3. Study of correlation effects for double ionization in Compton scattering

It is only recently that calculations of the Compton scattering ionization ratio R_C [28,29,30,31,32,33,34,35] have become available. For finite energies (up to 20 keV) these calculations, performed for He, disagree significantly. However in the high energy limit all these theoretical results appear to be in agreement, as discussed by Suri, Pisk and Pratt [20]. We quote here our recent description [2] of these Compton calculations.

Suri et al [32,3], using the generalized shake-off model or impulse approximation (IA), have derived a high energy limit formula, analogous to Eq. (2) for photoeffect, for the double to total ionization ratio for Compton scattering:

$$R_{C}=1-\sum_{B}\int d^{3}{\rm r}_{1}\left\vert\int\Phi_{B}^{... ..._{i}({\bf r}_{1},{\bf r}_{2})d^{3}{\rm r}_{2}\right\vert^{2} .$$ (6)

Here, as in Eq. (2), $\Psi_{i}({\bf r_{1}},{\bf r_{2}})$ 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 a highly correlated initial state wave function Suri et al obtain R_C=0.008 $(0.8\%)$ for He. At finite energies their results with IA are approaching the limiting shake-off value from below. The shake-off ratio for Compton scattering, given by Eq. (6), differs from the photoeffect shake-off ratio, given by Eq. (2), because different regions of the initial state contribute to these two processes, even though in both cases a high energy incident photon is assumed. The photoeffect shake-off ratio will be determined exclusively by the region where one of the electrons is at the nucleus (or, in the relativistic case, at small-Compton wavelength-distances), as required for the ejection of a single electron when the photon is absorbed. 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.

Andersson and Burgdörfer[29] had 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 a 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. Originally [30] they thought that their results reached an asymptotic limit of around $R_{C}=1.7 \%$ somewhat above 10 keV. But in their later work [31], considering energies up to 20 keV, they found that their ratio was beginning to decrease and that their component matrix elements had not at all reached an asymptotic limit. They concluded their results were consistent with the high energy limits of Andersson and Burgdörfer[29] and Suri et al [32].

Kornberg and Miraglia [35] have 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. (6) ( R_C=0.008 for Helium). The L-form results, like those of Andersson and Burgdörfer (who used the same form of the A² operator, but more correlated initial state and final states) give an asymptotic limit which 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 apparent contrast to the above results, in a calculation using the lowest order MBPT based on hydrogenic wave functions and introducing an effective nuclear charge, Amusia and Mikhailov [33,34] interpreted their result as giving R_C=0.0168 for Helium at high energy. However, Suri, Pisk and Pratt [20] afterwards showed that the choice of effective charge for the calculation was unjustified. If bare charge is used in the calculation, as is in fact more appropriate, the result would be consistent with other high energy predictions for R_C.

The experimental data which now exist for Compton scattering are suggestive but not yet entirely 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 [36] and by Spielberger et al[26], than are the other predictions. At higher energies, 15-20 keV, the measurements of Levin et al [36] seem to be in quite good agreement with the predictions of Bergstrom, Hino and Macek [31], which are higher then the results of Andersson and Burgdörfer by as much as 50%. Finally, recent measurement by Spielberger et al [37] 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 [38] performed with 57 keV photons is less conclusive. They found $R_{C}=(1.25 \pm 0.3)\%$.

As one considers higher energy regions, it is clear relativistic effects must be examined. Recently, Suri et al have extended their theoretical analysis of double ionization by Compton scattering, based on sudden approximation [3], to relativistic regions. They have demonstrated explicitly 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 [19]).

