<i>Ab initio</i> calculation of hyperfine-structure constant <i>A</i> of Fr and evaluation of magnetic dipole moments of Fr isotopes

Lou Bing-Qiong; Li Fang; Wang Pei-Yan; Wang Li-Ming; Tang Yong-Bo

doi:10.7498/aps.68.20190113

Abstract

As the heaviest atom in alkali-metal elements, Fr atom has been regarded as a candidate for the search of the permanent electric dipole moment of the electron and of parity-nonconservation effects. Accurate knowledge of Fr atomic properties is of great interest. In this work, we use a relativistic coupled-cluster method to calculate the magnetic dipole hyperfine structure constants for nS (n = 7－12), nP (n = 7－12) and nD (n = 6－11) states of ²¹²Fr. A finite B-spline basis set is used to expand the Dirac radial function, including completely the single and double excitation in correlation calculation. Our results are compared with available theoretical and experimental values. The comparison shows that our method can offer accurate calculation of magnetic dipole hyperfine structure constant. For 7P state the differences between our results and experimental values are within 1%. The magnetic dipole hyperfine structure constants for 12S, nP (n = 9－12) and nD (n = 10－11) states are reported for the first time, which are very useful as benchmarks for experimental measurements and calculations by other theoretical methods of these quantities. In the relativistic coupled-cluster theoretical framework, we study the electron correlation effect on hyperfine-structure constant A for the S, P, and D states of Fr. We observe that the electron correlation effect is very important for hyperfine-structure constant properties. The D state has a considerable correlation effect. At the same time, we also investigate contribution trends of individual electron correlation effects involving direct, core-polarization and pair-correlation ones in S, P, and D Rydberg series. It is found that the dominant contributions for the S_1/2, P_1/2,3/2 and nD_3/2 (n = 7－11) states are to from the direct effect; however, the dominant contributions for the 6D_3/2, and nD_5/2 (n = 6－11) states are due to the pair-correlation and the core-polarization, respectively. For D_5/2 states, there is very strong cancellation among these individual correlation effects. The knowledge of these correlation trends is useful for studying the permanent electric dipole moment and parity-nonconservation effect of Fr in future. Moreover, the magnetic dipole moment $ {\mu}$ for each of isotopes ^{207−213,220−228}Fr is determined by combining with experimental values for magnetic dipole hyperfine structure constant of 7P state. For each of isotope ^207−213Fr, our magnetic dipole moment $ {\mu}$ is perfectly consistent with the experimental value, and our uncertainties are twice smaller than those in the experiments . For each of isotope ^220−228Fr, our magnetic dipole moment $ {\mu}$ has a larger uncertainty, but is still in agreement with the experimental magnetic dipole moment $ {\mu}$.

Keywords:

Authors and contacts

1.
College of Physics and Materials Science, Henan Normal University, Xinxiang 453000, China
2.
Faculty of Arts and Sciences, Shenzhen Technology University, Shenzhen 518118, China

Corresponding author: Tang Yong-Bo, ybtang@whu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 1154094, 11774080).

