Energy-dependent cross ection and mean free path

[tom.stoer]

The mean free path is usually determined via the scattering cross section σ; one starts with a differential equation for the intensity

$$dI(x)= -n\,\sigma\,I(x)\,dx$$

Are there generalizations for this derivation for energy-dependent cross sections σ(E)?

 

[mathman]

You didn't say what particle you are scattering. If it is neutron or photon, then the formula still holds, but only between collisions. This is one area where Monte Carlo method is used.

 

[tom.stoer]

You didn't say what particle you are scattering. If it is neutron or photon, then the formula still holds, but only between collisions. This is one area where Monte Carlo method is used.

I am not looking for a specific problem, but for the general ansatz. First one would have to modify it like

$$dI(x)= -n\,\sigma(E)\,I(x)\,dx$$

but then one has to take into account that

$$\langle E \rangle = f(x)$$

i.e. a typical particle at x has a typical energy E. I don't see how to formulate that problem. It's clear that one can use Monte Carlo Simulation, but there should be a general ansatz

 

[mathman]

Since you need Monte Carlo for the specific problems I cited (happens to be an area where I have worked), I doubt if there is something general as you are looking for.

Using Monte Carlo the average energy isn't used.

 

[tom.stoer]

I am only looking for the ansatz, a (small set of) formula(s) I can write down. I don't care about the solution for the moment ;-)

Many problems which you can't solve analytically have a rigorous definition; you can't solve QCD analytically, but you can write down the lattice QCD lagrangian

 

[mathman]

There are two basic formulas. The transmission through media, which follows an exponential law, and the interaction when a particle hits something. The latter is dependent on what particles you are considering and the medium.

I don't have any references handy.