Relativistic calculations of the ground and inner-L-shell excited energy   levels of berylliumlike ions

M. Y. Kaygorodov; Y. S. Kozhedub; I. I. Tupitsyn; A. V. Malyshev; D.; A. Glazov; G. Plunien; and V. M. Shabaev

arXiv:1901.02083·physics.atom-ph·March 29, 2019

Relativistic calculations of the ground and inner-L-shell excited energy levels of berylliumlike ions

M. Y. Kaygorodov, Y. S. Kozhedub, I. I. Tupitsyn, A. V. Malyshev, D., A. Glazov, G. Plunien, and V. M. Shabaev

PDF

TL;DR

This paper presents comprehensive relativistic calculations of energy levels for berylliumlike ions across a wide atomic number range, incorporating quantum electrodynamics and other corrections for high accuracy.

Contribution

It introduces a large-scale relativistic configuration-interaction combined with many-body perturbation theory for precise energy level calculations of berylliumlike ions.

Findings

01

Accurate energy levels for ions with Z=10 to Z=92.

02

Quantified uncertainties in the calculations.

03

Included QED, recoil, and Breit corrections.

Abstract

Large-scale relativistic configuration-interaction method combined with many-body perturbation theory is consistently applied to calculations of the energy levels of the ground and inner-L-shell excited states of berylliumlike ions in the range $10 \leq Z \leq 92$ . The quantum electrodynamics, nuclear recoil, and frequency-dependent Breit corrections are taken into account. The obtained results are supplemented with the systematical estimation of the uncertainties.

Tables5

Table 1. Table 1: The contributions of the second order of the perturbation theory to the Dirac - Coulomb - Breit ground-state binding energy of Ne 6+ , in a.u.. The calculations are performed for the LDF potential including the single and double excitations. The values listed after the first row are the increments obtained by adding successively the configurations while the maximal value of the orbital quantum number L max subscript 𝐿 max L_{\mathrm{max}} is increased. The columns from left to right correspond to the enlargement of the size of the one-electron basis set, and N max subscript 𝑁 max N_{\mathrm{max}} labels the number of the one-electron virtual orbitals for a particular L max subscript 𝐿 max L_{\mathrm{max}} . The last line shows the total results of the calculations with the particular basis sets. The final value is obtained by the extrapolation to the case N max → ∞ → subscript 𝑁 max N_{\rm max}\to\infty .

$L_{\max}$	$N_{\max} = 10$	$N_{\max} = 15$	$N_{\max} = 20$	$N_{\max} = 25$	$N_{\max} = 30$	$N_{\max} \to \infty$
1	$- 0.232 241$	$- 0.232 472$	$- 0.232 481$	$- 0.232 483$	$- 0.232 484$
2	$- 0.011 581$	$- 0.011 632$	$- 0.011 636$	$- 0.011 638$	$- 0.011 638$
3	$- 0.003 028$	$- 0.003 063$	$- 0.003 069$	$- 0.003 070$	$- 0.003 071$
4	$- 0.001 111$	$- 0.001 144$	$- 0.001 149$	$- 0.001 151$	$- 0.001 151$
5	$- 0.000 495$	$- 0.000 521$	$- 0.000 526$	$- 0.000 526$	$- 0.000 528$
6	$- 0.000 247$	$- 0.000 268$	$- 0.000 273$	$- 0.000 276$	$- 0.000 275$
7	$- 0.000 138$	$- 0.000 152$	$- 0.000 157$	$- 0.000 158$	$- 0.000 159$
8	$- 0.000 082$	$- 0.000 091$	$- 0.000 095$	$- 0.000 097$	$- 0.000 098$
9	$- 0.000 052$	$- 0.000 059$	$- 0.000 062$	$- 0.000 063$	$- 0.000 064$
10	$- 0.000 034$	$- 0.000 039$	$- 0.000 042$	$- 0.000 043$	$- 0.000 043$
$11 - \infty$	$- 0.000 117 (24)$	$- 0.000 133 (21)$	$- 0.000 144 (22)$	$- 0.000 151 (22)$	$- 0.000 156 (23)$
Total	$- 0.249 126 (24)$	$- 0.249 574 (21)$	$- 0.249 634 (22)$	$- 0.249 656 (22)$	$- 0.249 667 (23)$	$- 0.249 685 (29)$

Table 2. Table 2: Contributions to the DCB energy of the ground and singly-excited n = 2 𝑛 2 n=2 states in Ne 6+ , Xe 50+ , and U 88+ , in a.u.. The calculations are performed for the LDF potential. See the text for details.

