Relativistic calculations of the isotope shifts in highly charged Li-like ions

Relativistic calculations of the isotope shifts of energy levels in highly charged Li-like ions are performed. The nuclear recoil (mass shift) contributions are calculated by merging the perturbative and large-scale configuration-interaction Dirac-Fock-Sturm (CI-DFS) methods. The nuclear size (field shift) contributions are evaluated by the CI-DFS method including the electron-correlation, Breit, and QED corrections. The nuclear deformation and nuclear polarization corrections to the isotope shifts in Li-like neodymium, thorium, and uranium are also considered. The results of the calculations are compared with the theoretical values obtained with other methods.


I. INTRODUCTION
In the last years a great progress was achieved in experimental studies of the isotope shifts in highly charged ions [1,2]. In Ref. [1] the isotope shift in B-like argon was measured employing laser spectroscopic methods at EBIT. This experiment provided first tests of the relativistic theory of the nuclear recoil effect with highly charged ions [3]. In Ref.
[2] the measurements of the isotope shifts in dielectronic recombination spectra for Li-like neodymium ions with A=142 and A=150 allowed determination of the nuclear charge radius difference. The accuracy of this experiment was also sensitive to the relativistic nuclear recoil contribution. Moreover, in Refs. [4][5][6] it was demonstrated that the DR experiments at GSI can be extended to radioactive isotopes with a lifetime longer than about 10 s. It is expected that with the new FAIR facilities [7] in Darmstadt the isotope shift measurements in heavy ions will be improved in accuracy by an order of magnitude. From the theoretical side, to meet this accuracy one needs to evaluate the nuclear size (field shift) and nuclear recoil (mass shift) contributions, including the relativistic and QED effects.
High-precision calculations of the mass shifts in highly charged Li-like ions were performed in Ref. [8], where the nuclear recoil contributions obtained within the Breit approximation (non-QED terms) were combined with the related terms obtained using the relativistic theory beyond the Breit approximation (QED terms). The QED contributions were evaluated to zeroth order of the 1/Z perturbation theory (Z is the nuclear charge number), while the Breit-approximation calculations were performed using the configuration-interaction Dirac-Fock-Sturm (CI-DFS) method [3]. An independent calculation of the non-QED mass shifts was presented in Ref. [9]. The results of this calculation, that was based on the multiconfiguration Dirac-Fock (MCDF) method, agree with those from Ref. [8] for low-and middle-Z ions. However, there is some discrepancy in the results for heavy Li-like ions. Therefore, it would be very important to calculate the relativistic nuclear recoil contributions using a different approach. To this end, in the present paper we develop a method which merges the perturbative and CI-DFS calculations. Namely, we calculate the nuclear recoil contributions within the Breit approximation to zeroth and first orders in 1/Z and add the related contributions of second and higher orders in 1/Z, obtained using the CI-DFS method. For checking purposes, we also perform the perturbative calculations starting with effective local potentials that partly include the electron-electron interaction effects. Although the calcu-lations of the 1/Z nuclear recoil contributions are restricted to the Breit approximation, the developed method can be straightforwardly extended beyond this approximation. The obtained non-QED results are combined with the corresponding QED contributions of the zeroth order in 1/Z to get the most accurate theoretical data for the mass shifts in highly charged Li-like ions. In addition, the field shifts are calculated in the framework of the Dirac-Coulomb-Breit Hamiltonian. These calculations, being performed by the CI-DFS method, are compared with the corresponding MCDF calculations of Ref. [9]. The QED corrections to the field shifts are also evaluated. In addition, we consider the nuclear deformation and nuclear polarization corrections to the isotope shifts for Li-like neodymium, thorium, and uranium. As the result, the most precise theoretical values of the isotope shifts for the 2p 1/2 − 2s and 2p 3/2 − 2s transitions in Li-like ions are presented.
The relativistic units ( = c = 1) are used throughout the paper.

