energy Levels of Mercury.
<br clear=right>
</p>
</td></table>

<h1>Procedure</h1>

<h2>Required Components</h2>

<p> For this experiment you will be using equipment provided by Leybold®. Go to https://www.leybold-shop.com/physics/physics-equipment/atomic-and-nuclear-physics/franck-hertz-experiments/mercury/franck-hertz-supply-unit-555880.html and click on <b>Related Documents</b>. Read through the instruction sheet for the Franck-Hertz Supply Unit (pay attention to sections 1-4, 5.1, 5.2 and 5.6) and the Experiment Descriptions for Hg (P6.2.4.1) and Ne (P6.2.4.3). These leaflets provide useful information on how to use the equipment and optimize the Franck-Hertz signal. THE EXPERIMENTAL SETUP OF THE TEMPERATURE PROBE IS CRITICAL - THE PROBE MUST BE INSERTED INTO THE BLIND HOLE OF THE COPPER TUBE of the oven. Ensure the temperature sensor is properly connected and IS NOT TOUCHING THE Hg Franck-Hertz tube. </p>

<h3>Observing the Signal</h3>

<p>Electrons liberated from the filament and accelerated to the detector plate which do not collide with an Hg atom will register as a current. This current is amplified by the supply unit and be viewed on the oscilloscope. Note that evidence of collisions with Hg atoms will result in a deficit of current at specific accelerating voltages. This will be observed as dips on the oscilloscope trace. It is the origin and properties of these dips which is the focus of this experiment.</p>

<p>When the temperature is stable you can record the current-voltage characteristic of the Franck-Hertz tube. The current-voltage trace can be observed using the oscilloscope in XY mode. Make sure the signal you observe does not have horizontal clipping (the peaks cut off); see the leaflets for guidance on how to optimize your signal. (What is the meaning of the vertical cut-off?)</p>

<p>Record an I-V curve for an initial temperature of about 180ºC. Set the oscilloscope display mode to XY and the persist to 2 seconds to best visually observe the oscilloscope signal on the screen. </p>

<h3>Saving the Scope Traces</h3>
<p>Use a USB to save your optimized Franck-Hertz signals using the following procedure. (For more information on saving in XY mode refer to the user manual for the oscilloscope.) </p>
<ol>
<li>Connect your USB device to the oscilloscope. </li>
<li>Switch the oscilloscope display mode from XY to YT.</li>
<li>Push the Save/Recall button on the oscilloscope to activate the save menu. </li>
<li>Push the "Print" button to save all files to your USB drive. (The "Print" button is set to "Save All Files". This will save waveforms on Ch.1 and Ch.2 and a picture of the waveforms. </li>
</ol>

<h2>Measurements</h2>

<p>Find as many values as you can of the excitation energy ("excitation potential") for Hg from your record. Repeat these measurements for 5 different temperature values ranging from 140ºC to 195ºC. <b>DO NOT EXCEED a setpoint temperature of 195ºC on the supply unit.</b> Comment on the effect of the Hg pressure in the tube. Perform a full error analysis and compare your results with the expected values.</p>
<p> The neon tube is operated at room temperature. Optimize and record the Ne Franck-Hertz curve and find the excitation energy values for Ne. Compare with the expected values. Can you see the luminous layers in the neon tube? (Hint: Use the MAN operating mode to manually adjust the accelerating voltage and turn off the lights in the room.)

<h1>Questions</h1>
<ol>
<li>Explain the effect of changing the grid-to-plate voltage (V<sub>1</sub>)?</li>
<li>Find out what is meant by "contact potential" in the Franck-Hertz tube and explain how it could be determined. Can you estimate it from your record?</li>
<li>Determine from simple classical mechanics (using a head-on collision with recoil at 180 degrees) what fraction of an electron's kinetic energy can be transferred to a mercury atom in an '''elastic''' collision. Derive an approximate value of the fraction. Repeat for a neon atom. </li>
<li>Why are the other levels not observed? (e.g. 6<sup>3</sup>P<sub>2</sub>, 6<sup>3</sup>P<sub>o</sub>, 6<sup>1</sup>P<sub>1</sub>.)</li>
</ol>

<h1>References</h1>
<ol>
<li>Brehm, J., Mullin W, ''Modern Physics'', p. 168</li>
<li>Halliday, D., Resnick, R., ''Physics I'', pp. 522-24.</li>
<li>Carpenter, K.H., [http://ajp.aapt.org/resource/1/ajpias/v43/i2/p190_s1| Amer. J. Phys. '''43''' (1975) 190].</li>
<li>Hanne, G. F. “What Really Happens in the Franck–Hertz Experiment with Mercury?” American journal of physics 56.8 (1988): 696–700. Web. https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/sqt9v/cdi_scitation_primary_10_1119_1_15503 </li>
<li>Huebner, J. S. “Comment on the Franck–Hertz Experiment.” American journal of physics 44.3 (1976): 302–303. Web.
https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/sqt9v/cdi_crossref_primary_10_1119_1_10596</li>
<li>Liu, F.H., ''“Franck-Hertz expt. with higher excitation level”'' [http://ajp.aapt.org/resource/1/ajpias/v55/i4/p366_s1 Am. J. Phys. '''55''' (1987) 366].</li>
<li>Preston, D., Dietz, E.,'' The Art of Experimental Physics'', pp. 197ff.</li>
</ol>

Main Page/PHYS 3220/new Excitation Potentials

2021-08-13T18:09:32Z

Gloria:

<h1>The Franck-Hertz Experiment: Excitation Potentials of Mercury and Neon</h1>

<h1>Introduction</h1>

<p>One of the most direct proofs of the existence of discrete energy states within the atom was first demonstrated in experiments on critical potentials, performed initially by Franck and Hertz in the early 1900's. Studying the way electrons lose energy in collisions with mercury vapour, they laid the basis for the quantum theory of atoms by observing that the electrons give energy to internal motion of mercury atoms in discrete units only.</p>
<p>The collision of a neutral atom with a fast particle (e.g., an electron) may result in the excitation or ionization of the atom. A slow electron in an elastic collision can give very little of its kinetic energy to the translational motion of a mercury atom (without changing the energy state of the atom) - just as a ping-pong ball cannot effectively move a billiard ball. If a moderately slow electron has enough kinetic energy to overcome an atomic excitation threshold (several eV) the collision may be inelastic and much of the energy of the electron can go into exciting a higher state of the atom. The energy in electron volts (eV) necessary to raise an atom from its normal ("ground") state to a given excited state is called the excitation potential for that state. For sufficiently high scattering energy of the impinging electron even ionization may occur.</p>
<p>The energy levels of mercury (Hg) are shown in Fig. 1; it is easy to see that the internal structure is complicated - a consequence of the many electrons in the atom. The diagram gives considerable information you need to know for this experiment. The numbers associated with the lines drawn between the energy levels are wavelengths (in Angstroms Å). In the present experiment we explore only the energy levels 6<sup>3</sup>P on the diagram, the first group of excited states. The electrons do not acquire enough energy to excite many of the other levels.</p>
<p>The Franck-Hertz apparatus consists of an evacuated glass envelope containing a cathode, screen, plate and a small drop of mercury, which can be vaporized by heating. The plate is always kept slightly negative with respect to the grid (that acts as an anode, i.e. accelerates the electrons) and both are set at various positive voltages with respect to the cathode. As the grid potential is raised, the plate current increases accordingly. For accelerating voltages below 5V all collisions with mercury atoms will be elastic (kinetic energy below about 5 eV). Hence, these electrons are energetic enough to overcome the negative plate-grid potential and are collected by the plate. The current flowing in the tube depends upon both the number of charged carriers (electrons) and their velocities (j = nev). Thus a significant change in the particle velocity can affect the size of the current. Once electrons with more than about 5eV energy excite a mercury atom, they slow down and the current flowing in the tube drops. If there is a larger voltage across the tube so that the electron can be re-accelerated to ~ 5 eV after giving it up once in the first collision, then we can see decreases in the current at higher voltages corresponding to a repeated inelastic collision. This process can yield a cyclic rise and fall of the current with the voltage.</p>

<table width=400 align=center><td>
<p align=justify>[[File:Fh-fig1.png|800px|border|center]]
<b>Figure 1 -</b> Energy Levels of Mercury.
<br clear=right>
</p>
</td></table>

<h1>Procedure</h1>

<h2>Required Components</h2>

<p> For this experiment you will be using equipment provided by Leybold®. Go to https://www.leybold-shop.com/physics/physics-equipment/atomic-and-nuclear-physics/franck-hertz-experiments/mercury/franck-hertz-supply-unit-555880.html and click on <b>Related Documents</b>. Read through the instruction sheet for the Franck-Hertz Supply Unit (pay attention to sections 1-4, 5.1, 5.2 and 5.6) and the Experiment Descriptions for Hg (P6.2.4.1) and Ne (P6.2.4.3). These leaflets provide useful information on how to use the equipment and optimize the Franck-Hertz signal. THE EXPERIMENTAL SETUP OF THE TEMPERATURE PROBE IS CRITICAL - THE PROBE MUST BE INSERTED INTO THE BLIND HOLE OF THE COPPER TUBE of the oven. Ensure the temperature sensor is properly connected and IS NOT TOUCHING THE Hg Franck-Hertz tube. </p>

<h3>Observing the Signal</h3>

<p>Electrons liberated from the filament and accelerated to the detector plate which do not collide with an Hg atom will register as a current. This current is amplified by the supply unit and be viewed on the oscilloscope. Note that evidence of collisions with Hg atoms will result in a deficit of current at specific accelerating voltages. This will be observed as dips on the oscilloscope trace. It is the origin and properties of these dips which is the focus of this experiment.</p>

<p>When the temperature is stable you can record the current-voltage characteristic of the Franck-Hertz tube. The current-voltage trace can be observed using the oscilloscope in XY mode. Make sure the signal you observe does not have horizontal clipping (the peaks cut off); see the leaflets for guidance on how to optimize your signal. (What is the meaning of the vertical cut-off?)</p>

<p>Record an I-V curve for an initial temperature of about 180ºC. Set the oscilloscope display mode to XY and the persist to 2 seconds to best visually observe the oscilloscope signal on the screen. </p>

<h3>Saving the Scope Traces</h3>
<p>Use a USB to save your optimized Franck-Hertz signals using the following procedure. (For more information on saving in XY mode refer to the user manual for the oscilloscope.) </p>
<ol>
<li>Connect your USB device to the oscilloscope. </li>
<li>Switch the oscilloscope display mode from XY to YT.</li>
<li>Push the Save/Recall button on the oscilloscope to activate the save menu. </li>
<li>Push the "Print" button to save all files to your USB drive. (The "Print" button is set to "Save All Files". This will save waveforms on Ch.1 and Ch.2 and a picture of the waveforms. </li>
</ol>

<h2>Measurements</h2>

<p>Find as many values as you can of the excitation energy ("excitation potential") for Hg from your record. Repeat these measurements for 5 different temperature values ranging from 140ºC to 195ºC. <b>DO NOT EXCEED a setpoint temperature of 195ºC on the supply unit.</b> Comment on the effect of the Hg pressure in the tube. Perform a full error analysis and compare your results with the expected values.</p>
<p> The neon tube is operated at room temperature. Optimize and record the Ne Franck-Hertz curve and find the excitation energy values for Ne. Compare with the expected values. Can you see the luminous layers in the neon tube? (Hint: Use the MAN operating mode to manually adjust the accelerating voltage and turn off the lights in the room.)

<h1>Questions</h1>
<ol>
<li>Explain the effect of changing the grid-to-plate voltage (V<sub>1</sub>)?</li>
<li>Find out what is meant by "contact potential" in the Franck-Hertz tube and explain how it could be determined. Can you estimate it from your record?</li>
<li>Determine from simple classical mechanics (using a head-on collision with recoil at 180 degrees) what fraction of an electron's kinetic energy can be transferred to a mercury atom in an '''elastic''' collision. Derive an approximate value of the fraction. Repeat for a neon atom. </li>
<li>Why are the other levels not observed? (e.g. 6<sup>3</sup>P<sub>2</sub>, 6<sup>3</sup>P<sub>o</sub>, 6<sup>1</sup>P<sub>1</sub>.)</li>
</ol>

<h1>References</h1>
<ol>
<li>Brehm, J., Mullin W, ''Modern Physics'', p. 168</li>
<li>Halliday, D., Resnick, R., ''Physics I'', pp. 522-24.</li>
<li>Carpenter, K.H., [http://ajp.aapt.org/resource/1/ajpias/v43/i2/p190_s1| Amer. J. Phys. '''43''' (1975) 190].</li>
<li>Hanne, G. F. “What Really Happens in the Franck–Hertz Experiment with Mercury?” American journal of physics 56.8 (1988): 696–700. Web. https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/sqt9v/cdi_scitation_primary_10_1119_1_15503 </li>
<li>Huebener, J.S., [http://ajp.aapt.org/resource/1/ajpias/v44/i3/p302_s1 Amer. J. Phys. '''44''' (1976) 302].</li>
<li>Liu, F.H., ''“Franck-Hertz expt. with higher excitation level”'' [http://ajp.aapt.org/resource/1/ajpias/v55/i4/p366_s1 Am. J. Phys. '''55''' (1987) 366].</li>
<li>Preston, D., Dietz, E.,'' The Art of Experimental Physics'', pp. 197ff.</li>
</ol>

Main Page/PHYS 3220/Radioactive Decays

2021-08-13T17:54:43Z

Gloria:

<h1>Radioactive Decays</h1>

<h1>Learning Outcomes</h1>
<ol>
<li>Three types of radioactivity
<li>Poisson statistics
<li>Radiation detection technology
</ol>

<h1>Introduction</h1>

<h2>Radioactive Decays</h2>
<p>Radioactive nuclear decays can be classified according to their decay mechanism: </p>

<ol style="list-style-type:lower-roman">
<li><p><b>α-decay:</b> heavy radionucleides often decay via the emission of a cluster composed of 2 protons and 2 neutrons, i.e., a <sub>2</sub>He<sup>4</sup> nucleus.</p></li>
<li><p><b>β-decay:</b> nuclei away from the line of stability N = Z, where N is the total number of neutrons, and Z the total number of protons, can lower their energy, and hence become more stable, by emitting either an electron or a positron. In the case of neutron-rich nuclei, a neutron converts into a proton, electron and antineutrino. The fast electron is emitted from the nucleus, corresponding to the β<sup>-</sup> decay of free neutrons (half-life 10.6 min.). For proton-rich nuclei, a proton is converted into a neutron, positron and a neutrino (β<sup>+</sup> decay). The fast positron emerges from the nucleus. This latter process may seem counterintuitive as it cannot occur for free protons (why?). The rest of the nuclear system supplies the energy necessary for the reaction to take place.</p></li>

<li><p><b>γ-decay:</b> the emission of photons with energies higher than X-rays (MeV-range) is the result of a nuclear transition from an excited to a lower state in complete analogy with photon emission from excited atoms (eV to keV-range). This decay almost always accompanies α- and β-decays, since these processes usually leave the daughter nucleus in an excited state.</p></li>

<li><p><b>spontaneous fission:</b>the emission of nuclear clusters bigger than α-particles is a rare process that has been studied recently in a systematic way at heavy ion facilities. It represents an alternative but rare decay mechanism, which provides insight into the nature of nuclear forces.</p></li>
</ol>

<p>All modern physics texts contain a chapter that describes nuclear phenomenology as well as a table of isotopes. Understand the basic principles (there will be no need to understand previous chapters of the book for this!). See, e.g., refs. 1-3. </p>

<h2>Detection of radiation</h2>

<p>The detection of nuclear radiation relies on the property that it ionizes the surrounding matter through which it passes. This statement is obvious for the charged α, and β particles. For γ particles the ionization arises through the photoeffect and Compton scattering (refs. 1-3). This ionization can be detected through the electric spark induced between condenser plates that are biased with a high voltage, resulting in a short burst of current. This is the principle of a Geiger-Müller (GM) tube. The efficiency of detection depends on the voltage applied to the gas-filled tube (why can’t one use a vacuum tube?). It is important to realize that the detector has a finite efficiency, i.e., it does not detect every single α, β, or γ particle entering the detector. In particular, the efficiency depends on the voltage applied with a threshold behaviour (around 900 V) followed by saturation. In small hand-held radiation counters the high voltage is produced by a DC-DC converter as used in electronic flashlights. Read the description of GM counters available in many texts (e.g., refs. 1,2,6), and include a concise description in your own words in your report.</p>
<p>Other detection mechanisms used for monitoring are: (i) exposure blackening of photographic film, e.g., in personal total dose monitors; (ii) scintillator counters; (iii) triggering of semiconductor devices; etc. </p>

<h2>Absorption of radiation</h2>
<p>Radiation is slowed down and eventually stopped as it passes through matter. This fact is exploited both in shielding and in applications of radiation for energy deposition (e.g., burning of cancer cells in radiation medicine). The absorption of the three different forms of radiation by matter is very different: α particles are heavy and doubly charged, therefore, they give up their energy readily in collisions with the nuclei of the surrounding matter; β particles are lighter and faster (as they emerge from the decay), and therefore pass more readily through matter until they are stopped; γ-rays have the best penetration characteristics, i.e., are the most difficult to shield. α particles, which have typical energies of 5 MeV, are stopped by a few centimeters of air, since they are doubly charged and slow compared to β particles. They are detected by GM counters only if they enter through a specially designed opening (transparent to them provided they are fast enough). </p>

<p>The stopping power and energy deposition is also a function that depends strongly on the kinetic energy of the ionizing particles. In radiation medicine this is exploited, e.g., by having fast particles penetrate healthy tissue with limited damage but sufficient slow-down such that energy deposition becomes efficient when the tissue to be destroyed is reached. Usually physicists with nuclear medicine training are in charge of designing a radiation plan for each patient depending on the location of the tissue to be destroyed, vicinity of vital organs, etc. This is a non-trivial process, since secondary radiation (e.g., production of electrons) contributes to the energy deposition and may diffuse the flux of radiation.</p>

<h2>Lifetimes of radioactive sources</h2>
<p>A proper understanding of nuclear decays on the basis of a nuclear shell model (in analogy to atomic structure of electronic energy levels) enables one to predict the energies of the emitted particles as well as the half-lifes. The lifetime is related to the broadening in energy of the decaying state and can be understood from Heisenberg's uncertainty principle. (As a function of time the number of decaying particles is described by an exponential decay law.) </p>

<p>The radioactive sources that we use in this experiment do not permit a measurement of the decay law, since they have long lifetimes (tens to thousands of years), i.e., it is impossible to observe the decrease in radioactivity over a reasonable time span. However, sources with a short lifetime can be produced by exposure of a sample to a high-flux source, e.g., a reactor, which results in the conversion of stable nuclei into unstable ones.</p>

<h2>Statistics of nuclear counting</h2>
<p>In the early studies of radioactivity it was not understood whether radioactivity was a purely random process, or whether the emission of one particle might effect the emission of others. One can prove that the observation of the number of independent decays per time interval (count rate) as a function of time should result in a Poissonian distribution (ref. 5). In the limit of high count rates the Poissonian distribution can be approximated by a Gaussian distribution. </p>

<p>Rutherford performed experiments that showed that the probability, ''P(n)'', of observing ''n'' counts in a fixed time interval followed the Poisson formula</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Rd-eqn1.png|150px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

<p>where the ''average'' number of counts per interval is calculated as </p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Rd-eqn2.png|280px|center]]
</p>
</td></table>

<p>For all the measurements in this experiment that are performed with computerized data acquisition and data analysis, the Poissonian character of the statistical distribution of decay events are to be investigated and verified. Since the computer program "Particle Tracking.vi" performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (ch. 11 in ref. 5).</p>

<p>To illustrate how one explicitly analyzes the data we include an example for your convenience.</p>

<ol>
<li>Let us say that you record the number of counts heard during 100 five-second intervals by entering a mark in the column appropriate for that number of counts (col. 2 in the table below).
<table width=420 align=center>
<tr><td width=120><b>Number of Counts in interval (n)</b></td><td width=120><b>Number of times Count occurs</b></td><td width=100><b>''P(n)''</b></td><td width=100><b>Total Counts</b></td></tr>
<tr><td>0</td> <td>I(1)</td> <td>0.01</td> <td>0x1=0</td></tr>
<tr><td>1</td> <td>III(3)</td> <td>0.03</td> <td>1x3=3</td></tr>
<tr><td>2</td> <td>IIII I(5)</td> <td>0.05</td> <td>2x5=10</td></tr>
<tr><td>etc..</td><td></td><td></td><td></td></tr>
</table>
</li>

<li><p>Now construct a bar graph for the results, showing ''P(n)'' vs ''n'', where ''P(n)'' is the probability for finding n counts:</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Rd-eqn3.png|260px|center]]
</p>
</td></table>
<p>Then, using the Poisson distribution (Eq. 1) evaluate ''P(n)'' and graph the theoretical distribution over the same range of values. To do this, you require the value of n-bar; this should be the mean number of counts in your measurement:</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Rd-eqn4.png|110px|center]]
</p>
</td></table>
<p>Thus, your theoretical distribution and your experimental results will have the same mean.</p>
</li>
<li><p>Now calculate the standard deviation of your data:</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Rd-eqn5.png|180px|center]]
</p>
</td></table>
</li>

<li><p>Compare this with the expected standard deviation from the theoretical probability distribution, which is (for a Poisson distribution):</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Rd-eqn6.png|140px|center]]
</p>
</td></table>
<p>Note that this simple relation between the '''mean''' and the standard deviation is not a property of all distributions.</p>
</li>

<li><p>To see if the numbers of counts obey Poisson statistics in a quantitative way, we use the Chi-squared (χ<sup>2</sup>) test (ch. 12 in ref. 5; an example is given on pg. 235). From the reduced χ<sup>2</sup> value one infers the agreement.</p>
</li>

</ol>

<h1>Experimental Procedure</h1>

<p>In this experiment a Geiger-Müller counter with a computer interface is used to detect the radiation coming from the natural background, as well as from some weak sources. The statistics of the decays is investigated to confirm the independence of the decay mechanism. The dependence of the count rate on the distance from the source is also investigated. Also, the Geiger-Müller method for detection of radioactivity will be investigated.</p>

<p>Familiarize yourself with the computer-interfaced GM counter and associated computer software.</p>

<h2>Required Components</h2>
<ol>
<li>[[Media:Radioactive-ACratemeter.JPG|AC Powered Table-Top GM Counter]]</li>
<li>[[Media:RDHandHeldGM.JPG|Hand-held GM Counter]]</li>
<li>[[Media:RDBeigeFiesta.JPG|Beige 'Fiesta' Ceramic Dish]]</li>
<li>[[Media:RDOrangeFiesta.JPG|Orange 'Fiesta' Ceramic Dish]]</li>
<li>[[Media:RDMantles.JPG|α,γ Source: <sub>90</sub>Th<sup>232</sup>, Lantern Mantles]]</li>
</ol>

<h2>Hardware instructions:</h2>
<p>The hand-held GM counter can be operated independent of the computer interface. You should use it in range I (up to 2000? counts per minute - cpm), and turn on the audio monitoring. The background rate should be in the range of up to a few counts per second. For sources we use a bag containing Coleman-type naphta lantern mantles and Fiesta plates. Original Coleman mantles used radioactive elements until 1990; the clones still use a <sub>90</sub>Th<sup>232</sup> α emitter to enhance fluorescence. Radioactive elements were also used in glazing for bathroom tiles and Fiesta plates (no longer on the market). Make sure that the sources are some distance away from the GM counter when measuring the background radiation.</p>

<h2>Computer Instruction</h2>

<p>Data will be collected using a program called "Particle Tracking.vi" located on the desktop.
This program uses the microphone input of the computer to monitor the counts from the "Radiation Alert- Monitor 4" detector. The operation of the program is is described below</p>

<table width=400 align=center>
<td>
<p align=justify>[[File:Rd-vi.png|800px|center]]
</p>
</td></table>

<p>''Note that the program displays a histogram of the results for you to see, but only the raw data of the counts is written to the output file.''</p>

<h2>Required Data</h2>
<ol>
<li>Test the statistics of nuclear background radiation. Note the direction in which the GM counter is pointing. Make sure that it is aiming at free space, and not at a potential radioactive source. Take at least two runs, one of which should be with a larger amount of data to observe an improvement in the fit to a Poissonian distribution. Comment on the χ<sup>2</sup> obtained, and quote the decay rate, with its standard error. Include histograms of the distributions. Repeat the longer run with the GM counter pointing in an orthogonal direction. Are the data consistent with the previous run? Should they be? What are some sources of background radiation? Save the data points for one of the long runs to a data file. Perform the Poisson statistics analysis explicitly as described in the example in the previous section.</li>

<li>Perform measurements similar to (1) while bringing the bag with lantern mantles (<sub>90</sub>Th<sup>232</sup> α,γ source) close to the opening of the GM counter. Comment on the obtained distribution. Use a detailed table of isotopes (with decay schemes) to identify the radionuclide of the thorium family (ref. 6). </li>

<li>Place the orange 'Fiesta' ceramic dish plate on the table. Mount the GM counter centered above the plate using a retort stand. Measure average count rates as a function of distance, e.g., 0.5 cm, 5 cm, 10 cm, 15 cm, 20 cm, 25 cm. Has the count rate at 25 cm reached the background count rate within errors? Plot the count rates after subtraction of the background rate as a function of distance. What functional behaviour do you find? Can you explain why the Geiger counter is responding when exposed to the Fiesta plate? Show relevant decay chain diagrams.</li>

<li>Turn on the AC powered table-top GM counter. Set the knob to HV and dial up an operating voltage not exceeding 1200 Volts. Set the knob to display count rate X1 (in counts per minute) and note the background radiation. Place the beige Fiesta dish close to the exposed GM tube (the aluminium shield can be rotated such that an opening appears). You may need to reduce the sensitivity of the meter by setting the knob to the X10 range. Then measure the count rate as a function of the operating voltage.</li>
</ol>

<p>Incorporate in your report an outline on the three nuclear decay mechanisms. The function of the GM counter should also be explained briefly in the report.</p>

<h1>References</h1>

<ol>
<li>Knoll, G.F., ''Radiation Detection and Measurement'', 2nd ed.</li>
<li>Tsoulfanidis, N., ''Measurement and Detection of Radiation''.</li>
<li>Rohlf, J.W., ''Modern Physics from α to Z<sup>0</sup>'', Wiley 1994 https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/q36jf8/alma991011387139705164</li>
<li>Thornton, Stephen T., and Andrew F. Rex. Modern Physics for Scientists and Engineers . 3rd ed. Belmont, CA: Thomson, Brooks/Cole, 2006. Print. https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/q36jf8/alma991026442989705164 </li>
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li>
<li>Taylor, J.R., ''An Introduction to Error Analysis'', University Science Books, 1997.</li>
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li>
</ol>

