We calculate 2s and 2p photoionization cross sections both in the one-configuration approximation and using the R-matrix technique based on the close coupling theory (Berrington et al. 1995).
In the one-configuration scheme, we used the Pauli-Fock core and continuous spectrum wavefunctions obtained by averaging the electrostatic interaction over the configuration (Kau et al. 1997). The calculations are performed in the relaxed-core approximation accounting for the rearrangement of atomic residue upon creation of an inner-shell vacancy (Amusia & Cherepkov 1975; Sukhorukov et al. 1979)
In the R-matrix approach, which concerns only single, SU and cSU cross
sections, configuration interaction (CI) is present both in the description
of the initial and residual term wavefunctions. Free channel coupling is
fully taken into account as well as exchange potentials (a priori
not negligible for low free l quantum number). Dipole channel coupling is
responsible for the cSU effects. The closed channel system Al+ + e- 2P
defines the Al ground state wavefunction, and 3
even allowed symmetries S, P, D, for the free final wave functions are
necessary for obtaining the residual term cross sections.
Although in such a model the same set of basic orbitals is used for both ground and 2s/2p vacancy states, the inclusion of CI between, for example, 2s2p63s23p and 4p improves the 2s vacancy state description. In a more thorough but tedious calculation, additional pseudo-orbitals could be introduced to obtain improved CI wavefunctions for the residual ion.
The cross sections are calculated with length formulae.
In the case of 2s-photoionization, to describe the free systems Al+ + e-, the products of soft X-ray photoionization leading to single, SU and cSU processes, we chose two sets of Al+ residual configurations. Ground configurations: 2s22p63s2, 3s3p, 3p2, 3s4s, 3s4p, 3s3d, and 2s-hole configurations: 2s2p63s23p, 4p, 2s2p63s3p4s, 2s2p63p3, 2s2p63s3p2 and 2s2p63s23d.
Two bound channels are present in the close-coupling expansion: 2s22p63s23p and 2s22p63p3. The first set of configurations is present partly to describe the neutral target 2s22p63s23p (closed channel system). The orbitals 1s, 2s, 2p, 3s, 3p, 3d, 4s and 4p are defined by their respective scaling parameters (Superstructure code of Eissner and Nussbaumer 1969): 1.4868, 1.1315, 1.0803, 1.1135, 1.1340, 1.1900, 1.1639 and 1.1150, respectively.
In the case of 2p-photoionization the set of configurations for the description of the ground state is the same as above, while the configurations for the states with 2p-vacancy were the following: 2s22p53s23p, 4p, 3p3, 3s3p4s, 3s3p2 and 3s23d. The scaling parameters for the orbitals 1s, 2s, 2p, 3s, 3p, 3d, 4s and 4p were 1.4728, 1.0617, 1.0440, 1.0064, 1.0051, 1.0493, 1.0171, 0.9985.
We calculate SU and SO processes in the frame of sudden perturbation theory (SPT) (Sachenko & Demekhin 1965). This approximation is valid at photon energies above the respective double excitation/ionization thresholds where SU and SO probabilities reach constant values. This saturation occurs normally at photon energies exceeding SU/SO threshold by about the binding energy of an electron being additionally excited (Kochur et al. 1990)
The probabilities of double monopole shake processes were calculated with the formula obtained by simple statistical modification of the traditional SPT formula as described by Kochur et al. (2002).
In order to carry out L vacancy Auger rates using the R-matrix code, we
consider the reverse scattering problem Al++ + e-
Al+for large incident energies.
To obtain the probabilities of electron capture leading to formation of highly excited inner-shell vacancy states Al+, we include in the close coupling expansion the L-vacancy bound channel configurations and eventually add several chosen closed channels.
Resonances will occur in the scattering matrix at energies in the vicinity of 2s- and 2p-vacancy states. By analyzing the resonant behaviour of the scattering matrix diagonal elements, we obtain positions and partial decay rates of those states (De Araújo & Petrini 1988). In this way the free final channel coupling is fully taken into account. This coupling is important for 2s- and 2p-vacancy states formed by free electrons with low (a few Rydbergs) energies: for the 2p-vacancy Auger-decaying into 2p63s, 2p63p, and for the 2s-vacancy decaying into 2p53s3p, 2p53s2 via Coster-Kronig transitions.
The a priori weak channel rates will be strongly affected by dipolar coupling.
For 2p vacancy decays of 2p53s23p states, we also evaluate the shake
up process associated with the Auger decay itself (Auger shake processes),
for example, the probabilities for the ejected Auger electron to excite the
residual ion: 2p53s23p
2p63p + e-(E)
2p63d + e-(E')
Although the processes of this kind are relatively weak, they give rise to additional line emission.
To describe the cascading decay of the states produced by photoionization one needs to know, at each decay step, the branching ratios, i.e. the probabilities for a given vacancy state to decay into each of possible final states. The decay branching ratios for various states produced during cascading decay following first-step decays were calculated in configuration-average approximation using the average partial transition widths. The latter quantities were calculated via the methods described by Kochur et al. (1994). Very briefly, the expressions for the transitions partial widths are factorized so that one term depends only on the symmetry of the atomic orbitals involved in the transition and on electron subshells occupation numbers; another term is expressed with radiation or radiationless transition integrals calculated with the wavefunctions optimized for an appropriate electron configuration.
Copyright ESO 2002