### 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 . Then the asymptotic constant 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 equal to one for simplicity.

The main difficulty in approximating (A.228) for large distances is the argument of the exponential in the fraction. It is not accurate enough to just say that is approximately equal to . You need the more accurate approximation

The final approximation is from taking a factor 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 in the -​direction.

It follows that

Also, in the second exponential, since ,

Writing out the complete expression (A.228) and comparing with (A.216) gives the constant and hence the differential cross section.