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}{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

Homework 4

1.)Compute[math] \frac{dE}{dx}[/math] for a heavy charged particle traveling through a liquid Hydrogen target. In clas 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.) Use GEANT4 to simulate the calcualtion in part a above and compare answers.

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}{m_e} \right ) ^2 \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:SPIM RangeInCopper.pdf Range of several Particles through Copper as a runction of energy

Homework 6

1.) You rneed 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} dx[/math]

2.) Alter GEANT4 example N02 to check your answer for problem 1 above. I expect you to hand in a scrren shot showing GEANT4 tacking 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.

Homework 9

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