Main Page/PHYS 3220/Radioactive Decays

2021-08-13T17:46:15Z

Gloria:

<h1>Radioactive Decays</h1>

<h1>Learning Outcomes</h1>
<ol>
<li>Three types of radioactivity
<li>Poisson statistics
<li>Radiation detection technology
</ol>

<h1>Introduction</h1>

<h2>Radioactive Decays</h2>
<p>Radioactive nuclear decays can be classified according to their decay mechanism: </p>

<ol style="list-style-type:lower-roman">
<li><p><b>α-decay:</b> heavy radionucleides often decay via the emission of a cluster composed of 2 protons and 2 neutrons, i.e., a <sub>2</sub>He<sup>4</sup> nucleus.</p></li>
<li><p><b>β-decay:</b> nuclei away from the line of stability N = Z, where N is the total number of neutrons, and Z the total number of protons, can lower their energy, and hence become more stable, by emitting either an electron or a positron. In the case of neutron-rich nuclei, a neutron converts into a proton, electron and antineutrino. The fast electron is emitted from the nucleus, corresponding to the β<sup>-</sup> decay of free neutrons (half-life 10.6 min.). For proton-rich nuclei, a proton is converted into a neutron, positron and a neutrino (β<sup>+</sup> decay). The fast positron emerges from the nucleus. This latter process may seem counterintuitive as it cannot occur for free protons (why?). The rest of the nuclear system supplies the energy necessary for the reaction to take place.</p></li>

<li><p><b>γ-decay:</b> the emission of photons with energies higher than X-rays (MeV-range) is the result of a nuclear transition from an excited to a lower state in complete analogy with photon emission from excited atoms (eV to keV-range). This decay almost always accompanies α- and β-decays, since these processes usually leave the daughter nucleus in an excited state.</p></li>

<li><p><b>spontaneous fission:</b>the emission of nuclear clusters bigger than α-particles is a rare process that has been studied recently in a systematic way at heavy ion facilities. It represents an alternative but rare decay mechanism, which provides insight into the nature of nuclear forces.</p></li>
</ol>

<p>All modern physics texts contain a chapter that describes nuclear phenomenology as well as a table of isotopes. Understand the basic principles (there will be no need to understand previous chapters of the book for this!). See, e.g., refs. 1-3. </p>

<h2>Detection of radiation</h2>

<p>The detection of nuclear radiation relies on the property that it ionizes the surrounding matter through which it passes. This statement is obvious for the charged α, and β particles. For γ particles the ionization arises through the photoeffect and Compton scattering (refs. 1-3). This ionization can be detected through the electric spark induced between condenser plates that are biased with a high voltage, resulting in a short burst of current. This is the principle of a Geiger-Müller (GM) tube. The efficiency of detection depends on the voltage applied to the gas-filled tube (why can’t one use a vacuum tube?). It is important to realize that the detector has a finite efficiency, i.e., it does not detect every single α, β, or γ particle entering the detector. In particular, the efficiency depends on the voltage applied with a threshold behaviour (around 900 V) followed by saturation. In small hand-held radiation counters the high voltage is produced by a DC-DC converter as used in electronic flashlights. Read the description of GM counters available in many texts (e.g., refs. 1,2,6), and include a concise description in your own words in your report.</p>
<p>Other detection mechanisms used for monitoring are: (i) exposure blackening of photographic film, e.g., in personal total dose monitors; (ii) scintillator counters; (iii) triggering of semiconductor devices; etc. </p>

<h2>Absorption of radiation</h2>
<p>Radiation is slowed down and eventually stopped as it passes through matter. This fact is exploited both in shielding and in applications of radiation for energy deposition (e.g., burning of cancer cells in radiation medicine). The absorption of the three different forms of radiation by matter is very different: α particles are heavy and doubly charged, therefore, they give up their energy readily in collisions with the nuclei of the surrounding matter; β particles are lighter and faster (as they emerge from the decay), and therefore pass more readily through matter until they are stopped; γ-rays have the best penetration characteristics, i.e., are the most difficult to shield. α particles, which have typical energies of 5 MeV, are stopped by a few centimeters of air, since they are doubly charged and slow compared to β particles. They are detected by GM counters only if they enter through a specially designed opening (transparent to them provided they are fast enough). </p>

<p>The stopping power and energy deposition is also a function that depends strongly on the kinetic energy of the ionizing particles. In radiation medicine this is exploited, e.g., by having fast particles penetrate healthy tissue with limited damage but sufficient slow-down such that energy deposition becomes efficient when the tissue to be destroyed is reached. Usually physicists with nuclear medicine training are in charge of designing a radiation plan for each patient depending on the location of the tissue to be destroyed, vicinity of vital organs, etc. This is a non-trivial process, since secondary radiation (e.g., production of electrons) contributes to the energy deposition and may diffuse the flux of radiation.</p>

<h2>Lifetimes of radioactive sources</h2>
<p>A proper understanding of nuclear decays on the basis of a nuclear shell model (in analogy to atomic structure of electronic energy levels) enables one to predict the energies of the emitted particles as well as the half-lifes. The lifetime is related to the broadening in energy of the decaying state and can be understood from Heisenberg's uncertainty principle. (As a function of time the number of decaying particles is described by an exponential decay law.) </p>

<p>The radioactive sources that we use in this experiment do not permit a measurement of the decay law, since they have long lifetimes (tens to thousands of years), i.e., it is impossible to observe the decrease in radioactivity over a reasonable time span. However, sources with a short lifetime can be produced by exposure of a sample to a high-flux source, e.g., a reactor, which results in the conversion of stable nuclei into unstable ones.</p>

<h2>Statistics of nuclear counting</h2>
<p>In the early studies of radioactivity it was not understood whether radioactivity was a purely random process, or whether the emission of one particle might effect the emission of others. One can prove that the observation of the number of independent decays per time interval (count rate) as a function of time should result in a Poissonian distribution (ref. 5). In the limit of high count rates the Poissonian distribution can be approximated by a Gaussian distribution. </p>

<p>Rutherford performed experiments that showed that the probability, ''P(n)'', of observing ''n'' counts in a fixed time interval followed the Poisson formula</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Rd-eqn1.png|150px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

<p>where the ''average'' number of counts per interval is calculated as </p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Rd-eqn2.png|280px|center]]
</p>
</td></table>

<p>For all the measurements in this experiment that are performed with computerized data acquisition and data analysis, the Poissonian character of the statistical distribution of decay events are to be investigated and verified. Since the computer program "Particle Tracking.vi" performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (ch. 11 in ref. 5).</p>

<p>To illustrate how one explicitly analyzes the data we include an example for your convenience.</p>

<ol>
<li>Let us say that you record the number of counts heard during 100 five-second intervals by entering a mark in the column appropriate for that number of counts (col. 2 in the table below).
<table width=420 align=center>
<tr><td width=120><b>Number of Counts in interval (n)</b></td><td width=120><b>Number of times Count occurs</b></td><td width=100><b>''P(n)''</b></td><td width=100><b>Total Counts</b></td></tr>
<tr><td>0</td> <td>I(1)</td> <td>0.01</td> <td>0x1=0</td></tr>
<tr><td>1</td> <td>III(3)</td> <td>0.03</td> <td>1x3=3</td></tr>
<tr><td>2</td> <td>IIII I(5)</td> <td>0.05</td> <td>2x5=10</td></tr>
<tr><td>etc..</td><td></td><td></td><td></td></tr>
</table>
</li>

<li><p>Now construct a bar graph for the results, showing ''P(n)'' vs ''n'', where ''P(n)'' is the probability for finding n counts:</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Rd-eqn3.png|260px|center]]
</p>
</td></table>
<p>Then, using the Poisson distribution (Eq. 1) evaluate ''P(n)'' and graph the theoretical distribution over the same range of values. To do this, you require the value of n-bar; this should be the mean number of counts in your measurement:</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Rd-eqn4.png|110px|center]]
</p>
</td></table>
<p>Thus, your theoretical distribution and your experimental results will have the same mean.</p>
</li>
<li><p>Now calculate the standard deviation of your data:</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Rd-eqn5.png|180px|center]]
</p>
</td></table>
</li>

<li><p>Compare this with the expected standard deviation from the theoretical probability distribution, which is (for a Poisson distribution):</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Rd-eqn6.png|140px|center]]
</p>
</td></table>
<p>Note that this simple relation between the '''mean''' and the standard deviation is not a property of all distributions.</p>
</li>

<li><p>To see if the numbers of counts obey Poisson statistics in a quantitative way, we use the Chi-squared (χ<sup>2</sup>) test (ch. 12 in ref. 5; an example is given on pg. 235). From the reduced χ<sup>2</sup> value one infers the agreement.</p>
</li>

</ol>

<h1>Experimental Procedure</h1>

<p>In this experiment a Geiger-Müller counter with a computer interface is used to detect the radiation coming from the natural background, as well as from some weak sources. The statistics of the decays is investigated to confirm the independence of the decay mechanism. The dependence of the count rate on the distance from the source is also investigated. Also, the Geiger-Müller method for detection of radioactivity will be investigated.</p>

<p>Familiarize yourself with the computer-interfaced GM counter and associated computer software.</p>

<h2>Required Components</h2>
<ol>
<li>[[Media:Radioactive-ACratemeter.JPG|AC Powered Table-Top GM Counter]]</li>
<li>[[Media:RDHandHeldGM.JPG|Hand-held GM Counter]]</li>
<li>[[Media:RDBeigeFiesta.JPG|Beige 'Fiesta' Ceramic Dish]]</li>
<li>[[Media:RDOrangeFiesta.JPG|Orange 'Fiesta' Ceramic Dish]]</li>
<li>[[Media:RDMantles.JPG|α,γ Source: <sub>90</sub>Th<sup>232</sup>, Lantern Mantles]]</li>
</ol>

<h2>Hardware instructions:</h2>
<p>The hand-held GM counter can be operated independent of the computer interface. You should use it in range I (up to 2000? counts per minute - cpm), and turn on the audio monitoring. The background rate should be in the range of up to a few counts per second. For sources we use a bag containing Coleman-type naphta lantern mantles and Fiesta plates. Original Coleman mantles used radioactive elements until 1990; the clones still use a <sub>90</sub>Th<sup>232</sup> α emitter to enhance fluorescence. Radioactive elements were also used in glazing for bathroom tiles and Fiesta plates (no longer on the market). Make sure that the sources are some distance away from the GM counter when measuring the background radiation.</p>

<h2>Computer Instruction</h2>

<p>Data will be collected using a program called "Particle Tracking.vi" located on the desktop.
This program uses the microphone input of the computer to monitor the counts from the "Radiation Alert- Monitor 4" detector. The operation of the program is is described below</p>

<table width=400 align=center>
<td>
<p align=justify>[[File:Rd-vi.png|800px|center]]
</p>
</td></table>

<p>''Note that the program displays a histogram of the results for you to see, but only the raw data of the counts is written to the output file.''</p>

<h2>Required Data</h2>
<ol>
<li>Test the statistics of nuclear background radiation. Note the direction in which the GM counter is pointing. Make sure that it is aiming at free space, and not at a potential radioactive source. Take at least two runs, one of which should be with a larger amount of data to observe an improvement in the fit to a Poissonian distribution. Comment on the χ<sup>2</sup> obtained, and quote the decay rate, with its standard error. Include histograms of the distributions. Repeat the longer run with the GM counter pointing in an orthogonal direction. Are the data consistent with the previous run? Should they be? What are some sources of background radiation? Save the data points for one of the long runs to a data file. Perform the Poisson statistics analysis explicitly as described in the example in the previous section.</li>

<li>Perform measurements similar to (1) while bringing the bag with lantern mantles (<sub>90</sub>Th<sup>232</sup> α,γ source) close to the opening of the GM counter. Comment on the obtained distribution. Use a detailed table of isotopes (with decay schemes) to identify the radionuclide of the thorium family (ref. 6). </li>

<li>Place the orange 'Fiesta' ceramic dish plate on the table. Mount the GM counter centered above the plate using a retort stand. Measure average count rates as a function of distance, e.g., 0.5 cm, 5 cm, 10 cm, 15 cm, 20 cm, 25 cm. Has the count rate at 25 cm reached the background count rate within errors? Plot the count rates after subtraction of the background rate as a function of distance. What functional behaviour do you find? Can you explain why the Geiger counter is responding when exposed to the Fiesta plate? Show relevant decay chain diagrams.</li>

<li>Turn on the AC powered table-top GM counter. Set the knob to HV and dial up an operating voltage not exceeding 1200 Volts. Set the knob to display count rate X1 (in counts per minute) and note the background radiation. Place the beige Fiesta dish close to the exposed GM tube (the aluminium shield can be rotated such that an opening appears). You may need to reduce the sensitivity of the meter by setting the knob to the X10 range. Then measure the count rate as a function of the operating voltage.</li>
</ol>

<p>Incorporate in your report an outline on the three nuclear decay mechanisms. The function of the GM counter should also be explained briefly in the report.</p>

<h1>References</h1>

<ol>
<li>Knoll, G.F., ''Radiation Detection and Measurement'', 2nd ed.</li>
<li>Tsoulfanidis, N., ''Measurement and Detection of Radiation''.</li>
<li>Rohlf, J.W., ''Modern Physics from α to Z<sup>0</sup>'', Wiley 1994 https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/q36jf8/alma991011387139705164</li>
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li>
<li>Taylor, J.R., ''An Introduction to Error Analysis'', University Science Books, 1997.</li>
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li>
</ol>

Main Page/PHYS 3220

2021-08-13T13:43:17Z

Gloria:

<h1>PHYS 3220 3.0 Experiments in Modern Physics </h1>
<p>A selection of experiments in fluid mechanics,
electromagnetism, optics, and atomic, nuclear, and
particle physics. Analysis of the data and detailed
write-ups are required. One lecture hour which is
devoted to techniques of data analysis and three
laboratory hours per week.</p>

<h1>Laboratory Manual</h1>
<p>'''IMPORTANT NOTE: Fall 2020 PHYS 3220 Lab Scheduling procedures will be announced via the Moodle eClass course website http://moodle.yorku.ca/ prior to the beginning of term.'''</p>

<ul>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Cavendish|Measurement of the Gravitation Constant G: The Cavendish Experiment]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Speed of Light|A Measurement of the Velocity of Light: The Foucault-Michelson Experiment]] </td><td> 209 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]] </td><td> 111 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Millikan|Determination of the Electric Charge Unit ''e'' : The Millikan Oil Drop Experiment]] </td><td> 111 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Thermionic|Thermionic Emission]]</td><td> 126 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 126 PSE </td><td></td></tr>
</table>
</ul>

Main Page/PHYS 3220

2021-08-11T20:22:20Z

Gloria:

<h1>PHYS 3220 3.0 Experiments in Modern Physics </h1>
<p>A selection of experiments in fluid mechanics,
electromagnetism, optics, and atomic, nuclear, and
particle physics. Analysis of the data and detailed
write-ups are required. One lecture hour which is
devoted to techniques of data analysis and three
laboratory hours per week.</p>

<h1>Laboratory Manual</h1>
<p>'''IMPORTANT NOTE: Fall 2020 PHYS 3220 Lab Scheduling procedures will be announced via the Moodle eClass course website http://moodle.yorku.ca/ prior to the beginning of term.'''</p>

<ul>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Cavendish|Measurement of the Gravitation Constant G: The Cavendish Experiment]]</td><td> 123 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Speed of Light|A Measurement of the Velocity of Light: The Foucault-Michelson Experiment]] </td><td> 209 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]] </td><td> 111 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Millikan|Determination of the Electric Charge Unit ''e'' : The Millikan Oil Drop Experiment]] </td><td> 111 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Thermionic|Thermionic Emission]]</td><td> 126 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 209 PSE </td><td></td></tr>
</table>
</ul>

Main Page/PHYS 3220/Digital Oscilloscope

2021-08-11T20:09:50Z

Gloria:

<h1>Digital Storage Oscilloscope</h1>

<p>In this experiment we use a function generator to produce rectangular pulses which are recorded and analyzed using a modern digital storage oscilloscope capable of performing a fast fourier transform (FFT) on a given signal. Then the behaviour of a simple RC (integrator) circuit fed by a square wave pulse is analyzed both in the steady-state and transient (turn-on) regimes. Finally the square-wave pulse is used to induce damped harmonic motion in an LC circuit.</p>

<h1>Introduction</h1>
<p>The behaviour of short time-varying signals can be investigated easily with a digital storage oscilloscope (DSO) that will allow you to trigger single events and to store them for any length of time. Periodic signals play an important role in many areas of physics. Periodic signals are conveniently analyzed in terms of harmonic (or frequency) content, either by means of a Fourier series or by a Fourier transform [1]. Typically, oscilloscopes display signals in the time domain. The DSO you are using here will allow you to process these signals so they can be displayed in the frequency domain by using an FFT signal processing module included in the DSO. For a finite wavetrain recorded at discrete time intervals two parameters impose practical limitations on acquiring knowledge about the frequency content of a pulse. The sampling rate ''Δt'' limits the maximum frequency that can be recorded (intuitively: a signal that changes sign at every ''t<sub>j</sub>'' = ''j Δt'' has the highest frequency that can be represented on the discrete time axis). The length of the recorded signal, ''T'', limits the frequency resolution, ''Δf'': the lowest frequency that can be recorded corresponds to a wave with a period that equals ''T''.</p>
<p>Thus, the Fourier transform that would be available in an ideal measurement (continuous sampling and infinite length of measurement)</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Dso-eqn1.png|220px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>this is replaced by a discrete Fourier transform: </p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Dso-eqn2.png|310px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

<p>where the length of the signal in the time domain is given by ''T'' = ''N Δt'' , the maximum frequency (called the Nyquist frequency) is actually one-half of the inverse sampling rate ''f<sub>c</sub>'' = 1/(2 ''Δt'') as the range of frequencies ranges in principle from -''f<sub>c</sub>'' to ''f<sub>c</sub>'' (but the answer is symmetric in f so we are only interested in the range [0, ''f<sub>c</sub>'']). The frequency resolution equals ''Δf'' = 1/(''N Δt''). There are numerous problems associated with the replacement of eq (1) by eq (2), these problems are discussed in detail in [2] (and hinted at in the manual for the Tektronix TBS 1052B-EDU oscilloscope). To minimize some of the errors one usually multiplies the time signal with a window function whose main purpose is to provide a smooth switching of the pulse and thereby to eliminate artificial high-frequency components that would be associated with chopping a periodic signal at ''t'' = 0 and ''t'' = ''T'' and assuming periodicity (wrap-around).</p>
<p>In a digital storage oscilloscope (as in computer programs that analyze recorded data) an FT function is provided by an efficient algorithm (fast FT = FFT) [2]. In a computer interfaced experiment one usually chooses the sampling rate and temporal record length freely. However, with the DSO these parameters are not independent and are actually controlled simultaneously by the DSO time base control. Furthermore, the number of points used in memory (for display on the screen and transfer to a computer) for a single trace is fixed. The FFT algorithm requires this number to be a power of 2 (typical numbers are 1024, 2048, etc. ). </p>
<p>Consider a rectangular pulse, and a square-wave in particular. It can be thought of as a superposition of a sine wave with the same period τ, given by admixtures of sine waves with higher frequencies (or periods that are simple fractions of τ). This is known from Fourier series expansions of this periodic function. Only sine waves with odd multiples of the base frequency contribute (odd-order harmonics), and the coefficients in the linear combination can be calculated [1]. In part 1 of the experiment the Fourier spectrum (which could detect any frequency components, and not just multiples of the base frequency) is taken for rectangular pulses with different duty cycles. The objective is to show that for a pulse with duty cycle (τ/''n'')/τ = 1:''n'' (cf.. Fig. 1) the ''n''<sup>th</sup> harmonic is absent. For a square-wave pulse (1:2) only odd orders appear.</p>

<table width=500 align=center><td>
<p align=justify>[[File:Dso-fig1.png|500px|border|center]]
<b>Figure 1 -</b> A rectangular pulse with a duty cycle of 1:3.
<br clear=right>
</p>
</td></table>

<p>For a square wave pulse the harmonic coefficients can be calculated from the Fourier series expansion to be (we assume a signal that ranges between 0 and 1)</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Dso-eqn3.png|300px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

<h1>Procedure</h1>
<h2>Part 1: Harmonic Analysis of Rectangular Pulses</h2>


<h3>The Tektronix TBS 1052B-EDU Oscilloscope</h3>
<p>Connect the 50 Ω output of the function generator (WAVETEK, model 184) to Ch.1 of the DSO. Turn on the power switch of the function generator and familiarize yourself with the effect of varying the frequency settings of the function generator, and learn how to use the digital scope. Links to the user manuals: [[Media:DSOManual.pdf| DSO User Manual]] and [[Media:Wavetek184FGManual.pdf| Wavetek FG User Manual]]. Refer to section 3 of the WAVETEK instruction manual for a description of the function generator operation controls. There are several menus that control the DSO operation. For this part you need to use the menu that controls the first Y channel (CH 1), and possibly the TRIGGER menu. In the TRIGGER menu it is important that the <u>Source</u> of triggering be specified as CH 1 and the <u>Mode</u> of triggering be set on Auto. (Only for the measurement of the transient behaviour of an RC circuit will you need to switch to Single Mode). The 5 buttons to the right of the LCD display are used to toggle through the options indicated on the DSO screen.</p>
<p>The display of channel 1 or 2 can be turned on and off using the <u>CH1</u> and <u>CH2</u> menu buttons. Once the menu for a given channel is activated the 5 buttons to the right of the LCD display can be used to control:
<ol style="list-style-type:lower-latin">
<li> the ''Coupling'' (DC, AC, Ground).</li>
<li>''Bandwidth limit'' (suppress high frequencies if desired).</li>
<li>''Volts/Division'': coarse or fine (this controls whether the knobs carry out the usual large incremental voltage scale steps (1V, 2V, 5V, etc. ) or fine steps that actually simulate an infinitely variable scale control on an analogue scope).</li>
<li>''Probe'' (10X if the Tektronix probe is attached, 1X if a simple coaxial cable is used (this setting alters the display of scale for Volts/Division being measured - it is needed as the supplied probes step down the voltage by a 1:10 ratio, i.e., 1 V applied to the probe results in 0.1 V at the coaxial input to the scope).</li>
<li>''Invert (on/off)''. When “on” this will invert the displayed signal.</li>
</ol></p>

<p>The <u>MEASURE</u> menu can be used to find period and frequency of a periodic signal, in this case a pulse. a) The top button controls whether the lower 4 (of the 5 to the right of the LCD) provide control over the source or the type of measurement. ''Sources'' can be CH1 or CH2; ''Types'' of measurement are: Frequency, Period, Mean (average voltage), Peak-to-peak voltage, Cyc RMS(?), Rise and Fall times for a pulse, as well as positive and negative width (this is particularly useful for a rectangular wave form).</p>

<p>The <u>CURSOR</u> menu will give you control over the position of a set either of two vertical or two horizontal cursors that span the display area. These are controlled by the multipurpose knob and selected using the lower two buttons to the right of the LCD screen. A useful feature of the cursors is that one can read off the cursor positions indicated on the right of the screen as digital numbers, for the horizontal positions the difference indicates a time with likewise a vertical difference indicates signal amplititude its inverse (as frequency).</p>

<p>The <u>MATH</u> menu is invoked by pressing the red button labelled ''M'' to the left of CH 1. In math mode one can add, subtract or multiply CH 1 and CH 2 signals.</p>

<p>The <u>FFT</u> menu is invoked by pressing the yellow button labelled ''FFT'' located just below the math menu button. With it on can carry out a Fourier transform FFT of CH 1 or CH 2 separately; within the FFT menu one can choose a windowing function [<u>Rectangular</u> implies no windowing, <u>Hanning</u> [2] a standard function that compromises between accurate measurement of frequency amplitudes and the accuracy of frequency measurements (the windowing function introduces an artificial width: a single-frequency signal is broadened in frequency content), <u>Flattop</u> provides more accurate amplitudes in the Fourier spectrum. Finally one can zoom in on the FFT spectrum, since the display actually only uses a few hundred pixels in the horizontal (and vertical) directions, whereas an FFT is calculated with over 4000 points. Note that the FFT is calculated for incoming data, i.e., it is not possible on the TBS 1052B-EDU to record a pulse and then carry out the FFT for that pulse. </p>

<p>The vertical scale used for the FFT is calibrated in decibels. The logarithmic decibel scale is explained in 1<sup>st</sup> year physics texts in the context of sound waves. It is a relative scale, i.e., it depends on some reference strength (which in the case of the TBS 1052B-EDU is chosen to be: 0 dB = 1 V RMS amplitude). A drop by 3 dB corresponds approximately to a reduction by a factor of 2 in amplitude.</p>
<p>With the <u>RUN/STOP</u> button (top right on the scope) you can capture the display (helpful if the signal is shaky), the <u>HARDCOPY</u> button provides a screenshot output to a printer connected to the parallel port, while <u>AUTOSET</u> helps to find settings for some acceptable screen display if you have no idea how to set the vertical amplification and the timebase (horizontal) for your particular signal. It is customary to start by pressing autoset, observe the settings chosen by the scope, and subsequently fine tune the settings as appropriate.</p>

<h3>Measurement of the rectangular pulses produced by the function generator outputs</h3>

<p>Begin detailed measurements by setting a (1:''n'') duty cycle at some frequency with the function generator. Carry out the harmonic analysis using the FFT menu functions. Save the data of the time signal and the FFT using a USB stick to include with your report. Provide your observations about the harmonic spectrum.</p>
<p>Measure rise and fall times for the square wave pulse. For this purpose start with a display of a few cycles on the screen. Now move one of the edges (first a rising, later a falling edge) to the center of the display. Turn the (horizontal) timebase switch to display shorter segments of the pulse until the initially vertical line acquires the characteristic shape for a charging capacitor (discharging in the case of a falling edge), i.e., it becomes an exponential function. Note the time scale at which this happens. Use the Measure menu to obtain a measurement of the rise and fall times (it will depend somewhat on the segment displayed on the screen, since it measures between 90 and 10 % of the signal displayed; cf. the manual for the TBS 1052B-EDU). You should spend some time thinking about clocks used in computers, how fast they have to be (PC chips run internally at speeds in the low GHz range these days, while the entire computer (the bus) can be clocked at up to 66 MHz), and how the rise and fall times are important since during those times the logical state (0 or 1) is really undetermined. There can be no ‘perfect’ square-wave or rectangular pulse.</p>

