The MPE features in the absorption spectra of Kr and Rb will be discussed following the order of the well-separated groups of coexcitations of consecutive subshells. The introductory analysis of the K edge reveals some details of configuration interaction and coupling useful in explanation of the weaker subshell MPE features. The region of the steeper slope is discussed separately. The model of the CR/PCI contribution extending over the entire MPE spectrum is described with a heuristic ansatz to provide a clearer view of the MPE.

Both deconvolution and comprehensive modeling of the features are used to sharpen the detail and to facilitate the identification of excitation channels. The model decomposition of features is shown in the simpler cases of deep MPE, while for the complex subvalence MPE (1s5s, 1s4p and 1s4s) the deconvoluted picture provides better insight.

In modeling, the MPE features are reconstructed by a superposition of three types of functional components: Lorentzian peaks

for resonant channels, cumulative Lorentzian distribution

for shake-up edges, and exponential saturation profiles

for shake-off channels.
The first two are characterized by three standard parameters, the amplitude A, the width , and the energy Eo. In exponential saturation, the energy parameter is the threshold of the shake-off channel, and the width is given by the range, i.e. the reciprocal of the saturation constant. For resonant lines, the intensity will be given as the product of the amplitude and the width so that a comparison between groups and elements is enabled.

For the shake-off channel another theoretical ansatz (Thomas profile) has been derived from a statistical description of the process [20]. The comparison with satellite emission data on Cu [19] shows that an exponential saturation profile can be adapted to a closer fit. Since this functional form also follows from the statistical description with a slightly different yet entirely physical assumption on the momentum density we adopt it in the present study. The range of the exponential is proportional with the excess energy, i.e. the binding energy of the shaken electron. The coefficient of proportionality in Cu data is 0.4, and the same value will be used here, in satisfactory agreement with the data.


The deconvolution is best exploited in the region of major absorption edges where the obscuring effect of strong and extensive tails of the edge and the accompanying resonance lines is removed and finer features become visible. In Kr and Rb, the deconvoluted K edge regions appear almost identical to those of the lighter homologues Ar and K (Figs 4 a,b) when intensity relative to the K edge jump is considered and the energy scale relative to the K edge is used. The span of edge features in Rb and Kr appears slightly contracted since the excited states are less tightly bound due to a weaker penetration of outer electrons to the core.

Fig. 4. Normalized Kr and Rb K-edge (solid) and some of the model components (dashes, dots) - see Table 1. The deconvoluted spectra (dash-dots) are compared, respectively, to argon and potassium K-edge (above). Dirac-Fock energies are indicated. Energy scale here and subsequently as explained in Table1.

The simple edge shape of Kr can be modeled with just three elements (Table 1 a), the resonant peak of the [1s]5p transition, its Rydberg follower [1s]6p, and the K edge [1s] itself. The position of the model edge is shifted from the DF estimate by 0.7 eV, owing to the accumulation of unresolved lines of the Rydberg series [42].

In Rb, considerable additional detail is introduced by the presence of the valence 5s electron (Table 1b). As in the case of potassium, the DF calculation shows that in the resonant [1s]5p state the coupling is pure (5s5p)1s, leading to a triplet-singlet splitting of 1.9 eV. Configuration interaction (CI) suppresses the splitting; the definite value is not known for convergence problems in the DF code. The extrapolation from DF models with artificially increased nuclear charge points to a value 1.4 eV, obtained at Z eff. = 37.15. The least-square value from experimental data is 1 0.3 eV, with the intensity ratio of the two components of 4,8 : 1.

The coexcitations of the 5s valence electron follow immediately above the [1s] edge. The resonant peak due to the [1s5s]5p6s transition is plainly visible in the measured spectrum. The deconvolution reveals also the Rydberg resonance [1s5s]6p6s and the shake-up edges [1s5s]5p and [1s5s]6s. The latter, stemming from the ordinary shake promotion of the valence electron in the 1s photoeffect, appears stronger than the reverse possibility of the valence monopole ejection accompanying a dipole 1s 5p excitation.


