In Figure 6.3 we show the corresponding forward emission cross section data for Z=47, E1=100 keV, E2=50keV. We see that for this lower Z the exact partial wave results are beginning to peak more sharply, but the Bethe-Heitler results still significantly underestimate the cross sections at large scattering angles and large momentum transfers. The broadening of the peak for increasing Z in the exact partial wave calculations was also seen in the results of Fink [18] and Tseng [16]. While we are showing a lower energy case than previously, the results are similar throughout the energy range considered here. In Figure 6.4 we show the cross section for fixed (at , the incident and outgoing electron are collinear) for the same energies and atomic number as in the case above. We see that for these angles or momentum transfers, the Bethe-Heitler and Elwert-Bethe-Heitler results overestimate the exact partial wave results.
In our calculations, including Z= 47, 53, 60, 68, and 79, with energies E1 from 50 keV to 450 keV, and k/T1 from 0.4 to 0.9, we find quite generally that the Bethe-Heitler results overestimate our calculations for small momentum transfers, empirically for , and underestimate our calculations for large momentum transfers. For non-forward photon emission angles, and particularly for large photon emission angles, the division into small and large momentum transfers is not as sharp as in the forward emission case, but the results are qualitatively the same. Integration over the outgoing electron angle , effectively integrating over kinematically allowed momentum transfers, ``averages out'' these differences to some extent, giving the lesser 10-50% differences seen by Fink in . Note we have found that inclusion of the Elwert factor provides mixed results for - causing greater overestimates in the small momentum transfer range but improving the agreement for large momentum transfers, for the energies considered here. In these energy ranges the Elwert factor improves the integrated result for .