HomeWork Simulations of Particle Interactions with Matter

Homework 1

1.) Mawell Boltzmann

Given the Maxwell -Boltzmann Distribution

[math]N(v) = 4 \pi \left ( \frac{m}{2\pi kT}\right)^{3/2} v^2 e^{-\frac{mv^2}{2kT}}[/math]

a.) Show <v>

Show that

[math]\lt v\gt = 4\pi \left ( \frac{m}{2 \pi kT}\right )^{3/2} \left( \frac{2kT}{m}\right)^2 \frac{\Gamma(2)}{2}[/math]

b.) Energy Fluctuation

Show that the energy fluctuation is

[math]\frac{1}{4} m \lt \left ( v^2 - \lt v^2\gt \right)^2\gt = \frac{3}{2} (kT)^2[/math]

Note

[math]\lt \left ( v - \lt v\gt \right)^2\gt = \lt v^2 - 2v\lt v\gt + \lt v\gt ^2\gt = \lt v^2\gt - (\lt v\gt )^2[/math]

[math]= \frac{3kT}{m} - \frac{8kT}{m}[/math] = velocity fluctuation

[math]\frac{m^2}{4} \lt \left ( v^2 - \lt v^2\gt \right)^2\gt = \frac{m^2}{4}\left ( \lt v^4\gt - (\lt v^2\gt )^2 \right )[/math]

[math]=\frac{1}{4} \left ( 15(kT)^2 - (3kT)^2\right)[/math]

2.) MC calculation of Pi

Calculate [math]\pi[/math] using the Monte Carlo method described in the Notes

Homework 2

1.) Derive Rutherford Formula

Derive the Rutherford Scattering formula.

2.) EXN02 in GEANT

a.) Compile and run the default version of ExN02 in GEANT4

You can use a computer screen shot to prove you did this.

Homework 3

Download and install your own version of GEANT4

Use a screen shot to prove you did it.

Homework 4

1.) Use GEANT4 to simulate the calculation the energy loss of a ion through LH2. In class I showed an example for an incident 10 MeV electron. You need to pick another particle (proton, pion, ...) and a different energy. Compare your answer with the Triumf curve.

2.) Compute[math] \frac{dE}{dx}[/math] for the heavy charged particle you chose to simulate in problem 1. Use the particle's energy at one of the tracking steps and compare to what GEANT4 found.

Homework 5

1.) Show that the maximum energy transfered to thin absorbers for a relativistic head on collision is

[math]W_{max} = \frac{(pc)^2}{\frac{1}{2} \left [ m_e c^2 + \left ( \frac{M^2 c^2}{m_e} \right ) \right ] + \sqrt{(pc)^2 + (Mc^2)^2}}[/math]

[math]p[/math] = momentum of incident heavy charged ion of mass [math]M[/math]

[math]m_e[/math] = mass of target electron initially at rest

2.) Use GEANT4 to determine the Range of the particle chosen in Homework 4 through liquid hydrogen as a function of Energy.

File:RangeInLH2.pdf

Homework 6

1.) You need to lower the beam energy of 600 MeV protons to 400 MeV using a slab of copper. The density of the copper is 8.962 [math]\frac{g}{cm^3}[/math]. Determine how thick the copper should be by integrating the stopping power curve:

[math]x = -\int_{600 MeV}^{400 MeV}\left [\frac{dE}{dx} \right ]^{-1} dE[/math]

Stopping Power of several particles through Copper as a function of energy is shown in this curve. File:StoppingPowerInCopper.pdf

2.) Alter GEANT4 example N02 to check your answer for problem 1 above. I expect you to hand in a screen shot showing GEANT4 tracking the proton from 600 MeV to 400 MeV.

3.) Find [math]\frac{\sigma_R}{R}[/math] using GEANT4 for a 600 MeV proton traveling through a slab of copper. You will need to make the copper thick enough to stop the proton. Then output the stopping distance to a file which you can read into ROOT using some of the software we used for Homework 1's RNG problem.

Homework 7

SPIM_Brem_Lab_Instructions

Media:SPIM_BremE-Spectrum-Tantalum.pdf

Media:SPIM_LaTex_TemplateFile.txt

Homework 8

1.) Write a Paragraph (4 - 5 sentences) desribing the Simulation you would like to perform as your Project for this class. You will need to write a title. You will need to specify the reaction you will be simulating.

Homework 9

SPIM_PhotElectricEffect_Lab

Homework 10

SPIM_ComptonScattering_Lab

Homework 11

There are 2 parts to this homework. First you will compare relative rates for the PhotoElectric, Compton, and pair production physics processes using the same target you used in Homework 11. Second you will write another section of your project which describes the experimental results you are going to compare to using GEANT4.

1.) Compare Photoelectric, Compton and pair production rates relative to eachother using the same target used in the last Homework assignment (#10).

a.) first turn on all three physics processes for a gamma particle in the physics list.

b.) add variables to the output which can be used to identify which physics process is responsible for the event being written to the output file.

c.) Run the simulation so the incident photon energy spans energies from 100 eV to 10 GeV.

d.) Use ROOT to plot a 3-D representation of the Process type on one axis, the incident photon energy on the other axis and the number of counts along the z-axis.

A bad example of such a plot for a 30 cm long Argon gas target is given in the file

Yours will have better labels

Hint:

->Draw("ProcesID:Egamma","","lego");

2.) Add another section to your project report which describes the experimental measurements you will be using to compare to GEANT4. I am expecting to see a plot and references.

Homework 13

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