HomeWork Simulations of Particle Interactions with Matter

Homework 1

1.) Mawell Boltzmann

Given the Maxwell -Boltzmann Distribution

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

Note

[math]\displaystyle{ < \left ( v - <v>\right)^2> = <v^2 - 2v<v> + <v>^2> = <v^2> - (<v>)^2 }[/math]

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

[math]\displaystyle{ \frac{m^2}{4} < \left ( v^2 - <v^2>\right)^2> = \frac{m}{4}\left ( <v^4> - (<v^2>)^2 \right ) }[/math]

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

2.) MC calculation of Pi

Calculate [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ p }[/math] = momentum of incident heavy charged ion of mass [math]\displaystyle{ M }[/math]

[math]\displaystyle{ 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]\displaystyle{ \frac{g}{cm^3} }[/math]. Determine how thick the copper should be by integrating the stopping power curve:

[math]\displaystyle{ 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]\displaystyle{ \frac{\sigma_R}{R} }[/math] using GEANT4 for a 600 MeV proton traveling through a slab of copper.

Homework 7

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

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