RRPA

To account for the interchannel coupling, or, which is the same, electron correlation in calculations of photoionization parameters, various many-body theories exist. In this paper, following Refs. [20,29,30,33], the focus is on results obtained in the framework of both the nonrelativistic random phase approximation with exchange (RPAE) [55] and its relativistic analogy the relativistic random phase approximation (RRPA) [56]. RPAE makes use of a nonrelativistic HF approximation as the zero-order approximation. RRPA is based upon the relativistic Dirac HF approximation as the zero-order basis, so that relativistic effects are included not as perturbations but explicitly. Both RPAE and RRPA implicitly sum up certain electron-electron perturbations, including the interelectron interaction between electrons from... 

The reader is referred to [55] for further details of RPAE and to [56] for details of RRPA. 

The RRPA calculated [29] valence ras photoionization cross sections for both the free and encaged atoms, at the frozen-cage approximation level, are displayed in Figures 22-24. To demonstrate the importance of electron... 

Figure 22 RRPA calculated data [29] for the Mg 3s photoionization cross section of free Mg and Mg C60 Mg, marked Mg, as a function of photoelectron energy. The data were obtained at two levels of the RRPA calculations, namely accounting for, (a) only two RRPA interacting channels ( 2ch ) and (b) nine RRPA channels ( 9ch ).

Table 1 RRPA calculated [29] photoelectron energy positions of the Cooper minima (au) for the free and encaged Ca, Sr and Ba...

Figure 27 Relativistic RPAE calculated results [30] of the 6s dipole photoelectron angular distribution parameter j06s(eo) from free Hg and <3>Hg, The RRPA calculations included interchannel coupling...

Figure 28 Relativistic RPAE calculated results [30] of the 6s dipole photoelectron angular distribution parameter of Hg at two different levels of truncation with regard to RRPA interchannel coupling (a) including channels from the 6s2 subshell alone, Aa, and (b) including channels from the 6s2 and 5d10 subshells of d>Hg, as in Figure 27. Confinement effects were accounted for in the A-potential model at the frozen-cage approximation level.

Figure 5.8 Angular distribution parameter / 2p for 2p photoionization in atomic magnesium as a function of photon energy. Experimental data points with error bars. Theoretical data (adapted to the experimental threshold for 2p ionization 2s - np resonances between 94 and 97 eV photon energy omitted) full curve, RRPA results [DMa83] within uncertainties of the drawing the saipe result is obtained in the MBPT approach [Alt89] broken curve, HF(1P) results [V6188] chain curve, Herman-Skillman results [DSa73]. From [KHL92],...

HS [DSa73] HFCP) [V6188] RRPA [DMa83] MBPT [Alt89] Exp. 

Next the results from the relativistic random-phase approximation (RRPA) and the many-body perturbation theory (MBPT), also shown in Table 5.1, will be discussed. Because both calculations include basically the same electron-electron interactions, rather good agreement exists, and it is sufficient to concentrate only on the RRPA model. 

The relativistic or non-relativistic random-phase approximation (RRPA or RPA)t is a generalized self-consistent field procedure which may be derived making the Dirac/Hartree-Fock equations time-dependent. Therefore, the approach is often called time-dependent Dirac/Hartree-Fock. The name random phase comes from the original application of this method to very large systems where it was argued that terms due to interactions between many alternative pairs of excited particles, so-called two-particle-two-hole interactions ((2p-2h) see below) tend to... 

Following this discussion of the RPA method the results for 2p photoionization in magnesium can be interpreted. From the foregoing discussion one expects that the RRPA method should give better results than the HF ( P) approach, because... 

In 2p photoionization in magnesium the partial cross section o2p is large compared to o3s and photoionization channel will be small. (In contrast, the effect of 2p photoionization on the 3s and 2s channels can be expected to be significant.) More important here is the continuum mixing of Fig. 5.10(a) with es and ed partial waves. Since Dd is larger than DS, it is essentially the Dd -amplitude which modifies the Ds -amplitude. This can be verified from Table 5.1 by comparing the RRPA results with those of the HFf P) calculation DS is increased while Dd remains nearly the same. 

After this necessary excursion into the broad field of electron-electron interactions as described in diagrammatic language and by certain approximations, the effects of electron correlations beyond the RRPA approach can be summarized by using the following rules (neglecting the photoelectron s self-energy diagram) ISCI and FISCI must be taken explicitly into account, and the theoretical ionization threshold has to be adapted to the experimental value. 

This is a very important result. It states that both dipole amplitudes from the RRPA calculation are modified by a common factor that reflects the influences of electron correlations in the initial and final ionic states which are beyond mean-field electron-electron interactions. The A0a0 2-value is called the spectroscopic factor (or the quasi-particle strength or the pole strength or the renormalization factor) and describes the weight given to the improved 2p photoionization cross section as compared to a calculation which does not include these specific electron correlations. The remaining intensity is transferred to satellite processes... 

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