Physics Wiki - User contributions [en]

Main Page/PHYS 4210/SonoluminescenceDev

2011-11-17T15:32:23Z

WorkStudy:

<h1>Sonoluminescence and Blackbody Radiation</h1>

<p>Sonoluminescence is the process by which a gas bubble trapped at the antinode of an ultrasonic standing wave emits visible radiation. This strange phenomenon will be the platform on which 3-dimensional standing waves and black-body radiation will be investigated.</p>

<h1>Introduction</h1>
<p> Single bubble sonoluminescence, hereafter abbreviated as SL, was discovered in the late 1980's and has received a great deal of attention. This remarkable process involves the trapping of gas bubble at the antinode of a ultrasonic standing wave in a liquid. </p>

<h3> 3D Standing Waves</h3>
<p>A plane wave in the z</hat> direction incident on a flat wall will reflect into the -z</hat> direction, and interfere with the incoming wave based on the superposition principle. at a particular distance from the wall,</p>

<p>In order for the trapping of gas bubbles, a 3-dimensional ultrasonic standing wave need to be attained. The resonant frequency of a the standing wave will occur if each of the 3 dimensions has a length which is equal to an integer number of half wavelengths of the acoustic wave. One will notice that the shape of the liquid cell is rectangular, having the length and width being equal. The height of the cell, and hence a value of great importance for this experiment, is set by the volume of water added to the cell. </p>

<h3> The Ultrasonic Horn</h3>
<p>The ultrasonic horn is used to deliver acoustic power to the volume of water. Internally, the horn contains a series of annular shaped disc transducers which are bolted into its base. The basic structure and shape of the horn is designed to efficiently couple the pressure waves generated from the transducers to the narrow stem of the horn. All of the transducers are the same. They consist of a ceramic material which has been prepared in such a manner as to have a permanent polarization. In other words,there are specialized capacitors. As a charged is placed across this capacitor there is a force generated across the ends, and the capacitor wants to separate. Since the transducers are compressed, this repulsive force does not physically expand the disc but does produce dynamic pressure. As the charge across the transducer is reversed, there is now an attractive force which results in a negative pressure amplitude. The capacitance of the particular tranducers used is 11 nF. In order to efficiently couple electrical power into the transducers, it is advantageous to connect an inductor in series to achieve electrical resonance as given by the formula:</p>

<h3> The Cell</h3>
<p> The cell is a plastic container onto which is epoxied a small ceramic transducer that serves as a microphone. Since this transducer is not compressed small fluctuations in its diameter produce a measurable signal. BY attaching this transducer to the bottom of the cell, one can easily detect when the pressure is the cell in in resonance by looking for large amplitude sine waves on a oscilloscope at the ultrasonic frequency. The cell is provided with a black shroud which serves which has a circular opening on one side to allow for light from the SL to enter the photomultiplier tube.</p>

<h3> The Photomultiplier Tube</h3>
<p> A photomultiplier tube (PMT) is a device which coverts one incident photon into a large pulse of electrons which can then be read out on an oscilloscope. The photon passes through the glass enclosure of the PMT an liberates an electron from a coated surface, this electron is then accelerated by static electric fields into dynode 1, and this initial electron free many more electrons. This bunch of electron then accelerates into dynode 2 freeing a larger bunch of electrons. This process continues many times such that even one incident photon can produce a measurable number of electrons at the output. '''A PMT is a very sensitive optical detector.''' Due to the high sensitivity, having too many photons entering the PMT can damage the device, and this should be avoided. Even with the black shroud covering the cell, you should not apply power to the PMT without also shutting of the room lights.</p>
<p>The Fluke 412B power supply will provide the high voltage required to bias the dynodes of the PMT. Typically, the PMT should be run at '''XXXX''' V. The PMT will output a negative going pulse with a sharp rise time and a relatively slow fall time. The amplitude of this pulse is related to the number of photons hitting the PMT during a particular pulse. In this experiment, the effect of Sonoluminescence only occurs once during an acoustic wave period, hence, the SL bubbles is flashing at around 27 kHz. '''The PMT will output pulses synchronized with the acoustic frequency whose amplitude is related to the brightness of the sonoluminescence.'''</p>

<p>The particular ultrasonic horn used has components which make it tunes for a frequency from 25kHz to 27kHz. Outside of those ranges, the acoustic energy available in the horn is
</p>

<h2>BlackBody Radiation</h2>

<h1> Procedure</h1>

<h2> Required Components</h2>
<ol>
<li>Flask</li>
<li>De-ionized Water</li>
<li>Vacuum Pump</li>
<li>[[Media:HP33258.JPG|HP33258 Function Generator]]</li>
<li>[[Media:TDS2014B.JPG|TDS2014B Digital Oscilloscope]]</li>
<li>[[Media:SL100B.JPG|SL100B Sonoluminescence Control Box]]</li>
<li>[[Media:SL100B2.JPG|Ultrasonic Horn]]</li>
<li>[[Media:SLPMT1P28.JPG|1P28 PhotoMultiplier Tube]]</li>
<li>[[Media:SLOpticalLenses.JPG|Various Optical Lenses]]</li>
<li>[[Media:SLPowerSupply.JPG|HV Power Supply]]</li>
</ol>

<h2> Degas the water</h2>
<p> In order for the water to support a sonoluminescence bubble it is necessary for the water to be partially degassed. An effective way to do this is to starting will deionized water in a flask, place a rubber stopper in the top of the flask with a tube passing though it, then attach a roughing pump to end of the tube. The boiling point of water is reduced to below room temperature if the pressure in the flask can be reduced below about 25 torr. The vacuum pump you are using can easily achieve that. You will notice rapid bubbling of the water inside the flask after the pump is turned on. Allow to pump to work for 30 minutes, gently shaking the flask while pumping will help ensure more thorough degassing. In addition, during the degassing procedure, have the flask sitting in a bath of ice water, as the sonoluminescence effect is greater when the liquid is at lower temperatures.<ref> G.E. Vazquez and S.J. Putterman, [http://prl.aps.org/abstract/PRL/v85/i14/p3037_1 Phys. Rev. Lett., '''85''', 3037 (1999)]</ref> </p>

<table width=500 align=center><td>
<p align=justify>[[File:SL_fig_degas.png|500px|border|center]]
<b>Figure XX -</b> Degassing the water.
<br clear=right>
</p>
</td></table>

<p>Once the degassing procedure is complete, turn off the vacuum pump, and slowly open the pressure relief wave. Then, ''gently'' pour the water into the cell being careful not to introduce more bubbles. Fill the cell to the correct amount to create standing waves of the desired order.</p>

<h2>Electrical Connections</h2>
<p> The basic setup for the experiment is shown below. The acoustic frequency is provided by a 1 Vpp sine wave from the HP3325A synthesizer/function generator. The appropriate resonance frequency, as determined by cell geometry is around 27kHz.
This sine wave is then amplifier by the Sonolumiscence control box, and fed into the ultrasonic horn. To electronically view the effects of sonoluminescence, the output of cell transducer is put to one input of the oscilloscope (you could trigger off of this), and the second input should be the High Freq. Output. This High Freq. Ouput filters the cell transducer signal with 150 kHz high-pass filter since the effect of a trapped bubble is to cause a high frequency response on the cell transducer. There are two connections made from the Sonoluminescence SL100B to the cell box- a multi-pin power/control cable, and cell tranducer input using coaxial connector.<p>

<p>When using the Photomulitplier
</p>

<h2> Observe trapped bubbles and Sonoluminescence</h2>
<p> Once the water is added, place the black shroud around the cell. Lower the ultrasonic horn into the cell such that the tip of the horn is 5mm - 10mm below the surface of the water. </p>

<h2> Detect the Sonoluminescence using a PMT </h2>
<p>With the HV power supply still switched off, adjust the height of the PMT and bring it close to the opening in the black shroud. Once you have a trapped, sonoluminescence bubble, close the front viewing flap. Turn on a small desk lamp, and shut off the main room light. Turn on the HV power supply to about 800V and look on the oscilloscope for pulses synchronous with the acoustic radiation. Sketch the result. Determine the rise time and decay time of the PMT pulses. Characterize the peak height. </p>

<h2>Determine the Temperature of the SL </h2>

<p> Using the varoius
</p>

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

Main Page/PHYS 4210/SonoluminescenceDev

2011-11-17T15:32:04Z

WorkStudy:

<h1>Sonoluminescence and Blackbody Radiation</h1>

<p>Sonoluminescence is the process by which a gas bubble trapped at the antinode of an ultrasonic standing wave emits visible radiation. This strange phenomenon will be the platform on which 3-dimensional standing waves and black-body radiation will be investigated.</p>

<h1>Introduction</h1>
<p> Single bubble sonoluminescence, hereafter abbreviated as SL, was discovered in the late 1980's and has received a great deal of attention. This remarkable process involves the trapping of gas bubble at the antinode of a ultrasonic standing wave in a liquid. </p>

<h3> 3D Standing Waves</h3>
<p>A plane wave in the z</hat> direction incident on a flat wall will reflect into the -z</hat> direction, and interfere with the incoming wave based on the superposition principle. at a particular distance from the wall,</p>

<p>In order for the trapping of gas bubbles, a 3-dimensional ultrasonic standing wave need to be attained. The resonant frequency of a the standing wave will occur if each of the 3 dimensions has a length which is equal to an integer number of half wavelengths of the acoustic wave. One will notice that the shape of the liquid cell is rectangular, having the length and width being equal. The height of the cell, and hence a value of great importance for this experiment, is set by the volume of water added to the cell. </p>

<h3> The Ultrasonic Horn</h3>
<p>The ultrasonic horn is used to deliver acoustic power to the volume of water. Internally, the horn contains a series of annular shaped disc transducers which are bolted into its base. The basic structure and shape of the horn is designed to efficiently couple the pressure waves generated from the transducers to the narrow stem of the horn. All of the transducers are the same. They consist of a ceramic material which has been prepared in such a manner as to have a permanent polarization. In other words,there are specialized capacitors. As a charged is placed across this capacitor there is a force generated across the ends, and the capacitor wants to separate. Since the transducers are compressed, this repulsive force does not physically expand the disc but does produce dynamic pressure. As the charge across the transducer is reversed, there is now an attractive force which results in a negative pressure amplitude. The capacitance of the particular tranducers used is 11 nF. In order to efficiently couple electrical power into the transducers, it is advantageous to connect an inductor in series to achieve electrical resonance as given by the formula:</p>

<h3> The Cell</h3>
<p> The cell is a plastic container onto which is epoxied a small ceramic transducer that serves as a microphone. Since this transducer is not compressed small fluctuations in its diameter produce a measurable signal. BY attaching this transducer to the bottom of the cell, one can easily detect when the pressure is the cell in in resonance by looking for large amplitude sine waves on a oscilloscope at the ultrasonic frequency. The cell is provided with a black shroud which serves which has a circular opening on one side to allow for light from the SL to enter the photomultiplier tube.</p>

<h3> The Photomultiplier Tube</h3>
<p> A photomultiplier tube (PMT) is a device which coverts one incident photon into a large pulse of electrons which can then be read out on an oscilloscope. The photon passes through the glass enclosure of the PMT an liberates an electron from a coated surface, this electron is then accelerated by static electric fields into dynode 1, and this initial electron free many more electrons. This bunch of electron then accelerates into dynode 2 freeing a larger bunch of electrons. This process continues many times such that even one incident photon can produce a measurable number of electrons at the output. '''A PMT is a very sensitive optical detector.''' Due to the high sensitivity, having too many photons entering the PMT can damage the device, and this should be avoided. Even with the black shroud covering the cell, you should not apply power to the PMT without also shutting of the room lights.</p>
<p>The Fluke 412B power supply will provide the high voltage required to bias the dynodes of the PMT. Typically, the PMT should be run at '''XXXX''' V. The PMT will output a negative going pulse with a sharp rise time and a relatively slow fall time. The amplitude of this pulse is related to the number of photons hitting the PMT during a particular pulse. In this experiment, the effect of Sonoluminescence only occurs once during an acoustic wave period, hence, the SL bubbles is flashing at around 27 kHz. '''The PMT will output pulses synchronized with the acoustic frequency whose amplitude is related to the brightness of the sonoluminescence.'''</p>

<p>The particular ultrasonic horn used has components which make it tunes for a frequency from 25kHz to 27kHz. Outside of those ranges, the acoustic energy available in the horn is
</p>

<h2>BlackBody Radiation</h2>

<h1> Procedure</h1>

<h2> Required Components</h2>
<ol>
Flask</li>
<li>De-ionized Water</li>
<li>Vacuum Pump</li>
<li>[[Media:HP33258.JPG|HP33258 Function Generator]]</li>
<li>[[Media:TDS2014B.JPG|TDS2014B Digital Oscilloscope]]</li>
<li>[[Media:SL100B.JPG|SL100B Sonoluminescence Control Box]]</li>
<li>[[Media:SL100B2.JPG|Ultrasonic Horn]]</li>
<li>[[Media:SLPMT1P28.JPG|1P28 PhotoMultiplier Tube]]</li>
<li>[[Media:SLOpticalLenses.JPG|Various Optical Lenses]]</li>
<li>[[Media:SLPowerSupply.JPG|HV Power Supply]]</li>
</ol>

<h2> Degas the water</h2>
<p> In order for the water to support a sonoluminescence bubble it is necessary for the water to be partially degassed. An effective way to do this is to starting will deionized water in a flask, place a rubber stopper in the top of the flask with a tube passing though it, then attach a roughing pump to end of the tube. The boiling point of water is reduced to below room temperature if the pressure in the flask can be reduced below about 25 torr. The vacuum pump you are using can easily achieve that. You will notice rapid bubbling of the water inside the flask after the pump is turned on. Allow to pump to work for 30 minutes, gently shaking the flask while pumping will help ensure more thorough degassing. In addition, during the degassing procedure, have the flask sitting in a bath of ice water, as the sonoluminescence effect is greater when the liquid is at lower temperatures.<ref> G.E. Vazquez and S.J. Putterman, [http://prl.aps.org/abstract/PRL/v85/i14/p3037_1 Phys. Rev. Lett., '''85''', 3037 (1999)]</ref> </p>

<table width=500 align=center><td>
<p align=justify>[[File:SL_fig_degas.png|500px|border|center]]
<b>Figure XX -</b> Degassing the water.
<br clear=right>
</p>
</td></table>

<p>Once the degassing procedure is complete, turn off the vacuum pump, and slowly open the pressure relief wave. Then, ''gently'' pour the water into the cell being careful not to introduce more bubbles. Fill the cell to the correct amount to create standing waves of the desired order.</p>

<h2>Electrical Connections</h2>
<p> The basic setup for the experiment is shown below. The acoustic frequency is provided by a 1 Vpp sine wave from the HP3325A synthesizer/function generator. The appropriate resonance frequency, as determined by cell geometry is around 27kHz.
This sine wave is then amplifier by the Sonolumiscence control box, and fed into the ultrasonic horn. To electronically view the effects of sonoluminescence, the output of cell transducer is put to one input of the oscilloscope (you could trigger off of this), and the second input should be the High Freq. Output. This High Freq. Ouput filters the cell transducer signal with 150 kHz high-pass filter since the effect of a trapped bubble is to cause a high frequency response on the cell transducer. There are two connections made from the Sonoluminescence SL100B to the cell box- a multi-pin power/control cable, and cell tranducer input using coaxial connector.<p>

<p>When using the Photomulitplier
</p>

<h2> Observe trapped bubbles and Sonoluminescence</h2>
<p> Once the water is added, place the black shroud around the cell. Lower the ultrasonic horn into the cell such that the tip of the horn is 5mm - 10mm below the surface of the water. </p>

<h2> Detect the Sonoluminescence using a PMT </h2>
<p>With the HV power supply still switched off, adjust the height of the PMT and bring it close to the opening in the black shroud. Once you have a trapped, sonoluminescence bubble, close the front viewing flap. Turn on a small desk lamp, and shut off the main room light. Turn on the HV power supply to about 800V and look on the oscilloscope for pulses synchronous with the acoustic radiation. Sketch the result. Determine the rise time and decay time of the PMT pulses. Characterize the peak height. </p>

<h2>Determine the Temperature of the SL </h2>

<p> Using the varoius
</p>

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

Main Page/PHYS 4210/SonoluminescenceDev

2011-11-17T15:31:33Z

WorkStudy:

<h1>Sonoluminescence and Blackbody Radiation</h1>

<p>Sonoluminescence is the process by which a gas bubble trapped at the antinode of an ultrasonic standing wave emits visible radiation. This strange phenomenon will be the platform on which 3-dimensional standing waves and black-body radiation will be investigated.</p>

<h1>Introduction</h1>
<p> Single bubble sonoluminescence, hereafter abbreviated as SL, was discovered in the late 1980's and has received a great deal of attention. This remarkable process involves the trapping of gas bubble at the antinode of a ultrasonic standing wave in a liquid. </p>

<h3> 3D Standing Waves</h3>
<p>A plane wave in the z</hat> direction incident on a flat wall will reflect into the -z</hat> direction, and interfere with the incoming wave based on the superposition principle. at a particular distance from the wall,</p>

<p>In order for the trapping of gas bubbles, a 3-dimensional ultrasonic standing wave need to be attained. The resonant frequency of a the standing wave will occur if each of the 3 dimensions has a length which is equal to an integer number of half wavelengths of the acoustic wave. One will notice that the shape of the liquid cell is rectangular, having the length and width being equal. The height of the cell, and hence a value of great importance for this experiment, is set by the volume of water added to the cell. </p>

<h3> The Ultrasonic Horn</h3>
<p>The ultrasonic horn is used to deliver acoustic power to the volume of water. Internally, the horn contains a series of annular shaped disc transducers which are bolted into its base. The basic structure and shape of the horn is designed to efficiently couple the pressure waves generated from the transducers to the narrow stem of the horn. All of the transducers are the same. They consist of a ceramic material which has been prepared in such a manner as to have a permanent polarization. In other words,there are specialized capacitors. As a charged is placed across this capacitor there is a force generated across the ends, and the capacitor wants to separate. Since the transducers are compressed, this repulsive force does not physically expand the disc but does produce dynamic pressure. As the charge across the transducer is reversed, there is now an attractive force which results in a negative pressure amplitude. The capacitance of the particular tranducers used is 11 nF. In order to efficiently couple electrical power into the transducers, it is advantageous to connect an inductor in series to achieve electrical resonance as given by the formula:</p>

<h3> The Cell</h3>
<p> The cell is a plastic container onto which is epoxied a small ceramic transducer that serves as a microphone. Since this transducer is not compressed small fluctuations in its diameter produce a measurable signal. BY attaching this transducer to the bottom of the cell, one can easily detect when the pressure is the cell in in resonance by looking for large amplitude sine waves on a oscilloscope at the ultrasonic frequency. The cell is provided with a black shroud which serves which has a circular opening on one side to allow for light from the SL to enter the photomultiplier tube.</p>

<h3> The Photomultiplier Tube</h3>
<p> A photomultiplier tube (PMT) is a device which coverts one incident photon into a large pulse of electrons which can then be read out on an oscilloscope. The photon passes through the glass enclosure of the PMT an liberates an electron from a coated surface, this electron is then accelerated by static electric fields into dynode 1, and this initial electron free many more electrons. This bunch of electron then accelerates into dynode 2 freeing a larger bunch of electrons. This process continues many times such that even one incident photon can produce a measurable number of electrons at the output. '''A PMT is a very sensitive optical detector.''' Due to the high sensitivity, having too many photons entering the PMT can damage the device, and this should be avoided. Even with the black shroud covering the cell, you should not apply power to the PMT without also shutting of the room lights.</p>
<p>The Fluke 412B power supply will provide the high voltage required to bias the dynodes of the PMT. Typically, the PMT should be run at '''XXXX''' V. The PMT will output a negative going pulse with a sharp rise time and a relatively slow fall time. The amplitude of this pulse is related to the number of photons hitting the PMT during a particular pulse. In this experiment, the effect of Sonoluminescence only occurs once during an acoustic wave period, hence, the SL bubbles is flashing at around 27 kHz. '''The PMT will output pulses synchronized with the acoustic frequency whose amplitude is related to the brightness of the sonoluminescence.'''</p>

<p>The particular ultrasonic horn used has components which make it tunes for a frequency from 25kHz to 27kHz. Outside of those ranges, the acoustic energy available in the horn is
</p>

<h2>BlackBody Radiation</h2>

<h1> Procedure</h1>

<h2> Required Components</h2>
<ol>
<il>Flask</li>
<li>De-ionized Water</li>
<li>Vacuum Pump</li>
<li>[[Media:HP33258.JPG|HP33258 Function Generator]]</li>
<li>[[Media:TDS2014B.JPG|TDS2014B Digital Oscilloscope]]</li>
<li>[[Media:SL100B.JPG|SL100B Sonoluminescence Control Box]]</li>
<li>[[Media:SL100B2.JPG|Ultrasonic Horn]]</li>
<li>[[Media:SLPMT1P28.JPG|1P28 PhotoMultiplier Tube]]</li>
<li>[[Media:SLOpticalLenses.JPG|Various Optical Lenses]]</li>
<li>[[Media:SLPowerSupply.JPG|HV Power Supply]]</li>
</ol>

<h2> Degas the water</h2>
<p> In order for the water to support a sonoluminescence bubble it is necessary for the water to be partially degassed. An effective way to do this is to starting will deionized water in a flask, place a rubber stopper in the top of the flask with a tube passing though it, then attach a roughing pump to end of the tube. The boiling point of water is reduced to below room temperature if the pressure in the flask can be reduced below about 25 torr. The vacuum pump you are using can easily achieve that. You will notice rapid bubbling of the water inside the flask after the pump is turned on. Allow to pump to work for 30 minutes, gently shaking the flask while pumping will help ensure more thorough degassing. In addition, during the degassing procedure, have the flask sitting in a bath of ice water, as the sonoluminescence effect is greater when the liquid is at lower temperatures.<ref> G.E. Vazquez and S.J. Putterman, [http://prl.aps.org/abstract/PRL/v85/i14/p3037_1 Phys. Rev. Lett., '''85''', 3037 (1999)]</ref> </p>

<table width=500 align=center><td>
<p align=justify>[[File:SL_fig_degas.png|500px|border|center]]
<b>Figure XX -</b> Degassing the water.
<br clear=right>
</p>
</td></table>

<p>Once the degassing procedure is complete, turn off the vacuum pump, and slowly open the pressure relief wave. Then, ''gently'' pour the water into the cell being careful not to introduce more bubbles. Fill the cell to the correct amount to create standing waves of the desired order.</p>

<h2>Electrical Connections</h2>
<p> The basic setup for the experiment is shown below. The acoustic frequency is provided by a 1 Vpp sine wave from the HP3325A synthesizer/function generator. The appropriate resonance frequency, as determined by cell geometry is around 27kHz.
This sine wave is then amplifier by the Sonolumiscence control box, and fed into the ultrasonic horn. To electronically view the effects of sonoluminescence, the output of cell transducer is put to one input of the oscilloscope (you could trigger off of this), and the second input should be the High Freq. Output. This High Freq. Ouput filters the cell transducer signal with 150 kHz high-pass filter since the effect of a trapped bubble is to cause a high frequency response on the cell transducer. There are two connections made from the Sonoluminescence SL100B to the cell box- a multi-pin power/control cable, and cell tranducer input using coaxial connector.<p>

<p>When using the Photomulitplier
</p>

<h2> Observe trapped bubbles and Sonoluminescence</h2>
<p> Once the water is added, place the black shroud around the cell. Lower the ultrasonic horn into the cell such that the tip of the horn is 5mm - 10mm below the surface of the water. </p>

<h2> Detect the Sonoluminescence using a PMT </h2>
<p>With the HV power supply still switched off, adjust the height of the PMT and bring it close to the opening in the black shroud. Once you have a trapped, sonoluminescence bubble, close the front viewing flap. Turn on a small desk lamp, and shut off the main room light. Turn on the HV power supply to about 800V and look on the oscilloscope for pulses synchronous with the acoustic radiation. Sketch the result. Determine the rise time and decay time of the PMT pulses. Characterize the peak height. </p>

<h2>Determine the Temperature of the SL </h2>

<p> Using the varoius
</p>

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

File:TDS2014B.JPG

2011-11-17T15:21:42Z

WorkStudy:

File:SLSetUp.JPG

2011-11-17T15:21:30Z

WorkStudy:

File:SLPowerSupply.JPG

2011-11-17T15:21:22Z

WorkStudy:

File:SLPMT1P28.JPG

2011-11-17T15:21:14Z

WorkStudy:

File:SLOpticalLenses.JPG

2011-11-17T15:21:03Z

WorkStudy:

File:SL100B2.JPG

2011-11-17T15:20:52Z

WorkStudy:

File:SL100B.JPG

2011-11-17T15:20:42Z

WorkStudy:

File:HP33258.JPG

2011-11-17T15:20:32Z

WorkStudy:

Main Page/PHYS 4210/Zeeman Effect

2011-11-01T15:34:24Z

WorkStudy:

<h1>Zeeman Effect</h1>
<p>In this classic experiment that predates the development of quantum mechanics one investigates the light emitted by atoms in the presence of a homogeneous magnetic field. Of particular interest is the observation that this light is polarized in the presence of a magnetic field. The high-resolution spectroscopy required to resolve the line splittings is performed with a multiple-beam interferometer called a Lummer-Gehrcke plate which is similar to a Fabry-Perot interferometer.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Magnetic Sublevels</li>
<li>Total Angular Momentum</li>
<li>TEM wave</li>
<li>Quantization Axis</li>
<li>Orbital Angular Momentum</li>
<li>Spin Angular Momentum</li>
</ul>
</td>

<td width=250>
<ul>
<li>Normal Zeeman Effect</li>
<li>Anomalous Zeeman Effect</li>
<li>Lummer-Gehrcke Plate</li>
<li>Multi-beam Interferometer</li>
<li>Quarter-wave Plate</li>
</ul>
</td>
</table>

<h2> Required Components</h2>

<ol>
<li>[[Media:ZEMagnetPower.JPG|Magnet Power Supply]]</li>
<li>[[Media:ZEElectromagnet.JPG|Electromagnet]]</li>
<li>[[Media:ZEDischargePower.JPG|Discharge Power Supply]]</li>
<li>[[Media:ZECCDCamera.JPG|CCD Camera]]</li>
<li>[[Media:ZELummer.JPG|Lummer-Gehrcke Plate]]</li>
<li>[[Media:ZEPolarizers.JPG|Polarizers and Waveplate]]</li>
</ol>

