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

  1. Nov 12, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the differential cross-section for small-angle scattering in a field
    U(r)=a/sqrt(b^2 + r^2)


    2. Relevant 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.

    3. 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.
     
  2. jcsd
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