D.47 Born differential cross section

This note derives the Born differential cross section of addendum {A.30}.

The general idea is to approximate (A.228) for large distances $r$. Then the asymptotic constant $C_{\rm {f}}$ in (A.216) can be identified, which gives the differential cross section according to (A.218). Note that the Born approximation took the asymptotic constant $C_{\rm {f}}^{\rm {l}}$ equal to one for simplicity.

The main difficulty in approximating (A.228) for large distances $r$ is the argument of the exponential in the fraction. It is not accurate enough to just say that $\vert{\skew0\vec r}-{\skew0\vec r}^{\,\prime}\vert$ is approximately equal to $r$. You need the more accurate approximation

\begin{displaymath}
\vert{\skew0\vec r}- {\skew0\vec r}^{\,\prime}\vert = \sqr...
...im r - \frac{{\skew0\vec r}}{r}\cdot{\skew0\vec r}^{\,\prime}
\end{displaymath}

The final approximation is from taking a factor $r^2$ out of the square root and then approximating the rest by a Taylor series. Note that the fraction in the final term is the unit vector ${\hat\imath}_r$ in the $r$-​direction.

It follows that

\begin{displaymath}
\frac{e^{{\rm i}p_\infty \vert{\skew0\vec r}-{\skew0\vec r...
...bar}
\qquad {\skew0\vec p}_\infty = p_\infty {\hat\imath}_r
\end{displaymath}

Also, in the second exponential, since $z'$ $\vphantom0\raisebox{1.5pt}{$\equiv$}$ ${\hat k}\cdot{\skew0\vec r}^{\,\prime}$,

\begin{displaymath}
e^{{\rm i}p_\infty z'/\hbar} = e^{{\rm i}{\skew0\vec p}_\i...
... \qquad {\skew0\vec p}_\infty^{\,\rm {l}} = p_\infty {\hat k}
\end{displaymath}

Writing out the complete expression (A.228) and comparing with (A.216) gives the constant $C_{\rm {f}}$ and hence the differential cross section.