Hyperfine interaction constants and Landé <i>g</i> factors of clock states of Al-like ions

Wang Xia; Jia Fang-Shi; Yao Ke; Yan Jun; Li Ji-Guang; Wu Yong; Wang Jian-Guo

doi:10.7498/aps.72.20230940

Abstract

The highly charged Al-like ions are the potential candidates for the next-generation atomic optical clocks, and their atomic parameters are also useful in plasma and nuclear physics. In the present work, the hyperfine interaction constants and Landé g factors of 3s²3p ²P_{1/2, 3/2} states in the ground configuration for Al-like ions in a range between Si⁺ and Kr²³⁺ ions are calculated by using the multi-configuration Dirac-Hartree-Fock method. Owing to the fact that hyperfine interaction constant is sensitive to electron correlation effects, we systematically investigate its influence on the hyperfine interaction constants, particularly for the high-order correlation related to the 2p electrons. According to this investigation and by taking into account the Breit interaction and QED corrections, we achieve the computational accuracy at a level of 1% and 10^–5 for the hyperfine interaction constants and Landé g factors, respectively, except for the Si⁺ ion. Furthermore, the electronic parts of hyperfine interaction constants and g factors are fitted with functions of atomic number. The deviations between these fitted formulas and the ab initio calculations are less than 2% and 10^–5 for the hyperfine interaction constants and the g factors, respectively. As a result, the hyperfine interaction constants and g factors of all isotopes can be determined for Al-like ions with 14 ≤ Z ≤ 54.

Keywords:

hyperfine structure constant /
Landé g factor /
multi-configuration Dirac-Hartree-Fock method /
electron correlation effect /
Al-like ions

Authors and contacts

1.
National Key Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
2.
Key Laboratory of Nuclear Physics and Ion-beam Application, Fudan University, Shanghai 200433, China
3.
Center for Applied Physics and Technology, College of Engineering, Peking University, Beijing 100871, China

Corresponding author: Li Ji-Guang, li_jiguang@iapcm.ac.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11874090).

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References

[1]	Safronova M S, Budker D, DeMille D, Kimball D F J, Derevianko A, Clark C W 2018 Rev. Mod. Phys. 90 025008 Google Scholar
[2]	Zhang B L, Huang Y, Zhang H Q, Hao Y M, Zeng M Y, Guan H, Gao K L 2020 Chin. Phys. B 29 074209 Google Scholar
[3]	Brewer S M, Chen J S, Hankin A M, Clements E R, Chou C W, Wineland D J, Hume D B, Leibrandt D R 2019 Phys. Rev. Lett. 123 033201 Google Scholar
[4]	Kozlov M G, Safronova M S, López-Urrutia J R C, Schmidt P O 2018 Rev. Mod. Phys. 90 045005 Google Scholar
[5]	Derevianko A, Dzuba V A, Flambaum V V 2012 Phys. Rev. Lett. 109 180801 Google Scholar
[6]	Dzuba V A, Derevianko A, Flambaum V V 2012 Phys. Rev. A 86 054501 Google Scholar
[7]	Dzuba V A, Flambaum V V 2017 Highly Charged Ions for Atomic Clocks and Search for Variation of the Fine Structure Constant (In: Wada M, Schury P, Ichikawa Y (eds) TCP 2014 Springer, Cham.) p79
[8]	Yudin V I, Taichenachev A V, Derevianko A 2014 Phys. Rev. Lett. 113 233003 Google Scholar
[9]	Yu Y, Sahoo B K 2016 Phys. Rev. A 94 062502 Google Scholar
[10]	Li J G, Godefroid M, Wang J G 2016 J. Phys. B: At. Mol. Opt. Phys. 49 115002 Google Scholar
[11]	Lu B, Zhang T X, Chang H, Li J G, Wu Y, Wang J G 2019 Phys. Rev. A 100 012504 Google Scholar
[12]	Goyal A, Khatri I, Singh A K, Mohan M, Sharma R, Singh N 2016 Atoms 4 22 Google Scholar
[13]	Beiersdorfer P, Träbert E, Pinnington E H 2003 Astrophys. J. 587 836 Google Scholar
[14]	Schiffmann S, Brage T, Judge P G, Paraschiv A R, Wang K 2021 Astrophys. J 923 186 Google Scholar
[15]	Booth A J, Blackwell D E 1983 Mon. Not. R. Astro. Soc. 204 777 Google Scholar
[16]	Roederer I U, Lawler J E 2021 Astrophys. J. 912 119 Google Scholar
[17]	Grant I P 2007 Relativistic Quantum Theory of Atoms and Molecules-Theory and Computation (New York: Springer) p393
[18]	Dyall K G, Grant I P, Johnson C T, Parpia F A, Plummer E P 1989 Comput. Phys. Commun. 55 425 Google Scholar
[19]	Jönsson P, Parpia F A, Fischer C F 1996 Comput. Phys. Commun. 96 301 Google Scholar
[20]	Cheng K T, Childs W J 1985 Phys. Rev. A 31 2775 Google Scholar
[21]	Froese C F, Gaigalas G, Jönsson P, Bierón J 2019 Comput. Phys. Commun. 237 184 Google Scholar
[22]	Zhang T X, Xie L Y, Li J G, Lu Z H 2017 Phys. Rev. A 96 012514 Google Scholar
[23]	Li J G, Jönsson P, Godefroid M, Dong C Z, Gaigalas G 2012 Phys. Rev. A 86 052523 Google Scholar
[24]	Kramida A, Ralchenko Yu, Reader J, NIST ASD Team 2022 NIST Atomic Spectra Database
[25]	Stone N J 2005 At. Data Nucl. Data Tables 90 75 Google Scholar
[26]	Pyykkö P 2008 Mol. Phys. 106 1965 Google Scholar

