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Homework Statement
Calculate the energy loss loss ##\Delta T## for protons, deuterons and ##\alpha##-particles between ##10## to ##200##MeV when they're passing through a 2mm thick plastic scintillator. Suppose ##Z/A=0.56##, ##I=65eV## and ##\rho = 1.10##g/cm^3.
Homework Equations
Bethe formula
##\frac{dE}{dx}=\left( \frac{e^2}{4 \pi \epsilon_0} \right)^2 \frac{4 \pi z^2 N_0 Z\rho}{mc^2\beta^2 A} \left[ \ln \left( \frac{ 2mc^2 \beta^2}{I}\right) - \ln (1-\beta^2)-\beta^2\right]##
##N_0## is Avogadro's constant, ##v = \beta c##, ##ze## the electric charge of the particle, ##Z, A## and ##\rho## the atomic number, atomic weight, and density of the stopping material and ##m## the electron mass.
Relativistic energy
##T = mc^2-\frac{mc^2}{\sqrt{1-v^2/c^2}} = mc^2-\frac{mc^2}{\sqrt{1-\beta^2}}##
Mass of proton ##m_p = 938.28##MeV/c^2
Mass of electron ##m = 0.511003##MeV/c^2
Electron charge ##1.60217662 \cdot 10^{-19}## coulomb
Avogadro's constant ##6.0221409\cdot 10^{23}##
permittivity of free space ##8.85418782\cdot 10^{-12}## F/m
The Attempt at a Solution
I started by just trying to compute ##\Delta T## for ##T = 10 MeV## for the proton.
The ##\beta## factor after some algebra is
##\beta = \sqrt{1-\left(\frac{1}{\frac{T}{mc^2}+1}\right)^2}##
For the rest of the values I converted everything to SI units and then input the formula into matlab. However I end up with an error in the order of ##10^2## if I do this and I think it may be since I'm using the formula with the wrong units. The ##Z/A## value in particular I have no unit of and I don't know if I should convert that to kg or not if it's in atomic mass units.
Here's the MATLAB code for the calculation:
Code:
e=1.60217662e-19; %electron charge
e0 = 8.85418782e-12; %permitivity of free space
ZA = 0.56; %unit??
I=65e-6; %MeV
rho = 1.10*1000; %converted to kg/m^3
N0 = 6.0221409e+23; %avogadro
d=0.002; %distance in meter
z=1; %proton
mc2 = 0.511003; %MeV electron
mc2p = 938.28; %MeV proton
T = 10; %MeV
beta=sqrt(1-1./(T./mc2p+1).^2);
k = (e^2/(4*pi*e0))^2*4*pi*z^2*N0*rho*ZA./(mc2*e*1e6); %convert to joule/meter
temp = 1/beta^2*(log(2*mc2*beta.^2./I)-log(1-beta.^2)-beta.^2);
dT = d*k.*temp/e/10^6 %convert to MeV
I realize this is quite a long post so I'm mostly hoping that someone is familiar with the formula to know which units I should be using since I suspect that is my error.