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References

[1]	Grant I P 2007 Relativistic Quantum Theory of Atoms and Molecules (New York: Springer) pp533−577
[2]	Fischer C F, Brage T, Jönsson P 1997 Computational Atomic Structure: An MCHF Approach (UK: Institute of Physics) pp1−67
[3]	Jönsson P, Gaigalas G, Bieroń J, Fishcher C F, Grant I 2013 Computer Physics Communications. 184 2197 Google Scholar
[4]	Jönsson P, He X, Fishcher C F, Grant I 2007 Computer Physics Communications. 177 597 Google Scholar
[5]	Dzuba V A, Flambaum V V, Kozlov M G 1996 Phys. Rev. A 54 3948 Google Scholar
[6]	Dzuba V A, Johnson W R 1998 Phys. Rev. A 57 2459 Google Scholar
[7]	Angstmann E J, Dzuba V A, Flambaum V V 2004 Phys. Rev. A 70 014102 Google Scholar
[8]	Dinh T H, Dzuba V A, Flambaum V V, Ginges J S M 2008 Phys. Rev. A 78 054501 Google Scholar
[9]	Kozlov M G, Porsev S G, Johnson W R 2001 Phys. Rev. A 64 052107 Google Scholar
[10]	Pal R, Safronova M S, Johnson W R, Derevianko A, Porsev S G 2007 Phys. Rev. A 75 042515 Google Scholar
[11]	Blundell S A, Johnson W R, Liu Z W, Sapirstein 1989 Phys. Rev. A 40 2233 Google Scholar
[12]	Eliav E, Vikas M J, Ishikawa Y, Kaldor U 2005 Chem. Phys. 311 163 Google Scholar
[13]	Mani B K, Angom D 2011 Phys. Rev. A 83 012501 Google Scholar
[14]	Kallay M, Nataraj H S, Sahoo B K, Das B P, Visscher L 2011 Phys. Rev. A 83 030503 Google Scholar
[15]	Nandy D K, Singh Y, Sahoo B K 2014 Phys. Rev. A 89 062509 Google Scholar
[16]	Borschevsky A, Eliav E, Vilkas M J, Ishikawa Y, Kaldor U 2007 Phys. Rev. A 75 042514 Google Scholar
[17]	Eliav E, Kaldor U, Ishikawa Y 1996 Phys. Rev. A 53 3050 Google Scholar
[18]	Chaudhuri R K, Chattopadhyay S, Mahapatra U S 2013 J. Phys. Chem. A 117 12616 Google Scholar
[19]	Tang Y B, Lou B Q, Shi T Y 2017 Phys. Rev. A 96 022513 Google Scholar
[20]	Tang Y B, Gao N N, Lou B Q, Shi T Y 2018 Phys. Rev. A 98 062511 Google Scholar
[21]	Byrnes T M R, Dzuba V A, Flambaum F F, Murray D W 1999 Phys. Rev. A 59 3082 Google Scholar
[22]	Mukherjee D, Sahoo B K, Nataraj H S, Das B P 2009 J. Phys. Chem. A 113 12549 Google Scholar
[23]	Sakemi Y, Harada K, Hayamizu T, Itoh M, Kawamura H, Liu S, Nataraj H S, Oikawa A, Saito M, Sato T 2011 J. Phys. Conf. Ser. 302 012051 Google Scholar
[24]	Sahoo B K, Aoki T, Das B P, Sakemi Y 2016 Phys. Rev. A 93 032520 Google Scholar
[25]	Atutov S N, Calabrese R, Corradi L, Dainelli A, Mauro C D, Khanbekyan A, Mariotti E, Minguzzi P, Moi L, Sanguinetti S, Stancari G, Tomassetti L 2008 Proc. SPIE 7027 70270C
[26]	Ekström C, Ingelman S, Wannberg G, Skarestad M 1978 Physica Scripta 18 51 Google Scholar
[27]	Coc A, Thibault C, Touchard F, Duong H T, Juncar P, Liberman S, Pinard J, Lermé J, Vialle J L, Büttgenbach S, Mueller A C, Pesnelle A, the ISOLDE Collaboration 1985 Phys. Lett. B 163 66 Google Scholar
[28]	Coc A, Thibault C, Touchard F, Duong H T, Juncar P, Liberman S, Pinard J, Carre M, Lermé J, Vialle J L, Büttgenbach S, Mueller A C, Pesnelle A, the ISOLDE Collaboration 1987 Nucl. Phys. A 468 1 Google Scholar
[29]	Arnold E, Borchers W, Duong H T, Juncar P, Lermé J, Lievens P, Neu W, Neugart R, Pellerin M, Pinard J, Vialle J L, Wendt K, the ISOLDE Collaboration 1990 J. Phys. B 23 3511 Google Scholar
[30]	Arnold E, Borchers W, Carré M, Duong H T, Juncar P, Lermé J, Liberman S, Neu W, Neugart R, Otten W, Pellerin M, Pinard J, Pesnelle A, Vialle J L, Wendt K, the ISOLDE Collaboration 1989 J. Phys. B 22 L391 Google Scholar
[31]	Bauche J, Duong H T, Juncar P, Liberman S, Pinard J, Coc A, Thibault C, Touchard F, Lermé J, Vialle J L, Büttgenbach S, Mueller A C, Pesnelle A, the ISOLDE Collaboration 1986 J. Phys. B 19 L593 Google Scholar
[32]	Grossman J S, Orozco L A, Simsarian J E, Sprouse G D, Zhao W Z 1999 Phys. Rev. Lett. 83 935 Google Scholar
[33]	Sansonetti J E 2007 J. Phys. Chem. Ref. Data 36 497 Google Scholar
[34]	Gomez E, Aubin S, Orozco L A, Sprouse G D, Iskrenova-Tchoukova E, Safronova M S 2008 Phys. Rev. Lett. 100 172502 Google Scholar
[35]	Dzuba V A, Flambaum V V, Sushkov O P 1984 J. Phys. B: At. Mol. Phys. 17 1953 Google Scholar
[36]	Owusu A, Dougherty R W, Gowri G, Das T P 1997 Phys. Rev. A 56 305 Google Scholar
[37]	Safronova M S, Johnson W R, Derevianko A 1999 Phys. Rev. A 60 4476 Google Scholar
[38]	Sahoo B K, Nandy D K, Das B P, Sakemi Y 2015 Phys. Rev. A 91 042507 Google Scholar
[39]	Duong H T, Juncar P, Liberman S, Mueller A C, Neugart R, Otten E W, Peuse B, Pinard J, Stoke H H, Thibault C, Touchard F, Vialle J L, Wendt K, the ISOLDE Collaboration 1987 Europhys. Lett. 3 175 Google Scholar
[40]	Barber Z W, Stalnaker J E, Lemke N D, Poli N, Oates C W, Fortier T M, Diddams S A, Hollberg L, Hoyt C W, Taichenachev A V, Yudin V I 2008 Phys. Rev. Lett. 100 103002 Google Scholar
[41]	Kien F L, Balykin V I, Hakuta K 2005 J. Phys. Soc. Jpn. 74 910 Google Scholar
[42]	Ingvar L 1978 Int. J. Quantum Chem. 12 33
[43]	Sinha D, Mukhopadhyay S, Mukherjee D 1986 Chem. Phys. Lett. 129 369 Google Scholar
[44]	Blundell S A, Johnson W R, Sapiratein J 1991 Phys. Rev. A 43 3407 Google Scholar
[45]	Porsev S G, Beloy K, Derevianko A 2010 Phys. Rev. D 82 036008 Google Scholar
[46]	Sahoo B K, Sur C, Beier T, Das B P, Chaudhuri R K, Mukherjee D 2007 Phys. Rev. A 75 042504 Google Scholar
[47]	Safronova M S, Safronova U I 2011 Phys. Rev. A 83 052508 Google Scholar

