Ground-state g factor of middle-Z boronlike ions

Ground-state g factor of middle- boronlike ions

V A Agababaev    D A Glazov    A V Volotka    D V Zinenko    V M Shabaev and G Plunien Department of Physics, Saint-Petersburg State University, 199034 Saint-Petersburg, Russia Saint-Petersburg State Electrotechnical University ‘‘LETI’’, 197376 Saint-Petersburg, Russia Helmholtz-Institut Jena, D-07743 Jena, Germany Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany
Abstract

Theoretical calculations of the interelectronic-interaction and QED corrections to the g factor of the ground state of boronlike ions are presented. The first-order interelectronic-interaction and the self-energy corrections are evaluated within the rigorous QED approach in the effective screening potential. The second-order interelectronic interaction is considered within the Breit approximation. The nuclear recoil effect is also taken into account. The results for the ground-state g factor of boronlike ions in the range =10–20 are presented and compared to the previous calculations.

The past two decades have been marked by intensive development of the g-factor studies in highly charged ions [1, 2]. Experimental precision has reached the level of for hydrogenlike and lithiumlike ions [3, 4, 5, 6, 7]. Cooperative experimental and theoretical work led to the most accurate up-to-date value of the electron mass [8]. The most stringent test of the many-electron QED effects in the presence of magnetic field has been achieved with middle- lithiumlike ions [7, 9, 10, 11]. Simultaneous high-precision g-factor measurement for two calcium isotopes [10] and the rigorous evaluation of the relativistic nuclear-recoil effect [12, 13] have opened perspective for testing bound-state QED effects beyond the Furry picture (external field approximation for the nucleus). Independent determination of the fine structure constant is possible in g-factor studies with high- [14] or middle- [15] boronlike, lithiumlike and hydrogenlike ions. The ARTEMIS experiment presently implemented at GSI aims at measurement of the g factors of both ground and first excited states of boronlike argon [16]. In this regard, the leading interelectronic-interaction, QED and recoil corrections to these g factors were calculated in Refs. [17, 18] employing the bound-state QED perturbation theory and the configuration-interaction Dirac-Fock-Sturm (CI-DFS) method. In Ref. [19] the GRASP2K program package based on the relativistic multi-configuration Dirac-Hartree-Fock method was used to obtain the energy levels, the hyperfine interaction constants and the g factors in beryllium-, boron-, carbon- and nitrogen-like ions in the range =8–42. In Ref. [20] the g factors of boronlike ions in the range =14–92 were evaluated within the multi-configuration Dirac-Fock method using the MCDFGME code. Significant difference between the results of Refs. [17, 18, 19, 20] motivated us to perform independent calculations within the framework of the bound-state QED perturbation theory. In this paper, we present the results for the ground-state g factor of boronlike ions in the range =10–20. The relativistic units () and the Heaviside charge unit () are used throughout the paper.

The total g-factor value of boronlike ion with spinless nucleus can be written as

(1)

where the leading contribution can be found analytically from the Dirac equation with the point-nucleus potential,

(2)

and , , and denote the interelectronic-interaction, QED, nuclear recoil and nuclear size corrections, respectively.

The correction due to the interelectronic interaction is considered within the perturbation theory. The term of the first order in is calculated within the rigorous QED approach, i.e., to all orders in . The second-order contribution is considered within the Breit approximation. In Refs. [9, 21] the two-photon-exchange corrections to the g factor and to the hyperfine splitting have been evaluated within the rigorous QED approach for lithiumlike ions. The formulae presented in Ref. [21] can be used to derive the corresponding expressions within the Breit approximation. A distinctive feature of the g-factor calculations is the necessity to account for the negative-energy-states contribution, since it is comparable in magnitude to the positive-energy counterpart.

In order to account approximately for the higher-order corrections, an effective screening potential is introduced in the Dirac equation. It leads to emergence of the zeroth-order contribution — difference between the g-factor values for the effective screening and the pure Coulomb potentials. The corresponding counterterms have to be taken into account in the first- and second-order contributions. We consider four different screening potentials — core-Hartree (CH), Dirac-Hartree (DH), Kohn-Sham (KS) and Dirac-Slater (DS). Explicit formulae for these potentials can be found e.g. in Refs. [22, 23]. We note that the evaluation of the two-photon-exchange contribution in the pure Coulomb nuclear potential is related to some numerical problems in case of the boronlike ions.

