Nuclear recoil corrections to the $2p_frac{3}{2}$ state energy of hydrogen-like and high $Z

4
(1) ∆E (1) ≡ ∆Ec + ∆Etr(1) =
(1)
m2 − ε2 a . 2M
(10)
The αZ -expansion of this term is given in [4]. (2) (2) The terms ∆Ec , ∆Etr(1) , and ∆Etr(2) are given in the form that allows one to use the relativistic Coulomb Green function for their calculation. (As was found by Pachucki and Grotch [7], this form is convenient for the αZ expansion calculations as well.) In [4] we transformed these equations to the form that is more convenient for calculations using the finite basis set methods [14] and calculated them for the 1s, 2s, and 2p 1 states by using the 2 B-spline method for the Dirac equation [15,16]. The calculation for the 2p 3 2 state considered here was carried out in the same way. Table 1 shows the results of the numerical calculation for the 2p 3 state 2 expressed in terms of the function P (αZ ) defined by
(1) (2)
(4) (5)
(6)
αl (l = 1, 2, 3) are the Dirac matrices, Dlm (ω ) is the transverse part of the photon propagator in the Coulomb gauge. In the coordinate representation it is Dik (ω, r) = − 1 exp (i|ω |r ) (exp (i|ω |r ) − 1) . δik + ∇i ∇k 4π r ω 2r i 2πM (8)
2
Hydrogen-like atoms
The complete αZ -dependence expressions for the nuclear recoil corrections to the energy levels of hydrogen-like atoms were first derived by one of the authors of the present paper [9] (a part of the expressions was found earlier by Braun [10]). These expressions consist of three contributions: the Coulomb contribution, the one-transverse-photon contribution, and the twotransverse-photon contribution. For a state a the Coulomb contribution is
The two-transverse-photon contributioa|D(ω )G(ω + εa )D(ω )|a .
−∞ (1)
(9)
(1) The terms ∆Ec and ∆Etr(1) are leading at low Z . These terms can easily be calculated by using the virial relations for the Dirac equation [11-13]. Such a calculation gives [9]
2
1
Introduction
Till recently accurate QED calculations of the nuclear recoil corrections to atomic energy levels were of interest mainly in connection with high precision measurements of the Lamb shift in hydrogen [1,2]. In this case the nuclear recoil corrections may be calculated in lowest orders in αZ (α is the fine structure constant, Z is the nuclear charge). However recent achievements in experimental investigations of highly charged ions [3] require calculations of the nuclear recoil corrections right up Z = 92. In the last case the parameter αZ can no longer be considered small and, so, the calculations without expansion in αZ are required. In our previous paper [4] we calculated the nuclear recoil corrections in all orders in αZ for the 1s, 2s, 2p 1 states of hydrogen-like atoms and the 2 (1s)2 2s, (1s)2 2p 1 states of high Z lithium-like atoms. We found that the 2 nuclear recoil contribution to the energy of the (1s)2 2p 1 - (1s)2 2s transi2 tion in lithium-like uranium constitutes −0.07 eV and, so, is comparable with the uncertainty of the experimental value of the transition energy: 280.59(10) eV [3]. We found also the nuclear recoil contribution, additional to Salpeter’s one, to the Lamb shift (n = 2) of hydrogen to be −1.32(6) kHz . This result is in good agreement with recent analytical calculations of the 2 m2 (αZ )6 ln (αZ ) and m (αZ )6 corrections [5-8], according to which the terms M M 2 (αZ )6 ln (αZ ) cancel each other [5,6], the contribution of order of order m M 2 m (αZ )6 is −0.77 kHz for the 2s state [7] and 0.58 kHz for the 2p 1 state [8]. M 2 In the present paper we extend the results of [4] to the case of the 2p 3 2 state of hydrogen-like and high Z lithium-like atoms. The relativistic units h ¯ = c = 1 are used in the paper.
Nuclear recoil corrections to the 2p 3 state energy of 2 hydrogen-like and high Z lithium-like atoms in all orders in αZ
arXiv:quant-ph/9510016v1 16 Oct 1995
A.N.Artemyev, V.M.Shabaev, and V.A.Yerokhin Department of Physics, St.Petersburg State University, Oulianovskaya 1, Petrodvorets, St.Petersburg 198904, Russia Nuclear recoil corrections ... PACS number: 3130J
3
given by
(1) (2) ∆Ec = ∆Ec + ∆Ec , 2 p (1) ∆Ec = a| |a , 2M 2πi ∞ 2 (2) dω δ+ (ω ) a|[p, Vc]G(ω + εa )[p, Vc ]|a , ∆Ec = M −∞
(1) (2) (3)
where |a is the unperturbed state of the Dirac electron in the Coulomb field is the Coulomb potential of the nucleus, p is the of the nucleus, Vc = − αZ r momentum operator, δ+ (ω ) = 2iπ (ω + i0)−1 , G(ω ) = (ω − H (1 − i0))−1 is the relativistic Coulomb Green function, H = αp + βm + Vc . The onetransverse-photon contribution is ∆Etr(1) = ∆Etr(1) + ∆Etr(1) , 1 (1) a| D(0)p + pD(0) |a , ∆Etr(1) = − 2M 1 ∞ (2) ∆Etr(1) = − dω δ+ (ω ) a| [p, Vc ]G(ω + εa )D(ω ) M −∞ −D(ω )G(ω + εa )[p, Vc ] |a , where Dm (ω ) = −4παZαl Dlm (ω ) , (7)