Cited By

图 1 ²¹²Fr原子S_1/2, P_1/2, P_3/2, D_3/2和D_5/2态磁偶极超精细结构常数中的电子关联效应

Figure 1. Electron correlation effects in hyperfine-structure constant A for S_1/2, P_1/2, P_3/2, D_3/2 and D_5/2 states of ²¹²Fr.

DownLoad: Full-Size Img PowerPoint

图 2 直接效应A_DF、核极化效应A_CP、对关联效应A_PC, 以及相对于CCSD的3种效应的总和A_T = A_DF + A_CP + A_PC, 针对主量子数n的S, P和D态的结果A的比率　(a) A_DF/A; (b) A_CP/A; (c) A_PC/A; (d) A_T/A

Figure 2. Ratios of direct effect A_DF, core polarization effect A_CP, pair correlation effect A_PC, and the total of the three effects A_T = A_DF + A_PC + A_CP to the CCSD, results A for S, P and D states against the principal quantum number n: (a) A_DF/A; (b) A_CP/A; (c) A_PC/A; (d) A_T/A.

DownLoad: Full-Size Img PowerPoint

表 1 ²¹²Fr原子S态的超精细结构常数A (单位: MHz)

Table 1. Hyperfine-structure constant A for the S states of ²¹²Fr in MHz.

能级	${A_{{\rm{DF}}}}$	${A_{{\rm{CCSD}}}}$	$\varDelta /$%	Ref.[38]	实验值
7S_1/2	6001.76	9403.56	30.93	9124(94)	9064.2(2)^[27]
					9064.4(1.5)^[39]
8S_1/2	1538.03	2014.10	17.37	1986(19)
9S_1/2	631.98	792.19	13.66	784(9)
10S_1/2	321.24	396.86	11.95	419(9)	401(5)^[29]
11S_1/2	185.41	225.71	11.10	212(9)	225(3)^[29]
12S_1/2	116.42	141.25	10.80

DownLoad: CSV

表 2 ²¹²Fr原子P态的超精细结构常数A (单位: MHz)

Table 2. Hyperfine-structure constant A for the P states of ²¹²Fr in MHz.

能级	${A_{{\rm{DF}}}}$	${A_{{\rm{CCSD}}}}$	$\varDelta /$%	Ref.[38]	实验值
7P_1/2	642.48	1198.10	41.96	1181(9)	1189.1(4.6)^[28]
					1187.1(6.8)^[39]
					1192.0(2)^[32]
8P_1/2	228.04	372.04	33.66	371(5)	373.0(1)^[39]
9P_1/2	106.78	167.21	30.88
10P_1/2	58.35	89.53	29.46
11P_1/2	35.26	53.38	28.51
12P_1/2	22.88	34.24	27.68
7P_3/2	51.05	97.88	43.55	96(3)	97.2(1)^[27]
					97.2(1)^[39]
8P_3/2	18.67	32.51	37.82	32(3)	32.8(1)^[39]
9P_3/2	8.89	15.00	35.83
10P_3/2	4.91	8.15	34.75
11P_3/2	3.00	4.92	34.01
12P_3/2	1.97	3.20	33.33

DownLoad: CSV

表 3 ²¹²Fr原子D态的超精细结构常数A (单位: MHz)

Table 3. Hyperfine-structure constant A for the D states of ²¹²Fr in MHz.

