$g$ factor of the $[(1s)^2(2s)^2 2p]~{}^2P_{3/2}$ state of middle-$Z$ boronlike ions

TL;DR

This paper presents precise theoretical calculations of the g-factor for the first excited state of boronlike ions with atomic numbers 10 to 20, incorporating QED, recoil, and interelectronic effects.

Contribution

It provides the first comprehensive QED-based g-factor calculations for this state across a range of Z, improving accuracy over previous models.

Findings

Calculated g-factors for Z=10 to 20 ions.

Quantified interelectronic and QED corrections.

Compared results with previous data.

Abstract

Theoretical \emph{g}-factor calculations for the first excited \exst state of boronlike ions in the range $Z$=10--20 are presented and compared to the previously published values. The first-order interelectronic-interaction contribution is evaluated within the rigorous QED approach in the effective screening potential. The second-order contribution is considered within the Breit approximation. The QED and nuclear recoil corrections are also taken into account.

Tables

Table 1. Table 1: Interelectronic-interaction correction to the g factor of boronlike ions in the P 3 / 2 2 superscript subscript 𝑃 3 2 2 {}^{2}P_{3/2}~{} state. The terms of the zeroth ( Δ ​ g int ( 0 ) Δ superscript subscript 𝑔 int 0 \Delta g_{\mathrm{int}}^{(0)} ), first ( Δ ​ g int ( 1 ) Δ superscript subscript 𝑔 int 1 \Delta g_{\mathrm{int}}^{(1)} ), and second ( Δ ​ g int ( 2 ) Δ superscript subscript 𝑔 int 2 \Delta g_{\mathrm{int}}^{(2)} ) orders of perturbation theory obtained with the core-Hartree (CH), Dirac-Hartree (DH), Kohn-Sham (KS), and Dirac-Slater (DS) screening potentials. First-order term is split into the contributions of the positive-energy ( Δ ​ g int ( 1 ) ​ [ + ] Δ superscript subscript 𝑔 int 1 delimited-[] \Delta g_{\mathrm{int}}^{(1)}[+] ) and negative-energy ( Δ ​ g int ( 1 ) ​ [ − ] Δ superscript subscript 𝑔 int 1 delimited-[] \Delta g_{\mathrm{int}}^{(1)}[-] ) spectra calculated within the Breit approximation and the QED part ( Δ ​ g int ( 1 ) ​ [ QED ] Δ superscript subscript 𝑔 int 1 delimited-[] QED \Delta g_{\mathrm{int}}^{(1)}[\mathrm{QED}] ). All numbers are in units of 10 − 6 superscript 10 6 10^{-6} .