Ion	Contribution	$1 s^{2} 2 s^{2}$ ${}^{1}S_{0}$	$1 s^{2} 2 s 2 p$ ${}^{3}P_{0}$	$1 s^{2} 2 s 2 p$ ${}^{3}P_{1}$	$1 s^{2} 2 s 2 p$ ${}^{1}P_{1}$	$1 s^{2} 2 s 2 p$ ${}^{3}P_{2}$
Ne⁶⁺	PT0	$- 91.402 481$	$- 90.768 504$	$- 90.768 504$	$- 90.760 454$	$- 90.760 454$
	PT1	$- 18.825 843$	$- 19.065 206$	$- 19.063 164$	$- 18.547 442$	$- 19.066 726$
	PT2 SD	$- 0.249 685 (29)$	$- 0.085 084 (13)$	$- 0.085 063 (13)$	$- 0.143 987 (71)$	$- 0.085 042 (15)$
	PT3 SD	$0.066 395 (74)$	$0.003 883 (9)$	$0.003 877 (8)$	$0.009 919 (59)$	$0.003 883 (8)$
	CI+PT3 SD	$- 0.006 567 (46)$	$0.000 008 (4)$	$0.000 008 (3)$	$0.000 064 (49)$	$0.000 008 (3)$
	CI SD	$- 0.000 399 (30)$	$- 0.000 193 (4)$	$- 0.000 190 (4)$	$- 0.000 244 (44)$	$- 0.000 190 (5)$
	CI+PT3 T	$- 0.001 555 (8)$	$- 0.000 042 (3)$	$- 0.000 042 (3)$	$- 0.000 349 (16)$	$- 0.000 042 (3)$
	CI T	$0.000 063 (9)$	$- 0.000 019 (3)$	$- 0.000 019 (2)$	$- 0.000 024 (7)$	$- 0.000 018 (2)$
	CI+PT3 Q	$- 0.002 688 (22)$	$- 0.000 002 (1)$	$- 0.000 002 (1)$	$- 0.000 010 (5)$	$0.000 002 (1)$
	CI Q	$0.000 069 (19)$	$0.000 000$	$0.000 000$	$0.000 000$	$0.000 000$
	Total	$- 110.422 691 (102)$	$- 109.915 159 (17)$	$- 109.913 099 (16)$	$- 109.442 527 (115)$	$- 109.908 582 (18)$
Xe⁵⁰⁺	PT0	$- 3597.287 490$	$- 3592.603 624$	$- 3592.603 624$	$- 3578.750 225$	$- 3578.750 225$
	PT1	$- 116.550 483$	$- 117.521 971$	$- 116.671 166$	$- 115.601 314$	$- 117.977 435$
	PT2 SD	$- 0.574 620 (38)$	$- 0.136 988 (42)$	$- 0.150 397 (38)$	$- 0.179 117 (55)$	$- 0.131 641 (39)$
	PT3 SD	$0.098 523 (39)$	$0.001 185 (13)$	$0.001 507 (18)$	$0.002 115 (23)$	$0.001 099 (12)$
	CI+PT3 SD	$- 0.008 335 (8)$	$0.000 001 (1)$	$- 0.000 041 (5)$	$0.000 043 (10)$	$0.000 001 (1)$
	CI SD	$- 0.000 041 (5)$	$- 0.000 009 (2)$	$- 0.000 012 (4)$	$- 0.000 009 (3)$	$- 0.000 009 (4)$
	CI+PT3 T	$- 0.000 836 (4)$	$- 0.000 002 (1)$	$- 0.000 005 (2)$	$- 0.000 013 (3)$	$- 0.000 002 (1)$
	CI T	$0.000 011 (4)$	$- 0.000 002 (1)$	$- 0.000 002 (2)$	$- 0.000 002 (2)$	$- 0.000 002 (1)$
	CI+PT3 Q	$- 0.001 627 (5)$	$0.000 000$	$0.000 000$	$0.000 000$	$0.000 000$
	CI Q	$0.000 010 (3)$	$0.000 000$	$0.000 000$	$0.000 000$	$0.000 000$
	Total	$- 3714.324 888 (56)$	$- 3710.261 411 (31)$	$- 3709.423 740 (42)$	$- 3694.528 520 (61)$	$- 3696.858 214 (41)$
U⁸⁸⁺	PT0	$- 11 796.004 585$	$- 11 785.830 399$	$- 11 785.830 399$	$- 11 631.048 631$	$- 11 631.048 631$
	PT1	$- 228.772 670$	$- 228.353 746$	$- 226.870 923$	$- 227.010 654$	$- 231.520 289$
	PT2 SD	$- 0.814 213 (47)$	$- 0.284 411 (44)$	$- 0.298 765 (37)$	$- 0.289 955 (50)$	$- 0.239 898 (48)$
	PT3 SD	$0.211 751 (48)$	$0.002 258 (19)$	$0.002 427 (22)$	$0.002 182 (28)$	$0.001 467 (17)$
	CI+PT3 SD	$- 0.053 448 (15)$	$- 0.000 001 (1)$	$- 0.000 005 (3)$	$0.000 008 (2)$	$0.000 001 (1)$
	CI SD	$- 0.000 043 (9)$	$- 0.000 016 (3)$	$- 0.000 016 (4)$	$- 0.000 011 (3)$	$- 0.000 010 (3)$
	CI+PT3 T	$- 0.001 041 (5)$	$- 0.000 003 (2)$	$- 0.000 005 (2)$	$- 0.000 007 (2)$	$- 0.000 001 (1)$
	CI T	$0.000 014 (2)$	$- 0.000 002$	$- 0.000 002 (2)$	$- 0.000 002 (1)$	$- 0.000 001 (1)$
	CI+PT3 Q	$- 0.001 802 (7)$	$0.000 000$	$0.000 000$	$0.000 000$	$0.000 000$
	CI Q	$0.000 008 (2)$	$0.000 000$	$0.000 000$	$0.000 000$	$0.000 000$
	Total	$- 12 025.436 029 (70)$	$- 12 014.466 319 (48)$	$- 12 012.997 688 (43)$	$- 11 858.347 070 (57)$	$- 11 862.807 362 (51)$

Table 3. Table 3: The DCB energies without and with the frequency-dependent Breit correction included of the ground and singly-excited n = 2 𝑛 2 n=2 states in Ne 6+ and U 88+ , in a.u.. The calculations are performed for the LDF and CH screening potentials.

$Z$	Contribution	Scr. pot.	$1 s^{2} 2 s^{2} {}^{1}S_{0}$	$1 s^{2} 2 s 2 p^{3} P_{0}$	$1 s^{2} 2 s 2 p^{3} P_{1}$	$1 s^{2} 2 s 2 p^{1} P_{1}$	$1 s^{2} 2 s 2 p^{3} P_{2}$	$\pm {(α Z)}^{3}$
10	DCB	LDF	$- 110.422 691 (102)$	$- 109.915 159 (17)$	$- 109.913 099 (16)$	$- 109.442 527 (115)$	$- 109.908 582 (18)$	$\pm 0.00039$
	DCB	CH	$- 110.422 734 (173)$	$- 109.915 150 (56)$	$- 109.913 083 (37)$	$- 109.442 502 (166)$	$- 109.908 561 (56)$
	DCB $+$ Breit $(ω)$	LDF	$- 110.422 690 (102)$	$- 109.915 159 (17)$	$- 109.913 100 (16)$	$- 109.442 528 (115)$	$- 109.908 584 (18)$
	DCB $+$ Breit $(ω)$	CH	$- 110.422 733 (173)$	$- 109.915 150 (56)$	$- 109.913 083 (37)$	$- 109.442 503 (166)$	$- 109.908 563 (56)$
92	DCB	LDF	$- 12 025.436 029 (70)$	$- 12 014.466 319 (48)$	$- 12 012.997 688 (43)$	$- 11 858.347 070 (57)$	$- 11 862.807 362 (51)$	$\pm 0.30259$
	DCB	CH	$- 12 025.439 460 (168)$	$- 12 014.468 848 (97)$	$- 12 013.000 236 (92)$	$- 11 858.350 558 (119)$	$- 11 862.810 878 (85)$
	DCB $+$ Breit $(ω)$	LDF	$- 12 025.389 517 (70)$	$- 12 014.433 221 (48)$	$- 12 012.964 639 (43)$	$- 11 858.550 937 (57)$	$- 11 863.010 939 (51)$
	DCB $+$ Breit $(ω)$	CH	$- 12 025.392 794 (168)$	$- 12 014.435 521 (97)$	$- 12 012.966 955 (92)$	$- 11 858.554 549 (119)$	$- 11 863.014 749 (85)$

Table 4. Table 4: The DCB energies for the ground and singly-excited n = 2 𝑛 2 n=2 states of Be-like ions with Z = 10 𝑍 10 Z=10 – 92 92 92 supplemented with the frequency-dependent Breit correction, in a.u.. The total energies are presented for the ground state, whereas for the other states the energies relative to the ground state are given. The calculations are performed for the LDF potential. The energies are compared with the results of the previous relativistic calculations.