Table 1. Best-fit model parameters of the normalized Kr and Rb K-edge and calculated Dirac-Fock energies of the corresponding electron transitions relative to the threshold ( = 14326.4 eV, 15205.6 eV respectively). The respective experimental values 14327.0 eV and 15206.9 eV are established from the positions of the pre-edge resonant peaks. The apparent edge in the spectrum is shifted downwards by the accumulation of unresolved lines of the Rydberg series. A common value for the linewidths of the model elements is 3.1 eV.
*at increased nuclear charge Z = 37.15 (see text).


Core relaxation and PCI

If quantitative agreement with experimental data is required even the theoretical reconstruction of the dominant single-electron photoabsorption channel without MPE contributions represents a difficult task, as shown in the analyses of the Kr K edge [5,8]. The asymptotic decrease of the photoelectric cross section far above the edge can be described sufficiently well by a Victoreen power formula , n around 3, but for the region of steeper slope immediately above the edge a quantitative model has not yet been given. In Fig. 1, the region seems to reach as far as the 1s3d MPE group. According to Tulkki [40], the cross section calculated with inclusion of core relaxation and PCI effects exceeds the unrelaxed cross section immediately above the edge but approaches it asymptotically. The contribution of the relaxation and PCI may thus be regarded as superposed onto the single electron cross section. Its overall effect in the Kr absorption spectrum has already been discussed by Deutsch et al. [5,11].

The presence of the CR/PCI contribution affects the definition of the edge amplitude. To maintain the consistency with tabulated absorption data we define the amplitude of the edge as the Victoreen extrapolation from values far above the edge. It is thus the photoabsorption cross section with saturated shake channels, extrapolated to the edge energy. The extrapolation, however, excludes the CR/PCI contribution since it dies out before the asymptotic region.

Fig. 5. a:- Kr absorption cross section normalized to the asymptotic Victoreen formula; an arbitrarily shifted exponential ansatz for PCI contribution; b: - reduced MPE spectrum of Kr and Rb after removal of the best-fit exponential. The inset shows an expanded view of the complex region just above the edge with 1s5s, 1s4p and 1s4s excitations. Labels A - C are discussed in text.

In analysis of MPE groups it proves useful to eliminate from the measured absorption spectrum the Victoreen trend since its slope affects the shapes of weaker MPE details. Following the above discussion, we divide the experimental cross section by the Victoreen cross section, determined in a least-square fit in the high-energy region above perceptible MPE features. The renormalized spectrum is seen as a slow decrease, interspersed with sharp MPE features, from the K edge to the asymptotic unit value (Fig 5a).

The smooth segments of the spectrum between MPE groups look as parts of a single continuum with steadily decreasing slope. In an attempt to describe the continuum with a simple function of energy, an exponential is found to fit surprisingly well. Indeed, the exponential determined to fit the most conspicuous smooth spectral region between 1s4s and 1s3d MPE groups remains roughly parallel also with the flat segments on either side. Its removal results in a reduced cross section (Fig. 5b) that increases stepwise to the asymptotic value of 1. The monotonous increase, if the narrow resonant contributions are momentarily disregarded, follows from the succession of the shake-up and shake-off channels. Several ambiguous MPE features as e.g. sharp spikes of the 1s4p group can now be clearly identified: after removing of additional slope the resonant and shake-up channels are easily discerned. Likewise, the saturation profile of shake-off channels is made evident.

The same procedure is successful for Rb. The reduced cross sections of Fig 5b will be used in the analysis of MPE in Kr and Rb, and the channel strengths will refer to the particular renormalization discussed above. The exponential ansatz, however heuristic and approximate, applies also in other similar cases, as in the high-resolution absorption spectra of hydrides of the 4p elements [43]. Even the 20 eV long stretch of smooth cross section between the edge and the valence MPE group (1s3p) in the spectra of Ar and K [27] fits an exponential well. The effective range of the exponential shows a distinct increase with atomic number Z (Table 2).

Table 2. Range of the exponential ansatz for some elements. The numbers in brackets indicate the uncertainty of the last digit.



Last change: 28-Jun-2006