View source for 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><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>\frac{1}{4} m < \left ( v^2 - <v^2>\right)^2> = \frac{3}{2} (kT)^2</math>


;Note
: <math>< \left ( v - <v>\right)^2>  = <v^2 - 2v<v> + <v>^2> = <v^2> - (<v>)^2</math> 
: <math>= \frac{3kT}{m} - \frac{8kT}{m}</math> = velocity fluctuation
: <math>\frac{m^2}{4} < \left ( v^2 - <v^2>\right)^2> = \frac{m^2}{4}\left ( <v^4> - (<v^2>)^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 [http://www.iac.isu.edu/mediawiki/index.php/Simulations_of_Particle_Interactions_with_Matter#Example_2_Calculation_of_.CF.80 Notes]

==3.) Histograms using ROOT==


=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.
===b.) Now make your own copy of it and change the target material===

=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.

[[Image: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. [[Image: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 =

= Homework 10 =

= Homework 11 =

= Homework 12 =

= Homework 13 =

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[http://www.iac.isu.edu/mediawiki/index.php/Simulations_of_Particle_Interactions_with_Matter Back to Notes]