<p>In the following exercise observe how a simple logic gate can be used to shape the pulse. Use the TTL output of the function generator on Ch. 2 and look at the same characteristics as with the 50 Ohm output. Compare the measured rise and fall times of the two outputs. You can overlap the two signals and zoom in to observe the two signals in detail. </p>


<p>To deepen your understanding of the idea of triggering on a signal it is interesting to display the signals from two uncorrelated generators: use the signal from the one of the function generator output (50 Ohm output or TTL output) on one channel and connect the internal 1 kHz generator to the other. (See TBS 1052B-EDU scope diagram – probe compensation terminal lugs located in the DSO manual.) From the TRIGGER menu choose to trigger on either CH1 or CH2. What do you observe? Use the RUN/STOP button repeatedly to capture still images and explain the behaviour.</p>

<h2>Part 2: Charging and Discharging a Capacitor</h2>

<p>Set the TTL output of the function generator to produce a square-wave signal of a given frequency (rectangular pulse with duty cycle 1:2). Connect an RC circuit such that the capacitor C is charged via the variable resistor box R. Measure the voltage as produced by the function generator (use a BNC Tee to split the TTL output) on CH1 and the voltage across C on CH2. Note that CH1 and CH2 have a common ground: think before making the connections. The physics of charging and discharging a capacitor is explained in first-year physics texts. Familiarize yourself with the material, we provide no equations here. You need to realize that during the ‘low’ output the square-wave generator acts as a short, i.e., it discharges the capacitor through the resistor R (the internal resistance of the function generator is often in the 50 Ohms range which is negligible if R is in the kΩ range).</p>
<p>For a fixed choice of R and C (calculate the time constant T) make measurements for three different settings of the square-wave frequency. Adjust the period τ  (to have a 1:2 duty cycle) such that the three cases cover a period τ less than ''T'', comparable to ''T'', and bigger than ''T''. Use DC coupling on both channels, and observe the steady-state behaviour of the sequence of periodically charging and discharging. Provide explanations to compare the three cases. Why is the voltage across the capacitor periodic? In the next section you will measure the transient or turn-on behaviour. Fig. 3 is provided as an illustration of the case where ''τ'' < ''T'' such that the capacitor does not fully charge or discharge during one of the half-periods τ/2. Each segment of the curve is obtained from the solution to the differential equation describing the charge or discharge regimes, and the correct initial condition is being applied (we assume no charges on the capacitor plates at ''t''=0).</p>
<table width=500 align=center><td>
<p align=justify>[[File:Dso-fig3.png|500px|border|center]]
<b>Figure 3 -</b> RC circuit response to a periodic signal.
<br clear=right>
</p>
</td></table>
<p>An interesting quantity to measure in the charging/discharging RC circuit is the voltage across the resistor, which according to Ohm’s law provides a measure of the current in the circuit. Thus, it is possible to observe how the current changes sign and jumps from some low (possibly near-zero) value at the end of a cycle to ±U/R. Mathematically the answer is discontinuous, and this is interesting to investigate in real life. From a measurement point of view the matter can be straightforward: if one uses an external function generator, one can simply connect a probe across R. Note, however, that one cannot also connect the other probe across the capacitor (or to measure the signal coming from the function generator), since CH1 and CH2 have a common ground (and the internal 1 kHz generator uses this ground as well). One trick, however, is to perform the same measurement as before on CH1 and CH2, and to display the difference between the two channels. This is done by using the CH1 + CH2 option on the MATH menu while inverting one of the channels. Perform such a measurement with several cycles shown on the display. There is an apparent discontinuity in the voltage across R (obtained as the difference between the voltage produced by the function generator and the voltage across C). While a discontinuity is located at the center of the screen zoom in using the TIMEBASE adjustment. At what times scale do you resolve the discontinuity, and why? </p>

<h2>Part 3: Transient Behaviour (cf.. Fig. 3)</h2>
<p>Use the same connections as in Part 2, i.e., consider the charging and discharging of a capacitor C through a resistor R. In this part we are interested in the first 10-20 cycles of a square-wave pulse to see how the RC circuit approaches its nearly-periodic behaviour observed in the previous part. For this purpose one has to set the triggering onto the mode single pulses and then adjust the trigger level for a small positive voltage. The RUN/STOP button is used to acquire the signal. The trigger level is set to such a value that when the square-wave generator is disconnected, the scope indicates readiness to record but is not triggered. As the square wave is applied by flipping a switch a single trace is recorded. It is possible to adjust the timing of the single trace with the horizontal adjustment, and by repeating the measurement.</p>
<p>Make observations for a settings of the square-wave time constants τ < ''T'', and comment on your results. Why is the other limit τ <'' T'' uninteresting?</p>

<h2>Part 4: Damped Oscillations in an LC circuit</h2>
<p>Here we use square-wave pulses to induce damped oscillations in a circuit that consists of an inductance L and a capacitance C connected in series. Consult a first-year physics text (e.g., [4]) on the physics of storing electrical energy in the form of an electric field in C and in the form of a magnetic field in L, and how they are exchanged in an RLC circuit. Note that a resistance is present even if no resistor is put into the circuit, as there is no ideal coil, i.e., it always has an Ohmic resistance. The differential equation that can be derived from Kirchoff’s law together with the properties of a capacitor, an inductance and a resistor is comparable to that of a damped harmonic oscillator. For the capacitor charge ''q(t)'' we obtain:</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Dso-eqn4.png|210px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>

<p>Here L plays the role of inertia (mass), R appears as a friction constant, and the capacitance C plays the role of the spring constant. Understand why the application of a square-wave pulse corresponds to kicking a damped harmonic oscillator: ''q''(0) = 0, ''q’''(0) ≠ 0.</p>
<p>In the case of weak damping (undercritical damping) the solution to eq. (3) has the form:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Dso-eqn5.png|280px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>
<p>Record time signals for an LC circuit with known capacitance and inductance, and compare with the solution to the differential equation. Is the measured frequency consistent with the solution? Estimate the resistance from the decrease in the envelope. You can measure the resistance R with a digital voltmeter.


Note, the commercial function generator has a typical internal resistance of 50 Ω. Perform measurements with a variable resistor box R included in the circuit and adjust R to find critical damping (in addition to the undercritically damped situation described above). Critical damping is obtained when the time constant of the damping ''T'' = 2L/R equals the inverse of the natural (circular) frequency ω.</p>

<h1>References</h1>
<ol>
<li>G.A. Arfken, ''Mathematical Methods for Physicists'', Academic Press.</li>
<li>W.P. Press, S. Teukolsky, Vetterling, Flannery, ''Numerical Recipes'', Chapter 12, Cambridge University Press.</li>
<li>Horowitz Hill, ''The Art of Electronics'', chapter 5.14, Cambridge University Press.</li>
<li>R. Wolfson, J.M. Pasachoff, ''Physics'', 2nd ed., chapter 33-3, Harper Collins, New York 1995.</li>
</ol>

Main Page/PHYS 3220/Digital Oscilloscope

2021-08-11T20:09:10Z

Gloria:

<h1>Digital Storage Oscilloscope</h1>

<p>In this experiment we use a function generator to produce rectangular pulses which are recorded and analyzed using a modern digital storage oscilloscope capable of performing a fast fourier transform (FFT) on a given signal. Then the behaviour of a simple RC (integrator) circuit fed by a square wave pulse is analyzed both in the steady-state and transient (turn-on) regimes. Finally the square-wave pulse is used to induce damped harmonic motion in an LC circuit.</p>

<h1>Introduction</h1>
<p>The behaviour of short time-varying signals can be investigated easily with a digital storage oscilloscope (DSO) that will allow you to trigger single events and to store them for any length of time. Periodic signals play an important role in many areas of physics. Periodic signals are conveniently analyzed in terms of harmonic (or frequency) content, either by means of a Fourier series or by a Fourier transform [1]. Typically, oscilloscopes display signals in the time domain. The DSO you are using here will allow you to process these signals so they can be displayed in the frequency domain by using an FFT signal processing module included in the DSO. For a finite wavetrain recorded at discrete time intervals two parameters impose practical limitations on acquiring knowledge about the frequency content of a pulse. The sampling rate ''Δt'' limits the maximum frequency that can be recorded (intuitively: a signal that changes sign at every ''t<sub>j</sub>'' = ''j Δt'' has the highest frequency that can be represented on the discrete time axis). The length of the recorded signal, ''T'', limits the frequency resolution, ''Δf'': the lowest frequency that can be recorded corresponds to a wave with a period that equals ''T''.</p>
<p>Thus, the Fourier transform that would be available in an ideal measurement (continuous sampling and infinite length of measurement)</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Dso-eqn1.png|220px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>this is replaced by a discrete Fourier transform: </p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Dso-eqn2.png|310px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

<p>where the length of the signal in the time domain is given by ''T'' = ''N Δt'' , the maximum frequency (called the Nyquist frequency) is actually one-half of the inverse sampling rate ''f<sub>c</sub>'' = 1/(2 ''Δt'') as the range of frequencies ranges in principle from -''f<sub>c</sub>'' to ''f<sub>c</sub>'' (but the answer is symmetric in f so we are only interested in the range [0, ''f<sub>c</sub>'']). The frequency resolution equals ''Δf'' = 1/(''N Δt''). There are numerous problems associated with the replacement of eq (1) by eq (2), these problems are discussed in detail in [2] (and hinted at in the manual for the Tektronix TBS 1052B-EDU oscilloscope). To minimize some of the errors one usually multiplies the time signal with a window function whose main purpose is to provide a smooth switching of the pulse and thereby to eliminate artificial high-frequency components that would be associated with chopping a periodic signal at ''t'' = 0 and ''t'' = ''T'' and assuming periodicity (wrap-around).</p>
<p>In a digital storage oscilloscope (as in computer programs that analyze recorded data) an FT function is provided by an efficient algorithm (fast FT = FFT) [2]. In a computer interfaced experiment one usually chooses the sampling rate and temporal record length freely. However, with the DSO these parameters are not independent and are actually controlled simultaneously by the DSO time base control. Furthermore, the number of points used in memory (for display on the screen and transfer to a computer) for a single trace is fixed. The FFT algorithm requires this number to be a power of 2 (typical numbers are 1024, 2048, etc. ). </p>
<p>Consider a rectangular pulse, and a square-wave in particular. It can be thought of as a superposition of a sine wave with the same period τ, given by admixtures of sine waves with higher frequencies (or periods that are simple fractions of τ). This is known from Fourier series expansions of this periodic function. Only sine waves with odd multiples of the base frequency contribute (odd-order harmonics), and the coefficients in the linear combination can be calculated [1]. In part 1 of the experiment the Fourier spectrum (which could detect any frequency components, and not just multiples of the base frequency) is taken for rectangular pulses with different duty cycles. The objective is to show that for a pulse with duty cycle (τ/''n'')/τ = 1:''n'' (cf.. Fig. 1) the ''n''<sup>th</sup> harmonic is absent. For a square-wave pulse (1:2) only odd orders appear.</p>

<table width=500 align=center><td>
<p align=justify>[[File:Dso-fig1.png|500px|border|center]]
<b>Figure 1 -</b> A rectangular pulse with a duty cycle of 1:3.
<br clear=right>
</p>
</td></table>

<p>For a square wave pulse the harmonic coefficients can be calculated from the Fourier series expansion to be (we assume a signal that ranges between 0 and 1)</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Dso-eqn3.png|300px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

<h1>Procedure</h1>
<h2>Part 1: Harmonic Analysis of Rectangular Pulses</h2>


<h3>The Tektronix TBS 1052B-EDU Oscilloscope</h3>
<p>Connect the 50 Ω output of the function generator (WAVETEK, model 184) to Ch.1 of the DSO. Turn on the power switch of the function generator and familiarize yourself with the effect of varying the frequency settings of the function generator, and learn how to use the digital scope. Links to the user manuals: [[Media:DSOManual.pdf| DSO User Manual]] and [[Media:Wavetek184FGManual.pdf| Wavetek FG User Manual]]. Refer to section 3 of the WAVETEK instruction manual for a description of the function generator operation controls. There are several menus that control the DSO operation. For this part you need to use the menu that controls the first Y channel (CH 1), and possibly the TRIGGER menu. In the TRIGGER menu it is important that the <u>Source</u> of triggering be specified as CH 1 and the <u>Mode</u> of triggering be set on Auto. (Only for the measurement of the transient behaviour of an RC circuit will you need to switch to Single Mode). The 5 buttons to the right of the LCD display are used to toggle through the options indicated on the DSO screen.</p>
<p>The display of channel 1 or 2 can be turned on and off using the <u>CH1</u> and <u>CH2</u> menu buttons. Once the menu for a given channel is activated the 5 buttons to the right of the LCD display can be used to control:
<ol style="list-style-type:lower-latin">
<li> the ''Coupling'' (DC, AC, Ground).</li>
<li>''Bandwidth limit'' (suppress high frequencies if desired).</li>
<li>''Volts/Division'': coarse or fine (this controls whether the knobs carry out the usual large incremental voltage scale steps (1V, 2V, 5V, etc. ) or fine steps that actually simulate an infinitely variable scale control on an analogue scope).</li>
<li>''Probe'' (10X if the Tektronix probe is attached, 1X if a simple coaxial cable is used (this setting alters the display of scale for Volts/Division being measured - it is needed as the supplied probes step down the voltage by a 1:10 ratio, i.e., 1 V applied to the probe results in 0.1 V at the coaxial input to the scope).</li>
<li>''Invert (on/off)''. When “on” this will invert the displayed signal.</li>
</ol></p>

<p>The <u>MEASURE</u> menu can be used to find period and frequency of a periodic signal, in this case a pulse. a) The top button controls whether the lower 4 (of the 5 to the right of the LCD) provide control over the source or the type of measurement. ''Sources'' can be CH1 or CH2; ''Types'' of measurement are: Frequency, Period, Mean (average voltage), Peak-to-peak voltage, Cyc RMS(?), Rise and Fall times for a pulse, as well as positive and negative width (this is particularly useful for a rectangular wave form).</p>

<p>The <u>CURSOR</u> menu will give you control over the position of a set either of two vertical or two horizontal cursors that span the display area. These are controlled by the multipurpose knob and selected using the lower two buttons to the right of the LCD screen. A useful feature of the cursors is that one can read off the cursor positions indicated on the right of the screen as digital numbers, for the horizontal positions the difference indicates a time with likewise a vertical difference indicates signal amplititude its inverse (as frequency).</p>

<p>The <u>MATH</u> menu is invoked by pressing the red button labelled ''M'' to the left of CH 1. In math mode one can add, subtract or multiply CH 1 and CH 2 signals.</p>

<p>The <u>FFT</u> menu is invoked by pressing the yellow button labelled ''FFT'' located just below the math menu button. With it on can carry out a Fourier transform FFT of CH 1 or CH 2 separately; within the FFT menu one can choose a windowing function [<u>Rectangular</u> implies no windowing, <u>Hanning</u> [2] a standard function that compromises between accurate measurement of frequency amplitudes and the accuracy of frequency measurements (the windowing function introduces an artificial width: a single-frequency signal is broadened in frequency content), <u>Flattop</u> provides more accurate amplitudes in the Fourier spectrum. Finally one can zoom in on the FFT spectrum, since the display actually only uses a few hundred pixels in the horizontal (and vertical) directions, whereas an FFT is calculated with over 4000 points. Note that the FFT is calculated for incoming data, i.e., it is not possible on the TBS 1052B-EDU to record a pulse and then carry out the FFT for that pulse. </p>

<p>The vertical scale used for the FFT is calibrated in decibels. The logarithmic decibel scale is explained in 1<sup>st</sup> year physics texts in the context of sound waves. It is a relative scale, i.e., it depends on some reference strength (which in the case of the TBS 1052B-EDU is chosen to be: 0 dB = 1 V RMS amplitude). A drop by 3 dB corresponds approximately to a reduction by a factor of 2 in amplitude.</p>
<p>With the <u>RUN/STOP</u> button (top right on the scope) you can capture the display (helpful if the signal is shaky), the <u>HARDCOPY</u> button provides a screenshot output to a printer connected to the parallel port, while <u>AUTOSET</u> helps to find settings for some acceptable screen display if you have no idea how to set the vertical amplification and the timebase (horizontal) for your particular signal. It is customary to start by pressing autoset, observe the settings chosen by the scope, and subsequently fine tune the settings as appropriate.</p>

<h3>Measurement of the rectangular pulses produced by the function generator outputs</h3>

<p>Begin detailed measurements by setting a (1:''n'') duty cycle at some frequency with the function generator. Carry out the harmonic analysis using the FFT menu functions. Save the data of the time signal and the FFT using a USB stick to include with your report. Provide your observations about the harmonic spectrum.</p>
<p>Measure rise and fall times for the square wave pulse. For this purpose start with a display of a few cycles on the screen. Now move one of the edges (first a rising, later a falling edge) to the center of the display. Turn the (horizontal) timebase switch to display shorter segments of the pulse until the initially vertical line acquires the characteristic shape for a charging capacitor (discharging in the case of a falling edge), i.e., it becomes an exponential function. Note the time scale at which this happens. Use the Measure menu to obtain a measurement of the rise and fall times (it will depend somewhat on the segment displayed on the screen, since it measures between 90 and 10 % of the signal displayed; cf. the manual for the TBS 1052B-EDU). You should spend some time thinking about clocks used in computers, how fast they have to be (PC chips run internally at speeds in the low GHz range these days, while the entire computer (the bus) can be clocked at up to 66 MHz), and how the rise and fall times are important since during those times the logical state (0 or 1) is really undetermined. There can be no ‘perfect’ square-wave or rectangular pulse.</p>

<p>In the following exercise observe how a simple logic gate can be used to shape the pulse. Use the TTL output of the function generator on Ch. 2 and look at the same characteristics as with the 50 Ohm output. Compare the measured rise and fall times of the two outputs. You can overlap the two signals and zoom in to observe the two signals in detail. </p>


<p>To deepen your understanding of the idea of triggering on a signal it is interesting to display the signals from two uncorrelated generators: use the signal from the one of the function generator output (50 Ohm output or TTL output) on one channel and connect the internal 1 kHz generator to the other. (See TBS 1052B-EDU scope diagram – probe compensation terminal lugs located in the DSO manual in the red binder.) From the TRIGGER menu choose to trigger on either CH1 or CH2. What do you observe? Use the RUN/STOP button repeatedly to capture still images and explain the behaviour.</p>

<h2>Part 2: Charging and Discharging a Capacitor</h2>

<p>Set the TTL output of the function generator to produce a square-wave signal of a given frequency (rectangular pulse with duty cycle 1:2). Connect an RC circuit such that the capacitor C is charged via the variable resistor box R. Measure the voltage as produced by the function generator (use a BNC Tee to split the TTL output) on CH1 and the voltage across C on CH2. Note that CH1 and CH2 have a common ground: think before making the connections. The physics of charging and discharging a capacitor is explained in first-year physics texts. Familiarize yourself with the material, we provide no equations here. You need to realize that during the ‘low’ output the square-wave generator acts as a short, i.e., it discharges the capacitor through the resistor R (the internal resistance of the function generator is often in the 50 Ohms range which is negligible if R is in the kΩ range).</p>
<p>For a fixed choice of R and C (calculate the time constant T) make measurements for three different settings of the square-wave frequency. Adjust the period τ  (to have a 1:2 duty cycle) such that the three cases cover a period τ less than ''T'', comparable to ''T'', and bigger than ''T''. Use DC coupling on both channels, and observe the steady-state behaviour of the sequence of periodically charging and discharging. Provide explanations to compare the three cases. Why is the voltage across the capacitor periodic? In the next section you will measure the transient or turn-on behaviour. Fig. 3 is provided as an illustration of the case where ''τ'' < ''T'' such that the capacitor does not fully charge or discharge during one of the half-periods τ/2. Each segment of the curve is obtained from the solution to the differential equation describing the charge or discharge regimes, and the correct initial condition is being applied (we assume no charges on the capacitor plates at ''t''=0).</p>
<table width=500 align=center><td>
<p align=justify>[[File:Dso-fig3.png|500px|border|center]]
<b>Figure 3 -</b> RC circuit response to a periodic signal.
<br clear=right>
</p>
</td></table>
<p>An interesting quantity to measure in the charging/discharging RC circuit is the voltage across the resistor, which according to Ohm’s law provides a measure of the current in the circuit. Thus, it is possible to observe how the current changes sign and jumps from some low (possibly near-zero) value at the end of a cycle to ±U/R. Mathematically the answer is discontinuous, and this is interesting to investigate in real life. From a measurement point of view the matter can be straightforward: if one uses an external function generator, one can simply connect a probe across R. Note, however, that one cannot also connect the other probe across the capacitor (or to measure the signal coming from the function generator), since CH1 and CH2 have a common ground (and the internal 1 kHz generator uses this ground as well). One trick, however, is to perform the same measurement as before on CH1 and CH2, and to display the difference between the two channels. This is done by using the CH1 + CH2 option on the MATH menu while inverting one of the channels. Perform such a measurement with several cycles shown on the display. There is an apparent discontinuity in the voltage across R (obtained as the difference between the voltage produced by the function generator and the voltage across C). While a discontinuity is located at the center of the screen zoom in using the TIMEBASE adjustment. At what times scale do you resolve the discontinuity, and why? </p>

<h2>Part 3: Transient Behaviour (cf.. Fig. 3)</h2>
<p>Use the same connections as in Part 2, i.e., consider the charging and discharging of a capacitor C through a resistor R. In this part we are interested in the first 10-20 cycles of a square-wave pulse to see how the RC circuit approaches its nearly-periodic behaviour observed in the previous part. For this purpose one has to set the triggering onto the mode single pulses and then adjust the trigger level for a small positive voltage. The RUN/STOP button is used to acquire the signal. The trigger level is set to such a value that when the square-wave generator is disconnected, the scope indicates readiness to record but is not triggered. As the square wave is applied by flipping a switch a single trace is recorded. It is possible to adjust the timing of the single trace with the horizontal adjustment, and by repeating the measurement.</p>
<p>Make observations for a settings of the square-wave time constants τ < ''T'', and comment on your results. Why is the other limit τ <'' T'' uninteresting?</p>

<h2>Part 4: Damped Oscillations in an LC circuit</h2>
<p>Here we use square-wave pulses to induce damped oscillations in a circuit that consists of an inductance L and a capacitance C connected in series. Consult a first-year physics text (e.g., [4]) on the physics of storing electrical energy in the form of an electric field in C and in the form of a magnetic field in L, and how they are exchanged in an RLC circuit. Note that a resistance is present even if no resistor is put into the circuit, as there is no ideal coil, i.e., it always has an Ohmic resistance. The differential equation that can be derived from Kirchoff’s law together with the properties of a capacitor, an inductance and a resistor is comparable to that of a damped harmonic oscillator. For the capacitor charge ''q(t)'' we obtain:</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Dso-eqn4.png|210px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>

<p>Here L plays the role of inertia (mass), R appears as a friction constant, and the capacitance C plays the role of the spring constant. Understand why the application of a square-wave pulse corresponds to kicking a damped harmonic oscillator: ''q''(0) = 0, ''q’''(0) ≠ 0.</p>
<p>In the case of weak damping (undercritical damping) the solution to eq. (3) has the form:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Dso-eqn5.png|280px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>
<p>Record time signals for an LC circuit with known capacitance and inductance, and compare with the solution to the differential equation. Is the measured frequency consistent with the solution? Estimate the resistance from the decrease in the envelope. You can measure the resistance R with a digital voltmeter.


Note, the commercial function generator has a typical internal resistance of 50 Ω. Perform measurements with a variable resistor box R included in the circuit and adjust R to find critical damping (in addition to the undercritically damped situation described above). Critical damping is obtained when the time constant of the damping ''T'' = 2L/R equals the inverse of the natural (circular) frequency ω.</p>

<h1>References</h1>
<ol>
<li>G.A. Arfken, ''Mathematical Methods for Physicists'', Academic Press.</li>
<li>W.P. Press, S. Teukolsky, Vetterling, Flannery, ''Numerical Recipes'', Chapter 12, Cambridge University Press.</li>
<li>Horowitz Hill, ''The Art of Electronics'', chapter 5.14, Cambridge University Press.</li>
<li>R. Wolfson, J.M. Pasachoff, ''Physics'', 2nd ed., chapter 33-3, Harper Collins, New York 1995.</li>
</ol>

Main Page/PHYS 3220/Cavendish

2021-08-11T18:53:01Z

Gloria:

<h1>Measurement of the Gravitational Constant ''G'' with a Torsion Balance: The Cavendish Experiment</h1>

<h1> Learning Outcomes</h1>
<ol>
<li>Rotational motion and torque
<li>Damped harmonic oscillations
<li>Error analysis
<li>Computational curve fitting
<li>Exposure to LabView
</ol>

<h1>Introduction</h1>
<p>Cavendish first performed the measurement of ''G'', one of the classic experiments in physics, in 1798. Since then, there have been many attempts <ref>J.W. Beams, <i>"Finding a Better Value for ''G''"</i>, [http://www.physicstoday.org/resource/1/phtoad/v24/i5/p34_s1 Physics Today, '''24''', 34 (1971)]</ref> to improve on this determination using variations of the same basic experiment. For example, see Gundlach<ref> J.H. Gundlach & S.M. Merkowtiz, <i>"Measurement of Newton's Constant Using a Torsion Balance with Angular Acceleration Feedback"</i>, [http://prl.aps.org/abstract/PRL/v85/i14/p2869_1 Phys. Rev. Lett., '''85''', 2869 (2000)]</ref> (2000) and Quinn<ref> T. Quinn, H. Parks, C. Speake & R. Davis, <i>"Improved Determination of ''G'' Using Two Methods"</i>, [http://prl.aps.org/abstract/PRL/v111/i10/e101102 Phys. Rev. Lett., '''111''', 101102 (2013)]</ref> (2013). The apparatus at your disposal is a modified form of that used by Cavendish and others. The high sensitivity of the apparatus demands patience, perseverance and care from the experimenter. This is a very delicate instrument so treat it gently.</p>
<p>A laser, labelled Source in Fig. 1, illuminates a small (effectively) massless mirror MM that is attached to a light horizontal rod holding two small lead balls of mass m at a separation of 10cm. The small balls and mirror are suspended from a 35cm bronze torsion wire (perpendicular to the page in Fig. 1). The entire apparatus is enclosed in a rigid case that is mounted securely on a wall or table.</p>
<p>When two massive lead balls (each with mass M = 1.5kg) are placed asymmetrically as illustrated, in position AA, a small torque acts on the torsion balance twisting the torsion wire and causing the image of the light source to swing through a measurable distance Δ' along the opposite wall. The swing gradually decays until equilibrium is reached. The massive balls are next placed at diametrically opposite points, BB, for further measurement. After the oscillations have died away once more, a second equilibrium position is reached at a distance Δ' with respect to the other side of the zero position. We will use this pattern to evaluate a value for G, the gravitational constant.</p>
<table width=400 align=center><td>
<p align=justify>[[File:Cav-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Experiment setup (Top View).
<br clear=right>
</p>
</td></table>

<h1>Theory</h1>
The torque generated by a force <b>F</b> acting on a mass ''m'' located at <b>r</b> is defined through the cross product <b>T</b> = '''r''' x '''F'''. For rotational motion in a plane described by an angle θ(t), angular velocity ω(t), and acceleration α(t) the torque has only a single non-zero component. It is perpendicular to the plane and is denoted by a scalar: T = T<sub>z</sub>. The combination of the definition of torque with Newton's law of motion, and the moment of inertia I results in the equation of motion T = I α, where T represents the sum of all torques acting on the system (in our case two small masses rotating about their centre of mass). The external torque in our case is provided by the angular form of Hooke's law, as applicable for the torsion of wires, and we ignore for the moment the internal friction in the wire (which, however, is crucial to obtain a steady-state solution after long intervals).