Ion	$1 s^{2} 2 s^{2} {}^{1}S_{0}$	$1 s^{2} 2 s 2 p^{3} P_{0}$	$1 s^{2} 2 s 2 p^{3} P_{1}$	$1 s^{2} 2 s 2 p^{1} P_{1}$	$1 s^{2} 2 s 2 p^{3} P_{2}$	Ref.
Ne⁶⁺	$- 110.42269 (10)$	$0.50753 (10)$	$0.50959 (10)$	$0.98016 (15)$	$0.51411 (10)$
		$0.5069$	$0.5090$	$0.9794$	$0.5135$	Cheng et al. (2008)
		$0.5063$	$0.5084$	$0.9732$	$0.5129$	Safronova et al. (1996)
Ar¹⁴⁺	$- 379.38562 (6)$	$1.04647 (7)$	$1.07912 (7)$	$2.06504 (9)$	$1.15558 (7)$
	$- 379.3853$	$1.0465$	$1.0791$	$2.0653$	$1.1556$	Yerokhin et al. (2015)
		$1.0462$	$1.0788$	$2.0615$	$1.1553$	Safronova et al. (1996)
Fe²²⁺	$- 812.59026 (6)$	$1.60359 (6)$	$1.74386 (6)$	$3.44430 (9)$	$2.16495 (6)$
	$- 812.5898$	$1.6037$	$1.7440$	$3.4448$	$2.1653$	Yerokhin et al. (2014)
		$1.6036$	$1.7439$	$3.4448$	$2.1649$	Cheng et al. (2008)
		$1.6034$	$1.7436$	$3.4419$	$2.1647$	Safronova et al. (1996)
Cu²⁵⁺	$- 1018.31580 (6)$	$1.82046 (6)$	$2.02396 (6)$	$4.12200 (9)$	$2.72516 (6)$
		$1.8203$	$2.0236$	$4.1199$	$2.7248$	Safronova et al. (1996)
Zn²⁶⁺	$- 1092.22806 (9)$	$1.89411 (11)$	$2.12087 (12)$	$4.37677 (14)$	$2.94268 (11)$
		$1.8941$	$2.1209$	$4.3772$	$2.9426$	Cheng et al. (2008)
		$1.89389$	$2.12053$	$4.37467$	$2.94233$	Safronova et al. (1996)
Kr³²⁺	$- 1592.72137 (6)$	$2.35438 (7)$	$2.73476 (7)$	$6.31245 (9)$	$4.66589 (7)$
		$2.3544$	$2.7349$	$6.3125$	$4.6658$	Cheng et al. (2008)
		$2.3566$	$2.7365$	$6.3134$	$4.6656$	Safronova (2000)
		$2.3542$	$2.7343$	$6.3105$	$4.6653$	Safronova et al. (1996)
Mo³⁸⁺	$- 2193.53880 (7)$	$2.85758 (8)$	$3.39847 (8)$	$9.21628 (10)$	$7.35801 (8)$
		$2.8577$	$3.3988$	$9.2163$	$7.3579$	Cheng et al. (2008)
		$2.8574$	$3.3980$	$9.2144$	$7.3572$	Safronova et al. (1996)
Xe⁵⁰⁺	$- 3714.32145 (7)$	$4.06066 (7)$	$4.89818 (7)$	$19.77947 (8)$	$17.44970 (7)$
	$- 3714.320$					Malyshev et al. (2014)
		$4.0601$	$4.8976$	$19.7766$	$17.4474$	Safronova et al. (1996)
Nd⁵⁶⁺	$- 4646.20449 (5)$	$4.79683 (6)$	$5.76626 (6)$	$28.64454 (7)$	$26.04788 (6)$
		$4.7971$	$5.7660$	$28.6411$	$26.0449$	Safronova et al. (1996)

Table 5. Table 5: The total binding energies for berylliumlike ions with Z = 10 − 92 𝑍 10 92 Z=10-92 , in a.u.. The DCB energy, the frequency-dependent Breit correction, QEDMOD and nuclear recoil corrections are shown separately. For the ground state the total energies are presented, whereas for all other states the energies relative to the ground state are given. The comparison with the recent relativistic calculations Cheng et al. ( 2008 ); Malyshev et al. ( 2014 ); Yerokhin et al. ( 2015 , 2014 ) , the data from the NIST Atomic Spectra Database Kramida et al. ( 2018 ) , and the available experimental values is presented.

