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3.1 Comparison with (Elwert-)Bethe-Heitler approximation

In Figure 6.2 we show predictions of the various theories for the reduced cross section $d^3\!\sigma/Z^2$, as a function of $\theta_2$, for the simple forward photon emission case of $\theta_1=0^\circ$. The case shown is for Gold (Z=79) at E1=300keV and E2=150 keV, an example of equal energy sharing between the photon and outgoing electron. The data was smoothed using a cubic spline to increase legibility. Due to the excessive amount of computation time involved, our calculations were made on a grid at $9^\circ$ increments (starting from $\theta_2=0^\circ$), with additional grid points in the peak region of the cross section. We see that the Bethe-Heitler and Elwert-Bethe-Heitler results significantly overestimate the cross section near the peak. Also, the peak of the Bethe-Heitler cross section is narrower than that in the exact partial wave result: at larger angles the Bethe-Heitler result is underestimating the cross section. Inclusion of the multiplicative Elwert factor somewhat improves the large angle behavior but worsens the results near the peak in the cross section. In this simple case, where the incident electron and outgoing photon momenta are collinear, the cross section is a single-valued function of momentum transfer. In Figure 6.2 we have indicated the momentum transfers as well as the scattering angles. The kinematically allowed momentum transfer range in this case is between $q_{\rm min}\approx 0.1\,mc$ and $q_{\rm max}\approx 1.8\,mc$ (m is the mass of the electron and c the velocity of light in vacuum), with the limits corresponding to the outgoing electron parallel or anti-parallel to the incident electron. We see that at low momentum transfers ( $q < 0.35\,mc$) the Bethe-Heitler cross sections are systematically larger than the exact partial wave results, even larger if using the Elwert factor. At larger momentum transfers the Bethe-Heitler results are systematically smaller than the exact partial wave results; use of the Elwert factor give some improvement.

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 $d^2\!\sigma$ 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 $\theta_2=-\theta_1=9^\circ$ (at $\phi_2=180^\circ$, 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 $q<0.6(Z\alpha)\,mc$, 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 $d\Omega_2$, effectively integrating over kinematically allowed momentum transfers, ``averages out'' these differences to some extent, giving the lesser 10-50% differences seen by Fink in $d^2\!\sigma$. Note we have found that inclusion of the Elwert factor provides mixed results for $d^3\!\sigma$ - 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 $d^2\!\sigma$.


next up previous
Next: 3.2 Comparison with the Up: 3. Results Previous: 3. Results
Eoin Carney
1999-06-14