<h1>Introduction</h1>
<p>The Zeeman effect is a powerful demonstration of the splittings of magnetic sublevels in an angular momentum multiplet. Many aspects of the emission of light by excited atoms, particularly when exposed to strong magnetic fields ('''B''') were understood by Lorentz in a classical model [1,2] before the advent of quantum mechanics. It is possible to understand the changes to classical electron orbits due to the Lorentz force in a 3D harmonic oscillator model. When one complements this with the idea that electromagnetic waves are transverse (the associated electric and magnetic fields of the EM wave oscillate in a plane perpendicular to the propagation direction of the wave), one can understand why circularly polarized light emerges as the atoms are observed in a direction longitudinal with the external '''B''' field, and why they appear as plane-polarized as viewed from the transverse direction. The understanding in the classical framework helps to build an intuition about the problem.</p>
<p>In the modern quantum mechanical description [2,5,6] one has to take into account that the presence of the '''B''' field singles out an axis. The additional interaction term between the magnetic moment of the electronic state (proportional to the ''z'' component of the total angular momentum) and '''B''' serves to split the magnetic sublevels of states with non-zero angular momentum. The additional interaction forces the use of this axis as a quantization axis. Without an external field one usually picks a ''z'' axis, but should arrive at results that are independent of this choice. To obtain the observed result that the light emanating from spontaneous transitions without an external field is unpolarized, one has to average over random orientations of the quantization axis. The observation that a definite orientation is singled out as quantization axis in the Zeeman effect is sometimes referred to as ‘space quantization’.</p>
<p>The problem can be illustrated by using pure orbital angular momentum states, i.e.,ignoring spin, and considering an np - ms transition. This transition is an allowed electric dipole transition, since a single unit of orbital angular momentum is changed, and this difference of one unit is carried away in the form of the spin for the spontaneously radiated photon. The important quantity to watch is the change in the projection of the orbital angular momentum, which can be +1, 0, -1 depending on the choice of the magnetic sublevel.</p>
<p>A transition 2p0 - 1s is associated with the emission of linearly polarized light with the oscillating electric field vector aligned with the z axis, with the wave propagation vector being orthogonal to this axis. This can be understood from the fact that the only non-vanishing matrix element for the dipole operator is <1s|''z''|2p<sub>0</sub>>. Similar calculations show that the transitions originating in the ''m'' = 1 and ''m'' = -1 sublevels result in circularly polarized light being emitted, which can propagate in the z direction only. One of the fascinating aspects of the Zeeman experiment is the following. For field-free atoms no axis is singled out, and thus, one has to include all possible orientations of the ''z'' axis, which results in the prediction that the light emitted from free atoms is unpolarized. However, once a homogeneous magnetic field is applied, an axis is singled out in space, which becomes the natural quantization axis. By probing the polarization of the spontaneously emitted light of atoms in the presence of a magnetic field one can verify that indeed the turn-on of the field causes a repopulation of the magnetic sublevels in a way that corresponds to the classical predictions of the Lorentz model. Thus, it is necessary to observe the light emitted longitudinally and transverse to the magnetic field.</p>
<p>The general theory of the Zeeman effect is complicated by the fact that the total angular momentum, i.e., added orbital and spin angular momentum of the active electron has to be considered. Based on orbital angular momentum alone the magnetic moment of an electron in a non-zero m sublevel is an integer multiple of the projection ''l<sub>z</sub>''. Once one couples ''l'' and ''s'' to form ''j'' = ''l'' + ''s'', the magnetic moment can be, but need not to be an integer multiple of ''j<sub>z</sub>'', and the proportionality is given by the Lande factor ''g''. The Lande factor can take on half-integer numbers for the initial and/or final states involved in the transition. One distinguishes between the normal and anomalous Zeeman effects depending on whether this complication arises or not. [3]. The anomalous effect is rather common in atomic transitions, but in this experiment a transition with the normal Zeeman effect has been selected. The red line in cadmium (643.8 nm), which is the equivalent of the yellow line in mercury, cf.. the Grotrian diagram shown in the appendix, and the level diagram in Fig. 1 (which is Fig. 7.3 from Melissinos [3]).
</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Level diagram for the Cd 643.8 nm transition with and without B field.
<br clear=right>
</p>
</td></table>
<p>From an experimental point of view a high demand is placed on the optical resolution of the interferometer. The idea is to inspect the interference pattern for a given line and to observe the quantitative changes in the pattern as the B field is applied to determine the wavelengths of the various components. The high resolution required can be obtained from multiple-beam interferometers, such as the Fabry-Perot (FP) interferometer [1,3]. Melissinos [3] discusses the analysis of the circular fringe pattern as produced by the FP. An easier alternative is provided by a special instrument that perfects the same method, called a Lummer-Gehrcke (LG) plate [1]. Since its interference pattern is more complicated to derive, you should concentrate on understanding the principles of multiple-beam interferometry using the FP and be aware of the analogies. Note that the FP has a wide range of applications in optics.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2a.png|500px|border|center]]
<b>Figure 2a -</b> Multiple reflection between the surfaces of a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2b.png|500px|border|center]]
<b>Figure 2b -</b> Apparatus for the Zeeman experiment with a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>

<table width=700 align=center><td>
<p align=justify>[[File:Zee_fig2c.png|700px|border|center]]
<b>Figure 2c -</b> Apparatus for the Zeeman experiment.
<br clear=right>
</p>
</td></table>

<p>The LG plate shown as part of the apparatus in Fig. 2 consists of a precisely polished quartz glass plate of given thickness d with a prism attached at one end so that light entering from the slit has an angle of incidence on the plate that is near the critical angle. This results in some refractive transmission and mostly reflection at the glass/air surface. The reflected light inside the glass plate undergoes multiple ‘bounces’ of this type (interior reflection and partial refractive transmission). Two different interference patterns emerge when looking at a grazing angle at the top or bottom of the LG plate. The pattern formed at the top shows sharp bright lines on a dark background. In contrast to a Michelson interferometer a multiple-beam interferometer such as the FP and LG can produce an uneven interference pattern [1,3].</p>
<p>The separation between the fringes that appear without the magnetic field depends on the angle of observation. This spacing ''ΔA'' has to be determined for the particular fringe chosen for observation. As a magnetic field is applied each bright fringe splits either into two or into three depending on the orientation with respect to the magnetic field. To obtain a quantitative measure of the Zeeman effect, one needs to determine the '''B''' dependent splitting ''ΔS'' relative to ''ΔA''. Making use of the ratio eliminates the need to know the optical magnification, observation angle and distance from the plate. The frequency splitting depends also on the LG plate thickness d (as in the FP case), and additionally on the index of refraction η of the quartz glass. In the FP case this would be equal to 1, but there are versions of the experiment where the evacuation of a sealed FP interferometer is used to produce a scanning effect in the fringe pattern [4]. The frequency splitting can be written as</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn1.png|160px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>The corresponding energy difference should equal</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn2.png|220px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
<p>One common method of determining the electron charge-to-mass ratio is through eq (2). Our interest is, however, to determine the energy splitting as a function of the magnetic field strength ''B''.</p>
<p>In our version of the Zeeman experiment the intense red Cd line at 643.8 nm is used. For optical transitions Cd acts as an effective two-electron system, i.e., it has a He-like configuration, as has Hg. Compare the Grotrian diagram shown in Fig. 1 to the one for mercury provided in the appendix (cf.. the Franck-Hertz experiment). In order to understand the selection rules for allowed electric dipole transitions follow the arguments given in ref. 4 in the context of the HeNe laser experiment. The two active electrons have combined orbital angular momentum ''L'' = l<sub>1</sub>+l<sub>2</sub>, spin S = s<sub>1</sub>+s<sub>2</sub>, and total angular momentum J = j<sub>1</sub>+j<sub>2</sub>. Allowed transitions require a change of one unit in ''L'' and ''J'' to make up for the spin of the photon, considering that spin flip is unlikely. Using <sup>(2''S''+1)</sup>''L<sub>J</sub>'' notation (with ''L'' = 0 denoted as S, ''L'' = 1 as P, ''L'' = 2 as D, etc) we have for the relevant line a <sup>1</sup>D<sub>2</sub> - <sup>1</sup>P<sub>1</sub> transition. This means that the spins are paired up (spin singlet) and that in nonrelativistic notation a 5s5d to 5s5p transition takes place. In Hg the equivalent line at the n = 6 level is the yellow line at 579 nm. The shift of the same line towards yellow is the result of having an additional electron shell in the core. What wavelength is associated with this transition in He?</p>
<p>For the advanced student: note that the initial level splits into 5 sublevels ''M<sub>J</sub>'' = -2,-1,0,1,2 , while the final state has ''M<sub>J</sub>'' = -1,0,1. They split equidistantly and one can group the nine possible transitions according to the allowed ''ΔM'' = -1,0,1 (cf.. Ref. 3 Fig. 7.3). To understand the polarization of the emitted light in the presence of the external B field note that for an electromagnetic wave its electric, magnetic fields and the wave propagation vector '''k''' form a right-handed coordinate system. Understand the validity of the dipole approximation (the wavelength λ is much longer than atomic dimensions) and how the electric field of the EM wave can be replaced by a constant vector times a temporary oscillatory factor (ref. [3,6]). Convince yourself why no linearly polarized light can be observed in the longitudinal direction as the magnetic field is turned on. Correspondingly understand why circularly polarized light as observed in the longitudinal direction must appear as linearly polarized when observed in a direction transverse to the ''B'' field. Why can all three components associated with the ''ΔM'' = -1,0,1 selection rule be observed in the transverse direction?</p>
<p>To verify these predictions about the polarization states of the light when the B field is turned on you need to recall some optical properties of polarizers and of quarter-wave plates (cf.. Ref.1). By placing these in the correct order you can verify the following:</p>
<p>Using a polarizer when observing in the transverse direction identify the polarization states of the three components with '''B''' turned on; what happens for ''B'' = 0?</p>
<p>Using a quarter-wave plate convert the circularly polarized (CP) light to two perpendicular
linearly polarized components corresponding to left- and right CP light respectively. Use a polarizer to extinguish each of these components separately; what happens for ''B'' = 0?</p>
<p>Collect sufficient data for both observation directions to demonstrate the linearity of the line
splitting with the magnetic field. You will need to perform a calibration of the magnetic field as a function of the current and should comment on possible saturation effects, i.e., a linear behaviour of the splitting with ''B'', but non-linear with the magnet current ''I'' at strong fields, where part of the electric energy may be converted to heat. Determine the gyromagnetic ratio (''g'') from your observations. </p>
<p>'''Warning:''' The Cd lamp emits ultraviolet light in addition to other lines such as the red Cd line. The apparatus contains a narrow-band red filter so that your eyes are protected when observing through the telescope. Avoid looking into the lamp itself (even though it is rated to be safe), i.e., cover the apparatus with a sheet of paper to reduce unnecessary eye contact with the lamp. The lamp takes several minutes to reach a proper operating temperature for the red line to be visible.</p>
<p><b>Obtain assistance when changing the observation direction from transverse to longitudinal or vice versa!</b></p>

<h3> Using the CCD Camera</h3>
<p>A CCD camera (AmScope MU300) is provided to allow for easy viewing of the lines. To operate the CCD, run the program "ToupView", then select ''Acquire''-> ''Live Capture''-> ''UCMOS03100KPA'' from the menu. A window will appear which is the live image being collected by the CCD. The brightness of the image can be changed using the ''Setup'' -> ''View Souce Properties'' -> ''Expose'' tab. A reasonable choice of values is an ''Exposure'' of 700ms with an ''Analog Gain'' of 3. Correct adjustment of the support system will allow you to clearly view the lines. Sliding of the CCD camera in and out will allow for focus. You can Save the image using ''Capture'' -> ''Capture a Frame'' command.</p>
<table width=500 align=center><td>
<p align=justify>[[File:ToupView.png|500px|border|center]]
<b>Figure 3-</b> ToupView CCD Camera Interface.
<br clear=right>
</p>
</td></table>
<p>Once you are able to see nice clear lines as shown in Figure 3, use the ''Region of Interest'' tool to focus in on a few lines in center. You can now use the ''Zoom'' to expand the image. One the left margin of the video image, there is a scale showing the pixel number. You can use this as a fixed reference point- as you increase the applied magnetic field, the line will split into sublevels, check the dial gauge, and then use the adjusting screw to place the shifted line back to the pixel number of the original line, then check the new reading of the dial gauge and record the measurement.</p>

<h1>References</h1>
<ol>
<li>Jenkins F.A., White H.E., ''Fundamentals of Optics'', McGraw-Hill</li>
<li>Brehm J.J., Mullin W.J., ''Introduction to the Structure of Matter'', Wiley</li>
<li>Melissinos A.C., ''Experiments in Modern Physics'', Academic Press</li>
<li>Preston D.W. Dietz E.R., ''The Art of Experimental Physics'',Wiley</li>
<li>Merzbacher E., ''Quantum Mechanics'', Wiley</li>
<li>Bethe H.A., Salpeter E.E., ''Quantum Mechanics of One- and Two-Electron Systems'', Springer</li>
<li>Radzig A.A., Smirnov B.M., ''Reference Data on Atoms Molecules and Ions'', Springer 1985.</li>
</ol>

<h1>Appendix</h1>
<ul>
<li><b>Data for the Lummer-Gehrcke plate</b> [from Leybold’s manual]: ''d'' = 4.04 mm η = 1.4567 </li>
<li>Grotrian diagrams for Cd, Hg, and He taken from ref. 7. A copy of these are in the binder in the laboratory.
<ul>
<li>[[Media:GrotrianH.pdf| Hydrogen]]</li>
<li>[[Media:GrotrianHe.pdf| Helium]]</li>
<li>[[Media:GrotrianHg.pdf| Mercury]]</li>
<li>[[Media:GrotrianHgCd.pdf| Cadmium and Mercury]]</li>
</ul>
</li>
</ul>

Main Page/PHYS 4210/Zeeman Effect

2011-11-01T15:33:24Z

WorkStudy:

<h1>Zeeman Effect</h1>
<p>In this classic experiment that predates the development of quantum mechanics one investigates the light emitted by atoms in the presence of a homogeneous magnetic field. Of particular interest is the observation that this light is polarized in the presence of a magnetic field. The high-resolution spectroscopy required to resolve the line splittings is performed with a multiple-beam interferometer called a Lummer-Gehrcke plate which is similar to a Fabry-Perot interferometer.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Magnetic Sublevels</li>
<li>Total Angular Momentum</li>
<li>TEM wave</li>
<li>Quantization Axis</li>
<li>Orbital Angular Momentum</li>
<li>Spin Angular Momentum</li>
</ul>
</td>

<td width=250>
<ul>
<li>Normal Zeeman Effect</li>
<li>Anomalous Zeeman Effect</li>
<li>Lummer-Gehrcke Plate</li>
<li>Multi-beam Interferometer</li>
<li>Quarter-wave Plate</li>
</ul>
</td>
</table>

<h2> Required Components</h2>

<ol>
<li>[[Media:ZEMagnetPower.JPG|Magnet Power Supply]]</li>
<li>[[Media:ZEElectromagnet.JPG|Electromagnet]]</li>
<li>[[Media:ZEDischargePower.JPG|Discharge Power Supply]]</li>
<li>[[Media:ZECCDCamera.JPG|CCD Camera]]</li>
<li>[[Media:ZELummer.JPG|Lummer-Gehrcke Plate]]</li>
<li>[[Media:ZEPolarizers.JPG|Polarizers and Waveplate]]</li>
</ol>

<h1>Introduction</h1>
<p>The Zeeman effect is a powerful demonstration of the splittings of magnetic sublevels in an angular momentum multiplet. Many aspects of the emission of light by excited atoms, particularly when exposed to strong magnetic fields ('''B''') were understood by Lorentz in a classical model [1,2] before the advent of quantum mechanics. It is possible to understand the changes to classical electron orbits due to the Lorentz force in a 3D harmonic oscillator model. When one complements this with the idea that electromagnetic waves are transverse (the associated electric and magnetic fields of the EM wave oscillate in a plane perpendicular to the propagation direction of the wave), one can understand why circularly polarized light emerges as the atoms are observed in a direction longitudinal with the external '''B''' field, and why they appear as plane-polarized as viewed from the transverse direction. The understanding in the classical framework helps to build an intuition about the problem.</p>
<p>In the modern quantum mechanical description [2,5,6] one has to take into account that the presence of the '''B''' field singles out an axis. The additional interaction term between the magnetic moment of the electronic state (proportional to the ''z'' component of the total angular momentum) and '''B''' serves to split the magnetic sublevels of states with non-zero angular momentum. The additional interaction forces the use of this axis as a quantization axis. Without an external field one usually picks a ''z'' axis, but should arrive at results that are independent of this choice. To obtain the observed result that the light emanating from spontaneous transitions without an external field is unpolarized, one has to average over random orientations of the quantization axis. The observation that a definite orientation is singled out as quantization axis in the Zeeman effect is sometimes referred to as ‘space quantization’.</p>
<p>The problem can be illustrated by using pure orbital angular momentum states, i.e.,ignoring spin, and considering an np - ms transition. This transition is an allowed electric dipole transition, since a single unit of orbital angular momentum is changed, and this difference of one unit is carried away in the form of the spin for the spontaneously radiated photon. The important quantity to watch is the change in the projection of the orbital angular momentum, which can be +1, 0, -1 depending on the choice of the magnetic sublevel.</p>
<p>A transition 2p0 - 1s is associated with the emission of linearly polarized light with the oscillating electric field vector aligned with the z axis, with the wave propagation vector being orthogonal to this axis. This can be understood from the fact that the only non-vanishing matrix element for the dipole operator is <1s|''z''|2p<sub>0</sub>>. Similar calculations show that the transitions originating in the ''m'' = 1 and ''m'' = -1 sublevels result in circularly polarized light being emitted, which can propagate in the z direction only. One of the fascinating aspects of the Zeeman experiment is the following. For field-free atoms no axis is singled out, and thus, one has to include all possible orientations of the ''z'' axis, which results in the prediction that the light emitted from free atoms is unpolarized. However, once a homogeneous magnetic field is applied, an axis is singled out in space, which becomes the natural quantization axis. By probing the polarization of the spontaneously emitted light of atoms in the presence of a magnetic field one can verify that indeed the turn-on of the field causes a repopulation of the magnetic sublevels in a way that corresponds to the classical predictions of the Lorentz model. Thus, it is necessary to observe the light emitted longitudinally and transverse to the magnetic field.</p>
<p>The general theory of the Zeeman effect is complicated by the fact that the total angular momentum, i.e., added orbital and spin angular momentum of the active electron has to be considered. Based on orbital angular momentum alone the magnetic moment of an electron in a non-zero m sublevel is an integer multiple of the projection ''l<sub>z</sub>''. Once one couples ''l'' and ''s'' to form ''j'' = ''l'' + ''s'', the magnetic moment can be, but need not to be an integer multiple of ''j<sub>z</sub>'', and the proportionality is given by the Lande factor ''g''. The Lande factor can take on half-integer numbers for the initial and/or final states involved in the transition. One distinguishes between the normal and anomalous Zeeman effects depending on whether this complication arises or not. [3]. The anomalous effect is rather common in atomic transitions, but in this experiment a transition with the normal Zeeman effect has been selected. The red line in cadmium (643.8 nm), which is the equivalent of the yellow line in mercury, cf.. the Grotrian diagram shown in the appendix, and the level diagram in Fig. 1 (which is Fig. 7.3 from Melissinos [3]).
</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Level diagram for the Cd 643.8 nm transition with and without B field.
<br clear=right>
</p>
</td></table>
<p>From an experimental point of view a high demand is placed on the optical resolution of the interferometer. The idea is to inspect the interference pattern for a given line and to observe the quantitative changes in the pattern as the B field is applied to determine the wavelengths of the various components. The high resolution required can be obtained from multiple-beam interferometers, such as the Fabry-Perot (FP) interferometer [1,3]. Melissinos [3] discusses the analysis of the circular fringe pattern as produced by the FP. An easier alternative is provided by a special instrument that perfects the same method, called a Lummer-Gehrcke (LG) plate [1]. Since its interference pattern is more complicated to derive, you should concentrate on understanding the principles of multiple-beam interferometry using the FP and be aware of the analogies. Note that the FP has a wide range of applications in optics.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2a.png|500px|border|center]]
<b>Figure 2a -</b> Multiple reflection between the surfaces of a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2b.png|500px|border|center]]
<b>Figure 2b -</b> Apparatus for the Zeeman experiment with a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>

<table width=500 align=center><td>
<p align=justify>[[File:Zee_fig2c.png|500px|border|center]]
<b>Figure 2c -</b> Apparatus for the Zeeman experiment with a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>

<p>The LG plate shown as part of the apparatus in Fig. 2 consists of a precisely polished quartz glass plate of given thickness d with a prism attached at one end so that light entering from the slit has an angle of incidence on the plate that is near the critical angle. This results in some refractive transmission and mostly reflection at the glass/air surface. The reflected light inside the glass plate undergoes multiple ‘bounces’ of this type (interior reflection and partial refractive transmission). Two different interference patterns emerge when looking at a grazing angle at the top or bottom of the LG plate. The pattern formed at the top shows sharp bright lines on a dark background. In contrast to a Michelson interferometer a multiple-beam interferometer such as the FP and LG can produce an uneven interference pattern [1,3].</p>
<p>The separation between the fringes that appear without the magnetic field depends on the angle of observation. This spacing ''ΔA'' has to be determined for the particular fringe chosen for observation. As a magnetic field is applied each bright fringe splits either into two or into three depending on the orientation with respect to the magnetic field. To obtain a quantitative measure of the Zeeman effect, one needs to determine the '''B''' dependent splitting ''ΔS'' relative to ''ΔA''. Making use of the ratio eliminates the need to know the optical magnification, observation angle and distance from the plate. The frequency splitting depends also on the LG plate thickness d (as in the FP case), and additionally on the index of refraction η of the quartz glass. In the FP case this would be equal to 1, but there are versions of the experiment where the evacuation of a sealed FP interferometer is used to produce a scanning effect in the fringe pattern [4]. The frequency splitting can be written as</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn1.png|160px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>The corresponding energy difference should equal</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn2.png|220px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
<p>One common method of determining the electron charge-to-mass ratio is through eq (2). Our interest is, however, to determine the energy splitting as a function of the magnetic field strength ''B''.</p>
<p>In our version of the Zeeman experiment the intense red Cd line at 643.8 nm is used. For optical transitions Cd acts as an effective two-electron system, i.e., it has a He-like configuration, as has Hg. Compare the Grotrian diagram shown in Fig. 1 to the one for mercury provided in the appendix (cf.. the Franck-Hertz experiment). In order to understand the selection rules for allowed electric dipole transitions follow the arguments given in ref. 4 in the context of the HeNe laser experiment. The two active electrons have combined orbital angular momentum ''L'' = l<sub>1</sub>+l<sub>2</sub>, spin S = s<sub>1</sub>+s<sub>2</sub>, and total angular momentum J = j<sub>1</sub>+j<sub>2</sub>. Allowed transitions require a change of one unit in ''L'' and ''J'' to make up for the spin of the photon, considering that spin flip is unlikely. Using <sup>(2''S''+1)</sup>''L<sub>J</sub>'' notation (with ''L'' = 0 denoted as S, ''L'' = 1 as P, ''L'' = 2 as D, etc) we have for the relevant line a <sup>1</sup>D<sub>2</sub> - <sup>1</sup>P<sub>1</sub> transition. This means that the spins are paired up (spin singlet) and that in nonrelativistic notation a 5s5d to 5s5p transition takes place. In Hg the equivalent line at the n = 6 level is the yellow line at 579 nm. The shift of the same line towards yellow is the result of having an additional electron shell in the core. What wavelength is associated with this transition in He?</p>
<p>For the advanced student: note that the initial level splits into 5 sublevels ''M<sub>J</sub>'' = -2,-1,0,1,2 , while the final state has ''M<sub>J</sub>'' = -1,0,1. They split equidistantly and one can group the nine possible transitions according to the allowed ''ΔM'' = -1,0,1 (cf.. Ref. 3 Fig. 7.3). To understand the polarization of the emitted light in the presence of the external B field note that for an electromagnetic wave its electric, magnetic fields and the wave propagation vector '''k''' form a right-handed coordinate system. Understand the validity of the dipole approximation (the wavelength λ is much longer than atomic dimensions) and how the electric field of the EM wave can be replaced by a constant vector times a temporary oscillatory factor (ref. [3,6]). Convince yourself why no linearly polarized light can be observed in the longitudinal direction as the magnetic field is turned on. Correspondingly understand why circularly polarized light as observed in the longitudinal direction must appear as linearly polarized when observed in a direction transverse to the ''B'' field. Why can all three components associated with the ''ΔM'' = -1,0,1 selection rule be observed in the transverse direction?</p>
<p>To verify these predictions about the polarization states of the light when the B field is turned on you need to recall some optical properties of polarizers and of quarter-wave plates (cf.. Ref.1). By placing these in the correct order you can verify the following:</p>
<p>Using a polarizer when observing in the transverse direction identify the polarization states of the three components with '''B''' turned on; what happens for ''B'' = 0?</p>
<p>Using a quarter-wave plate convert the circularly polarized (CP) light to two perpendicular
linearly polarized components corresponding to left- and right CP light respectively. Use a polarizer to extinguish each of these components separately; what happens for ''B'' = 0?</p>
<p>Collect sufficient data for both observation directions to demonstrate the linearity of the line
splitting with the magnetic field. You will need to perform a calibration of the magnetic field as a function of the current and should comment on possible saturation effects, i.e., a linear behaviour of the splitting with ''B'', but non-linear with the magnet current ''I'' at strong fields, where part of the electric energy may be converted to heat. Determine the gyromagnetic ratio (''g'') from your observations. </p>
<p>'''Warning:''' The Cd lamp emits ultraviolet light in addition to other lines such as the red Cd line. The apparatus contains a narrow-band red filter so that your eyes are protected when observing through the telescope. Avoid looking into the lamp itself (even though it is rated to be safe), i.e., cover the apparatus with a sheet of paper to reduce unnecessary eye contact with the lamp. The lamp takes several minutes to reach a proper operating temperature for the red line to be visible.</p>
<p><b>Obtain assistance when changing the observation direction from transverse to longitudinal or vice versa!</b></p>

<h3> Using the CCD Camera</h3>
<p>A CCD camera (AmScope MU300) is provided to allow for easy viewing of the lines. To operate the CCD, run the program "ToupView", then select ''Acquire''-> ''Live Capture''-> ''UCMOS03100KPA'' from the menu. A window will appear which is the live image being collected by the CCD. The brightness of the image can be changed using the ''Setup'' -> ''View Souce Properties'' -> ''Expose'' tab. A reasonable choice of values is an ''Exposure'' of 700ms with an ''Analog Gain'' of 3. Correct adjustment of the support system will allow you to clearly view the lines. Sliding of the CCD camera in and out will allow for focus. You can Save the image using ''Capture'' -> ''Capture a Frame'' command.</p>
<table width=500 align=center><td>
<p align=justify>[[File:ToupView.png|500px|border|center]]
<b>Figure 3-</b> ToupView CCD Camera Interface.
<br clear=right>
</p>
</td></table>
<p>Once you are able to see nice clear lines as shown in Figure 3, use the ''Region of Interest'' tool to focus in on a few lines in center. You can now use the ''Zoom'' to expand the image. One the left margin of the video image, there is a scale showing the pixel number. You can use this as a fixed reference point- as you increase the applied magnetic field, the line will split into sublevels, check the dial gauge, and then use the adjusting screw to place the shifted line back to the pixel number of the original line, then check the new reading of the dial gauge and record the measurement.</p>

