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Born approx.

  1. Dec 26, 2009 #1

    MathematicalPhysicist

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    1. The problem statement, all variables and given/known data
    Consider a non-relativistic scattering of a particle of mass m and charge e from a fixed distibution of charge [tex]\rho(r)[/tex]. Assume that the charge distribution is neutral, [tex]\int d^3r \rho(r) =0[/tex], it's spherically symmetric, and the second moment is defined as:
    [tex]A=\int d^3r r^2\rho(r)[/tex].
    Use the Born approximation to derive the differential cross section for the scattering of a particle of wave vector k.


    2. Relevant equations
    Let [tex]q=2ksin(\theta/2)[/tex] and the amplitude of the differential cross section to be:
    [tex]f(\theta)=(-2m/(q\hbar^2))\int_{0}^{\infty} rV(r)sin(qr)dr[/tex]
    (for a spherically symmetric potential).
    N.B
    theta is the scattering angle.

    3. The attempt at a solution
    To do the calculation all I need to know is what is the potential,
    now the constant A has magnitude of charge times displacement squared, which means eA/r^3 will give units of potetnial energy, but when I insert this I get that the integral doesnt converge, am I wrong here?
    If it were the first moment then the integral will converge (the known intgral of sin(x)/x on the etire real line).
    Any suggestions here?

    Thanks in advance.
     
  2. jcsd
  3. Dec 29, 2009 #2

    MathematicalPhysicist

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