Luminosity of an elastic proton-proton scatter

Question

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! :)

Answer 1

You do not need the volume. You need the number density,

$$ n=\frac NV=\frac p{kT} $$

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 $e^{-n\ell\sigma}$.

Answer 2

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.