Construction of the energy matrix for complex atoms

Eur. Phys. J. Plus (2016) 131: 429 DOI 10.1140/epjp/i2016-16429-3 THE EUROPEAN PHYSICAL JOURNAL PLUS Regular Article Construction of the energy matrix for complex atoms Part VI: Core polarization effects Magdalena Elantkowska1,a , Jaroslaw Ruczkowski2 , and Jerzy Dembczyński2 1 Institute of Materials Research and Quantum Engineering, Faculty of Technical Physics, Poznan University of Technology, Piotrowo 3, 60-965 Poznań, Poland 2 Institute of Control and Information Engineering, Faculty of Electrical Engineering, Poznan University of Technology, Piotrowo 3A, 60-965 Poznań, Poland Received: 3 October 2016 Published online: 15 December 2016 c The Author(s) 2016. This article is published with open access at Springerlink.com  Abstract. The continuation of series of papers concerning the construction of the energy matrix for complex atoms is presented. The second-order perturbation theory contributions originating from core polarization effects in the hyperfine structure are considered. Fifteen new formulae for angular coefficients of core polarization parameters are given. The complete set of corrections up to the second-order perturbation theory was taken into account and the accuracy of the wave functions in the intermediate coupling scheme, on the example of the lanthanum atom, was checked. 1 Introduction In the first part of our series of publications entitled Construction of the energy matrix for complex atoms, a method of semi-empirical analysis of complex atoms was introduced in general [1]. In the subsequent works of this series, an exhaustive description of electrostatic interaction up to second-order perturbation theory, electrostatically correlated spin-orbit interactions (CSO) and electrostatically correlated hyperfine structure interactions (CHFS) was presented [2– 5]. In each of these publications, the explicit form of analytical formulae, derived in our research group, was given. The aim of this paper is a description of the effects of configuration interaction on the atomic hyperfine structure, known as core polarization effects, in the case of nlN , nlN n1 l1N1 and nlN n1 l1N1 n2 l2N2 configurations. Important differences appear in our approach compared to previous works on the effects of configuration interactions by other authors [6–19] and can be summarized as follows: – we replace the description of the configuration interaction with effective operators through direct expressions for matrix elements; – we expand the considered configuration base from nlN +nlN −1 n1 l1 to nlN +nlN −N1 n1 l1N1 +nlN −N1 −N2 n1 l1N1 n2 l2N2 ; – we include in the consideration the interactions between the configurations under study. The next section of the current paper contains a short summary of the studies on the hyperfine structure of free atoms. Section 3 contains the description of a hyperfine structure many-body parametrization method. Section 4 contains the explanation of the symbols used in this work and fifteen explicit formulae for electrostatically correlated hyperfine interactions. An example of the application of new parameters for the multi-configurations system of the lanthanum atom is presented in sect. 5. 2 Effects of configuration interaction on atomic hyperfine structure The hyperfine structure of the atomic spectra is usually interpreted in the framework of the effective operator formalism proposed by Sandars and Beck [20]. This theory assumes three radial parameters for each open shell and for each kind of multipole interaction, which should be handled as free adjustable parameters to take into account relativistic and a e-mail: Page 2 of 15 Eur. Phys. J. Plus (2016) 131: 429 configuration interaction (CI) effects. The influence of CI on the hyperfine structure has been studied theoretically, especially by Judd [21, 22]. For the first time, Bauche and Judd [23] showed, in the hyperfine structure analysis of atomic plutonium, the need to consider the effects of perturbation hyperfine structure through the interaction with the configurations arising from excitation of one electron belonging to a closed shell n0 s0 to an empty shell n s . The authors introduced the name of the effect as “hfs core-polarization effect”. In the following years, the extensive research on the configuration interaction effects originated from closed shells to empty shells excitations, were conducted by Judd [21, 22], Sandars [24], Bauche-Arnould [25, 26], Armstrong [27], Lindgren and Rosen [28] and Büttgenbach [29]. A short summary of these works was presented in our previous works [1, 5, 30]. In 1985, Dembczyński [31] proposed a new method of hyperfine structure parametrization, which took into consideration simultaneously one- and two-body interactions in (3d+4s)N +2 configurations system. This approach was applied successfully to the interpretation of the spectra of iron-group elements [32–35] and, after the generalization, to the elements with three open electronic shells [36–38]. Detailed discussion on the interpretation of accurate measurements of hyperfine structure splittings in neutral and singly ionised complex atoms was presented in our papers [39, 40]. Another problem that should be considered in the interpretation of hyperfine structure is the inclusion of the offdiagonal excitation between configurations. For the first time, in the paper from 1977, Bauche and Bauche-Arnould [41] have shown that the far configuration mixing effect perturbs strongly the off-diagonal spin-dipole hfs matrix elements between 3dN +1 4s and 3dN 4s2 in the case of (3dN 4s2 )3 F Ti I and (3dN 4s2 )2 D Sc I. Empirically this effect has been found to be significant only by Himmel [42] in the case of OsI 5d6 6s2 . Usually the hyperfine interaction between configurations is neglected, because at first order, the only contribution to the magnetic hfs operator is due to the spin-dipole part. By Hartree-Fock calculations very small values are found for the corresponding radial integrals (∼ 4s|r−3 |3d). In 1981 Dembczyński et al. [43], using the atomic beam magnetic resonance detected by the laser-induced resonance fluorescence method (ABMR-LIRF), found experimental evidence of an extremely strong far configuration mixing effect on off-diagonal matrix elements between configurations, which can be explained only by taking into account the two-body core polarization effect, which screens the ordinary one-body core polarization parameter a10 3d . Moreover, they showed that the influence of the off-diagonal spin-dipole part a12 3d4s , which was discussed by Bauche-Arnoult, was insignificant. Later, Dembczyński presented the appropriate formulae for off-diagonal matrix elements in the case (3d + 4s)N +2 configuration system [31]. 3 Parametrization of the configuration interaction effects on the hyperfine structure In 2010 [30] we published a new approach to the hyperfine structure many-body parametrization. In the configuration system (5d + 6s)N of the lanthanum atom, we conducted an alternative analysis of the second-order contributions, ba (...truncated)