# Luminosity of an elastic proton-proton scatter

Suppose you have a proton beam ($$E=500\,\textrm{MeV})$$ with a current of $$6 \cdot 10^{-12}\,\textrm{A}$$ that is colliding with a gaseous hydrogen target which is $$3\,\textrm{cm}$$ thick. The total cross section is $$20\,\textrm{mb}$$. (Further suppose the target pressure and temperature are $$p=1013\,\textrm{hPa}$$ and $$T=273\,\textrm{K}$$.)

Now I want to calculate the Luminosity which is defined as proton flux times number of target elements.

I tried to calculate the number of target elements $$N$$ by using the law of ideal gas: $$p \cdot V = N \cdot k_B \cdot T$$

But my Problem is that I don’t know how to find the Volume of the gas target? Can someone help?

Thanks! :)

$$n=\frac NV=\frac p{kT}$$
Then the product $$n\ell$$, with $$\ell$$ the known thickness of your target, is the number of scatterers per unit area, and the transmission is $$e^{-n\ell\sigma}$$.
You don't need the Volume $$V$$ to calculate the luminosity, just the density $$n=N/V=p/(k_B T)$$ and length of the target, plus the beam flux which you can calculate from the beam current and the proton electric charge.