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图 1 类铝等电子序列3s²3p ²P_{1/2, 3/2}能级的(a)超精细结构常数电子部分矩阵元A_el和B_el以及(b)朗德g因子随原子序数Z的变化关系. 图中实线表示由拟合公式得出的结果, 散点表示用MCDHF方法从头计算的结果

Figure 1. (a) Electronic parts of hyperfine structure constants and (b) Landé g factors of 3s²3p ²P_{1/2, 3/2} states of Al-like isoelectronic sequence ions as functions of atomic number. The solid line represents these results obtained by from numerical fitting formula, and the discrete point represents these results obtained by our ab initio calculation using MCDHF method.

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表 1 Si⁺和Co¹⁴⁺离子3s²3p ²P_{1/2, 3/2}能级的激发能ΔE (cm^–1)随组态空间扩大的收敛趋势. DHF代表单组态计算模型. AO和VO分别代表在每个计算模型下允许被激发的占据轨道和新添加的关联轨道. N_CSF代表相应的组态波函数数目

Table 1. Excitation energies ΔE (in cm^–1) of 3s²3p ²P_{1/2, 3/2} states of Si⁺ and Co¹⁴⁺ ions as functions of various computational models. DHF stands for the single configuration approximation model. AO and VO represent the occupied orbitals allowed to be replaced and the added virtual orbitals in each computational model, respectively. N_CSF represents the corresponding numbers of CSFs.

Model		AO	VO	N_CSF	ΔE
Model		AO	VO	N_CSF	Si⁺	Co¹⁴⁺
DHF		—	—	2	305	23957
VV + CV	1	{3s, 3p}	{4s, 4p, 3d, 4f, 5g}	204	293	23471
	2	{2s, 2p, 3s, 3p}	{5s, 5p, 4d, 5f, 6g}	8577	305	23588
	3	{1s, 2s, 2p, 3s, 3p}	{6s, 6p, 5d, 6f, 6g}	20491	304	23598
	4	{1s, 2s, 2p, 3s, 3p}	{7s, 7p, 6d, 7f, 6g}	33508	304	23605
	5	{1s, 2s, 2p, 3s, 3p}	{8s, 8p, 7d, 7f, 6g}	44508	305	23609
	6	{1s, 2s, 2p, 3s, 3p}	{9s, 9p, 8d, 7f, 6g}	57218	305	23610
	7	{1s, 2s, 2p, 3s, 3p}	{10s, 10p, 9d, 7f, 6g}	71638	305	23611
	8	{1s, 2s, 2p, 3s, 3p}	{11s, 11p, 10d, 7f, 6g}	87768	305	23611
	9	{1s, 2s, 2p, 3s, 3p}	{12s, 12p, 11d, 7f, 6g}	105608	305	23611
	10	{1s, 2s, 2p, 3s, 3p}	{13s, 13p, 12d, 7f, 6g}	125158	305	23611

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表 2 Si⁺和Co¹⁴⁺离子3s²3p ²P_{1/2, 3/2}能级超精细结构常数电子部分矩阵元A_el (MHz/μ_N)和B_el (MHz/b)以及朗德g因子随组态空间扩展的收敛情况. DHF为单组态近似模型

Table 2. Electronic parts of hyperfine structure constants A_el(MHz/μ_N) and B_el(MHz/b) and Landé g factors of 3s²3p ²P_{1/2, 3/2} states in Si⁺ and Co¹⁴⁺ ions as functions of various computational models. DHF stands for the single configuration approximation model.

