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Relativistic recoil corrections to the atomic energy levels

The quantum electrodynamic theory of the nuclear recoil effect in atoms to all orders in \alpha Z and to first order in m/M is considered. The complete \alpha Z - dependence formulas for the relativistic recoil corrections to the atomic energy levels are derived in a simple way. The results of numerical calculations of the recoil effect to all orders in \alpha Z are presented for hydrogenlike and lithiumlike atoms. These results are compared with analytical results obtained to lowest orders in \alpha Z. It is shown that even for hydrogen the numerical calculations to all orders in \alpha Z provide most precise theoretical predictions for the relativistic recoil correction of first order in m/M.


I. INTRODUCTION
In the non-relativistic quantum mechanics the nuclear recoil effect for a hydrogenlike atom is easily taken into account by using the reduced mass µ = mM/(m + M ) instead of the electron mass m (M is the nuclear mass). It means that to account for the nuclear recoil effect to first order in m/M we must simply replace the binding energy E by E(1 − m/M ).
Let us consider now a relativistic hydrogenlike atom. In the infinite nucleus mass approximation a hydrogenlike atom is described by the Dirac equation (h = c = 1) where V C is the Coulomb potential of the nucleus. For the point-nucleus case, analytical solution of this equation yields the well known formula for the energy of a bound state: where n is the principal quantum number and j is the total angular momentum of the electron. The main problem we will discuss in this paper is the following: what is the recoil correction to this formula? It is known that to the lowest order in αZ the relativistic recoil correction to the energy levels can be derived from the Breit equation. Such a derivation was made by Breit and Brown in 1948 [1] (see also [2]). They found that the relativistic recoil correction to the lowest order in αZ consists of two terms. The first term reduces the fine structure splitting by the factor (1 − m/M ). The second term does not affect the fine structure splitting and is equal to −(αZ) 4 m 2 /(8M n 4 ). Calculations of the recoil effect to higher orders in αZ demand using QED beyond the Breit approximation. In quantum electrodynamics a two-body system is generally treated by the Bethe-Salpeter method [3] or by one of versions of the quasipotential method proposed first by Logunov and Tavkhelidze [4]. In Ref. [5] (see also [6]), using the Bethe-Salpeter equation, Salpeter calculated the recoil correction of order (αZ) 5 m 2 /M to the energy levels of a hydrogenlike atom. This correction gives a contribution of 359 kHz to the 2s -2p 1/2 splitting in hydrogen. The current uncertainties of the Lamb and isotopic shift measurements are much smaller than this value (see, e.g., [7]) and, therefore, calculations of the recoil corrections of higher orders in αZ are required. In addition, for the last decade a great progress was made in high precision measurements of the Lamb shifts in high-Z few-electron ions [8][9][10]. In these systems, the parameter αZ is not small and, therefore, calculations of the relativistic recoil corrections to all orders in αZ are needed.

II. RELATIVISTIC FORMULA FOR THE RECOIL CORRECTION
First attempts to derive formulas for the relativistic recoil corrections to all orders in αZ were undertaken in [11,12]. As a result of these attempts, only a part of the desired expressions was found in [12] (see Ref. [13] for details). The complete αZ-dependence formula for the relativistic recoil effect in the case of a hydrogenlike atom was derived in [14].
The derivation of [14] was based on using a quasipotential equation in which the heavy particle is put on the mass shell [15,16]. According to [14], the relativistic recoil correction to the energy of a state a is the sum of a lower-order term ∆E L and a higher-order term ∆E H : Here |a is the unperturbed state of the Dirac electron in the Coulomb field V C (r) = −αZ/r, p = −i∇ is the momentum operator, G(ω) = [ω − H(1 − i0)] −1 is the relativistic Coulomb-Green function, H = α · p + βm + V C , α l (l = 1, 2, 3) are the Dirac matrices, ε a is the unperturbed Dirac-Coulomb energy,

