# Derive an expression for the rate of interactions in a fixed target experiment

1. Jan 31, 2012

### blueyellow

1. The problem statement, all variables and given/known data

Derive an expression for the rate of interactions in a fixed target experiment, for which the beam of incoming protons has a current I, the target density is ρ(g/cm$^{3}$), the target thickness is d and the interaction cross-section is σ. You assume that the beam particles are highly relativistic (i.e. v≈c) and that the target is thin (i.e. there is negligible attenuation of the beam in the target itself. You may need to use other physical constants in you expression.

Use your expression to calculate the interaction rate for a 1mA beam of high-energy protons on a 1mm thick liquid Hydrogen target.

3. The attempt at a solution

density=mass/volume
=$\frac{m_{p}\times number-of-protons-per-second-per-solid-angle\times t\times Ω}{\sigma\times d}$
number of electrons per second per solid angle=$\frac{density\times σ\times d}{m_{p}\times t\times Ω}$

=$\frac{dσ}{dΩ}$$\frac{density\times d}{m_{p}\times t}$
=$\frac{dσ}{dΩ}$$\frac{density\times d\times I}{m_{p}\times Q}$

number of protons per second=$\int$$\frac{dσ}{dΩ}$$\frac{density\times d\times I}{m_{p}\times Q}$dΩ

But what formula does one use for d/dΩ?

the Rutherford cross-section? The Mott cross-section?

Last edited: Jan 31, 2012
2. Feb 4, 2012

### BruceW

why would you use the differential cross-section? They give you the cross-section, and the question doesn't want to know anything about the angle of scattering. I think you need to use the equation giving the interaction rate in terms of the cross-section.