$Z$	Term	DCB	Breit $(ω)$	QEDMOD	Recoil	Total	Other theory	NIST	Experiment
10	${}^{1}S_{0}$	$- 110.42269 (10) (0)$	$0.00000$	$0.01096 (21)$	$0.00305$	$- 110.40868 (45)$		$- 110.4087 (5)$
	${}^{3}P_{0}$	$0.50753 (10) (0)$	$0.00000$	$- 0.00044 (26)$	$- 0.00013$	$0.50696 (62)$	$0.5069$ ^a	$0.50691 (2)$
	${}^{3}P_{1}$	$0.50959 (10) (0)$	$0.00000$	$- 0.00043 (26)$	$- 0.00013$	$0.50903 (62)$	$0.5090$ ^a	$0.50898 (2)$
	${}^{3}P_{2}$	$0.51411 (10) (0)$	$0.00000$	$- 0.00046 (26)$	$- 0.00013$	$0.51356 (62)$	$0.5135$ ^a	$0.51351 (2)$
	${}^{1}P_{1}$	$0.98016 (15) (0)$	$0.00000$	$- 0.00041 (27)$	$- 0.00014$	$0.97956 (63)$	$0.9794$ ^a	$0.979392 (6)$
18	${}^{1}S_{0}$	$- 379.38563 (6) (1)$	$0.00001$	$0.0880 (13)$	$0.00577$	$- 379.2919 (25)$	$- 379.2918 (8)$ ^b
							$- 379.2909 (15)$ ^c
	${}^{3}P_{0}$	$1.04649 (7) (0)$	$- 0.00001$	$- 0.0042 (16)$	$- 0.00028$	$1.0420 (35)$	$1.0422 (4)$ ^b		$1.0419 (9)$ ^d
							$1.0419$ ^a
	${}^{3}P_{1}$	$1.07914 (7) (0)$	$- 0.00002$	$- 0.0041 (16)$	$- 0.00028$	$1.0747 (35)$	$1.0749 (6)$ ^b		$1.07466 (5)$ ^d
							$1.0747$ ^a
	${}^{3}P_{2}$	$1.15563 (7) (0)$	$- 0.00005$	$- 0.0039 (17)$	$- 0.00028$	$1.1514 (35)$	$1.1516 (10)$ ^b		$1.1513 (6)$ ^d
							$1.1513$ ^a
	${}^{1}P_{1}$	$2.06508 (9) (0)$	$- 0.00004$	$- 0.0042 (17)$	$- 0.00030$	$2.0605 (35)$	$2.0611 (4)$ ^b		$2.06048 (9)$ ^d
							$2.0604$ ^a
26	${}^{1}S_{0}$	$- 812.59034 (6) (4)$	$0.00008$	$0.3109 (29)$	$0.00790$	$- 812.2714 (74)$	$- 812.2695 (30)$ ^e	$- 812.24 (7)$
							$- 812.2688 (17)$ ^c
	${}^{3}P_{0}$	$1.60367 (6) (0)$	$- 0.00008$	$- 0.0162 (36)$	$- 0.00041$	$1.587 (10)$	$1.5875 (45)$ ^e	$1.5865 (15)$
							$1.5870$ ^a
	${}^{3}P_{1}$	$1.74399 (6) (0)$	$- 0.00014$	$- 0.0159 (36)$	$- 0.00041$	$1.728 (10)$	$1.7280 (45)$ ^e	$1.7274 (3)$
							$1.7275$ ^a
	${}^{3}P_{2}$	$2.16534 (6) (0)$	$- 0.00038$	$- 0.0148 (37)$	$- 0.00041$	$2.150 (10)$	$2.1505 (45)$ ^e	$2.1496 (3)$
							$2.1496$ ^a
	${}^{1}P_{1}$	$3.44463 (9) (0)$	$- 0.00033$	$- 0.0156 (38)$	$- 0.00042$	$3.428 (10)$	$3.4290 (45)$ ^e	$3.4282 (3)$
							$3.4282$ ^a
29	${}^{1}S_{0}$	$- 1018.31594 (6) (1)$	$0.00014$	$0.4501 (40)$	$0.00878$	$- 1017.857 (10)$		$- 1017.81 (8)$
	${}^{3}P_{0}$	$1.82060 (6) (0)$	$- 0.00014$	$- 0.0240 (50)$	$- 0.00046$	$1.796 (14)$	$1.7947$ ^a	$1.795 (5)$
	${}^{3}P_{1}$	$2.02418 (6) (0)$	$- 0.00022$	$- 0.0237 (50)$	$- 0.00046$	$2.000 (14)$	$2.0001$ ^a	$2.0001 (5)$
	${}^{3}P_{2}$	$2.72584 (6) (0)$	$- 0.00068$	$- 0.0218 (50)$	$- 0.00047$	$2.703 (14)$	$2.7032$ ^a	$2.7033 (5)$
	${}^{1}P_{1}$	$4.12261 (9) (0)$	$- 0.00060$	$- 0.0228 (51)$	$- 0.00048$	$4.099 (14)$	$4.0979$ ^a	$4.0979 (5)$
30	${}^{1}S_{0}$	$- 1092.22823 (6) (1)$	$0.00017$	$0.5047 (44)$	$0.00926$	$- 1091.714 (11)$	$- 1091.7106 (17)$ ^c	$- 1091.67 (9)$
	${}^{3}P_{0}$	$1.89427 (6) (0)$	$- 0.00016$	$- 0.0271 (54)$	$- 0.00048$	$1.866 (16)$	$1.8663$ ^a
	${}^{3}P_{1}$	$2.12112 (6) (0)$	$- 0.00026$	$- 0.0268 (54)$	$- 0.00049$	$2.094 (16)$	$2.0934$ ^a
	${}^{3}P_{2}$	$2.94350 (6) (0)$	$- 0.00081$	$- 0.0246 (55)$	$- 0.00050$	$2.918 (16)$	$2.9172$ ^a
	${}^{1}P_{1}$	$4.37750 (9) (0)$	$- 0.00073$	$- 0.0257 (56)$	$- 0.00051$	$4.351 (16)$	$4.3515$ ^a

Equations17

H_{DCB} Ψ = E Ψ,

H_{DCB} Ψ = E Ψ,

H_{DCB} = Λ^{(+)} [H_{0} + V_{int}] Λ^{(+)},

H_{DCB} = Λ^{(+)} [H_{0} + V_{int}] Λ^{(+)},

H_{0} = i \sum N h_{D} (i), V_{int} = i < j \sum N [V_{C} (i, j) + V_{B} (i, j)],

h_{D} = α \cdot p + (β - 1) m + V_{nucl} (r),

h_{D} = α \cdot p + (β - 1) m + V_{nucl} (r),

V_{C} (i, j) = \frac{α}{r _{ij}},

V_{C} (i, j) = \frac{α}{r _{ij}},

V_{B} (i, j) = - \frac{α}{2 r _{ij}} [α_{i} \cdot α_{j} + \frac{( α _{i} \cdot r _{ij} ) ( α _{j} \cdot r _{ij} )}{r _{ij}^{2}}] .

V_{B} (i, j) = - \frac{α}{2 r _{ij}} [α_{i} \cdot α_{j} + \frac{( α _{i} \cdot r _{ij} ) ( α _{j} \cdot r _{ij} )}{r _{ij}^{2}}] .

\tilde{h}_{D} = α \cdot p + (β - 1) m + V_{nucl} (r) + V_{scr} (r),

\tilde{h}_{D} = α \cdot p + (β - 1) m + V_{nucl} (r) + V_{scr} (r),

H_{MS} = \frac{1}{2 M} i, j \sum {p_{i} \cdot p_{j} - \frac{α Z}{r _{i}} [α_{i} + \frac{( α _{i} \cdot r _{i} ) r _{i}}{r _{i}^{2}}] \cdot p_{j}} .