If a torsion balance is twisted by a torque (couple) through a small angle θ (radians), the restoring torque is proportional to the angular displacement and is oppositely directed,
<table width=200 align=center><td>
<p align=justify>[[File:Cav-eqn1.png|80px|center]]
</p>
</td></table>

where C is the torque constant (cf. spring constant) of the wire. (What are the units of C?) We combine this with the equation of motion to obtain
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:Cav-eqn2.png|110px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

Equation (1) is a homogeneous linear ordinary differential equation with constant coefficients. The friction responsible for the damping of the oscillations has been omitted here. Solve the equation and show that the period P is given by
<table width=200 align=center><tr><td>
<p align=justify>[[File:Cav-eqn3.png|110px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

Neglecting the material between the two small spheres, the moment of inertia I of the torsion balance about the axis of rotation is
<table width=200 align=center><tr><td>
<p align=justify>[[File:Cav-eqn4.png|80px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

where m is the mass of each ball and 2d is the distance between them.

Thus by measuring the period P and calculating I, the torque constant C may be found from equation (2). If Δ' is the deflection of the light beam from the rest position upon rotation of the torsion wire through θ radians, (Fig. 1) then
<table width=200 align=center><td>
<p align=justify>[[File:Cav-eqn3b.png|80px|center]]
</p>
</td></table>

<p>where ''D'' is the distance between the mirror and the recording medium.</p>
<p>Once the large spheres are moved to their alternate asymmetrical position, the total deflection Δ produced (after the oscillations have died out) is (see Fig. 2)</p>
<table width=200 align=center><tr><td>
<p align=justify>[[File:Cav-eqn5.png|130px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>

<p>The torque (couple) exerted by the large masses is</p>
<table width=200 align=center><tr><td>
<p align=justify>[[File:Cav-eqn6.png|160px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>
<p>Here ''F'' is the magnitude of the gravitational force of attraction between the small and large masses, and is given by</p>
<table width=200 align=center><td>
<p align=justify>[[File:Cav-eqn7.png|90px|center]]
</p>
</td></table>
<p>where ''b'' is the distance between the centres of the small (''m'') and large (''M'') masses at equilibrium.</p>

<h1>Method</h1>
<p><b>IMPORTANT: Treat the equipment gently. For example, slamming the door or bumping the apparatus will considerably lengthen the experimental observation time. </b></p>
<p>In order for the balance to operate correctly, the torsion wire must be precisely vertical and must be able to move freely. The centres of the four balls must lie in one plane as well.(The balance is aligned and is ready for data collection.)</p>
<p> A computerized data collection method will be used in this experiment. The position of the laser beam can be monitored on the opposite wall. Rotate the large masses about 180 degrees so that they almost touch the case. You should make this adjustment carefully to ensure that the heavy balls do not knock the case. If the case is accidentally knocked, the mirror will be set into a large amplitude oscillation that will take about 1 hour to decay. You can therefore save yourself a lot of time (and effort) by making this adjustment delicately in the first place. </p>
<p>If this step has been completed successfully the position of the laser spot on the opposite wall will change slowly with time. The light spot will undergo damped oscillations about an equilibrium position Δ'. Determine the mean period of oscillation P from your plot of displacement vs. time. Allow about 45 min – 60 min for these oscillations to damp out and measure the new equilibrium position x<sub>1</sub> (see Fig. 2).</p>
<table width=400 align=center><td>
<p align=justify>[[File:Cav-fig2.png|400px|border|center]]
<b>Figure 2 -</b> Graph of small-mass oscillations.
<br clear=right>
</p>
</td></table>
<p>Carefully reverse the position of the heavy balls and repeat the procedure recording position vs. time and the new position at equilibrium x<sub>2</sub>. Now repeat this once more and obtain an average value for Δ. Find the separation of the centres of gravity of the two balls using your data and the apparatus parameters given in the table below. What approximation(s) does this involve?</p>
<p> Part of the laser beam is reflected onto a 128-segment linear photodiode array. The output of the photodiode array is monitored on an oscilloscope, and a computer connected to the oscilloscope downloads the data for processing. A control program written in Labview collects and processes the data, so you obtain data points which form a graph similar to what would be manually collected by observing the motion of the laser beam on the opposite wall.</p>

<h1>Procedure</h1>

<table width=400 align=center><td>
<p align=justify>[[File:Cav-fig3new.png|500px|border|center]]
<b>Figure 3 -</b> Experimental Setup.
<br clear=right>
</p>
</td></table>

<h2>Data Collection</h2>
<ol>
<li>Run the program "Labview 2014", and open the vi called '''"Cavendish v2.vi"'''.</li>
<li>Click the white arrow located at the top left of the toolbar to start the program. When the program runs, it leads you through the steps of turning on the power supply, oscilloscope and laser. Follow the directions carefully, as the program needs to obtain a background light level with the laser off.</li>
<li>When prompted, turn on the oscilloscope and power supply only using the power buttons. The power supply should be giving +5V, and the settings should not be touched. Click OK in Labview.</li>
<li>Next, you will be prompted to turn on the laser. To operate the laser, simply flip up the toggle switch of the laser power supply. The laser will appear a few seconds afterwards. CLick OK in Labview.</li>
<li>Data collection will begin and continue until the "STOP" button is pressed.
<li>When prompted enter a simple short file name (no spaces or characters).
<li>The data will be saved in the folder "student data" located on the desktop as two lists of numbers - one is the centre pixel number, and the other is the time. You need to convert centre pixel number to a displacement in order to calculate G.</li>

<table width=400 align=center><td>
<p align=justify>[[File:Cavendishv2_vi.png|500px|border|center]]
<b>Figure 4 -</b> The Labview control program.
<br clear=right>
</p>
</td></table>

</ol>

<h2>Parameters of Apparatus</h2>
<table width=600>
<tr><td width=500> Diameter of large spherical mass</td><td width=100> 6.386 cm</td></tr>
<tr><td> Diameter of small spherical mass</td><td> n.a. </td></tr>
<tr><td> Mass of large spherical mass</td><td>1500g</td></tr>
<tr><td> Mass of small spherical mass</td><td>20g</td></tr>
<tr><td> Thickness of Cavendish box enclosure</td><td>3.01cm</td></tr>
<tr><td> Separation between the centre of the mirror and the centre of the small spherical mass</td><td>5.0cm</td></tr>
<tr><td><b> Details of the Photodiode Array </b></td><td></td></tr>
<tr><td> Number of pixels</td><td>128</td></tr>
<tr><td> Active area length</td><td> 10.2cm</td></tr>
<tr><td> Pixel size</td><td> 0.8mm x 0.8mm</td></tr>

</table>

<h2>Tasks</h2>
<ol>
<li>Derive ''G'' from the above equations in terms of the period of oscillation.</li>
<li>Discuss the effect of the attraction of the distant 1.5 kg sphere for the small balls.</li>
<li>Enumerate and discuss other possible sources of error in the measurement of ''G'' using this apparatus.</li>
<li>When the beam of light oscillates about its final position, it slowly damps out. Assume that the damping force is proportional to the (angular) velocity to find the equation of motion. From the data, find the damping constant. See textbook references on damped harmonic motion<ref>For a discussion on damping see: A.P. French, ''Vibration and Waves'', Norton, pp.62-70</ref>.</li>
<li>Find the expression for the damping force at any time and compare the frequency of the motion without damping to that with damping. Comment on the difference(s) between the two.</li>
<li>Why do you need to measure the period of oscillation?</li>
<li>This experiment makes efficient use of real-time data acquisition and analysis. This is accomplished using the programming language LabView. What logical steps is the program following to convert what it downloads from the oscilloscope into a reasonable estimation of the photodiode pixel number which has been illuminated by the laser? You will need to look at the "block diagram" of the program. (You can inspect the block diagram while the program is running). You are only required to understand the general algorithm, not the details. You may find <ref>[http://www.ni.com/gettingstarted/labviewbasics/ Getting Started with LabView]</ref> a useful starting point for understanding LabView. </li>
</ol>

<table width=400 align=center><td>
<p align=justify>[[File:Cav-fig4.png|400px|border|center]]
<b>Figure 5 -</b> Cavendish Beam Schematic.
<br clear=right>
</p>
</td></table>

<table width=400 align=center><td>
<p align=justify>[[File:Cav-fig5.jpg|400px|border|center]]
<b>Figure 6 -</b> Cavendish Beam.
<br clear=right>
</p>
</td></table>

<h1>References</h1>
<references/>

Main Page/PHYS 3220/new Excitation Potentials

2021-08-11T18:50:51Z

Gloria:

<h1>The Franck-Hertz Experiment: Excitation Potentials of Mercury and Neon</h1>

<h1>Introduction</h1>

<p>One of the most direct proofs of the existence of discrete energy states within the atom was first demonstrated in experiments on critical potentials, performed initially by Franck and Hertz in the early 1900's. Studying the way electrons lose energy in collisions with mercury vapour, they laid the basis for the quantum theory of atoms by observing that the electrons give energy to internal motion of mercury atoms in discrete units only.</p>
<p>The collision of a neutral atom with a fast particle (e.g., an electron) may result in the excitation or ionization of the atom. A slow electron in an elastic collision can give very little of its kinetic energy to the translational motion of a mercury atom (without changing the energy state of the atom) - just as a ping-pong ball cannot effectively move a billiard ball. If a moderately slow electron has enough kinetic energy to overcome an atomic excitation threshold (several eV) the collision may be inelastic and much of the energy of the electron can go into exciting a higher state of the atom. The energy in electron volts (eV) necessary to raise an atom from its normal ("ground") state to a given excited state is called the excitation potential for that state. For sufficiently high scattering energy of the impinging electron even ionization may occur.</p>
<p>The energy levels of mercury (Hg) are shown in Fig. 1; it is easy to see that the internal structure is complicated - a consequence of the many electrons in the atom. The diagram gives considerable information you need to know for this experiment. The numbers associated with the lines drawn between the energy levels are wavelengths (in Angstroms Å). In the present experiment we explore only the energy levels 6<sup>3</sup>P on the diagram, the first group of excited states. The electrons do not acquire enough energy to excite many of the other levels.</p>
<p>The Franck-Hertz apparatus consists of an evacuated glass envelope containing a cathode, screen, plate and a small drop of mercury, which can be vaporized by heating. The plate is always kept slightly negative with respect to the grid (that acts as an anode, i.e. accelerates the electrons) and both are set at various positive voltages with respect to the cathode. As the grid potential is raised, the plate current increases accordingly. For accelerating voltages below 5V all collisions with mercury atoms will be elastic (kinetic energy below about 5 eV). Hence, these electrons are energetic enough to overcome the negative plate-grid potential and are collected by the plate. The current flowing in the tube depends upon both the number of charged carriers (electrons) and their velocities (j = nev). Thus a significant change in the particle velocity can affect the size of the current. Once electrons with more than about 5eV energy excite a mercury atom, they slow down and the current flowing in the tube drops. If there is a larger voltage across the tube so that the electron can be re-accelerated to ~ 5 eV after giving it up once in the first collision, then we can see decreases in the current at higher voltages corresponding to a repeated inelastic collision. This process can yield a cyclic rise and fall of the current with the voltage.</p>

<table width=400 align=center><td>
<p align=justify>[[File:Fh-fig1.png|800px|border|center]]
<b>Figure 1 -</b> Energy Levels of Mercury.
<br clear=right>
</p>
</td></table>

<h1>Procedure</h1>

<h2>Required Components</h2>

<p> For this experiment you will be using equipment provided by Leybold®. Go to https://www.leybold-shop.com/physics/physics-equipment/atomic-and-nuclear-physics/franck-hertz-experiments/mercury/franck-hertz-supply-unit-555880.html and click on <b>Related Documents</b>. Read through the instruction sheet for the Franck-Hertz Supply Unit (pay attention to sections 1-4, 5.1, 5.2 and 5.6) and the Experiment Descriptions for Hg (P6.2.4.1) and Ne (P6.2.4.3). These leaflets provide useful information on how to use the equipment and optimize the Franck-Hertz signal. THE EXPERIMENTAL SETUP OF THE TEMPERATURE PROBE IS CRITICAL - THE PROBE MUST BE INSERTED INTO THE BLIND HOLE OF THE COPPER TUBE of the oven. Ensure the temperature sensor is properly connected and IS NOT TOUCHING THE Hg Franck-Hertz tube. </p>

<h3>Observing the Signal</h3>

<p>Electrons liberated from the filament and accelerated to the detector plate which do not collide with an Hg atom will register as a current. This current is amplified by the supply unit and be viewed on the oscilloscope. Note that evidence of collisions with Hg atoms will result in a deficit of current at specific accelerating voltages. This will be observed as dips on the oscilloscope trace. It is the origin and properties of these dips which is the focus of this experiment.</p>

<p>When the temperature is stable you can record the current-voltage characteristic of the Franck-Hertz tube. The current-voltage trace can be observed using the oscilloscope in XY mode. Make sure the signal you observe does not have horizontal clipping (the peaks cut off); see the leaflets for guidance on how to optimize your signal. (What is the meaning of the vertical cut-off?)</p>

<p>Record an I-V curve for an initial temperature of about 180ºC. Set the oscilloscope display mode to XY and the persist to 2 seconds to best visually observe the oscilloscope signal on the screen. </p>

<h3>Saving the Scope Traces</h3>
<p>Use a USB to save your optimized Franck-Hertz signals using the following procedure. (For more information on saving in XY mode refer to the user manual for the oscilloscope.) </p>
<ol>
<li>Connect your USB device to the oscilloscope. </li>
<li>Switch the oscilloscope display mode from XY to YT.</li>
<li>Push the Save/Recall button on the oscilloscope to activate the save menu. </li>
<li>Push the "Print" button to save all files to your USB drive. (The "Print" button is set to "Save All Files". This will save waveforms on Ch.1 and Ch.2 and a picture of the waveforms. </li>
</ol>

<h2>Measurements</h2>

<p>Find as many values as you can of the excitation energy ("excitation potential") for Hg from your record. Repeat these measurements for 5 different temperature values ranging from 140ºC to 195ºC. <b>DO NOT EXCEED a setpoint temperature of 195ºC on the supply unit.</b> Comment on the effect of the Hg pressure in the tube. Perform a full error analysis and compare your results with the expected values.</p>
<p> The neon tube is operated at room temperature. Optimize and record the Ne Franck-Hertz curve and find the excitation energy values for Ne. Compare with the expected values. Can you see the luminous layers in the neon tube? (Hint: Use the MAN operating mode to manually adjust the accelerating voltage and turn off the lights in the room.)

<h1>Questions</h1>
<ol>
<li>Explain the effect of changing the grid-to-plate voltage (V<sub>1</sub>)?</li>
<li>Find out what is meant by "contact potential" in the Franck-Hertz tube and explain how it could be determined. Can you estimate it from your record?</li>
<li>Determine from simple classical mechanics (using a head-on collision with recoil at 180 degrees) what fraction of an electron's kinetic energy can be transferred to a mercury atom in an '''elastic''' collision. Derive an approximate value of the fraction. Repeat for a neon atom. </li>
<li>Why are the other levels not observed? (e.g. 6<sup>3</sup>P<sub>2</sub>, 6<sup>3</sup>P<sub>o</sub>, 6<sup>1</sup>P<sub>1</sub>.)</li>
</ol>

<h1>References</h1>
<ol>
<li>Brehm, J., Mullin W, ''Modern Physics'', p. 168</li>
<li>Halliday, D., Resnick, R., ''Physics I'', pp. 522-24.</li>
<li>Carpenter, K.H., [http://ajp.aapt.org/resource/1/ajpias/v43/i2/p190_s1| Amer. J. Phys. '''43''' (1975) 190].</li>
<li>Hanne, G.F.,'' “What really happens in the Franck-Hertz experiment with mercury?”'' [http://ajp.aapt.org/resource/1/ajpias/v56/i8/p696_s1 Am. J. Phys. '''56''' (1988) 696].</li>
<li>Huebener, J.S., [http://ajp.aapt.org/resource/1/ajpias/v44/i3/p302_s1 Amer. J. Phys. '''44''' (1976) 302].</li>
<li>Liu, F.H., ''“Franck-Hertz expt. with higher excitation level”'' [http://ajp.aapt.org/resource/1/ajpias/v55/i4/p366_s1 Am. J. Phys. '''55''' (1987) 366].</li>
<li>Preston, D., Dietz, E.,'' The Art of Experimental Physics'', pp. 197ff.</li>
</ol>

Main Page/PHYS 3220/new Excitation Potentials

2021-08-11T15:55:28Z

Gloria:

<h1>The Franck-Hertz Experiment: Excitation Potentials of Mercury and Neon</h1>

<h1>Introduction</h1>

<p>One of the most direct proofs of the existence of discrete energy states within the atom was first demonstrated in experiments on critical potentials, performed initially by Franck and Hertz in the early 1900's. Studying the way electrons lose energy in collisions with mercury vapour, they laid the basis for the quantum theory of atoms by observing that the electrons give energy to internal motion of mercury atoms in discrete units only.</p>
<p>The collision of a neutral atom with a fast particle (e.g., an electron) may result in the excitation or ionization of the atom. A slow electron in an elastic collision can give very little of its kinetic energy to the translational motion of a mercury atom (without changing the energy state of the atom) - just as a ping-pong ball cannot effectively move a billiard ball. If a moderately slow electron has enough kinetic energy to overcome an atomic excitation threshold (several eV) the collision may be inelastic and much of the energy of the electron can go into exciting a higher state of the atom. The energy in electron volts (eV) necessary to raise an atom from its normal ("ground") state to a given excited state is called the excitation potential for that state. For sufficiently high scattering energy of the impinging electron even ionization may occur.</p>
<p>The energy levels of mercury (Hg) are shown in Fig. 1; it is easy to see that the internal structure is complicated - a consequence of the many electrons in the atom. The diagram gives considerable information you need to know for this experiment. The numbers associated with the lines drawn between the energy levels are wavelengths (in Angstroms Å). In the present experiment we explore only the energy levels 6<sup>3</sup>P on the diagram, the first group of excited states. The electrons do not acquire enough energy to excite many of the other levels.</p>
<p>The Franck-Hertz apparatus consists of an evacuated glass envelope containing a cathode, screen, plate and a small drop of mercury, which can be vaporized by heating. The plate is always kept slightly negative with respect to the grid (that acts as an anode, i.e. accelerates the electrons) and both are set at various positive voltages with respect to the cathode. As the grid potential is raised, the plate current increases accordingly. For accelerating voltages below 5V all collisions with mercury atoms will be elastic (kinetic energy below about 5 eV). Hence, these electrons are energetic enough to overcome the negative plate-grid potential and are collected by the plate. The current flowing in the tube depends upon both the number of charged carriers (electrons) and their velocities (j = nev). Thus a significant change in the particle velocity can affect the size of the current. Once electrons with more than about 5eV energy excite a mercury atom, they slow down and the current flowing in the tube drops. If there is a larger voltage across the tube so that the electron can be re-accelerated to ~ 5 eV after giving it up once in the first collision, then we can see decreases in the current at higher voltages corresponding to a repeated inelastic collision. This process can yield a cyclic rise and fall of the current with the voltage.</p>

<table width=400 align=center><td>
<p align=justify>[[File:Fh-fig1.png|800px|border|center]]
<b>Figure 1 -</b> Energy Levels of Mercury.
<br clear=right>
</p>
</td></table>

<h1>Procedure</h1>

<h2>Required Components</h2>

<p> For this experiment you will be using equipment provided by Leybold®. Go to https://www.leybold-shop.com/physics/physics-equipment/atomic-and-nuclear-physics/franck-hertz-experiments/mercury/franck-hertz-supply-unit-555880.html and click on <b>Related Documents</b>. Read through sections 1-4, 5.1, 5.2 and 5.6 of the instruction sheet for the Franck-Hertz Supply Unit and the Experiment Descriptions for Hg (P6.2.4.1) and Ne (P6.2.4.3). These leaflets provide useful information on how to use the equipment and optimize the Franck-Hertz signal.</p>

<h3>Observing the Signal</h3>

<p>Electrons liberated from the filament and accelerated to the detector plate which do not collide with an Hg atom will register as a current. This current is amplified by the supply unit and be viewed on the oscilloscope. Note that evidence of collisions with Hg atoms will result in a deficit of current at specific accelerating voltages. This will be observed as dips on the oscilloscope trace. It is the origin and properties of these dips which is the focus of this experiment.</p>

<p>When the temperature is stable you can record the current-voltage characteristic of the Franck-Hertz tube. The current-voltage trace can be observed using the oscilloscope in XY mode. Make sure the signal you observe does not have horizontal clipping (the peaks cut off); see the leaflets for guidance on how to optimize your signal. (What is the meaning of the vertical cut-off?)</p>

<p>Record an I-V curve for an initial temperature of about 180ºC. Set the oscilloscope display mode to XY and the persist to 2 seconds to best visually observe the oscilloscope signal on the screen. </p>

<h3>Saving the Scope Traces</h3>
<p>Use a USB to save your optimized Franck-Hertz signals using the following procedure. (For more information on saving in XY mode refer to the user manual for the oscilloscope.) </p>
<ol>
<li>Connect your USB device to the oscilloscope. </li>
<li>Switch the oscilloscope display mode from XY to YT.</li>
<li>Push the Save/Recall button on the oscilloscope to activate the save menu. </li>
<li>Push the "Print" button to save all files to your USB drive. (The "Print" button is set to "Save All Files". This will save waveforms on Ch.1 and Ch.2 and a picture of the waveforms. </li>
</ol>

<h2>Measurements</h2>

<p>Find as many values as you can of the excitation energy ("excitation potential") for Hg from your record. Repeat these measurements for 5 different temperature values ranging from 140ºC to 195ºC. <b>DO NOT EXCEED a setpoint temperature of 195ºC on the supply unit.</b> Comment on the effect of the Hg pressure in the tube. Perform a full error analysis and compare your results with the expected values.</p>
<p> The neon tube is operated at room temperature. Optimize and record the Ne Franck-Hertz curve and find the excitation energy values for Ne. Compare with the expected values. Can you see the luminous layers in the neon tube? (Hint: Use the MAN operating mode to manually adjust the accelerating voltage and turn off the lights in the room.)