II. RELATIVISTIC NUCLEAR RECOIL EFFECT
Full relativistic theory of the nuclear recoil effect can be formulated only within quantum electrodynamics [10][11][12][13][14][15]. However, the lowest-order relativistic nuclear recoil corrections can be calculated within the Breit approximation employing the operator [10,11,16]: where the indices i and k numerate the atomic electrons, p is the momentum operator, α incorporates the Dirac matrices, and D is given by: The nuclear recoil operator (1) can be written as a sum: where is normal mass shift (NMS) operator, is relativistic normal mass shift (RNMS) operator, is specific mass shift (SMS) operator, and is relativistic specific mass shift (RSMS) operator.
Analytical calculations of the expectation values of the operators H NMS and H RNMS with the Dirac-Coulomb wave functions were performed in Ref. [10]. In Ref. [17], the operator H M was used to evaluate the lowest-order relativistic nuclear recoil corrections to energy levels of He-and Li-like ions to zeroth order in 1/Z (that corresponds to independent electron approximation). Nowadays, this operator is widely used in relativistic calculations of the nuclear recoil effect using the configuration-interaction and multiconfiguration Dirac-Fock methods [3,8,9,[18][19][20][21]. It is known, however, that these methods can have a rather poor convergence in calculations of the specific mass shift. Moreover, the CI-DF and MCDF methods can not be adopted to account for the QED nuclear recoil contribution, which becomes very significant for heavy ions (see, e.g., Ref. [8]). In the present paper we develop the perturbative approach to calculations of the interelectronic-interaction corrections to the mass shifts. Although the perturbative calculations presented below are restricted to the Breit approximation, the developed approach has a potential to be extended to the full relativistic treatment.
To derive the nuclear recoil contributions to the binding energies of Li-like ions by perturbation theory, we use the two-time Green's function method [22] with the (1s) 2 shell regarded as belonging to a redefined vacuum. The energy shift of a level a (valence state) due to all perturbative interactions is given by where ∆g aa (E) = g aa (E) − g aa (E), g aa (E) is the Fourier transform of the two-time Green function, projected on the unperturbed state a, g a is the unperturbed energy of the a state, which in the case under consideration is simply equal to the Dirac energy of the valence electron: E (0) a = ε a . The contour Γ surrounds the level a and keeps outside all other singularities of ∆g aa (E). It is oriented anticlockwise. The Green function g aa (E) is constructed by perturbation theory according to Feynman's rules given in Refs. [14,22]. Since we restrict our calculation to the Breit approximation, we consider all the photon propagators in the Coulomb gauge at zero energy transfer (ω = 0) and restrict the summations over the intermediate electron states to the positive energy spectrum. In addition, we neglect the two-transverse photon nuclear recoil contributions [14].
To zeroth order in 1/Z, the nuclear recoil corrections are defined by diagrams presented in Fig. 1. In these diagrams, in accordance with Refs. [14,22], the dotted line ended by bold wiles at both sides denotes the "Coulomb recoil" interaction that leads to the NMS and SMS contributions. The dashed line ended by a bold wile at one side designates the "one-transverse-photon recoil" interaction that leads to the RNMS and RSMS contributions.
For the Coulomb recoil diagram (Fig. 1) one easily finds where η n = ε n − ε F and ε F is the Fermi energy, which is chosen to be higher than the oneelectron closed-shell energies and lower than the energies of the one-electron valence states.
Using the identity 1 and formula (8) to the first order, we get This expression is conveniently divided into one and two-electron parts: These formulas give the exact value of the Coulomb-recoil contribution within the full relativistic approach to zeroth order in 1/Z. To separate the Breit-approximation term, we represent the one-electron contribution as follows The first term in this equation gives the normal mass shift to zeroth order in 1/Z, while the second term determines the QED part of the one-electron Coulomb-recoil contribution.
The expression ∆E (two−el) Coul defines the specific mass shift of zeroth order in 1/Z. Therefore, we can write where the upper index (0) corresponds to the zeroth order in 1/Z. Performing similar calculations of the one-transverse-photon recoil contributions ( Fig. 1 ) and keeping only the terms which correspond to the Breit approximation, we get Let us consider the electron-electron interaction corrections to the nuclear recoil effect.
To first order in 1/Z, the interelectronic-interaction corrections to the NMS and SMS contributions are defined by Feynman's diagrams presented in Fig. 2. In these diagrams, the wavy line indicates the electron-electron interaction taken in the Breit approximation: In interelectronic-interaction corrections to the RNMS and RSMS contributions. The calculation of the diagram a and the related partners using formula (8) leads to the following expressions: where the scalar products of the vectors are implicit. For the other diagrams (b-h), we give the explicit expressions for the NMS and SMS contributions only:

∆E
(1,g) kinetic-balance (DKB) finite basis set method [23] with the basis functions constructed from B-splines [24]. The calculations have been carried out for extended nuclei. The Fermi model was used to describe the nuclear charge distribution and the nuclear charge radii were taken from Refs. [25,26].
To evaluate the nuclear recoil corrections of the second and higher orders in 1/Z, we used the CI-DFS method. With this method we calculated the total nuclear recoil contributions within the Breit approximation, including the Coulomb and Breit electron-electron interaction projected on the positive energy states. This was done by evaluating the expectation value of the nuclear recoil operator (1) with the CI-DFS wave function. To separate terms of different orders in 1/Z, the electron-electron interaction operator was taken in a form: where V is given by equation (21) and λ is a scaling parameter. For small λ, the nuclear recoil contribution can be expanded in powers of λ: where The second-and higher-order contribution where the terms E 0 and E 1 are determined numerically according to equation (42).
Finally, one should consider the nuclear recoil contributions beyond the Breit approximation (QED nuclear recoil terms). The QED calculations of the nuclear recoil effect for highly charged ions to zeroth order in 1/Z were performed in Refs. [13,15,27] for point charged nuclei and in Refs. [8,28,29] for extended nuclei. In the present paper, to get the QED nuclear recoil corrections, we interpolated the corresponding data from Ref. [8].

III. FINITE NUCLEAR SIZE EFFECT
The finite size of atomic nuclei leads to the field shifts of the energy levels. The nuclear charge distribution is usually approximated by the spherically-symmetric Fermi model: where the parameter a is generally fixed to be a = 2.3/(4ln3) fm and the parameters N and c are determined using the given value of the root-mean-square (rms) nuclear charge radius R = r 2 1/2 and the normalization condition: d rρ(r, R) = 1. The potential induced by the nuclear charge distribution ρ(r, R) is defined as where r > = max(r, r ). The isotope field shift within the Breit approximation can be obtained by solving the Dirac-Coulomb-Breit equation with the potential (45) for two different isotopes and taking the corresponding energy difference.
Since the finite nuclear size effect is mainly determined by the rms nuclear charge radius (see, e.g., Ref. [30]), the energy difference between two isotopes can be approximated as where F is the field shift factor and δ r 2 is the mean-square charge radius difference. In accordance with this definition, the F -factor can be calculated by or, using the Hellmann-Feynman theorem, by where ψ is the wave function of the state under consideration and the index i runs over all atomic electrons. If we neglect the variation of the electronic density inside the nucleus, we get (see, e.g., Refs. [9,21,31]): In what follows, the values of F calculated by formulas (47), (48), and (49) will be referred as obtained by methods 1, 2, and 3, respectively. In addition to the FS evaluated with the DCB Hamiltonian, one should account for the QED corrections to the field shift. Approximately, these corrections can be evaluated using analytical formulas from Ref. [32]. The results obtained by these formulas for s and p 1/2 states are in a fair agreement with the accurate numerical calculations performed for H-like ions in Ref. [33].

