**3. Illustrative Examples**

As a first example, consider the dipole photoelectron angular distribution resulting from the photoionization of the 5s subshell of the closed-shell Xe atom. The general form of the dipole angular distribution for linearly polarized incident radiation for subshell *i* is given as the differential photoionization cross section by [25]

$$\frac{d\sigma\_i}{d\Omega} = \frac{\sigma\_i}{4\pi} [1 + \beta\_i P\_2(\cos\theta)],\tag{84}$$

where *σi* is the total subshell cross section, *<sup>P</sup>*2(*x*)=(<sup>3</sup>*x*<sup>2</sup> − 1)/2, *θ* is the angle between photon polarization and photoelectron momentum directions, and *βi* is the dipole angular distribution asymmetry parameter. Nonrelativistically, for a closed shell atom, *βi* = 2 and is independent of energy for an initial ns subshell [24]. From a physical point of view, this occurs because, nonrelativistically, there is only a single final state partial wave, characterized by ns → *ε*p, so that there is nothing to interfere with, and the angular distribution is just determined by the symmetry of the *ε*p-wave. When relativistic interactions are included, the situation is changed in that there are then two possible photoionizing transitions ns → *ε*p1/2 and ns → *ε*p3/2, and these can interfere with one another, giving rise to both a deviation of *βi* from the value two, and an energy dependence. This behavior will be most evident near Cooper minima [26] owing to the fact that the two relativistic channels exhibit the minima at slightly different energies [27], so that in this region the matrix elements for the two relativistic channels can be vastly different, both in magnitude and phase.

The Xe 5s photoionization cross section and *β* parameter calculated using RRPA, including correlation in the form of interchannel coupling with 5p and 4d relativistic photoionization channels, are depicted in Figure 4. The cross section exhibits a deep minimum, indicative of the Cooper minima in the 5s → *ε*p1/2 and 5s→ *ε*p3/2 channels. Correspondingly, the *β* parameter shows a deep minimum around that energy, and over a significant energy range, *β* is energy-dependent. It is to be emphasized, that, while the nonrelativistic RPAE does pretty well on the cross section [1], the *β* parameter predicted by the nonrelativistic calculation is constant and equal to two at all energies. Thus, it is evident that the addition of relativistic effects to the original nonrelativistic RPAE brings out additional physical effects. It should also be noted that, although the RRPA result for the cross section is rather good in the Cooper minimum region, the predictions for *β* are not, owing to the mission of satellite photoionization channels in the RRPA [28].

**Figure 4.** Xe 5s photoionization cross section (upper curve) and *β* parameter vs. photon energy, *ω*, calculated using RRPA [29].

The next example is the phenomenon of spin-orbit-activated interaction interchannel coupling (SOIAIC). Basically, this results from the interchannel coupling among the photoionization channels emanating from the two members of a spin-orbit doublet, nl, with *j* = *l* ± 12 . The phenomenon was discovered experimentally in the photoionization of the 3d subshell of Xe [30] and subsequently explained theoretically [31]. The explanation given in [31] was based upon the nonrelativistic RPAE that was artificially made to include the spin-orbit splitting of the Xe 3d spin-orbit split levels. Subsequently, RRPA was applied and verified the explanation [32]. The results, both experimental and theoretical, are shown in Figure 5.

**Figure 5.** Photoionization cross sections for the 3d subshell of xenon. Solid lines represent calculated 13-channel RRPA (with relaxation) cross sections for the respective 3d5/2 and 3d3/2 subshells [32]. The calculated cross sections in length and velocity gauge differ by at most 3%, so that the corresponding curves are almost indistinguishable in the scale of the figure. Dashed lines represent the results of SPRPAE results of [31]. Dotted-dashed lines represent results of the ASFA calculations [30]. Closed and open circles denote the respective experimental results for 3d5/2and 3d3/2subshells [30].

The key point here is that both the 3d5/2 and 3d3/2 cross sections exhibit shape resonances [33] just above their respective thresholds. As a result, the 3d3/2 cross section is significantly larger than the 3d5/2 cross section just above the 3d3/2 threshold. As a general rule, when a large photoionization cross section is degenerate with a small one, the small cross section is altered owing to interchannel coupling (configuration interaction in the continuum) [34]. This is exactly what is seen here in the form of the structure in the 3d5/2 cross section just above the 3d3/2 threshold. It is evident that the relativistic spinorbit splitting of the 3d thresholds is crucial to the existence of this SOIAIC phenomenon, and it is also clear that SOIAIC is likely to be in evidence for the photoionization of inner subshells that have a significant spin-orbit splitting and exhibit a near-threshold shape resonance. Furthermore, as a result of the fact that the photoionization cross sections of inner shells of confined atoms often have significant near-threshold maxima [35], owing to the phenomenon of confinement resonances [36], the SOIAIC should be much more generally exhibited [37].

