Evaluation of the anomalous X-ray energy in VIP experiment: some values from Dirac-Fock method

The goal of the VIP experiment was to improve the limit on the Pauli Exclusion Principle through the search of 'anomalous' X-ray in copper atoms, produced by 'fresh' electrons (introduced through electrical current that crosses our sample and thus 'new' respect to the existing ones in the copper, that had already all the time to perform the allowed and 'prohibited' transitions) which have the probability to do the Pauli-forbidden transition [1] [2]. Here we calculated the energy shift of the anomalous X-ray for three possible decay channels: direct transition to 1s (K-edge) and 2p (3p) capture with a subsequent Kα(Kβ) emission [3]. Since we are considering a non-antisymmetrized electron then we can follow the case of a muonic atom. The reported values were obtained from Dirac-Fock calculations, as previously performed for a variety of muonic atoms by Mallow et al. [4]. Dirac-Fock approximation takes into account the relativistic corrections that are relevant for heavy ions, as we have considered our sample (ZCu = 29). Thanks to a numerical code it is possible to perform relativistic calculations for atoms, using the 'multiconfiguration' approximation for the N-electron wave function. This scheme proceeds through the optimization of the parameters during a self-consistent process [5]. Muon and electron can be treated in analogous way through the self-consistent field theory, to obtain wave functions and energies. The most commonly used approximation to treat the Hamiltonian of an N-electron system is the so called 'no pair' approximation that explicity excludes electron-positron pairs. The effects of the Breit operator, the Lamb shift and all sort of radiative corrections, as perturbations, are included.