<h1>Questions</h1>
<ol>
<li>Explain the effect of changing the grid-to-plate voltage (V<sub>1</sub>)?</li>
<li>Find out what is meant by "contact potential" in the Franck-Hertz tube and explain how it could be determined. Can you estimate it from your record?</li>
<li>Determine from simple classical mechanics (using a head-on collision with recoil at 180 degrees) what fraction of an electron's kinetic energy can be transferred to a mercury atom in an '''elastic''' collision. Derive an approximate value of the fraction. Repeat for a neon atom. </li>
<li>Why are the other levels not observed? (e.g. 6<sup>3</sup>P<sub>2</sub>, 6<sup>3</sup>P<sub>o</sub>, 6<sup>1</sup>P<sub>1</sub>.)</li>
</ol>

<h1>References</h1>
<ol>
<li>Brehm, J., Mullin W, ''Modern Physics'', p. 168</li>
<li>Halliday, D., Resnick, R., ''Physics I'', pp. 522-24.</li>
<li>Carpenter, K.H., [http://ajp.aapt.org/resource/1/ajpias/v43/i2/p190_s1| Amer. J. Phys. '''43''' (1975) 190].</li>
<li>Hanne, G.F.,'' “What really happens in the Franck-Hertz experiment with mercury?”'' [http://ajp.aapt.org/resource/1/ajpias/v56/i8/p696_s1 Am. J. Phys. '''56''' (1988) 696].</li>
<li>Huebener, J.S., [http://ajp.aapt.org/resource/1/ajpias/v44/i3/p302_s1 Amer. J. Phys. '''44''' (1976) 302].</li>
<li>Liu, F.H., ''“Franck-Hertz expt. with higher excitation level”'' [http://ajp.aapt.org/resource/1/ajpias/v55/i4/p366_s1 Am. J. Phys. '''55''' (1987) 366].</li>
<li>Preston, D., Dietz, E.,'' The Art of Experimental Physics'', pp. 197ff.</li>
</ol>

Main Page/PHYS 4210/Coaxial Cable

2021-01-15T18:52:35Z

Gloria:

<h1>Coaxial Cable</h1>

<p>Carefully read Experiment 1 in The Art of Experimental Physics, Preston and Dietz, [[Media:Coaxial-AEP-1991.pdf|available here]]<ref>[https://www.library.yorku.ca/find/Record/1038893 The Art of Experimental Physics], John Wiley and Sons,1991.</ref>. This write-up is meant to supplement that material.
</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>TEM Mode</li>
<li>Inductance per unit Length</li>
<li>Resistance per unit Length</li>
<li>Capacitance per unit Length</li>
<li>Siemens per unit Length</li>
<li>Propagation Constant</li>
<li>Attenuation Constant</li>
</ul>
</td>
<td width=250>
<ul>
<li>Phase Constant</li>
<li>Characteristic Impedance</li>
<li>Reflection Coefficient</li>
<li>Load Impedance</li>
<li>Source Impedance</li>
<li>Impedance Matching</li>
</ul>
</td>
</table>

<h1>Theory</h1>
<p>A coaxial cable (coax) is one of the most popular means for transmitting the electromagnetic power from a generator to a receiver. It consists of an inner and outer conducting lines (usually made of copper) separated by a dielectric layer (usually teflon). The electromagnetic wave can propagate in a coax in the form of TE, TM, and TEM modes, and the latter is the one that is most commonly used. In this experiment, the effect of a coaxial cable on the propagating signal and some physical characteristics of a coax are studied.</p>
<p>There are two levels of description of electromagnetic phenomena. The more fundamental one relies on Maxwell’s equations combined with material equations. Within this approach, one uses the language of electric and magnetic field strengths, potentials, dielectric susceptibility and magnetic permeability, etc. Engineers, however, prefer a simpler language of currents, voltages and lump elements, that is, resistance, capacitance, conductance, and inductance. In order for this simplified description to be applicable to the problem considered, the characteristic sizes of lump elements must be much smaller than the wavelength of an electromagnetic wave. Nevertheless, with a small modification, the approach of engineers allows us to obtain some meaningful results even when this requirement is not met. We will adopt the language of engineers in the following discussion.</p>
<p>The idea of the modification is that, instead of using the net inductance, capacitance, resistance and conductance, we use their values per unit length. Thus, a coax is viewed as a system with distributed parameters. Let us write down the expressions for the unit-length characteristics of a coax. A current in the central conductor sets up a magnetic field; hence, the line has inductance whose value per unit length is</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn1.png|90px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>where μ is magnetic permeability of the dielectric layer between the two conductors, and ''a'' and ''b'' are the radii of the inner and outer conductors, respectively.</p>
<p>Next, the capacitance per unit length between the two conductors is:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn2.png|110px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
<p>where ε is the dielectric permittivity of the dielectric layer. </p>
<p>Since both inner and outer conductors are not ideal, the line has a resistance per unit length</p>
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn3.png|150px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>
<p>Here, μ<sub>c</sub> and σ<sub>c</sub> are the magnetic permeability and conductivity of the conductors, respectively. The resistance value (3) is determined by the skin depth that measures the characteristic distance of penetration of the electromagnetic wave into the conductors. It is within the skin layer that the absorption of the electromagnetic power is most intense. The skin depth, in turn, depends on the frequency, ν, of the wave, hence R depends on ν as well.</p>
<p>Finally, the dielectric separating the two conductors is not ideal, hence there always is a leakage current when the conductors are kept at different potentials. This effect is accounted for by introducing the conductance per unit length of the line</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn4.png|150px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>
<p>σ being the conductivity of the dielectric layer.</p>

<p><b>Exercise:</b> Derive (1) and (2). Hint: to derive (1), suppose that some current ''J'' flows through the inner conductor of the line. Find the value of the magnetic field produced by this current, and then the magnetic field energy stored in a unit length element of the line. This energy also equals ''LJ<sup>2''</sup>/2, from the definition of inductance per unit length. To derive (2), assume that the inner and outer conductors have charges per unit length equal to +''q'' and –''q''. Find the potential difference between the two conductors. From the definition of a unit length capacitance, the potential difference equals to ''q/C''.</p>
<p>A coax can be thought of as a collection of infinitely many infinitesimal elements, each of which can be presented with an equivalent circuit shown in Fig. 1 (similar to Fig. 1.4 of Preston). </p>
<table width=400 align=center><td>
<p align=justify>[[File:Coax-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Coaxial cable cross sectional view.
<br clear=right>
</p>
</td></table>
<p>As the current travels along the inner conductor, its total decrease on the element Δz is the sum of the leakage currents through the capacitance and displacement current. The former equals to ''GΔzV'' , and the latter to ''CΔz(dV/dt)''. Considering the node that joins the inductance, conductance, and capacitance, the first Kirchoff’s law yields:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn5.png|230px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>
<p>Rearranging the terms and taking the limit ''Δz → 0'' , we obtain</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn6.png|140px|center]]
</p>
</td><td> <b>(6)</b></td></tr></table>
<p>Similarly, a voltage drop on this element equals the electromotive force of self-induction and the voltage developed across the resistance:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn7.png|140px|center]]
</p>
</td><td> <b>(7)</b></td></tr></table>
<p><b>Exercise:</b> Derive (7). Use the second Kirchoff’s law to find the voltage drop across Δz.</p>
<p>Differentiating the last two equations with respect to z, we obtain:</p>
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn8.png|270px|center]]
</p>
</td><td> <b>(8)</b></td></tr></table>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn9.png|270px|center]]
</p>
</td><td> <b>(9)</b></td></tr></table>
<p><b>Exercise:</b> Show this.</p>
<p>These equations are called telegraph equations. We will look for their solution in the harmonic form</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn10.png|270px|center]]
</p>
</td><td> <b>(10)</b></td></tr></table>
<p>Substitution of these expressions into the original telegraph equations (8) yields:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn11.png|180px|center]]
</p>
</td><td> <b>(11)</b></td></tr></table>
<p>where the total series impedance and shunt conductance are</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn12.png|220px|center]]
</p>
</td><td> <b>(12)</b></td></tr></table>
<p>Equations (11) and (12) allow solutions in the form</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn13.png|300px|center]]
</p>
</td><td> <b>(13)</b></td></tr></table>
<p>where </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn14.png|300px|center]]
</p>
</td><td> <b>(14)</b></td></tr></table>
<p>is the propagation constant. It is a complex number whose real part, α, is called attenuation constant, and the imaginary part, β, is the phase constant. The two terms in Eqs. (13) represent the waves travelling in the positive and negative z-directions.</p>

<p><b>Exercise:</b> Which one is which? Hint: The wave attenuates in the direction of propagation.</p>
<p>Since the telegraph equations for voltage and current are of the same form, it can be concluded that the relationship between the amplitudes of currents and voltages propagating to the right and to the left is linear. The proportionality constant relating these quantities is called the characteristic impedance of the line, and its value equals to</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn15.png|210px|center]]
</p>
</td><td> <b>(15)</b></td></tr></table>
<p>as can be verified from Eqs.(6, 7). It then follows that</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn16.png|180px|center]]
</p>
</td><td> <b>(16)</b></td></tr></table>
<p>In the high-frequency limit,''ωL >> R'', ''ωC >> G'' (or, alternatively, for an ideal line with perfect inner and outer conductors and a perfectly insulating dielectric layer in between), the expression for the characteristic impedance of the line simplifies to</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn17.png|170px|center]]
</p>
</td><td> <b>(17)</b></td></tr></table>
<p>Note that the characteristic impedance of an ideal line is a frequency-independent quantity whose imaginary part equals to zero.</p>

<p><b>Exercise:</b> Derive the expression for ''Z<sub>c</sub>''. Hint: Use Eq. (6) or (7) to find the relationship between current and voltage amplitudes from the general solution (13).</p>

<p>Suppose that one end of the line is terminated at ''z'' = ''L'' with a load of impedance ''Z<sub>L</sub>'', so that the relationship between current and voltage on the load is</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn18.png|100px|center]]
</p>
</td><td> <b>(18)</b></td></tr></table>
<p>Substitution of Eqs. (16) for currents yields:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn19.png|200px|center]]
</p>
</td><td> <b>(19)</b></td></tr></table>
<p>The amplitudes ''V<sub>1</sub>'' and ''V<sub>2</sub>'' can be interpreted as the amplitudes of the waves reflected from and incident on the load, respectively. It then follows that the ratio of the reflected to incident voltage at ''z'' = ''L'', called reflection coefficient from the output, is:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn20.png|280px|center]]
</p>
</td><td> <b>(20)</b></td></tr></table>
<p>For currents,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn21.png|120px|center]]
</p>
</td><td> <b>(21)</b></td></tr></table>
<p><b>Exercise:</b> Why?</p>

<p>It is evident from Eq. (20) that Γ<sub>L</sub> changes from +1 (open end, ''Z<sub>L</sub>'' = ∞) to –1 (shorted, ''Z<sub>L</sub>'' = 0), depending on the value of load impedance. Impedance matching of the load to the line occurs when ''Z<sub>L</sub>'' = ''Z<sub>c</sub>''. In this case, Γ<sub>L</sub> = 0 and there is no reflected wave.</p>
<p>Let us finally recall that the generator itself has a finite input impedance ''Z<sub>0</sub>''. Redefining the voltage amplitudes ''V<sub>1</sub>'' and ''V<sub>2</sub>'' as those of the waves incident on and reflected from the input, we find the input reflection coefficient</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn22.png|100px|center]]
</p>
</td><td> <b>(22)</b></td></tr></table>
<p>Again, input impedance matching occurs when ''Z<sub>0</sub>'' = ''Z<sub>c</sub>'' and Γ<sub>0</sub> = 0.</p>

<h1>Experiment</h1>
<p>In this experiment, the following equipment will be used:</p>
<ul>
<li>A coaxial cable of 20 to 30 m in length</li>
<li>Rectangular pulse generator</li>
<li>Oscilloscope</li>
<li>Differentiating RC-circuit</li>
<li>Multimeter</li>
<li>Adjustable resistor</li>
</ul>

<h2>Part 1</h2>

<p>Connect the circuit as shown in Fig. 2. Rectangular pulses produced by the generator are differentiated by the RC-circuit, so that voltage entering the coaxial line represents a series of spikes, as shown in the figure. Set the generator frequency to 10 kHz. With infinite resistance of the load, observe the multireflected pulse on the oscilloscope. Explain the shape of the observed signal. Next, connect the inner and outer conductors of the line at the output end with a wire. Explain the difference of the observed multireflected pulse from the previous case. With ''Z<sub>L</sub>''= ∞ , calculate the velocity of pulse propagation along the line. Determine the dielectric constant of teflon from the measured velocity. Use the vacuum value for the magnetic permeability of teflon, μ=4πx10<sup>-7</sup> NA<sup>-2</sup>.</p>
<table width=400 align=center><td>
<p align=justify>[[File:Coax-fig2.png|500px|border|center]]
<b>Figure 2 -</b> Circuit to study pulses propagating along the line.
<br clear=right>
</p>
</td></table>
<p>Measure directly the capacitance of the coax with a multimeter. Determine the dielectric constant and compare it to the previously obtained value. Comment on the possible discrepancy.</p>

<h2>Part 2</h2>
<p>Assuming the ideal line, calculate its characteristic impedance from Eq. (17). Again, use the vacuum value for the magnetic permeability of teflon.</p>
<p>Attach a variable resistance to the output end of the line and measure the reflection coefficient at the output as a function of load resistance. The following procedure is suggested.</p>
<p>First, reconnect the circuit as shown in Fig. 2 and turn on the generator to produce the multireflected pulse. The heights of the n<sup>th</sup> and (n+1)<sup>th</sup> spikes are related to each other as:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn23.png|100px|center]]
</p>
</td><td> <b>(23)</b></td></tr></table>

<p><b>Exercise:</b> What is the proportionality constant ''k'' equal to in terms of the attenuation constant and the input reflection coefficient?</p>

<p>At each value of ''Z<sub>L</sub>'', choose two consecutive spikes and measure the ratio of their heights. Note that the first and the second spikes are not a very good choice because the first pulse coming directly from the generator via the differentiating circuit gets absorbed not only in the coaxial line, but also in various connections between the coax, the differentiator, and the scope. Feel free to adjust the generator frequency and oscilloscope settings to get an optimal scope trace. In order to find the proportionality constant ''k'' in (23), you can use the fact that at infinite load resistance (open output end of the line), the coefficient of reflection at the output equals one:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn24.png|130px|center]]
</p>
</td><td> <b>(24)</b></td></tr></table>
<p>Plot ''Γ<sub>L</sub>'' vs. ''Z<sub>L</sub>''. Fit this plot with Eq. (20) to determine the characteristic impedance of the line. Does this value of ''Z<sub>c</sub>'' agree with your previously obtained estimate?</p>

<h2>Part 3</h2>
<p>With the circuit connected as in Fig. 2, determine the line’s attenuation constant from the observed multireflected pulse. Is it safe to assume that the input reflection coefficient equals one? Justify this assumption experimentally and theoretically by estimating the input reflection coefficient. Note that the impedance of the scope is about 1 MΩ, and the typical impedance of the generator is 600 Ω. Assume for simplicity that the capacitance has zero resistance.</p>

<h1>References</h1>
<references/>

<p> 2. J.M. Serra ''et. al.'', "''A wave lab inside a coaxial cable''", [http://ajp.aapt.org/resource/1/ajpias/v55/i4/p366_s1 Eur. J. Phys. '''25''' (2004) 581]</p>

Main Page/PHYS 4210/Coaxial Cable

2021-01-15T18:51:54Z

Gloria:

<h1>Coaxial Cable</h1>

<p>Carefully read Experiment 1 in The Art of Experimental Physics, Preston and Dietz, [[Media:Lasers-AEP-1991.pdf|available here]]<ref>[https://www.library.yorku.ca/find/Record/1038893 The Art of Experimental Physics], John Wiley and Sons,1991.</ref>. This write-up is meant to supplement that material.
</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>TEM Mode</li>
<li>Inductance per unit Length</li>
<li>Resistance per unit Length</li>
<li>Capacitance per unit Length</li>
<li>Siemens per unit Length</li>
<li>Propagation Constant</li>
<li>Attenuation Constant</li>
</ul>
</td>
<td width=250>
<ul>
<li>Phase Constant</li>
<li>Characteristic Impedance</li>
<li>Reflection Coefficient</li>
<li>Load Impedance</li>
<li>Source Impedance</li>
<li>Impedance Matching</li>
</ul>
</td>
</table>

<h1>Theory</h1>
<p>A coaxial cable (coax) is one of the most popular means for transmitting the electromagnetic power from a generator to a receiver. It consists of an inner and outer conducting lines (usually made of copper) separated by a dielectric layer (usually teflon). The electromagnetic wave can propagate in a coax in the form of TE, TM, and TEM modes, and the latter is the one that is most commonly used. In this experiment, the effect of a coaxial cable on the propagating signal and some physical characteristics of a coax are studied.</p>
<p>There are two levels of description of electromagnetic phenomena. The more fundamental one relies on Maxwell’s equations combined with material equations. Within this approach, one uses the language of electric and magnetic field strengths, potentials, dielectric susceptibility and magnetic permeability, etc. Engineers, however, prefer a simpler language of currents, voltages and lump elements, that is, resistance, capacitance, conductance, and inductance. In order for this simplified description to be applicable to the problem considered, the characteristic sizes of lump elements must be much smaller than the wavelength of an electromagnetic wave. Nevertheless, with a small modification, the approach of engineers allows us to obtain some meaningful results even when this requirement is not met. We will adopt the language of engineers in the following discussion.</p>
<p>The idea of the modification is that, instead of using the net inductance, capacitance, resistance and conductance, we use their values per unit length. Thus, a coax is viewed as a system with distributed parameters. Let us write down the expressions for the unit-length characteristics of a coax. A current in the central conductor sets up a magnetic field; hence, the line has inductance whose value per unit length is</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn1.png|90px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>where μ is magnetic permeability of the dielectric layer between the two conductors, and ''a'' and ''b'' are the radii of the inner and outer conductors, respectively.</p>
<p>Next, the capacitance per unit length between the two conductors is:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn2.png|110px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
<p>where ε is the dielectric permittivity of the dielectric layer. </p>
<p>Since both inner and outer conductors are not ideal, the line has a resistance per unit length</p>
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn3.png|150px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>
<p>Here, μ<sub>c</sub> and σ<sub>c</sub> are the magnetic permeability and conductivity of the conductors, respectively. The resistance value (3) is determined by the skin depth that measures the characteristic distance of penetration of the electromagnetic wave into the conductors. It is within the skin layer that the absorption of the electromagnetic power is most intense. The skin depth, in turn, depends on the frequency, ν, of the wave, hence R depends on ν as well.</p>
<p>Finally, the dielectric separating the two conductors is not ideal, hence there always is a leakage current when the conductors are kept at different potentials. This effect is accounted for by introducing the conductance per unit length of the line</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn4.png|150px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>
<p>σ being the conductivity of the dielectric layer.</p>

<p><b>Exercise:</b> Derive (1) and (2). Hint: to derive (1), suppose that some current ''J'' flows through the inner conductor of the line. Find the value of the magnetic field produced by this current, and then the magnetic field energy stored in a unit length element of the line. This energy also equals ''LJ<sup>2''</sup>/2, from the definition of inductance per unit length. To derive (2), assume that the inner and outer conductors have charges per unit length equal to +''q'' and –''q''. Find the potential difference between the two conductors. From the definition of a unit length capacitance, the potential difference equals to ''q/C''.</p>
<p>A coax can be thought of as a collection of infinitely many infinitesimal elements, each of which can be presented with an equivalent circuit shown in Fig. 1 (similar to Fig. 1.4 of Preston). </p>
<table width=400 align=center><td>
<p align=justify>[[File:Coax-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Coaxial cable cross sectional view.
<br clear=right>
</p>
</td></table>
<p>As the current travels along the inner conductor, its total decrease on the element Δz is the sum of the leakage currents through the capacitance and displacement current. The former equals to ''GΔzV'' , and the latter to ''CΔz(dV/dt)''. Considering the node that joins the inductance, conductance, and capacitance, the first Kirchoff’s law yields:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn5.png|230px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>
<p>Rearranging the terms and taking the limit ''Δz → 0'' , we obtain</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn6.png|140px|center]]
</p>
</td><td> <b>(6)</b></td></tr></table>
<p>Similarly, a voltage drop on this element equals the electromotive force of self-induction and the voltage developed across the resistance:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn7.png|140px|center]]
</p>
</td><td> <b>(7)</b></td></tr></table>
<p><b>Exercise:</b> Derive (7). Use the second Kirchoff’s law to find the voltage drop across Δz.</p>
<p>Differentiating the last two equations with respect to z, we obtain:</p>
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn8.png|270px|center]]
</p>
</td><td> <b>(8)</b></td></tr></table>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn9.png|270px|center]]
</p>
</td><td> <b>(9)</b></td></tr></table>
<p><b>Exercise:</b> Show this.</p>
<p>These equations are called telegraph equations. We will look for their solution in the harmonic form</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn10.png|270px|center]]
</p>
</td><td> <b>(10)</b></td></tr></table>
<p>Substitution of these expressions into the original telegraph equations (8) yields:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn11.png|180px|center]]
</p>
</td><td> <b>(11)</b></td></tr></table>
<p>where the total series impedance and shunt conductance are</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn12.png|220px|center]]
</p>
</td><td> <b>(12)</b></td></tr></table>
<p>Equations (11) and (12) allow solutions in the form</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn13.png|300px|center]]
</p>
</td><td> <b>(13)</b></td></tr></table>
<p>where </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn14.png|300px|center]]
</p>
</td><td> <b>(14)</b></td></tr></table>
<p>is the propagation constant. It is a complex number whose real part, α, is called attenuation constant, and the imaginary part, β, is the phase constant. The two terms in Eqs. (13) represent the waves travelling in the positive and negative z-directions.</p>

<p><b>Exercise:</b> Which one is which? Hint: The wave attenuates in the direction of propagation.</p>
<p>Since the telegraph equations for voltage and current are of the same form, it can be concluded that the relationship between the amplitudes of currents and voltages propagating to the right and to the left is linear. The proportionality constant relating these quantities is called the characteristic impedance of the line, and its value equals to</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn15.png|210px|center]]
</p>
</td><td> <b>(15)</b></td></tr></table>
<p>as can be verified from Eqs.(6, 7). It then follows that</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn16.png|180px|center]]
</p>
</td><td> <b>(16)</b></td></tr></table>
<p>In the high-frequency limit,''ωL >> R'', ''ωC >> G'' (or, alternatively, for an ideal line with perfect inner and outer conductors and a perfectly insulating dielectric layer in between), the expression for the characteristic impedance of the line simplifies to</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn17.png|170px|center]]
</p>
</td><td> <b>(17)</b></td></tr></table>
<p>Note that the characteristic impedance of an ideal line is a frequency-independent quantity whose imaginary part equals to zero.</p>

<p><b>Exercise:</b> Derive the expression for ''Z<sub>c</sub>''. Hint: Use Eq. (6) or (7) to find the relationship between current and voltage amplitudes from the general solution (13).</p>

<p>Suppose that one end of the line is terminated at ''z'' = ''L'' with a load of impedance ''Z<sub>L</sub>'', so that the relationship between current and voltage on the load is</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn18.png|100px|center]]
</p>
</td><td> <b>(18)</b></td></tr></table>
<p>Substitution of Eqs. (16) for currents yields:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn19.png|200px|center]]
</p>
</td><td> <b>(19)</b></td></tr></table>
<p>The amplitudes ''V<sub>1</sub>'' and ''V<sub>2</sub>'' can be interpreted as the amplitudes of the waves reflected from and incident on the load, respectively. It then follows that the ratio of the reflected to incident voltage at ''z'' = ''L'', called reflection coefficient from the output, is:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn20.png|280px|center]]
</p>
</td><td> <b>(20)</b></td></tr></table>
<p>For currents,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn21.png|120px|center]]
</p>
</td><td> <b>(21)</b></td></tr></table>
<p><b>Exercise:</b> Why?</p>

<p>It is evident from Eq. (20) that Γ<sub>L</sub> changes from +1 (open end, ''Z<sub>L</sub>'' = ∞) to –1 (shorted, ''Z<sub>L</sub>'' = 0), depending on the value of load impedance. Impedance matching of the load to the line occurs when ''Z<sub>L</sub>'' = ''Z<sub>c</sub>''. In this case, Γ<sub>L</sub> = 0 and there is no reflected wave.</p>
<p>Let us finally recall that the generator itself has a finite input impedance ''Z<sub>0</sub>''. Redefining the voltage amplitudes ''V<sub>1</sub>'' and ''V<sub>2</sub>'' as those of the waves incident on and reflected from the input, we find the input reflection coefficient</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn22.png|100px|center]]
</p>
</td><td> <b>(22)</b></td></tr></table>
<p>Again, input impedance matching occurs when ''Z<sub>0</sub>'' = ''Z<sub>c</sub>'' and Γ<sub>0</sub> = 0.</p>

<h1>Experiment</h1>
<p>In this experiment, the following equipment will be used:</p>
<ul>
<li>A coaxial cable of 20 to 30 m in length</li>
<li>Rectangular pulse generator</li>
<li>Oscilloscope</li>
<li>Differentiating RC-circuit</li>
<li>Multimeter</li>
<li>Adjustable resistor</li>
</ul>

<h2>Part 1</h2>

<p>Connect the circuit as shown in Fig. 2. Rectangular pulses produced by the generator are differentiated by the RC-circuit, so that voltage entering the coaxial line represents a series of spikes, as shown in the figure. Set the generator frequency to 10 kHz. With infinite resistance of the load, observe the multireflected pulse on the oscilloscope. Explain the shape of the observed signal. Next, connect the inner and outer conductors of the line at the output end with a wire. Explain the difference of the observed multireflected pulse from the previous case. With ''Z<sub>L</sub>''= ∞ , calculate the velocity of pulse propagation along the line. Determine the dielectric constant of teflon from the measured velocity. Use the vacuum value for the magnetic permeability of teflon, μ=4πx10<sup>-7</sup> NA<sup>-2</sup>.</p>
<table width=400 align=center><td>
<p align=justify>[[File:Coax-fig2.png|500px|border|center]]
<b>Figure 2 -</b> Circuit to study pulses propagating along the line.
<br clear=right>
</p>
</td></table>
<p>Measure directly the capacitance of the coax with a multimeter. Determine the dielectric constant and compare it to the previously obtained value. Comment on the possible discrepancy.</p>

<h2>Part 2</h2>
<p>Assuming the ideal line, calculate its characteristic impedance from Eq. (17). Again, use the vacuum value for the magnetic permeability of teflon.</p>
<p>Attach a variable resistance to the output end of the line and measure the reflection coefficient at the output as a function of load resistance. The following procedure is suggested.</p>
<p>First, reconnect the circuit as shown in Fig. 2 and turn on the generator to produce the multireflected pulse. The heights of the n<sup>th</sup> and (n+1)<sup>th</sup> spikes are related to each other as:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn23.png|100px|center]]
</p>
</td><td> <b>(23)</b></td></tr></table>

<p><b>Exercise:</b> What is the proportionality constant ''k'' equal to in terms of the attenuation constant and the input reflection coefficient?</p>

<p>At each value of ''Z<sub>L</sub>'', choose two consecutive spikes and measure the ratio of their heights. Note that the first and the second spikes are not a very good choice because the first pulse coming directly from the generator via the differentiating circuit gets absorbed not only in the coaxial line, but also in various connections between the coax, the differentiator, and the scope. Feel free to adjust the generator frequency and oscilloscope settings to get an optimal scope trace. In order to find the proportionality constant ''k'' in (23), you can use the fact that at infinite load resistance (open output end of the line), the coefficient of reflection at the output equals one:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn24.png|130px|center]]
</p>
</td><td> <b>(24)</b></td></tr></table>
<p>Plot ''Γ<sub>L</sub>'' vs. ''Z<sub>L</sub>''. Fit this plot with Eq. (20) to determine the characteristic impedance of the line. Does this value of ''Z<sub>c</sub>'' agree with your previously obtained estimate?</p>

<h2>Part 3</h2>
<p>With the circuit connected as in Fig. 2, determine the line’s attenuation constant from the observed multireflected pulse. Is it safe to assume that the input reflection coefficient equals one? Justify this assumption experimentally and theoretically by estimating the input reflection coefficient. Note that the impedance of the scope is about 1 MΩ, and the typical impedance of the generator is 600 Ω. Assume for simplicity that the capacitance has zero resistance.</p>

<h1>References</h1>
<references/>

<p> 2. J.M. Serra ''et. al.'', "''A wave lab inside a coaxial cable''", [http://ajp.aapt.org/resource/1/ajpias/v55/i4/p366_s1 Eur. J. Phys. '''25''' (2004) 581]</p>

Main Page/PHYS 4210/Coaxial Cable

2021-01-15T18:44:05Z

Gloria:

<h1>Coaxial Cable</h1>

<p>Carefully read Experiment 1 in The Art of Experimental Physics, Preston and Dietz, [[Media:Lasers-AEP-1991.pdf|available here]]<ref>[https://www.library.yorku.ca/find/Record/1038893 The Art of Experimental Physics], John Wiley and Sons,1991.</ref>. This write-up is meant to supplement that material.
</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>TEM Mode</li>
<li>Inductance per unit Length</li>
<li>Resistance per unit Length</li>
<li>Capacitance per unit Length</li>
<li>Siemens per unit Length</li>
<li>Propagation Constant</li>
<li>Attenuation Constant</li>
</ul>
</td>
<td width=250>
<ul>
<li>Phase Constant</li>
<li>Characteristic Impedance</li>
<li>Reflection Coefficient</li>
<li>Load Impedance</li>
<li>Source Impedance</li>
<li>Impedance Matching</li>
</ul>
</td>
</table>

<h1>Theory</h1>
<p>A coaxial cable (coax) is one of the most popular means for transmitting the electromagnetic power from a generator to a receiver. It consists of an inner and outer conducting lines (usually made of copper) separated by a dielectric layer (usually teflon). The electromagnetic wave can propagate in a coax in the form of TE, TM, and TEM modes, and the latter is the one that is most commonly used. In this experiment, the effect of a coaxial cable on the propagating signal and some physical characteristics of a coax are studied.</p>
<p>There are two levels of description of electromagnetic phenomena. The more fundamental one relies on Maxwell’s equations combined with material equations. Within this approach, one uses the language of electric and magnetic field strengths, potentials, dielectric susceptibility and magnetic permeability, etc. Engineers, however, prefer a simpler language of currents, voltages and lump elements, that is, resistance, capacitance, conductance, and inductance. In order for this simplified description to be applicable to the problem considered, the characteristic sizes of lump elements must be much smaller than the wavelength of an electromagnetic wave. Nevertheless, with a small modification, the approach of engineers allows us to obtain some meaningful results even when this requirement is not met. We will adopt the language of engineers in the following discussion.</p>
<p>The idea of the modification is that, instead of using the net inductance, capacitance, resistance and conductance, we use their values per unit length. Thus, a coax is viewed as a system with distributed parameters. Let us write down the expressions for the unit-length characteristics of a coax. A current in the central conductor sets up a magnetic field; hence, the line has inductance whose value per unit length is</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn1.png|90px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>where μ is magnetic permeability of the dielectric layer between the two conductors, and ''a'' and ''b'' are the radii of the inner and outer conductors, respectively.</p>
<p>Next, the capacitance per unit length between the two conductors is:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn2.png|110px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
<p>where ε is the dielectric permittivity of the dielectric layer. </p>
<p>Since both inner and outer conductors are not ideal, the line has a resistance per unit length</p>
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn3.png|150px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>
<p>Here, μ<sub>c</sub> and σ<sub>c</sub> are the magnetic permeability and conductivity of the conductors, respectively. The resistance value (3) is determined by the skin depth that measures the characteristic distance of penetration of the electromagnetic wave into the conductors. It is within the skin layer that the absorption of the electromagnetic power is most intense. The skin depth, in turn, depends on the frequency, ν, of the wave, hence R depends on ν as well.</p>
<p>Finally, the dielectric separating the two conductors is not ideal, hence there always is a leakage current when the conductors are kept at different potentials. This effect is accounted for by introducing the conductance per unit length of the line</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn4.png|150px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>
<p>σ being the conductivity of the dielectric layer.</p>

<p><b>Exercise:</b> Derive (1) and (2). Hint: to derive (1), suppose that some current ''J'' flows through the inner conductor of the line. Find the value of the magnetic field produced by this current, and then the magnetic field energy stored in a unit length element of the line. This energy also equals ''LJ<sup>2''</sup>/2, from the definition of inductance per unit length. To derive (2), assume that the inner and outer conductors have charges per unit length equal to +''q'' and –''q''. Find the potential difference between the two conductors. From the definition of a unit length capacitance, the potential difference equals to ''q/C''.</p>
<p>A coax can be thought of as a collection of infinitely many infinitesimal elements, each of which can be presented with an equivalent circuit shown in Fig. 1 (similar to Fig. 1.4 of Preston). </p>
<table width=400 align=center><td>
<p align=justify>[[File:Coax-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Coaxial cable cross sectional view.
<br clear=right>
</p>
</td></table>
<p>As the current travels along the inner conductor, its total decrease on the element Δz is the sum of the leakage currents through the capacitance and displacement current. The former equals to ''GΔzV'' , and the latter to ''CΔz(dV/dt)''. Considering the node that joins the inductance, conductance, and capacitance, the first Kirchoff’s law yields:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn5.png|230px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>
<p>Rearranging the terms and taking the limit ''Δz → 0'' , we obtain</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn6.png|140px|center]]
</p>
</td><td> <b>(6)</b></td></tr></table>
<p>Similarly, a voltage drop on this element equals the electromotive force of self-induction and the voltage developed across the resistance:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn7.png|140px|center]]
</p>
</td><td> <b>(7)</b></td></tr></table>
<p><b>Exercise:</b> Derive (7). Use the second Kirchoff’s law to find the voltage drop across Δz.</p>
<p>Differentiating the last two equations with respect to z, we obtain:</p>
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn8.png|270px|center]]
</p>
</td><td> <b>(8)</b></td></tr></table>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn9.png|270px|center]]
</p>
</td><td> <b>(9)</b></td></tr></table>
<p><b>Exercise:</b> Show this.</p>
<p>These equations are called telegraph equations. We will look for their solution in the harmonic form</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn10.png|270px|center]]
</p>
</td><td> <b>(10)</b></td></tr></table>
<p>Substitution of these expressions into the original telegraph equations (8) yields:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn11.png|180px|center]]
</p>
</td><td> <b>(11)</b></td></tr></table>
<p>where the total series impedance and shunt conductance are</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn12.png|220px|center]]
</p>
</td><td> <b>(12)</b></td></tr></table>
<p>Equations (11) and (12) allow solutions in the form</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn13.png|300px|center]]
</p>
</td><td> <b>(13)</b></td></tr></table>
<p>where </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn14.png|300px|center]]
</p>
</td><td> <b>(14)</b></td></tr></table>
<p>is the propagation constant. It is a complex number whose real part, α, is called attenuation constant, and the imaginary part, β, is the phase constant. The two terms in Eqs. (13) represent the waves travelling in the positive and negative z-directions.</p>

<p><b>Exercise:</b> Which one is which? Hint: The wave attenuates in the direction of propagation.</p>
<p>Since the telegraph equations for voltage and current are of the same form, it can be concluded that the relationship between the amplitudes of currents and voltages propagating to the right and to the left is linear. The proportionality constant relating these quantities is called the characteristic impedance of the line, and its value equals to</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn15.png|210px|center]]
</p>
</td><td> <b>(15)</b></td></tr></table>
<p>as can be verified from Eqs.(6, 7). It then follows that</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn16.png|180px|center]]
</p>
</td><td> <b>(16)</b></td></tr></table>
<p>In the high-frequency limit,''ωL >> R'', ''ωC >> G'' (or, alternatively, for an ideal line with perfect inner and outer conductors and a perfectly insulating dielectric layer in between), the expression for the characteristic impedance of the line simplifies to</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn17.png|170px|center]]
</p>
</td><td> <b>(17)</b></td></tr></table>
<p>Note that the characteristic impedance of an ideal line is a frequency-independent quantity whose imaginary part equals to zero.</p>

<p><b>Exercise:</b> Derive the expression for ''Z<sub>c</sub>''. Hint: Use Eq. (6) or (7) to find the relationship between current and voltage amplitudes from the general solution (13).</p>

<p>Suppose that one end of the line is terminated at ''z'' = ''L'' with a load of impedance ''Z<sub>L</sub>'', so that the relationship between current and voltage on the load is</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn18.png|100px|center]]
</p>
</td><td> <b>(18)</b></td></tr></table>
<p>Substitution of Eqs. (16) for currents yields:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn19.png|200px|center]]
</p>
</td><td> <b>(19)</b></td></tr></table>
<p>The amplitudes ''V<sub>1</sub>'' and ''V<sub>2</sub>'' can be interpreted as the amplitudes of the waves reflected from and incident on the load, respectively. It then follows that the ratio of the reflected to incident voltage at ''z'' = ''L'', called reflection coefficient from the output, is:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn20.png|280px|center]]
</p>
</td><td> <b>(20)</b></td></tr></table>
<p>For currents,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn21.png|120px|center]]
</p>
</td><td> <b>(21)</b></td></tr></table>
<p><b>Exercise:</b> Why?</p>

<p>It is evident from Eq. (20) that Γ<sub>L</sub> changes from +1 (open end, ''Z<sub>L</sub>'' = ∞) to –1 (shorted, ''Z<sub>L</sub>'' = 0), depending on the value of load impedance. Impedance matching of the load to the line occurs when ''Z<sub>L</sub>'' = ''Z<sub>c</sub>''. In this case, Γ<sub>L</sub> = 0 and there is no reflected wave.</p>
<p>Let us finally recall that the generator itself has a finite input impedance ''Z<sub>0</sub>''. Redefining the voltage amplitudes ''V<sub>1</sub>'' and ''V<sub>2</sub>'' as those of the waves incident on and reflected from the input, we find the input reflection coefficient</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn22.png|100px|center]]
</p>
</td><td> <b>(22)</b></td></tr></table>
<p>Again, input impedance matching occurs when ''Z<sub>0</sub>'' = ''Z<sub>c</sub>'' and Γ<sub>0</sub> = 0.</p>

<h1>Experiment</h1>
<p>In this experiment, the following equipment will be used:</p>
<ul>
<li>A coaxial cable of 20 to 30 m in length</li>
<li>Rectangular pulse generator</li>
<li>Oscilloscope</li>
<li>Differentiating RC-circuit</li>
<li>Multimeter</li>
<li>Adjustable resistor</li>
</ul>

<h2>Part 1</h2>

<p>Connect the circuit as shown in Fig. 2. Rectangular pulses produced by the generator are differentiated by the RC-circuit, so that voltage entering the coaxial line represents a series of spikes, as shown in the figure. Set the generator frequency to 10 kHz. With infinite resistance of the load, observe the multireflected pulse on the oscilloscope. Explain the shape of the observed signal. Next, connect the inner and outer conductors of the line at the output end with a wire. Explain the difference of the observed multireflected pulse from the previous case. With ''Z<sub>L</sub>''= ∞ , calculate the velocity of pulse propagation along the line. Determine the dielectric constant of teflon from the measured velocity. Use the vacuum value for the magnetic permeability of teflon, μ=4πx10<sup>-7</sup> NA<sup>-2</sup>.</p>
<table width=400 align=center><td>
<p align=justify>[[File:Coax-fig2.png|500px|border|center]]
<b>Figure 2 -</b> Circuit to study pulses propagating along the line.
<br clear=right>
</p>
</td></table>
<p>Measure directly the capacitance of the coax with a multimeter. Determine the dielectric constant and compare it to the previously obtained value. Comment on the possible discrepancy.</p>

<h2>Part 2</h2>
<p>Assuming the ideal line, calculate its characteristic impedance from Eq. (17). Again, use the vacuum value for the magnetic permeability of teflon.</p>
<p>Attach a variable resistance to the output end of the line and measure the reflection coefficient at the output as a function of load resistance. The following procedure is suggested.</p>
<p>First, reconnect the circuit as shown in Fig. 2 and turn on the generator to produce the multireflected pulse. The heights of the n<sup>th</sup> and (n+1)<sup>th</sup> spikes are related to each other as:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn23.png|100px|center]]
</p>
</td><td> <b>(23)</b></td></tr></table>

<p><b>Exercise:</b> What is the proportionality constant ''k'' equal to in terms of the attenuation constant and the input reflection coefficient?</p>

<p>At each value of ''Z<sub>L</sub>'', choose two consecutive spikes and measure the ratio of their heights. Note that the first and the second spikes are not a very good choice because the first pulse coming directly from the generator via the differentiating circuit gets absorbed not only in the coaxial line, but also in various connections between the coax, the differentiator, and the scope. Feel free to adjust the generator frequency and oscilloscope settings to get an optimal scope trace. In order to find the proportionality constant ''k'' in (23), you can use the fact that at infinite load resistance (open output end of the line), the coefficient of reflection at the output equals one:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn24.png|130px|center]]
</p>
</td><td> <b>(24)</b></td></tr></table>
<p>Plot ''Γ<sub>L</sub>'' vs. ''Z<sub>L</sub>''. Fit this plot with Eq. (20) to determine the characteristic impedance of the line. Does this value of ''Z<sub>c</sub>'' agree with your previously obtained estimate?</p>

<h2>Part 3</h2>
<p>With the circuit connected as in Fig. 2, determine the line’s attenuation constant from the observed multireflected pulse. Is it safe to assume that the input reflection coefficient equals one? Justify this assumption experimentally and theoretically by estimating the input reflection coefficient. Note that the impedance of the scope is about 1 MΩ, and the typical impedance of the generator is 600 Ω. Assume for simplicity that the capacitance has zero resistance.</p>

<h1>References</h1>
<references/>
<ol>
<li>D. W. Preston and E.R. Dietz, [https://www.library.yorku.ca/find/Record/1038893 ''The Art of Experimental Physics''] , (Wiley) Experiment 1.</li>
<li>J.M. Serra ''et. al.'', "''A wave lab inside a coaxial cable''", [http://ajp.aapt.org/resource/1/ajpias/v55/i4/p366_s1 Eur. J. Phys. '''25''' (2004) 581]</li>
</ol>

Main Page/PHYS 4210/Coaxial Cable

2021-01-15T18:42:15Z

Gloria:

<h1>Coaxial Cable</h1>

<p>Carefully read Experiment 1 in The Art of Experimental Physics, Preston and Dietz, [[Media:Lasers-AEP-1991.pdf|available here]]<ref>[https://www.library.yorku.ca/find/Record/1038893 The Art of Experimental Physics], John Wiley and Sons,1991.</ref>. This write-up is meant to supplement that material.
</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>TEM Mode</li>
<li>Inductance per unit Length</li>
<li>Resistance per unit Length</li>
<li>Capacitance per unit Length</li>
<li>Siemens per unit Length</li>
<li>Propagation Constant</li>
<li>Attenuation Constant</li>
</ul>
</td>
<td width=250>
<ul>
<li>Phase Constant</li>
<li>Characteristic Impedance</li>
<li>Reflection Coefficient</li>
<li>Load Impedance</li>
<li>Source Impedance</li>
<li>Impedance Matching</li>
</ul>
</td>
</table>

<h1>Theory</h1>
<p>A coaxial cable (coax) is one of the most popular means for transmitting the electromagnetic power from a generator to a receiver. It consists of an inner and outer conducting lines (usually made of copper) separated by a dielectric layer (usually teflon). The electromagnetic wave can propagate in a coax in the form of TE, TM, and TEM modes, and the latter is the one that is most commonly used. In this experiment, the effect of a coaxial cable on the propagating signal and some physical characteristics of a coax are studied.</p>
<p>There are two levels of description of electromagnetic phenomena. The more fundamental one relies on Maxwell’s equations combined with material equations. Within this approach, one uses the language of electric and magnetic field strengths, potentials, dielectric susceptibility and magnetic permeability, etc. Engineers, however, prefer a simpler language of currents, voltages and lump elements, that is, resistance, capacitance, conductance, and inductance. In order for this simplified description to be applicable to the problem considered, the characteristic sizes of lump elements must be much smaller than the wavelength of an electromagnetic wave. Nevertheless, with a small modification, the approach of engineers allows us to obtain some meaningful results even when this requirement is not met. We will adopt the language of engineers in the following discussion.</p>
<p>The idea of the modification is that, instead of using the net inductance, capacitance, resistance and conductance, we use their values per unit length. Thus, a coax is viewed as a system with distributed parameters. Let us write down the expressions for the unit-length characteristics of a coax. A current in the central conductor sets up a magnetic field; hence, the line has inductance whose value per unit length is</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn1.png|90px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>where μ is magnetic permeability of the dielectric layer between the two conductors, and ''a'' and ''b'' are the radii of the inner and outer conductors, respectively.</p>
<p>Next, the capacitance per unit length between the two conductors is:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn2.png|110px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
<p>where ε is the dielectric permittivity of the dielectric layer. </p>
<p>Since both inner and outer conductors are not ideal, the line has a resistance per unit length</p>
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn3.png|150px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>
<p>Here, μ<sub>c</sub> and σ<sub>c</sub> are the magnetic permeability and conductivity of the conductors, respectively. The resistance value (3) is determined by the skin depth that measures the characteristic distance of penetration of the electromagnetic wave into the conductors. It is within the skin layer that the absorption of the electromagnetic power is most intense. The skin depth, in turn, depends on the frequency, ν, of the wave, hence R depends on ν as well.</p>
<p>Finally, the dielectric separating the two conductors is not ideal, hence there always is a leakage current when the conductors are kept at different potentials. This effect is accounted for by introducing the conductance per unit length of the line</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn4.png|150px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>
<p>σ being the conductivity of the dielectric layer.</p>

<p><b>Exercise:</b> Derive (1) and (2). Hint: to derive (1), suppose that some current ''J'' flows through the inner conductor of the line. Find the value of the magnetic field produced by this current, and then the magnetic field energy stored in a unit length element of the line. This energy also equals ''LJ<sup>2''</sup>/2, from the definition of inductance per unit length. To derive (2), assume that the inner and outer conductors have charges per unit length equal to +''q'' and –''q''. Find the potential difference between the two conductors. From the definition of a unit length capacitance, the potential difference equals to ''q/C''.</p>
<p>A coax can be thought of as a collection of infinitely many infinitesimal elements, each of which can be presented with an equivalent circuit shown in Fig. 1 (similar to Fig. 1.4 of Preston). </p>
<table width=400 align=center><td>
<p align=justify>[[File:Coax-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Coaxial cable cross sectional view.
<br clear=right>
</p>
</td></table>
<p>As the current travels along the inner conductor, its total decrease on the element Δz is the sum of the leakage currents through the capacitance and displacement current. The former equals to ''GΔzV'' , and the latter to ''CΔz(dV/dt)''. Considering the node that joins the inductance, conductance, and capacitance, the first Kirchoff’s law yields:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn5.png|230px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>
<p>Rearranging the terms and taking the limit ''Δz → 0'' , we obtain</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn6.png|140px|center]]
</p>
</td><td> <b>(6)</b></td></tr></table>
<p>Similarly, a voltage drop on this element equals the electromotive force of self-induction and the voltage developed across the resistance:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn7.png|140px|center]]
</p>
</td><td> <b>(7)</b></td></tr></table>
<p><b>Exercise:</b> Derive (7). Use the second Kirchoff’s law to find the voltage drop across Δz.</p>
<p>Differentiating the last two equations with respect to z, we obtain:</p>
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn8.png|270px|center]]
</p>
</td><td> <b>(8)</b></td></tr></table>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn9.png|270px|center]]
</p>
</td><td> <b>(9)</b></td></tr></table>
<p><b>Exercise:</b> Show this.</p>
<p>These equations are called telegraph equations. We will look for their solution in the harmonic form</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn10.png|270px|center]]
</p>
</td><td> <b>(10)</b></td></tr></table>
<p>Substitution of these expressions into the original telegraph equations (8) yields:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn11.png|180px|center]]
</p>
</td><td> <b>(11)</b></td></tr></table>
<p>where the total series impedance and shunt conductance are</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn12.png|220px|center]]
</p>
</td><td> <b>(12)</b></td></tr></table>
<p>Equations (11) and (12) allow solutions in the form</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn13.png|300px|center]]
</p>
</td><td> <b>(13)</b></td></tr></table>
<p>where </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn14.png|300px|center]]
</p>
</td><td> <b>(14)</b></td></tr></table>
<p>is the propagation constant. It is a complex number whose real part, α, is called attenuation constant, and the imaginary part, β, is the phase constant. The two terms in Eqs. (13) represent the waves travelling in the positive and negative z-directions.</p>

<p><b>Exercise:</b> Which one is which? Hint: The wave attenuates in the direction of propagation.</p>
<p>Since the telegraph equations for voltage and current are of the same form, it can be concluded that the relationship between the amplitudes of currents and voltages propagating to the right and to the left is linear. The proportionality constant relating these quantities is called the characteristic impedance of the line, and its value equals to</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn15.png|210px|center]]
</p>
</td><td> <b>(15)</b></td></tr></table>
<p>as can be verified from Eqs.(6, 7). It then follows that</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn16.png|180px|center]]
</p>
</td><td> <b>(16)</b></td></tr></table>
<p>In the high-frequency limit,''ωL >> R'', ''ωC >> G'' (or, alternatively, for an ideal line with perfect inner and outer conductors and a perfectly insulating dielectric layer in between), the expression for the characteristic impedance of the line simplifies to</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn17.png|170px|center]]
</p>
</td><td> <b>(17)</b></td></tr></table>
<p>Note that the characteristic impedance of an ideal line is a frequency-independent quantity whose imaginary part equals to zero.</p>

<p><b>Exercise:</b> Derive the expression for ''Z<sub>c</sub>''. Hint: Use Eq. (6) or (7) to find the relationship between current and voltage amplitudes from the general solution (13).</p>

<p>Suppose that one end of the line is terminated at ''z'' = ''L'' with a load of impedance ''Z<sub>L</sub>'', so that the relationship between current and voltage on the load is</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn18.png|100px|center]]
</p>
</td><td> <b>(18)</b></td></tr></table>
<p>Substitution of Eqs. (16) for currents yields:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn19.png|200px|center]]
</p>
</td><td> <b>(19)</b></td></tr></table>
<p>The amplitudes ''V<sub>1</sub>'' and ''V<sub>2</sub>'' can be interpreted as the amplitudes of the waves reflected from and incident on the load, respectively. It then follows that the ratio of the reflected to incident voltage at ''z'' = ''L'', called reflection coefficient from the output, is:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn20.png|280px|center]]
</p>
</td><td> <b>(20)</b></td></tr></table>
<p>For currents,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn21.png|120px|center]]
</p>
</td><td> <b>(21)</b></td></tr></table>
<p><b>Exercise:</b> Why?</p>

<p>It is evident from Eq. (20) that Γ<sub>L</sub> changes from +1 (open end, ''Z<sub>L</sub>'' = ∞) to –1 (shorted, ''Z<sub>L</sub>'' = 0), depending on the value of load impedance. Impedance matching of the load to the line occurs when ''Z<sub>L</sub>'' = ''Z<sub>c</sub>''. In this case, Γ<sub>L</sub> = 0 and there is no reflected wave.</p>
<p>Let us finally recall that the generator itself has a finite input impedance ''Z<sub>0</sub>''. Redefining the voltage amplitudes ''V<sub>1</sub>'' and ''V<sub>2</sub>'' as those of the waves incident on and reflected from the input, we find the input reflection coefficient</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn22.png|100px|center]]
</p>
</td><td> <b>(22)</b></td></tr></table>
<p>Again, input impedance matching occurs when ''Z<sub>0</sub>'' = ''Z<sub>c</sub>'' and Γ<sub>0</sub> = 0.</p>

<h1>Experiment</h1>
<p>In this experiment, the following equipment will be used:</p>
<ul>
<li>A coaxial cable of 20 to 30 m in length</li>
<li>Rectangular pulse generator</li>
<li>Oscilloscope</li>
<li>Differentiating RC-circuit</li>
<li>Multimeter</li>
<li>Adjustable resistor</li>
</ul>

<h2>Part 1</h2>

<p>Connect the circuit as shown in Fig. 2. Rectangular pulses produced by the generator are differentiated by the RC-circuit, so that voltage entering the coaxial line represents a series of spikes, as shown in the figure. Set the generator frequency to 10 kHz. With infinite resistance of the load, observe the multireflected pulse on the oscilloscope. Explain the shape of the observed signal. Next, connect the inner and outer conductors of the line at the output end with a wire. Explain the difference of the observed multireflected pulse from the previous case. With ''Z<sub>L</sub>''= ∞ , calculate the velocity of pulse propagation along the line. Determine the dielectric constant of teflon from the measured velocity. Use the vacuum value for the magnetic permeability of teflon, μ=4πx10<sup>-7</sup> NA<sup>-2</sup>.</p>
<table width=400 align=center><td>
<p align=justify>[[File:Coax-fig2.png|500px|border|center]]
<b>Figure 2 -</b> Circuit to study pulses propagating along the line.
<br clear=right>
</p>
</td></table>
<p>Measure directly the capacitance of the coax with a multimeter. Determine the dielectric constant and compare it to the previously obtained value. Comment on the possible discrepancy.</p>

<h2>Part 2</h2>
<p>Assuming the ideal line, calculate its characteristic impedance from Eq. (17). Again, use the vacuum value for the magnetic permeability of teflon.</p>
<p>Attach a variable resistance to the output end of the line and measure the reflection coefficient at the output as a function of load resistance. The following procedure is suggested.</p>
<p>First, reconnect the circuit as shown in Fig. 2 and turn on the generator to produce the multireflected pulse. The heights of the n<sup>th</sup> and (n+1)<sup>th</sup> spikes are related to each other as:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn23.png|100px|center]]
</p>
</td><td> <b>(23)</b></td></tr></table>

<p><b>Exercise:</b> What is the proportionality constant ''k'' equal to in terms of the attenuation constant and the input reflection coefficient?</p>

<p>At each value of ''Z<sub>L</sub>'', choose two consecutive spikes and measure the ratio of their heights. Note that the first and the second spikes are not a very good choice because the first pulse coming directly from the generator via the differentiating circuit gets absorbed not only in the coaxial line, but also in various connections between the coax, the differentiator, and the scope. Feel free to adjust the generator frequency and oscilloscope settings to get an optimal scope trace. In order to find the proportionality constant ''k'' in (23), you can use the fact that at infinite load resistance (open output end of the line), the coefficient of reflection at the output equals one:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Coax-eqn24.png|130px|center]]
</p>
</td><td> <b>(24)</b></td></tr></table>
<p>Plot ''Γ<sub>L</sub>'' vs. ''Z<sub>L</sub>''. Fit this plot with Eq. (20) to determine the characteristic impedance of the line. Does this value of ''Z<sub>c</sub>'' agree with your previously obtained estimate?</p>

<h2>Part 3</h2>
<p>With the circuit connected as in Fig. 2, determine the line’s attenuation constant from the observed multireflected pulse. Is it safe to assume that the input reflection coefficient equals one? Justify this assumption experimentally and theoretically by estimating the input reflection coefficient. Note that the impedance of the scope is about 1 MΩ, and the typical impedance of the generator is 600 Ω. Assume for simplicity that the capacitance has zero resistance.</p>