Model		Si⁺					Co¹⁴⁺
		A_el		B_el	g		A_el		B_el	g
		²P_1/2	²P_3/2	²P_3/2	²P_1/2	²P_3/2	²P_1/2	²P_3/2	²P_3/2	²P_1/2	²P_3/2
DHF		682.2	134.4	248.7	0.6658188	1.3340409	28653	5391	10060	0.6645201	1.3329968
VV + CV	1	647.7	154.5	237.1	0.6658181	1.3340408	27658	5879	9804	0.6645062	1.3330011
	2	741.9	175.0	289.1	0.6658123	1.3340361	28843	6071	10282	0.6644682	1.3329818
	3	783.5	172.2	296.2	0.6658100	1.3340348	29191	5814	10276	0.6644659	1.3329820
	4	799.1	160.4	298.3	0.6658098	1.3340346	29287	5766	10290	0.6644616	1.3329826
	5	799.8	163.2	299.2	0.6658099	1.3340344	29307	5799	10273	0.6644639	1.3329808
	6	800.9	163.4	297.3	0.6658093	1.3340345	29319	5788	10255	0.6644639	1.3329812
	7	801.1	163.9	297.7	0.6658094	1.3340345	29321	5797	10255	0.6644622	1.3329818
	8	802.7	163.7	295.6	0.6658094	1.3340343	29326	5796	10239	0.6644634	1.3329815
	9	802.8	164.0	295.8	0.6658092	1.3340344	29326	5799	10244	0.6644630	1.3329810
	10	802.6	163.9	295.5	0.6658088	1.3340346	29321	5796	10245	0.6644622	1.3329812

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表 3 不同计算模型下Si⁺与Co¹⁴⁺离子3s²3p ²P_{1/2, 3/2}能级的激发能ΔE (cm^–1)、超精细结构常数电子部分矩阵元A_el (MHz/μ_N)和B_el (MHz/b)以及朗德g因子

Table 3. Excitation energies ΔE (cm^–1), electronic parts of hyperfine structure constants A_el (MHz/μ_N) and B_el (MHz/b) and Landé g factors of 3s²3p ²P_{1/2, 3/2} states in Si⁺ and Co¹⁴⁺ ions as functions of various computational models.

Model	Si⁺						Co¹⁴⁺
	ΔE	A_el		B_el	g		ΔE	A_el		B_el	g
	²P_1/2→ ²P_3/2	²P_1/2	²P_3/2	²P_3/2	²P_1/2	²P_3/2	²P_1/2→ ²P_3/2	²P_1/2	²P_3/2	²P_3/2	²P_1/2	²P_3/2
DHF	305	682	134	249	0.665819	1.334041	23957	28653	5391	10060	0.664520	1.332997
VV+CV-10	305	803	164	295	0.665809	1.334035	23611	29321	5796	10245	0.664462	1.332981
+MR₁	305	803	164	296	0.665809	1.334035	23611	29319	5797	10245	0.664462	1.332981
+CC_2p	310	785	150	288	0.665811	1.334036	23656	29287	5757	10232	0.664465	1.332983
+TQ_2p	307	785	160	292	0.665810	1.334036	23618	29245	5784	10229	0.664464	1.332983
+BQ	291	785	160	292	0.665810	1.334036	23044	29230	5788	10211	0.664464	1.332983
CCSD^[9]	—	—	—	—	—	—	22932(13)	—	—	—	—	—
NIST^[24]	287	—	—	—	—	—	22979	—	—	—	—	—

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表 4 类铝等电子序列3s²3p ²P_{1/2, 3/2}能级的超精细结构常数电子部分矩阵元A_el(MHz/μ_N), B_el (MHz/b)和朗德g因子. 括号内的数字表示计算结果相应的不确定度

Table 4. Electronic parts of hyperfine structure constants A_el (MHz/μ_N) and B_el (MHz/b) and Landé g factors of 3s²3p ²P_{1/2, 3/2} states of Al-like isoelectronic sequence ions. Numbers in parentheses represent the computational errors.

