Explore BrainMass

Explore BrainMass

    Born approximation problem

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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.

    © BrainMass Inc. brainmass.com February 24, 2021, 2:35 pm ad1c9bdddf


    Solution Preview

    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' = ...

    Solution Summary

    A detailed solution is given. The Born approximation for symmetric potentials is valid, that the cross section is determined.