H_{MS} = \frac{1}{2 M} i, j \sum {p_{i} \cdot p_{j} - \frac{α Z}{r _{i}} [α_{i} + \frac{( α _{i} \cdot r _{i} ) r _{i}}{r _{i}^{2}}] \cdot p_{j}} .

Δ E_{MS} = \expectationvalue H_{MS} Ψ .

Δ E_{MS} = \expectationvalue H_{MS} Ψ .

Peer Reviews

No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.

Videos

No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.

Full text

Relativistic calculations of the ground and inner-L-shell excited energy levels of berylliumlike ions

M. Y. Kaygorodov1, Y. S. Kozhedub1,2, I. I. Tupitsyn1,3, A. V. Malyshev1,2, D. A. Glazov1, G. Plunien4, and V. M. Shabaev1

1Department of Physics, St. Petersburg State University, 7/9 Universitetskaya nab., 199034 St. Petersburg, Russia

2 NRC “Kurchatov Institute”, Academician Kurchatov 1, 123182 Moscow, Russia

3 Center for Advanced Studies, Peter the Great St. Petersburg Polytechnic University, Polytechnicheskaya 29, 195251 St. Petersburg, Russia

4Institut für Theoretische Physik, TU Dresden, Mommsenstrasse 13, D-01062 Dresden, Germany

Abstract

Large-scale relativistic configuration-interaction method combined with many-body perturbation theory is consistently applied to calculations of the energy levels of the ground and inner-L-shell excited states of berylliumlike ions in the range $10\leq Z\leq 92$ . The quantum electrodynamics, nuclear recoil, and frequency-dependent Breit corrections are taken into account. The obtained results are supplemented with the systematical estimation of the uncertainties.

I Introduction

The development of high-precision physics of few-electron heavy ions in recent decades was motivated by testing quantum electrodynamics (QED) at strong fields, precise determination of the fundamental constants and searches for their time variation, provision of astrophysical investigations and other applications, see, e.g., the recent reviews Safronova et al. (2018); Kozlov et al. (2018); Shabaev et al. (2018) and references therein. The most advanced calculations of the binding energies of highly-charged ions are performed within ab initio QED approach and include all the corrections up to the second-order QED contributions. To date, these state-of-the-art calculations have been done for the ground and low-lying excited states of He-like Persson et al. (1996); Yerokhin et al. (1997); Artemyev et al. (2005); Malyshev et al. (2019), Li-like Yerokhin et al. (2001); Kozhedub et al. (2010); Sapirstein and Cheng (2011), and B-like Artemyev et al. (2007, 2013); Malyshev et al. (2017) ions. For Be-like ions, however, the QED calculations of this kind have been accomplished for the ground state only Malyshev et al. (2014, 2015). The energies of the excited states of Be-like ions were studied previously within the various relativistic methods Safronova et al. (1996); Chen and Cheng (1997); Safronova (2000); Majumder and Das (2000); Gu (2005); Ho et al. (2006); Cheng et al. (2008); Jian-Hui et al. (2008); Sampaio et al. (2013); Yerokhin et al. (2014, 2015).

The main goal of the present work is a high-precision evaluation of the electron-electron correlation effects for the ground and inner-L-shell excited states of Be-like ions within the Breit approximation. Berylliumlike ions are the simplest examples of atomic systems where both intrashell and intershell interactions are important. High-precision theoretical predictions require accurate evaluation of both kinds of interaction. To perform the calculations we employ the large-scale relativistic configuration-interaction (CI) method combined with many-body perturbation theory (PT) in the basis of Dirac-Sturm orbitals. Special attention is payed to a systematic estimation of the uncertainties of the obtained results. In order to obtain the total energy-level values, the CI energies are supplemented by the frequency-dependent Breit and nuclear recoil corrections. The radiative QED effects are taken into account by means of the model QED operator approach Shabaev et al. (2013, 2015). The calculations are carried out along the beryllium isoelectronic sequence with the nuclear charge number in the range $10\leq Z\leq 92$ . All these data yield predictive results, i.e., the results with systematical estimations of numerical errors and uncertainties due to uncalculated effects. The obtained theoretical predictions of the transition energies provide a straightforward way for improving as soon as the higher-order QED corrections will be available.

The paper is organized as follows. In the next section, we give a brief outline of our approach and computation method. Section III presents the numerical results and the comparison with the previous theoretical calculations and available experimental data. The relativistic units $(\hbar=c=1)$ and the Heaviside charge unit $(\alpha=e^{2}/4\pi,\,e<0)$ are used throughout the paper.

II Methods of calculations

II.1 Electron correlations

Within the lowest-order relativistic approximation, the energy $E$ of an $N$ -electron atom is given by the eigenvalue of the Dirac-Coulomb-Breit (DCB) equation

[TABLE]

where $\Psi$ is the many-electron wave function. The DCB Hamiltonian in the “no-pair” approximation is conventionally defined by

[TABLE]

where the indices $i$ and $j$ numerate the electrons, $h_{\rm D}$ is the one-particle Dirac Hamiltonian,

[TABLE]

$\boldsymbol{\alpha}$ and $\beta$ are the Dirac matrices, $\boldsymbol{p}$ is the momentum operator, $m$ is the electron mass, $V_{\rm nucl}$ is the binding potential of the nucleus, $V_{\mathrm{C}}$ and $V_{\mathrm{B}}$ are the Coulomb part and the frequency-independent Breit part of the electron-electron interaction, respectively,

[TABLE]

Here $\alpha$ is the fine structure constant, $r_{ij}=|\boldsymbol{r}_{ij}|=|\boldsymbol{r}_{i}-\boldsymbol{r}_{j}|$ , $\boldsymbol{r}_{i}$ is the position vector of the $i$ -th electron. $\Lambda^{(+)}$ in Eq. (2) is the product of the one-electron projectors on the positive-energy states which correspond to some one-particle Dirac Hamiltonian $\tilde{h}_{\rm D}$ , which does not necessarily coincide with (4). In principle, the Hamiltonian $\tilde{h}_{\rm D}$ may include even nonlocal parts, e.g., the Dirac-Fock-Sturm operator. However, having in mind that the results of the present work are supposed to be merged with the anticipated rigorous QED calculations, in the present work we define the operator $\tilde{h}_{\rm D}$ as follows

[TABLE]

where $V_{\rm scr}(r)$ is the local screening potential which models the interelectronic interaction. Varying the screening potential $V_{\rm scr}(r)$ in Eq. (7) modifies the definition of the positive-energy-states projection operator $\Lambda^{(+)}$ and, therefore, slightly influences the solutions of the DCB equation (1). This small ambiguity in the DCB results is determined by the arbitrariness of the realization of the “no-pair” approximation and can be fully eliminated only within the rigorous bound-state QED approach. The uncertainty related to the ambiguity is kept under control in the present work.

