3. Results

We summarize our results in Table 7.1. While a fully comprehensive comparison of these theories has not been undertaken, we have a significant amount of cross section data, which we believe is representative, over a broad range of parameters, for bremsstrahlung accompanying scattering from targets with $47\le Z\le 79$. We have limited our calculations to energies considered in existing or anticipated experiments (E₁=50-450 keV, 0.4 < k/T₁ < 0.9where T₁ is the electron's initial kinetic energy). We have calculated cross section data on a discrete grid over a full $4\pi$solid angle for many energies and several Z within these ranges. We have performed calculations in the non-coplanar geometry, as well as in the coplanar geometry, in anticipation of experimental data [14,15]. Here we compare our results with those of the simpler theories for these ranges of parameters. Comparisons of these theories have been previously made for the case of the doubly differential cross section [16,17]. We emphasize here when comparisons of the triply differential cross section contrast with, or agree with, the conclusions from the comparisons of $d^2\!\sigma$. In all cases, our integrated cross section data agree with reported $d^2\!\sigma$ data where it exists, and we do not reach any conclusions for $d^2\!\sigma$ which contradict those of Tseng and Pratt [16] or Fink [17].

We find, however, that for $d^3\!\sigma$, 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, $\theta_1 > 45^\circ$, 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 $d^2\!\sigma$ [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 $d^2\!\sigma$ 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 $d^2\!\sigma$, we find (as already indicated and as we describe in more detail below) the following results for $d^3\!\sigma$: 1) We find more extreme differences than in $d^2\!\sigma$ 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.