A more detailed study of nonrelativistic Compton scattering (more then just the total cross section), at high energies, has recently been presented by Wang et al[39]. These authors report calculations of the ratio $R_{C}(\epsilon)$ between double and single ionization singly differential cross sections (with respect to energy transfer $\epsilon=\omega_{i}-\omega_{f}$; $\omega_{i}$ is incident photon energy and $\omega_{f}$ scattered photon energy) by Compton scattering. In their numerical calculations they used only the A² term from the complete photon-electron interaction Hamiltonian. (See our discussion below of the appropriateness of this assumption). These calculations show that, except for small energy transfers (where sudden approximation is not supposed to be good) this ratio $R_{C}(\epsilon)$ is constant, (equalling the shake-off constant R_C of total cross sections) up to the free Compton scattering edge (maximum energy transfer in Compton scattering from free electrons at rest, $\epsilon_{free}^{max}=\omega_{i}/[1+m/(2\omega_{i})]$). This is consistent with the usual Compton shake-off picture. For larger energy transfers (which contribute negligibly to the total cross section) the ratio changes. It first slightly decreases, then increases toward a maximum [their value of $R_{C}(\epsilon)$ at this peak is between 3.6 % and 4.2 % for incoming photon energies between 20 keV and 90 keV], and then decreases toward a value above the shake-off ratio (for 20 keV photons this value is around 2.6 %, according to Wang et al). They find that at relatively low incident photon energies (such as 20 keV) an additional term contributes significantly to the shake-off term even at the largest allowed energy transfers. They call this term an ``exchange shake-off term''. The authors argue that for larger incident photon energies the ``exchange shake-off term'' becomes small and, therefore, the ratio $R_{C}(\epsilon)$ should approach the photoeffect ratio R_P exactly, for large energy transfers. (The authors assume that the high energy photoeffect ratio is the shake-off ratio R_P^s.o.=1.66%). This behavior of the ratio $R_{C}(\epsilon)$ reflects the fact that large energy transfers, as for photoeffect, are only possible when large momenta are transfered to another particle in the atomic system.

This (nonconstant) behavior of the ratio $R_{C}(\epsilon)$ with energy transfer $\epsilon$ can qualitatively be understood within the shake-off picture. The utility of a shake-off calculation of $R_{C}(\epsilon)$ for high energy transfers was analyzed by Suri et al [40], taking into account the complete electron-photon interaction Hamiltonian, i. e. in the nonrelativistic formulation including both A² and ${\bf p}\cdot{\bf A}$interactions. The shake-off contribution to the differential ratio $R_{C}(\epsilon)$ is obtained if most energy transfer is taken by one of the two electrons. Then, necessarily, large momenta are transfered to the nucleus and not to another electron, which only shakes off. This can be experimentally distinguished by observing the recoil of the nucleus which, for large energy transfers, becomes large.

For large energy transfers the dominant part of the photon-bound electron interaction which determines Compton scattering becomes the ${\bf p}\cdot{\bf A}$ term, not the A² term considered by Wang et al. It can be shown, using the low energy theorem, that for high energy transfers (low outgoing photon energies) the Compton scattering matrix element becomes proportional to the photoabsorption matrix element. As earlier discussed, photoabsorption at these energies is determined by two mechanisms, the shake-off and quasi-equal-sharing mechanisms. Both of these mechanisms should influence the Compton ratio $R_{C}(\epsilon)$ when $\epsilon \rightarrow \omega_{i}$. However the shake-off ratio does not depend on the choice of interaction as long as the same region of the matrix element is being emphasized. Therefore, as discussed by Suri et al, for very large energy transfers ( $\epsilon \rightarrow \omega_{i}$) the Compton ratio $R_{C}(\epsilon)$ can be understood in terms of these mechanisms. In the situation when the shake-off assumption is valid (we can experimentally distinguish such events) the Compton ratio $R_{C}(\epsilon)$ approaches the photoeffect shake-off ratio R_P^s.o., as suggested by Wang et al.

The shake-off picture is valid for those Compton events in which one electron takes most of the energy transfer. (Judging by the required energy for the shake-off assumption in photoabsorption to be valid, this energy transfer should be larger then approximately 3 keV [40], which would require a far higher incident photon energy). In the region between roughly 3 keV energy transfer and the free Compton scattering edge one bound electron takes most energy transfer through the free Compton mechanism (no momentum transfer to another particle is necessary). The differential ratio in this region is nearly constant, corresponding to the total ratio R_C. We are probing some average correlation between electrons in the initial state, as in the case of studying the total ratio R_C. Note that, to obtain 3 keV electrons through this free Compton mechanism (which dominates the total cross section), incident photons should have about 30 keV. To have the shake-off picture valid for the total cross section for the double ionization by Compton scattering most electrons should have energy above 3 keV, which means that photon energies above 50-60 keV are required. For higher energy transfers, momentum transfer (by the electron) to another particle is necessary. In the case of the shake-off mechanism this particle is the nucleus. In order to transfer large momenta to the nucleus an electron must approach it and as energy transfer increases, it must be closer, so that in double ionization we are probing the correlation between the two electrons when one electron is near the nucleus. For very large energy transfers we are probing the same region as in the case of photoeffect with shake-off. Therefore, in the shake-off region for large energy transfer $\epsilon$, the differential ratio $R_{C}(\epsilon)$ approaches the photoeffect shake-off ratio R_P^s.o as $\epsilon \rightarrow \omega_{i}$. Thus different regions of energy transfer give us information about different regions of the ground state, even when the shake-off approach remains valid. This example shows that even when the shake-off assumption remains valid, the behavior of $R_{C}(\epsilon)$ for differential cross sections is more complex, dependent on $\epsilon$ but not $\omega_{i}$, than for R_P (a constant), independent of $\omega_{i}$ ( which is the energy transfer $\epsilon$ in photoeffect). Study of $R_{C}(\epsilon)$ with energetic photons probes electron-electron correlations in various regions of the ground state wave function.