<h1>References</h1>
<ol>
<li>D. W. Preston and E.R. Dietz, [https://www.library.yorku.ca/find/Record/1038893 ''The Art of Experimental Physics''] , (Wiley) Experiment 1.</li>
<li>J.M. Serra ''et. al.'', "''A wave lab inside a coaxial cable''", [http://ajp.aapt.org/resource/1/ajpias/v55/i4/p366_s1 Eur. J. Phys. '''25''' (2004) 581]</li>
</ol>

File:Coaxial-AEP-1991.pdf

2021-01-15T18:39:39Z

Gloria:

Main Page/PHYS 4210/Electron Spin Resonance

2021-01-15T18:16:30Z

Gloria:

<h1>Electron Spin Resonance</h1>

<p>The objective of this experiment is to investigate the electron's internal angular momentum called spin. <b>A sample containing a substance with a 'free' electron in an s-state</b> is placed in a homogeneous magnetic field of strength B, which splits the energy levels for electrons with spin projections aligned with the field compared to the counter-aligned ones. A radio-frequency generator is used to resonantly pump electrons from the spin state with lower energy to the upper state. Tracing the resonance as a function of B provides the means for determining the gyro-magnetic ratio of the electron.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Free Electron</li>
<li>S-state</li>
<li>Spin ½</li>
<li>Fermions</li>
<li>Angular momentum</li>
<li>Lande-g factor</li>

</ul>
</td>
<td width=250>
<ul>
<li>Helmholtz Coils</li>
<li>Gyromagnetic Ratio</li>
<li>Anomalous Zeeman Effect</li>
<li>Normal Zeeman Effect</li>
<li>Magnetic Dipole</li>
<li>Axis of Quantization</li>
</ul>
</td>
</table>

<h1>Introduction</h1>
<p>In this experiment we obtain direct evidence for the unusual property of electrons, namely that while being point particles they exhibit the property of having two spin states. Most introductory texts on modern physics contain discussions of the historical developments leading to the discovery of this intrinsic angular momentum variable by Goudsmit and Uhlenbeck. Today we know that all elementary particles are spin - 1/2 objects called fermions, while the mediators of interactions are spin-1 objects called bosons (photons, gluons, W and Z gauge bosons for weak interactions). This experiment is also related to the study of electronic structure in magnetic and disordered systems; the 1977 Nobel Prize in physics went to P.W. Anderson, N.F. Mott, and J.H. Van Vleck for a number of achievements in this area. In chemistry ESR is applied as an important analysis technique to determine the properties of outer electrons responsible for covalent bonding in molecules. The twin concept to ESR is NMR - nuclear magnetic resonance - a technique known commonly as a non-intrusive medical diagnostic tool: it detects proton densities via their spin. It is also closely related to FMR – ferromagnetic resonance, and the Zeeman effect.</p>
<p>ESR is also known as EPR – electron paramagnetic resonance, since it deals only with paramagnetic materials. The atoms in a paramagnetic material have unpaired electrons, and therefore a net magnetic moment, but the internal interactions are not strong enough and do not create areas of similarly oriented magnetic moments, or ferromagnetism. The paramagnetic material is therefore a collection of randomly directed magnetic moments, all of which align the same direction due to the torque created by an external magnetic field. The paramagnetic material used in this experiment is an organic chemical called a free radical: diphnyl picryl hydrazyl ([http://en.wikipedia.org/wiki/DPPH DPPH]). The outer shell electron of this substance acts as though it was free, or almost so, resulting in a g-factor of 2.0023, which has a fractional difference of less than one part in a thousand from a perfectly free electron .</p>
<p>The idea in the spin resonance experiment is the following: if we place a macroscopic sample of material containing appropriate electrons in the vicinity of a radiating antenna, while an external magnetic field is present to make the energy splitting of the spin-up and spin-down states coincide with the energy of the radio wave quanta, we should be able to observe the fact that energy gets absorbed by the electrons in the sample. This observation is possible by detecting a change in a receiver placed in the vicinity, but also by observing the current that the radio generator passes through the coil (which acts as the antenna). In the microwave case one uses the former technique, while in the radiofrequency case (our experiment) the latter is preferred. The presence of the sample with resonating electrons (electrons in spin states of lower energy get transferred to the less favourable state, and decay later spontaneously) leads not only to a measurable change in the DC current through the coil, but also to a detuning of the frequency (the inductivity of the coil changes). For a discussion of both the radio wave and microwave methods read the descriptions of nuclear magnetic resonance and electron spin resonance in <ref>A.C. Melissinos , [https://www.library.yorku.ca/find/Record/1641963 ''Experiments in Modern Physics''], Academic Press, (2003). </ref>.</p>

<p>According to Quantum theory, an electron has ‘spin’, a property which behaves like angular motion, resulting in an angular momentum and consequently, as any ‘moving’ charge, a magnetic moment:</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn1.png|110px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

<p>where ''µ<sub>B</sub>'' is the Bohr magneton and ''g'' is the Landé factor. But since quantum theory permits only two values for the spin ''S<sub>z</sub>''= ± hbar/2 , the magnetic moment results in only two possible projections:</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn2.png|110px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

<p>and the resulting energy levels are then:</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn3.png|110px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

<p>Since the coupling of a magnetic moment with an external magnetic field B results in a torque aligning the magnetic moment with the field. The application of electromagnetic radiation to such a system, with a carefully set energy to match the splitting between the + and – values, results in the absorption of energy from the field, which is observable and therefore of interest to us:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn4.png|200px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>

<p>here ω is the frequency of the applied electromagnetic field (What is the expected energy scale?). Since this result is for only a single free electron, while we’re dealing with a macroscopic sample, a statistical correction has to be made. Using the Boltzmann distribution for the energy levels N<sub>+</sub> and N<sub>-</sub>:</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn5.png|170px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>

<p>where ''k'' is the Boltzmann constant and ''T'' is the temperature. It is important to note that a long natural lifetime of the upper state in a magnetic field is shortened at room temperature (''kT''=0.025eV), resulting in an equilibration of the spin level populations in a fraction of a second.</p>

<p>For most transition lines a fractional value of ''g'' was observed, a phenomenon highly important for the development of quantum theory. To unravel the complicated behavior of fractional ''g'' values in atomic spectroscopy, physicists had to figure out (before the advent of modern quantum mechanics) that the electron must have an internal angular momentum with an unusual value of ''g''=2 (all classical charge distributions have ''g''=1), and that this internal angular momentum when coupled to the orbital angular momentum (which has the classical value of ''g''=1) leads to fractional ''g'' values for most optical transitions. Thus quantum mechanics was subject to tough testing with the Schrödinger-Pauli theory incorporating spin in an ad-hoc way, and explaining atomic spectroscopy to a high degree of accuracy. Later Dirac showed how the marriage of quantum mechanics and special relativity leads to a prediction of ''g''=2 for fermions. Today the g value of the electron is one of the best-known quantities in physics (measured to 14 decimal places)<ref>D. Hanneke, S. Fogwell & G. Gabrielse, ''New Measurement of the Electron Magnetic Moment and the Fine Structure Constant'' [http://prl.aps.org/abstract/PRL/v100/i12/e120801 Phys. Rev. Lett. '''100''',120801 (2008)]</ref>. The correction of the value of ''g''=2 is fully explained by quantum field theory. </p>

<p>In the present experiment we investigate this internal angular momentum of the electron without the added complication of orbital angular momentum. The spin degree of freedom - in contrast to orbital angular momentum - has nothing to do with the orbiting motion of the electrons. The classical analogy of a spinning particle is also inappropriate, since there is not much meaning to a spinning point particle. As explained above, for the intrinsic spin angular momentum variable the interaction energy with a magnetic field requires a special choice for the gyro-magnetic ratio, namely ''g''=2. For magnetic fields that can be created in the laboratory (sub-Tesla range) the interaction energies given by eq. (4) correspond to radiofrequencies ranging up to microwaves as ''B'' is increased (verify this. i.e. calculate frequencies and wavelengths for a few examples). Thus, we need a mechanism to perform the excitation, and a way to measure the relationship between the transition frequency and the magnetic field strength. Electron spin resonance solves both problems in an elegant way, we measure the energy splitting between the two spin levels in the ground state of the spatial motion directly.</p>

<p>To make the experiment work one modulates the external magnetic field with a low-frequency AC signal component (60 cycles). For a fixed radiofrequency one is then able to sweep through a range of B values, such that at a particular phase (time) the resonance condition is met, and this condition will happen periodically. By observing the DC current through the transmitting coil in coincidence with the voltage used to generate the external magnetic field one can find out at what strength of the magnetic field the resonance condition is met. For this technique to work, one has to know that the relaxation times for the electrons are shorter than the time constant of the modulating field. If this is not the case, then less than half of the electrons in the sample would be available for excitation, and the signal might deteriorate too rapidly for detection to occur. </p>

<p> A note about the directions of fields and polarizations used in this experiment; The transition that we are driving in this experiment is a magnetic-dipole transition- meaning that it is the magnetic field component of the radio-frequency radiation which is driving the transition. Also, note that we are transitioning from an m<sub>s</sub>=-1/2 to m<sub>s</sub>=+1/2 state. This type of transition (Δm<sub>s</sub>=1) requires a circularly polarized driving field. The axis of quantization in this experiment is defined by the DC magnetic field created by the Helmholtz coils, so we need an RF magnetic field circularly polarized with respect to the DC magnetic field. The RF-coil used generates a linearly-polarized magnetic field along its axis. If the coil were aligned such that the RF-coil was along the axis of the DC-magnetic field from the Helmholtz coil, we would never drive the transition of interest. However, if the RF-coil was perpendicular to the DC magnetic field, then it can be shown that the a linearly-polarized field aligned perpendicularly to an axis of quantization is equivalent to a circularly-polarized field with respect to the axis of quantization. Therefore, in this experiment, the RF-coil is aligned perpendicular to the axis of the DC-magnetic field. (Be sure you understand this concept).</p>

<h1>Procedure</h1>

<p>To prepare for the experimental setup read through [[Media:514571e.pdf|ESR Control Unit]], [[Media:P6262_e.pdf|Electron spin resonance at DPPH]] and [[Media:P6263_e.pdf|Passive RF Oscillator]].

<h2>Part I: Verify the detection technique for the resonance condition. </h2>

<p>A radiofrequency generator is connected to a probe with an inductance (three exchangeable coils with different inductances) ''L'', and a variable capacitor ''C''. Changing the capacitance results in an adjustable frequency range for each coil allowing us to scan between approximately 20 and 100 MHz. A digital frequency meter is built into the control box that contains the radio-oscillator, as well as the power supply for the Helmholtz coils for the magnetic field. Note that the oscillator is split into two parts with the electronics residing in the control box, and the frequency-determining components ''L'' and ''C'' (and an amplitude control potentiometer) residing in the probe. </p>

<p>A receiver circuit made up of a coil and a variable capacitance is brought close to the radiating coil. It is connected to an AC voltmeter that indicates the strength of the received signal. For fixed transmitter frequency one varies the receiver's capacitor until the receiver circuit matches the transmitter (i.e. achieve resonance). Observe how the receiver shows a strong signal on resonance, how the DC current through the transmitting coil drops on resonance, and how the frequency of the transmitting circuit detunes. Compare this to the resonance problem of classical mechanics (damped driven harmonic oscillator). Detuning is related to the fact that the resonance condition has a slight dependence on the damping constant. Describe your observations in the write-up. This part demonstrates that a transmitter (oscillator) reacts to the conditions of the space surrounding it (mathematically speaking: you are changing the boundary conditions in the Maxwell equations, and creating a load that affects the transmitting coil). The damping effect is not much different from the one obtained by putting an iron core into a coil, thereby forcing the alternating B field to do work on the electron spins in the iron core which would introduce damping in the oscillator circuit.</p>

<h2>Part II: Calibrate the magnetic field produced by the Helmholtz coils for a known DC current. </h2>

<p>Note that you have to place the two coils accurately at a separation a that corresponds to their radius ''R''. Use the result from Biot-Savart's law for a single coil for the magnetic field strength for a given current along its axis as a function of the separation from the coil ''d'':</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn6.png|200px|center]]
</p>
</td><td> <b>(6)</b></td></tr></table>

<p>To obtain the field generated by two coils separated at distance ''a''=''R'' (many first-year physics texts have this problem, e.g., Fishbane, Gasiorowicz, Thornton 2nd ed. Q 65 on p. 835). <b> Here ''μ<sub>0</sub>'' is not the same ''μ'' as in eq. (1)</b>! Note that the current in Biot-Savart law is the current through each coil and not necessarily the total current in the circuit (You can use the Leybold manual to check whether your numbers are in the right range). Check the homogeneity of the field for a=R. How does your calibration compare to the Biot-Savart result? The number of turns is noted on the coils as n=320. Calibrate the Gaussmeter to zero for zero current. Does the Earth's magnetic field come into play in this experiment?</p>

<h2>Part III: Perform the electron spin resonance experiment. </h2>

<p><b>Read through the [[Media:P6262_e.pdf|<i>Electron spin resonance at DPPH (P6.2.6.2)</i>]] Physics Leaflet provided by Leybold. Connect the control box to the oscilloscope as outlined in the instructions provided by Leybold using the dual-channel set-up while triggering on the channel displaying the driving voltage for the Helmholtz coils.</b> Understand the function of the controls of the DC current through the Helmholtz coils and the AC modulation by observing the voltage applied to the coil displayed on trace 1 (used to trigger the scope). This voltage is not coincident in time with the strength of the magnetic field. To adjust for that, one can shift the phase with a third control. This will be a one-time adjustment once the resonance condition is met. A phase adjustment is necessary since AC current and voltage are out of phase by about 90 degrees for an inductance. The second trace displays a voltage that is proportional to the DC current passing through the transmitting coil. Without a sample the trace should be a straight line.</p>

<p>Set the oscillator to a given frequency, set the DC current to some value (e.g. total current = 1A), pick a midrange value for the modulating magnetic field. Place the DPPH sample inside the transmitting coil. Vary the magnetic field modulation until dips in the trace representing the DC current through the transmitting coil appear. Due to the modulation in the field two resonant points should occur as the magnetic field passes through a maximum in its sinusoidal shape. Use the phase delay control in the control box to adjust the signal such that the resonant points occur symmetrically with respect to the maximum. This adjustment should be performed only once. <b>It would not be necessary if we used a Gauss probe to measure the magnetic field in AC mode </b>(try this out).</p>

<p>Now reduce the magnetic field modulation to a small value (you will need to decrease the DC component to maintain resonance). You can adjust the DC current through the Helmholtz coils so that the resonance occurs at those points in time where the AC component vanishes. Record the DC current through the Helmholtz coils at this point, since the resonance now occurs at the magnetic field strength corresponding to the DC current alone. Observe the frequency change (detuning) as you remove and re-insert the sample, and record it.</p>

<p>Repeat this measurement for about 15 settings of frequency in 5 MHz intervals. This requires a change in coils. Notice the different amount of detuning for the different coils. Perform a linear least squares fit with estimate of uncertainties. Does the fit pass through zero energy splitting (frequency) at zero current? If not, why not? Determine your value of the gyromagnetic ratio for the 'free' electron in DPPH. </p>

<h1>References</h1>
<references/>

Main Page/PHYS 4210/Electron Spin Resonance

2021-01-15T18:15:57Z

Gloria:

<h1>Electron Spin Resonance</h1>

<p>The objective of this experiment is to investigate the electron's internal angular momentum called spin. <b>A sample containing a substance with a 'free' electron in an s-state</b> is placed in a homogeneous magnetic field of strength B, which splits the energy levels for electrons with spin projections aligned with the field compared to the counter-aligned ones. A radio-frequency generator is used to resonantly pump electrons from the spin state with lower energy to the upper state. Tracing the resonance as a function of B provides the means for determining the gyro-magnetic ratio of the electron.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Free Electron</li>
<li>S-state</li>
<li>Spin ½</li>
<li>Fermions</li>
<li>Angular momentum</li>
<li>Lande-g factor</li>

</ul>
</td>
<td width=250>
<ul>
<li>Helmholtz Coils</li>
<li>Gyromagnetic Ratio</li>
<li>Anomalous Zeeman Effect</li>
<li>Normal Zeeman Effect</li>
<li>Magnetic Dipole</li>
<li>Axis of Quantization</li>
</ul>
</td>
</table>

<h1>Introduction</h1>
<p>In this experiment we obtain direct evidence for the unusual property of electrons, namely that while being point particles they exhibit the property of having two spin states. Most introductory texts on modern physics contain discussions of the historical developments leading to the discovery of this intrinsic angular momentum variable by Goudsmit and Uhlenbeck. Today we know that all elementary particles are spin - 1/2 objects called fermions, while the mediators of interactions are spin-1 objects called bosons (photons, gluons, W and Z gauge bosons for weak interactions). This experiment is also related to the study of electronic structure in magnetic and disordered systems; the 1977 Nobel Prize in physics went to P.W. Anderson, N.F. Mott, and J.H. Van Vleck for a number of achievements in this area. In chemistry ESR is applied as an important analysis technique to determine the properties of outer electrons responsible for covalent bonding in molecules. The twin concept to ESR is NMR - nuclear magnetic resonance - a technique known commonly as a non-intrusive medical diagnostic tool: it detects proton densities via their spin. It is also closely related to FMR – ferromagnetic resonance, and the Zeeman effect.</p>
<p>ESR is also known as EPR – electron paramagnetic resonance, since it deals only with paramagnetic materials. The atoms in a paramagnetic material have unpaired electrons, and therefore a net magnetic moment, but the internal interactions are not strong enough and do not create areas of similarly oriented magnetic moments, or ferromagnetism. The paramagnetic material is therefore a collection of randomly directed magnetic moments, all of which align the same direction due to the torque created by an external magnetic field. The paramagnetic material used in this experiment is an organic chemical called a free radical: diphnyl picryl hydrazyl ([http://en.wikipedia.org/wiki/DPPH DPPH]). The outer shell electron of this substance acts as though it was free, or almost so, resulting in a g-factor of 2.0023, which has a fractional difference of less than one part in a thousand from a perfectly free electron .</p>
<p>The idea in the spin resonance experiment is the following: if we place a macroscopic sample of material containing appropriate electrons in the vicinity of a radiating antenna, while an external magnetic field is present to make the energy splitting of the spin-up and spin-down states coincide with the energy of the radio wave quanta, we should be able to observe the fact that energy gets absorbed by the electrons in the sample. This observation is possible by detecting a change in a receiver placed in the vicinity, but also by observing the current that the radio generator passes through the coil (which acts as the antenna). In the microwave case one uses the former technique, while in the radiofrequency case (our experiment) the latter is preferred. The presence of the sample with resonating electrons (electrons in spin states of lower energy get transferred to the less favourable state, and decay later spontaneously) leads not only to a measurable change in the DC current through the coil, but also to a detuning of the frequency (the inductivity of the coil changes). For a discussion of both the radio wave and microwave methods read the descriptions of nuclear magnetic resonance and electron spin resonance in <ref>A.C. Melissinos , [https://www.library.yorku.ca/find/Record/1641963 ''Experiments in Modern Physics''], Academic Press, (2003). </ref>.</p>

<p>According to Quantum theory, an electron has ‘spin’, a property which behaves like angular motion, resulting in an angular momentum and consequently, as any ‘moving’ charge, a magnetic moment:</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn1.png|110px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

<p>where ''µ<sub>B</sub>'' is the Bohr magneton and ''g'' is the Landé factor. But since quantum theory permits only two values for the spin ''S<sub>z</sub>''= ± hbar/2 , the magnetic moment results in only two possible projections:</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn2.png|110px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

<p>and the resulting energy levels are then:</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn3.png|110px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

<p>Since the coupling of a magnetic moment with an external magnetic field B results in a torque aligning the magnetic moment with the field. The application of electromagnetic radiation to such a system, with a carefully set energy to match the splitting between the + and – values, results in the absorption of energy from the field, which is observable and therefore of interest to us:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn4.png|200px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>

<p>here ω is the frequency of the applied electromagnetic field (What is the expected energy scale?). Since this result is for only a single free electron, while we’re dealing with a macroscopic sample, a statistical correction has to be made. Using the Boltzmann distribution for the energy levels N<sub>+</sub> and N<sub>-</sub>:</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn5.png|170px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>

<p>where ''k'' is the Boltzmann constant and ''T'' is the temperature. It is important to note that a long natural lifetime of the upper state in a magnetic field is shortened at room temperature (''kT''=0.025eV), resulting in an equilibration of the spin level populations in a fraction of a second.</p>

<p>For most transition lines a fractional value of ''g'' was observed, a phenomenon highly important for the development of quantum theory. To unravel the complicated behavior of fractional ''g'' values in atomic spectroscopy, physicists had to figure out (before the advent of modern quantum mechanics) that the electron must have an internal angular momentum with an unusual value of ''g''=2 (all classical charge distributions have ''g''=1), and that this internal angular momentum when coupled to the orbital angular momentum (which has the classical value of ''g''=1) leads to fractional ''g'' values for most optical transitions. Thus quantum mechanics was subject to tough testing with the Schrödinger-Pauli theory incorporating spin in an ad-hoc way, and explaining atomic spectroscopy to a high degree of accuracy. Later Dirac showed how the marriage of quantum mechanics and special relativity leads to a prediction of ''g''=2 for fermions. Today the g value of the electron is one of the best-known quantities in physics (measured to 14 decimal places)<ref>D. Hanneke, S. Fogwell & G. Gabrielse, ''New Measurement of the Electron Magnetic Moment and the Fine Structure Constant'' [http://prl.aps.org/abstract/PRL/v100/i12/e120801 Phys. Rev. Lett. '''100''',120801 (2008)]</ref>. The correction of the value of ''g''=2 is fully explained by quantum field theory. </p>

<p>In the present experiment we investigate this internal angular momentum of the electron without the added complication of orbital angular momentum. The spin degree of freedom - in contrast to orbital angular momentum - has nothing to do with the orbiting motion of the electrons. The classical analogy of a spinning particle is also inappropriate, since there is not much meaning to a spinning point particle. As explained above, for the intrinsic spin angular momentum variable the interaction energy with a magnetic field requires a special choice for the gyro-magnetic ratio, namely ''g''=2. For magnetic fields that can be created in the laboratory (sub-Tesla range) the interaction energies given by eq. (4) correspond to radiofrequencies ranging up to microwaves as ''B'' is increased (verify this. i.e. calculate frequencies and wavelengths for a few examples). Thus, we need a mechanism to perform the excitation, and a way to measure the relationship between the transition frequency and the magnetic field strength. Electron spin resonance solves both problems in an elegant way, we measure the energy splitting between the two spin levels in the ground state of the spatial motion directly.</p>

<p>To make the experiment work one modulates the external magnetic field with a low-frequency AC signal component (60 cycles). For a fixed radiofrequency one is then able to sweep through a range of B values, such that at a particular phase (time) the resonance condition is met, and this condition will happen periodically. By observing the DC current through the transmitting coil in coincidence with the voltage used to generate the external magnetic field one can find out at what strength of the magnetic field the resonance condition is met. For this technique to work, one has to know that the relaxation times for the electrons are shorter than the time constant of the modulating field. If this is not the case, then less than half of the electrons in the sample would be available for excitation, and the signal might deteriorate too rapidly for detection to occur. </p>

<p> A note about the directions of fields and polarizations used in this experiment; The transition that we are driving in this experiment is a magnetic-dipole transition- meaning that it is the magnetic field component of the radio-frequency radiation which is driving the transition. Also, note that we are transitioning from an m<sub>s</sub>=-1/2 to m<sub>s</sub>=+1/2 state. This type of transition (Δm<sub>s</sub>=1) requires a circularly polarized driving field. The axis of quantization in this experiment is defined by the DC magnetic field created by the Helmholtz coils, so we need an RF magnetic field circularly polarized with respect to the DC magnetic field. The RF-coil used generates a linearly-polarized magnetic field along its axis. If the coil were aligned such that the RF-coil was along the axis of the DC-magnetic field from the Helmholtz coil, we would never drive the transition of interest. However, if the RF-coil was perpendicular to the DC magnetic field, then it can be shown that the a linearly-polarized field aligned perpendicularly to an axis of quantization is equivalent to a circularly-polarized field with respect to the axis of quantization. Therefore, in this experiment, the RF-coil is aligned perpendicular to the axis of the DC-magnetic field. (Be sure you understand this concept).</p>

<h1>Procedure</h1>

<p>To better prepare for the experimental setup read through [[Media:514571e.pdf|ESR Control Unit]], [[Media:P6262_e.pdf|Electron spin resonance at DPPH]] and [[Media:P6263_e.pdf|Passive RF Oscillator]].