	CH	DH	KS	DS
$Z=10$
$\Delta g_{\mathrm{int}}^{\left(0\right)}$	302.983	376.227	312.152	276.157
$\Delta g_{\mathrm{int}}^{\left(1\right)}[+]$	$-$22.611	$-$117.851	$-$34.824	17.678
$\Delta g_{\mathrm{int}}^{\left(1\right)}[-]$	$-$22.371	$-$2.806	$-$19.623	$-$29.741
$\Delta g_{\mathrm{int}}^{\left(1\right)}\left[\mathrm{QED}\right]$	$-$0.141	$-$0.111	$-$0.141	$-$0.158
$\Delta g_{\mathrm{int}}^{\left(2\right)}$	6.256	6.889	6.580	$-$0.758
$\Delta g_{\mathrm{int}}$	264.116	262.348	264.143	263.178
$Z=12$
$\Delta g_{\mathrm{int}}^{\left(0\right)}$	368.346	462.044	379.402	334.176
$\Delta g_{\mathrm{int}}^{\left(1\right)}[+]$	$-$27.393	$-$148.063	$-$41.786	22.005
$\Delta g_{\mathrm{int}}^{\left(1\right)}[-]$	$-$29.402	$-$4.746	$-$26.187	$-$38.614
$\Delta g_{\mathrm{int}}^{\left(1\right)}\left[\mathrm{QED}\right]$	$-$0.281	$-$0.233	$-$0.282	$-$0.308
$\Delta g_{\mathrm{int}}^{\left(2\right)}$	6.288	7.275	6.400	$-$0.441
$\Delta g_{\mathrm{int}}$	317.559	316.277	317.547	316.819
$Z=14$
$\Delta g_{\mathrm{int}}^{\left(0\right)}$	433.852	547.842	446.756	392.322
$\Delta g_{\mathrm{int}}^{\left(1\right)}[+]$	$-$32.164	$-$178.105	$-$48.754	26.389
$\Delta g_{\mathrm{int}}^{\left(1\right)}[-]$	$-$36.370	$-$6.680	$-$32.691	$-$47.407
$\Delta g_{\mathrm{int}}^{\left(1\right)}\left[\mathrm{QED}\right]$	$-$0.489	$-$0.420	$-$0.492	$-$0.528
$\Delta g_{\mathrm{int}}^{\left(2\right)}$	6.306	7.516	6.289	$-$0.265
$\Delta g_{\mathrm{int}}$	371.135	370.154	371.108	370.511
$Z=16$
$\Delta g_{\mathrm{int}}^{\left(0\right)}$	499.514	633.710	514.250	450.613
$\Delta g_{\mathrm{int}}^{\left(1\right)}[+]$	$-$36.888	$-$208.003	$-$55.691	30.839
$\Delta g_{\mathrm{int}}^{\left(1\right)}[-]$	$-$43.298	$-$8.610	$-$39.151	$-$56.142
$\Delta g_{\mathrm{int}}^{\left(1\right)}\left[\mathrm{QED}\right]$	$-$0.780	$-$0.686	$-$0.783	$-$0.833
$\Delta g_{\mathrm{int}}^{\left(2\right)}$	6.315	7.682	6.204	$-$0.166
$\Delta g_{\mathrm{int}}$	424.863	424.092	424.828	424.311
$Z=18$
$\Delta g_{\mathrm{int}}^{\left(0\right)}$	565.355	719.702	581.912	509.070
$\Delta g_{\mathrm{int}}^{\left(1\right)}[+]$	$-$41.544	$-$237.766	$-$62.572	35.370
$\Delta g_{\mathrm{int}}^{\left(1\right)}[-]$	$-$50.195	$-$10.541	$-$45.575	$-$64.829
$\Delta g_{\mathrm{int}}^{\left(1\right)}\left[\mathrm{QED}\right]$	$-$1.167	$-$1.043	$-$1.171	$-$1.235
$\Delta g_{\mathrm{int}}^{\left(2\right)}$	6.316	7.804	6.132	$-$0.116
$\Delta g_{\mathrm{int}}$	478.765	478.155	478.726	478.259
$Z=20$
$\Delta g_{\mathrm{int}}^{\left(0\right)}$	631.397	805.864	532.824	567.713
$\Delta g_{\mathrm{int}}^{\left(1\right)}[+]$	$-$46.116	$-$267.395	$-$69.382	39.996
$\Delta g_{\mathrm{int}}^{\left(1\right)}[-]$	$-$57.065	$-$12.472	$-$51.964	$-$73.472
$\Delta g_{\mathrm{int}}^{\left(1\right)}\left[\mathrm{QED}\right]$	$-$1.661	$-$1.507	$-$1.668	$-$1.747
$\Delta g_{\mathrm{int}}^{\left(2\right)}$	6.311	7.896	6.065	$-$0.101
$\Delta g_{\mathrm{int}}$	532.866	532.386	532.824	532.134

Table 2. Table 2: Individual contributions to the g factor of the P 3 / 2 2 superscript subscript 𝑃 3 2 2 {}^{2}P_{3/2}~{} state of boronlike ions in the range Z 𝑍 Z =10–20. The values obtained with the Kohn-Sham potential are used for the interelectronic-interaction correction Δ ​ g int Δ subscript 𝑔 int \Delta g_{\mathrm{int}} (see Table  LABEL:tab:g-int ). The g -factor values from Refs.  20 , 21 , 22 , 24 are given for comparison.

