Partial Wave Analysis of Scattering *

We can take a quick look at scattering from a potential in 3DWe assume that \bgroup\color{black}$V=0$\egroup far from the origin so the incoming and outgoing waves can be written in terms of our solutions for a constant potential.

In fact, an incoming plane wave along the $z$ direction can be expanded in Bessel functions.

\bgroup\color{black}$\displaystyle e^{ikz}=e^{ikr\cos\theta}=\sum\limits_{\ell=0}^\infty\sqrt{4\pi(2\ell+1)}i^\ell j_\ell(kr)Y_{\ell 0} $\egroup
Each angular momentum \bgroup\color{black}$(\ell)$\egroup term is called a partial wave. The scattering for each partial wave can be computed independently.

For large \bgroup\color{black}$r$\egroup the Bessel function becomes

\begin{displaymath}\bgroup\color{black} j_\ell(\rho)\rightarrow -{1\over 2ikr}\left(e^{-i(kr-\ell\pi/2)}-e^{i(kr-\ell\pi/2)}\right) ,\egroup\end{displaymath}

so our plane wave becomes

\begin{displaymath}\bgroup\color{black}e^{ikz}\rightarrow -\sum\limits_{\ell=0}^...
...{-i(kr-\ell\pi/2)}-e^{i(kr-\ell\pi/2)}\right)Y_{\ell 0} \egroup\end{displaymath}

The scattering potential will modify the plane wave, particularly the outgoing part. To maintain the outgoing flux equal to the incoming flux, the most the scattering can do is change the relative phase of the incoming an outgoing waves.

\begin{displaymath}\bgroup\color{black}R_\ell(r)\rightarrow-{1\over 2ikr}\left(e...
...l\pi/2)}-e^{2i\delta_\ell(k)}e^{i(kr-\ell\pi/2)}\right) \egroup\end{displaymath}

\begin{displaymath}\bgroup\color{black}={\sin(kr-\ell\pi/2+\delta_\ell(k))\over kr}e^{i\delta_\ell(k)} \egroup\end{displaymath}

The \bgroup\color{black}$\delta_\ell(k)$\egroup is called the phase shift for the partial wave of angular momentum \bgroup\color{black}$\ell$\egroup. We can compute the differential cross section for scattering

\begin{displaymath}\bgroup\color{black}{d\sigma\over d\Omega}\equiv{{\rm scattered flux into }d\Omega\over{\rm incident flux}} \egroup\end{displaymath}

in terms of the phase shifts.
\bgroup\color{black}$\displaystyle {d\sigma\over d\Omega}={1\over k^2}\left\brac...
...1)e^{i\delta_\ell}\sin(\delta_\ell)P_\ell(\cos\theta)\right\bracevert^2 $\egroup
The phase shifts must be computed by actually solving the problem for the particular potential.

In fact, for low energy scattering and short range potentials, the first term \bgroup\color{black}$\ell=0$\egroup is often enough to solve the problem.

Only the low \bgroup\color{black}$\ell$\egroup partial waves get close enough to the origin to be affected by the potential.

Jim Branson 2013-04-22