In table 1 the breakdown of the interelectronic-interaction correction is given in terms of the g-factor contributions multiplied by . The first-order term is split into three parts: the positive-energy-states () and negative-energy-states () contributions and the QED contribution (). The two former are obtained within the Breit approximation. The latter is found as the difference between the rigorous QED result and the Breit-approximation result. The total value of is found as a sum of the evaluated contributions,

(3)

where

(4)

We choose the result for the Kohn-Sham potential as the final one. The total value of would not depend on the effective potential, if all orders of the perturbation theory were taken into account rigorously. Thus the spread of the results for different potentials can serve as an estimation of the uncertainty due to the unknown higher-order contributions. As one can see from the table, the maximal difference of the values of varies between for =10 and for =20. Interelectronic-interaction corrections of the third and higher orders have been evaluated for lithiumlike ions within the CI-DFS [9] and CI [11] methods. The results obtained in these papers suggest that this estimation of the uncertainty is quite reliable. We can also estimate the unknown QED part of the two-photon-exchange correction as not more than based on the results of Ref. [9].

One-loop QED correction is given by the sum of the self-energy and the vacuum-polarization contributions,

(5)

The self-energy correction for the states was calculated to all orders in in Ref. [24]. The numerical approach was based on the Dirac-Coulomb Green’s function in order to achieve rather high accuracy, which is especially difficult for low nuclear charge. Instead, we use the approach developed in Refs. [23, 25], which is based on the DKB finite basis set [26]. Although, it is generally less accurate, it allows one to easily incorporate arbitrary spherically symmetric binding potential. In order to account approximately for the many-electron QED effects we use effective screening potentials, the same ones that we use for evaluation of : core-Hartree, Dirac-Hartree, Kohn-Sham and Dirac-Slater. The results of the calculations are given in table 2.

The one-electron vacuum-polarization correction is negligible for the state in the considered range of . The dominant effect of the vacuum polarization arises from the two-electron diagrams, where the and electrons of the closed shells come into play. Still, it is much smaller than the total theoretical uncertainty: for =18 it was estimated as in Ref. [17]. The two-loop QED contributions are taken into account to the zeroth order in according to Ref. [27].

The nuclear-recoil contribution was calculated for boronlike argon in Ref. [17] including the leading relativistic corrections and the screening effect. In Ref. [18] the first-order interelectronic-interaction correction was considered using the nonrelativistic approximation for the recoil operator. Recently, the nuclear recoil effect to the g factor of boronlike ions has been evaluated with the relativistic recoil operator in the zeroth and first orders in [28]. These results are used in the present compilation. The finite-nuclear-size correction for state to the leading order in can be written as [29]

(6)

where is the nuclear root-mean-square radius. For =10–20 equation (6) gives the values of the order , i.e., much smaller than the total theoretical uncertainty.

In table 3 we present the individual contributions and the total values of the g factor of boronlike ions in the range =10–20. The Kohn-Sham values of (see table 1) and (see table 2) are employed. Despite the different approach to evaluation of the second- and higher-order interelectronic-interaction effects, our results for argon are in agreement with Ref. [18]. For comparison we present also the data from Ref. [19] and Ref. [20]. One can see that the difference between the values of Verdebout \etaland of the present work grows monotonically from for =10 to for =20. The corresponding difference with the values of Marques \etalranges from for =14 to for =20. At present, we can not clearly identify the source of this disagreement. However, we suppose that the contribution of the negative-energy states was not completely taken into account in Refs. [19, 20].

We note also that the nonlinear contributions in magnetic field are important in boronlike ions [16, 17]. Recently, the second- and third-order effects have been evaluated within the fully relativistic approach for the wide range of [30]. While the second-order effect is not observable in the ground-state Zeeman splitting, the third-order effect has to be taken into account. Its relative contribution amounts to for =10 and for =20 at the field strength of 1 Tesla and it scales as .

In conclusion, the g factor of boronlike ions in the range =10–20 has been evaluated with an uncertainty on the level of . The leading interelectronic-interaction and QED effects have been calculated to all orders in . The higher-order interelectronic-interaction and nuclear-recoil effects have been taken into account within the Breit approximation.

