Next: 2.2 The Elwert-Haug approximation
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The Bethe-Heitler approximation [2] 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
[2] and a number of texts (see
[7], 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
[2]
|
(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
[8]
|
(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
[3].
Next: 2.2 The Elwert-Haug approximation
Up: 2. Discussion of theories
Previous: 2. Discussion of theories
Eoin Carney
1999-06-14