	${}_{10}{}^{}{}_{}{}^{20}$Ne5+		${}_{12}{}^{}{}_{}{}^{24}$Mg7+
Dirac value $g_D$	1.	332 623 079	1.	332 310 417
Interelectronic interaction $\Delta g_{\mathrm{int}}$	0.	000 264 1 (18)	0.	000 317 5 (13)
One-loop QED $\Delta g_{\mathrm{QED}}^{\left(1\right)}$	0.	000 775 7 (5)	0.	000 776 3 (7)
Two-loop QED $\Delta g_{\mathrm{QED}}^{\left(2\right)}$	$-$0.	000 001 2	$-$0.	000 001 2
Nuclear recoil $\Delta g_{\mathrm{rec}}$	$-$0.	000 008 9 (15)	$-$0.	000 007 8 (11)
Total value $g$	1.	333 652 8 (23)	1.	333 394 1 (17)
$g$ from Ref. 20	1.	333 695	1.	333 448
	${}_{14}{}^{}{}_{}{}^{28}$Si9+		${}_{16}{}^{}{}_{}{}^{32}$S11+
Dirac value $g_D$	1.	331 940 789	1.	331 514 136
Interelectronic interaction $\Delta g_{\mathrm{int}}$	0.	000 371 1 (10)	0.	000 424 8 (8)
One-loop QED $\Delta g_{\mathrm{QED}}^{\left(1\right)}$	0.	000 777 2 (9)	0.	000 778 2 (10)
Two-loop QED $\Delta g_{\mathrm{QED}}^{\left(2\right)}$	$-$0.	000 001 2	$-$0.	000 001 2
Nuclear recoil $\Delta g_{\mathrm{rec}}$	$-$0.	000 006 8 (8)	$-$0.	000 006 1 (6)
Total value $g$	1.	333 081 1 (16)	1.	332 709 8 (14)
$g$ from Ref. 20	1.	333 143	1.	332 783
$g$ from Ref. 22	1.	333 148 (7)	1.	332 788 (8)
	${}_{18}{}^{}{}_{}{}^{40}$Ar13+		${}_{20}{}^{}{}_{}{}^{40}$Ca15+
Dirac value $g_D$	1.	331 030 389	1.	330 489 471
Interelectronic interaction $\Delta g_{\mathrm{int}}$	0.	000 478 7 (6)	0.	000 532 8 (7)
One-loop QED $\Delta g_{\mathrm{QED}}^{\left(1\right)}$	0.	000 779 5 (12)	0.	000 780 9 (13)
Two-loop QED $\Delta g_{\mathrm{QED}}^{\left(2\right)}$	$-$0.	000 001 2 (1)	$-$0.	000 001 2 (1)
Nuclear recoil $\Delta g_{\mathrm{rec}}$	$-$0.	000 004 9 (4)	$-$0.	000 004 9 (4)
Total value $g$	1.	332 282 5 (14)	1.	331 797 1 (15)
$g$ from Ref. 20	1.	332 365	1.	331 891
$g$ from Ref. 22	1.	332 372 (1)	1.	331 899 (7)
$g$ from Ref. 21	1.	332 282 (3)
$g$ from Ref. 24	1.	332 286

Equations

$$g=g_{\mathrm{D}}+\Delta g_{\mathrm{int}}+\Delta g_{\mathrm{QED}}+\Delta g_{\mathrm{rec}}+\Delta g_{\mathrm{NS}},$$

$$g_{\mathrm{D}}=\frac{4}{15}\bigg{[}2\sqrt{4-(\alpha Z)^{2}}+1\bigg{]}=\frac{4}{3}-\frac{2}{15}(\alpha Z)^{2}-\dots\,.$$

$$\Delta g_{\mathrm{int}}=\Delta g_{\mathrm{int}}^{(0)}+\Delta g_{\mathrm{int}}^{(1)}+\Delta g_{\mathrm{int}}^{(2)},$$