These ideas can be further extended in studying correlation effects, using double ionization doubly differential cross sections (with respect to outgoing photon energy and angles) at high momentum transfers. As is well known, ordinary Compton scattering became an important tool in the study of the momentum distribution of bound electrons, through utilization of the impulse approximation. This momentum distribution is experimentally extracted from the Compton profile, which includes, in the case of helium-like systems, both single and double ionization (single ionization being dominant). However, we may define a double ionization Compton profile by specifying the state of ionization of the ion.

Using the impulse approximation formula obtained by Suri et al [32] the doubly differential cross section for double ionization Compton scattering from a two-electron atom is

$$\frac{d^{2}\sigma_{C}^{++}}{d\omega_{f}d\Omega_{f}}= \left( \... ...\frac{\omega_{f}}{\omega_{i}} \right) S^{++}(\omega,{\bf k}) ,$$ (7)

with the expression S⁺⁺ given by

$$S^{++}(\omega,{\bf k})=\int d^{3}{\rm p} \delta (\omega-\frac... ...si_{i}({\bf p},{\bf q}\ )d^{3}{\rm q}\right\vert^{2} \right] .$$ (8)

Here $(d\sigma/d \Omega_{f})_0$ is Thomson scattering cross section (for two electrons) and the meaning of other symbols is as in Eq. (6), except that the wave functions are represented in momentum space.

The structure of the double ionization Compton profile is similar to the structure of the Compton profile in ordinary Compton scattering [3] and the position of the peak is the same. As in ordinary Compton scattering, different regions of the Compton profile probe different regions of the initial bound two-electron state momentum distribution. However, in the ordinary Compton profile the correlation effects (we are primarily interested in correlation effects beyond the Hartree-Fock model) are relatively small (of the order of 1% or less for the helium-like atom which we discuss here). Contrary to this, in the double ionization Compton profile the correlation effects (beyond Hartree-Fock) are much more pronounced [3], as one would expect. Therefore, the study of the double ionization Compton profile can provide a more precise tool for the study of correlation than the ordinary Compton scattering. For simple systems, like Helium, these type of experiment seems to be within reach. Experimentally one may detect photons in coincidence with a doubly ionized ion. Detection of recoil ions produced in double ionization Compton scattering has already been successfully accomplished. Some comments on the validity of impulse approximation for the calculation of the double ionization Compton profile are in order. The utility of impulse approximation in calculating the ordinary Compton profile is well established, even in circumstances which stretch the assumptions of its derivation [41,42,43,44,45]. In the case of the Helium atom the theoretical predictions have been successfully compared with experimental Compton profiles at photon energies around 15 keV [41] for backward angles (to assure large energy transfers). These energies, which in fact push the parameters assumed in its derivation, may not be sufficient for double ionization Compton profile and larger energies may be required.

This can be seen from the study of the triply differential cross section of the single ionization Compton scattering. It has been found by Z. Kaliman et al [46] that for the triply differential cross section (when the outgoing electron is also observed) the impulse approximation approach is less justified at these energies. We may view the double ionization Compton profile as a less averaged observable (as is the triply differential cross section in single ionization), and therefore larger momentum transfers, corresponding to the assumptions underlying impulse approximation, are in fact required for its validity in describing non-averaged observables. These requirements correspond to the region of validity of the shake-off description. As already discussed, the shake-off assumption is valid when the fast outgoing electron has energy above approximately 3 keV. In order to reach this region for backward angles at least 30 keV photons are required.