IV. RESULTS AND DISCUSSIONS
The nuclear recoil contributions are conveniently expressed in terms of the K-factor defined by It follows that the isotope mass shift is given by where δM = M 1 − M 2 is the nuclear mass difference. In Table I  [ 8,9]. The total MS values including the QED recoil contributions are also presented. From Tables II and III it can be seen that the present results obtained using the perturbative approach are in perfect agreement with the calculations based on the CI-DFS method [8].
As to comparison with the MCDF calculations of Ref. [9], there exists some discrepancy for heavy ions. We note that this discrepancy is larger than the contribution of the second and higher orders in 1/Z.
For checking purposes, we have also performed the perturbative calculations starting with an effective potential, which includes both the Coulomb nuclear potential and the screening potential that partly accounts for the electron-electron interaction. In Table V  To evaluate the FS constant with the DCB Hamiltonian we used the CI-DFS method.
In Table VI we compare the non-QED F -factor, obtained by equations (47), (48) and (49) (methods 1, 2, and 3, respectively), for Li-like titanium (Z = 22), neodymium (Z = 60), and thorium (Z = 90). The results of Ref. [9], where the method 3 was employed, are also presented. It can be seen that the last method leads to a rather poor accuracy for heavy ions. In case of Li-like thorium the discrepancy between the most precise result obtained by methods 1 and 2 and that obtained by method 3 amounts to about 10 %. The discrepancy is much larger than the uncertainty due to neglecting the non-linear corrections to formula (46). This is confirmed by the data presented in Table VII. In this table the non-QED FS contributions to the isotope shift obtained by the direct calculation: where R 1 and R 2 are the nuclear charge radii of the isotopes taken from Ref. [26], are compared with the corresponding results calculated using the F -factor.  [32,33]. This was done by multiplying the s-state QED correction factor ∆ s [32,33] with the nuclear size effect on the total three-electron binding energy. The uncertainty of this evaluation was determined by comparing the obtained results for the vacuum-polarization correction with the related direct calculation and assuming the relative uncertainty of the total QED correction to be by 50 % larger. These calculations demonstrate rather large values of the QED contributions to the field shift for heavy ions. In Tables IX and X we present our total values of the F S constant for the 2p 1/2 −2s and 2p 3/2 −2s transitions in Li-like ions in the range Z=4-92. The uncertainty was evaluated as a quadratic sum of the uncertainty due to a variation of the nuclear charge radius value taken from Ref. [26], the uncertainty due to the determination of the QED contributions discussed above, and the uncertainty due to a variation of the nuclear charge distribution which was estimated as the difference between the results obtained for the Fermi and homogeneously charged sphere models.
Table XI presents individual contributions to the isotope shifts of the 2p 1/2 −2s and 2p 3/2 − 2s transitions in Li-like A Nd 57+ ions with A=142 and A=150, which were measured in Ref. [2]. In addition to the mass and field shifts, one has to account for the nuclear polarization [45][46][47][48] and nuclear deformation [25] effects. To evaluate the nuclear polarization effect, we used the approach [45,46] in which the many-body theory for virtual nuclear excitations was incorporated with the bound-state QED for the atomic electrons. For low-lying rotational and vibrational levels the nuclear excitation energies and transition probabilities for 142 Nd and 150 Nd nuclei were taken from Refs. [49,50], respectively. The contributions from the nuclear giant resonances were evaluated utilizing phenomenological energy-weighted sum rules. To calculate the nuclear deformation correction, the standard spherically-symmetric Fermi model of the nuclear charge distribution (44) must be replaced by [25]: where ρ( r) is the deformed Fermi distribution: Y 20 (Θ) is the spherical function and β 20 is the quadrupole deformation parameter. In accordance with Ref. [51] we take β 20 =0 for A=142 and β 20 =0.28 (5) for A=150, that leads to a non-zero nuclear deformation effect for the 150 Nd isotope only. The nuclear deformation correction is given by the difference between the nuclear size contributions evaluated with non-zero and zero values of β 20 for the same nuclear parameters r 2 1/2 and the atomic mass numbers. As it can be seen from Table XI, the nuclear polarization and deformation contributions to the isotope shifts are comparable with the QED corrections. The perfect agreement of the theoretical value of the isotope shift for the 2p 1/2 − 2s transition with the experiment [2] should not be surprising since the mean-square charge radius difference δ r 2 =1.36(1)(3) fm 2 was determined from this comparison [2].
In Table XII we present the isotope shifts of the 2p j − 2s transitions in Li-like thorium with atomic numbers A=232 and A=230, and in Li-like uranium for two pairs of even-even isotopes, 238 U 89+ − 236 U 89+ and 238 U 89+ − 234 U 89+ . The values of δ r 2 1/2 are taken from Ref. [26]. The mass and field shifts, including the QED corrections, are calculated as in the neodymium case. The nuclear polarization effect for thorium and uranium was evaluated in Refs. [45,46]. The nuclear deformation effect was calculated as in Ref. [25], using the experimental [52,53] and theoretical [54] data for the nuclear deformation parameters. The total uncertainty is mainly determined by the uncertainties of the nuclear deformation and polarization effects.

V. CONCLUSION
We presented relativistic calculations of the isotope shifts in Li-like ions. The calculations of the mass shifts were performed by merging the perturbative approach with the CI-DFS method. These calculations confirm our previous results obtained by the CI-DFS method [8] and agree with the related MCDF calculations by Li et al. [9] for low-and middle-Z systems. The perturbative method developed in the paper has a potential to be applied for              (33) a The uncertainty of δ r 2 is not included.