4)
D lm (ω) is the transverse part of the photon propagator in the Coulomb gauge. In the coordinate representation it is The scalar product is implicit in the equation (2.3). In Refs. [17,18], the formulas (2.1)-(2.3) were rederived by other methods and in [17] it was noticed that ∆E can be written in the following compact form: (2.6) However, the representation (2.1)-(2.3) is more convenient for practical calculations. The term ∆E L can easily be calculated by using the virial relations for the Dirac equation [19,20]. Such a calculation gives [14] This simple formula contains all the recoil corrections within the (αZ) 4 m 2 /M approximation. The term ∆E H taken to the lowest order in αZ gives the Salpeter correction [5]. Evaluation of this term to all orders in αZ will be discussed below.
The complete αZ-dependence formulas for the nuclear recoil corrections in high Z few-electron atoms were derived in Ref. [21]. As it follows from these formulas, within the (αZ) 4 m 2 /M approximation the nuclear recoil corrections can be obtained by averaging the operator with the Dirac wave functions. This operator can also be used for relativistic calculations of the nuclear recoil effect in neutral atoms. An independent derivation of this operator was done in [22]. The operator (2.8) was employed in [23] to calculate the (αZ) 4 m 2 /M corrections to the energy levels of two-and three-electron multicharged ions.

III. SIMPLE APPROACH TO THE RECOIL EFFECT IN ATOMS
As was shown in [13], to include the relativistic recoil corrections in calculations of the energy levels, we must add to the standard Hamiltonian of the electron-positron field interacting with the quantized electromagnetic field and with the Coulomb field of the nucleus V C , taken in the Coulomb gauge, the following term This operator acts only on the electron-positron and electromagnetic field variables. The normal ordered form of H M taken in the interaction representation must be added to the interaction Hamiltonian. It gives additional elements to the Feynman rules for the Green function. In the Furry picture, in addition to the standard Feynman rules in the energy representation (see [24,13]), the following verteces and lines appear (we assume that the Coulomb gauge is used) 1. Coulomb contribution.
An additional line ("Coulomb-recoil" line) appears to be q q q q q q q q q q q q q q q s s This line joins two vertices each of which corresponds to where p = −i∇ x and k = 1, 2, 3.

One-transverse-photon contribution.
An additional vertex on an electron line appears to be The transverse photon line attached to this vertex (at the point x) is At the point y this line is to be attached to an usual vertex in which we have −2πieγ where α l (l = 1, 2, 3) are the usual Dirac matrices.
An additional line ("two-transverse-photon-recoil" line) appears to be This line joins usual vertices (see the previous item).
Let as apply this formalism to the case of a single level a in a one-electron atom. To find the Coulomb nuclear recoil correction we have to calculate the contribution of the diagram shown in Fig. 1. A simple calculation of this diagram yields (see Ref. [13] for details) The one-transverse-photon nuclear recoil correction corresponds to the diagrams shown in Fig. 2. One easily obtains The two-transverse-photon nuclear recoil correction is defined by the diagram shown in Fig. 3. We find The sum of the contributions (3.2)-(3.4) is This exactly coincides with formula (2.6).
Consider now a high-Z two-electron atom. For simplicity, we will assume that the unperturbed wave function is a one-determinant function The nuclear recoil correction is the sum of the one-electron and two-electron contributions. The one-electron contribution is the sum of the expressions (3.5) for the a and b states. The two-electron contributions are defined by the diagrams shown in Figs. 4-6. A simple calculation of these diagrams yields The formula (3.7) was first derived by the quasipotential method in [21].