$$\Delta g_{\mathrm{int}}^{(1)}=\Delta g_{\mathrm{int}}^{(1)}[+]+\Delta g_{\mathrm{int}}^{(1)}[-]+\Delta g_{\mathrm{int}}^{(1)}[\mathrm{QED}].$$

$$\Delta g_{\mathrm{QED}}^{(1)}=\Delta g_{\mathrm{SE}}+\Delta g_{\mathrm{VP}}.$$

$$\Delta g_{\mathrm{SE}}=\frac{\alpha }{\pi }\left[b_{00}+\frac{(\alpha Z{)}^2}{4}{b}_{20}+\frac{(\alpha Z{)}^4}{8}\left\{\ln [(\alpha Z{)}^{-2}]b_{41}+b_{40}\right\}\right].$$

$$\Delta g_{\mathrm{NS}}=\frac{(\alpha Z{)}^6}{90}{m}^4⟨r^4⟩.$$

Full text

††articletype: Conference Proceedings

1 Introduction

Significant progress in the g-factor studies in highly charged ions has been achieved in the last two decades 1, 2. Contemporary experiments have reached the precision of $10^{-9}$–$10^{-11}$ for hydrogenlike and lithiumlike ions 3, 4, 5, 6, 7. One of the highlights in this field is the most accurate determination of the electron mass from the combined experimental and theoretical studies of the $g$ factor of hydrogenlike ions 8. Extension of these studies to lithiumlike ions has provided the stringent test of the many-electron QED effects 7, 9, 10, 11. The high-precision $g$-factor measurement of the two isotopes of lithiumlike calcium 10 and the most elaborate evaluation of the nuclear recoil effect for this system 12 have demonstrated a possibility to study the bound-state QED effects beyond the Furry picture in the strong field regime 13. It is expected that g-factor studies in few-electron ions will be able to provide an independent determination of the fine structure constant $\alpha$ 14, 15, 16.

The ALPHATRAP experiment at the Max-Planck-Institut für Kernphysik (MPIK) is capable of the ground-state g-factor measurements for wide range of few-electron ions, including boronlike ones 1. The ARTEMIS project at GSI implements the laser-microwave double-resonance spectroscopy of the Zeeman splitting in both ground $[(1s)^{2}(2s)^{2}2p]~{}{}^{2}P_{1/2}$ and first excited $[(1s)^{2}(2s)^{2}2p]~{}{}^{2}P_{3/2}$ states of middle-$Z$ boronlike ions 17, 18. In particular, boronlike argon is chosen as the first candidate for these measurements. Theoretical investigations of the $g$ factor of boronlike ions were performed recently in Refs. 19, 20, 21, 22, 23, 24. Various methods have been used in these works for evaluation of the interelectronic-interaction contribution, including the large-scale configuration-interaction approach in the basis of the Dirac-Fock-Sturm orbitals (CI-DFS) 19, 21, the GRASP2K 20 and MCDFGME 22 packages based on relativistic multi-configuration Dirac-Hartree-Fock (MCDHF) method, the second-order perturbation theory (PT) in effective screening potential 21, 23, and the high order coupled cluster (CC) method 24. For the ground-state $g$ factor of boronlike argon the results of the CI-DFS, PT, and CC approaches are in agreement on the level of $10^{-6}$, while both MCDHF results reveal a deviation on the level of $10^{-4}$. In the present work, we extend the second-order perturbation-theory calculations to the ${}^{2}P_{3/2}~{}$state. The QED and nuclear recoil corrections are also taken into account. The results for boronlike ions in the range $Z$=10–20 are presented and compared to the previously published values 19, 20, 21, 22, 24. We use the relativistic units ($\hbar=c=1$) and the Heaviside charge unit ($\alpha=e^{2}/(4\pi)$, $e<0$) throughout the paper.