The asymptotic branching ratios of spin-orbit doublets have been a topic of interest since the 1960s. Earlier, it was expected that far above thresholds, the ratio of cross sections for the members of a spin-orbit doublet, <sup>n</sup>*l*, with *j* = *l* ± 1 2 , respectively, should approach (*l* + <sup>1</sup>)/*l*, known as the statistical value, at asymptotically high energies [38]. However, it was later shown that relativistic effects on the initial state wave functions would cause the ratio to drop below the statistical [39,40]. Recently, experimental technology has improved to the point that this prediction has been verified experimentally [41]. In addition, it had been found that, in the vicinity of inner shell thresholds, there can be very large swings of the branching ratio over relatively small energy ranges [41,42]. However, all of this phenomenology cannot occur within a nonrelativistic framework; relativistic interactions are required.

As a particular example, the calculated branching ratio for Xe 5p using the RRPA with coupling among relativistic channels from all subshells except 1s [42] is depicted in Figure 6 over a very large energy range, from threshold to 500 a.u. (approximately 13.6 keV), where it is seen that, at the highest energies, that branching ratio is about 1.6, well below the statistical value for an initial np doublet, in keeping with the earlier predictions. In addition, there are seen to be significant excursions from monotone decreasing results in the vicinity of the n = 4, n = 3 and n = 2 thresholds. These excursions are the result of interchannel coupling between the relativistic photoionization channels resulting from the 5p*j* initial states and the channels from the inner shells. The fact that there are significant excursions of the branching ratios in these energy ranges means that the actual interchannel coupling matrix elements are themselves dependent upon relativistic interactions.

Figure 6 also shows the results of RRPA calculation including only coupling among the five 5p relativistic photoionization channels. It is evident that the truncated RRPA results agree quite well with the fully coupled branching ratios away from the inner shell thresholds but do not reproduce the significant excursions from the smooth curve in the vicinity of the inner shell thresholds, thereby showing conclusively that these excursions are the result of interchannel coupling with the inner shell photoionization channels.

The Cooper minimum [26] mentioned above in connection with Xe 5s photoionization, is a ubiquitous phenomenon that pervades the photoionization of outer and/or near-outer subshells of all of the elements of the periodic table [43,44]. Among the interesting facets of the influence of relativistic interactions which cause a single nonrelativistic Cooper minimum to be split into several relativistic Cooper minima dependent upon the total angular momentum, *j*, is of the initial and final states of the relativistic photoionizing transition [45,46]. In addition, the locations of Cooper minima depend very sensitively on many-body correlations, in addition to relativistic interactions; as a matter of fact, the Xe 5s Cooper minimum is below threshold in the discrete region at the level of single particle calculations [47] but appear in the continuum in calculations that include significant many-body effects [5]. Thus, RRPA is an ideal formalism to study these relativistic effects in Cooper minima. For completeness, it should be pointed out here that in the neighborhood of the Xe 5s Cooper minimum, quadrupole effects become important in the angular

distribution, although the total subshell cross section is virtually unaffected. The interference between dipole and quadrupole photoionizing transitions leads to extra terms in the expression for the differential cross section, but *β* remains unaffected to the first order [29]. In addition, calculations of the effects of quadrupole transitions on the differential cross section (photoelectron angular distribution) using the RRPA methodology has been shown to be in good agreemen<sup>t</sup> with the experiment [48].

**Figure 6.** Photoionization cross section branching ratios for Xe 5p3/2/5p1/2 calculated using RRPA with full coupling (red dots) and with only coupling among the 5p photoionization channels (blue squares) [41,42]. The vertical dashed lines indicate the thresholds.

Recent work on the splitting of Cooper minima (CM) for heavy and superheavy atoms has illustrated the importance of including correlation in the calculations [49]. As an example, given in Table 1 are the positions of the various 6s CM for six elements obtained at three levels of calculation: independent particle Dirac–Fock (DF), two-channel RRPA coupling on the two relativistic channels arising from 6s photoionization, and RRPA with full coupling of all the channels that might affect the result. To begin with, there are huge differences between the positions of the 6s → *ε*p3/2 CM and the 6s → *ε*p1/2 CM, and the differences increase with Z to an astounding degree in Og—a splitting of more than 4.5 keV. This comes about owing to the spin-orbit force which is attractive for the 6s → *ε*p1/2 final state but repulsive for the 6s → *ε*p3/2 final state. In addition, it is clear from Table 1 that correlation in the form of interchannel coupling induces rather significant changes in the location of 6s → *ε*p3/2 CM compared to the two-channel and DF results, changes that generally increase with Z, indicating the crucial importance of the interchannel coupling in the determination of the position of the CM in these heavy and superheavy elements.


**Table 1.** Positions of the Cooper minima (CM) in Dirac–Fock (DF), two-channel RRPA and RRPA with full coupling in photoelectron energy in (a.u.) [5].