Ions	A_el		B_el	g
Ions	²P_1/2	²P_3/2	²P_3/2	²P_1/2	²P_3/2
Si⁺	785(8)	160(8)	292(3)	0.66581	1.33404
P²⁺	1387(14)	290(3)	512(5)	0.66576	1.33399
S³⁺	2172(22)	457(5)	797(8)	0.66570	1.33394
Cl⁴⁺	3160(32)	664(7)	1152(12)	0.66562	1.33389
Ar⁵⁺	4372(44)	914(9)	1585(16)	0.66555	1.33383
K⁶⁺	5828(58)	1212(12)	2104(21)	0.66546	1.33376
Ca⁷⁺	7553(76)	1561(16)	2714(27)	0.66536	1.33368
Sc⁸⁺	9569(96)	1965(20)	3425(34)	0.66526	1.33360
Ti⁹⁺	11902(119)	2430(24)	4243(42)	0.66515	1.33352
V¹⁰⁺	14576(146)	2957(30)	5175(52)	0.66503	1.33342
Cr¹¹⁺	17618(176)	3553(36)	6230(62)	0.66490	1.33332
Mn¹²⁺	21055(211)	4220(42)	7416(74)	0.66476	1.33322
Fe¹³⁺	24916(249)	4964(50)	8740(87)	0.66462	1.33310
Co¹⁴⁺	29230(292)	5788(58)	10211(102)	0.66446	1.33298
Ni¹⁵⁺	34028(340)	6696(67)	11836(118)	0.66430	1.33286
Cu¹⁶⁺	39344(393)	7693(77)	13625(136)	0.66413	1.33273
Zn¹⁷⁺	45209(452)	8783(88)	15587(156)	0.66395	1.33259
Ga¹⁸⁺	51659(517)	9970(100)	17729(177)	0.66377	1.33244
Ge¹⁹⁺	58734(587)	11259(113)	20059(201)	0.66357	1.33229
As²⁰⁺	66469(665)	12654(127)	22588(226)	0.66337	1.33214
Se²¹⁺	74908(749)	14160(142)	25326(253)	0.66316	1.33197
Br²²⁺	84092(841)	15779(158)	28281(283)	0.66294	1.33180
Kr²³⁺	94066(941)	17519(175)	31462(315)	0.66271	1.33162

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表 5 类铝等电子序列3s²3p ²P_{1/2, 3/2}能级的超精细结构常数A, B (MHz)和朗德g因子. 所有核参数μ (μ_N)和Q (mb)均来自于文献[25, 26]. 星号表示用CCSD方法计算的结果^[9]. 括号内的数字表示计算结果相应的不确定度

Table 5. Hyperfine structure constants and g factors of 3s²3p ²P_{1/2, 3/2} states of Al-like isoelectronic sequence ions. Nuclear parameters μ (μ_N) and Q (mb) are taken from Ref. [25, 26]. Asterisk represents these results calculated by CCSD method^[9]. Numbers in parentheses represent the computational uncertainties.

Ion	I	μ	Q	A		B	g
Ion	I	μ	Q	²P_1/2	²P_3/2	²P_3/2	²P_1/2	²P_3/2
⁵¹V¹⁰⁺	7/2	5.1464	–52	21433(214)	4349(43)	–269(3)	0.66503	1.33342
				21456(146)^*	4342(68)^*	–222(6)^*	0.665196^*	1.333460^*
⁵³Cr¹¹⁺	3/2	–0.4743	–150	–5571(56)	–1123(11)	–935(9)	0.66490	1.33332
				–5578(30)^*	–1122(14)^*	–964(10)^*	0.665081^*	1.333363^*
⁵⁵Mn¹²⁺	5/2	3.4669	330	29198(292)	5853(59)	2447(24)	0.66476	1.33322
				29096(3)^*	5821(35)^*	3162(20)^*	0.664957^*	1.333258^*
⁵⁷Fe¹³⁺	1/2	0.09064	160	4517(45)	900(9)	1398(14)	0.66462	1.33310
				4509(39)^*	897(2)^*	961(10)^*	0.664825^*	1.333148^*
⁵⁹Co¹⁴⁺	7/2	4.615	420	38542(39)	7632(76)	4289(43)	0.66446	1.33298
				5245(42)^*	1037(47)^*	3603(40)^*	0.664684^*	1.333032^*
⁶¹Ni¹⁵⁺	3/2	–0.7497	162	–17006(170)	–3348(33)	1917(19)	0.66430	1.33286
				–17016(66)^*	–3345(27)^*	1918(20)^*	0.664536^*	1.332909^*
⁶³Cu¹⁶⁺	3/2	2.2259	–220	58383(584)	11416(114)	–2998(30)	0.66413	1.33273
				58412(254)^*	11416(511)^*	–3012(60)^*	0.664379^*	1.332779^*