2 Methods and results

The total g-factor value of boronlike ion with zero nuclear spin can be written as

[TABLE]

where $\Delta g_{\mathrm{int}}$, $\Delta g_{\mathrm{QED}}$, $\Delta g_{\mathrm{rec}}$, and $\Delta g_{\mathrm{NS}}$ are the interelectronic-interaction, QED, nuclear recoil, and nuclear size corrections, respectively. The Dirac value $g_{\mathrm{D}}$ for the $2p_{3/2}$ state is

[TABLE]

The interelectronic-interaction correction is considered within the perturbation theory. The first-order term $\Delta g_{\mathrm{int}}^{(1)}$ (one-photon exchange) is calculated within the rigorous QED approach, i.e., to all orders in $\alpha Z$. The second-order term $\Delta g_{\mathrm{int}}^{(2)}$ (two-photon exchange) is considered within the Breit approximation. The general formulae for this contribution can be found from the complete quantum electrodynamical formulae for the two-photon-exchange diagrams presented in Ref. 25. Care should be taken to account properly for the contribution of the negative-energy states, since it is comparable in magnitude to the positive-energy counter-part.

We incorporate the effective screening potential in the zeroth-order approximation. This improves the convergence of the perturbation theory and provides a reliable estimation of the higher-order remainder. The corresponding counter-terms should be considered in calculations of the first- and second-order contributions. The difference between the g-factor values in the screening and pure Coulomb potentials is termed as the zeroth-order contribution $\Delta g_{\mathrm{int}}^{(0)}$. We use the following well-known screening potentials: core-Hartree (CH), Dirac-Hartree (DH), Kohn-Sham (KS), and Dirac-Slater (DS), see, e.g., Ref. 26 for more details.

In Table LABEL:tab:g-int we present the interelectronic-interaction contributions to the g-factor multiplied by $10^{6}$. The total value of $\Delta g_{\mathrm{int}}$ is found as,

[TABLE]

where the first-order correction $\Delta g_{\mathrm{int}}^{(1)}$ is divided into the following three parts:

[TABLE]

The positive-energy-states ($\Delta g_{\mathrm{int}}^{(1)}[+]$) and negative-energy-states ($\Delta g_{\mathrm{int}}^{(1)}[-]$) contributions are calculated in the Breit approximation. The QED contribution ($\Delta g_{\mathrm{int}}^{(1)}[\mathrm{QED}]$) is the difference between the rigorous QED and the Breit-approximation values.

As the final results for $\Delta g_{\mathrm{int}}$, we take the values calculated in the Kohn-Sham potential. The uncertainty due to unknown higher-order contributions can be estimated as the spread of the obtained results for different potentials. As one can see from the Table LABEL:tab:g-int, the maximal difference of the values of $\Delta g_{\mathrm{int}}$ varies between $1.8\times 10^{-6}$ for $Z$=10 and $0.8\times 10^{-6}$ for $Z$=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.

The one-loop QED correction $\Delta g_{\mathrm{QED}}^{(1)}$ is given by the sum of the self-energy and vacuum-polarization contributions,

[TABLE]

The self-energy correction was calculated to all orders in $\alpha Z$ for both $2p_{1/2}$ and $2p_{3/2}$ states in the range $Z$=1–12 in Ref. 27. These values can be extrapolated to a good accuracy by the following $\alpha Z$-expansion 27, 28,

[TABLE]

The values $b_{00}(2p_{1/2})=-1/3$ and $b_{00}(2p_{3/2})=1/3$ have long been known 29, 30. The values $b_{20}(2p_{1/2})=0.48429$ and $b_{20}(2p_{3/2})=0.59214$ have been found in Ref. 28. Our fitting procedure based on the least squares method reproduces these coefficients on the level of $10^{-5}$ if they are taken as unknown, which serves as a check of its consistency. In this way we extrapolate the results of Ref. 27 up to $Z$=20. In addition, we estimate the screening correction for the $2p_{3/2}$ state employing the effective nuclear charge $Z_{\text{eff}}$ instead of $Z$ in Eq. (6). The effective nuclear charge $Z_{\text{eff}}$ is found from our rigorous calculations of the self-energy correction for the $2p_{1/2}$ state with an effective screening potential 23: Eq. (6) with $Z_{\text{eff}}$ should reproduce the result obtained with the Kohn-Sham potential. The screening shift $Z-Z_{\text{eff}}$ lies in the range 1.3–1.7 for the ions under consideration. We ascribe the 100% uncertainty to the screening correction obtained in this rather approximate way.

