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 = \sqrt...
...sim 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}...
...{\,\prime}/\hbar}
\qquad \vp_\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}\vp_\infty^{\,\rm ...
...ime}/\hbar}
\qquad \vp_\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.