<h1>References</h1>
<ol>
<li>Jenkins F.A., White H.E., ''Fundamentals of Optics'', McGraw-Hill</li>
<li>Brehm J.J., Mullin W.J., ''Introduction to the Structure of Matter'', Wiley</li>
<li>Melissinos A.C., ''Experiments in Modern Physics'', Academic Press</li>
<li>Preston D.W. Dietz E.R., ''The Art of Experimental Physics'',Wiley</li>
<li>Merzbacher E., ''Quantum Mechanics'', Wiley</li>
<li>Bethe H.A., Salpeter E.E., ''Quantum Mechanics of One- and Two-Electron Systems'', Springer</li>
<li>Radzig A.A., Smirnov B.M., ''Reference Data on Atoms Molecules and Ions'', Springer 1985.</li>
</ol>

<h1>Appendix</h1>
<ul>
<li><b>Data for the Lummer-Gehrcke plate</b> [from Leybold’s manual]: ''d'' = 4.04 mm η = 1.4567 </li>
<li>Grotrian diagrams for Cd, Hg, and He taken from ref. 7. A copy of these are in the binder in the laboratory.
<ul>
<li>[[Media:GrotrianH.pdf| Hydrogen]]</li>
<li>[[Media:GrotrianHe.pdf| Helium]]</li>
<li>[[Media:GrotrianHg.pdf| Mercury]]</li>
<li>[[Media:GrotrianHgCd.pdf| Cadmium and Mercury]]</li>
</ul>
</li>
</ul>

File:Zee fig2c.png

2011-11-01T15:32:39Z

WorkStudy:

Main Page/PHYS 4210/Zeeman Effect

2011-11-01T15:24:49Z

WorkStudy:

<h1>Zeeman Effect</h1>
<p>In this classic experiment that predates the development of quantum mechanics one investigates the light emitted by atoms in the presence of a homogeneous magnetic field. Of particular interest is the observation that this light is polarized in the presence of a magnetic field. The high-resolution spectroscopy required to resolve the line splittings is performed with a multiple-beam interferometer called a Lummer-Gehrcke plate which is similar to a Fabry-Perot interferometer.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Magnetic Sublevels</li>
<li>Total Angular Momentum</li>
<li>TEM wave</li>
<li>Quantization Axis</li>
<li>Orbital Angular Momentum</li>
<li>Spin Angular Momentum</li>
</ul>
</td>

<td width=250>
<ul>
<li>Normal Zeeman Effect</li>
<li>Anomalous Zeeman Effect</li>
<li>Lummer-Gehrcke Plate</li>
<li>Multi-beam Interferometer</li>
<li>Quarter-wave Plate</li>
</ul>
</td>
</table>

<h2> Required Components</h2>

<ol>
<li>[[Media:ZEMagnetPower.JPG|Magnet Power Supply]]</li>
<li>[[Media:ZEElectromagnet.JPG|Electromagnet]]</li>
<li>[[Media:ZEDischargePower.JPG|Discharge Power Supply]]</li>
<li>[[Media:ZECCDCamera.JPG|CCD Camera]]</li>
<li>[[Media:ZELummer.JPG|Lummer-Gehrcke Plate]]</li>
<li>[[Media:ZEPolarizers.JPG|Polarizers and Waveplate]]</li>
</ol>

<h1>Introduction</h1>
<p>The Zeeman effect is a powerful demonstration of the splittings of magnetic sublevels in an angular momentum multiplet. Many aspects of the emission of light by excited atoms, particularly when exposed to strong magnetic fields ('''B''') were understood by Lorentz in a classical model [1,2] before the advent of quantum mechanics. It is possible to understand the changes to classical electron orbits due to the Lorentz force in a 3D harmonic oscillator model. When one complements this with the idea that electromagnetic waves are transverse (the associated electric and magnetic fields of the EM wave oscillate in a plane perpendicular to the propagation direction of the wave), one can understand why circularly polarized light emerges as the atoms are observed in a direction longitudinal with the external '''B''' field, and why they appear as plane-polarized as viewed from the transverse direction. The understanding in the classical framework helps to build an intuition about the problem.</p>
<p>In the modern quantum mechanical description [2,5,6] one has to take into account that the presence of the '''B''' field singles out an axis. The additional interaction term between the magnetic moment of the electronic state (proportional to the ''z'' component of the total angular momentum) and '''B''' serves to split the magnetic sublevels of states with non-zero angular momentum. The additional interaction forces the use of this axis as a quantization axis. Without an external field one usually picks a ''z'' axis, but should arrive at results that are independent of this choice. To obtain the observed result that the light emanating from spontaneous transitions without an external field is unpolarized, one has to average over random orientations of the quantization axis. The observation that a definite orientation is singled out as quantization axis in the Zeeman effect is sometimes referred to as ‘space quantization’.</p>
<p>The problem can be illustrated by using pure orbital angular momentum states, i.e.,ignoring spin, and considering an np - ms transition. This transition is an allowed electric dipole transition, since a single unit of orbital angular momentum is changed, and this difference of one unit is carried away in the form of the spin for the spontaneously radiated photon. The important quantity to watch is the change in the projection of the orbital angular momentum, which can be +1, 0, -1 depending on the choice of the magnetic sublevel.</p>
<p>A transition 2p0 - 1s is associated with the emission of linearly polarized light with the oscillating electric field vector aligned with the z axis, with the wave propagation vector being orthogonal to this axis. This can be understood from the fact that the only non-vanishing matrix element for the dipole operator is <1s|''z''|2p<sub>0</sub>>. Similar calculations show that the transitions originating in the ''m'' = 1 and ''m'' = -1 sublevels result in circularly polarized light being emitted, which can propagate in the z direction only. One of the fascinating aspects of the Zeeman experiment is the following. For field-free atoms no axis is singled out, and thus, one has to include all possible orientations of the ''z'' axis, which results in the prediction that the light emitted from free atoms is unpolarized. However, once a homogeneous magnetic field is applied, an axis is singled out in space, which becomes the natural quantization axis. By probing the polarization of the spontaneously emitted light of atoms in the presence of a magnetic field one can verify that indeed the turn-on of the field causes a repopulation of the magnetic sublevels in a way that corresponds to the classical predictions of the Lorentz model. Thus, it is necessary to observe the light emitted longitudinally and transverse to the magnetic field.</p>
<p>The general theory of the Zeeman effect is complicated by the fact that the total angular momentum, i.e., added orbital and spin angular momentum of the active electron has to be considered. Based on orbital angular momentum alone the magnetic moment of an electron in a non-zero m sublevel is an integer multiple of the projection ''l<sub>z</sub>''. Once one couples ''l'' and ''s'' to form ''j'' = ''l'' + ''s'', the magnetic moment can be, but need not to be an integer multiple of ''j<sub>z</sub>'', and the proportionality is given by the Lande factor ''g''. The Lande factor can take on half-integer numbers for the initial and/or final states involved in the transition. One distinguishes between the normal and anomalous Zeeman effects depending on whether this complication arises or not. [3]. The anomalous effect is rather common in atomic transitions, but in this experiment a transition with the normal Zeeman effect has been selected. The red line in cadmium (643.8 nm), which is the equivalent of the yellow line in mercury, cf.. the Grotrian diagram shown in the appendix, and the level diagram in Fig. 1 (which is Fig. 7.3 from Melissinos [3]).
</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Level diagram for the Cd 643.8 nm transition with and without B field.
<br clear=right>
</p>
</td></table>
<p>From an experimental point of view a high demand is placed on the optical resolution of the interferometer. The idea is to inspect the interference pattern for a given line and to observe the quantitative changes in the pattern as the B field is applied to determine the wavelengths of the various components. The high resolution required can be obtained from multiple-beam interferometers, such as the Fabry-Perot (FP) interferometer [1,3]. Melissinos [3] discusses the analysis of the circular fringe pattern as produced by the FP. An easier alternative is provided by a special instrument that perfects the same method, called a Lummer-Gehrcke (LG) plate [1]. Since its interference pattern is more complicated to derive, you should concentrate on understanding the principles of multiple-beam interferometry using the FP and be aware of the analogies. Note that the FP has a wide range of applications in optics.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2a.png|500px|border|center]]
<b>Figure 2a -</b> Multiple reflection between the surfaces of a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2b.png|500px|border|center]]
<b>Figure 2b -</b> Apparatus for the Zeeman experiment with a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>

<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2c.png|500px|border|center]]
<b>Figure 2c -</b> Apparatus for the Zeeman experiment with a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>

<p>The LG plate shown as part of the apparatus in Fig. 2 consists of a precisely polished quartz glass plate of given thickness d with a prism attached at one end so that light entering from the slit has an angle of incidence on the plate that is near the critical angle. This results in some refractive transmission and mostly reflection at the glass/air surface. The reflected light inside the glass plate undergoes multiple ‘bounces’ of this type (interior reflection and partial refractive transmission). Two different interference patterns emerge when looking at a grazing angle at the top or bottom of the LG plate. The pattern formed at the top shows sharp bright lines on a dark background. In contrast to a Michelson interferometer a multiple-beam interferometer such as the FP and LG can produce an uneven interference pattern [1,3].</p>
<p>The separation between the fringes that appear without the magnetic field depends on the angle of observation. This spacing ''ΔA'' has to be determined for the particular fringe chosen for observation. As a magnetic field is applied each bright fringe splits either into two or into three depending on the orientation with respect to the magnetic field. To obtain a quantitative measure of the Zeeman effect, one needs to determine the '''B''' dependent splitting ''ΔS'' relative to ''ΔA''. Making use of the ratio eliminates the need to know the optical magnification, observation angle and distance from the plate. The frequency splitting depends also on the LG plate thickness d (as in the FP case), and additionally on the index of refraction η of the quartz glass. In the FP case this would be equal to 1, but there are versions of the experiment where the evacuation of a sealed FP interferometer is used to produce a scanning effect in the fringe pattern [4]. The frequency splitting can be written as</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn1.png|160px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>The corresponding energy difference should equal</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn2.png|220px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
<p>One common method of determining the electron charge-to-mass ratio is through eq (2). Our interest is, however, to determine the energy splitting as a function of the magnetic field strength ''B''.</p>
<p>In our version of the Zeeman experiment the intense red Cd line at 643.8 nm is used. For optical transitions Cd acts as an effective two-electron system, i.e., it has a He-like configuration, as has Hg. Compare the Grotrian diagram shown in Fig. 1 to the one for mercury provided in the appendix (cf.. the Franck-Hertz experiment). In order to understand the selection rules for allowed electric dipole transitions follow the arguments given in ref. 4 in the context of the HeNe laser experiment. The two active electrons have combined orbital angular momentum ''L'' = l<sub>1</sub>+l<sub>2</sub>, spin S = s<sub>1</sub>+s<sub>2</sub>, and total angular momentum J = j<sub>1</sub>+j<sub>2</sub>. Allowed transitions require a change of one unit in ''L'' and ''J'' to make up for the spin of the photon, considering that spin flip is unlikely. Using <sup>(2''S''+1)</sup>''L<sub>J</sub>'' notation (with ''L'' = 0 denoted as S, ''L'' = 1 as P, ''L'' = 2 as D, etc) we have for the relevant line a <sup>1</sup>D<sub>2</sub> - <sup>1</sup>P<sub>1</sub> transition. This means that the spins are paired up (spin singlet) and that in nonrelativistic notation a 5s5d to 5s5p transition takes place. In Hg the equivalent line at the n = 6 level is the yellow line at 579 nm. The shift of the same line towards yellow is the result of having an additional electron shell in the core. What wavelength is associated with this transition in He?</p>
<p>For the advanced student: note that the initial level splits into 5 sublevels ''M<sub>J</sub>'' = -2,-1,0,1,2 , while the final state has ''M<sub>J</sub>'' = -1,0,1. They split equidistantly and one can group the nine possible transitions according to the allowed ''ΔM'' = -1,0,1 (cf.. Ref. 3 Fig. 7.3). To understand the polarization of the emitted light in the presence of the external B field note that for an electromagnetic wave its electric, magnetic fields and the wave propagation vector '''k''' form a right-handed coordinate system. Understand the validity of the dipole approximation (the wavelength λ is much longer than atomic dimensions) and how the electric field of the EM wave can be replaced by a constant vector times a temporary oscillatory factor (ref. [3,6]). Convince yourself why no linearly polarized light can be observed in the longitudinal direction as the magnetic field is turned on. Correspondingly understand why circularly polarized light as observed in the longitudinal direction must appear as linearly polarized when observed in a direction transverse to the ''B'' field. Why can all three components associated with the ''ΔM'' = -1,0,1 selection rule be observed in the transverse direction?</p>
<p>To verify these predictions about the polarization states of the light when the B field is turned on you need to recall some optical properties of polarizers and of quarter-wave plates (cf.. Ref.1). By placing these in the correct order you can verify the following:</p>
<p>Using a polarizer when observing in the transverse direction identify the polarization states of the three components with '''B''' turned on; what happens for ''B'' = 0?</p>
<p>Using a quarter-wave plate convert the circularly polarized (CP) light to two perpendicular
linearly polarized components corresponding to left- and right CP light respectively. Use a polarizer to extinguish each of these components separately; what happens for ''B'' = 0?</p>
<p>Collect sufficient data for both observation directions to demonstrate the linearity of the line
splitting with the magnetic field. You will need to perform a calibration of the magnetic field as a function of the current and should comment on possible saturation effects, i.e., a linear behaviour of the splitting with ''B'', but non-linear with the magnet current ''I'' at strong fields, where part of the electric energy may be converted to heat. Determine the gyromagnetic ratio (''g'') from your observations. </p>
<p>'''Warning:''' The Cd lamp emits ultraviolet light in addition to other lines such as the red Cd line. The apparatus contains a narrow-band red filter so that your eyes are protected when observing through the telescope. Avoid looking into the lamp itself (even though it is rated to be safe), i.e., cover the apparatus with a sheet of paper to reduce unnecessary eye contact with the lamp. The lamp takes several minutes to reach a proper operating temperature for the red line to be visible.</p>
<p><b>Obtain assistance when changing the observation direction from transverse to longitudinal or vice versa!</b></p>

<h3> Using the CCD Camera</h3>
<p>A CCD camera (AmScope MU300) is provided to allow for easy viewing of the lines. To operate the CCD, run the program "ToupView", then select ''Acquire''-> ''Live Capture''-> ''UCMOS03100KPA'' from the menu. A window will appear which is the live image being collected by the CCD. The brightness of the image can be changed using the ''Setup'' -> ''View Souce Properties'' -> ''Expose'' tab. A reasonable choice of values is an ''Exposure'' of 700ms with an ''Analog Gain'' of 3. Correct adjustment of the support system will allow you to clearly view the lines. Sliding of the CCD camera in and out will allow for focus. You can Save the image using ''Capture'' -> ''Capture a Frame'' command.</p>
<table width=500 align=center><td>
<p align=justify>[[File:ToupView.png|500px|border|center]]
<b>Figure 3-</b> ToupView CCD Camera Interface.
<br clear=right>
</p>
</td></table>
<p>Once you are able to see nice clear lines as shown in Figure 3, use the ''Region of Interest'' tool to focus in on a few lines in center. You can now use the ''Zoom'' to expand the image. One the left margin of the video image, there is a scale showing the pixel number. You can use this as a fixed reference point- as you increase the applied magnetic field, the line will split into sublevels, check the dial gauge, and then use the adjusting screw to place the shifted line back to the pixel number of the original line, then check the new reading of the dial gauge and record the measurement.</p>

<h1>References</h1>
<ol>
<li>Jenkins F.A., White H.E., ''Fundamentals of Optics'', McGraw-Hill</li>
<li>Brehm J.J., Mullin W.J., ''Introduction to the Structure of Matter'', Wiley</li>
<li>Melissinos A.C., ''Experiments in Modern Physics'', Academic Press</li>
<li>Preston D.W. Dietz E.R., ''The Art of Experimental Physics'',Wiley</li>
<li>Merzbacher E., ''Quantum Mechanics'', Wiley</li>
<li>Bethe H.A., Salpeter E.E., ''Quantum Mechanics of One- and Two-Electron Systems'', Springer</li>
<li>Radzig A.A., Smirnov B.M., ''Reference Data on Atoms Molecules and Ions'', Springer 1985.</li>
</ol>

<h1>Appendix</h1>
<ul>
<li><b>Data for the Lummer-Gehrcke plate</b> [from Leybold’s manual]: ''d'' = 4.04 mm η = 1.4567 </li>
<li>Grotrian diagrams for Cd, Hg, and He taken from ref. 7. A copy of these are in the binder in the laboratory.
<ul>
<li>[[Media:GrotrianH.pdf| Hydrogen]]</li>
<li>[[Media:GrotrianHe.pdf| Helium]]</li>
<li>[[Media:GrotrianHg.pdf| Mercury]]</li>
<li>[[Media:GrotrianHgCd.pdf| Cadmium and Mercury]]</li>
</ul>
</li>
</ul>

Main Page/PHYS 4210/Zeeman Effect

2011-11-01T15:21:06Z

WorkStudy:

<h1>Zeeman Effect</h1>
<p>In this classic experiment that predates the development of quantum mechanics one investigates the light emitted by atoms in the presence of a homogeneous magnetic field. Of particular interest is the observation that this light is polarized in the presence of a magnetic field. The high-resolution spectroscopy required to resolve the line splittings is performed with a multiple-beam interferometer called a Lummer-Gehrcke plate which is similar to a Fabry-Perot interferometer.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Magnetic Sublevels</li>
<li>Total Angular Momentum</li>
<li>TEM wave</li>
<li>Quantization Axis</li>
<li>Orbital Angular Momentum</li>
<li>Spin Angular Momentum</li>
</ul>
</td>

<td width=250>
<ul>
<li>Normal Zeeman Effect</li>
<li>Anomalous Zeeman Effect</li>
<li>Lummer-Gehrcke Plate</li>
<li>Multi-beam Interferometer</li>
<li>Quarter-wave Plate</li>
</ul>
</td>
</table>

<h2> Required Components</h2>

<ol>
<li>[[Media:ZEMagnetPower.JPG|Magnet Power Supply]]</li>
<li>[[Media:ZEElectromagnet.JPG|Electromagnet]]</li>
<li>[[Media:ZEDischargePower.JPG|Discharge Power Supply]]</li>
<li>[[Media:ZECCDCamera.JPG|CCD Camera]]</li>
<li>[[Media:ZELummer.JPG|Lummer-Gehrcke Plate]]</li>
<li>[[Media:ZEPolarizers.JPG|Polarizers and Waveplate]]</li>
</ol>

<h1>Introduction</h1>
<p>The Zeeman effect is a powerful demonstration of the splittings of magnetic sublevels in an angular momentum multiplet. Many aspects of the emission of light by excited atoms, particularly when exposed to strong magnetic fields ('''B''') were understood by Lorentz in a classical model [1,2] before the advent of quantum mechanics. It is possible to understand the changes to classical electron orbits due to the Lorentz force in a 3D harmonic oscillator model. When one complements this with the idea that electromagnetic waves are transverse (the associated electric and magnetic fields of the EM wave oscillate in a plane perpendicular to the propagation direction of the wave), one can understand why circularly polarized light emerges as the atoms are observed in a direction longitudinal with the external '''B''' field, and why they appear as plane-polarized as viewed from the transverse direction. The understanding in the classical framework helps to build an intuition about the problem.</p>
<p>In the modern quantum mechanical description [2,5,6] one has to take into account that the presence of the '''B''' field singles out an axis. The additional interaction term between the magnetic moment of the electronic state (proportional to the ''z'' component of the total angular momentum) and '''B''' serves to split the magnetic sublevels of states with non-zero angular momentum. The additional interaction forces the use of this axis as a quantization axis. Without an external field one usually picks a ''z'' axis, but should arrive at results that are independent of this choice. To obtain the observed result that the light emanating from spontaneous transitions without an external field is unpolarized, one has to average over random orientations of the quantization axis. The observation that a definite orientation is singled out as quantization axis in the Zeeman effect is sometimes referred to as ‘space quantization’.</p>
<p>The problem can be illustrated by using pure orbital angular momentum states, i.e.,ignoring spin, and considering an np - ms transition. This transition is an allowed electric dipole transition, since a single unit of orbital angular momentum is changed, and this difference of one unit is carried away in the form of the spin for the spontaneously radiated photon. The important quantity to watch is the change in the projection of the orbital angular momentum, which can be +1, 0, -1 depending on the choice of the magnetic sublevel.</p>
<p>A transition 2p0 - 1s is associated with the emission of linearly polarized light with the oscillating electric field vector aligned with the z axis, with the wave propagation vector being orthogonal to this axis. This can be understood from the fact that the only non-vanishing matrix element for the dipole operator is <1s|''z''|2p<sub>0</sub>>. Similar calculations show that the transitions originating in the ''m'' = 1 and ''m'' = -1 sublevels result in circularly polarized light being emitted, which can propagate in the z direction only. One of the fascinating aspects of the Zeeman experiment is the following. For field-free atoms no axis is singled out, and thus, one has to include all possible orientations of the ''z'' axis, which results in the prediction that the light emitted from free atoms is unpolarized. However, once a homogeneous magnetic field is applied, an axis is singled out in space, which becomes the natural quantization axis. By probing the polarization of the spontaneously emitted light of atoms in the presence of a magnetic field one can verify that indeed the turn-on of the field causes a repopulation of the magnetic sublevels in a way that corresponds to the classical predictions of the Lorentz model. Thus, it is necessary to observe the light emitted longitudinally and transverse to the magnetic field.</p>
<p>The general theory of the Zeeman effect is complicated by the fact that the total angular momentum, i.e., added orbital and spin angular momentum of the active electron has to be considered. Based on orbital angular momentum alone the magnetic moment of an electron in a non-zero m sublevel is an integer multiple of the projection ''l<sub>z</sub>''. Once one couples ''l'' and ''s'' to form ''j'' = ''l'' + ''s'', the magnetic moment can be, but need not to be an integer multiple of ''j<sub>z</sub>'', and the proportionality is given by the Lande factor ''g''. The Lande factor can take on half-integer numbers for the initial and/or final states involved in the transition. One distinguishes between the normal and anomalous Zeeman effects depending on whether this complication arises or not. [3]. The anomalous effect is rather common in atomic transitions, but in this experiment a transition with the normal Zeeman effect has been selected. The red line in cadmium (643.8 nm), which is the equivalent of the yellow line in mercury, cf.. the Grotrian diagram shown in the appendix, and the level diagram in Fig. 1 (which is Fig. 7.3 from Melissinos [3]).
</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Level diagram for the Cd 643.8 nm transition with and without B field.
<br clear=right>
</p>
</td></table>
<p>From an experimental point of view a high demand is placed on the optical resolution of the interferometer. The idea is to inspect the interference pattern for a given line and to observe the quantitative changes in the pattern as the B field is applied to determine the wavelengths of the various components. The high resolution required can be obtained from multiple-beam interferometers, such as the Fabry-Perot (FP) interferometer [1,3]. Melissinos [3] discusses the analysis of the circular fringe pattern as produced by the FP. An easier alternative is provided by a special instrument that perfects the same method, called a Lummer-Gehrcke (LG) plate [1]. Since its interference pattern is more complicated to derive, you should concentrate on understanding the principles of multiple-beam interferometry using the FP and be aware of the analogies. Note that the FP has a wide range of applications in optics.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2a.png|500px|border|center]]
<b>Figure 2a -</b> Multiple reflection between the surfaces of a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2b.png|500px|border|center]]
<b>Figure 2b -</b> Apparatus for the Zeeman experiment with a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<p>The LG plate shown as part of the apparatus in Fig. 2 consists of a precisely polished quartz glass plate of given thickness d with a prism attached at one end so that light entering from the slit has an angle of incidence on the plate that is near the critical angle. This results in some refractive transmission and mostly reflection at the glass/air surface. The reflected light inside the glass plate undergoes multiple ‘bounces’ of this type (interior reflection and partial refractive transmission). Two different interference patterns emerge when looking at a grazing angle at the top or bottom of the LG plate. The pattern formed at the top shows sharp bright lines on a dark background. In contrast to a Michelson interferometer a multiple-beam interferometer such as the FP and LG can produce an uneven interference pattern [1,3].</p>
<p>The separation between the fringes that appear without the magnetic field depends on the angle of observation. This spacing ''ΔA'' has to be determined for the particular fringe chosen for observation. As a magnetic field is applied each bright fringe splits either into two or into three depending on the orientation with respect to the magnetic field. To obtain a quantitative measure of the Zeeman effect, one needs to determine the '''B''' dependent splitting ''ΔS'' relative to ''ΔA''. Making use of the ratio eliminates the need to know the optical magnification, observation angle and distance from the plate. The frequency splitting depends also on the LG plate thickness d (as in the FP case), and additionally on the index of refraction η of the quartz glass. In the FP case this would be equal to 1, but there are versions of the experiment where the evacuation of a sealed FP interferometer is used to produce a scanning effect in the fringe pattern [4]. The frequency splitting can be written as</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn1.png|160px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>The corresponding energy difference should equal</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn2.png|220px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
<p>One common method of determining the electron charge-to-mass ratio is through eq (2). Our interest is, however, to determine the energy splitting as a function of the magnetic field strength ''B''.</p>
<p>In our version of the Zeeman experiment the intense red Cd line at 643.8 nm is used. For optical transitions Cd acts as an effective two-electron system, i.e., it has a He-like configuration, as has Hg. Compare the Grotrian diagram shown in Fig. 1 to the one for mercury provided in the appendix (cf.. the Franck-Hertz experiment). In order to understand the selection rules for allowed electric dipole transitions follow the arguments given in ref. 4 in the context of the HeNe laser experiment. The two active electrons have combined orbital angular momentum ''L'' = l<sub>1</sub>+l<sub>2</sub>, spin S = s<sub>1</sub>+s<sub>2</sub>, and total angular momentum J = j<sub>1</sub>+j<sub>2</sub>. Allowed transitions require a change of one unit in ''L'' and ''J'' to make up for the spin of the photon, considering that spin flip is unlikely. Using <sup>(2''S''+1)</sup>''L<sub>J</sub>'' notation (with ''L'' = 0 denoted as S, ''L'' = 1 as P, ''L'' = 2 as D, etc) we have for the relevant line a <sup>1</sup>D<sub>2</sub> - <sup>1</sup>P<sub>1</sub> transition. This means that the spins are paired up (spin singlet) and that in nonrelativistic notation a 5s5d to 5s5p transition takes place. In Hg the equivalent line at the n = 6 level is the yellow line at 579 nm. The shift of the same line towards yellow is the result of having an additional electron shell in the core. What wavelength is associated with this transition in He?</p>
<p>For the advanced student: note that the initial level splits into 5 sublevels ''M<sub>J</sub>'' = -2,-1,0,1,2 , while the final state has ''M<sub>J</sub>'' = -1,0,1. They split equidistantly and one can group the nine possible transitions according to the allowed ''ΔM'' = -1,0,1 (cf.. Ref. 3 Fig. 7.3). To understand the polarization of the emitted light in the presence of the external B field note that for an electromagnetic wave its electric, magnetic fields and the wave propagation vector '''k''' form a right-handed coordinate system. Understand the validity of the dipole approximation (the wavelength λ is much longer than atomic dimensions) and how the electric field of the EM wave can be replaced by a constant vector times a temporary oscillatory factor (ref. [3,6]). Convince yourself why no linearly polarized light can be observed in the longitudinal direction as the magnetic field is turned on. Correspondingly understand why circularly polarized light as observed in the longitudinal direction must appear as linearly polarized when observed in a direction transverse to the ''B'' field. Why can all three components associated with the ''ΔM'' = -1,0,1 selection rule be observed in the transverse direction?</p>
<p>To verify these predictions about the polarization states of the light when the B field is turned on you need to recall some optical properties of polarizers and of quarter-wave plates (cf.. Ref.1). By placing these in the correct order you can verify the following:</p>
<p>Using a polarizer when observing in the transverse direction identify the polarization states of the three components with '''B''' turned on; what happens for ''B'' = 0?</p>
<p>Using a quarter-wave plate convert the circularly polarized (CP) light to two perpendicular
linearly polarized components corresponding to left- and right CP light respectively. Use a polarizer to extinguish each of these components separately; what happens for ''B'' = 0?</p>
<p>Collect sufficient data for both observation directions to demonstrate the linearity of the line
splitting with the magnetic field. You will need to perform a calibration of the magnetic field as a function of the current and should comment on possible saturation effects, i.e., a linear behaviour of the splitting with ''B'', but non-linear with the magnet current ''I'' at strong fields, where part of the electric energy may be converted to heat. Determine the gyromagnetic ratio (''g'') from your observations. </p>
<p>'''Warning:''' The Cd lamp emits ultraviolet light in addition to other lines such as the red Cd line. The apparatus contains a narrow-band red filter so that your eyes are protected when observing through the telescope. Avoid looking into the lamp itself (even though it is rated to be safe), i.e., cover the apparatus with a sheet of paper to reduce unnecessary eye contact with the lamp. The lamp takes several minutes to reach a proper operating temperature for the red line to be visible.</p>
<p><b>Obtain assistance when changing the observation direction from transverse to longitudinal or vice versa!</b></p>