A. Hydrogenlike atoms
According to equations (2.1)-(2.3) the recoil correction is the sum of the low-order and higher-order terms. The low-order term ∆E L is given by equation (2.7). The higher order term ∆E H was calculated to all orders in αZ in [25][26][27]. The results of these calculations expressed in terms of the function P (αZ) defined as  Comparing the function P (αZ) from Table 1 with the lowest order contributions (4.2)-(4.4) shows that for high Z the complete αZ-dependence results differ considerably from Salpeter's ones.
In the case of hydrogen, the difference ∆P = P − P S amounts to -0.01616 (3), -0.01617(5), and 0.00772 for the 1s, 2s, and 2p 1/2 states, respectively. Table 2 displays the relativistic recoil corrections, beyond the Salpeter ones, to the hydrogen energy levels. These values include also the corresponding correction from the low-order term (2.7) which is calculated by The results of Refs. [25,27] which are exact in αZ are compared with the related corrections obtained to the lowest order in αZ. In [28,29] it was found that the (αZ) 6 log (αZ)m 2 /M corrections cancel each other. The (αZ) 6 m 2 /M correction was derived in [18] for s-states and in [30] for p-states. The (αZ) 7 log 2 (αZ)m 2 /M correction was recently evaluated in Refs. [31,32]. The uncertainty of the calculation based on the expansion in αZ is defined by uncalculated terms of order (αZ) 7 m 2 /M and is expected to be about 1 kHz for the 1s state. It follows that the results of the complete αZ-dependence calculations are in a good agreement with the results obtained to lowest orders in αZ but are of much higher accuracy. As it follows from Ref. [13], the formulas (2.1)-(2.3) will incorporate partially the nuclear size corrections to the recoil effect if V C (r) is taken to be the potential of an extended nucleus. In particular, this replacement allows one to account for the nuclear size corrections to the Coulomb part of the recoil effect. In Ref. [33], where the calculations of the recoil effect for extended nuclei were performed, it was found that, in the case of hydrogen, the leading relativistic nuclear size correction to the Coulomb low-order part is comparable with the total value of the (αZ) 6 m 2 /M correction but is cancelled by the nuclear size correction to the Coulomb higher-order part.
One of the main goals of the calculations of Refs. [25,26,33] was to evaluate the nuclear recoil correction for highly charged ions. In the case of the ground state of hydrogenlike uranium these calculations yield -0.51 eV for the point nucleus case [25] and -0.46 eV for the extended nucleus case [33]. This correction is big enough to be included in the current theoretical prediction for the 1s Lamb shift in hydrogenlike uranium [34] but is small compared with the present experimental uncertainty which amounts to 13 eV [10]. However, a much higher precision was obtained in experiments with heavy lithiumlike ions [8,9]. In this connection in Refs. [25,26] the nuclear recoil corrections for lithiumlike ions were calculated as well.

B. Lithiumlike ions
In lithiumlike ions, in addition to the one-electron contributions, we must evaluate the two-electron contributions. In the case of one electron over the (1s) 2 shell the total two-electron contribution to the zeroth order in 1/Z is given by the expression where D is defined by equation (2.4). Calculation of this term causes no problem [25,26]. For the 2p 1/2 and 2p 3/2 states, the results of this calculation expressed in terms of the function Q(αZ) defined by are presented in Table 3. For the s-states the two-electron contribution is equal zero. To the lowest orders in αZ the function Q(αZ) is given by [23]  The expressions (4.9)-(4.10) serve as a good approximation for the Q(αZ) function even for very high Z. For low Z, in addition to the corrections considered here, the Coulomb interelectronic interaction effect on the nonrelativistic nuclear recoil correction must be taken into account. It contributes on the level of order (1/Z)(αZ) 2 m 2 /M .
To date, the highest precision in experiments with heavy ions was obtained for the 2p 3/2 −2s transition in lithiumlike bismuth [9]. The transition energy measured in this experiment amounts to (2788.14±0.04) eV. In [8] the energy of the 2p 1/2 − 2s transition in lithiumlike uranium was measured to be (280.59 ± 0.10) eV. In both cases the recoil correction amounts to -0.07 eV and, therefore, is comparable with the experimental uncertainty. At present, the uncertainty of the theoretical predictions for these transition energies is defined by uncalculated contributions of second order in α (see Refs. [34,35]). When calculations of these contributions are completed, it will be possible to probe the recoil effect in high-Z few-electron systems. This will provide a unique possibility for testing the quantum electrodynamics in the region of strong coupling (αZ ∼ 1) beyond the external field approximation since in calculations of all other QED corrections in heavy ions the nucleus is considered only as a stationary source of the classical electromagnetic field.

V. CONCLUSION
In this paper the relativistic theory of the recoil effect in atoms is considered. It is shown that the complete αZdependence calculation of the recoil correction provides the highest precision even in the case of hydrogen. The recoil corrections to the energy levels of highly charged ions contribute on the level of the present experimental accuracy. It provides good perspectives for testing the quantum electrodynamics in the region of strong coupling (αZ ∼ 1) beyond the external field approximation.