Second-order correction to the low-frequency approximation for scattering in a laser field

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Abstract
It has been known for some time how to construct an approximation to the amplitude for non- relativistic potential scattering in a low-frequency laser field that is correct to first order in the frequency and that depends only on the on-shell field-free scattering amplitude. The next term in the expansion, of second order in the frequency, is determined here. In addition to the physical field-free scattering amplitude, a knowledge of the single-photon spontaneous bremsstrahlung amplitude (up to terms of first order in the frequency) is required. The improved low-frequency approximation obtained here is used to derive an improved sum rule. It is found that when the cross section (correct to second order) is summed over final photon states, the result depends only on the physical field-free cross section in a form which is identical to that obtained from a classical description of the motion of the particle in the field, with the collision taking place instantaneously and without influence from the field. A relativistic version of this sum rule was given earlier by Brown and Goble [Phys. Rev. 173, 1505 (1968)], who based their analysis on general field-theoretic considerations.