<h3> Using the CCD Camera</h3>
<p>A CCD camera (AmScope MU300) is provided to allow for easy viewing of the lines. To operate the CCD, run the program "ToupView", then select ''Acquire''-> ''Live Capture''-> ''UCMOS03100KPA'' from the menu. A window will appear which is the live image being collected by the CCD. The brightness of the image can be changed using the ''Setup'' -> ''View Souce Properties'' -> ''Expose'' tab. A reasonable choice of values is an ''Exposure'' of 700ms with an ''Analog Gain'' of 3. Correct adjustment of the support system will allow you to clearly view the lines. Sliding of the CCD camera in and out will allow for focus. You can Save the image using ''Capture'' -> ''Capture a Frame'' command.</p>
<table width=500 align=center><td>
<p align=justify>[[File:ToupView.png|500px|border|center]]
<b>Figure 3-</b> ToupView CCD Camera Interface.
<br clear=right>
</p>
</td></table>
<p>Once you are able to see nice clear lines as shown in Figure 3, use the ''Region of Interest'' tool to focus in on a few lines in center. You can now use the ''Zoom'' to expand the image. One the left margin of the video image, there is a scale showing the pixel number. You can use this as a fixed reference point- as you increase the applied magnetic field, the line will split into sublevels, check the dial gauge, and then use the adjusting screw to place the shifted line back to the pixel number of the original line, then check the new reading of the dial gauge and record the measurement.</p>

<h1>References</h1>
<ol>
<li>Jenkins F.A., White H.E., ''Fundamentals of Optics'', McGraw-Hill</li>
<li>Brehm J.J., Mullin W.J., ''Introduction to the Structure of Matter'', Wiley</li>
<li>Melissinos A.C., ''Experiments in Modern Physics'', Academic Press</li>
<li>Preston D.W. Dietz E.R., ''The Art of Experimental Physics'',Wiley</li>
<li>Merzbacher E., ''Quantum Mechanics'', Wiley</li>
<li>Bethe H.A., Salpeter E.E., ''Quantum Mechanics of One- and Two-Electron Systems'', Springer</li>
<li>Radzig A.A., Smirnov B.M., ''Reference Data on Atoms Molecules and Ions'', Springer 1985.</li>
</ol>

<h1>Appendix</h1>
<ul>
<li><b>Data for the Lummer-Gehrcke plate</b> [from Leybold’s manual]: ''d'' = 4.04 mm η = 1.4567 </li>
<li>Grotrian diagrams for Cd, Hg, and He taken from ref. 7. A copy of these are in the binder in the laboratory.
<ul>
<li>[[Media:GrotrianH.pdf| Hydrogen]]</li>
<li>[[Media:GrotrianHe.pdf| Helium]]</li>
<li>[[Media:GrotrianHg.pdf| Mercury]]</li>
<li>[[Media:GrotrianHgCd.pdf| Cadmium and Mercury]]</li>
</ul>
</li>
</ul>

File:ZESetup.JPG

2011-11-01T15:19:37Z

WorkStudy:

File:ZEPolarizers.JPG

2011-11-01T15:19:06Z

WorkStudy:

File:ZEMagnetPower.JPG

2011-11-01T15:18:25Z

WorkStudy:

File:ZELummer.JPG

2011-11-01T15:17:59Z

WorkStudy:

File:ZEElectromagnet.JPG

2011-11-01T15:17:37Z

WorkStudy:

File:ZEDischargePower.JPG

2011-11-01T15:17:17Z

WorkStudy:

File:ZECCDCamera.JPG

2011-11-01T15:16:55Z

WorkStudy:

Main Page/PHYS 3220/Radioactive Decays

2011-11-01T14:32:58Z

WorkStudy:

<h1>Radioactive Decays</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>

<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 fast electron is emitted from the nucleus, thus converting a neutron into a proton (and an electron + antineutrino) - corresponding to the β<sup>-</sup> decay of free neutrons (half-life  10.6 min.). For proton-rich nuclei a fast positron emerges from the nucleus (β<sup>+</sup> decay) whereby a proton is converted into a neutron and a neutrino. 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 higher energies than hard 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 understood 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 (ref. 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. ref. 1,2,6), and provide a concise description inyour own words with 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 by passage through matter. This is used both in shielding, as well as 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, therefore, 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. Gamma rays have the best penetration characteristics, i.e., are hardest 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 an especially 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 used, e.g., by having fast particles penetrating 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. Read the detailed write-up for the “Absorption of Radiation” experiment.</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 the question had to be resolved 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 goes over into a Gaussian. </p>
<p>Rutherford performed experiments which 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 ''Radiation Counter'' <b>***CHANGE THIS PROGRAM REFERENCE****</b> performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (cf.. also chapter 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>II(2)</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 (cf.. chpt. 12 in ref. 5, an example is given on p.235). From the reduced χ<sup>2</sup> value one infers the agreement.</p>
</li>

</ol>

<h1>Experimental Procedure</h1>

<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 - 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 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>1. 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 chi-squared obtained, and quote the decay rate, with its standard error. Include print-outs 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. How do these results compare to the results from the computer program?</li>

<li>2. 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>3. 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>4. 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 outlilne 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</li>
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li>
<li>Taylor, J.R., ''An Introduction to Error Analysis''.</li>
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li>
</ol>

Main Page/PHYS 3220/Radioactive Decays

2011-11-01T14:31:37Z

WorkStudy:

<h1>Radioactive Decays</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>

<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 fast electron is emitted from the nucleus, thus converting a neutron into a proton (and an electron + antineutrino) - corresponding to the β<sup>-</sup> decay of free neutrons (half-life  10.6 min.). For proton-rich nuclei a fast positron emerges from the nucleus (β<sup>+</sup> decay) whereby a proton is converted into a neutron and a neutrino. 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 higher energies than hard 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 understood 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 (ref. 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. ref. 1,2,6), and provide a concise description inyour own words with 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 by passage through matter. This is used both in shielding, as well as 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, therefore, 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. Gamma rays have the best penetration characteristics, i.e., are hardest 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 an especially 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 used, e.g., by having fast particles penetrating 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. Read the detailed write-up for the “Absorption of Radiation” experiment.</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 the question had to be resolved 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 goes over into a Gaussian. </p>
<p>Rutherford performed experiments which 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 ''Radiation Counter'' <b>***CHANGE THIS PROGRAM REFERENCE****</b> performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (cf.. also chapter 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>II(2)</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 (cf.. chpt. 12 in ref. 5, an example is given on p.235). From the reduced χ<sup>2</sup> value one infers the agreement.</p>
</li>

</ol>

<h1>Experimental Procedure</h1>

<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 - 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 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>1. 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 chi-squared obtained, and quote the decay rate, with its standard error. Include print-outs 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. How do these results compare to the results from the computer program?</li>

<li>2. 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>3. 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>4. 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 outlilne 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</li>
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li>
<li>Taylor, J.R., ''An Introduction to Error Analysis''.</li>
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li>
</ol>

Main Page/PHYS 4210/Zeeman Effect

2011-11-01T14:29:39Z

WorkStudy:

<h1>Zeeman Effect</h1>
<p>In this classic experiment that predates the development of quantum mechanics one investigates the light emitted by atoms in the presence of a homogeneous magnetic field. Of particular interest is the observation that this light is polarized in the presence of a magnetic field. The high-resolution spectroscopy required to resolve the line splittings is performed with a multiple-beam interferometer called a Lummer-Gehrcke plate which is similar to a Fabry-Perot interferometer.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Magnetic Sublevels</li>
<li>Total Angular Momentum</li>
<li>TEM wave</li>
<li>Quantization Axis</li>
<li>Orbital Angular Momentum</li>
<li>Spin Angular Momentum</li>
</ul>
</td>

<td width=250>
<ul>
<li>Normal Zeeman Effect</li>
<li>Anomalous Zeeman Effect</li>
<li>Lummer-Gehrcke Plate</li>
<li>Multi-beam Interferometer</li>
<li>Quarter-wave Plate</li>
</ul>
</td>
</table>

<h2> Required Components</h2>

<ol>
<li>[[Media:.JPG|Magnet Power Supply]]</li>
<li>[[Media:.JPG|Electromagnet]]</li>
<li>[[Media:.JPG|Discharge Power Supply]]</li>
<li>[[Media:.JPG|CCD Camera]]</li>
<li>[[Media:.JPG|Lummer-Gehrcke Plate]]</li>
<li>[[Media:.JPG|Polarizers and Waveplate]]</li>
</ol>

<h1>Introduction</h1>
<p>The Zeeman effect is a powerful demonstration of the splittings of magnetic sublevels in an angular momentum multiplet. Many aspects of the emission of light by excited atoms, particularly when exposed to strong magnetic fields ('''B''') were understood by Lorentz in a classical model [1,2] before the advent of quantum mechanics. It is possible to understand the changes to classical electron orbits due to the Lorentz force in a 3D harmonic oscillator model. When one complements this with the idea that electromagnetic waves are transverse (the associated electric and magnetic fields of the EM wave oscillate in a plane perpendicular to the propagation direction of the wave), one can understand why circularly polarized light emerges as the atoms are observed in a direction longitudinal with the external '''B''' field, and why they appear as plane-polarized as viewed from the transverse direction. The understanding in the classical framework helps to build an intuition about the problem.</p>
<p>In the modern quantum mechanical description [2,5,6] one has to take into account that the presence of the '''B''' field singles out an axis. The additional interaction term between the magnetic moment of the electronic state (proportional to the ''z'' component of the total angular momentum) and '''B''' serves to split the magnetic sublevels of states with non-zero angular momentum. The additional interaction forces the use of this axis as a quantization axis. Without an external field one usually picks a ''z'' axis, but should arrive at results that are independent of this choice. To obtain the observed result that the light emanating from spontaneous transitions without an external field is unpolarized, one has to average over random orientations of the quantization axis. The observation that a definite orientation is singled out as quantization axis in the Zeeman effect is sometimes referred to as ‘space quantization’.</p>
<p>The problem can be illustrated by using pure orbital angular momentum states, i.e.,ignoring spin, and considering an np - ms transition. This transition is an allowed electric dipole transition, since a single unit of orbital angular momentum is changed, and this difference of one unit is carried away in the form of the spin for the spontaneously radiated photon. The important quantity to watch is the change in the projection of the orbital angular momentum, which can be +1, 0, -1 depending on the choice of the magnetic sublevel.</p>
<p>A transition 2p0 - 1s is associated with the emission of linearly polarized light with the oscillating electric field vector aligned with the z axis, with the wave propagation vector being orthogonal to this axis. This can be understood from the fact that the only non-vanishing matrix element for the dipole operator is <1s|''z''|2p<sub>0</sub>>. Similar calculations show that the transitions originating in the ''m'' = 1 and ''m'' = -1 sublevels result in circularly polarized light being emitted, which can propagate in the z direction only. One of the fascinating aspects of the Zeeman experiment is the following. For field-free atoms no axis is singled out, and thus, one has to include all possible orientations of the ''z'' axis, which results in the prediction that the light emitted from free atoms is unpolarized. However, once a homogeneous magnetic field is applied, an axis is singled out in space, which becomes the natural quantization axis. By probing the polarization of the spontaneously emitted light of atoms in the presence of a magnetic field one can verify that indeed the turn-on of the field causes a repopulation of the magnetic sublevels in a way that corresponds to the classical predictions of the Lorentz model. Thus, it is necessary to observe the light emitted longitudinally and transverse to the magnetic field.</p>
<p>The general theory of the Zeeman effect is complicated by the fact that the total angular momentum, i.e., added orbital and spin angular momentum of the active electron has to be considered. Based on orbital angular momentum alone the magnetic moment of an electron in a non-zero m sublevel is an integer multiple of the projection ''l<sub>z</sub>''. Once one couples ''l'' and ''s'' to form ''j'' = ''l'' + ''s'', the magnetic moment can be, but need not to be an integer multiple of ''j<sub>z</sub>'', and the proportionality is given by the Lande factor ''g''. The Lande factor can take on half-integer numbers for the initial and/or final states involved in the transition. One distinguishes between the normal and anomalous Zeeman effects depending on whether this complication arises or not. [3]. The anomalous effect is rather common in atomic transitions, but in this experiment a transition with the normal Zeeman effect has been selected. The red line in cadmium (643.8 nm), which is the equivalent of the yellow line in mercury, cf.. the Grotrian diagram shown in the appendix, and the level diagram in Fig. 1 (which is Fig. 7.3 from Melissinos [3]).
</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Level diagram for the Cd 643.8 nm transition with and without B field.
<br clear=right>
</p>
</td></table>
<p>From an experimental point of view a high demand is placed on the optical resolution of the interferometer. The idea is to inspect the interference pattern for a given line and to observe the quantitative changes in the pattern as the B field is applied to determine the wavelengths of the various components. The high resolution required can be obtained from multiple-beam interferometers, such as the Fabry-Perot (FP) interferometer [1,3]. Melissinos [3] discusses the analysis of the circular fringe pattern as produced by the FP. An easier alternative is provided by a special instrument that perfects the same method, called a Lummer-Gehrcke (LG) plate [1]. Since its interference pattern is more complicated to derive, you should concentrate on understanding the principles of multiple-beam interferometry using the FP and be aware of the analogies. Note that the FP has a wide range of applications in optics.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2a.png|500px|border|center]]
<b>Figure 2a -</b> Multiple reflection between the surfaces of a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2b.png|500px|border|center]]
<b>Figure 2b -</b> Apparatus for the Zeeman experiment with a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<p>The LG plate shown as part of the apparatus in Fig. 2 consists of a precisely polished quartz glass plate of given thickness d with a prism attached at one end so that light entering from the slit has an angle of incidence on the plate that is near the critical angle. This results in some refractive transmission and mostly reflection at the glass/air surface. The reflected light inside the glass plate undergoes multiple ‘bounces’ of this type (interior reflection and partial refractive transmission). Two different interference patterns emerge when looking at a grazing angle at the top or bottom of the LG plate. The pattern formed at the top shows sharp bright lines on a dark background. In contrast to a Michelson interferometer a multiple-beam interferometer such as the FP and LG can produce an uneven interference pattern [1,3].</p>
<p>The separation between the fringes that appear without the magnetic field depends on the angle of observation. This spacing ''ΔA'' has to be determined for the particular fringe chosen for observation. As a magnetic field is applied each bright fringe splits either into two or into three depending on the orientation with respect to the magnetic field. To obtain a quantitative measure of the Zeeman effect, one needs to determine the '''B''' dependent splitting ''ΔS'' relative to ''ΔA''. Making use of the ratio eliminates the need to know the optical magnification, observation angle and distance from the plate. The frequency splitting depends also on the LG plate thickness d (as in the FP case), and additionally on the index of refraction η of the quartz glass. In the FP case this would be equal to 1, but there are versions of the experiment where the evacuation of a sealed FP interferometer is used to produce a scanning effect in the fringe pattern [4]. The frequency splitting can be written as</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn1.png|160px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>The corresponding energy difference should equal</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn2.png|220px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
<p>One common method of determining the electron charge-to-mass ratio is through eq (2). Our interest is, however, to determine the energy splitting as a function of the magnetic field strength ''B''.</p>
<p>In our version of the Zeeman experiment the intense red Cd line at 643.8 nm is used. For optical transitions Cd acts as an effective two-electron system, i.e., it has a He-like configuration, as has Hg. Compare the Grotrian diagram shown in Fig. 1 to the one for mercury provided in the appendix (cf.. the Franck-Hertz experiment). In order to understand the selection rules for allowed electric dipole transitions follow the arguments given in ref. 4 in the context of the HeNe laser experiment. The two active electrons have combined orbital angular momentum ''L'' = l<sub>1</sub>+l<sub>2</sub>, spin S = s<sub>1</sub>+s<sub>2</sub>, and total angular momentum J = j<sub>1</sub>+j<sub>2</sub>. Allowed transitions require a change of one unit in ''L'' and ''J'' to make up for the spin of the photon, considering that spin flip is unlikely. Using <sup>(2''S''+1)</sup>''L<sub>J</sub>'' notation (with ''L'' = 0 denoted as S, ''L'' = 1 as P, ''L'' = 2 as D, etc) we have for the relevant line a <sup>1</sup>D<sub>2</sub> - <sup>1</sup>P<sub>1</sub> transition. This means that the spins are paired up (spin singlet) and that in nonrelativistic notation a 5s5d to 5s5p transition takes place. In Hg the equivalent line at the n = 6 level is the yellow line at 579 nm. The shift of the same line towards yellow is the result of having an additional electron shell in the core. What wavelength is associated with this transition in He?</p>
<p>For the advanced student: note that the initial level splits into 5 sublevels ''M<sub>J</sub>'' = -2,-1,0,1,2 , while the final state has ''M<sub>J</sub>'' = -1,0,1. They split equidistantly and one can group the nine possible transitions according to the allowed ''ΔM'' = -1,0,1 (cf.. Ref. 3 Fig. 7.3). To understand the polarization of the emitted light in the presence of the external B field note that for an electromagnetic wave its electric, magnetic fields and the wave propagation vector '''k''' form a right-handed coordinate system. Understand the validity of the dipole approximation (the wavelength λ is much longer than atomic dimensions) and how the electric field of the EM wave can be replaced by a constant vector times a temporary oscillatory factor (ref. [3,6]). Convince yourself why no linearly polarized light can be observed in the longitudinal direction as the magnetic field is turned on. Correspondingly understand why circularly polarized light as observed in the longitudinal direction must appear as linearly polarized when observed in a direction transverse to the ''B'' field. Why can all three components associated with the ''ΔM'' = -1,0,1 selection rule be observed in the transverse direction?</p>
<p>To verify these predictions about the polarization states of the light when the B field is turned on you need to recall some optical properties of polarizers and of quarter-wave plates (cf.. Ref.1). By placing these in the correct order you can verify the following:</p>
<p>Using a polarizer when observing in the transverse direction identify the polarization states of the three components with '''B''' turned on; what happens for ''B'' = 0?</p>
<p>Using a quarter-wave plate convert the circularly polarized (CP) light to two perpendicular
linearly polarized components corresponding to left- and right CP light respectively. Use a polarizer to extinguish each of these components separately; what happens for ''B'' = 0?</p>
<p>Collect sufficient data for both observation directions to demonstrate the linearity of the line
splitting with the magnetic field. You will need to perform a calibration of the magnetic field as a function of the current and should comment on possible saturation effects, i.e., a linear behaviour of the splitting with ''B'', but non-linear with the magnet current ''I'' at strong fields, where part of the electric energy may be converted to heat. Determine the gyromagnetic ratio (''g'') from your observations. </p>
<p>'''Warning:''' The Cd lamp emits ultraviolet light in addition to other lines such as the red Cd line. The apparatus contains a narrow-band red filter so that your eyes are protected when observing through the telescope. Avoid looking into the lamp itself (even though it is rated to be safe), i.e., cover the apparatus with a sheet of paper to reduce unnecessary eye contact with the lamp. The lamp takes several minutes to reach a proper operating temperature for the red line to be visible.</p>
<p><b>Obtain assistance when changing the observation direction from transverse to longitudinal or vice versa!</b></p>

<h3> Using the CCD Camera</h3>
<p>A CCD camera (AmScope MU300) is provided to allow for easy viewing of the lines. To operate the CCD, run the program "ToupView", then select ''Acquire''-> ''Live Capture''-> ''UCMOS03100KPA'' from the menu. A window will appear which is the live image being collected by the CCD. The brightness of the image can be changed using the ''Setup'' -> ''View Souce Properties'' -> ''Expose'' tab. A reasonable choice of values is an ''Exposure'' of 700ms with an ''Analog Gain'' of 3. Correct adjustment of the support system will allow you to clearly view the lines. Sliding of the CCD camera in and out will allow for focus. You can Save the image using ''Capture'' -> ''Capture a Frame'' command.</p>
<table width=500 align=center><td>
<p align=justify>[[File:ToupView.png|500px|border|center]]
<b>Figure 3-</b> ToupView CCD Camera Interface.
<br clear=right>
</p>
</td></table>
<p>Once you are able to see nice clear lines as shown in Figure 3, use the ''Region of Interest'' tool to focus in on a few lines in center. You can now use the ''Zoom'' to expand the image. One the left margin of the video image, there is a scale showing the pixel number. You can use this as a fixed reference point- as you increase the applied magnetic field, the line will split into sublevels, check the dial gauge, and then use the adjusting screw to place the shifted line back to the pixel number of the original line, then check the new reading of the dial gauge and record the measurement.</p>

<h1>References</h1>
<ol>
<li>Jenkins F.A., White H.E., ''Fundamentals of Optics'', McGraw-Hill</li>
<li>Brehm J.J., Mullin W.J., ''Introduction to the Structure of Matter'', Wiley</li>
<li>Melissinos A.C., ''Experiments in Modern Physics'', Academic Press</li>
<li>Preston D.W. Dietz E.R., ''The Art of Experimental Physics'',Wiley</li>
<li>Merzbacher E., ''Quantum Mechanics'', Wiley</li>
<li>Bethe H.A., Salpeter E.E., ''Quantum Mechanics of One- and Two-Electron Systems'', Springer</li>
<li>Radzig A.A., Smirnov B.M., ''Reference Data on Atoms Molecules and Ions'', Springer 1985.</li>
</ol>

<h1>Appendix</h1>
<ul>
<li><b>Data for the Lummer-Gehrcke plate</b> [from Leybold’s manual]: ''d'' = 4.04 mm η = 1.4567 </li>
<li>Grotrian diagrams for Cd, Hg, and He taken from ref. 7. A copy of these are in the binder in the laboratory.
<ul>
<li>[[Media:GrotrianH.pdf| Hydrogen]]</li>
<li>[[Media:GrotrianHe.pdf| Helium]]</li>
<li>[[Media:GrotrianHg.pdf| Mercury]]</li>
<li>[[Media:GrotrianHgCd.pdf| Cadmium and Mercury]]</li>
</ul>
</li>
</ul>

Main Page/PHYS 3220/Radioactive Decays

2011-11-01T14:26:48Z

WorkStudy:

<h1>Radioactive Decays</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>

<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 fast electron is emitted from the nucleus, thus converting a neutron into a proton (and an electron + antineutrino) - corresponding to the β<sup>-</sup> decay of free neutrons (half-life  10.6 min.). For proton-rich nuclei a fast positron emerges from the nucleus (β<sup>+</sup> decay) whereby a proton is converted into a neutron and a neutrino. 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 higher energies than hard 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 understood 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 (ref. 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. ref. 1,2,6), and provide a concise description inyour own words with 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 by passage through matter. This is used both in shielding, as well as 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, therefore, 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. Gamma rays have the best penetration characteristics, i.e., are hardest 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 an especially 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 used, e.g., by having fast particles penetrating 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. Read the detailed write-up for the “Absorption of Radiation” experiment.</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 the question had to be resolved 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 goes over into a Gaussian. </p>
<p>Rutherford performed experiments which 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 ''Radiation Counter'' <b>***CHANGE THIS PROGRAM REFERENCE****</b> performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (cf.. also chapter 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>II(2)</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 (cf.. chpt. 12 in ref. 5, an example is given on p.235). From the reduced χ<sup>2</sup> value one infers the agreement.</p>
</li>

</ol>

<h1>Experimental Procedure</h1>

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

<h2>Required Components</h2>
<ol>
<li>[[Media:.JPG|AC powered table-top GM counter]]</li>
<li>[[Media:.JPG|hand-held GM counter]]</li>
<li>[[Media:.JPG|beige 'Fiesta' ceramic dish]]</li>
<li>[[Media:.JPG|orange 'Fiesta' ceramic dish]]</li>
<li>[[Media:.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 - 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 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>1. 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 chi-squared obtained, and quote the decay rate, with its standard error. Include print-outs 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. How do these results compare to the results from the computer program?</li>

<li>2. 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>3. 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>4. 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 outlilne 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</li>
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li>
<li>Taylor, J.R., ''An Introduction to Error Analysis''.</li>
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li>
</ol>

Main Page/PHYS 4210/Zeeman Effect

2011-11-01T14:24:48Z

WorkStudy:

<h1>Zeeman Effect</h1>
<p>In this classic experiment that predates the development of quantum mechanics one investigates the light emitted by atoms in the presence of a homogeneous magnetic field. Of particular interest is the observation that this light is polarized in the presence of a magnetic field. The high-resolution spectroscopy required to resolve the line splittings is performed with a multiple-beam interferometer called a Lummer-Gehrcke plate which is similar to a Fabry-Perot interferometer.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Magnetic Sublevels</li>
<li>Total Angular Momentum</li>
<li>TEM wave</li>
<li>Quantization Axis</li>
<li>Orbital Angular Momentum</li>
<li>Spin Angular Momentum</li>
</ul>
</td>

<td width=250>
<ul>
<li>Normal Zeeman Effect</li>
<li>Anomalous Zeeman Effect</li>
<li>Lummer-Gehrcke Plate</li>
<li>Multi-beam Interferometer</li>
<li>Quarter-wave Plate</li>
</ul>
</td>
</table>

<h2> Required Components</h2>

<ol>
<li>Magnet Power Supply</li>
<li>Electromagnet</li>
<li>Discharge Power Supply</li>
<li>CCD Camera</li>
<li>Lummer-Gehrcke Plate</li>
<li>Polarizers and Waveplate</li>
</ol>

<h1>Introduction</h1>
<p>The Zeeman effect is a powerful demonstration of the splittings of magnetic sublevels in an angular momentum multiplet. Many aspects of the emission of light by excited atoms, particularly when exposed to strong magnetic fields ('''B''') were understood by Lorentz in a classical model [1,2] before the advent of quantum mechanics. It is possible to understand the changes to classical electron orbits due to the Lorentz force in a 3D harmonic oscillator model. When one complements this with the idea that electromagnetic waves are transverse (the associated electric and magnetic fields of the EM wave oscillate in a plane perpendicular to the propagation direction of the wave), one can understand why circularly polarized light emerges as the atoms are observed in a direction longitudinal with the external '''B''' field, and why they appear as plane-polarized as viewed from the transverse direction. The understanding in the classical framework helps to build an intuition about the problem.</p>
<p>In the modern quantum mechanical description [2,5,6] one has to take into account that the presence of the '''B''' field singles out an axis. The additional interaction term between the magnetic moment of the electronic state (proportional to the ''z'' component of the total angular momentum) and '''B''' serves to split the magnetic sublevels of states with non-zero angular momentum. The additional interaction forces the use of this axis as a quantization axis. Without an external field one usually picks a ''z'' axis, but should arrive at results that are independent of this choice. To obtain the observed result that the light emanating from spontaneous transitions without an external field is unpolarized, one has to average over random orientations of the quantization axis. The observation that a definite orientation is singled out as quantization axis in the Zeeman effect is sometimes referred to as ‘space quantization’.</p>
<p>The problem can be illustrated by using pure orbital angular momentum states, i.e.,ignoring spin, and considering an np - ms transition. This transition is an allowed electric dipole transition, since a single unit of orbital angular momentum is changed, and this difference of one unit is carried away in the form of the spin for the spontaneously radiated photon. The important quantity to watch is the change in the projection of the orbital angular momentum, which can be +1, 0, -1 depending on the choice of the magnetic sublevel.</p>
<p>A transition 2p0 - 1s is associated with the emission of linearly polarized light with the oscillating electric field vector aligned with the z axis, with the wave propagation vector being orthogonal to this axis. This can be understood from the fact that the only non-vanishing matrix element for the dipole operator is <1s|''z''|2p<sub>0</sub>>. Similar calculations show that the transitions originating in the ''m'' = 1 and ''m'' = -1 sublevels result in circularly polarized light being emitted, which can propagate in the z direction only. One of the fascinating aspects of the Zeeman experiment is the following. For field-free atoms no axis is singled out, and thus, one has to include all possible orientations of the ''z'' axis, which results in the prediction that the light emitted from free atoms is unpolarized. However, once a homogeneous magnetic field is applied, an axis is singled out in space, which becomes the natural quantization axis. By probing the polarization of the spontaneously emitted light of atoms in the presence of a magnetic field one can verify that indeed the turn-on of the field causes a repopulation of the magnetic sublevels in a way that corresponds to the classical predictions of the Lorentz model. Thus, it is necessary to observe the light emitted longitudinally and transverse to the magnetic field.</p>
<p>The general theory of the Zeeman effect is complicated by the fact that the total angular momentum, i.e., added orbital and spin angular momentum of the active electron has to be considered. Based on orbital angular momentum alone the magnetic moment of an electron in a non-zero m sublevel is an integer multiple of the projection ''l<sub>z</sub>''. Once one couples ''l'' and ''s'' to form ''j'' = ''l'' + ''s'', the magnetic moment can be, but need not to be an integer multiple of ''j<sub>z</sub>'', and the proportionality is given by the Lande factor ''g''. The Lande factor can take on half-integer numbers for the initial and/or final states involved in the transition. One distinguishes between the normal and anomalous Zeeman effects depending on whether this complication arises or not. [3]. The anomalous effect is rather common in atomic transitions, but in this experiment a transition with the normal Zeeman effect has been selected. The red line in cadmium (643.8 nm), which is the equivalent of the yellow line in mercury, cf.. the Grotrian diagram shown in the appendix, and the level diagram in Fig. 1 (which is Fig. 7.3 from Melissinos [3]).
</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Level diagram for the Cd 643.8 nm transition with and without B field.
<br clear=right>
</p>
</td></table>
<p>From an experimental point of view a high demand is placed on the optical resolution of the interferometer. The idea is to inspect the interference pattern for a given line and to observe the quantitative changes in the pattern as the B field is applied to determine the wavelengths of the various components. The high resolution required can be obtained from multiple-beam interferometers, such as the Fabry-Perot (FP) interferometer [1,3]. Melissinos [3] discusses the analysis of the circular fringe pattern as produced by the FP. An easier alternative is provided by a special instrument that perfects the same method, called a Lummer-Gehrcke (LG) plate [1]. Since its interference pattern is more complicated to derive, you should concentrate on understanding the principles of multiple-beam interferometry using the FP and be aware of the analogies. Note that the FP has a wide range of applications in optics.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2a.png|500px|border|center]]
<b>Figure 2a -</b> Multiple reflection between the surfaces of a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2b.png|500px|border|center]]
<b>Figure 2b -</b> Apparatus for the Zeeman experiment with a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<p>The LG plate shown as part of the apparatus in Fig. 2 consists of a precisely polished quartz glass plate of given thickness d with a prism attached at one end so that light entering from the slit has an angle of incidence on the plate that is near the critical angle. This results in some refractive transmission and mostly reflection at the glass/air surface. The reflected light inside the glass plate undergoes multiple ‘bounces’ of this type (interior reflection and partial refractive transmission). Two different interference patterns emerge when looking at a grazing angle at the top or bottom of the LG plate. The pattern formed at the top shows sharp bright lines on a dark background. In contrast to a Michelson interferometer a multiple-beam interferometer such as the FP and LG can produce an uneven interference pattern [1,3].</p>
<p>The separation between the fringes that appear without the magnetic field depends on the angle of observation. This spacing ''ΔA'' has to be determined for the particular fringe chosen for observation. As a magnetic field is applied each bright fringe splits either into two or into three depending on the orientation with respect to the magnetic field. To obtain a quantitative measure of the Zeeman effect, one needs to determine the '''B''' dependent splitting ''ΔS'' relative to ''ΔA''. Making use of the ratio eliminates the need to know the optical magnification, observation angle and distance from the plate. The frequency splitting depends also on the LG plate thickness d (as in the FP case), and additionally on the index of refraction η of the quartz glass. In the FP case this would be equal to 1, but there are versions of the experiment where the evacuation of a sealed FP interferometer is used to produce a scanning effect in the fringe pattern [4]. The frequency splitting can be written as</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn1.png|160px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>The corresponding energy difference should equal</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn2.png|220px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
<p>One common method of determining the electron charge-to-mass ratio is through eq (2). Our interest is, however, to determine the energy splitting as a function of the magnetic field strength ''B''.</p>
<p>In our version of the Zeeman experiment the intense red Cd line at 643.8 nm is used. For optical transitions Cd acts as an effective two-electron system, i.e., it has a He-like configuration, as has Hg. Compare the Grotrian diagram shown in Fig. 1 to the one for mercury provided in the appendix (cf.. the Franck-Hertz experiment). In order to understand the selection rules for allowed electric dipole transitions follow the arguments given in ref. 4 in the context of the HeNe laser experiment. The two active electrons have combined orbital angular momentum ''L'' = l<sub>1</sub>+l<sub>2</sub>, spin S = s<sub>1</sub>+s<sub>2</sub>, and total angular momentum J = j<sub>1</sub>+j<sub>2</sub>. Allowed transitions require a change of one unit in ''L'' and ''J'' to make up for the spin of the photon, considering that spin flip is unlikely. Using <sup>(2''S''+1)</sup>''L<sub>J</sub>'' notation (with ''L'' = 0 denoted as S, ''L'' = 1 as P, ''L'' = 2 as D, etc) we have for the relevant line a <sup>1</sup>D<sub>2</sub> - <sup>1</sup>P<sub>1</sub> transition. This means that the spins are paired up (spin singlet) and that in nonrelativistic notation a 5s5d to 5s5p transition takes place. In Hg the equivalent line at the n = 6 level is the yellow line at 579 nm. The shift of the same line towards yellow is the result of having an additional electron shell in the core. What wavelength is associated with this transition in He?</p>
<p>For the advanced student: note that the initial level splits into 5 sublevels ''M<sub>J</sub>'' = -2,-1,0,1,2 , while the final state has ''M<sub>J</sub>'' = -1,0,1. They split equidistantly and one can group the nine possible transitions according to the allowed ''ΔM'' = -1,0,1 (cf.. Ref. 3 Fig. 7.3). To understand the polarization of the emitted light in the presence of the external B field note that for an electromagnetic wave its electric, magnetic fields and the wave propagation vector '''k''' form a right-handed coordinate system. Understand the validity of the dipole approximation (the wavelength λ is much longer than atomic dimensions) and how the electric field of the EM wave can be replaced by a constant vector times a temporary oscillatory factor (ref. [3,6]). Convince yourself why no linearly polarized light can be observed in the longitudinal direction as the magnetic field is turned on. Correspondingly understand why circularly polarized light as observed in the longitudinal direction must appear as linearly polarized when observed in a direction transverse to the ''B'' field. Why can all three components associated with the ''ΔM'' = -1,0,1 selection rule be observed in the transverse direction?</p>
<p>To verify these predictions about the polarization states of the light when the B field is turned on you need to recall some optical properties of polarizers and of quarter-wave plates (cf.. Ref.1). By placing these in the correct order you can verify the following:</p>
<p>Using a polarizer when observing in the transverse direction identify the polarization states of the three components with '''B''' turned on; what happens for ''B'' = 0?</p>
<p>Using a quarter-wave plate convert the circularly polarized (CP) light to two perpendicular
linearly polarized components corresponding to left- and right CP light respectively. Use a polarizer to extinguish each of these components separately; what happens for ''B'' = 0?</p>
<p>Collect sufficient data for both observation directions to demonstrate the linearity of the line
splitting with the magnetic field. You will need to perform a calibration of the magnetic field as a function of the current and should comment on possible saturation effects, i.e., a linear behaviour of the splitting with ''B'', but non-linear with the magnet current ''I'' at strong fields, where part of the electric energy may be converted to heat. Determine the gyromagnetic ratio (''g'') from your observations. </p>
<p>'''Warning:''' The Cd lamp emits ultraviolet light in addition to other lines such as the red Cd line. The apparatus contains a narrow-band red filter so that your eyes are protected when observing through the telescope. Avoid looking into the lamp itself (even though it is rated to be safe), i.e., cover the apparatus with a sheet of paper to reduce unnecessary eye contact with the lamp. The lamp takes several minutes to reach a proper operating temperature for the red line to be visible.</p>
<p><b>Obtain assistance when changing the observation direction from transverse to longitudinal or vice versa!</b></p>

<h3> Using the CCD Camera</h3>
<p>A CCD camera (AmScope MU300) is provided to allow for easy viewing of the lines. To operate the CCD, run the program "ToupView", then select ''Acquire''-> ''Live Capture''-> ''UCMOS03100KPA'' from the menu. A window will appear which is the live image being collected by the CCD. The brightness of the image can be changed using the ''Setup'' -> ''View Souce Properties'' -> ''Expose'' tab. A reasonable choice of values is an ''Exposure'' of 700ms with an ''Analog Gain'' of 3. Correct adjustment of the support system will allow you to clearly view the lines. Sliding of the CCD camera in and out will allow for focus. You can Save the image using ''Capture'' -> ''Capture a Frame'' command.</p>
<table width=500 align=center><td>
<p align=justify>[[File:ToupView.png|500px|border|center]]
<b>Figure 3-</b> ToupView CCD Camera Interface.
<br clear=right>
</p>
</td></table>
<p>Once you are able to see nice clear lines as shown in Figure 3, use the ''Region of Interest'' tool to focus in on a few lines in center. You can now use the ''Zoom'' to expand the image. One the left margin of the video image, there is a scale showing the pixel number. You can use this as a fixed reference point- as you increase the applied magnetic field, the line will split into sublevels, check the dial gauge, and then use the adjusting screw to place the shifted line back to the pixel number of the original line, then check the new reading of the dial gauge and record the measurement.</p>

<h1>References</h1>
<ol>
<li>Jenkins F.A., White H.E., ''Fundamentals of Optics'', McGraw-Hill</li>
<li>Brehm J.J., Mullin W.J., ''Introduction to the Structure of Matter'', Wiley</li>
<li>Melissinos A.C., ''Experiments in Modern Physics'', Academic Press</li>
<li>Preston D.W. Dietz E.R., ''The Art of Experimental Physics'',Wiley</li>
<li>Merzbacher E., ''Quantum Mechanics'', Wiley</li>
<li>Bethe H.A., Salpeter E.E., ''Quantum Mechanics of One- and Two-Electron Systems'', Springer</li>
<li>Radzig A.A., Smirnov B.M., ''Reference Data on Atoms Molecules and Ions'', Springer 1985.</li>
</ol>

<h1>Appendix</h1>
<ul>
<li><b>Data for the Lummer-Gehrcke plate</b> [from Leybold’s manual]: ''d'' = 4.04 mm η = 1.4567 </li>
<li>Grotrian diagrams for Cd, Hg, and He taken from ref. 7. A copy of these are in the binder in the laboratory.
<ul>
<li>[[Media:GrotrianH.pdf| Hydrogen]]</li>
<li>[[Media:GrotrianHe.pdf| Helium]]</li>
<li>[[Media:GrotrianHg.pdf| Mercury]]</li>
<li>[[Media:GrotrianHgCd.pdf| Cadmium and Mercury]]</li>
</ul>
</li>
</ul>

Main Page/PHYS 4210/Zeeman Effect

2011-11-01T14:24:09Z

WorkStudy:

<h1>Zeeman Effect</h1>
<p>In this classic experiment that predates the development of quantum mechanics one investigates the light emitted by atoms in the presence of a homogeneous magnetic field. Of particular interest is the observation that this light is polarized in the presence of a magnetic field. The high-resolution spectroscopy required to resolve the line splittings is performed with a multiple-beam interferometer called a Lummer-Gehrcke plate which is similar to a Fabry-Perot interferometer.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Magnetic Sublevels</li>
<li>Total Angular Momentum</li>
<li>TEM wave</li>
<li>Quantization Axis</li>
<li>Orbital Angular Momentum</li>
<li>Spin Angular Momentum</li>
</ul>
</td>

<td width=250>
<ul>
<li>Normal Zeeman Effect</li>
<li>Anomalous Zeeman Effect</li>
<li>Lummer-Gehrcke Plate</li>
<li>Multi-beam Interferometer</li>
<li>Quarter-wave Plate</li>
</ul>
</td>
</table>

<h2> Required Components</h2>

<ol>
<li>magnet power supply</li>
<li>electromagnet</li>
<li>discharge power supply</li>
<li>CCD camera</li>
<li>Lummer-Gehrcke Plate</li>
<li>polarizers and waveplate</li>
</ol>

<h1>Introduction</h1>
<p>The Zeeman effect is a powerful demonstration of the splittings of magnetic sublevels in an angular momentum multiplet. Many aspects of the emission of light by excited atoms, particularly when exposed to strong magnetic fields ('''B''') were understood by Lorentz in a classical model [1,2] before the advent of quantum mechanics. It is possible to understand the changes to classical electron orbits due to the Lorentz force in a 3D harmonic oscillator model. When one complements this with the idea that electromagnetic waves are transverse (the associated electric and magnetic fields of the EM wave oscillate in a plane perpendicular to the propagation direction of the wave), one can understand why circularly polarized light emerges as the atoms are observed in a direction longitudinal with the external '''B''' field, and why they appear as plane-polarized as viewed from the transverse direction. The understanding in the classical framework helps to build an intuition about the problem.</p>
<p>In the modern quantum mechanical description [2,5,6] one has to take into account that the presence of the '''B''' field singles out an axis. The additional interaction term between the magnetic moment of the electronic state (proportional to the ''z'' component of the total angular momentum) and '''B''' serves to split the magnetic sublevels of states with non-zero angular momentum. The additional interaction forces the use of this axis as a quantization axis. Without an external field one usually picks a ''z'' axis, but should arrive at results that are independent of this choice. To obtain the observed result that the light emanating from spontaneous transitions without an external field is unpolarized, one has to average over random orientations of the quantization axis. The observation that a definite orientation is singled out as quantization axis in the Zeeman effect is sometimes referred to as ‘space quantization’.</p>
<p>The problem can be illustrated by using pure orbital angular momentum states, i.e.,ignoring spin, and considering an np - ms transition. This transition is an allowed electric dipole transition, since a single unit of orbital angular momentum is changed, and this difference of one unit is carried away in the form of the spin for the spontaneously radiated photon. The important quantity to watch is the change in the projection of the orbital angular momentum, which can be +1, 0, -1 depending on the choice of the magnetic sublevel.</p>
<p>A transition 2p0 - 1s is associated with the emission of linearly polarized light with the oscillating electric field vector aligned with the z axis, with the wave propagation vector being orthogonal to this axis. This can be understood from the fact that the only non-vanishing matrix element for the dipole operator is <1s|''z''|2p<sub>0</sub>>. Similar calculations show that the transitions originating in the ''m'' = 1 and ''m'' = -1 sublevels result in circularly polarized light being emitted, which can propagate in the z direction only. One of the fascinating aspects of the Zeeman experiment is the following. For field-free atoms no axis is singled out, and thus, one has to include all possible orientations of the ''z'' axis, which results in the prediction that the light emitted from free atoms is unpolarized. However, once a homogeneous magnetic field is applied, an axis is singled out in space, which becomes the natural quantization axis. By probing the polarization of the spontaneously emitted light of atoms in the presence of a magnetic field one can verify that indeed the turn-on of the field causes a repopulation of the magnetic sublevels in a way that corresponds to the classical predictions of the Lorentz model. Thus, it is necessary to observe the light emitted longitudinally and transverse to the magnetic field.</p>
<p>The general theory of the Zeeman effect is complicated by the fact that the total angular momentum, i.e., added orbital and spin angular momentum of the active electron has to be considered. Based on orbital angular momentum alone the magnetic moment of an electron in a non-zero m sublevel is an integer multiple of the projection ''l<sub>z</sub>''. Once one couples ''l'' and ''s'' to form ''j'' = ''l'' + ''s'', the magnetic moment can be, but need not to be an integer multiple of ''j<sub>z</sub>'', and the proportionality is given by the Lande factor ''g''. The Lande factor can take on half-integer numbers for the initial and/or final states involved in the transition. One distinguishes between the normal and anomalous Zeeman effects depending on whether this complication arises or not. [3]. The anomalous effect is rather common in atomic transitions, but in this experiment a transition with the normal Zeeman effect has been selected. The red line in cadmium (643.8 nm), which is the equivalent of the yellow line in mercury, cf.. the Grotrian diagram shown in the appendix, and the level diagram in Fig. 1 (which is Fig. 7.3 from Melissinos [3]).
</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Level diagram for the Cd 643.8 nm transition with and without B field.
<br clear=right>
</p>
</td></table>
<p>From an experimental point of view a high demand is placed on the optical resolution of the interferometer. The idea is to inspect the interference pattern for a given line and to observe the quantitative changes in the pattern as the B field is applied to determine the wavelengths of the various components. The high resolution required can be obtained from multiple-beam interferometers, such as the Fabry-Perot (FP) interferometer [1,3]. Melissinos [3] discusses the analysis of the circular fringe pattern as produced by the FP. An easier alternative is provided by a special instrument that perfects the same method, called a Lummer-Gehrcke (LG) plate [1]. Since its interference pattern is more complicated to derive, you should concentrate on understanding the principles of multiple-beam interferometry using the FP and be aware of the analogies. Note that the FP has a wide range of applications in optics.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2a.png|500px|border|center]]
<b>Figure 2a -</b> Multiple reflection between the surfaces of a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2b.png|500px|border|center]]
<b>Figure 2b -</b> Apparatus for the Zeeman experiment with a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<p>The LG plate shown as part of the apparatus in Fig. 2 consists of a precisely polished quartz glass plate of given thickness d with a prism attached at one end so that light entering from the slit has an angle of incidence on the plate that is near the critical angle. This results in some refractive transmission and mostly reflection at the glass/air surface. The reflected light inside the glass plate undergoes multiple ‘bounces’ of this type (interior reflection and partial refractive transmission). Two different interference patterns emerge when looking at a grazing angle at the top or bottom of the LG plate. The pattern formed at the top shows sharp bright lines on a dark background. In contrast to a Michelson interferometer a multiple-beam interferometer such as the FP and LG can produce an uneven interference pattern [1,3].</p>
<p>The separation between the fringes that appear without the magnetic field depends on the angle of observation. This spacing ''ΔA'' has to be determined for the particular fringe chosen for observation. As a magnetic field is applied each bright fringe splits either into two or into three depending on the orientation with respect to the magnetic field. To obtain a quantitative measure of the Zeeman effect, one needs to determine the '''B''' dependent splitting ''ΔS'' relative to ''ΔA''. Making use of the ratio eliminates the need to know the optical magnification, observation angle and distance from the plate. The frequency splitting depends also on the LG plate thickness d (as in the FP case), and additionally on the index of refraction η of the quartz glass. In the FP case this would be equal to 1, but there are versions of the experiment where the evacuation of a sealed FP interferometer is used to produce a scanning effect in the fringe pattern [4]. The frequency splitting can be written as</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn1.png|160px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>The corresponding energy difference should equal</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn2.png|220px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
<p>One common method of determining the electron charge-to-mass ratio is through eq (2). Our interest is, however, to determine the energy splitting as a function of the magnetic field strength ''B''.</p>
<p>In our version of the Zeeman experiment the intense red Cd line at 643.8 nm is used. For optical transitions Cd acts as an effective two-electron system, i.e., it has a He-like configuration, as has Hg. Compare the Grotrian diagram shown in Fig. 1 to the one for mercury provided in the appendix (cf.. the Franck-Hertz experiment). In order to understand the selection rules for allowed electric dipole transitions follow the arguments given in ref. 4 in the context of the HeNe laser experiment. The two active electrons have combined orbital angular momentum ''L'' = l<sub>1</sub>+l<sub>2</sub>, spin S = s<sub>1</sub>+s<sub>2</sub>, and total angular momentum J = j<sub>1</sub>+j<sub>2</sub>. Allowed transitions require a change of one unit in ''L'' and ''J'' to make up for the spin of the photon, considering that spin flip is unlikely. Using <sup>(2''S''+1)</sup>''L<sub>J</sub>'' notation (with ''L'' = 0 denoted as S, ''L'' = 1 as P, ''L'' = 2 as D, etc) we have for the relevant line a <sup>1</sup>D<sub>2</sub> - <sup>1</sup>P<sub>1</sub> transition. This means that the spins are paired up (spin singlet) and that in nonrelativistic notation a 5s5d to 5s5p transition takes place. In Hg the equivalent line at the n = 6 level is the yellow line at 579 nm. The shift of the same line towards yellow is the result of having an additional electron shell in the core. What wavelength is associated with this transition in He?</p>
<p>For the advanced student: note that the initial level splits into 5 sublevels ''M<sub>J</sub>'' = -2,-1,0,1,2 , while the final state has ''M<sub>J</sub>'' = -1,0,1. They split equidistantly and one can group the nine possible transitions according to the allowed ''ΔM'' = -1,0,1 (cf.. Ref. 3 Fig. 7.3). To understand the polarization of the emitted light in the presence of the external B field note that for an electromagnetic wave its electric, magnetic fields and the wave propagation vector '''k''' form a right-handed coordinate system. Understand the validity of the dipole approximation (the wavelength λ is much longer than atomic dimensions) and how the electric field of the EM wave can be replaced by a constant vector times a temporary oscillatory factor (ref. [3,6]). Convince yourself why no linearly polarized light can be observed in the longitudinal direction as the magnetic field is turned on. Correspondingly understand why circularly polarized light as observed in the longitudinal direction must appear as linearly polarized when observed in a direction transverse to the ''B'' field. Why can all three components associated with the ''ΔM'' = -1,0,1 selection rule be observed in the transverse direction?</p>
<p>To verify these predictions about the polarization states of the light when the B field is turned on you need to recall some optical properties of polarizers and of quarter-wave plates (cf.. Ref.1). By placing these in the correct order you can verify the following:</p>
<p>Using a polarizer when observing in the transverse direction identify the polarization states of the three components with '''B''' turned on; what happens for ''B'' = 0?</p>
<p>Using a quarter-wave plate convert the circularly polarized (CP) light to two perpendicular
linearly polarized components corresponding to left- and right CP light respectively. Use a polarizer to extinguish each of these components separately; what happens for ''B'' = 0?</p>
<p>Collect sufficient data for both observation directions to demonstrate the linearity of the line
splitting with the magnetic field. You will need to perform a calibration of the magnetic field as a function of the current and should comment on possible saturation effects, i.e., a linear behaviour of the splitting with ''B'', but non-linear with the magnet current ''I'' at strong fields, where part of the electric energy may be converted to heat. Determine the gyromagnetic ratio (''g'') from your observations. </p>
<p>'''Warning:''' The Cd lamp emits ultraviolet light in addition to other lines such as the red Cd line. The apparatus contains a narrow-band red filter so that your eyes are protected when observing through the telescope. Avoid looking into the lamp itself (even though it is rated to be safe), i.e., cover the apparatus with a sheet of paper to reduce unnecessary eye contact with the lamp. The lamp takes several minutes to reach a proper operating temperature for the red line to be visible.</p>
<p><b>Obtain assistance when changing the observation direction from transverse to longitudinal or vice versa!</b></p>

