QED effects in heavy few-electron ions

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Abstract

Accurate calculations of the binding energies, the hyperfine splitting, the bound-electron g-factor, and the parity nonconservation effects in heavy few-electron ions are considered. The calculations include the relativistic, quantum electrodynamic (QED), electron-correlation, and nuclear effects. The theoretical results are compared with available experimental data. A special attention is focused on tests of QED in a strong Coulomb field.

Introduction

Accurate calculations of heavy few-electron ions must be performed in the framework of the rigorous quantum electrodynamic (QED) formalism. The basic methods of quantum electrodynamics were formulated to the beginning of 1930s, almost immediately after creation of quantum mechanics. This theory provided description of such low-order processes as emission and absorption of photons and creation and annihilation of electron-positron pairs. However, application of this theory to some higher-order effects gave infinite results. This problem remained unsolved until the late 1940’s when Lamb and Retherford discovered the 2s2p1/2 splitting (Lamb shift) in hydrogen. This discovery stimulated theorists to complete the creation of QED since it was believed that this splitting is of quantum electrodynamic origin. First evaluation of the Lamb shift was performed by Bethe who used Kramer’s idea of the mass renormalization. A consequent QED formalism was developed by Dyson, Feynman, Schwinger, and Tomonaga. They showed that all infinities can be removed from the theory by so-called renormalization procedure. The basic idea of this procedure is the following. The electron mass and the electron charge, which originally occur in the theory, are not directly measurable quantities. All physical quantities calculated within QED become finite if they are expressed in terms of the physical electron mass and charge, parameters which can directly be measured in experiment. All calculations in QED are based on the perturbation theory in the fine structure constant α1/137.036. The individual terms of the perturbation series are conveniently represented by so-called Feynman diagrams.

Before the beginning of 1970s investigations of QED effects in atomic systems were mainly restricted to low-Z atoms such as hydrogen or helium (here and below, Z is the nuclear charge number). In these systems, in addition to the small parameter α, there is another small parameter, which is αZ. For this reason, all calculations of low-Z atoms were based on the expansion in α and αZ.

A great progress in experimental investigations of heavy few-electron ions, which was made for the last decades (see [1], [2] and references therein), has required accurate QED calculations for these systems. Investigations of heavy few-electron ions play a special role in tests of quantum electrodynamics. This is due to two reasons. First, in contrast to low-Z atoms, the parameter αZ is not small and, therefore, the calculations must be performed without any expansion in αZ [3]. Second, in contrast to heavy neutral atoms, the electron-correlation effects can be calculated to high accuracy using pertubation theory in the parameter 1/Z. For this reason, the QED effects are not masked by large electron-correlation effects, as is the case in neutral atoms. This provides an excellent opportunity to test QED at strong electric fields.

The calculations of high-Z few-electron ions are generally based on perturbation theory. To zeroth order, one can consider that electrons interact only with the Coulomb field of the nucleus. The interelectronic-interaction and QED effects are accounted for by perturbation theory in the parameters 1/Z and α, respectively. This leads to quantum electrodynamics in the Furry picture. To formulate the perturbation theory for calculations of the energy levels, transition and scattering amplitudes, it is convenient to use the two-time Green function method [4]. For very heavy ions the parameter 1/Z becomes comparable with α and, therefore, all the corrections may be classified by the parameter α only. In the present paper, we consider the current status of these calculations.

Relativistic units (=c=1) are used in the paper.

Section snippets

H-like ions

To calculate the binding energy in a hydrogenlike ion, we may start with the Dirac equation,(αp+βm+VC(r))ψ(r)=Eψ(r),where VC(r) is the Coulomb potential induced by the nucleus. For the point-nucleus case, this equation leads to the binding energyEnjmc2=(αZ)22ν221+(αZ/ν)2+1+(αZ/ν)2mc2,where ν=n+(j+1/2)2(αZ)2(j+1/2), n is the principal quantum number, and j is the total angular momentum. To get the binding energy to a higher accuracy, we have to evaluate the quantum electrodynamic and

Hyperfine splitting in heavy ions

High-precision measurements of the hyperfine splitting (HFS) in heavy hydrogen-like ions [68], [69], [70], [71], [72] have triggered a great interest to theoretical calculations of this effect. The ground-state hyperfine splitting of a hydrogen-like ion is conveniently written as [73]:ΔEμ=43α(αZ)3μμNmmp2I+12Imc2×{A(αZ)(1δ)(1ε)+xrad}.Here mp is the proton mass, μ is the nuclear magnetic moment, μN is the nuclear magneton, and I is the nuclear spin. A(αZ) denotes the relativistic factorA(αZ)=1γ(

Bound-electron g-factor

The g-factor of a hydrogenlike ion with a spinless nucleus can be defined asg(e)=JMJ|μz(e)|JMJμBMJ,where μ(e) is the operator of the magnetic moment of electron and μB is the Bohr magneton. For the 1s state, a simple relativistic calculation based on the Dirac equation yieldsgD=243(11(αZ)2).The total g-factor value can be written asg(e)=gD+ΔgQED+Δgrec+ΔgNS,where ΔgQED is the QED correction, Δgrec is the nuclear recoil correction, and ΔgNS is the finite-nuclear-size correction.

Parity nonconservation effects with heavy ions

Investigations of parity nonconservation (PNC) effects in atomic systems play a prominent role in tests of the Standard Model (SM) [100]. The well-known cesium experiment by Wieman’s group [101], compared to the most elaborated theoretical result (see Ref. [102] and references therein), provided the most accurate test of electroweak theory at the low-energy regime. Further improvement of tests of the Standard Model with neutral atoms, from theoretical side, is mainly limited by difficulties

Conclusion

In this paper we have considered the present status of calculations of the QED effects in heavy few-electron ions. Calculations of the PNC effects for heavy ions have also been discussed.

To date, the most accurate tests of QED effects on binding energies in a strong Coulomb field have been accomplished in heavy Li-like ions: on a 0.5% accuracy level to first order in α and on a 15% accuracy level to second order in α. An improvement of the experimental accuracy by an order of magnitude

Acknowledgements

This work was supported in part by RFBR (Grant No. 04-02-17574) and by INTAS-GSI (Grant No. 03-54-3604). A.N.A., S.S.B., Y.S.K., N.S.O., and V.A.Y acknowledge the support by the “Dynasty” foundation.

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