2. Study of correlation effects by double ionization in photoeffect

Until recently it was believed that the energy dependence of R_P for Helium was well understood over the whole nonrelativistic energy region. The theoretical predictions were consistent with measurements showing that R_P increases with energy, reaching a maximum (R_P=0.05) at photon energies of $\omega=150-250$ eV, and then decreases to the constant shake-off value R_P=0.017 for energies above roughly 3 keV. However, there are new developments both in the high energy region (our concern here) and at low energies.

New measurements [11], performed using cold target recoil spectroscopy, have found the low energy peak of R_P to be smaller than previously believed, by about 25 %. These new data are in agreement with with some newer calculations [12,13] and they are also consistent with calculations [6] relating the photoeffect ratio to that for fast charged particle impact.

The shake-off value of the ratio R_P for high energy photoeffect is given in nonrelativistic calculations [14,15,16,17] as

$$R_{P}^{s.o.}=1-\frac{\sum_{B}\left\vert\int\Phi_{B}^{\ast} ({... ...vert\Psi_{i}({\bf r}_{1},0)\right\vert^{2}d^{3} {\rm r}_{1}} .$$ (2)

Here $\Psi_{i}({\bf r_{1}},{\bf r_{2}})$ 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. The shake-off formula Eq. (2) has usually been derived in the nonrelativistic dipole approximation without retardation. It reflects the fact that, in the dipole approximation, the dominant mechanism of photoabsorption is the one in which the nucleus takes almost all additional momentum required by energy and momentum conservation and that, at high energy, almost all energy transfer is to one electron (the shake-off situation). [Note that one of the electrons in Eq. (2) is removed from near the nucleus, ${\bf r}_{2}=0$.] In the dipole approximation a two-electron system, in the absence of the nuclear field, cannot absorb a photon.

Most theoretical analysis of photoeffect uses the dipole approximation. However, the shake-off result Eq. (2) is not changed, presuming the dominance of the shake-off mechanism, if retardation effects (i.e. higher multipole effects) are included, given that for low Z two-electron systems retardation has little effect on the ground state wave function $\Psi_{i}$ and hydrogenic bound states $\Phi_{B}$. In explicit calculations, using wave functions consistent with shake-off assumptions, Kornberg and Miraglia [18] have shown that, even when retardation alters the cross sections for single and double photoionization by significant factors, the same factor enters both processes, and so the ratio R_P is unchanged.

We may likewise expect that relativistic effects do not alter the photoeffect ratio R_P, assuming the continued validity of the shake-off argument. The shake-off process, when applied to double photoionization, can be viewed as a two-step process. In the first step one electron is suddenly removed, corresponding to the single ionization process. Here we may allow very large, even relativistic, energies and therefore relativistic effects can be important. The second step is shake-off of the other electron, which is most often a shaken-off slow nonrelativistic electron, irregardless of the energies in the first step. Calculating the ratio between double and single photoionization cross sections, under the assumption of the validity of the sudden approximation, the contributions of the first step (the single ionization cross section), including relativistic as well as retardation effects, cancel out completely, leaving R_P as in Eq. (2) [19]. This observable, which we may call the shake-off ratio R_P^s.o., gives us information about the initial state electron-electron correlation when one of the electrons is near the nucleus. As shown by Suri, Pisk and Pratt [20], this correlation can be described by a photoeffect effective shake-off charge ( $Z_{eff} \sim Z-0.53$). This effective charge is determined from the monopole term of the electron-electron interaction in the ground state of the helium-like atom (the first term in the multipole expansion of the electron-electron interaction, i.e. $1/\vert {\bf r}_{1}-{\bf r}_{2}\vert$ term of the exact Hamiltonian of a helium-like system is replaced by 1/r_> where r_> is the larger of r₁ and r₂) [21].

At high energies, where the shake-off mechanism has been generally assumed to be valid, it now appears that another mechanism may contribute significantly to the double to single photoabsorption ratio. Recently, Drukarev [4] has called attention to this ``quasi-equal-energy sharing'' mechanism [22,23,24] for double photoeffect, which would first manifest itself as a linear rise of the ratio with energy, causing a roughly 10% correction to the shake-off limit for R<sub>P</sub> by about 12 keV. The prediction awaits experimental verification. In this, alternative, mechanism a large momentum is transfered to another electron (instead of to the nucleus as in the shake-off mechanism) and, therefore, both electrons leave the ion with similar large energies. (This would be forbidden in dipole approximation). This leads to a mechanism for double ionization which is not available for single ionization. Amusia et al[22] and Drukarev [4] performed an analysis, initially within nonrelativistic quantum mechanics, which did not assume dipole approximation, therefore allowing two electrons to absorb a photon even in the absence of the atomic nucleus. A pair of photoelectrons produced through this alternative mechanism share the photon energy nearly equally (and so we may call this mechanism an ``quasi-equal-energy sharing mechanism''). The residual ion is left with small momentum. This is different from the case when photoelectrons are produced through the shake-off mechanism, in which one electron takes almost all the photon energy and the residual ion takes substantial transfered momentum.