<h2>Part I: Verify the detection technique for the resonance condition. </h2>

<p>A radiofrequency generator is connected to a probe with an inductance (three exchangeable coils with different inductances) ''L'', and a variable capacitor ''C''. Changing the capacitance results in an adjustable frequency range for each coil allowing us to scan between approximately 20 and 100 MHz. A digital frequency meter is built into the control box that contains the radio-oscillator, as well as the power supply for the Helmholtz coils for the magnetic field. Note that the oscillator is split into two parts with the electronics residing in the control box, and the frequency-determining components ''L'' and ''C'' (and an amplitude control potentiometer) residing in the probe. </p>

<p>A receiver circuit made up of a coil and a variable capacitance is brought close to the radiating coil. It is connected to an AC voltmeter that indicates the strength of the received signal. For fixed transmitter frequency one varies the receiver's capacitor until the receiver circuit matches the transmitter (i.e. achieve resonance). Observe how the receiver shows a strong signal on resonance, how the DC current through the transmitting coil drops on resonance, and how the frequency of the transmitting circuit detunes. Compare this to the resonance problem of classical mechanics (damped driven harmonic oscillator). Detuning is related to the fact that the resonance condition has a slight dependence on the damping constant. Describe your observations in the write-up. This part demonstrates that a transmitter (oscillator) reacts to the conditions of the space surrounding it (mathematically speaking: you are changing the boundary conditions in the Maxwell equations, and creating a load that affects the transmitting coil). The damping effect is not much different from the one obtained by putting an iron core into a coil, thereby forcing the alternating B field to do work on the electron spins in the iron core which would introduce damping in the oscillator circuit.</p>

<h2>Part II: Calibrate the magnetic field produced by the Helmholtz coils for a known DC current. </h2>

<p>Note that you have to place the two coils accurately at a separation a that corresponds to their radius ''R''. Use the result from Biot-Savart's law for a single coil for the magnetic field strength for a given current along its axis as a function of the separation from the coil ''d'':</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn6.png|200px|center]]
</p>
</td><td> <b>(6)</b></td></tr></table>

<p>To obtain the field generated by two coils separated at distance ''a''=''R'' (many first-year physics texts have this problem, e.g., Fishbane, Gasiorowicz, Thornton 2nd ed. Q 65 on p. 835). <b> Here ''μ<sub>0</sub>'' is not the same ''μ'' as in eq. (1)</b>! Note that the current in Biot-Savart law is the current through each coil and not necessarily the total current in the circuit (You can use the Leybold manual to check whether your numbers are in the right range). Check the homogeneity of the field for a=R. How does your calibration compare to the Biot-Savart result? The number of turns is noted on the coils as n=320. Calibrate the Gaussmeter to zero for zero current. Does the Earth's magnetic field come into play in this experiment?</p>

<h2>Part III: Perform the electron spin resonance experiment. </h2>

<p><b>Read through the [[Media:P6262_e.pdf|<i>Electron spin resonance at DPPH (P6.2.6.2)</i>]] Physics Leaflet provided by Leybold. Connect the control box to the oscilloscope as outlined in the instructions provided by Leybold using the dual-channel set-up while triggering on the channel displaying the driving voltage for the Helmholtz coils.</b> Understand the function of the controls of the DC current through the Helmholtz coils and the AC modulation by observing the voltage applied to the coil displayed on trace 1 (used to trigger the scope). This voltage is not coincident in time with the strength of the magnetic field. To adjust for that, one can shift the phase with a third control. This will be a one-time adjustment once the resonance condition is met. A phase adjustment is necessary since AC current and voltage are out of phase by about 90 degrees for an inductance. The second trace displays a voltage that is proportional to the DC current passing through the transmitting coil. Without a sample the trace should be a straight line.</p>

<p>Set the oscillator to a given frequency, set the DC current to some value (e.g. total current = 1A), pick a midrange value for the modulating magnetic field. Place the DPPH sample inside the transmitting coil. Vary the magnetic field modulation until dips in the trace representing the DC current through the transmitting coil appear. Due to the modulation in the field two resonant points should occur as the magnetic field passes through a maximum in its sinusoidal shape. Use the phase delay control in the control box to adjust the signal such that the resonant points occur symmetrically with respect to the maximum. This adjustment should be performed only once. <b>It would not be necessary if we used a Gauss probe to measure the magnetic field in AC mode </b>(try this out).</p>

<p>Now reduce the magnetic field modulation to a small value (you will need to decrease the DC component to maintain resonance). You can adjust the DC current through the Helmholtz coils so that the resonance occurs at those points in time where the AC component vanishes. Record the DC current through the Helmholtz coils at this point, since the resonance now occurs at the magnetic field strength corresponding to the DC current alone. Observe the frequency change (detuning) as you remove and re-insert the sample, and record it.</p>

<p>Repeat this measurement for about 15 settings of frequency in 5 MHz intervals. This requires a change in coils. Notice the different amount of detuning for the different coils. Perform a linear least squares fit with estimate of uncertainties. Does the fit pass through zero energy splitting (frequency) at zero current? If not, why not? Determine your value of the gyromagnetic ratio for the 'free' electron in DPPH. </p>

<h1>References</h1>
<references/>

Main Page/PHYS 4210/Electron Spin Resonance

2021-01-15T18:13:44Z

Gloria:

<h1>Electron Spin Resonance</h1>

<p>The objective of this experiment is to investigate the electron's internal angular momentum called spin. <b>A sample containing a substance with a 'free' electron in an s-state</b> is placed in a homogeneous magnetic field of strength B, which splits the energy levels for electrons with spin projections aligned with the field compared to the counter-aligned ones. A radio-frequency generator is used to resonantly pump electrons from the spin state with lower energy to the upper state. Tracing the resonance as a function of B provides the means for determining the gyro-magnetic ratio of the electron.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Free Electron</li>
<li>S-state</li>
<li>Spin ½</li>
<li>Fermions</li>
<li>Angular momentum</li>
<li>Lande-g factor</li>

</ul>
</td>
<td width=250>
<ul>
<li>Helmholtz Coils</li>
<li>Gyromagnetic Ratio</li>
<li>Anomalous Zeeman Effect</li>
<li>Normal Zeeman Effect</li>
<li>Magnetic Dipole</li>
<li>Axis of Quantization</li>
</ul>
</td>
</table>

<h1>Introduction</h1>
<p>In this experiment we obtain direct evidence for the unusual property of electrons, namely that while being point particles they exhibit the property of having two spin states. Most introductory texts on modern physics contain discussions of the historical developments leading to the discovery of this intrinsic angular momentum variable by Goudsmit and Uhlenbeck. Today we know that all elementary particles are spin - 1/2 objects called fermions, while the mediators of interactions are spin-1 objects called bosons (photons, gluons, W and Z gauge bosons for weak interactions). This experiment is also related to the study of electronic structure in magnetic and disordered systems; the 1977 Nobel Prize in physics went to P.W. Anderson, N.F. Mott, and J.H. Van Vleck for a number of achievements in this area. In chemistry ESR is applied as an important analysis technique to determine the properties of outer electrons responsible for covalent bonding in molecules. The twin concept to ESR is NMR - nuclear magnetic resonance - a technique known commonly as a non-intrusive medical diagnostic tool: it detects proton densities via their spin. It is also closely related to FMR – ferromagnetic resonance, and the Zeeman effect.</p>
<p>ESR is also known as EPR – electron paramagnetic resonance, since it deals only with paramagnetic materials. The atoms in a paramagnetic material have unpaired electrons, and therefore a net magnetic moment, but the internal interactions are not strong enough and do not create areas of similarly oriented magnetic moments, or ferromagnetism. The paramagnetic material is therefore a collection of randomly directed magnetic moments, all of which align the same direction due to the torque created by an external magnetic field. The paramagnetic material used in this experiment is an organic chemical called a free radical: diphnyl picryl hydrazyl ([http://en.wikipedia.org/wiki/DPPH DPPH]). The outer shell electron of this substance acts as though it was free, or almost so, resulting in a g-factor of 2.0023, which has a fractional difference of less than one part in a thousand from a perfectly free electron .</p>
<p>The idea in the spin resonance experiment is the following: if we place a macroscopic sample of material containing appropriate electrons in the vicinity of a radiating antenna, while an external magnetic field is present to make the energy splitting of the spin-up and spin-down states coincide with the energy of the radio wave quanta, we should be able to observe the fact that energy gets absorbed by the electrons in the sample. This observation is possible by detecting a change in a receiver placed in the vicinity, but also by observing the current that the radio generator passes through the coil (which acts as the antenna). In the microwave case one uses the former technique, while in the radiofrequency case (our experiment) the latter is preferred. The presence of the sample with resonating electrons (electrons in spin states of lower energy get transferred to the less favourable state, and decay later spontaneously) leads not only to a measurable change in the DC current through the coil, but also to a detuning of the frequency (the inductivity of the coil changes). For a discussion of both the radio wave and microwave methods read the descriptions of nuclear magnetic resonance and electron spin resonance in <ref>A.C. Melissinos , [https://www.library.yorku.ca/find/Record/1641963 ''Experiments in Modern Physics''], Academic Press, (2003). </ref>.</p>

<p>According to Quantum theory, an electron has ‘spin’, a property which behaves like angular motion, resulting in an angular momentum and consequently, as any ‘moving’ charge, a magnetic moment:</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn1.png|110px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

<p>where ''µ<sub>B</sub>'' is the Bohr magneton and ''g'' is the Landé factor. But since quantum theory permits only two values for the spin ''S<sub>z</sub>''= ± hbar/2 , the magnetic moment results in only two possible projections:</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn2.png|110px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

<p>and the resulting energy levels are then:</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn3.png|110px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

<p>Since the coupling of a magnetic moment with an external magnetic field B results in a torque aligning the magnetic moment with the field. The application of electromagnetic radiation to such a system, with a carefully set energy to match the splitting between the + and – values, results in the absorption of energy from the field, which is observable and therefore of interest to us:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn4.png|200px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>

<p>here ω is the frequency of the applied electromagnetic field (What is the expected energy scale?). Since this result is for only a single free electron, while we’re dealing with a macroscopic sample, a statistical correction has to be made. Using the Boltzmann distribution for the energy levels N<sub>+</sub> and N<sub>-</sub>:</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn5.png|170px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>

<p>where ''k'' is the Boltzmann constant and ''T'' is the temperature. It is important to note that a long natural lifetime of the upper state in a magnetic field is shortened at room temperature (''kT''=0.025eV), resulting in an equilibration of the spin level populations in a fraction of a second.</p>

<p>For most transition lines a fractional value of ''g'' was observed, a phenomenon highly important for the development of quantum theory. To unravel the complicated behavior of fractional ''g'' values in atomic spectroscopy, physicists had to figure out (before the advent of modern quantum mechanics) that the electron must have an internal angular momentum with an unusual value of ''g''=2 (all classical charge distributions have ''g''=1), and that this internal angular momentum when coupled to the orbital angular momentum (which has the classical value of ''g''=1) leads to fractional ''g'' values for most optical transitions. Thus quantum mechanics was subject to tough testing with the Schrödinger-Pauli theory incorporating spin in an ad-hoc way, and explaining atomic spectroscopy to a high degree of accuracy. Later Dirac showed how the marriage of quantum mechanics and special relativity leads to a prediction of ''g''=2 for fermions. Today the g value of the electron is one of the best-known quantities in physics (measured to 14 decimal places)<ref>D. Hanneke, S. Fogwell & G. Gabrielse, ''New Measurement of the Electron Magnetic Moment and the Fine Structure Constant'' [http://prl.aps.org/abstract/PRL/v100/i12/e120801 Phys. Rev. Lett. '''100''',120801 (2008)]</ref>. The correction of the value of ''g''=2 is fully explained by quantum field theory. </p>

<p>In the present experiment we investigate this internal angular momentum of the electron without the added complication of orbital angular momentum. The spin degree of freedom - in contrast to orbital angular momentum - has nothing to do with the orbiting motion of the electrons. The classical analogy of a spinning particle is also inappropriate, since there is not much meaning to a spinning point particle. As explained above, for the intrinsic spin angular momentum variable the interaction energy with a magnetic field requires a special choice for the gyro-magnetic ratio, namely ''g''=2. For magnetic fields that can be created in the laboratory (sub-Tesla range) the interaction energies given by eq. (4) correspond to radiofrequencies ranging up to microwaves as ''B'' is increased (verify this. i.e. calculate frequencies and wavelengths for a few examples). Thus, we need a mechanism to perform the excitation, and a way to measure the relationship between the transition frequency and the magnetic field strength. Electron spin resonance solves both problems in an elegant way, we measure the energy splitting between the two spin levels in the ground state of the spatial motion directly.</p>

<p>To make the experiment work one modulates the external magnetic field with a low-frequency AC signal component (60 cycles). For a fixed radiofrequency one is then able to sweep through a range of B values, such that at a particular phase (time) the resonance condition is met, and this condition will happen periodically. By observing the DC current through the transmitting coil in coincidence with the voltage used to generate the external magnetic field one can find out at what strength of the magnetic field the resonance condition is met. For this technique to work, one has to know that the relaxation times for the electrons are shorter than the time constant of the modulating field. If this is not the case, then less than half of the electrons in the sample would be available for excitation, and the signal might deteriorate too rapidly for detection to occur. </p>

<p> A note about the directions of fields and polarizations used in this experiment; The transition that we are driving in this experiment is a magnetic-dipole transition- meaning that it is the magnetic field component of the radio-frequency radiation which is driving the transition. Also, note that we are transitioning from an m<sub>s</sub>=-1/2 to m<sub>s</sub>=+1/2 state. This type of transition (Δm<sub>s</sub>=1) requires a circularly polarized driving field. The axis of quantization in this experiment is defined by the DC magnetic field created by the Helmholtz coils, so we need an RF magnetic field circularly polarized with respect to the DC magnetic field. The RF-coil used generates a linearly-polarized magnetic field along its axis. If the coil were aligned such that the RF-coil was along the axis of the DC-magnetic field from the Helmholtz coil, we would never drive the transition of interest. However, if the RF-coil was perpendicular to the DC magnetic field, then it can be shown that the a linearly-polarized field aligned perpendicularly to an axis of quantization is equivalent to a circularly-polarized field with respect to the axis of quantization. Therefore, in this experiment, the RF-coil is aligned perpendicular to the axis of the DC-magnetic field. (Be sure you understand this concept).</p>

<h1>Procedure</h1>

<p>To better prepare for the experimental setup read through [[Media:514571e.pdf|Instruction Manual]], [[Media:P6262_e.pdf|Electron spin resonance at DPPH]] and [[Media:P6263_e.pdf|Passive Coil]].

<h2>Part I: Verify the detection technique for the resonance condition. </h2>

<p>A radiofrequency generator is connected to a probe with an inductance (three exchangeable coils with different inductances) ''L'', and a variable capacitor ''C''. Changing the capacitance results in an adjustable frequency range for each coil allowing us to scan between approximately 20 and 100 MHz. A digital frequency meter is built into the control box that contains the radio-oscillator, as well as the power supply for the Helmholtz coils for the magnetic field. Note that the oscillator is split into two parts with the electronics residing in the control box, and the frequency-determining components ''L'' and ''C'' (and an amplitude control potentiometer) residing in the probe. </p>

<p>A receiver circuit made up of a coil and a variable capacitance is brought close to the radiating coil. It is connected to an AC voltmeter that indicates the strength of the received signal. For fixed transmitter frequency one varies the receiver's capacitor until the receiver circuit matches the transmitter (i.e. achieve resonance). Observe how the receiver shows a strong signal on resonance, how the DC current through the transmitting coil drops on resonance, and how the frequency of the transmitting circuit detunes. Compare this to the resonance problem of classical mechanics (damped driven harmonic oscillator). Detuning is related to the fact that the resonance condition has a slight dependence on the damping constant. Describe your observations in the write-up. This part demonstrates that a transmitter (oscillator) reacts to the conditions of the space surrounding it (mathematically speaking: you are changing the boundary conditions in the Maxwell equations, and creating a load that affects the transmitting coil). The damping effect is not much different from the one obtained by putting an iron core into a coil, thereby forcing the alternating B field to do work on the electron spins in the iron core which would introduce damping in the oscillator circuit.</p>

<h2>Part II: Calibrate the magnetic field produced by the Helmholtz coils for a known DC current. </h2>

<p>Note that you have to place the two coils accurately at a separation a that corresponds to their radius ''R''. Use the result from Biot-Savart's law for a single coil for the magnetic field strength for a given current along its axis as a function of the separation from the coil ''d'':</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Esr-eqn6.png|200px|center]]
</p>
</td><td> <b>(6)</b></td></tr></table>

<p>To obtain the field generated by two coils separated at distance ''a''=''R'' (many first-year physics texts have this problem, e.g., Fishbane, Gasiorowicz, Thornton 2nd ed. Q 65 on p. 835). <b> Here ''μ<sub>0</sub>'' is not the same ''μ'' as in eq. (1)</b>! Note that the current in Biot-Savart law is the current through each coil and not necessarily the total current in the circuit (You can use the Leybold manual to check whether your numbers are in the right range). Check the homogeneity of the field for a=R. How does your calibration compare to the Biot-Savart result? The number of turns is noted on the coils as n=320. Calibrate the Gaussmeter to zero for zero current. Does the Earth's magnetic field come into play in this experiment?</p>

<h2>Part III: Perform the electron spin resonance experiment. </h2>

<p><b>Read through the [[Media:P6262_e.pdf|<i>Electron spin resonance at DPPH (P6.2.6.2)</i>]] Physics Leaflet provided by Leybold. Connect the control box to the oscilloscope as outlined in the instructions provided by Leybold using the dual-channel set-up while triggering on the channel displaying the driving voltage for the Helmholtz coils.</b> Understand the function of the controls of the DC current through the Helmholtz coils and the AC modulation by observing the voltage applied to the coil displayed on trace 1 (used to trigger the scope). This voltage is not coincident in time with the strength of the magnetic field. To adjust for that, one can shift the phase with a third control. This will be a one-time adjustment once the resonance condition is met. A phase adjustment is necessary since AC current and voltage are out of phase by about 90 degrees for an inductance. The second trace displays a voltage that is proportional to the DC current passing through the transmitting coil. Without a sample the trace should be a straight line.</p>

<p>Set the oscillator to a given frequency, set the DC current to some value (e.g. total current = 1A), pick a midrange value for the modulating magnetic field. Place the DPPH sample inside the transmitting coil. Vary the magnetic field modulation until dips in the trace representing the DC current through the transmitting coil appear. Due to the modulation in the field two resonant points should occur as the magnetic field passes through a maximum in its sinusoidal shape. Use the phase delay control in the control box to adjust the signal such that the resonant points occur symmetrically with respect to the maximum. This adjustment should be performed only once. <b>It would not be necessary if we used a Gauss probe to measure the magnetic field in AC mode </b>(try this out).</p>

<p>Now reduce the magnetic field modulation to a small value (you will need to decrease the DC component to maintain resonance). You can adjust the DC current through the Helmholtz coils so that the resonance occurs at those points in time where the AC component vanishes. Record the DC current through the Helmholtz coils at this point, since the resonance now occurs at the magnetic field strength corresponding to the DC current alone. Observe the frequency change (detuning) as you remove and re-insert the sample, and record it.</p>

<p>Repeat this measurement for about 15 settings of frequency in 5 MHz intervals. This requires a change in coils. Notice the different amount of detuning for the different coils. Perform a linear least squares fit with estimate of uncertainties. Does the fit pass through zero energy splitting (frequency) at zero current? If not, why not? Determine your value of the gyromagnetic ratio for the 'free' electron in DPPH. </p>

<h1>References</h1>
<references/>

File:P6263 e.pdf

2021-01-15T18:08:02Z

Gloria:

File:P6262 e.pdf

2021-01-15T18:07:47Z

Gloria:

File:514571e.pdf

2021-01-15T18:07:22Z

Gloria:

Main Page/PHYS 4210/He-Ne Lasers

2021-01-15T16:40:12Z

Gloria:

<h1>He-Ne Lasers</h1>

<p>In this experiment we first align an open-ended laser. Then we set up some transverse mode patterns, and perform further exercises and experiments to understand how a laser works.</p>

<h2> Key Concepts</h2>
<table width=500>
<td width=250><ul>
<li>Stimulated Emission</li>
<li>Spontaneous Emission</li>
<li>Stimulated Emission</li>
<li>Incoherent/Coherent Radiation</li>
<li>Einstein Coefficients</li>
<li>Population Inversion</li>
<li>Forbidden Transitions</li>
<li>Metastable States</li>
<li>LS coupling</li>
<li>Electric dipole selection rules</li>
</ul>
</td>
<td width=250>
<ul>
<li>Axial Modes</li>
<li>TEM modes</li>
<li>Spectral Width</li>
<li>Atomic Lineshape</li>
<li>Loss/Gain Coefficient</li>
<li>Index of Refraction</li>
<li>Brewster’s Angle</li>
<li>Malus’s Law</li>
<li>Fresnel-Arago Law</li>
<li>Q-Switch</li>
</ul>
</td>
</table>

<h1>Reading and Exercises</h1>
<ul>
<li>Read pages 94 to 105 and pages 100 to 112 from Preston-Dietz, [[Media:Lasers-AEP-1991.pdf|available here]]<ref>[https://www.library.yorku.ca/find/Record/1038893 The Art of Experimental Physics], John Wiley and Sons,1991.</ref>. </li>
<li>Carry out '''Exercise 1''' (pg. 100), '''Exercise 2''' (pg. 103), '''Exercise 3''' (pg. 104), and '''Exercise 4''' (pg. 104). </li>
<li> Complete '''Exercise 1''' (pp 111-112). Do not forget to answer the last question of the exercise: Calculate the frequency difference between two adjacent axial modes TEM<sub>oom</sub> & TEM<sub>oo(m+1)</sub>.</li>
<li> Submit all exercises as part of your report, either in the introduction or as an appendix as you deem appropriate. </li>
</ul>

<h1>Experiments</h1>
<h2>Aligning the laser</h2>
<p>Align the laser until it begins lasing. The TA will discuss techniques to accomplish this.</p>
<p>When you are successful with the aligning process, and lasing is achieved, try varying the distances between the mirrors (using the adjusting screws on the laser mount) that still supports lasing. Record, in your lab book, the aligning process used, maximum, minimum and 'best' distances between the mirrors.</p>
<p>What the radius of curvature of the mirrors which form the optical cavity? </p>
<p>Use the polarizers to determine the polarization of laser.</p>
<p>Assume that the He-Ne produces 3 mW of laser output power and that the electrical data given applies to your gas discharge tube. Compute the efficiency, in percent, for converting electrical energy to red laser light energy with this He-Ne laser. Discuss your results.</p>

<h2>Brewster's Angle</h2>
<p>Using the glass plate provided, find an approximate value for the Brewster's angle. You can determine this by rotating the glass plate until lasing stops. Only at the Brewster's angle does lasing resume.</p>
<p>Knowing that the tangent of the Brewster's angle is the ratio of the refractive indices of the lasing medium to air, determine the refractive index of the glass medium.</p>
<p>Every resonant laser cavity has a characteristic quality factor or Q that measures the internal losses. The higher the Q, the lower the losses.</p>
<p>A Q-switch pulse can be made by blocking one end of the mirror, then exciting the medium and then quickly unblocking the mirror. Stimulated emission will quickly drain the stored laser energy from the cavity in a short pulse with peak power much higher than the laser can produce. One can think of a Q-switch as a device that quickly switches from absorbing to transmitting, suddenly reducing cavity losses.</p>
<p>The Q-switch pulse length is given by</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn1a.png|180px|center]]
</p>
</td></table>
<p>where t is the round trip time (back and forth in the cavity), and R is the output mirror reflectivity ( >98% ).</p>
<p>Therefore</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn2.png|210px|center]]
</p>
</td></table>
<p>where ''L'' is the distance between the mirrors, ''n<sub>1</sub>'' is the refractive index of the medium, and ''c'' is the speed of light. Pulse length can then be written as</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn3.png|180px|center]]
</p>
</td></table>
<p>Using the data from your laser, what is the theoretical value for the pulse length?</p>

<h2>TEM Modes</h2>
<p>Set up a camera with the screen at about 1 to 2 meters from the output mirror. Adjust the output coupler screws (or any other adjustments) to produce the TEM<sub>00m</sub>, TEM<sub>10m</sub>, TEM<sub>01m</sub>, .... modes. Photograph or sketch a few of them.</p>

<h2>Beam Profile of the TEM<sub>oom</sub> and the TEM<sub>10m</sub> modes</h2>
<p>Realign the beam to produce the TEM<sub>oom</sub> mode.</p>
<p>You will use a rotating mirror and a photodiode monitored on an oscilloscope to observe the profile of the laser beam.</p>
<p>Be sure to ensure the photodiode is not saturating when the laser is aligned onto it. If it is, switch the scale of the photodiode amplifier to a lower gain setting.</p>
<p>Repeat your observations for the TEM<sub>o1m</sub> mode. Remember that photodetectors are square-law detectors, i.e., the current density J is proportional to the square of the electric field. (See Preston for details). Sketch the beam profiles for both modes.</p>

<h2>Beam Profile or Shape</h2>
<p>A laser beam has a certain profile with most energy concentrated at the center. The beam has the following form</p>

<table width=400 align=center><td>
<p align=justify>[[File:HeNe-fig1v2.jpg|400px|border|center]]
<b>Figure 1 -</b> Amplitude distribution across laser beam oscillating in the TEM<sub>oo</sub> mode.
<br clear=right>
</p>
</td></table>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn4.png|120px|center]]
</p>
</td></table>

<p>where ''w'' is the radius of the beam. The Gaussian function, exp [- (''r''/''w'')<sup>2</sup> ] falls to 1/e, when ''r'' = ''w'', i.e.,</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn5.png|90px|center]]
</p>
</td></table>
<p>Since the energy is proportional to the square of the amplitude, the beam radius, or SPOT SIZE, ''w'', is defined as that distance from the axis where the power has dropped to 1/e<sup>2</sup> of its value at the center. Twice that distance, 2''w'', is the beam diameter.</p>
<p>The beam radius, ''w'', is the function of distance along the axis. If we call ''x'' the axial distance measured from the midpoint between the two (concave) mirrors, then the parameter ''w'' is given by</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn6.png|160px|center]]
</p>
</td></table>
<p>where λ is the wavelength and ''w<sub>0</sub>'' is the minimum beam radius between mirrors.</p>
<table width=400 align=center><td>
<p align=justify>[[File:HeNe-fig2v2.jpg|400px|border|center]]
<br clear=right>
</p>
</td></table>

<p>Note that from ''w<sub>x</sub>'' above; at ''x'' = 0, ''w<sub>x</sub>'' = ''w<sub>o</sub>''.</p>
<p>From Preston (equation 21, p. 102),</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn7.png|160px|center]]
</p>
</td></table>
<p>Calculate ''w<sub>o</sub>'' and ''w<sub>x</sub>''.</p>
<p>From your observations of the beam profile for the TEM<sub>oom</sub>, determine ''w<sub>x</sub>'', the beam radius. How does your calculated value compare with the experimental value? Explain any differences.</p>

<h2>Malus's Law</h2>
<p>Malus’ law states that when a linearly polarized light beam of intensity ''I<sub>0</sub>'' passes through a linear polarizer with its axis rotated by angle ''A'' from the light beam polarization, the emergent intensity ''I'' is given by</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn8.png|90px|center]]
</p>
</td></table>
<p>Use the rotatable polarizer and photodiode detector to verify this law quantitatively. Make detector readings at several values of angle ''A'' and record them in a neat table in your notebook. Graph your data to demonstrate the expected cos<sup>2</sup>''A''dependence.</p>

<h2>Verification of the Fresnel-Arago Law</h2>
<p>Fresnel-Arago law state that two coherent light rays which are polarized right angles to each other will not mutually interfere. </p>
<p>Use the laser to set up the Michelson interferometer as shown below to form an interference pattern. </p>
<p>Insert polarizers P<sub>1</sub> and P<sub>2</sub> such that their axes of polarization is in the same direction. You may have to make slight adjustments to retain the interference pattern. Now, rotate ONE of the polarizers through 90º. Verify that the law is true. Try to take pictures of the resulting effect and include them with your report.</p>

<h1>References</h1>
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