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[1]	Safronova M S, Budker D, DeMille D, Kimball D F J, Derevianko A, Clark C W 2018 Rev. Mod. Phys. 90 025008 Google Scholar
[2]	Zhang B L, Huang Y, Zhang H Q, Hao Y M, Zeng M Y, Guan H, Gao K L 2020 Chin. Phys. B 29 074209 Google Scholar
[3]	Brewer S M, Chen J S, Hankin A M, Clements E R, Chou C W, Wineland D J, Hume D B, Leibrandt D R 2019 Phys. Rev. Lett. 123 033201 Google Scholar
[4]	Kozlov M G, Safronova M S, López-Urrutia J R C, Schmidt P O 2018 Rev. Mod. Phys. 90 045005 Google Scholar
[5]	Derevianko A, Dzuba V A, Flambaum V V 2012 Phys. Rev. Lett. 109 180801 Google Scholar
[6]	Dzuba V A, Derevianko A, Flambaum V V 2012 Phys. Rev. A 86 054501 Google Scholar
[7]	Dzuba V A, Flambaum V V 2017 Highly Charged Ions for Atomic Clocks and Search for Variation of the Fine Structure Constant (In: Wada M, Schury P, Ichikawa Y (eds) TCP 2014 Springer, Cham.) p79
[8]	Yudin V I, Taichenachev A V, Derevianko A 2014 Phys. Rev. Lett. 113 233003 Google Scholar
[9]	Yu Y, Sahoo B K 2016 Phys. Rev. A 94 062502 Google Scholar
[10]	Li J G, Godefroid M, Wang J G 2016 J. Phys. B: At. Mol. Opt. Phys. 49 115002 Google Scholar
[11]	Lu B, Zhang T X, Chang H, Li J G, Wu Y, Wang J G 2019 Phys. Rev. A 100 012504 Google Scholar
[12]	Goyal A, Khatri I, Singh A K, Mohan M, Sharma R, Singh N 2016 Atoms 4 22 Google Scholar
[13]	Beiersdorfer P, Träbert E, Pinnington E H 2003 Astrophys. J. 587 836 Google Scholar
[14]	Schiffmann S, Brage T, Judge P G, Paraschiv A R, Wang K 2021 Astrophys. J 923 186 Google Scholar
[15]	Booth A J, Blackwell D E 1983 Mon. Not. R. Astro. Soc. 204 777 Google Scholar
[16]	Roederer I U, Lawler J E 2021 Astrophys. J. 912 119 Google Scholar
[17]	Grant I P 2007 Relativistic Quantum Theory of Atoms and Molecules-Theory and Computation (New York: Springer) p393
[18]	Dyall K G, Grant I P, Johnson C T, Parpia F A, Plummer E P 1989 Comput. Phys. Commun. 55 425 Google Scholar
[19]	Jönsson P, Parpia F A, Fischer C F 1996 Comput. Phys. Commun. 96 301 Google Scholar
[20]	Cheng K T, Childs W J 1985 Phys. Rev. A 31 2775 Google Scholar
[21]	Froese C F, Gaigalas G, Jönsson P, Bierón J 2019 Comput. Phys. Commun. 237 184 Google Scholar
[22]	Zhang T X, Xie L Y, Li J G, Lu Z H 2017 Phys. Rev. A 96 012514 Google Scholar
[23]	Li J G, Jönsson P, Godefroid M, Dong C Z, Gaigalas G 2012 Phys. Rev. A 86 052523 Google Scholar
[24]	Kramida A, Ralchenko Yu, Reader J, NIST ASD Team 2022 NIST Atomic Spectra Database
[25]	Stone N J 2005 At. Data Nucl. Data Tables 90 75 Google Scholar
[26]	Pyykkö P 2008 Mol. Phys. 106 1965 Google Scholar

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Vol. 72, Issue 22 - 2023-11-20

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Hyperfine interaction constants and Landé g factors of clock states of Al-like ions

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Hyperfine interaction constants and Landé g factors of clock states of Al-like ions

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Corresponding author: Li Ji-Guang, li_jiguang@iapcm.ac.cn

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