As noted by Suri, Pisk and Pratt [2], in the experimental situation in which photoeffect events are distinguished from Compton events by measuring the recoil of the residual ion [25,26], the events produced through the quasi-equal-energy sharing mechanism are not counted in photoeffect events, but with Compton events. These experiments count as photoeffect events only those produced in the shake-off regime with large momentum transfered to the ion (one electron takes almost all available energy while the other shakes off), and therefore such experiments would not observe the term in R_P caused by the quasi-equal-energy sharing mechanism, involving small momentum transfer to the nucleus. The largest photon energy used in experiments which distinguish photoeffect from Compton scattering was 7 keV, by Spielberger et al [26]. The linear term in the photoeffect ratio would contribute approximately 5% at these energies, according to Drukarev. However, as explained above, even increasing the accuracy of their experimental data (which is about 10%) they would not detect this rise in the ratio. The lower energy photoabsorption experiments, which are more precise, and which do not distinguish between photoeffect and Compton scattering (photoeffect however dominates the total cross sections) report the photoeffect double to single ratio to approach a constant above roughly 3 keV. At these energies the contribution of the quasi-equal-energy sharing mechanism (which would be included in some non-shake calculations such as MBPT) would be around 2.5 %, which is of the order of the Compton scattering contribution. At such a level of experimental accuracy both of these effects need to be included in comparison with theory.

Other, more accurate calculations should be able to reproduce this linear rise in the photoabsorption ratio R_P providing that the initial state used in the calculations describes correctly the correlation between the electrons when they are close to each other [27]. Many-body perturbation theory calculations should reproduce this behavior; the approach of Amusia et al is an example.

According to Amusia et al [22] and Drukarev [4], different regions of the spectrum of outgoing electrons (differential cross section for double ionization photoabsorption with respect to an electron energy) are dominated by different photoabsorption mechanisms. These authors distinguish three different regions in the spectrum. (1) In the region where one electron accepts almost all photon energy (the other electron has energy of the order of binding energy E_B) the dominant mechanism is shake-off as previously discussed. One may use criteria in terms of electron velocities (which is intuitively connected with the shake-off picture), requiring that one electron has a much smaller velocity then the other. This criteria and the criteria of Amusia et al [22] and Drukarev [4] are equivalent. (2) As the energy of the other electron increases (both electrons have energies much bigger then E_Bbut the energy sharing is still highly asymmetric, $\vert E_{1}-E_{2} \vert/\omega\sim 1$ [4]) the influence of the final state interaction becomes more important and eventually dominates. Here for example velocities are comparable. In the total cross section, the contribution of this region, for high energy photons, is negligible compared to the shake-off contribution. (3) As the energy sharing becomes nearly equal, the quasi-equal-energy sharing mechanism becomes important. For large incident photon energies, in the region of the spectrum defined by

$$\frac{\vert E_{1}-E_{2} \vert}{\omega}\leq (\frac{\omega}{m})^{\frac{1}{2}} ,$$ (3)

the quasi-equal-energy sharing mechanism highly dominates over the other two mechanisms. At high photon energies this mechanism, unlike the final state interaction mechanism, gives a significant contribution to the total double ionization cross section. According to Drukarev and Karpeshin [23], who calculated the effect of this mechanism on the photoeffect double to single ratio at relativistic energies, this mechanism dominates the double ionization at relativistic energies. At ultrarelativistic energies it saturates and the ratio approaches the constant R_P=0.59/Z² (while both single and double cross section decrease as $\omega^{-1}$), much different from the shake-off constant ( R_P^s.o.=0.093/Z² if using the same model).

This non-shake mechanism of high energy photoabsorption can be observed by detecting electrons which, for nonrelativistic energies, share energy nearly equally and have nearly opposite directions of their momenta. Following the analysis of Amusia et al and Drukarev, we may write the unpolarized differential cross section for double ionization through the quasi-equal-energy sharing mechanism in terms of relative electron momenta p [27], keeping only the dominant term in $\vert {\bf k}\vert=\omega$

$$\frac{d\sigma^{++}_{e.s.m}}{d\Omega} \sim \frac{(1-\cos^{2}\vartheta) \cos^{2}\vartheta}{p^{7}}\frac{\omega}{m} ,$$ (4)

where $\vartheta$ is the angle between electron relative momentum p and photon momentum k. This term is of electric quadrupole character, as one could expect, since the mechanism is forbidden in dipole approximation. [If the photon interaction with electron spin is included the magnetic dipole term will also contribute. This term has been calculated and discussed by Amusia et al [22]. For nearly exact energy sharing this term is dominant (for exactly equal energy sharing the quadrupole term Eq. (4) is exactly zero). However, the contribution of this term to the total double ionization cross section is small compared to the contribution of the electric quadrupole term [22].] Since (at nonrelativistic energies) electrons are ejected with nearly equal and opposite momenta (small momentum transfer to the nucleus) $\vartheta$ is nearly equal to the angle between an electron momentum and the direction of photon momentum. We may therefore compare Eq. (4) with the corresponding expression for double ionization through the shake-off mechanism, keeping also only the dominant term in $\vert {\bf k}\vert=\omega$, which in this case means neglecting retardation and keeping only the dipole term

$$\frac{d\sigma^{++}_{s.o.}}{d\Omega_{1}} \sim \frac{(1-\cos^{2}\vartheta_{1})} {p_{1}^{7}},$$ (5)

where label 1 denotes the fast electron. >From these expressions we see that in the quasi-equal-energy sharing region, unlike in the shake-off region, the electrons, which are opposite in momentum, are leaving the ion mostly in directions between the direction of photon polarization and the direction of the photon beam; the maximum in Eq. (4) is at $\vartheta_{1}=\pi/4$. The maximum in Eq. (5) appears at $\vartheta_{1}=\pi/2$. If retardation effects were included this maximum would shift toward forward angles. Consequences of higher order retardation corrections in Eq. (4) have not been examined.