In the present study, we treat the correlations within the framework of the large-scale CI approach with the configuration-state-function space which includes the quadruple excitations from the reference state(s). In order to obtain the high-precision DCB energies, we combine this approach with the many-body PT and the CI+PT Dzuba et al. (1996); Kozlov et al. (2015) methods. High-precision calculations must take into account all types of the correlations: the valence-valence, the core-valence, and the core-core ones. Within the CI+PT method, the configuration-interaction calculations are carried out for the valence electrons while the residual core-valence and core-core interaction is treated by means of the many-body perturbation theory taking into account the contributions up to the third order. In contrast to the CI approach, for a given set of the configuration-state functions the PT and CI+PT methods provide only an approximate solution of Eq. (1), treating the interelectronic-interaction effects incompletely. However, PT and CI+PT calculations can be performed employing the much more saturated basis sets and, therefore, provide corrections, which can be used to improve the final CI results. The PT series in the present work are constructed employing the initial approximation, which is chosen to be consistent with the definition of the Hamiltonian (7). That is, the zeroth-order approximation in PT calculations corresponds to the Hamiltonian $\sum_{i}^{N}\tilde{h}_{\rm D}{(i)}$ . The numerical scheme is discussed in more details in Sec. III.

Another important feature of the present evaluations is a systematic estimation of the numerical uncertainties of the obtained results. For each particular state and each nuclear charge, we perform the calculations with a large number of the different sets of the configuration-state functions. By analyzing the successive increments of the results obtained with the increasing basis sets, we conclude how well the PT, CI+PT and the final CI results converge.

II.2 QED effects

Accurate theoretical predictions of the energy levels demand a supplementation of the relativistic energies obtained from the DCB Hamiltonian by the QED corrections. Because of significant technical difficulties, ab initio calculations of the many-electron QED effects, in particular, for berylliumlike ions, are mainly restricted to the ground state. For excited states, especially for quasi-degenerate states, or more complex atoms (ions) one has to rely on a simplifed treatment of the QED effects.

First, we should take into account the frequency-dependent Breit correction. This correction can be obtained by using the full Coulomb-gauge QED interelectronic-interaction operator (see, e.g. Ref. Shabaev et al. (2013) and references therein) instead of the Coulomb $V_{\mathrm{C}}$ and standard Breit $V_{\mathrm{B}}$ interactions in Eq. (2). Following Ref. Yerokhin and Surzhykov (2012), in order to avoid the spurious effects, we perform this substitution only for the DCB Hamiltonian matrix elements with the reference-state wave functions. The uncertainty due to the omitted higher-order QED electron-correlation contributions is conservatively estimated to be $\pm\alpha^{2}(\alpha Z)^{3}m$ r.u. or $\pm(\alpha Z)^{3}$ a.u.. According to the available data for helium-, lithium-, and berylliumlike ions this estimation appears to be reliable.

The self-energy and vacuum-polarization corrections are treated by means of the model QED (QEDMOD) operator $h_{\mathrm{QED}}$ Shabaev et al. (2013, 2015) included into the DCB Hamiltonian (2). Taking the difference between the CI energies evaluated with and without $h_{\mathrm{QED}}$ , we obtain the QED correction to the total energy. If the evaluation is carried out with the Dirac-Coulomb basis set, this approach takes into account the QED effects to zeroth order in $1/Z$ completely. The screened QED contributions of the first and higher orders in $1/Z$ are treated in this way only approximately. In order to estimate the accuracy of the method we compared the screening effects evaluated employing the QEDMOD approach with the rigorous QED calculations of the first order in the interelectronic interaction for He-like ions Artemyev et al. (2005), Li-like ions Kozhedub et al. (2010) and the available states of Be-like ions Malyshev et al. (2014, 2015); Yerokhin et al. (2014, 2015). Finally, we estimate the uncertainty of the screened QED effects calculated to be at the level of $15\%$ for neon and growing linearly up to $30\%$ for uranium. This also includes the uncertainty due to omitted two-loop one-electron QED effects.

II.3 Nuclear recoil effect

The nuclear recoil correction to the DCB energy accounts for the finite mass of the nucleus. The fully relativistic theory of the recoil effect can be formulated only within QED Shabaev (1985, 1988, 1998); Adkins et al. (2007). In the lowest-order relativistic approximation and in the first order in the electron-to-nucleus mass ratio $m/M$ the nuclear recoil effect on the binding energy can be treated exploiting the following mass shift (MS) Hamiltonian Shabaev (1985, 1988); Palmer (1987)

[TABLE]

In the present paper, the energy shifts due to the nuclear recoil effect are evaluated as the expectation value of the MS Hamiltonian (8) with the solutions of the DCB equation (1) Tupitsyn et al. (2003),

[TABLE]

The QED nuclear recoil corrections can be comparable in magnitude to the results obtained within the lowest-order relativistic approximation for high- $Z$ ions, see, e.g., Ref. Artemyev et al. (1995a, b); Shabaev et al. (1999); Malyshev et al. (2018). In the present work, we omit these corrections since they are negligible compared to the uncertainty of the other QED contributions. The QED nuclear recoil effect on the binding energies of Be-like ions, in principle, can be estimated within the independent-electron approximation using the results presented in Ref. Malyshev et al. (2018).

III Results and discussion

In the present work, the numerical procedure of solving the DCB equation (1) is based on the CI method in the basis of the four-component Dirac-Sturm (DS) orbitals Tupitsyn et al. (2003, 2005). The employed one-electron basis set consists of the eigenfunctions of the Dirac equation with the Hamiltonian (7) for the K and L shells and the solutions of the corresponding Dirac-Sturm equation for the other orbitals. Then, the projection operator $\Lambda^{(+)}$ in Eq. (2) is constructed from the positive-energy eigenvectors obtained by the diagonalization of the one-electron Dirac Hamiltonian $\tilde{h}_{\rm D}$ in this basis. All the calculations are performed with the local Dirac-Fock (LDF) Shabaev et al. (2005) screening potential included into the Dirac and Dirac-Sturm equations. Introducing a screening potential in the zeroth-order approximation implies that the PT series are constructed for the residual interelectronic interaction. The LDF potential is designed by the inversion of the radial Dirac equation with the radial wave functions obtained in the Dirac-Fock approximation. To describe the nuclear charge distribution, the standard Fermi model with the nuclear radii taken from Ref. Angeli and Marinova (2013) is used. We remind that the key condition for merging the CI results with ab initio QED calculations is the use of the consistent projection operator $\Lambda^{(+)}$ , while the employed one-electron basis set could be obtained for any other screening potential or, e.g., nonlocal Dirac-Fock-Sturm operator.

