   Next: 2.2 The Elwert-Haug approximation Up: 2. Discussion of theories Previous: 2. Discussion of theories

## 2.1 Bethe-Heitler approximation

The Bethe-Heitler approximation  is the simplest of the three theories discussed here. It is also the most widely used, since the cross sections , and for radiation in scattering from a Coulomb point charge can be expressed in closed form in terms of elementary functions. The Bethe-Heitler formula results from the application of the Born approximation for the interaction of the scattering electron with the potential, retaining only the first non-zero term. Detailed discussions of the Bethe-Heitler equation for can be found in  and a number of texts (see , for example). Note that, since it is derived from a first order perturbation for scattering in the potential, the Z dependence of the Bethe-Heitler result factors; is Z independent. When scattering from a neutral or partially ionized atom, the Bethe-Heitler result includes multiplication of the matrix element by the form factor (as follows from the first Born treatment), introducing a more complicated Z dependence. The form factor (1)

where  is the charge density and is the magnitude of the momentum transfer. In our calculations of the predictions of this theory form factors are obtained by numerically integrating Equation (1), using the charge density obtained from bound state wavefunctions in the Dirac-Slater potential discussed earlier, but with a Latter tail. Corrections to the Bethe-Heitler cross sections are of relative order , where is the initial or final electron velocity, so the condition for the validity of this approximation is frequently written . This condition must be satisfied for both the initial and final electron.

One of the most significant problems with the Bethe-Heitler result is that it predicts a zero cross section in the limit that the photon takes all of the initial electron's energy (the tip'' region of the spectrum). Approaching this limit, the outgoing electron has very little energy, and the condition for validity of the Born approximation is violated. The behavior in this limit can be fairly well corrected, at least for the spectrum , by multiplying the Bethe-Heitler result by the Elwert factor (2)

where and are the initial and final velocity of the electron in units of the speed of light in vacuum. While originally derived in a comparison of non-relativistic Born and exact Coulomb results, the form (2) is usually taken as the relativistic generalization. The relativistic Elwert factor is simply the square of the ratio of the normalizations (N2/N1)2 of the Sommerfeld-Maue wavefunctions. Multiplying the Bethe-Heitler cross section by the Elwert factor gives a nonzero cross section in the tip (hard photon) region and has no effect in the soft photon limit. A more detailed discussion of the Elwert factor can be found in .   Next: 2.2 The Elwert-Haug approximation Up: 2. Discussion of theories Previous: 2. Discussion of theories
Eoin Carney
1999-06-14