CH DH KS DS
379.092 470.808 390.491 345.422
30.899 91.371 39.453 6.168
1.820 38.936 4.897 15.864
0.148 0.118 0.149 0.166
10.139 14.568 10.531 0.003
356.364 354.951 356.523 354.956
461.050 578.458 474.753 418.092
38.052 111.255 47.404 8.707
3.321 53.249 7.622 18.987
0.291 0.245 0.294 0.320
10.119 14.802 10.141 0.343
429.505 428.509 429.573 428.395
543.283 686.232 559.217 490.972
45.268 131.208 55.478 11.194
4.725 67.395 10.212 22.206
0.506 0.440 0.509 0.546
10.104 14.963 9.899 0.519
502.888 502.151 502.915 501.957
625.826 794.263 643.939 564.093
52.499 151.206 63.582 13.599
6.043 81.416 12.694 25.515
0.802 0.713 0.809 0.857
10.088 15.083 9.718 0.601
576.570 576.011 576.572 575.752
708.721 902.650 728.969 637.488
59.722 171.243 71.670 15.902
7.275 95.330 15.078 28.914
1.194 1.080 1.204 1.266
10.068 15.180 9.566 0.622
650.598 650.177 650.584 649.855
792.014 1011.470 814.355 711.194
66.922 191.319 79.712 18.082
8.417 109.147 17.365 32.408
1.695 1.552 1.708 1.785
10.043 15.261 9.429 0.597
725.023 724.714 724.998 724.332
Table 1: Interelectronic-interaction correction to the g factor of boronlike ions in terms of . The contributions of the zeroth (), first () and second () orders of the perturbation theory obtained with the core-Hartree (CH), Dirac-Hartree (DH), Kohn-Sham (KS) and Dirac-Slater (DS) screening potentials. The first-order term is split into the contributions of the positive-energy () and negative-energy () spectra calculated within the Breit approximation and the QED part ().
CH DH KS DS
773.05 773.06 772.99 772.95
772.43 772.49 772.36 772.29
771.61 771.70 771.53 771.44
770.60 770.71 770.50 770.39
769.39 769.51 769.26 769.13
767.95 768.10 767.81 767.65
Table 2: Self-energy correction to the g factor of boronlike ions obtained with the core-Hartree (CH), Dirac-Hartree (DH), Kohn-Sham (KS) and Dirac-Slater (DS) screening potentials in terms of .
Ne Mg
Dirac value 0. 665 777 663 0. 665 385 559
Interelectronic interaction 0. 000 356 5 (16) 0. 000 429 6 (12)
One-loop QED 0. 000 773 0 (4) 0. 000 772 4 (5)
Two-loop QED 0. 000 001 2 0. 000 001 2
Nuclear recoil 0. 000 015 2 (12) 0. 000 013 6 (7)
Total value 0. 665 347 2 (20) 0. 665 030 4 (15)
Total value [19] 0. 665 392 0. 665 084
Si S
Dirac value 0. 664 921 417 0. 664 384 860
Interelectronic interaction 0. 000 502 9 (10) 0. 000 576 6 (8)
One-loop QED 0. 000 771 5 (6) 0. 000 770 5 (8)
Two-loop QED 0. 000 001 2 0. 000 001 2
Nuclear recoil 0. 000 012 3 (4) 0. 000 011 1 (3)
Total value 0. 664 641 7 (12) 0. 664 181 1 (12)
Total value [19] 0. 664 704 0. 664 252
Total value [20] 0. 664 829 (40) 0. 664 400 (46)
Ar Ca
Dirac value 0. 663 775 447 0. 663 092 678
Interelectronic interaction 0. 000 650 6 (7) 0. 000 725 0 (7)
One-loop QED 0. 000 769 3 (9) 0. 000 767 8 (10)
Two-loop QED 0. 000 001 2 (1) 0. 000 001 2 (1)
Nuclear recoil 0. 000 009 1 (2) 0. 000 009 3 (2)
Total value 0. 663 648 8 (12) 0. 663 041 8 (12)
Total value [19] 0. 663 728 0. 663 130
Total value [20] 0. 663 899 (2) 0. 663 325 (56)
Total value [18] 0. 663 647 7 (7)
Table 3: Ground-state g factor of boronlike ions in the range =10–20. The values obtained with the Kohn-Sham potential are used for the interelectronic-interaction correction (see table 1) and the one-loop QED correction (see table 2). The g-factor values from Refs. [18, 19, 20] are given for comparison.
\ack

The work was supported in part by RFBR (Grant No. 16-02-00334), by DFG (Grant No. VO 1707/1-3), by SPbSU-DFG (Grant No. 11.65.41.2017 and No. STO 346/5-1) and by SPbSU (Grant No. 11.40.538.2017). V.A.A. acknowledges the support by the German-Russian Interdisciplinary Science Center (G-RISC). The numerical computations were performed at the St. Petersburg State University Computing Center.

