Main Page/PHYS 3220/Particle Tracking
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Particle Tracking Chamber Simulation
Chamber Works is a software package that is used as a tool to introduce some phenomena and methods of particle physics. Particle production and the subsequent spontaneous decays, and associated lifetimes are some basic concepts. Such decays are constrained by the laws of conservation of momentum and energy. In addition, at high energies, relativistic kinematics will have to be employed. The aim of this simulated tracking device experiment is to understand the process of generation and decay of the following particles: the neutral pion, charged pions and muons.
Introduction
There are three basic parts to the ChamberWorks program: (1) The Trackmaker is an interactive simulator through which you will learn how different values of the magnetic field or velocity of an incoming particle affect the motion of the particle(s) in the chamber. You should be able to understand the trajectory as a direct result of the Lorentz force. (2) The second part is the particle detection facility, which is a three dimensional chamber that allows projections and rotations of the particle to be observed as it traverses the chamber. The primary particle is allowed to decay, and the charged products of the decay then follow paths specified by their energy and momentum. (3) The third part is a Reminder utility that can be used to remind the student of useful equations, such as the Lorentz force, or the expression for the momentum of a charged particle in a magnetic field perpendicular to its plane of motion.
The ChamberWorks program allows you to develop an understanding of spontaneous decays. In the microscopic domain, quantum mechanics does not permit us to think of particles as having well defined trajectories. Similarly, for spontaneous decays (cf.. radioactivity) we cannot predict at what moment any single particle will decay. We are only allowed statistical information for an ensemble of particles. ChamberWorks uses a random number generator to simulate decay events according to the known lifetimes of the particles involved. In fact, it even knows about branching ratios (relative probabilities), if particles have different modes of decay.
Spontaneous decays follow an exponential decay law. The lifetime (τ) of a particle is defined as the time at which the number of states or particles surviving is e-1 of the number present initially. In contrast, the half life (τ1/2) is the time at which the number of particles surviving is 1/2 of that present initially, where τ1/2 = (ln 2) τ = 0.693 τ.
Experimental Procedure
Familiarise yourself with the write-up in the Chamber Works User's Guide. A quick reading of the manual will help you to get oriented, but you will need to review the material more carefully as you perform the experiment. Prior to starting the experiment, refresh your knowledge of relativistic kinematics, (e.g. the chapter on Relativity in ref 2) and the motion of charged particles in a magnetic field. Understand the units for mass and energy as used in this experiment.
Understand the tools available in Trackmaker. ACTIVATE this part with the mouse. Play with the tools to gain expertise.
The options available are to SELECT a particle type (electron or positron), or the incoming speed (v, 2v, 3v), or the incoming direction (left to right, right to left). You may choose the field to be IN (negative B field) or OUT (positive B field) of the screen, with values ranging from zero to 1.0 on a relative scale. Learn how the radius of the path changes as the B field changes or the incoming velocity changes, e.g. change the velocity and see how the radius of curvature changes. Change the magnetic field to twice or three times a chosen value. Does the curvature change as predicted by the equation? Use the Tape measure to measure a chord. How would you measure a radius of curvature knowing the chord? You cannot obtain a momentum in this exercise since the magnetic field is not absolute (e.g. in kGauss) but only relative.
Calculate and show in your report the velocity (in units of c) of an electron (mass 0.511 MeV) with a kinetic energy of 1 MeV and 10 MeV. What does this tell you about the velocity of this very light particle compared to the velocity of light, i.e. can it be considered relativistic?
Change the values of the B field, and confirm that the momentum is what you expect for an electron injected with the given velocity v, 2v, 3v. What effect does a positive vs. a negative B field have on the electron? Use the right hand rule to confirm your findings. Remember that the magnetic field is perpendicular to the plane of incidence for the incoming particle. Try changing the magnetic field to capture the particle on the screen. This is an interactive package, so use this feature to learn about particle motion!
Repeat your experiment for positrons, and verify to yourself that electrons and positrons have exactly the same behaviour except they are oppositely charged particles. Knowing the direction of the B field, you should be able to tell from the curvature whether the particle is an electron or positron. See page 22 of the User's Guide to see if you fulfilled all the goals of this section using Trackmaker.
