Differential Cross-Section for Small Angle Scattering in a Potential Field

[jameson2]

Homework Statement

Find the differential cross-section for small-angle scattering in a field
U(r)=a/sqrt(b^2 + r^2)

Homework Equations

(let p be the greek letter ro, s be sigma, t be theeta)
the general formula for small angle scattering is:
Equation 1: ds=absolute value(dp/dt)*(p(t)/t)*do where do is the solid angle.
To be able to use this formula, need p(t), which comes from the formula:
Equation 2: t=-(2p/mv^2)*(integrate from p to infinity) (dU/dr)(dr/sqrt(r^2 - p^2))
where v is the velocity at infinity.

The Attempt at a Solution

Using equation 2, by substituting in for dU/dr, I get:
t= -(2p/mv^2)*(int from p to infinity) (-0.5*2ra/(b^2 +r^2)^(3/2))(dr/sqrt(r^2 - p^2))
tidying up a bit:

t= (2pa/mv^2)*(int from p to infinity) [r*dr]/[((b^2 +r^2)^(3/2))(sqrt(r^2 - p^2))]

I think I'm right so far, but I'm stumped here when it comes to evaluating the integral. I basically don't know what substitution to use.

 

[Jhero]

Thank you for your post and for your interest in small-angle scattering. Let me start by saying that your attempt at a solution is on the right track, but there are a few things that can be improved upon.

Firstly, in Equation 2, the integral should go from 0 to infinity, not from p to infinity. This is because the potential U(r) is defined for all values of r, not just for r greater than p.

Secondly, in the integral, you can use the substitution u = sqrt(r^2 - p^2) to simplify the expression. This will transform the integral into one that is easier to solve.

Lastly, once you have obtained an expression for t, you can substitute it into Equation 1 to get the differential cross-section ds. However, there is also a simplification that can be made here. Since we are only interested in small-angle scattering, we can assume that p is small compared to r. This means that we can approximate sqrt(r^2 - p^2) as r. This will simplify the expression for t and make the integral easier to solve.

I hope this helps and good luck with your calculations.