3. Results

We find, however, that for
,
in general, the
Bethe-Heitler results and the Elwert-Bethe-Heitler results are
frequently more than a factor of two too small or too large in
comparison to the exact partial wave calculations. For these higher
*Z* elements no substantial region of systematic agreement with either
version of this theory was found in this work, though one can see
trends of improving agreement as *Z* decreases. For these energies we
find that the Elwert-Haug predictions are significantly closer to ours
than are Bethe-Heitler predictions, even for *Z*=79, for small
momentum transfers. However, for large momentum transfers, or for
large emission angles,
,
we find that the
Elwert-Haug results approach the Elwert-Bethe-Heitler results and,
correspondingly, significantly underestimate our results.

We may summarize the results of Fink [18] and Tseng and
Pratt [16] on the agreement of the simpler theories with
the exact partial wave treatment for
[17]
for heavier elements (*Z*>60) at energies similar to those considered
here: 1) Bethe-Heitler results away from the tip show differences
(10-50%) from exact partial wave results, typically underestimating.
2) Use of the Elwert factor typically improves the Bethe-Heitler
result. 3) For this energy range (though not at higher energy)
Elwert-Haug results underestimate the exact partial wave results,
deviating as significantly as Elwert-Bethe-Heitler results. For
the Elwert-Haug result was generally no improvement on
Bethe-Heitler. See the schematic summary in Table
7.1. A somewhat more detailed discussion of the agreement
between the theories can be found in [17,13].

In contrast to these results for
,
we find (as already
indicated and as we describe in more detail below) the following
results for
:
1) We find more extreme differences than
in
between the Bethe-Heitler and exact partial wave
results (100-400%), and we find that the sign of the difference
between the two theories changes as a function of the momentum
transfer *q*. 2) We find that for these energies use of the Elwert
factor as defined in Eq. (2) is inappropriate at
small momentum transfers. 3) We find some region of agreement with
the Elwert-Haug theory at small momentum transfers, particularly for
high incident and outgoing electron energies, but not for large
momentum transfers. In the following subsections we show
representative data and compare these theories in more detail. See
also the schematic summary in Table 7.1.

- 3.1 Comparison with (Elwert-)Bethe-Heitler approximation
- 3.2 Comparison with the Elwert-Haug approximation