X-ray Transition Energies Database Main Page

Theoretical Transition Energies

In this section, we present a schematic description of the procedures that are used to generate the theory portion of this database [88].

Uncorrelated Dirac-Fock Energies

The first step in the calculation minimizes the energy (with relativistic corrections) in the independent electron approximation, for each hole state. This provides a suitable starting point for adding many-body and QED contributions. We follow the Dirac-Fock method, which allows treatment of arbitrary atoms with arbitrary structure [89,90]. We have included full exchange and relaxation (to account for inactive orbital rearrangement due to the hole presence). The electron-electron interaction used in this program contains all magnetic and retardation effects, which is very important for large Z. The magnetic interaction is treated on an equal footing with the Coulomb interaction, to account for higher order effects in the wavefunction (that are also useful for evaluating radiative corrections to the electron-electron interaction). All of these calculations must be done with proper nuclear charge models, to account for finite-nuclear-size corrections to all contributions. Nuclear deformations must be included for heavy nuclei [91,92]. For all elements for which experiments have been performed, we used experimental nuclear charge radii. For others we used a formula from Johnson & Soff, corrected for nuclear deformations for Z > 90 [93]. Contributions of deformation to the rms radius (the only parameter of importance to the atomic calculation) are roughly constant (0.11 fm) for Z > 90. There is an unknown region, between Bi and Th (83 < Z < 90) where deformation effects start to be important, but for which they are not known. When experiments are available for a particular isotope, we calculated separately the energies for each isotope.

There are special difficulties involved when dealing with atoms with open outer shells (obviously this is the most common case). Computing all energies EJ for total angular momentum J would be both impossible and useless. The Dirac-Fock method circumvents this difficulty. One can evaluate directly an average energy that corresponds to the barycenter of all EJ with weight (2J + 1). There are still a few cases for which the average calculation cannot converge (when the open shells have identical symmetry). In these cases the outer electrons have been rearranged in an identical fashion for all hole states of the atom, to minimize possible shifts due to this procedure.

Correlation and Auger Shifts

Once the Dirac-Fock energy is obtained, many-body effects beyond Dirac-Fock relaxation must be taken into account. These include relaxation beyond the spherical average, correlation (due to both Coulomb and magnetic interaction), and corrections due to the autoionizing nature of hole states (Auger shift). Since the many body generalization of the Dirac-Fock method, the so-called Multiconfiguation Dirac-Fock (MCDF), is very inefficient for hole states, we turned to relativistic many-body perturbation theory (RMBPT) to evaluate these quantities. These many-body effects contribute very significantly to the final value [94-97]. As these calculations are very time consuming, they are performed only for selected Z and interpolated. Since the Auger shifts do not always have a smooth Z-dependence, care has been taken to evaluate them at as many different Zs as practical to ensure a good reproduction of irregularities.

QED Corrections

The QED corrections originate in the quantum nature of both the electromagnetic and electron fields. They can be divided into two categories, radiative and non-radiative. The first one includes one-electron and electron-electron radiative corrections. The one-electron radiative corrections (self-energy and vacuum polarization) scale as Z4/n3 (n being the principal quantum number) and are thus very important for inner shells and high Z. For the self-energy and Z > 10 one must use all-order calculations [98-104]. Vacuum polarization can be evaluated at the Uehling [105] and Wichmann and Kroll level [106]. The electron-electron radiative correction scales as Z3/n3. Ab initio calculations have been performed only for a few-electron ions [107,108]. Here we use the Welton approximation that has been shown to reproduce very closely ab initio results in all examples that have been calculated [91 and 109-112].

The second category (non-radiative) is composed of corrections to the electron-electron interaction that cannot be accounted for by RMBPT and MCDF. These corrections start at the two-photon interaction and include three-body effects. The two-photon, non-radiative QED contribution has been calculated recently only for the ground state of two-electron ions and cannot be evaluated in practice for atoms with more than two or three electrons [113,114].

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