The dominant contribution of the vacuum polarization is given by the two-electron diagrams where the vacuum-polarization potential acts on the $1s$ and $2s$ electrons. This contribution was estimated as $5.5\times 10^{-9}$ for $Z$=18 in Ref. 19, which is much smaller than the total theoretical uncertainty. The two-loop contribution $\Delta g_{\mathrm{QED}}^{(2)}$ is represented by its zeroth-order term of the $\alpha Z$-expansion 30.

The nuclear recoil effect in boronlike argon was calculated in Refs. 19, 21 within the Breit approximation to zeroth and first orders in $1/Z$. Systematic calculations of this effect for the $2p_{1/2}$ state in the range $Z$=10–20 were performed in Ref. 31. Recently, these calculations have been extended to $Z$=20–92 including the leading-order QED contributions beyond the Breit approximation 32. In the present paper, we evaluate this effect for the $2p_{3/2}$ state with the relativistic recoil operators to zeroth order in $1/Z$ with the Kohn-Sham effective screening potential. The leading-order term of the finite-nuclear-size correction can be written as 33

[TABLE]

For $Z$=10–20 it gives the values of the order $10^{-17}$–$10^{-15}$ which is negligible at the present level of accuracy.

The individual contributions and the total g-factor values for the $2p_{3/2}$ state of boronlike ions in the range $Z$=10–20 are presented in Table LABEL:tab:g-int. The values of $\Delta g_{\mathrm{int}}$ calculated in the Kohn-Sham potential are used. Our results for argon are in agreement with the PT results from Refs. 19, 21 and with the CC results from Ref. 24. The difference between the data from Ref. 20 and those of the present work ranges from $0.000\,042$ for $Z$=10 to $0.000\,094$ for $Z$=20. The difference between the data from Ref. 22 and those of the present work ranges from $0.000\,067$ for $Z$=14 to $0.000\,102$ for $Z$=20. The origin of this disagreement is not clear at present. We suppose that the negative-energy-states contribution was not taken into account completely in Refs. 20, 22.

Zeeman splitting of the $2p_{j}$ states acquires significant nonlinear contributions. In particular, the second- and third-order terms in magnetic field can be observed in forthcoming measurements for boronlike argon 17, 19. Recently, the systematic calculations of these terms for the wide range of boronlike ions have been presented by our group 34. The most important contribution for the $2p_{3/2}$ state is the shift of the levels with $m_{j}=\pm 1/2$ proportional to $B^{2}$. It can be represented as the $m_{j}$-dependent $g$-factor contribution varying from $\pm 1.52\times 10^{-4}$ for $Z$=10 to $\pm 5.68\times 10^{-6}$ for $Z$=20 at the field of 1 T (it scales linearly with $B$). For more detailed description of the second- and third-order contributions see Ref. 34.

3 Conclusion

In conclusion, the g factor of the ${}^{2}P_{3/2}~{}$state of boronlike ions in the range $Z$=10–20 has been evaluated with an uncertainty on the level of $10^{-6}$. The leading interelectronic-interaction correction has been calculated to all orders in $\alpha Z$. The higher-order interelectronic-interaction and nuclear-recoil effects have been taken into account within the Breit approximation. The one-loop self-energy correction has been found from extrapolation of the previously published high-precision results for $Z$=1–12 with an approximate account for screening.

Acknowledgements

The work was supported in part by RFBR (Grants No. 16-02-00334 and 19-02-00974), 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 (COLLAB 2018: 34824940). 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.