In our set-up, the geometry of the device allows the maximum path length to lie close to the PMT. This is ideal because we can first find the maximum path length and then use this to find the theoretical value for the maximum amount of energy reaching the PMT from a cosmic ray.

1. Find the maximum path length from measurements and geometry. Using trigonometry, q =16° . Using similar triangles and more trig, d/x = cos q . This yields a value of 11.2 cm for x.

 

 
 
 


 

2. Find the maximum energy emitted as photons over this path length. We know that x is 11.2 cm. From The Particle Physics Booklet, dE/dx = 1.936 MeV/(g cm²) and D = 1.032 g/cm³ for polystyrene scintillator. Therefore,

 

 
 
 





3. Find the fraction of the photons reaching the PMT. As the ray passes through the scintillator, the photons are emitted radially. Only a cone of photons will reach the PMT. To simplify calculations, we assume all of the energy is emmited at the midpoint of the path in the scintillator. The fraction reaching the PMT is approximately equal to the area of the base of the cone (i.e. the end of the scintillator) divided by the surface area of the sphere.

 

 
 
 







4. Calculate the energy deposited by the fraction of photons reaching the PMT. Assuming our path found is the path of the maximum energy, the peak should be given by

            To compare with our results . . .