Relativistic calculations of the nuclear recoil effect in highly charged Li-like ions

As is known, in nonrelativistic atomic theory the one-electron nuclear recoil effect can be easily evaluated using the reduced electron mass. Full relativistic theory of the recoil effect can be formulated only within quantum electrodynamics [1–4]. However, the lowest-order relativistic recoil corrections can be calculated within the Breit approximation by employing a relativistic nuclear recoil operator. The main goal of this paper consists of evaluating the one-electron nuclear recoil operator contributions for highly charged Li-like ions within the Breit approximation. To perform such calculations, we use perturbation theory in the parameter 1/Z, where Z is the nuclear charge number. Alternatively, the interelectronic-interaction corrections to the recoil effect can be evaluated by employing the configuration-interaction and multiconfiguration Dirac–Fock methods [6–10]. In these methods, the recoil operator is averaged with the many-electron wave function. In this paper, we consider calculations of the one-electron nuclear recoil operator contributions for highly charged Li-like ions by perturbation theory to zeroth and first orders in the parameter 1/Z and compare the results with those from [8, 10].

Within the Breit approximation, the nuclear recoil effect can be described by the operator [1, 2, 5]

$$\begin{equation} H = \frac{1}{2M}\sum_{i,k}\left[ {\boldsymbol{p}}_{\boldsymbol{i}} {\boldsymbol{p}}_{\boldsymbol{k}} - \frac{\alpha Z}{r_i}\left({\boldsymbol{\alpha}}_{\boldsymbol{i}}+\frac{({\boldsymbol{\alpha}}_{\boldsymbol{i}}{\boldsymbol{r}}_{\boldsymbol{i}}){\boldsymbol{r}}_{\boldsymbol{i}}}{r_i^2}\right){\boldsymbol{p}}_{\boldsymbol{k}}\right], \end{equation} \tag{ 1 }$$

where p is the momentum operator and α incorporates the Dirac matrices. To generate the perturbation series, it is convenient to use the quantum field formalism with the closed (1s)² shell regarded as belonging to a redefined vacuum [11]. With this technique, we have derived all the one- and two-electron nuclear recoil contributions to the binding energies of Li-like ions to zeroth and first orders in 1/Z. In this paper, however, the numerical calculations have been performed for the one-electron recoil operator contributions, i = k in equation (1), only. These contributions are given by the sum of the normal mass shift (NMS) and the relativistic NMS (RNMS):

$$\begin{equation} H_{\rm NMS} = \frac{1}{2M}\sum_{i}{\boldsymbol{p}}_{\boldsymbol{i}}^{2}, \end{equation} \tag{ 2 }$$

$$\begin{equation} H_{\rm RNMS} = -\frac{1}{2M}\sum_{i}\frac{\alpha Z}{r_i} \left({\boldsymbol{\alpha}}_{\boldsymbol{i}}+\frac{({\boldsymbol{\alpha}}_{\boldsymbol{i}}{\boldsymbol{r}}_{\boldsymbol{i}}){\boldsymbol{r}}_{\boldsymbol{i}}}{r_i^2}\right){\boldsymbol{p}}_{\boldsymbol{i}}. \end{equation} \tag{ 3 }$$

The calculations of these expressions to zeroth and first orders in 1/Z are performed using the dual kinetic balance method for the Dirac equation [12] with the basis functions constructed from B-splines [13].

The results of the calculations are conveniently expressed in terms of the factor K defined by

$$\begin{equation} \Delta E = \langle{\psi}| H_{M}| {\psi}\rangle \equiv K/M, \end{equation} \tag{ 4 }$$

where |ψ〉 is the eigenvector of the Dirac–Coulomb–Breit Hamiltonian. The values of K = K_NMS + K_RNMS (in GHz u) for the 2p_1/2–2s transition energy in Li-like ions are presented in table 1. In the point-nucleus case, the one-electron recoil operator part of the zeroth order in 1/Z is easily found by employing the analytical formula from [1]. According to this formula, the sum of the zeroth-order NMS and RNMS contributions is equal to zero for the 2p_1/2–2s transition. These results are presented in the second column. In the next column, the corresponding extended-nucleus results obtained for the Fermi model of the nuclear charge distribution are given.

The one-electron recoil contributions of first order in 1/Z can be obtained by calculations of the one-photon exchange diagrams, in which one of the external electron lines is modified by including the interaction with the one-electron recoil operators:

$$\begin{equation} |a \rangle \to |a \rangle + \sum_{n}^{{\varepsilon}_n \ne {\varepsilon}_a} \frac{|n \rangle \langle n| H_{\rm NMS}+H_{\rm RNMS} |a \rangle}{{\varepsilon}_a - {\varepsilon}_n}. \end{equation} \tag{ 5 }$$

The results of these calculations are listed in the fourth column of the table.

In the fifth column, our total results, given by the sum of the (1/Z)⁰ and (1/Z)¹ terms, are compared with the related theoretical data from [8, 10]. The latter were obtained using the configuration-interaction and multiconfiguration Dirac–Fock methods and, therefore, should additionally include the (1/Z)² and higher-order terms. In the case of Be⁺, the large deviation of our results from those of [8, 10] is due to the (1/Z)² and higher-order terms, which are rather significant. For middle- and high-Z ions, where the (1/Z)² terms are strongly suppressed, the results are in reasonable agreement with each other. However, because the total contribution changes the sign at Z ≈ 75, the higher-order terms become relatively significant. This could be a reason for the 10% difference between the perturbative and multiconfiguration Dirac–Fock results in the case of Hg⁷⁷⁺. Since for high-Z ions the electron–electron interaction and relativistic contributions to the recoil effect are smaller than the QED recoil contribution [3, 8], the perturbative methods, which can incorporate the QED contributions as well, seem rather promising.

For checking purposes, in the last column of the table we present the results obtained by using the reduced mass in the calculations of the binding energy up to the (1/Z)⁰ and (1/Z)¹ terms. This means that, to get the recoil effect, the sum of the one-electron Dirac binding energies and the one-photon exchange contributions is multiplied by the factor −m/M. As mentioned above, such an evaluation corresponds to the non-relativistic theory of the normal mass shift. It can be seen from the table (columns five and six) that, as expected, the corresponding results are in good agreement with each other in the low-Z (non-relativistic) limit. For high-Z ions, however, the results of the direct relativistic calculation deviate significantly from the results obtained by the use of the reduced mass.

To summarize, we presented the relativistic calculations of the one-electron recoil operator contributions for highly charged Li-like ions by perturbation theory to zeroth and first orders in the parameter 1/Z. For middle- and high-Z ions, the obtained results agree well with the related calculations performed using the configuration-interaction and multiconfiguration Dirac–Fock methods. To obtain the total value of the one-electron recoil operator contribution, the present results must be combined with the (1/Z)² and higher-order recoil contributions, which are important for low-Z ions, and with the QED recoil contributions, which become dominant for high-Z ions. Similar contributions should be evaluated with the two-electron recoil operator.

This work was supported by RFBR (grant no. 13-02-00630), DFG, GSI and the Ministry of Education and Science of the Russian Federation (project 8420). NAZ acknowledges financial support from the Helmholtz Association under grant agreement IK-RU-002, the Dynasty Foundation and G-RISC.