When you have completed this section, DEACTIVATE this part of the experiment.
In the next section, you will study the decay of charged and neutral pions, and charged muons. There are five windows to manipulate the displays and the kinematics of the generated event.
The SHOW window contains the Chamber Controls. The magnetic field (B) may be varied from 1.0 to +1.0 kGauss, the kinetic energy (T) from 0.1 to 1.0 keV, and the chamber size (L) from 1.0 m to 10 m. If the chamber size is small, you have a better chance of observing and measuring the primary decay properly. However, if there are multiple decays, a longer chamber will allow the secondary decay to occur and you can analyze this decay in the chamber.
There is also a selection for Chamber Projection, which projects the tracks in X Y, Y Z, X Z views. The X Y view shows the motion perpendicular to the B field, which is along the z axis and shown as a green line. Try all the views in order to think in 3-D. The Zoom option allows onefold, twofold or threefold magnification. The Projection Window is the most useful for making measurements of track length or curvature of the path. The tools to Grab a section of the track or to measure its length with a Tape Measure are also in this menu.
The SETTINGS window allows you to see the motion slowly. You can monitor the motion in the chamber itself or in the projected view (2 dimensional view).
The EVENT window allows you to select the type of particle decay. The decays are not identified, since you are to determine which of these interactions are associated with muons (both signs) or pions (both signs and neutral), which then decay according to their lifetime in the laboratory (what is the effect of the time dilation?).
The ACTIONS window allows the selection of a new event, or a replay of the old one. You can also use the power of 3-D graphics to rotate the view around any chosen axis. Reset the orientation when you wish to return to the normal setting.
The FILE window allows you to Open a new event file to Store your interesting events, or Print the events. Finally, it allows you to Quit the program gracefully.
Procedure to Analyze Events
The aim is to study and categorise each type of decay and to find the mean lifetime of the particles. The pions are a type of particle known as hadrons, while the electrons, positrons, and muons belong to a class known as leptons. Masses and lifetimes for some particles are provided in the table below. The charged pion decays primarily into a charged muon and a muon-neutrino (or anti-neutrino, depending on the charge of the muon). The neutral pion decays electromagnetically into two photons. A negative muon decays into three particles: an electron, an electron antineutrino, and a muon neutrino. Similarly, a positive muon decays into a positron, an electron neutrino, and a muon antineutrino. The different forces (electromagnetic vs. weak) responsible for the decays are reflected in the lifetimes of the particles: the electromagnetic interaction that is responsible for the π0 decay to two photons (quark-antiquark annihilation) is much stronger than the weak forces that govern the decays of π+/- and μ+/-.
| Particle | Mass(MeV) | Mean Lifetime(s) |
| π0 | 134.97 | 8.4 x 10-17 |
| π+/- | 139.57 | 26.03 x 10-9 |
| μ+/- | 105.66 | 2.20 x 10-6 |
| e+/- | 0.511 | stable |
Neutrinos and photons, in addition to being massless, are also chargeless, so they do not directly leave traces from the results of ionisation along their path in the chamber (read the section on the Basic Ideas Behind Particle Detection Chambers). We have to infer their presence from energy momentum conservation laws. However, there is a finite probability that a photon will convert into an electron positron pair in the presence of an atomic field (this is incorporated in the simulation). Again, by measuring the curvature of the electron positron pair we are able to obtain information about the primary interaction.
To learn more about elementary particles and the forces with which they interact, consult elementary particle physics textbooks. A fun introduction to the subject can be found in Ref. 3, and the references therein. A more comprehensive review of the properties of the known elementary particles can be found in Ref 4 and 5.
The incident particle is injected and confined to a plane perpendicular to the B field, and hence the transverse momentum is the only component of the total momentum of the particle. You can calculate the momentum of the particle by measuring the radius of curvature, using the Tape Measure. However, depending on the kinematics, you may not always have a fully contained circular path on your screen to measure the diameter of the circle. Knowing the length of a chord (d), and the distance from the centre of the chord to the point of closest distance (s) at the circumference, one can obtain the radius of curvature (R). Derive that this is given by R = s/2 + d2/(8s). Thus, knowing R and the incident B field, you can obtain the transverse momentum of the particle.
