Hyperfine Interactions 146/147: 127–131, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

127

Correlation and Relativistic Effects on Landé gJ

Factors of Atomic Ions

P. INDELICATO1, A.-M. MÅRTENSSON-PENDRILL2, W. QUINT3 andJ.-P. DESCLAUX4

1Laboratoire Kastler-Brossel, Unité Mixte de Recherche du CNRS n◦ C8552, École NormaleSupérieure et Université Pierre et Marie Curie, Case 74, 4 place Jussieu, F-75252 Paris CEDEX05, France; e-mail: paul.indelicato@spectro.jussieu.fr2Physics, Göteborg University and Chalmers University of Technology SE-41296 Göteborg, Sweden3GSI, Planckstrafle 1, Darmstadt, D-64291, Germany415 Chemin du Billery, F-38360, Sassenage, France

Abstract. We investigate relativistic effects on the Landé gJ factors of atomic ions using Multicon-figuration Dirac–Fock technique and Relativistic Many-Body perturbation theory. The role of Breitinteraction, negative energy continuum and correlation effects is studied in Li-like, B-like, N-likeions and Ca+ ground-states. We also investigate Ti-like ions which have a long-lived excited statewith J = 4. Those ions are all good candidates for employing the continuous Stern–Gerlach effectto measure their gJ -factor and provide accurate tests of relativistic many-body calculations.

Key words: Landé factors, highly-charged ions.

1. Introduction

The use of Penning traps has provided new experimental possibilities to do high-accuracy measurements of Landé gJ -factors. The ground-state Landé factor ofhydrogen-like carbon has been measured to an accuracy of 1 ppb by microwavespectroscopy via the continuous Stern–Gerlach effect [7]. Other ions have beenstudied like Ca+, Ba+ (including metastable states) with laser-microwave doubleresonance spectroscopy. In this paper we investigate the size of different relativis-tic and correlation contributions to the gJ -factor of several ions in the groundstate, and to the gJ -factor of Ti-like ions in a very metastable state with a high-angular momentum [13]. We discuss the possibility of employing the continuousStern–Gerlach effect for a measurements of the gJ -factor of those highly chargedfew-electron systems.

2. Theoretical considerations

Performing relativistic many-body calculations is much more complicated than theequivalent non-relativistic case. The only fully clean way of doing those calculationis to use quantum electrodynamics (QED). Yet because of the difficulties of the

128 P. INDELICATO ET AL.

evaluation of QED diagrams, and the large number of them that contributes to cor-relation at low-Z (perturbation parameter for correlation is 1/Z), such a calculationcannot be realized in practice for light elements. One must then use an approximateHamiltonian, derived from QED, and use either Relativistic Many-Body Perturba-tion theory (RMBPT) or a variational method like Multiconfiguration Dirac–Fock(MCDF). For evaluating quantities other than energy levels, like Landé gJ factors,an additional difficulty is that the approximate Hamiltonian derived from QEDis a no-pair approximation [5, 8], that is required to calculate energies. Yet thisapproximation is not good enough for some one-body operators, e.g., like the op-erator used for evaluating M1 transition probabilities [9]. Also the magnetic andretardation part of the electron–electron interaction could play an important role,which complicates even more the picture.

Here we investigate correlation and relativistic effects in several atoms and ionsto investigate the importance of different effects and motivate new experiments.Up to now the evaluation of QED radiative corrections to Landé factors has beenperformed only for hydrogen-like ions [1, 14] where they lead to extraordinarilygood tests of QED when compared to experiment [6, 7]. In this work we haveused the MCDF code developed by two of us (J.P.D. and P.I.) to evaluate Landéfactors following the formalism described in [2]. We also performed RMBPT cal-culations following the method described in [11] to evaluate the accuracy of theapproximations.

3. Results and discussion

3.1. LITHIUM-LIKE IONS

We have performed Multiconfiguration Dirac–Fock calculations of the Landé gJ -factors of three-electron ions. Relativistic effects are included ab-initio. Full self-consistent treatment of the Breit interaction allows us to include the Breit in-teraction effects to all orders. In this calculation we have been able to obtain afully-optimized wave function including all single, double and triple excitations toall levels with n � 6 and � � 5, which represents 4152 jj configurations. Note thatlimiting the calculation to double excitations leads to only 468 jj configurations.The results for Z = 4 are represented in Figure 1. We compare results obtainedwith an offspring of GRASP [3] by Bieron (Coulomb only, configuration activespace approximation [10]), recent high-accuracy variational calculations [20], ex-periment [19] and our MCDF calculations (Coulomb, full Breit, pair or no-pairapproximation) and RMBPT results [11]. The results for the MCDF calculationwith full account of the Breit interaction, the high-precision variational calculationand the RMBPT values are all in agreement with experiment. Calculations for Z

up to 20 are being performed. They will provide high-quality tests of QED andmany-body effects for three-electron ions when experiments using the continuousStern–Gerlach effect [6, 7] will be performed.