<h3> Using the CCD Camera</h3>
<p>A CCD camera (AmScope MU300) is provided to allow for easy viewing of the lines. To operate the CCD, run the program "ToupView", then select ''Acquire''-> ''Live Capture''-> ''UCMOS03100KPA'' from the menu. A window will appear which is the live image being collected by the CCD. The brightness of the image can be changed using the ''Setup'' -> ''View Souce Properties'' -> ''Expose'' tab. A reasonable choice of values is an ''Exposure'' of 700ms with an ''Analog Gain'' of 3. Correct adjustment of the support system will allow you to clearly view the lines. Sliding of the CCD camera in and out will allow for focus. You can Save the image using ''Capture'' -> ''Capture a Frame'' command.</p>
<table width=500 align=center><td>
<p align=justify>[[File:ToupView.png|500px|border|center]]
<b>Figure 3-</b> ToupView CCD Camera Interface.
<br clear=right>
</p>
</td></table>
<p>Once you are able to see nice clear lines as shown in Figure 3, use the ''Region of Interest'' tool to focus in on a few lines in center. You can now use the ''Zoom'' to expand the image. One the left margin of the video image, there is a scale showing the pixel number. You can use this as a fixed reference point- as you increase the applied magnetic field, the line will split into sublevels, check the dial gauge, and then use the adjusting screw to place the shifted line back to the pixel number of the original line, then check the new reading of the dial gauge and record the measurement.</p>

<h1>References</h1>
<ol>
<li>Jenkins F.A., White H.E., ''Fundamentals of Optics'', McGraw-Hill</li>
<li>Brehm J.J., Mullin W.J., ''Introduction to the Structure of Matter'', Wiley</li>
<li>Melissinos A.C., ''Experiments in Modern Physics'', Academic Press</li>
<li>Preston D.W. Dietz E.R., ''The Art of Experimental Physics'',Wiley</li>
<li>Merzbacher E., ''Quantum Mechanics'', Wiley</li>
<li>Bethe H.A., Salpeter E.E., ''Quantum Mechanics of One- and Two-Electron Systems'', Springer</li>
<li>Radzig A.A., Smirnov B.M., ''Reference Data on Atoms Molecules and Ions'', Springer 1985.</li>
</ol>

<h1>Appendix</h1>
<ul>
<li><b>Data for the Lummer-Gehrcke plate</b> [from Leybold’s manual]: ''d'' = 4.04 mm η = 1.4567 </li>
<li>Grotrian diagrams for Cd, Hg, and He taken from ref. 7. A copy of these are in the binder in the laboratory.
<ul>
<li>[[Media:GrotrianH.pdf| Hydrogen]]</li>
<li>[[Media:GrotrianHe.pdf| Helium]]</li>
<li>[[Media:GrotrianHg.pdf| Mercury]]</li>
<li>[[Media:GrotrianHgCd.pdf| Cadmium and Mercury]]</li>
</ul>
</li>
</ul>

Main Page/PHYS 4210/Zeeman Effect

2011-11-01T14:23:22Z

WorkStudy:

<h1>Zeeman Effect</h1>
<p>In this classic experiment that predates the development of quantum mechanics one investigates the light emitted by atoms in the presence of a homogeneous magnetic field. Of particular interest is the observation that this light is polarized in the presence of a magnetic field. The high-resolution spectroscopy required to resolve the line splittings is performed with a multiple-beam interferometer called a Lummer-Gehrcke plate which is similar to a Fabry-Perot interferometer.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Magnetic Sublevels</li>
<li>Total Angular Momentum</li>
<li>TEM wave</li>
<li>Quantization Axis</li>
<li>Orbital Angular Momentum</li>
<li>Spin Angular Momentum</li>
</ul>
</td>

<td width=250>
<ul>
<li>Normal Zeeman Effect</li>
<li>Anomalous Zeeman Effect</li>
<li>Lummer-Gehrcke Plate</li>
<li>Multi-beam Interferometer</li>
<li>Quarter-wave Plate</li>
</ul>
</td>
</table>

<h2> Required Components</h2>
<ol>
<li>magnet power supply</li>
<li>electromagnet</li>
<li>discharge power supply</li>
<li>CCD camera</li>
<li>Lummer Gerke Plate</li>
<li>polarizers and waveplate</li>
</ol>

<h1>Introduction</h1>
<p>The Zeeman effect is a powerful demonstration of the splittings of magnetic sublevels in an angular momentum multiplet. Many aspects of the emission of light by excited atoms, particularly when exposed to strong magnetic fields ('''B''') were understood by Lorentz in a classical model [1,2] before the advent of quantum mechanics. It is possible to understand the changes to classical electron orbits due to the Lorentz force in a 3D harmonic oscillator model. When one complements this with the idea that electromagnetic waves are transverse (the associated electric and magnetic fields of the EM wave oscillate in a plane perpendicular to the propagation direction of the wave), one can understand why circularly polarized light emerges as the atoms are observed in a direction longitudinal with the external '''B''' field, and why they appear as plane-polarized as viewed from the transverse direction. The understanding in the classical framework helps to build an intuition about the problem.</p>
<p>In the modern quantum mechanical description [2,5,6] one has to take into account that the presence of the '''B''' field singles out an axis. The additional interaction term between the magnetic moment of the electronic state (proportional to the ''z'' component of the total angular momentum) and '''B''' serves to split the magnetic sublevels of states with non-zero angular momentum. The additional interaction forces the use of this axis as a quantization axis. Without an external field one usually picks a ''z'' axis, but should arrive at results that are independent of this choice. To obtain the observed result that the light emanating from spontaneous transitions without an external field is unpolarized, one has to average over random orientations of the quantization axis. The observation that a definite orientation is singled out as quantization axis in the Zeeman effect is sometimes referred to as ‘space quantization’.</p>
<p>The problem can be illustrated by using pure orbital angular momentum states, i.e.,ignoring spin, and considering an np - ms transition. This transition is an allowed electric dipole transition, since a single unit of orbital angular momentum is changed, and this difference of one unit is carried away in the form of the spin for the spontaneously radiated photon. The important quantity to watch is the change in the projection of the orbital angular momentum, which can be +1, 0, -1 depending on the choice of the magnetic sublevel.</p>
<p>A transition 2p0 - 1s is associated with the emission of linearly polarized light with the oscillating electric field vector aligned with the z axis, with the wave propagation vector being orthogonal to this axis. This can be understood from the fact that the only non-vanishing matrix element for the dipole operator is <1s|''z''|2p<sub>0</sub>>. Similar calculations show that the transitions originating in the ''m'' = 1 and ''m'' = -1 sublevels result in circularly polarized light being emitted, which can propagate in the z direction only. One of the fascinating aspects of the Zeeman experiment is the following. For field-free atoms no axis is singled out, and thus, one has to include all possible orientations of the ''z'' axis, which results in the prediction that the light emitted from free atoms is unpolarized. However, once a homogeneous magnetic field is applied, an axis is singled out in space, which becomes the natural quantization axis. By probing the polarization of the spontaneously emitted light of atoms in the presence of a magnetic field one can verify that indeed the turn-on of the field causes a repopulation of the magnetic sublevels in a way that corresponds to the classical predictions of the Lorentz model. Thus, it is necessary to observe the light emitted longitudinally and transverse to the magnetic field.</p>
<p>The general theory of the Zeeman effect is complicated by the fact that the total angular momentum, i.e., added orbital and spin angular momentum of the active electron has to be considered. Based on orbital angular momentum alone the magnetic moment of an electron in a non-zero m sublevel is an integer multiple of the projection ''l<sub>z</sub>''. Once one couples ''l'' and ''s'' to form ''j'' = ''l'' + ''s'', the magnetic moment can be, but need not to be an integer multiple of ''j<sub>z</sub>'', and the proportionality is given by the Lande factor ''g''. The Lande factor can take on half-integer numbers for the initial and/or final states involved in the transition. One distinguishes between the normal and anomalous Zeeman effects depending on whether this complication arises or not. [3]. The anomalous effect is rather common in atomic transitions, but in this experiment a transition with the normal Zeeman effect has been selected. The red line in cadmium (643.8 nm), which is the equivalent of the yellow line in mercury, cf.. the Grotrian diagram shown in the appendix, and the level diagram in Fig. 1 (which is Fig. 7.3 from Melissinos [3]).
</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Level diagram for the Cd 643.8 nm transition with and without B field.
<br clear=right>
</p>
</td></table>
<p>From an experimental point of view a high demand is placed on the optical resolution of the interferometer. The idea is to inspect the interference pattern for a given line and to observe the quantitative changes in the pattern as the B field is applied to determine the wavelengths of the various components. The high resolution required can be obtained from multiple-beam interferometers, such as the Fabry-Perot (FP) interferometer [1,3]. Melissinos [3] discusses the analysis of the circular fringe pattern as produced by the FP. An easier alternative is provided by a special instrument that perfects the same method, called a Lummer-Gehrcke (LG) plate [1]. Since its interference pattern is more complicated to derive, you should concentrate on understanding the principles of multiple-beam interferometry using the FP and be aware of the analogies. Note that the FP has a wide range of applications in optics.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2a.png|500px|border|center]]
<b>Figure 2a -</b> Multiple reflection between the surfaces of a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2b.png|500px|border|center]]
<b>Figure 2b -</b> Apparatus for the Zeeman experiment with a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<p>The LG plate shown as part of the apparatus in Fig. 2 consists of a precisely polished quartz glass plate of given thickness d with a prism attached at one end so that light entering from the slit has an angle of incidence on the plate that is near the critical angle. This results in some refractive transmission and mostly reflection at the glass/air surface. The reflected light inside the glass plate undergoes multiple ‘bounces’ of this type (interior reflection and partial refractive transmission). Two different interference patterns emerge when looking at a grazing angle at the top or bottom of the LG plate. The pattern formed at the top shows sharp bright lines on a dark background. In contrast to a Michelson interferometer a multiple-beam interferometer such as the FP and LG can produce an uneven interference pattern [1,3].</p>
<p>The separation between the fringes that appear without the magnetic field depends on the angle of observation. This spacing ''ΔA'' has to be determined for the particular fringe chosen for observation. As a magnetic field is applied each bright fringe splits either into two or into three depending on the orientation with respect to the magnetic field. To obtain a quantitative measure of the Zeeman effect, one needs to determine the '''B''' dependent splitting ''ΔS'' relative to ''ΔA''. Making use of the ratio eliminates the need to know the optical magnification, observation angle and distance from the plate. The frequency splitting depends also on the LG plate thickness d (as in the FP case), and additionally on the index of refraction η of the quartz glass. In the FP case this would be equal to 1, but there are versions of the experiment where the evacuation of a sealed FP interferometer is used to produce a scanning effect in the fringe pattern [4]. The frequency splitting can be written as</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn1.png|160px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>The corresponding energy difference should equal</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn2.png|220px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
<p>One common method of determining the electron charge-to-mass ratio is through eq (2). Our interest is, however, to determine the energy splitting as a function of the magnetic field strength ''B''.</p>
<p>In our version of the Zeeman experiment the intense red Cd line at 643.8 nm is used. For optical transitions Cd acts as an effective two-electron system, i.e., it has a He-like configuration, as has Hg. Compare the Grotrian diagram shown in Fig. 1 to the one for mercury provided in the appendix (cf.. the Franck-Hertz experiment). In order to understand the selection rules for allowed electric dipole transitions follow the arguments given in ref. 4 in the context of the HeNe laser experiment. The two active electrons have combined orbital angular momentum ''L'' = l<sub>1</sub>+l<sub>2</sub>, spin S = s<sub>1</sub>+s<sub>2</sub>, and total angular momentum J = j<sub>1</sub>+j<sub>2</sub>. Allowed transitions require a change of one unit in ''L'' and ''J'' to make up for the spin of the photon, considering that spin flip is unlikely. Using <sup>(2''S''+1)</sup>''L<sub>J</sub>'' notation (with ''L'' = 0 denoted as S, ''L'' = 1 as P, ''L'' = 2 as D, etc) we have for the relevant line a <sup>1</sup>D<sub>2</sub> - <sup>1</sup>P<sub>1</sub> transition. This means that the spins are paired up (spin singlet) and that in nonrelativistic notation a 5s5d to 5s5p transition takes place. In Hg the equivalent line at the n = 6 level is the yellow line at 579 nm. The shift of the same line towards yellow is the result of having an additional electron shell in the core. What wavelength is associated with this transition in He?</p>
<p>For the advanced student: note that the initial level splits into 5 sublevels ''M<sub>J</sub>'' = -2,-1,0,1,2 , while the final state has ''M<sub>J</sub>'' = -1,0,1. They split equidistantly and one can group the nine possible transitions according to the allowed ''ΔM'' = -1,0,1 (cf.. Ref. 3 Fig. 7.3). To understand the polarization of the emitted light in the presence of the external B field note that for an electromagnetic wave its electric, magnetic fields and the wave propagation vector '''k''' form a right-handed coordinate system. Understand the validity of the dipole approximation (the wavelength λ is much longer than atomic dimensions) and how the electric field of the EM wave can be replaced by a constant vector times a temporary oscillatory factor (ref. [3,6]). Convince yourself why no linearly polarized light can be observed in the longitudinal direction as the magnetic field is turned on. Correspondingly understand why circularly polarized light as observed in the longitudinal direction must appear as linearly polarized when observed in a direction transverse to the ''B'' field. Why can all three components associated with the ''ΔM'' = -1,0,1 selection rule be observed in the transverse direction?</p>
<p>To verify these predictions about the polarization states of the light when the B field is turned on you need to recall some optical properties of polarizers and of quarter-wave plates (cf.. Ref.1). By placing these in the correct order you can verify the following:</p>
<p>Using a polarizer when observing in the transverse direction identify the polarization states of the three components with '''B''' turned on; what happens for ''B'' = 0?</p>
<p>Using a quarter-wave plate convert the circularly polarized (CP) light to two perpendicular
linearly polarized components corresponding to left- and right CP light respectively. Use a polarizer to extinguish each of these components separately; what happens for ''B'' = 0?</p>
<p>Collect sufficient data for both observation directions to demonstrate the linearity of the line
splitting with the magnetic field. You will need to perform a calibration of the magnetic field as a function of the current and should comment on possible saturation effects, i.e., a linear behaviour of the splitting with ''B'', but non-linear with the magnet current ''I'' at strong fields, where part of the electric energy may be converted to heat. Determine the gyromagnetic ratio (''g'') from your observations. </p>
<p>'''Warning:''' The Cd lamp emits ultraviolet light in addition to other lines such as the red Cd line. The apparatus contains a narrow-band red filter so that your eyes are protected when observing through the telescope. Avoid looking into the lamp itself (even though it is rated to be safe), i.e., cover the apparatus with a sheet of paper to reduce unnecessary eye contact with the lamp. The lamp takes several minutes to reach a proper operating temperature for the red line to be visible.</p>
<p><b>Obtain assistance when changing the observation direction from transverse to longitudinal or vice versa!</b></p>

<h3> Using the CCD Camera</h3>
<p>A CCD camera (AmScope MU300) is provided to allow for easy viewing of the lines. To operate the CCD, run the program "ToupView", then select ''Acquire''-> ''Live Capture''-> ''UCMOS03100KPA'' from the menu. A window will appear which is the live image being collected by the CCD. The brightness of the image can be changed using the ''Setup'' -> ''View Souce Properties'' -> ''Expose'' tab. A reasonable choice of values is an ''Exposure'' of 700ms with an ''Analog Gain'' of 3. Correct adjustment of the support system will allow you to clearly view the lines. Sliding of the CCD camera in and out will allow for focus. You can Save the image using ''Capture'' -> ''Capture a Frame'' command.</p>
<table width=500 align=center><td>
<p align=justify>[[File:ToupView.png|500px|border|center]]
<b>Figure 3-</b> ToupView CCD Camera Interface.
<br clear=right>
</p>
</td></table>
<p>Once you are able to see nice clear lines as shown in Figure 3, use the ''Region of Interest'' tool to focus in on a few lines in center. You can now use the ''Zoom'' to expand the image. One the left margin of the video image, there is a scale showing the pixel number. You can use this as a fixed reference point- as you increase the applied magnetic field, the line will split into sublevels, check the dial gauge, and then use the adjusting screw to place the shifted line back to the pixel number of the original line, then check the new reading of the dial gauge and record the measurement.</p>

<h1>References</h1>
<ol>
<li>Jenkins F.A., White H.E., ''Fundamentals of Optics'', McGraw-Hill</li>
<li>Brehm J.J., Mullin W.J., ''Introduction to the Structure of Matter'', Wiley</li>
<li>Melissinos A.C., ''Experiments in Modern Physics'', Academic Press</li>
<li>Preston D.W. Dietz E.R., ''The Art of Experimental Physics'',Wiley</li>
<li>Merzbacher E., ''Quantum Mechanics'', Wiley</li>
<li>Bethe H.A., Salpeter E.E., ''Quantum Mechanics of One- and Two-Electron Systems'', Springer</li>
<li>Radzig A.A., Smirnov B.M., ''Reference Data on Atoms Molecules and Ions'', Springer 1985.</li>
</ol>

<h1>Appendix</h1>
<ul>
<li><b>Data for the Lummer-Gehrcke plate</b> [from Leybold’s manual]: ''d'' = 4.04 mm η = 1.4567 </li>
<li>Grotrian diagrams for Cd, Hg, and He taken from ref. 7. A copy of these are in the binder in the laboratory.
<ul>
<li>[[Media:GrotrianH.pdf| Hydrogen]]</li>
<li>[[Media:GrotrianHe.pdf| Helium]]</li>
<li>[[Media:GrotrianHg.pdf| Mercury]]</li>
<li>[[Media:GrotrianHgCd.pdf| Cadmium and Mercury]]</li>
</ul>
</li>
</ul>

Main Page/PHYS 4210/Zeeman Effect

2011-11-01T14:22:07Z

WorkStudy:

<h1>Zeeman Effect</h1>
<p>In this classic experiment that predates the development of quantum mechanics one investigates the light emitted by atoms in the presence of a homogeneous magnetic field. Of particular interest is the observation that this light is polarized in the presence of a magnetic field. The high-resolution spectroscopy required to resolve the line splittings is performed with a multiple-beam interferometer called a Lummer-Gehrcke plate which is similar to a Fabry-Perot interferometer.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Magnetic Sublevels</li>
<li>Total Angular Momentum</li>
<li>TEM wave</li>
<li>Quantization Axis</li>
<li>Orbital Angular Momentum</li>
<li>Spin Angular Momentum</li>
</ul>
</td>

<td width=250>
<ul>
<li>Normal Zeeman Effect</li>
<li>Anomalous Zeeman Effect</li>
<li>Lummer-Gehrcke Plate</li>
<li>Multi-beam Interferometer</li>
<li>Quarter-wave Plate</li>
</ul>
</td>
</table>

<h2> Required Components</h2>

<li>magnet power supply</li>
<li>electromagnet</li>
<li>discharge power supply</li>
<li>CCD camera</li>
<li>Lummer Gerke Plate</li>
<li>polarizers and waveplate</li>

<h1>Introduction</h1>
<p>The Zeeman effect is a powerful demonstration of the splittings of magnetic sublevels in an angular momentum multiplet. Many aspects of the emission of light by excited atoms, particularly when exposed to strong magnetic fields ('''B''') were understood by Lorentz in a classical model [1,2] before the advent of quantum mechanics. It is possible to understand the changes to classical electron orbits due to the Lorentz force in a 3D harmonic oscillator model. When one complements this with the idea that electromagnetic waves are transverse (the associated electric and magnetic fields of the EM wave oscillate in a plane perpendicular to the propagation direction of the wave), one can understand why circularly polarized light emerges as the atoms are observed in a direction longitudinal with the external '''B''' field, and why they appear as plane-polarized as viewed from the transverse direction. The understanding in the classical framework helps to build an intuition about the problem.</p>
<p>In the modern quantum mechanical description [2,5,6] one has to take into account that the presence of the '''B''' field singles out an axis. The additional interaction term between the magnetic moment of the electronic state (proportional to the ''z'' component of the total angular momentum) and '''B''' serves to split the magnetic sublevels of states with non-zero angular momentum. The additional interaction forces the use of this axis as a quantization axis. Without an external field one usually picks a ''z'' axis, but should arrive at results that are independent of this choice. To obtain the observed result that the light emanating from spontaneous transitions without an external field is unpolarized, one has to average over random orientations of the quantization axis. The observation that a definite orientation is singled out as quantization axis in the Zeeman effect is sometimes referred to as ‘space quantization’.</p>
<p>The problem can be illustrated by using pure orbital angular momentum states, i.e.,ignoring spin, and considering an np - ms transition. This transition is an allowed electric dipole transition, since a single unit of orbital angular momentum is changed, and this difference of one unit is carried away in the form of the spin for the spontaneously radiated photon. The important quantity to watch is the change in the projection of the orbital angular momentum, which can be +1, 0, -1 depending on the choice of the magnetic sublevel.</p>
<p>A transition 2p0 - 1s is associated with the emission of linearly polarized light with the oscillating electric field vector aligned with the z axis, with the wave propagation vector being orthogonal to this axis. This can be understood from the fact that the only non-vanishing matrix element for the dipole operator is <1s|''z''|2p<sub>0</sub>>. Similar calculations show that the transitions originating in the ''m'' = 1 and ''m'' = -1 sublevels result in circularly polarized light being emitted, which can propagate in the z direction only. One of the fascinating aspects of the Zeeman experiment is the following. For field-free atoms no axis is singled out, and thus, one has to include all possible orientations of the ''z'' axis, which results in the prediction that the light emitted from free atoms is unpolarized. However, once a homogeneous magnetic field is applied, an axis is singled out in space, which becomes the natural quantization axis. By probing the polarization of the spontaneously emitted light of atoms in the presence of a magnetic field one can verify that indeed the turn-on of the field causes a repopulation of the magnetic sublevels in a way that corresponds to the classical predictions of the Lorentz model. Thus, it is necessary to observe the light emitted longitudinally and transverse to the magnetic field.</p>
<p>The general theory of the Zeeman effect is complicated by the fact that the total angular momentum, i.e., added orbital and spin angular momentum of the active electron has to be considered. Based on orbital angular momentum alone the magnetic moment of an electron in a non-zero m sublevel is an integer multiple of the projection ''l<sub>z</sub>''. Once one couples ''l'' and ''s'' to form ''j'' = ''l'' + ''s'', the magnetic moment can be, but need not to be an integer multiple of ''j<sub>z</sub>'', and the proportionality is given by the Lande factor ''g''. The Lande factor can take on half-integer numbers for the initial and/or final states involved in the transition. One distinguishes between the normal and anomalous Zeeman effects depending on whether this complication arises or not. [3]. The anomalous effect is rather common in atomic transitions, but in this experiment a transition with the normal Zeeman effect has been selected. The red line in cadmium (643.8 nm), which is the equivalent of the yellow line in mercury, cf.. the Grotrian diagram shown in the appendix, and the level diagram in Fig. 1 (which is Fig. 7.3 from Melissinos [3]).
</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Level diagram for the Cd 643.8 nm transition with and without B field.
<br clear=right>
</p>
</td></table>
<p>From an experimental point of view a high demand is placed on the optical resolution of the interferometer. The idea is to inspect the interference pattern for a given line and to observe the quantitative changes in the pattern as the B field is applied to determine the wavelengths of the various components. The high resolution required can be obtained from multiple-beam interferometers, such as the Fabry-Perot (FP) interferometer [1,3]. Melissinos [3] discusses the analysis of the circular fringe pattern as produced by the FP. An easier alternative is provided by a special instrument that perfects the same method, called a Lummer-Gehrcke (LG) plate [1]. Since its interference pattern is more complicated to derive, you should concentrate on understanding the principles of multiple-beam interferometry using the FP and be aware of the analogies. Note that the FP has a wide range of applications in optics.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2a.png|500px|border|center]]
<b>Figure 2a -</b> Multiple reflection between the surfaces of a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2b.png|500px|border|center]]
<b>Figure 2b -</b> Apparatus for the Zeeman experiment with a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<p>The LG plate shown as part of the apparatus in Fig. 2 consists of a precisely polished quartz glass plate of given thickness d with a prism attached at one end so that light entering from the slit has an angle of incidence on the plate that is near the critical angle. This results in some refractive transmission and mostly reflection at the glass/air surface. The reflected light inside the glass plate undergoes multiple ‘bounces’ of this type (interior reflection and partial refractive transmission). Two different interference patterns emerge when looking at a grazing angle at the top or bottom of the LG plate. The pattern formed at the top shows sharp bright lines on a dark background. In contrast to a Michelson interferometer a multiple-beam interferometer such as the FP and LG can produce an uneven interference pattern [1,3].</p>
<p>The separation between the fringes that appear without the magnetic field depends on the angle of observation. This spacing ''ΔA'' has to be determined for the particular fringe chosen for observation. As a magnetic field is applied each bright fringe splits either into two or into three depending on the orientation with respect to the magnetic field. To obtain a quantitative measure of the Zeeman effect, one needs to determine the '''B''' dependent splitting ''ΔS'' relative to ''ΔA''. Making use of the ratio eliminates the need to know the optical magnification, observation angle and distance from the plate. The frequency splitting depends also on the LG plate thickness d (as in the FP case), and additionally on the index of refraction η of the quartz glass. In the FP case this would be equal to 1, but there are versions of the experiment where the evacuation of a sealed FP interferometer is used to produce a scanning effect in the fringe pattern [4]. The frequency splitting can be written as</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn1.png|160px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>The corresponding energy difference should equal</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn2.png|220px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
<p>One common method of determining the electron charge-to-mass ratio is through eq (2). Our interest is, however, to determine the energy splitting as a function of the magnetic field strength ''B''.</p>
<p>In our version of the Zeeman experiment the intense red Cd line at 643.8 nm is used. For optical transitions Cd acts as an effective two-electron system, i.e., it has a He-like configuration, as has Hg. Compare the Grotrian diagram shown in Fig. 1 to the one for mercury provided in the appendix (cf.. the Franck-Hertz experiment). In order to understand the selection rules for allowed electric dipole transitions follow the arguments given in ref. 4 in the context of the HeNe laser experiment. The two active electrons have combined orbital angular momentum ''L'' = l<sub>1</sub>+l<sub>2</sub>, spin S = s<sub>1</sub>+s<sub>2</sub>, and total angular momentum J = j<sub>1</sub>+j<sub>2</sub>. Allowed transitions require a change of one unit in ''L'' and ''J'' to make up for the spin of the photon, considering that spin flip is unlikely. Using <sup>(2''S''+1)</sup>''L<sub>J</sub>'' notation (with ''L'' = 0 denoted as S, ''L'' = 1 as P, ''L'' = 2 as D, etc) we have for the relevant line a <sup>1</sup>D<sub>2</sub> - <sup>1</sup>P<sub>1</sub> transition. This means that the spins are paired up (spin singlet) and that in nonrelativistic notation a 5s5d to 5s5p transition takes place. In Hg the equivalent line at the n = 6 level is the yellow line at 579 nm. The shift of the same line towards yellow is the result of having an additional electron shell in the core. What wavelength is associated with this transition in He?</p>
<p>For the advanced student: note that the initial level splits into 5 sublevels ''M<sub>J</sub>'' = -2,-1,0,1,2 , while the final state has ''M<sub>J</sub>'' = -1,0,1. They split equidistantly and one can group the nine possible transitions according to the allowed ''ΔM'' = -1,0,1 (cf.. Ref. 3 Fig. 7.3). To understand the polarization of the emitted light in the presence of the external B field note that for an electromagnetic wave its electric, magnetic fields and the wave propagation vector '''k''' form a right-handed coordinate system. Understand the validity of the dipole approximation (the wavelength λ is much longer than atomic dimensions) and how the electric field of the EM wave can be replaced by a constant vector times a temporary oscillatory factor (ref. [3,6]). Convince yourself why no linearly polarized light can be observed in the longitudinal direction as the magnetic field is turned on. Correspondingly understand why circularly polarized light as observed in the longitudinal direction must appear as linearly polarized when observed in a direction transverse to the ''B'' field. Why can all three components associated with the ''ΔM'' = -1,0,1 selection rule be observed in the transverse direction?</p>
<p>To verify these predictions about the polarization states of the light when the B field is turned on you need to recall some optical properties of polarizers and of quarter-wave plates (cf.. Ref.1). By placing these in the correct order you can verify the following:</p>
<p>Using a polarizer when observing in the transverse direction identify the polarization states of the three components with '''B''' turned on; what happens for ''B'' = 0?</p>
<p>Using a quarter-wave plate convert the circularly polarized (CP) light to two perpendicular
linearly polarized components corresponding to left- and right CP light respectively. Use a polarizer to extinguish each of these components separately; what happens for ''B'' = 0?</p>
<p>Collect sufficient data for both observation directions to demonstrate the linearity of the line
splitting with the magnetic field. You will need to perform a calibration of the magnetic field as a function of the current and should comment on possible saturation effects, i.e., a linear behaviour of the splitting with ''B'', but non-linear with the magnet current ''I'' at strong fields, where part of the electric energy may be converted to heat. Determine the gyromagnetic ratio (''g'') from your observations. </p>
<p>'''Warning:''' The Cd lamp emits ultraviolet light in addition to other lines such as the red Cd line. The apparatus contains a narrow-band red filter so that your eyes are protected when observing through the telescope. Avoid looking into the lamp itself (even though it is rated to be safe), i.e., cover the apparatus with a sheet of paper to reduce unnecessary eye contact with the lamp. The lamp takes several minutes to reach a proper operating temperature for the red line to be visible.</p>
<p><b>Obtain assistance when changing the observation direction from transverse to longitudinal or vice versa!</b></p>

