Compton Scattering w/Moving Electron: Turner's Eq 5.29

[Ken Miller]

I beg your pardon for not writing out the math explicitly in the following. I started to do so, and realized that it would take me hours to debug my attemps at Latex! And I hope that the my explanation is clear enough that it's not needed.

I've been working through R.E. Turner's "Relativity Physics," and have a question about the example he gives of Compton scattering, in which the quantity of interest is the post-collision gamma particle. All the examples I've seen elsewhere assume that the electron is at rest before the collision. In Turner's example, the electron is in motion before the collision. When Turner expands the right-hand side of the 4-vector conservation equation (equation 5.29 if you happen to have a copy), he gets the following terms:

--the dot-product of the pre-collision electron with itself,
--the energy/time-like term from the dot-product of the electron with the pre-collision gamma particle
--the energy/time-like term from the dot-product of the electron with the post-collision gamma particle
--the energy/time-like term from the dot-product of pre- and post-collision gamma particles
--the dot-product of the momenta of pre-collision electron and the post-collision gamma ray

But since both the pre-collision electron and gamma ray have momentum, shouldn't there also be a term for the dot-product of the momenta of these two particles?

Similarly, shouldn't there be a term for the dot-product of the pre-collision electron and the post-collision gamma ray?

 

[robphy]

From the limited preview on Google books, that passage considers the usual case when the target electron is initially at rest, although the diagram is for the more general case with the target electron is initially in motion (which is Exercise 4... not viewable to me).

So, it seems that terms after Eq. 5.29 that are zero (because the electron’s initial spatial momentum is zero) have been dropped.

 

[Ken Miller]

Oh, of course. I was so focused on the diagram that I didn't even see that the text mentioned "electron at rest". Thank you for saving me a lot of puzzlement!