EFFECTS ON LANDE GJ FACTORS OF ATOMIC IONS 129

Figure 1. Lande gJ factors for Be+. FBSC: full Breit interaction used in the SCF process (thiswork). CSC: only the Coulomb interaction is used in the self-consistent field (this work). When np isadded, no pair approximation has been used to obtain the wave function. GRASP: MCDF calculationusing Oxford General Relativistic Atomic Structure Package [3, 10]. Exp: Experimental value from[19]. Z. C. Yan: Hylleraas-type non-relativistic calculations from [20].

3.2. EVALUATION OF THE Ca+ LANDÉ FACTOR

The Landé factor of Ca+ has been measured with high-accuracy at Mainz in aPenning trap [16]. We have performed a RMBPT calculation including Relativis-tic Random Phase Approximation (RPA) contributions. We also did a RelativisticConfiguration Interaction (RCI) calculation starting from a Dirac–Fock wave func-tion and adding single excitations of s and p orbitals (d orbital effects were foundnegligible). Using the Dirac–Fock potential we generated a basis set using B-splines, and included (discretized) continuum wave functions in the set of RCIorbitals. Both methods provide similar results.

Higher-order correlation (double excitation for RCI, beyond RPA for RMBPT)and Breit correlation remain to be investigated. The results are presented in Ta-ble I.

3.3. OTHER IONS

Highly-charged ions can exhibit interesting properties in their Landé factors. Onecan study cases in which the active electron is not an s electron, and investigaterelativistic effects. We study B-like ions (p1/2 ground state), N-like ions (p3/2

ground state) and a long-lived metastable state [13] of Ti-like (3d4, J = 4) ions, avery interesting sequence [4, 12, 15, 17, 18]. The results are presented in Table II.As it is only an exploratory work, we performed only Dirac–Fock calculations. Yetin the case of the N-like sequence we took into account the intra-shell correlation(i.e., using 1s22s22p + 1s22p3 configurations), which is expected to give a largeeffect.

130 P. INDELICATO ET AL.

Table I. Different approximations to the Ca+ Lande factor

Method RCI basis set size gJ − gfree gJ

Coulomb only

Dirac–Fock −4.851 × 10−5 2.0022708

RCI 20 −5.342 × 10−5 2.0022659

RCI 25 −5.305 × 10−5 2.0022662

RCI 30 −5.399 × 10−5 2.0022653

RMBPT −5.280 × 10−5 2.0022665

with Breit contribution

Dirac–Fock −5.159 × 10−5 2.0022677

Experiment [16] −6.263(9) × 10−5 2.00225664(9)

Table II. Lande factors in B-like ([B]), N-like ([N]) and Ti-like ([Ti]) ions. IS: intra-shell correlation.The change of sign in the QED corrections comes from the fact that p1/2 and p3/2 orbitals haveopposite relativistic angular momentum number κ – see, e.g., [2]

B J = 1/2 [B] Xe J = 1/2 N J = 3/2 [N] Xe J = 3/2 [Ti] Ta J = 4

Coulomb 0.666623 0.641814 1.999861 1.357132 1.060885

Breit contr. 0.000037 0.000750 −0.000022 0.002431 0.002074

QED (Coul.) −0.000773 −0.000773 0.002319 0.000871 0.000167

QED (Breit) 0.000000 0.000000 0.000000 0.000005 0.000004

IS Coul. −0.000016 −0.001404

IS Breit −0.000025 −0.000467

Total 0.665846 0.639920 2.002159 1.360439 1.063131

4. Conclusion

We have demonstrated a number of relativistic correlation effects on Landé factorsand provided some accurate predictions for some of them. Many interesting studieson few-electron ions or ions with complex structure can be undertaken, that will bea challenge for theory, but should lead to new insight on the relativistic many-body problem. The continuous Stern–Gerlach effect is well suited to investigateexperimentally all those ions, including some of them in a long-lived metastablestate.

Acknowledgements

This research was supported in part by the TMR Network Eurotraps ContractNumber ERBFMRXCT970144.

EFFECTS ON LANDE GJ FACTORS OF ATOMIC IONS 131

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