The direct application of the CI method for Be-like ions leads to time-consuming computational task which demands significant resources to be used. Meanwhile, the major part of the interelectronic interaction can be captured by means of the lowest orders of the many-body PT which allows one to perform the calculations with the larger basis sets at the same cost. Therefore, in order to reach high accuracy, in the present work the CI method was combined with the many-body PT (up to the third order) and the CI+PT approaches Dzuba et al. (1996). The numerical scheme to evaluate the DCB energies can be briefly described as follows. At the first stage, we perform the calculations of the second order of the PT taking into account all kinds of the single (S) and double (D) excitations (PT2 SD) (the zeroth (PT0) and first (PT1) orders are trivial). In Table 1, the contributions of the second order of the PT to the ground-state energy of berylliumlike neon ( $Z=10$ ) are presented as an example. We carry out a series of calculations with different bases and analyze the successive increments as the basis is enlarged. In our analysis, we study the dependence of the obtained results on the parameter $L_{\mathrm{max}}$ , which is the maximum value of the orbital quantum number for the DS orbitals, and the parameter $N_{\rm max}$ , which is the number of the DS orbitals for each $L$ . By extending the basis set and taking the differences between the results, we identify the contributions of the individual partial waves and check the stability of the results for each orbital number $L$ with regard to the number of the basis functions. The $L_{\rm max}\to\infty$ extrapolation is done by polynomial least-square fitting in $1/L$ . Finally, the extrapolation to the case $N_{\rm max}\to\infty$ is performed in the same way.

In Table 2, in order to illustrate the employed numerical scheme we present the detailed results for the calculations of the DCB energies for the ground and inner-L-shell excited states in Ne6+, Xe50+, and U88+ ions. The PT0 values are given in the first line for each ion. All the other rows represent various corrections to the PT0 values. These corrections are obtained by the successive application of the more and more sophisticated approaches to treat the correlation effects. The notations for the corresponding methods are indicated in the second column of Table 2. In the following, we use the same notations for the methods and the corrections obtained employing these methods.

For instance, the term labeled with “PT3 SD” in Table II is obtained by performing the PT calculations up to the third order including the SD excitations and subtracting the PT0, PT1 and PT2 SD values evaluated using the same basis. Since the Rayleigh-Schrödinger PT is employed, the PT3 SD term is just the PT contribution of the third order. Having performed the PT calculations, at the next stage we apply the CI and CI+PT methods in order to take into account the higher-order interelectronic-interaction effects. First, we restrict the configuration space to the SD excitations only. In the CI+PT method the configuration space is divided into a smaller subspace for the CI calculations and its orthogonal complement which is treated perturbatively. When the smaller CI subspace is constructed, only the excitations from the $n=2$ orbitals of the reference state(s) into the virtual shells with $2\leqslant n\leqslant 6$ are considered. Having completed the CI calculations, the interaction with the orthogonal complement is evaluated by the PT taking into account all the contributions up to the third order. To extract the correction to the contributions calculated at the previous stage, we take the difference between the results obtained for the identical basis within the CI+PT3 SD and PT3 SD approaches. The corresponding corrections are labeled “CI+PT3 SD” in Table 2. The CI+PT3 SD term provides the correction to the PT energy which is induced by the most important configurations. After this, employing the identical configuration spaces and considering the difference of the values calculated within the CI and CI+PT3 approaches, we extract the CI contribution for the configurations which hold the highly-excited virtual orbitals. The latter correction is indicated as “CI SD” in Table 2. Then we gradually expand the configuration space of the initial problem by considering the triple (T) and quadruple (Q) excitations. Following the same procedure, we obtain the CI+PT3 T, CI T, CI+PT3 Q, and CI Q corrections, see Table 2. Finally, summing up all the contributions listed in Table 2 we obtain the total value of the DCB energy for a given state.

In our CI+PT calculations, namely the CI+PT3 SD, CI+PT3 T, and CI+PT3 Q, we limit the basis set with ( $L_{\mathrm{max}}=5$ , $N_{\rm max}=25$ ). Since the nonperturbative treatment of the $1s$ orbitals and the highly-excited virtual orbitals considerably increases the configuration space we restrict the basis sets parameters for CI T and CI Q calculations to ( $L_{\mathrm{max}}=3$ , $N_{\rm max}=25$ ) and ( $L_{\mathrm{max}}=2$ , $N_{\rm max}=25$ ), respectively. From Table 2 one can see that the triple and quadruple excitations are important only for the ground state and are almost negligible for the inner-L-shell excited states. The uncertainties of all the contributions are mainly determined by the estimation of the convergence of the results with respect to the maximum value of the orbital quantum number $L_{\mathrm{max}}$ for the DS orbitals.

In order to illustrate the dependence of the obtained DCB energies on the realization of the “no-pair” approximation (see the discussion after Eq. (2)) we have performed the calculations for Be-like neon and uranium starting from the different initial approximations. In addition to the LDF potential introduced in Eq. (7), we have evaluated the DCB energies for the core-Hartree potential induced by the $1s^{2}$ closed shell as well. The results for the DCB energies without and with the frequency-dependent Breit correction are presented in Table 3. One can see that for high- $Z$ ions the scatter of the results obtained for the different projection operators $\Lambda^{(+)}$ in Eq. (2) is higher that the numerical uncertainty. For low- $Z$ ions this is not the case and the results obtained for the different initial approximations lie within the estimated error bar. From Table 3, it is seen that the frequency-dependent Breit correction can not eliminate this discrepancy between the results. This issue can be solved only by considering the excitations into the negative-energy continuum within the rigorous QED approach. In the following, this ambiguity of the DCB results is kept under control and covered by our estimation of the higher-order QED electron-correlation contributions which is shown in the last column of Table 3.