<h3> Using the CCD Camera</h3>
<p>A CCD camera (AmScope MU300) is provided to allow for easy viewing of the lines. To operate the CCD, run the program "ToupView", then select ''Acquire''-> ''Live Capture''-> ''UCMOS03100KPA'' from the menu. A window will appear which is the live image being collected by the CCD. The brightness of the image can be changed using the ''Setup'' -> ''View Souce Properties'' -> ''Expose'' tab. A reasonable choice of values is an ''Exposure'' of 700ms with an ''Analog Gain'' of 3. Correct adjustment of the support system will allow you to clearly view the lines. Sliding of the CCD camera in and out will allow for focus. You can Save the image using ''Capture'' -> ''Capture a Frame'' command.</p>
<table width=500 align=center><td>
<p align=justify>[[File:ToupView.png|500px|border|center]]
<b>Figure 3-</b> ToupView CCD Camera Interface.
<br clear=right>
</p>
</td></table>
<p>Once you are able to see nice clear lines as shown in Figure 3, use the ''Region of Interest'' tool to focus in on a few lines in center. You can now use the ''Zoom'' to expand the image. One the left margin of the video image, there is a scale showing the pixel number. You can use this as a fixed reference point- as you increase the applied magnetic field, the line will split into sublevels, check the dial gauge, and then use the adjusting screw to place the shifted line back to the pixel number of the original line, then check the new reading of the dial gauge and record the measurement.</p>

<h1>References</h1>
<ol>
<li>Jenkins F.A., White H.E., ''Fundamentals of Optics'', McGraw-Hill</li>
<li>Brehm J.J., Mullin W.J., ''Introduction to the Structure of Matter'', Wiley</li>
<li>Melissinos A.C., ''Experiments in Modern Physics'', Academic Press</li>
<li>Preston D.W. Dietz E.R., ''The Art of Experimental Physics'',Wiley</li>
<li>Merzbacher E., ''Quantum Mechanics'', Wiley</li>
<li>Bethe H.A., Salpeter E.E., ''Quantum Mechanics of One- and Two-Electron Systems'', Springer</li>
<li>Radzig A.A., Smirnov B.M., ''Reference Data on Atoms Molecules and Ions'', Springer 1985.</li>
</ol>

<h1>Appendix</h1>
<ul>
<li><b>Data for the Lummer-Gehrcke plate</b> [from Leybold’s manual]: ''d'' = 4.04 mm η = 1.4567 </li>
<li>Grotrian diagrams for Cd, Hg, and He taken from ref. 7. A copy of these are in the binder in the laboratory.
<ul>
<li>[[Media:GrotrianH.pdf| Hydrogen]]</li>
<li>[[Media:GrotrianHe.pdf| Helium]]</li>
<li>[[Media:GrotrianHg.pdf| Mercury]]</li>
<li>[[Media:GrotrianHgCd.pdf| Cadmium and Mercury]]</li>
</ul>
</li>
</ul>

Main Page/PHYS 4210/Zeeman Effect

2011-11-01T14:19:53Z

WorkStudy:

<h1>Zeeman Effect</h1>
<p>In this classic experiment that predates the development of quantum mechanics one investigates the light emitted by atoms in the presence of a homogeneous magnetic field. Of particular interest is the observation that this light is polarized in the presence of a magnetic field. The high-resolution spectroscopy required to resolve the line splittings is performed with a multiple-beam interferometer called a Lummer-Gehrcke plate which is similar to a Fabry-Perot interferometer.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Magnetic Sublevels</li>
<li>Total Angular Momentum</li>
<li>TEM wave</li>
<li>Quantization Axis</li>
<li>Orbital Angular Momentum</li>
<li>Spin Angular Momentum</li>
</ul>
</td>

<td width=250>
<ul>
<li>Normal Zeeman Effect</li>
<li>Anomalous Zeeman Effect</li>
<li>Lummer-Gehrcke Plate</li>
<li>Multi-beam Interferometer</li>
<li>Quarter-wave Plate</li>
</ul>
</td>
</table>

<h2> Required Components</h2>

magnet power supply
electromagnet
discharge power supply
CCD camera
Lummer Gerke Plate
polarizers and waveplate

<h1>Introduction</h1>
<p>The Zeeman effect is a powerful demonstration of the splittings of magnetic sublevels in an angular momentum multiplet. Many aspects of the emission of light by excited atoms, particularly when exposed to strong magnetic fields ('''B''') were understood by Lorentz in a classical model [1,2] before the advent of quantum mechanics. It is possible to understand the changes to classical electron orbits due to the Lorentz force in a 3D harmonic oscillator model. When one complements this with the idea that electromagnetic waves are transverse (the associated electric and magnetic fields of the EM wave oscillate in a plane perpendicular to the propagation direction of the wave), one can understand why circularly polarized light emerges as the atoms are observed in a direction longitudinal with the external '''B''' field, and why they appear as plane-polarized as viewed from the transverse direction. The understanding in the classical framework helps to build an intuition about the problem.</p>
<p>In the modern quantum mechanical description [2,5,6] one has to take into account that the presence of the '''B''' field singles out an axis. The additional interaction term between the magnetic moment of the electronic state (proportional to the ''z'' component of the total angular momentum) and '''B''' serves to split the magnetic sublevels of states with non-zero angular momentum. The additional interaction forces the use of this axis as a quantization axis. Without an external field one usually picks a ''z'' axis, but should arrive at results that are independent of this choice. To obtain the observed result that the light emanating from spontaneous transitions without an external field is unpolarized, one has to average over random orientations of the quantization axis. The observation that a definite orientation is singled out as quantization axis in the Zeeman effect is sometimes referred to as ‘space quantization’.</p>
<p>The problem can be illustrated by using pure orbital angular momentum states, i.e.,ignoring spin, and considering an np - ms transition. This transition is an allowed electric dipole transition, since a single unit of orbital angular momentum is changed, and this difference of one unit is carried away in the form of the spin for the spontaneously radiated photon. The important quantity to watch is the change in the projection of the orbital angular momentum, which can be +1, 0, -1 depending on the choice of the magnetic sublevel.</p>
<p>A transition 2p0 - 1s is associated with the emission of linearly polarized light with the oscillating electric field vector aligned with the z axis, with the wave propagation vector being orthogonal to this axis. This can be understood from the fact that the only non-vanishing matrix element for the dipole operator is <1s|''z''|2p<sub>0</sub>>. Similar calculations show that the transitions originating in the ''m'' = 1 and ''m'' = -1 sublevels result in circularly polarized light being emitted, which can propagate in the z direction only. One of the fascinating aspects of the Zeeman experiment is the following. For field-free atoms no axis is singled out, and thus, one has to include all possible orientations of the ''z'' axis, which results in the prediction that the light emitted from free atoms is unpolarized. However, once a homogeneous magnetic field is applied, an axis is singled out in space, which becomes the natural quantization axis. By probing the polarization of the spontaneously emitted light of atoms in the presence of a magnetic field one can verify that indeed the turn-on of the field causes a repopulation of the magnetic sublevels in a way that corresponds to the classical predictions of the Lorentz model. Thus, it is necessary to observe the light emitted longitudinally and transverse to the magnetic field.</p>
<p>The general theory of the Zeeman effect is complicated by the fact that the total angular momentum, i.e., added orbital and spin angular momentum of the active electron has to be considered. Based on orbital angular momentum alone the magnetic moment of an electron in a non-zero m sublevel is an integer multiple of the projection ''l<sub>z</sub>''. Once one couples ''l'' and ''s'' to form ''j'' = ''l'' + ''s'', the magnetic moment can be, but need not to be an integer multiple of ''j<sub>z</sub>'', and the proportionality is given by the Lande factor ''g''. The Lande factor can take on half-integer numbers for the initial and/or final states involved in the transition. One distinguishes between the normal and anomalous Zeeman effects depending on whether this complication arises or not. [3]. The anomalous effect is rather common in atomic transitions, but in this experiment a transition with the normal Zeeman effect has been selected. The red line in cadmium (643.8 nm), which is the equivalent of the yellow line in mercury, cf.. the Grotrian diagram shown in the appendix, and the level diagram in Fig. 1 (which is Fig. 7.3 from Melissinos [3]).
</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Level diagram for the Cd 643.8 nm transition with and without B field.
<br clear=right>
</p>
</td></table>
<p>From an experimental point of view a high demand is placed on the optical resolution of the interferometer. The idea is to inspect the interference pattern for a given line and to observe the quantitative changes in the pattern as the B field is applied to determine the wavelengths of the various components. The high resolution required can be obtained from multiple-beam interferometers, such as the Fabry-Perot (FP) interferometer [1,3]. Melissinos [3] discusses the analysis of the circular fringe pattern as produced by the FP. An easier alternative is provided by a special instrument that perfects the same method, called a Lummer-Gehrcke (LG) plate [1]. Since its interference pattern is more complicated to derive, you should concentrate on understanding the principles of multiple-beam interferometry using the FP and be aware of the analogies. Note that the FP has a wide range of applications in optics.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2a.png|500px|border|center]]
<b>Figure 2a -</b> Multiple reflection between the surfaces of a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2b.png|500px|border|center]]
<b>Figure 2b -</b> Apparatus for the Zeeman experiment with a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<p>The LG plate shown as part of the apparatus in Fig. 2 consists of a precisely polished quartz glass plate of given thickness d with a prism attached at one end so that light entering from the slit has an angle of incidence on the plate that is near the critical angle. This results in some refractive transmission and mostly reflection at the glass/air surface. The reflected light inside the glass plate undergoes multiple ‘bounces’ of this type (interior reflection and partial refractive transmission). Two different interference patterns emerge when looking at a grazing angle at the top or bottom of the LG plate. The pattern formed at the top shows sharp bright lines on a dark background. In contrast to a Michelson interferometer a multiple-beam interferometer such as the FP and LG can produce an uneven interference pattern [1,3].</p>
<p>The separation between the fringes that appear without the magnetic field depends on the angle of observation. This spacing ''ΔA'' has to be determined for the particular fringe chosen for observation. As a magnetic field is applied each bright fringe splits either into two or into three depending on the orientation with respect to the magnetic field. To obtain a quantitative measure of the Zeeman effect, one needs to determine the '''B''' dependent splitting ''ΔS'' relative to ''ΔA''. Making use of the ratio eliminates the need to know the optical magnification, observation angle and distance from the plate. The frequency splitting depends also on the LG plate thickness d (as in the FP case), and additionally on the index of refraction η of the quartz glass. In the FP case this would be equal to 1, but there are versions of the experiment where the evacuation of a sealed FP interferometer is used to produce a scanning effect in the fringe pattern [4]. The frequency splitting can be written as</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn1.png|160px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>The corresponding energy difference should equal</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn2.png|220px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
<p>One common method of determining the electron charge-to-mass ratio is through eq (2). Our interest is, however, to determine the energy splitting as a function of the magnetic field strength ''B''.</p>
<p>In our version of the Zeeman experiment the intense red Cd line at 643.8 nm is used. For optical transitions Cd acts as an effective two-electron system, i.e., it has a He-like configuration, as has Hg. Compare the Grotrian diagram shown in Fig. 1 to the one for mercury provided in the appendix (cf.. the Franck-Hertz experiment). In order to understand the selection rules for allowed electric dipole transitions follow the arguments given in ref. 4 in the context of the HeNe laser experiment. The two active electrons have combined orbital angular momentum ''L'' = l<sub>1</sub>+l<sub>2</sub>, spin S = s<sub>1</sub>+s<sub>2</sub>, and total angular momentum J = j<sub>1</sub>+j<sub>2</sub>. Allowed transitions require a change of one unit in ''L'' and ''J'' to make up for the spin of the photon, considering that spin flip is unlikely. Using <sup>(2''S''+1)</sup>''L<sub>J</sub>'' notation (with ''L'' = 0 denoted as S, ''L'' = 1 as P, ''L'' = 2 as D, etc) we have for the relevant line a <sup>1</sup>D<sub>2</sub> - <sup>1</sup>P<sub>1</sub> transition. This means that the spins are paired up (spin singlet) and that in nonrelativistic notation a 5s5d to 5s5p transition takes place. In Hg the equivalent line at the n = 6 level is the yellow line at 579 nm. The shift of the same line towards yellow is the result of having an additional electron shell in the core. What wavelength is associated with this transition in He?</p>
<p>For the advanced student: note that the initial level splits into 5 sublevels ''M<sub>J</sub>'' = -2,-1,0,1,2 , while the final state has ''M<sub>J</sub>'' = -1,0,1. They split equidistantly and one can group the nine possible transitions according to the allowed ''ΔM'' = -1,0,1 (cf.. Ref. 3 Fig. 7.3). To understand the polarization of the emitted light in the presence of the external B field note that for an electromagnetic wave its electric, magnetic fields and the wave propagation vector '''k''' form a right-handed coordinate system. Understand the validity of the dipole approximation (the wavelength λ is much longer than atomic dimensions) and how the electric field of the EM wave can be replaced by a constant vector times a temporary oscillatory factor (ref. [3,6]). Convince yourself why no linearly polarized light can be observed in the longitudinal direction as the magnetic field is turned on. Correspondingly understand why circularly polarized light as observed in the longitudinal direction must appear as linearly polarized when observed in a direction transverse to the ''B'' field. Why can all three components associated with the ''ΔM'' = -1,0,1 selection rule be observed in the transverse direction?</p>
<p>To verify these predictions about the polarization states of the light when the B field is turned on you need to recall some optical properties of polarizers and of quarter-wave plates (cf.. Ref.1). By placing these in the correct order you can verify the following:</p>
<p>Using a polarizer when observing in the transverse direction identify the polarization states of the three components with '''B''' turned on; what happens for ''B'' = 0?</p>
<p>Using a quarter-wave plate convert the circularly polarized (CP) light to two perpendicular
linearly polarized components corresponding to left- and right CP light respectively. Use a polarizer to extinguish each of these components separately; what happens for ''B'' = 0?</p>
<p>Collect sufficient data for both observation directions to demonstrate the linearity of the line
splitting with the magnetic field. You will need to perform a calibration of the magnetic field as a function of the current and should comment on possible saturation effects, i.e., a linear behaviour of the splitting with ''B'', but non-linear with the magnet current ''I'' at strong fields, where part of the electric energy may be converted to heat. Determine the gyromagnetic ratio (''g'') from your observations. </p>
<p>'''Warning:''' The Cd lamp emits ultraviolet light in addition to other lines such as the red Cd line. The apparatus contains a narrow-band red filter so that your eyes are protected when observing through the telescope. Avoid looking into the lamp itself (even though it is rated to be safe), i.e., cover the apparatus with a sheet of paper to reduce unnecessary eye contact with the lamp. The lamp takes several minutes to reach a proper operating temperature for the red line to be visible.</p>
<p><b>Obtain assistance when changing the observation direction from transverse to longitudinal or vice versa!</b></p>

<h3> Using the CCD Camera</h3>
<p>A CCD camera (AmScope MU300) is provided to allow for easy viewing of the lines. To operate the CCD, run the program "ToupView", then select ''Acquire''-> ''Live Capture''-> ''UCMOS03100KPA'' from the menu. A window will appear which is the live image being collected by the CCD. The brightness of the image can be changed using the ''Setup'' -> ''View Souce Properties'' -> ''Expose'' tab. A reasonable choice of values is an ''Exposure'' of 700ms with an ''Analog Gain'' of 3. Correct adjustment of the support system will allow you to clearly view the lines. Sliding of the CCD camera in and out will allow for focus. You can Save the image using ''Capture'' -> ''Capture a Frame'' command.</p>
<table width=500 align=center><td>
<p align=justify>[[File:ToupView.png|500px|border|center]]
<b>Figure 3-</b> ToupView CCD Camera Interface.
<br clear=right>
</p>
</td></table>
<p>Once you are able to see nice clear lines as shown in Figure 3, use the ''Region of Interest'' tool to focus in on a few lines in center. You can now use the ''Zoom'' to expand the image. One the left margin of the video image, there is a scale showing the pixel number. You can use this as a fixed reference point- as you increase the applied magnetic field, the line will split into sublevels, check the dial gauge, and then use the adjusting screw to place the shifted line back to the pixel number of the original line, then check the new reading of the dial gauge and record the measurement.</p>

<h1>References</h1>
<ol>
<li>Jenkins F.A., White H.E., ''Fundamentals of Optics'', McGraw-Hill</li>
<li>Brehm J.J., Mullin W.J., ''Introduction to the Structure of Matter'', Wiley</li>
<li>Melissinos A.C., ''Experiments in Modern Physics'', Academic Press</li>
<li>Preston D.W. Dietz E.R., ''The Art of Experimental Physics'',Wiley</li>
<li>Merzbacher E., ''Quantum Mechanics'', Wiley</li>
<li>Bethe H.A., Salpeter E.E., ''Quantum Mechanics of One- and Two-Electron Systems'', Springer</li>
<li>Radzig A.A., Smirnov B.M., ''Reference Data on Atoms Molecules and Ions'', Springer 1985.</li>
</ol>

<h1>Appendix</h1>
<ul>
<li><b>Data for the Lummer-Gehrcke plate</b> [from Leybold’s manual]: ''d'' = 4.04 mm η = 1.4567 </li>
<li>Grotrian diagrams for Cd, Hg, and He taken from ref. 7. A copy of these are in the binder in the laboratory.
<ul>
<li>[[Media:GrotrianH.pdf| Hydrogen]]</li>
<li>[[Media:GrotrianHe.pdf| Helium]]</li>
<li>[[Media:GrotrianHg.pdf| Mercury]]</li>
<li>[[Media:GrotrianHgCd.pdf| Cadmium and Mercury]]</li>
</ul>
</li>
</ul>

Main Page/PHYS 4210/Zeeman Effect

2011-11-01T14:14:07Z

WorkStudy:

<h1>Zeeman Effect</h1>
<p>In this classic experiment that predates the development of quantum mechanics one investigates the light emitted by atoms in the presence of a homogeneous magnetic field. Of particular interest is the observation that this light is polarized in the presence of a magnetic field. The high-resolution spectroscopy required to resolve the line splittings is performed with a multiple-beam interferometer called a Lummer-Gehrcke plate which is similar to a Fabry-Perot interferometer.</p>

<h2>Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Magnetic Sublevels</li>
<li>Total Angular Momentum</li>
<li>TEM wave</li>
<li>Quantization Axis</li>
<li>Orbital Angular Momentum</li>
<li>Spin Angular Momentum</li>

<h2> Required Components</h2>

magnet power supply
electromagnet
discharge power supply
CCD camera

</ul>
</td>
<td width=250>
<ul>
<li>Normal Zeeman Effect</li>
<li>Anomalous Zeeman Effect</li>
<li>Lummer-Gehrcke Plate</li>
<li>Multi-beam Interferometer</li>
<li>Quarter-wave Plate</li>
</ul>
</td>
</table>

<h1>Introduction</h1>
<p>The Zeeman effect is a powerful demonstration of the splittings of magnetic sublevels in an angular momentum multiplet. Many aspects of the emission of light by excited atoms, particularly when exposed to strong magnetic fields ('''B''') were understood by Lorentz in a classical model [1,2] before the advent of quantum mechanics. It is possible to understand the changes to classical electron orbits due to the Lorentz force in a 3D harmonic oscillator model. When one complements this with the idea that electromagnetic waves are transverse (the associated electric and magnetic fields of the EM wave oscillate in a plane perpendicular to the propagation direction of the wave), one can understand why circularly polarized light emerges as the atoms are observed in a direction longitudinal with the external '''B''' field, and why they appear as plane-polarized as viewed from the transverse direction. The understanding in the classical framework helps to build an intuition about the problem.</p>
<p>In the modern quantum mechanical description [2,5,6] one has to take into account that the presence of the '''B''' field singles out an axis. The additional interaction term between the magnetic moment of the electronic state (proportional to the ''z'' component of the total angular momentum) and '''B''' serves to split the magnetic sublevels of states with non-zero angular momentum. The additional interaction forces the use of this axis as a quantization axis. Without an external field one usually picks a ''z'' axis, but should arrive at results that are independent of this choice. To obtain the observed result that the light emanating from spontaneous transitions without an external field is unpolarized, one has to average over random orientations of the quantization axis. The observation that a definite orientation is singled out as quantization axis in the Zeeman effect is sometimes referred to as ‘space quantization’.</p>
<p>The problem can be illustrated by using pure orbital angular momentum states, i.e.,ignoring spin, and considering an np - ms transition. This transition is an allowed electric dipole transition, since a single unit of orbital angular momentum is changed, and this difference of one unit is carried away in the form of the spin for the spontaneously radiated photon. The important quantity to watch is the change in the projection of the orbital angular momentum, which can be +1, 0, -1 depending on the choice of the magnetic sublevel.</p>
<p>A transition 2p0 - 1s is associated with the emission of linearly polarized light with the oscillating electric field vector aligned with the z axis, with the wave propagation vector being orthogonal to this axis. This can be understood from the fact that the only non-vanishing matrix element for the dipole operator is <1s|''z''|2p<sub>0</sub>>. Similar calculations show that the transitions originating in the ''m'' = 1 and ''m'' = -1 sublevels result in circularly polarized light being emitted, which can propagate in the z direction only. One of the fascinating aspects of the Zeeman experiment is the following. For field-free atoms no axis is singled out, and thus, one has to include all possible orientations of the ''z'' axis, which results in the prediction that the light emitted from free atoms is unpolarized. However, once a homogeneous magnetic field is applied, an axis is singled out in space, which becomes the natural quantization axis. By probing the polarization of the spontaneously emitted light of atoms in the presence of a magnetic field one can verify that indeed the turn-on of the field causes a repopulation of the magnetic sublevels in a way that corresponds to the classical predictions of the Lorentz model. Thus, it is necessary to observe the light emitted longitudinally and transverse to the magnetic field.</p>
<p>The general theory of the Zeeman effect is complicated by the fact that the total angular momentum, i.e., added orbital and spin angular momentum of the active electron has to be considered. Based on orbital angular momentum alone the magnetic moment of an electron in a non-zero m sublevel is an integer multiple of the projection ''l<sub>z</sub>''. Once one couples ''l'' and ''s'' to form ''j'' = ''l'' + ''s'', the magnetic moment can be, but need not to be an integer multiple of ''j<sub>z</sub>'', and the proportionality is given by the Lande factor ''g''. The Lande factor can take on half-integer numbers for the initial and/or final states involved in the transition. One distinguishes between the normal and anomalous Zeeman effects depending on whether this complication arises or not. [3]. The anomalous effect is rather common in atomic transitions, but in this experiment a transition with the normal Zeeman effect has been selected. The red line in cadmium (643.8 nm), which is the equivalent of the yellow line in mercury, cf.. the Grotrian diagram shown in the appendix, and the level diagram in Fig. 1 (which is Fig. 7.3 from Melissinos [3]).
</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Level diagram for the Cd 643.8 nm transition with and without B field.
<br clear=right>
</p>
</td></table>
<p>From an experimental point of view a high demand is placed on the optical resolution of the interferometer. The idea is to inspect the interference pattern for a given line and to observe the quantitative changes in the pattern as the B field is applied to determine the wavelengths of the various components. The high resolution required can be obtained from multiple-beam interferometers, such as the Fabry-Perot (FP) interferometer [1,3]. Melissinos [3] discusses the analysis of the circular fringe pattern as produced by the FP. An easier alternative is provided by a special instrument that perfects the same method, called a Lummer-Gehrcke (LG) plate [1]. Since its interference pattern is more complicated to derive, you should concentrate on understanding the principles of multiple-beam interferometry using the FP and be aware of the analogies. Note that the FP has a wide range of applications in optics.</p>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2a.png|500px|border|center]]
<b>Figure 2a -</b> Multiple reflection between the surfaces of a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:Zee-fig2b.png|500px|border|center]]
<b>Figure 2b -</b> Apparatus for the Zeeman experiment with a Lummer-Gehrcke plate.
<br clear=right>
</p>
</td></table>
<p>The LG plate shown as part of the apparatus in Fig. 2 consists of a precisely polished quartz glass plate of given thickness d with a prism attached at one end so that light entering from the slit has an angle of incidence on the plate that is near the critical angle. This results in some refractive transmission and mostly reflection at the glass/air surface. The reflected light inside the glass plate undergoes multiple ‘bounces’ of this type (interior reflection and partial refractive transmission). Two different interference patterns emerge when looking at a grazing angle at the top or bottom of the LG plate. The pattern formed at the top shows sharp bright lines on a dark background. In contrast to a Michelson interferometer a multiple-beam interferometer such as the FP and LG can produce an uneven interference pattern [1,3].</p>
<p>The separation between the fringes that appear without the magnetic field depends on the angle of observation. This spacing ''ΔA'' has to be determined for the particular fringe chosen for observation. As a magnetic field is applied each bright fringe splits either into two or into three depending on the orientation with respect to the magnetic field. To obtain a quantitative measure of the Zeeman effect, one needs to determine the '''B''' dependent splitting ''ΔS'' relative to ''ΔA''. Making use of the ratio eliminates the need to know the optical magnification, observation angle and distance from the plate. The frequency splitting depends also on the LG plate thickness d (as in the FP case), and additionally on the index of refraction η of the quartz glass. In the FP case this would be equal to 1, but there are versions of the experiment where the evacuation of a sealed FP interferometer is used to produce a scanning effect in the fringe pattern [4]. The frequency splitting can be written as</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn1.png|160px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>The corresponding energy difference should equal</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Zee-eqn2.png|220px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
<p>One common method of determining the electron charge-to-mass ratio is through eq (2). Our interest is, however, to determine the energy splitting as a function of the magnetic field strength ''B''.</p>
<p>In our version of the Zeeman experiment the intense red Cd line at 643.8 nm is used. For optical transitions Cd acts as an effective two-electron system, i.e., it has a He-like configuration, as has Hg. Compare the Grotrian diagram shown in Fig. 1 to the one for mercury provided in the appendix (cf.. the Franck-Hertz experiment). In order to understand the selection rules for allowed electric dipole transitions follow the arguments given in ref. 4 in the context of the HeNe laser experiment. The two active electrons have combined orbital angular momentum ''L'' = l<sub>1</sub>+l<sub>2</sub>, spin S = s<sub>1</sub>+s<sub>2</sub>, and total angular momentum J = j<sub>1</sub>+j<sub>2</sub>. Allowed transitions require a change of one unit in ''L'' and ''J'' to make up for the spin of the photon, considering that spin flip is unlikely. Using <sup>(2''S''+1)</sup>''L<sub>J</sub>'' notation (with ''L'' = 0 denoted as S, ''L'' = 1 as P, ''L'' = 2 as D, etc) we have for the relevant line a <sup>1</sup>D<sub>2</sub> - <sup>1</sup>P<sub>1</sub> transition. This means that the spins are paired up (spin singlet) and that in nonrelativistic notation a 5s5d to 5s5p transition takes place. In Hg the equivalent line at the n = 6 level is the yellow line at 579 nm. The shift of the same line towards yellow is the result of having an additional electron shell in the core. What wavelength is associated with this transition in He?</p>
<p>For the advanced student: note that the initial level splits into 5 sublevels ''M<sub>J</sub>'' = -2,-1,0,1,2 , while the final state has ''M<sub>J</sub>'' = -1,0,1. They split equidistantly and one can group the nine possible transitions according to the allowed ''ΔM'' = -1,0,1 (cf.. Ref. 3 Fig. 7.3). To understand the polarization of the emitted light in the presence of the external B field note that for an electromagnetic wave its electric, magnetic fields and the wave propagation vector '''k''' form a right-handed coordinate system. Understand the validity of the dipole approximation (the wavelength λ is much longer than atomic dimensions) and how the electric field of the EM wave can be replaced by a constant vector times a temporary oscillatory factor (ref. [3,6]). Convince yourself why no linearly polarized light can be observed in the longitudinal direction as the magnetic field is turned on. Correspondingly understand why circularly polarized light as observed in the longitudinal direction must appear as linearly polarized when observed in a direction transverse to the ''B'' field. Why can all three components associated with the ''ΔM'' = -1,0,1 selection rule be observed in the transverse direction?</p>
<p>To verify these predictions about the polarization states of the light when the B field is turned on you need to recall some optical properties of polarizers and of quarter-wave plates (cf.. Ref.1). By placing these in the correct order you can verify the following:</p>
<p>Using a polarizer when observing in the transverse direction identify the polarization states of the three components with '''B''' turned on; what happens for ''B'' = 0?</p>
<p>Using a quarter-wave plate convert the circularly polarized (CP) light to two perpendicular
linearly polarized components corresponding to left- and right CP light respectively. Use a polarizer to extinguish each of these components separately; what happens for ''B'' = 0?</p>
<p>Collect sufficient data for both observation directions to demonstrate the linearity of the line
splitting with the magnetic field. You will need to perform a calibration of the magnetic field as a function of the current and should comment on possible saturation effects, i.e., a linear behaviour of the splitting with ''B'', but non-linear with the magnet current ''I'' at strong fields, where part of the electric energy may be converted to heat. Determine the gyromagnetic ratio (''g'') from your observations. </p>
<p>'''Warning:''' The Cd lamp emits ultraviolet light in addition to other lines such as the red Cd line. The apparatus contains a narrow-band red filter so that your eyes are protected when observing through the telescope. Avoid looking into the lamp itself (even though it is rated to be safe), i.e., cover the apparatus with a sheet of paper to reduce unnecessary eye contact with the lamp. The lamp takes several minutes to reach a proper operating temperature for the red line to be visible.</p>
<p><b>Obtain assistance when changing the observation direction from transverse to longitudinal or vice versa!</b></p>

