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Born approximation problem

Demonstrate, for a particle scattering from a finite-range, spherically symmetric potential V(r), which is weak enough so that the Born approximation for symmetric potentials is valid, that the total cross section, at very low energies, is a linear function of the energy σ(E) = σ0(1+αE), where σ0 is related to the volume integral of the potential... Please see attached for full question.


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The differential cross section is given by:

dsigma/dOmega = m^2/((2 pi)^2 h-bar^4)|T(k',k)|^2 (1)

k' is the wave vector after scattering, k before.

In the Born approximation T(k',k) is given by:

T(k',k)= Integral[d^3r Exp[i (k-k') dot r] V(r)] (2)

The potential is of short range, so only r within some small region can contribute to the integral. When the energy goes to zero, the k vectors go to zero to, so the term (k-k') dot r in the exponential can be regarded as a small parameter which we are allowed to expand in. It is convenient to put d = k - k'. We will need the magnitude of d, so let's calculate this first.

In elastic collisions |k| = |k'|. It thus follows that:

d^2 = (k - k')^2 = k^2 + k'^2 - 2 k dot k' = ...

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