In Table 4 we present the results for the DCB energies of the ground and inner-L-shell excited states of Be-like ions with the frequency-dependent Breit correction included (the DCB energies without this correction can be found in third column of Table 5). For the ground state the total energies are listed whereas for the excited states the energies relative to the ground state are given. The presented uncertainties are of purely numerical origin. Sometimes the behavior of the uncertainties is not smooth enough when nuclear charge $Z$ changes. As it was noted above, the error bar is determined mainly by the convergence of the PT, which is a little bit better for some $Z$ and states and a little bit worse for the others. We intentionally do not use any correction procedure and suppose that the uncertainty is reliable anyway. There exist many nonrelativistic and relativistic calculations of the correlation effects for berylliumlike ions in the literature Safronova et al. (1996); Chen and Cheng (1997); Safronova (2000); Majumder and Das (2000); Gu (2005); Ho et al. (2006); Cheng et al. (2008); Jian-Hui et al. (2008); Sampaio et al. (2013); Li et al. (2017). We compare our results with the most accurate ones: the CI calculations by Yerokhin et al. for Be-like argon Yerokhin et al. (2015) and iron Yerokhin et al. (2014), the CI calculations by Cheng et al. Cheng et al. (2008), the many-body PT calculations by Safronova et al. Safronova et al. (1996); Safronova (2000), and our previous ab initio QED evaluation of the ground-state energy Malyshev et al. (2014). We find our results to be in reasonable agreement with the previous ones, and have better numerical accuracy. Since the authors use different “no-pair” approximations, one can not expect to obtain the agreement better than the corresponding ambiguity.

Finally, in Table 5 we collect the DCB, frequency-dependent Breit, QEDMOD, and nuclear recoil contributions to the total energies of the ground and inner-L-shell excited states of Be-like ions. As in Table 4, for the ground state the total energies are given while for the excited states the energies relative to the ground state are shown. The DCB energies are presented with the numerical uncertainty (in the first brackets) as well as with the uncertainty due to the finite nuclear size effect (in the second brackets). The last one is conservatively estimated by adding quadratically two contributions. The first contribution is obtained by varying the root-mean-square nuclear radius within the error bars given in Ref. Angeli and Marinova (2013), and the second one is obtained by changing the model of the nuclear-charge distribution from the Fermi one to the homogeneously-charged-sphere model. In the case of uranium, we also take into account the nuclear deformation effect in accordance with the results of Ref. Kozhedub et al. (2008). The column labeled “Breit( $\omega)$ ” represents the frequency-dependent Breit contribution. The next two columns contain the QEDMOD and nuclear recoil contributions, respectively. The final uncertainty is mainly determined by the omitted higher-order QED electron-correlation contributions, see the discussion in subsection II.2. Its estimation is included in the uncertainty of the “Total” values in addition to the previously discussed ones. The obtained results are in reasonable agreement with the previous calculations, the data from the NIST Atomic Spectra Database Kramida et al. (2018), and the available experimental values Edlén (1983); Bernhardt et al. (2015). We consider our calculations as the best ones available, and our conservative estimation of the uncertainty can only be reduced by rigorous evaluation of the higher-order QED contributions.

IV Conclusion

To summarize, we have performed the systematic relativistic calculations of the ground and inner-L-shell excited energy levels of berylliumlike ions from neon to uranium. The main feature of the work is the high-precision evaluation of the Dirac-Coulomb-Breit contribution based on the large-scale relativistic configuration-interaction method combined with the perturbation theory and are supplemented with the systematic estimation of the uncertainties. In order to estimate the uncertainties of the CI energies, we have thoroughly analyzed the successive increments of the results obtained with the sets of configuration-state functions increased in all possible directions. The DCB energies are combined with the separately computed corrections: frequency-dependent Breit, QED (using the model QED operator), and nuclear recoil (using the relativistic mass shift operator). All our theoretical predictions are given with the uncertainties that include the estimation of the effects omitted in the present theoretical treatment. For the energy of the ground state, our results agree well with the benchmark theoretical data available in the literature Malyshev et al. (2014). For the transition energies, our theoretical approach provides the most precise values for berylliumlike ions in the range $Z=10$ – $92$ . For Be-like argon and iron our estimation of the omitted QED contributions is performed in a more conservative way than in Refs. Yerokhin et al. (2014, 2015). The total accuracy for all $Z$ is determined by the uncalculated contributions of the higher-order QED effects and can be improved as soon as these corrections are calculated.

V Acknowledgment

We thank Vladimir Yerokhin for valuable discussions. This work was supported by RFBR (Grants No. 16-02-00334, No. 18-03-01220), SPbSU-DFG (Grants No. 11.65.41.2017 and No. STO 346/5-1) and by Ministry of Education and Science of the Russian Federation (Grant No. 3.1463.2017/4.6). Y.S.K. and A.V.M. acknowledge the financial support from the FAIR-Russia Research Center. The work of Y.S.K. and D.A.G. was also supported by SPbSU (COLLAB 2018 id 34827923 and id 34824940). M.Y.K., A.V.M., and V.M.S. acknowledge the financial support from the Foundation for the advancement of theoretical physics and mathematics “BASIS”. The research was carried out using the computational resources provided by the St. Petersburg State University Computing Center.

Bibliography49

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1Safronova et al. (2018) M. S. Safronova, D. Budker, D. De Mille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, Rev. Mod. Phys. 90 , 025008 (2018).
2Kozlov et al. (2018) M. G. Kozlov, M. S. Safronova, J. R. Crespo López-Urrutia, and P. O. Schmidt, Rev. Mod. Phys. 90 , 045005 (2018).
3Shabaev et al. (2018) V. M. Shabaev, A. I. Bondarev, D. A. Glazov, M. Y. Kaygorodov, Y. S. Kozhedub, I. A. Maltsev, A. V. Malyshev, R. V. Popov, I. I. Tupitsyn, and N. A. Zubova, Hyperfine Interact. 239 , 60 (2018).
4Persson et al. (1996) H. Persson, S. Salomonson, P. Sunnergren, and I. Lindgren, Phys. Rev. Lett. 76 , 204 (1996).
5Yerokhin et al. (1997) V. A. Yerokhin, A. N. Artemyev, and V. M. Shabaev, Phys. Lett. A 234 , 361 (1997).
6Artemyev et al. (2005) A. N. Artemyev, V. M. Shabaev, V. A. Yerokhin, G. Plunien, and G. Soff, Phys. Rev. A 71 , 062104 (2005).
7Malyshev et al. (2019) A. V. Malyshev, Y. S. Kozhedub, D. A. Glazov, I. I. Tupitsyn, and V. M. Shabaev, Phys. Rev. A (2019), in press.
8Yerokhin et al. (2001) V. A. Yerokhin, A. N. Artemyev, V. M. Shabaev, M. M. Sysak, O. M. Zherebtsov, and G. Soff, Phys. Rev. A 64 , 032109 (2001).