3. Compton ratio R_C

There are now several calculations of the Compton scattering ionization ratio R_C [22,23,24,25,26,27,28,29]. For finite energies (up to 20 keV) these calculations, performed for He, disagree significantly. However in the high energy limit these theoretical results appear to be in agreement, as discussed by Suri, Pisk and Pratt [18].

Andersson and Burgdörfer[23] performed Compton calculations for helium using highly correlated initial state wave functions and approximately correlated or uncorrelated final state wave functions. Their various calculations all predicted the ratio decreasing toward the same constant ratio of $R_{C}=0.8 \%$. Hino, Bergstrom and Macek performed lowest order MBPT calculations for He with a basis set constructed using the V^N-1 potential. In their first paper [24] they asserted that their results reached an asymptotic limit of around $R_{C}=1.7 \%$ somewhat above 10 keV. However in their later work [25], performed with energies up to 20 keV, they found that their results for the ratio were decreasing and had not reached an asymptotic limit. They concluded their results were consistent with the high energy limits of Andersson and Burgdörfer[23] and Suri et al. [26]. Suri et al. [26,2], using generalized shake-off model or impulse approximation (IA) have derived a high energy limit formula for the double to total ionization ratio for Compton scattering

$$R_{C}=1-\sum_{B}\int d^{3}{\rm r}\left\vert\int\Phi_{B}^{\ast... ...\Psi_{i}({\bf r},{\bf r}_{1})d^{3}{\rm r}_{1}\right\vert^{2} .$$ (2)

Here $\Psi_{i}({\bf r},{\bf r}_{1})$ represents the initial state wave function, $\Phi_{B}({\bf r}_{1})$ is a bound state hydrogen like electron wave function (in the potential of charge Z), and the summation is over all bound states. Using highly correlated initial state wave function they obtain R_C=0.008 $(0.8\%)$ for He. At finite energies their results with IA approach the limiting shake-off value from below. The shake-off ratio for Compton scattering, given by Eq.(2), differs from the photoeffect shake-off ratio, given by Eq.(1), because different regions of the initial state contribute to these two processes. Namely, when the incoming photon energy goes to infinity the photoeffect shake-off ratio will be determined exclusively by the region where one of the electrons is at the nucleus. By contrast, the Compton process can occur on free electrons, and all regions contribute in proportion to the probability amplitude that an electron can be found there.

Using the lowest order MBPT based on hydrogenic wave functions and introducing an effective nuclear charge, Amusia and Mikhailov [27,28] obtained R_C=0.0168 for helium. Subsequently, Suri, Pisk and Pratt [18] showed that this introduction of effective charge in the calculation was unjustified. If bare charge is used in the calculation, as is more appropriate, the result would be consistent with other high energy predictions for R_C.

Kornberg and Miraglia [29] studied the energy dependence of R_C using L and V form of the A² interaction operator, correlated initial state and uncorrelated final electron states. Their results are form dependent for finite energies, but, for both forms, approach the limit predicted by Eq. (2) ( R_C=0.008 for helium). The L-form results support the results of Andersson and Burgdörfer (who used the same form of the A² operator, but more correlated initial state and final states) that the asymptotic limit is approached from above. However, the V-form results approach the asymptotic limit from below, as do the IA results of Suri et al..

In the energy region between 6 keV and 20 keV, experimental data are not yet definitely conclusive. At energies between 6 keV and 11 keV the theoretical predictions of Andersson and Burgdörfer seem to be in better agreement with measurements, performed by Levin et al. [30] and by Spielberger et al.[17], than are the other predictions. At higher energies, 15-20 keV, the measurements of Levin et al. [30] seem to be in quite good agreement with the predictions of Bergstrom, Hino and Macek [25], which are higher then the results of Andersson and Burgdörfer as much as 50%. Recent measurement by Spielberger et al. [31] with 58 keV photons found $R_{C}=(0.84_{-.11}^{+.08})\%$ in good agreement with the theoretical predictions for the shake-off limit. Another measurement by Wehlitz et al. [32] performed with 57 keV photons is less conclusive. They found $R_{C}=(1.25 \pm 0.3)\%$.

More extensive study, in a wide energy range, is clearly required in order to understand the energy dependence of R_C. There does seem to be agreement among the theoretical predictions that the high energy limit of R_C is given by Eq. (2), and this is supported by the measurement of Spielberger et al. [31]. However, more measurements in this and even higher energy regions are required. As stated by Spielberger et al [31], the experiments with photons above 100 keV up to `....150 keV are clearly within reach'. This is based on the facts that Compton scattering cross section slowly varies with energy, in this energy region, and on the availability of new-generation high energy synchrotron radiation facilities.

As one considers this higher energy region, it is clear relativistic effects must be examined. (All existing predictions are nonrelativistic). Recently, we have extended our theoretical analysis of double ionization by Compton scattering, based on sudden approximation [2], to relativistic regions. This approach should be appropriate for the relativistic treatment of double ionization Compton scattering from He-like systems. We have demonstrated that for low Z elements relativistic effects do not alter the high energy limit of the Compton double to single ratio R_C, although these effects do affect single and double ionization Compton scattering separately. (The shake-off argument that relativistic effects do not alter the ratio R_C is the same as for the shake-off for photoeffect).

Justification of the sudden approximation for double ionization by Compton scattering at high energies comes from the observation that, in single ionization Compton scattering at high incident photon energies, most outgoing electrons have large energy. Figure (5.1) illustrates that, in high energy Compton scattering, most electrons do have enough energy to apply the sudden approximation. We show the singly differential cross section (with respect to outgoing electron energy) in single ionization Compton scattering from helium, calculated using using an exact numerical approach or IA (which can not be distinguished on this figure for these energies). Results for incident photon energies of 25 keV and 150 keV are displayed. We see that most electrons have energies below the edge determined by the Compton energy for scattering from free electrons at rest. Between zero energy and this edge, the differential cross section is a slowly varying function of energy, and all these energies are, approximately, equally likely. If we assume that 3 keV outgoing electron energy is required to apply the sudden approximation (the shake-off ratio R_P in photoeffect, for helium, is approached at this energy) then we see that at 150 keV this approximation is valid for most of the Compton events.

Assuming the validity of sudden approximation we have concluded that relativistic effects are the same in the double and single ionization cross sections. Therefore, in treating double ionization cross sections by Compton scattering at high, relativistic energies it is appropriate to apply the shake-off factor to the single ionization cross sections. At these relativistic energies, and for low Z elements, the single ionization cross sections can be treated, within a high degree of accuracy, using impulse approximation. This is well established [33,34,35,36] for the treatment of cross sections up to the level of detail of doubly differential cross section (with respect to outgoing photon angles and energy) [37]. Therefore, in evaluating such differential cross sections, in this energy region we may use IA. Conversely, Suri et al. have verified that if one calculates double and single ionization singly differential cross sections the ratio is 0.008 (for He) for all, except very small angles, where both sudden approximation and IA are not valid.

An interesting application of shake-off related ideas in nonrelativistic Compton scattering has been presented by Wang et al.[38]. These authors present calculations of the ratio between double and single ionization singly differential cross sections (with respect to energy transfer) by Compton scattering. These calculations show that, except for small energy transfers (where sudden approximation is not supposed to be good) this ratio is constant (the shake-off constant R_C) up to the free Compton scattering edge. This is consistent with with the usual Compton shake-off picture. However, for larger energy transfers (which contribute negligibly to the total cross section) the ratio changes and rises toward the shake-off ratio for photoeffect R_P, reflecting the fact that large energy transfers, as for photoeffect, are only possible when large momenta are transfered to the atomic nucleus.