1. Introduction

Our present understanding of complex systems in various fields of physics is to a high degree based on independent particle approximation. But as one wants and needs to go beyond this simple approximation the study of the correlation effects among the various constituents of a system becomes very important. In atomic physics, unlike some other fields, the constituent particles and quite exact interactions among them are well known, based on quantum electrodynamics. Yet, our understanding of correlation effects even for the simplest many body atomic systems is far from satisfactory. In this paper we focus our attention on our understanding of one of the simplest correlation effects in one of the simplest atomic systems: high energy double photoionization of two electron (helium-like) atoms and ions.

The dominant photoionization processes, generally resulting in ejection of one electron, are photoeffect (incident photon absorbed) and Compton scattering (incident photon scattered). Photoeffect dominates the total cross section at lower photon energies and its relative contribution in the total photon atom interaction decreases as photon energy increases. For a He-like system, at energies for which the incoming photon momentum k becomes comparable to the average momentum of the bound electrons p_av (of a two electron atom or ion in its ground state), photoeffect and Compton scattering are comparable. For higher energies Compton scattering dominates. In the case of Helium Compton scattering dominates above, roughly, 6.5 keV.

Double ionization of two electron atoms and ions by one photon is now attracting significant attention, both experimentally and theoretically. The interest in the use of photons in ionizing two electron systems (particularly Helium) is based on the observation that the interaction between electrons and the radiation field is described by a one-body operator. This means that, in the lowest order for a given ionization process, a photon interacts with only one electron. Hence the simultaneous ejection of two electrons is due to the electron-electron interaction, and therefore double ionization by a photon provides a clear probe of electron-electron correlation. In a two electron system no other correlations are present.

For this simplest many-body system we hope to develop more exact theoretical descriptions and perform more complete experiments. Ultimately we wish to understand all correlation effects in this relatively simple system, with the purpose of thereby better understanding such effects in more complex systems. However, as a first step, further simplification in the study of this system may be obtained by using high energy photons. It has been shown, for both Compton scattering and photoeffect, that using high energy photons can provide conditions in which final state interactions (interactions between the two electrons subsequent to the photon-atom interaction) may be neglected; the processes reflect only the correlation effects in the initial state. This situation arises when one electron takes almost all energy transfered to the atomic system and is rapidly ejected, while the other electron just ``shakes off'' in the readjustment of the residual ion. The situation is particularly interesting for its simplicity in description and, since it generally dominates the total cross sections for the processes, it is reasonably accessible to experiment. We will here discuss the correlation effects which persist in photoionization with high energy photons.

The simplest observable to discuss, for both experimental and theoretical treatments of the correlation effects in these processes, has been the ratio R between probabilities for double and single photoionization. This ratio is nonzero because of electron-electron correlation; its value depends on incident photon energy. Also, the value depends on the process under consideration and the inclusiveness of its observation. Therefore we talk, for total cross sections, about the Compton double to single ionization ratio R_C and the corresponding photoeffect ionization ratio R_P. The energy dependence of these observables, and particularly their high energy limits, are presently subjects of extensive studies. In the last several years significant progress has been achieved [1]. However, the relatively large disagreements between theoretical predictions, as well as among experimental data, indicate that we still do not entirely understand, for either process, all aspects relevant in determining the ratio R.

Much understanding of these processes, and their ratios R, can be obtained within the shake-off approximation, i.e. a sudden approximation. Present experimental results using high energy photons can be understood within the shake-off picture (in fact some measured high energy photoabsorption ratios R_P implicitly assume a shake-off picture [2]). By shake-off approximation we mean the assumption that the final state electron-electron interaction can be neglected, since one electron accepts almost all energy transfer from the incident photon and moves away much faster than the other[3], which is generally ejected at nonrelativistic velocities consistent with the momentum distribution in the initial bound state. With such an assumption, we consider only a part (perhaps dominant) of the full kinematically allowed region and, therefore, study only (aspects of) the correlation effects in the initial atomic or ionic system.

In the case of photoabsorption, assuming the validity of shake-off approximation (only initial state correlation), this means that we are restricting our study of such correlation effects in the initial state to the region when one electron is close to the nucleus (required if it is to be ejected with high energy). Even if considering more differential observables, not a total cross section, we will see later that in photoeffect the shake-off assumption restricts us to correlation effects in which one electron is near the nucleus. This means that, assuming shake-off, double ionization differential cross sections (in the fast ejected electron angle) are also simply proportional to single ionization differential cross sections, with R_P(0.017 in the case of Helium) being the constant of proportionality for differential as well as for total cross sections. However, it is now being more widely appreciated that there seems to be another, non-shake, mechanism for high energy photoabsorption which also contributes significantly to the total as well as to the differential double ionization cross section [4]. The study of this ``quasi-equal-energy sharing mechanism'', if experimentally feasible, would reveal the correlation when two electrons, in the initial state, are close to each other, not necessarily close to the nucleus. (Like shake-off this quasi-equal-energy sharing mechanism does not involve final state correlations.)

In the case of Compton scattering, unlike the double photoabsorption, the shake-off mechanism does seem to be the only dominant mechanism for the double ionization total cross section at high energies. As one considers more differential observables other, non-shake, mechanisms may contribute or dominate. However, even when the shake-off assumption remains valid, the behavior of the ratio R_Cfor differential cross sections is more complex then for R_P (a constant). Here, even assuming shake-off, in studying differential cross sections we are still studying various regions of the initial state wave function. Therefore corresponding differential cross sections for double ionization and single ionization are not simply connected by a single shake-off ratio (as is the case for photoabsorption assuming shake-off). This will be discussed later, considering singly differential cross section with respect to energy transfer in scattering and obtaining, for high incident energy, a ratio R_C which is a function of the energy transfer.

Here we mention two other atomic processes, related to the photoionization processes, which also probe correlation effects at high energies. These processes, which have also attracted considerable attention both experimentally and theoretically, are (1) the double ionization of a helium-like atom by charged particles and (2) double-electron capture in heavy (highly-charged) ion-atom collisions. For high incident ion velocities the double to single ionization ratio (of total cross sections) for charged particle ionization is related to the photoabsorption ratios R_P at small photon energies [5,6]. For high energy transfers $\epsilon$ the ratio of differential cross sections (with respect to energy transfer) for ionization by charged particles is related to double photoionization processes (photoabsorption and Compton scattering) for high energy transfers [7,8]:

$$R_{Z}(\epsilon)=f_{Z,d}(\epsilon)R_{P}+(1-f_{Z,d}(\epsilon))R_{C,nd}(\epsilon) .$$ (1)

Here, $R_{Z}(\epsilon)$ is the ratio of double to single ionization differential cross sections for ionization by a charged particle, R_C,nd is the ratio of non-dipole cross sections for Compton scattering and $f_{Z,d}(\epsilon)$ is the dipole fraction of the differential cross section for single ionization by a charged particle.

Another interesting related process is double-electron capture in heavy (highly-charged) ion-atom collisions, in which a single high energy photon is emitted. In the target frame this primarily involves two quasi free electrons of similar momenta, which at relativistic velocities can be stopped with the emission of one photon. The correlation effects when electrons are close to each other may be important (these effects cause a non-shake contribution to the double to single photoabsorption ratio). For low Z ions the theoretical predictions for this process are reported to be in agreement with experimental data [9]. However, for highly charged ions a surprisingly enhanced cross section for double electron capture is found [10].