<h3> Using the CCD Camera</h3>
<p>A CCD camera (AmScope MU300) is provided to allow for easy viewing of the lines. To operate the CCD, run the program "ToupView", then select ''Acquire''-> ''Live Capture''-> ''UCMOS03100KPA'' from the menu. A window will appear which is the live image being collected by the CCD. The brightness of the image can be changed using the ''Setup'' -> ''View Souce Properties'' -> ''Expose'' tab. A reasonable choice of values is an ''Exposure'' of 700ms with an ''Analog Gain'' of 3. Correct adjustment of the support system will allow you to clearly view the lines. Sliding of the CCD camera in and out will allow for focus. You can Save the image using ''Capture'' -> ''Capture a Frame'' command.</p>
<table width=500 align=center><td>
<p align=justify>[[File:ToupView.png|500px|border|center]]
<b>Figure 3-</b> ToupView CCD Camera Interface.
<br clear=right>
</p>
</td></table>
<p>Once you are able to see nice clear lines as shown in Figure 3, use the ''Region of Interest'' tool to focus in on a few lines in center. You can now use the ''Zoom'' to expand the image. One the left margin of the video image, there is a scale showing the pixel number. You can use this as a fixed reference point- as you increase the applied magnetic field, the line will split into sublevels, check the dial gauge, and then use the adjusting screw to place the shifted line back to the pixel number of the original line, then check the new reading of the dial gauge and record the measurement.</p>

<h1>References</h1>
<ol>
<li>Jenkins F.A., White H.E., ''Fundamentals of Optics'', McGraw-Hill</li>
<li>Brehm J.J., Mullin W.J., ''Introduction to the Structure of Matter'', Wiley</li>
<li>Melissinos A.C., ''Experiments in Modern Physics'', Academic Press</li>
<li>Preston D.W. Dietz E.R., ''The Art of Experimental Physics'',Wiley</li>
<li>Merzbacher E., ''Quantum Mechanics'', Wiley</li>
<li>Bethe H.A., Salpeter E.E., ''Quantum Mechanics of One- and Two-Electron Systems'', Springer</li>
<li>Radzig A.A., Smirnov B.M., ''Reference Data on Atoms Molecules and Ions'', Springer 1985.</li>
</ol>

<h1>Appendix</h1>
<ul>
<li><b>Data for the Lummer-Gehrcke plate</b> [from Leybold’s manual]: ''d'' = 4.04 mm η = 1.4567 </li>
<li>Grotrian diagrams for Cd, Hg, and He taken from ref. 7. A copy of these are in the binder in the laboratory.
<ul>
<li>[[Media:GrotrianH.pdf| Hydrogen]]</li>
<li>[[Media:GrotrianHe.pdf| Helium]]</li>
<li>[[Media:GrotrianHg.pdf| Mercury]]</li>
<li>[[Media:GrotrianHgCd.pdf| Cadmium and Mercury]]</li>
</ul>
</li>
</ul>

Main Page/PHYS 3220/Rutherford I

2011-10-20T16:17:09Z

WorkStudy:

<h1>Rutherford Scattering I</h1>

<h1>Introduction</h1>
<p>The Rutherford scattering experiment, where alpha particles (doubly charged Helium nuclei) were scattered off of a target (gold, aluminum, etc.) represents one of the most important experiments of this century. While the bulk of the alpha particles were scattered at small angles, indicating a soft collision process, a finite number of alpha particles however did scatter at very large angles. This could only have occurred though a collision with a massive object. From the distance of closest approach of the alpha with this object, and using information on the size of the whole atom, we came to know that the atom was mostly empty space. The results of this experiment formed the basis of subatomic structure, as we know it today – that the atom has a hard central core consisting of a tiny but massive core called the nucleus, surrounded by electrons, forming an electrically neutral system. </p>

<p>In this experiment we reproduce the results of Rutherfordby allowing alpha particles from a radioactive source (Am-241) to impinge on thin gold foil. We then compare the experimentally observed differential cross section (related to the number of detected alpha particles), as a function of the angle of scatter of the alpha particles off the target atoms. By comparing these particles to the theoretical expectations of elastic scattering of two particles, we can confirm that the alpha predominantly scatter off the nuclear core of the atom, and hence the structure of the atom is as Rutherford suggested.</p>

<h1>Theory</h1>

<table width=400 align=center><td>
<p align=justify>[[File:RuthI-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Particle scattering.
<br clear=right>
</p>
</td></table>

<p>The theoretical analysis of the scattering cross section can be based on classical or quantum mechanics. In classical mechanics the number of particles scattered at a certain angle (θ) is a unique function of the impact parameter (b). We assume that a pure Coulomb potential is valid, but the appropriateness of this assumption shall be discussed later. To obtain the scattering cross section classically, first one solves Newton's equation of motion to obtain the relationship between impact parameter and scattering angle, and the results is that <b>b α 1/θ</b>. Small impact parameters thus lead to close encounters of the two charged objects, and thus large scattering angles. Conversely, distant collisions lead to small scattering angles. In quantum mechanics this relationship is not unique, but interestingly, a probability distribution arises for the particles to reach deflection angles θ that are synonymous with the classical answer (this is a special feature of the Coulomb potential). Although this relation tells us we are on the right track intuitively, unfortunately it is not very useful since we cannot measure b in any given interaction. We have to relate the scattered angle to something we can measure: the number of alpha particles scattered into the solid angle of the detecting device. </p>

<p>The number of particles scattered at a given angle depends on the change of the impact parameter with the scattering angle, and this can be verified from any book on modern or subatomic physics (e.g. ref 3,4). If the incident charged particle has charge Z<sub>1</sub> and energy E, and hits a target of charge Z<sub>2</sub>, the Rutherford differential cross section can be written as</p>

<table width=500 align=center>
<tr><td>
<p align=justify>[[File:RuthI-eqn1.png|350px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

<p><b>Q:</b> Calculate the total cross section (hint: it diverges!!!). How is it that we continue to use this formalism when the prediction is infinity?</p>

<p>We have thus related the theoretical expectation (labelled Rutherford) to the measured variables. It is useful to review the article in Melissinos [1] as one can appreciate how careful experimental procedure is so crucial, and also the various factors that can affect the final result. There are several subtle (and some not so subtle) points one has to take into consideration for a meaningful comparison. </p>

<h1>Apparatus</h1>

<p><b>Read the Leybold manual first to make sure you understand the apparatus.</b></p>

<p>The apparatus consists of the '''scattering chamber''', the '''vacuum system''', the '''detector''', and the '''data acquisition system'''. The components inside the scattering chamber necessary to perform the experiment are '''the source''', two types of '''target foil''' (gold and aluminium) and '''two slits''' (1 mm and 5 mm wide). See the data sheets provided by Leybold for further details.</p>

<table width=400 align=center><td>
<p align=justify>[[File:RuthI-fig2.png|500px|border|center]]
<b>Figure 2 -</b> Schematic of experiment.
<br clear=right>
</p>
</td></table>

<h2>The source and slits</h2>

<p>To install a foil please contact the TA or Lab Technologist.</p>

<p>Treat this source as you would any '''radioactive source''': with care. Wear gloves when handling the source or when inserting the slits and target, and do not take it out of its protective container without supervision. The radiation from the source is emitted at all possible angles within the geometry of the container. To turn the radioactive source into a well-defined beam of particles one has to collimate the beam by using apertures. The source is placed in a metal holder so that the alpha particles emanate in a cone. A collimating slit is placed at a distance of about 2.8 cm from the front face of the metal holder, and a metal foil of a few microns thickness is placed in a holder in front of the slit, directly against it (the drawing is exaggerated), i.e. further away from the source than the slit. Here “front” is defined by the direction of the alpha particles if they were unimpeded in their direction of travel from the source through the collimator. The detector is 2.7 cm in front of the foil. The source-foil assembly can be rotated about an axis that passes through the foil, while the detector is fixed. This is equivalent to a fixed source-target assembly and rotating the detector. The central axis of the detector should point along this axis of the foil-source assembly at zero degrees. In practice, any errors of this may cause an overall shift in the result, as we shall discuss later.</p>

<h2>The detector</h2>

<p>The detector is a silicon (solid-state) device, and has a slit on its face of height ( height 7mm x width 1.5mm. The detector is connected to a pre-amplifier (to amplify the weak signal) and discriminator, and these are set to generate definite pulses when a pulse generated in the detector exceeds a certain threshold (as defined by the discriminator). A A digital counter records the shaped pulses. This way one can suppress unwanted background sources (although light hitting the detector could also trigger events). If one sets the threshold too high, one suppresses of course some events of interest, but this should at worst lead to a reduced count rate, but not necessarily affect the outcome of the experiment. Can the dis. threshold be adjusted?</p>

<h2>The vacuum system</h2>

<p>There are several sets of measurements using the chamber. In between each stage, the chamber must be returned to normal atmospheric pressure so that the lid may be removed. Hence the process of evacuating the chamber, and returning it to normal pressure, are described at first. Practice with no foil in the chamber and test that the lid of the chamber is secure when the chamber is evacuated.</p>

<table width=400 align=center><td>
<p align=justify>[[File:RuthI-fig3.png|400px|border|center]]
<b>Figure 3 -</b> Vacuum system.
<br clear=right>
</p>
</td></table>

<h2>Evacuating the scattering chamber</h2>

<p>Please be gentle with the valves at all times the '''foil is delicate and very expensive''' to replace. '''Never touch the foil directly as it is easily perforated''', and take it or out of the holder very gently. Carry out the vacuum operation slowly. Always replace the foil in its container bag in between steps.</p>

<ol>
<li>Open B check for secure lid slowly (if it is not open). Close A.</li>
<li>Turn on the roughing pump until the vacuum reaches 70 mm of Mercury.</li>
<li>Slowly close B. </li>
<li>Turn off the pump. </li>
<li>NOTE this is very important! '''Open A (this is to avoid back pressure from the pump and oil from entering into the system).'''</li>
</ol>

<h2>Releasing the chamber to atmospheric pressure</h2>

<p><b>If the foil is in the chamber, make sure that the foil is perpendicular to the chamber valve when returning the vacuum to the normal pressure or evacuating the chamber.</b></p>

<ol>
<li>Close A. Open B slowly.</li>
<li>Open A slowly.</li>
</ol>

<h2>Data Acquisition System</h2>

<p>Note: There are two ways of recording the data: “rate” and “counts”. </p>

<p>A computer interfaced data acquisition system is provided, which can display on-line the Poisson statistical analysis of the recorded events. One might think that measuring the count rate 10 times (for a given time interval), and deducing the average and deviation would be sufficient. However, due to the statistical nature of the radioactive decay process we do not have a constant beam of particles; also the scattering itself is a probabilistic process. Such random events are obeying Poisson statistics. The computerized data acquisition system allows one to collect data in 10-second or 60-second intervals and to assemble a histogram (to be compared with a Poisson fit) and to display graphically the collected count rate. A function is provided to deduce the average count rate and deviation from the data. </p>

<h2>Required Components</h2>
<ol>
<li>[[Media:RSAngle.JPG|Protractor]]</li>
<li>[[Media:RSController.JPG|Control Box]]</li>
<li>[[Media:RSCounter.JPG|Counter]]</li>
<li>[[Media:RSDetector.JPG|Radioactive Detector]]</li>
<li>[[Media:RSGage.JPG|Vacuum Gage]]</li>
<li>[[Media:RSSlits.JPG|Slits]]</li>
<li>[[Media:RSSource.JPG|Source]]</li>
<li>[[Media:RSVacuum.JPG|Vacuum]]</li>
</ol>

<h1>Procedure</h1>

<p><b>We stress again that you must understand how the vacuum pump operates. Misuse can lead to destruction of the whole apparatus (oil flooding, tearing of the foils etc. ). </b></p>

<ol>
<li>Using a '''5mm slit''' and '''no foil''', find the “zero angle” (θ<sub>0</sub> ) of the apparatus by recording the counting rate of the alpha particles for several angles (-10<sup>o</sup> to +10<sup>o</sup>), starting from the zero on the dial angle indicator (chamber lid). Take measurements for negative and positive angles in increments of 2.5<sup>o</sup> (half of the smallest division on the dial).</li>

<li>Place the '''Gold foil''' together with the '''5mm slit''' on the mount and acquire measurements of counts for the following angles 10, 20, 30, 40, 50, 60 degrees with respect to θ<sub>0</sub>. Make sure you obtain at least 10 counts for each of these angles.</li>

<li>Repeat the above using the '''Aluminium foil''' with the '''5mm slit'''.</li>

<li>Use the '''5mm slit''' to take '''background''' measurements ('''do not place any of the foils''') for the same angles as in step 3 and use this date to correct for the above measurements. The small angles (<30<sup>o</sup>) should not take too long. However, as you reach the large angles, the time necessary to obtain enough readings can be substantial. Make sure you take at least 1 count for each of the angles (for the small angles you can probably take a few hundreds in a matter of seconds).</li>
</ol>

<p>Plot the log(rate) vs. log(sin<sup>4</sup>θ/2) with appropriate conversion to radians. Derive the relationship between the observed rate and the cross section. What do you expect for the behaviour of this graph? Using the relationship between the rate and the Rutherford cross section formula (equation1), find the ratio of atomic numbers between Gold and Aluminium.</p>

<h1>References</h1>

<ol>
<li>Melissinos, ''Experiments in Modern Physics'', Academic Press.</li>
<li>Preston and Deitz, ''The Art of Experimental Physics'', Wiley and Sons.</li>
<li>H. Frauenfelder and E. Henley, ''Subatomic Physics'', Prentice Hall.</li>
<li>A. Das and T. Ferbel, ''Introduction to Nuclear and Particle Physics'', J. Wiley.</li>
</ol>

File:RSVacuum.JPG

2011-10-20T16:12:09Z

WorkStudy:

File:RSSource.JPG

2011-10-20T16:11:31Z

WorkStudy:

File:RSSlits.JPG

2011-10-20T16:11:09Z

WorkStudy:

File:RSGage.JPG

2011-10-20T16:10:47Z

WorkStudy:

File:RSDetector.JPG

2011-10-20T16:10:22Z

WorkStudy:

File:RSCounter.JPG

2011-10-20T16:09:59Z

WorkStudy:

File:RSController.JPG

2011-10-20T16:09:24Z

WorkStudy:

File:RSAngle.JPG

2011-10-20T16:09:07Z

WorkStudy:

Main Page/PHYS 3220/Radioactive Decays

2011-10-20T15:33:17Z

WorkStudy:

<h1>Radioactive Decays</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>

<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 fast electron is emitted from the nucleus, thus converting a neutron into a proton (and an electron + antineutrino) - corresponding to the β<sup>-</sup> decay of free neutrons (half-life  10.6 min.). For proton-rich nuclei a fast positron emerges from the nucleus (β<sup>+</sup> decay) whereby a proton is converted into a neutron and a neutrino. 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 higher energies than hard 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 understood 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 (ref. 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. ref. 1,2,6), and provide a concise description inyour own words with 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 by passage through matter. This is used both in shielding, as well as 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, therefore, 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. Gamma rays have the best penetration characteristics, i.e., are hardest 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 an especially 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 used, e.g., by having fast particles penetrating 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. Read the detailed write-up for the “Absorption of Radiation” experiment.</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 the question had to be resolved 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 goes over into a Gaussian. </p>
<p>Rutherford performed experiments which 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 ''Radiation Counter'' <b>***CHANGE THIS PROGRAM REFERENCE****</b> performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (cf.. also chapter 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>II(2)</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 (cf.. chpt. 12 in ref. 5, an example is given on p.235). From the reduced χ<sup>2</sup> value one infers the agreement.</p>
</li>

</ol>

<h1>Experimental Procedure</h1>

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

<h2>Required Components</h2>
<ol>
<li>[[Media:RDTableTopGM.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 - 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 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>1. 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 chi-squared obtained, and quote the decay rate, with its standard error. Include print-outs 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. How do these results compare to the results from the computer program?</li>

<li>2. 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>3. 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>4. 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 outlilne 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</li>
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li>
<li>Taylor, J.R., ''An Introduction to Error Analysis''.</li>
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li>
</ol>

Main Page/PHYS 3220/Radioactive Decays

2011-10-20T15:32:17Z

WorkStudy:

<h1>Radioactive Decays</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>

<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 fast electron is emitted from the nucleus, thus converting a neutron into a proton (and an electron + antineutrino) - corresponding to the β<sup>-</sup> decay of free neutrons (half-life  10.6 min.). For proton-rich nuclei a fast positron emerges from the nucleus (β<sup>+</sup> decay) whereby a proton is converted into a neutron and a neutrino. 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 higher energies than hard 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 understood 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 (ref. 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. ref. 1,2,6), and provide a concise description inyour own words with 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 by passage through matter. This is used both in shielding, as well as 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, therefore, 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. Gamma rays have the best penetration characteristics, i.e., are hardest 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 an especially 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 used, e.g., by having fast particles penetrating 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. Read the detailed write-up for the “Absorption of Radiation” experiment.</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 the question had to be resolved 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 goes over into a Gaussian. </p>
<p>Rutherford performed experiments which 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 ''Radiation Counter'' <b>***CHANGE THIS PROGRAM REFERENCE****</b> performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (cf.. also chapter 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>II(2)</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 a value of the mean; 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 (cf.. chpt. 12 in ref. 5, an example is given on p.235). From the reduced χ<sup>2</sup> value one infers the agreement.</p>
</li>

</ol>

<h1>Experimental Procedure</h1>

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

<h2>Required Components</h2>
<ol>
<li>[[Media:RDTableTopGM.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 - 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 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>1. 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 chi-squared obtained, and quote the decay rate, with its standard error. Include print-outs 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. How do these results compare to the results from the computer program?</li>

<li>2. 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>3. 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>4. 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 outlilne 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</li>
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li>
<li>Taylor, J.R., ''An Introduction to Error Analysis''.</li>
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li>
</ol>

Main Page/PHYS 3220/Radioactive Decays

2011-10-20T15:29:30Z

WorkStudy:

<h1>Radioactive Decays</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>

<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 fast electron is emitted from the nucleus, thus converting a neutron into a proton (and an electron + antineutrino) - corresponding to the β<sup>-</sup> decay of free neutrons (half-life  10.6 min.). For proton-rich nuclei a fast positron emerges from the nucleus (β<sup>+</sup> decay) whereby a proton is converted into a neutron and a neutrino. 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 higher energies than hard 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 understood 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 (ref. 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. ref. 1,2,6), and provide a concise description inyour own words with 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 by passage through matter. This is used both in shielding, as well as 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, therefore, 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. Gamma rays have the best penetration characteristics, i.e., are hardest 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 an especially 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 used, e.g., by having fast particles penetrating 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. Read the detailed write-up for the “Absorption of Radiation” experiment.</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 the question had to be resolved 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 goes over into a Gaussian. </p>
<p>Rutherford performed experiments which 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 ''Radiation Counter'' <b>***CHANGE THIS PROGRAM REFERENCE****</b> performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (cf.. also chapter 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>II(2)</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 a value of ; 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 (cf.. chpt. 12 in ref. 5, an example is given on p.235). From the reduced χ<sup>2</sup> value one infers the agreement.</p>
</li>

</ol>

<h1>Experimental Procedure</h1>

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

<h2>Required Components</h2>
<ol>
<li>[[Media:RDTableTopGM.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 - 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 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>1. 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 chi-squared obtained, and quote the decay rate, with its standard error. Include print-outs 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. How do these results compare to the results from the computer program?</li>

<li>2. 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>3. 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>4. 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 outlilne 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</li>
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li>
<li>Taylor, J.R., ''An Introduction to Error Analysis''.</li>
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li>
</ol>

File:RDBeigeFiesta.JPG

2011-10-20T15:26:52Z

WorkStudy:

File:RDHandHeldGM.JPG

2011-10-20T15:26:33Z

WorkStudy:

File:RDMantles.JPG

2011-10-20T15:26:06Z

WorkStudy:

File:RDOrangeFiesta.JPG

2011-10-20T15:25:37Z

WorkStudy:

Main Page/PHYS 3220/Radioactive Decays

2011-10-20T15:23:44Z

WorkStudy:

<h1>Radioactive Decays</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>

<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 fast electron is emitted from the nucleus, thus converting a neutron into a proton (and an electron + antineutrino) - corresponding to the β<sup>-</sup> decay of free neutrons (half-life  10.6 min.). For proton-rich nuclei a fast positron emerges from the nucleus (β<sup>+</sup> decay) whereby a proton is converted into a neutron and a neutrino. 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 higher energies than hard 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 understood 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 (ref. 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. ref. 1,2,6), and provide a concise description inyour own words with 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 by passage through matter. This is used both in shielding, as well as 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, therefore, 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. Gamma rays have the best penetration characteristics, i.e., are hardest 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 an especially 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 used, e.g., by having fast particles penetrating 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. Read the detailed write-up for the “Absorption of Radiation” experiment.</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 the question had to be resolved 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 goes over into a Gaussian. </p>
<p>Rutherford performed experiments which 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 ''Radiation Counter'' <b>***CHANGE THIS PROGRAM REFERENCE****</b> performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (cf.. also chapter 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>II(2)</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 a value of ; 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 (cf.. chpt. 12 in ref. 5, an example is given on p.235). From the reduced χ<sup>2</sup> value one infers the agreement.</p>
</li>

</ol>

<h1>Experimental Procedure</h1>

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

<h2>Required Components</h2>
<ol>
<li>[[Media:RDTableTopGM.JPG|AC powered table-top GM counter]]</li>
<li>hand-held GM counter</li>
<li>beige 'Fiesta' ceramic dish</li>
<li>orange 'Fiesta' ceramic dish</li>
<li>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 - 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 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>1. 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 chi-squared obtained, and quote the decay rate, with its standard error. Include print-outs 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. How do these results compare to the results from the computer program?</li>

<li>2. 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>3. 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>4. 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 outlilne 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</li>
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li>
<li>Taylor, J.R., ''An Introduction to Error Analysis''.</li>
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li>
</ol>