# Scattering Amplitude in second Born Approximation for the Yukawa potential

Does anyone know where I can find the analytical expressions of the scattering amplitude in second Born Approximation for the Yukawa potential? I need it for the both cases of the method of partial waves and the Born series as defined to solve the Lippmann-Schwinger equation. In general the second term is unimportant being the Born Approximation very good in high-energy physics but it is not my case as I consider collisions in semiconductors. I would not like to waste time to calculate it as I presume that probably someone else has already done.

-
I don't have the expression off hand - I would have to work it out as well, but I remember the general rule of thumb (which may not apply in your case): if the second order term in the Born series matters, then probably all orders matter and you really should use resummed propagators - perhaps mean field or some other nonperturbative approximation. –  Michael Brown Aug 17 '13 at 2:05
You may find the answer in this 1956 paper: prola.aps.org/abstract/PR/v102/i2/p537_1 –  Luboš Motl Aug 17 '13 at 15:09
I am not concern about the convergence of the Born series for the Yukawa potential. I need the second term only to ascertain that including it the differential cross-section becomes smaller than the one calculated in first Born Approximation. That would mean that if the Born approximation is not satisfied then the differential cross-section in first Born Approximation overstimates the scattering probability. –  Caute Aug 22 '13 at 17:37
Lubos, I checked the paper you suggested. It is useless in my case as it considered relativistic scattering. Curiously it is the first time I saw the relativistic scattering treated in that way. As far as I know the standard scattering theory does not include relativist equations. Clearly there are QED or QFT methods to treat the relativistic cases. –  Caute Aug 22 '13 at 17:45