Coulomb Scattering Qm

Mott and Massey give the structure factor for Coulomb scattering

(1)
\begin{align} f(\theta) = \frac{Z^2 e^2}{mv^2}\frac{1}{\sin^2\frac{\theta}{2}}e^{i\alpha\log(1-\cos\theta)+2i\eta_0+i\pi} \end{align}

The wave function itself is determined by

(2)
\begin{align} \psi \approx e^{ikz} + \frac{f(\theta)}{r}e^{ikr} \end{align}

And the cross section is determined by

(3)
\begin{align} I(\theta)d\omega =|f(2\theta)+f(\pi-2\theta)|^2 4\cos\theta d\omega \end{align}

For the case where there is spin, M&M give

(4)
\begin{align} P_T = \frac{1}{4}(1-\cos\Theta) P_S + \frac{1}{4}(3+\cos\Theta)P_A \end{align}

where

(5)
\begin{align} P_s = |\psi(1,2)+\psi(2,1)|^2 \end{align}
(6)
\begin{align} P_A = |\psi(1,2)-\psi(2,1)|^2 \end{align}

In deriving this, M&M start with finding the probability of finding particle 1 in r1 and particle 2 in r2. And the result is

(7)
\begin{align} |\psi(1,2)|^2 + |\psi(2,1)|^2 - \frac{1}{2}[\psi(1,2)\psi^*(2,1)+\psi(2,1)\psi^*(1,2)](\cos\Theta +1) \end{align}

where

(8)
\begin{align} \cos\Theta = \bf{1}\cdot \bf{n} \end{align}

Applying this to the Coulomb interaction, one gets for a total probability

(9)
\begin{align} P_T = 16 \left( \frac{4}{\sin^4(2\theta)}-\frac{3+S^2}{\sin^2(2\theta)}\right ) \cos\theta \end{align}

where we used

(10)
\begin{align} <cos\Theta> = S^2 \end{align}

Equation 11 in Lee, Choi, Kang is

(11)
\begin{align} \frac{d\sigma}{d\Omega} = \frac{4r_e^2}{\left(\frac{v}{c}\right)^4}\left[\frac{4}{\sin^4\theta} -\frac{3+S^2}{\sin^2(\theta)}\right] \end{align}
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