References

References

  • [1] Sturm S, Vogel M, Köhler-Langes F, Quint W, Blaum K and Werth G 2017 Atoms 5 4
  • [2] Shabaev V M, Glazov D A, Plunien G and Volotka A V 2015 J. Phys. Chem. Ref. Data 44 031205
  • [3] Häffner H, Beier T, Hermanspahn N, Kluge H J, Quint W, Stahl S, Verdú J and Werth G 2000 Phys. Rev. Lett. 85 5308
  • [4] Verdú J, Djekić S, Stahl S, Valenzuela T, Vogel M, Werth G, Beier T, Kluge H J and Quint W 2004 Phys. Rev. Lett. 92 093002
  • [5] Sturm S, Wagner A, Schabinger B, Zatorski J, Harman Z, Quint W, Werth G, Keitel C H and Blaum K 2011 Phys. Rev. Lett. 107 023002
  • [6] Sturm S, Wagner A, Kretzschmar M, Quint W, Werth G and Blaum K 2013 Phys. Rev. A 87 030501
  • [7] Wagner A, Sturm S, Köhler F, Glazov D A, Volotka A V, Plunien G, Quint W, Werth G, Shabaev V M and Blaum K 2013 Phys. Rev. Lett. 110 033003
  • [8] Sturm S, Köhler F, Zatorski J, Wagner A, Harman Z, Werth G, Quint W, Keitel C H and Blaum K 2014 Nature 506 467
  • [9] Volotka A V, Glazov D A, Shabaev V M, Tupitsyn I I and Plunien G 2014 Phys. Rev. Lett. 112 253004
  • [10] Köhler F, Blaum K, Block M, Chenmarev S, Eliseev S, Glazov D A, Goncharov M, Hou J, Kracke A, Nesterenko D A, Novikov Y N, Quint W, Minaya Ramirez E, Shabaev V M, Sturm S, Volotka A V and Werth G 2016 Nat. Commun. 7 10246
  • [11] Yerokhin V A, Pachucki K, Puchalski M, Harman Z and Keitel C H 2017 Phys. Rev. A 95 062511
  • [12] Shabaev V M, Glazov D A, Malyshev A V and Tupitsyn I I 2017 Phys. Rev. Lett. 119 263001
  • [13] Malyshev A V, Shabaev V M, Glazov D A and Tupitsyn I I 2017 JETP Letters 106 765
  • [14] Shabaev V M, Glazov D A, Oreshkina N S, Volotka A V, Plunien G, Kluge H J and Quint W 2006 Phys. Rev. Lett. 96 253002
  • [15] Yerokhin V A, Berseneva E, Harman Z, Tupitsyn I I and Keitel C H 2016 Phys. Rev. Lett. 116 100801
  • [16] von Lindenfels D, Wiesel M, Glazov D A, Volotka A V, Sokolov M M, Shabaev V M, Plunien G, Quint W, Birkl G, Martin A and Vogel M 2013 Phys. Rev. A 87 023412
  • [17] Glazov D A, Volotka A V, Schepetnov A A, Sokolov M M, Shabaev V M, Tupitsyn I I and Plunien G 2013 Phys. Scr. T156 014014
  • [18] Shchepetnov A A, Glazov D A, Volotka A V, Shabaev V M, Tupitsyn I I and Plunien G 2015 J. Phys. Conf. Ser. 583 012001
  • [19] Verdebout S, Nazé C, Jönsson P, Rynkun P, Godefroid M and Gaigalas G 2014 At. Data Nucl. Data Tables 100 1111
  • [20] Marques J P, Indelicato P, Parente F, Sampaio J M and Santos J P 2016 Phys. Rev. A 94 042504
  • [21] Volotka A V, Glazov D A, Andreev O V, Shabaev V M, Tupitsyn I I and Plunien G 2012 Phys. Rev. Lett. 108 073001
  • [22] Sapirstein J and Cheng K T 2002 Phys. Rev. A 66 042501
  • [23] Glazov D A, Volotka A V, Shabaev V M, Tupitsyn I I and Plunien G 2006 Phys. Lett. A 357 330
  • [24] Yerokhin V A and Jentschura U D 2010 Phys. Rev. A 81 012502
  • [25] Volotka A V, Glazov D A, Plunien G, Shabaev V M and Tupitsyn I I 2006 Eur. Phys. J. D 38 293
  • [26] Shabaev V M, Tupitsyn I I, Yerokhin V A, Plunien G and Soff G 2004 Phys. Rev. Lett. 93 130405
  • [27] Grotch H and Kashuba R 1973 Phys. Rev. A 7 78
  • [28] Glazov D A, Malyshev A V, Shabaev V M and Tupitsin I I 2018 Opt. Spectrosc. 124 457
  • [29] Glazov D A and Shabaev V M 2002 Phys. Lett. A 297 408
  • [30] Varentsova A S, Agababaev V A, Glazov D A, Volchkova A M, Volotka A V, Shabaev V M and Plunien G 2018 Phys. Rev. A 97 043402
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