A nonrelativistic particle of charge ze, mass m, and initial speed v0 is incident on a fixed charge Ze at an impact parameter b that is large enough to ensure that the particle's deflection in the course of the collision is very small.
(a) Using the Larmor power formula and Newton's second law, calculate the total energy radiated, assuming (after you have computed the acceleration) that the particle's trajectory is a straight line at constant speed:
ΔW=3m2c3v0πz4Z2e6b31
(b) The expression found in part a is an approximation that fails at small enough impact parameter. For a repulsive potential the closest distance of approach at zero impact parameter, rc=2zZe2/mv02, serves as a length against which to measure b. The approximation will be valid for b≫rc. Compare the result of replacing b by rc in part a with the answer of Problem 14.5 for a head-on collision.
(c) A radiation cross section χ (with dimensions of energy times area) can be defined classically by multiplying ΔW(b) by 2πbdb and integrating over all impact parameters. Because of the divergence of the expression at small b, one must cut off the integration at some b=bmin. If, as in Chapter 13, the uncertainty principle is used to specify the minimum impact parameter, one may expect to obtain an approximation to the quantum-mechanical result. Compute such a cross section with the expression from part a. Compare your result with the Bethe–Heitler formula [N−1 times (15.30)].