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

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

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

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

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

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

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