能级	${A_{{\rm{DF}}}}$	${A_{{\rm{CCSD}}}}$	$\varDelta / $%	Ref.[38]	实验值
6D_3/2	33.25	92.91	61.27	79(5)
7D_3/2	16.82	30.17	39.65	29(3)
8D_3/2	8.65	13.81	32.20	13(1)	13.0(6)^[29]
9D_3/2	4.93	7.50	28.75	7(1)	7.1(7)^[29]
10D_3/2	3.06	4.52	26.64
11D_3/2	2.03	2.93	24.97
6D_5/2	13.14	–53.92	126.38	–54(5)
7D_5/2	6.32	–13.64	150.21	–15(3)
8D_5/2	3.20	–5.67	161.12	–6(1)	–7.1(6)^[29]
9D_5/2	1.81	–2.96	166.96	–3.3(6)	–3.6(4)^[29]
10D_5/2	1.12	–1.72	170.59
11D_5/2	0.74	–1.10	173.16

DownLoad: CSV

表 4 Fr原子同位素的磁偶极矩$\mu$

Table 4. Magnetic dipole moment $\mu$ of Fr isotope.

同位素	核自旋	7P_1/2		7P_3/2		${\mu}$
同位素	核自旋	${A_{{\rm{expt}}.}}$^[33]	${{\mu} _{1/2}}$	${A_{{\rm{expt}}.}}$^[33]	${{\mu} _{3/2}}$	${{\mu} _{{\rm{present}}}}$	${{\mu} _{{\rm{expt}}{\rm{.}}}}$^[33]
²⁰⁷Fr	9/2			90.7(6)	3.85(3)	3.85(3)	3.89(9)
²⁰⁸Fr	7	874.8(3)	4.723(2)	72.4(5)	4.784(33)	4.753(33)	4.75(10)
²⁰⁹Fr	9/2	1127.9(2)	3.914(1)	93.3(5)	3.963(21)	3.939(22)	3.95(8)
²¹⁰Fr	6	946.3(3)	4.379(1)	78.0(2)	4.418(11)	4.399(20)	4.40(9)
²¹¹Fr	9/2	1142.1(2)	3.964(1)	94.9(3)	4.031(13)	3.998(34)	4.00(8)
²¹²Fr	5	1187(7)	4.577(26)	97.2(1)	4.588(5)	4.583(30)	4.62(9)
²¹³Fr	9/2	1150(8)	3.991(28)	95.3(3)	4.047(13)	4.019(30)	4.02(8)
²²⁰Fr	1			–73.2(5)	–0.691(5)	–0.691(5)	–0.67(1)
²²¹Fr	5/2	808(12)	1.558(23)	65.5(6)	1.545(14)	1.552(25)	1.58(3)
²²²Fr	2			33(1)	0.623(19)	0.623(19)	0.63(1)
²²³Fr	3/2			83.3(9)	1.179(13)	1.179(13)	1.17(2)
²²⁴Fr	1			42.1(7)	0.397(7)	0.397(7)	0.40(1)
²²⁵Fr	3/2			77(3)	1.090(42)	1.090(42)	1.07(2)
²²⁶Fr	1			7(1)	0.066(9)	0.066(9)	0.071(2)
²²⁷Fr	1/2			316(2)	1.491(9)	1.491(9)	1.50(3)
²²⁸Fr	2			–41(2)	–0.77(4)	–0.77(4)	–0.76(2)

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[1]	Grant I P 2007 Relativistic Quantum Theory of Atoms and Molecules (New York: Springer) pp533−577
[2]	Fischer C F, Brage T, Jönsson P 1997 Computational Atomic Structure: An MCHF Approach (UK: Institute of Physics) pp1−67
[3]	Jönsson P, Gaigalas G, Bieroń J, Fishcher C F, Grant I 2013 Computer Physics Communications. 184 2197 Google Scholar
[4]	Jönsson P, He X, Fishcher C F, Grant I 2007 Computer Physics Communications. 177 597 Google Scholar
[5]	Dzuba V A, Flambaum V V, Kozlov M G 1996 Phys. Rev. A 54 3948 Google Scholar
[6]	Dzuba V A, Johnson W R 1998 Phys. Rev. A 57 2459 Google Scholar
[7]	Angstmann E J, Dzuba V A, Flambaum V V 2004 Phys. Rev. A 70 014102 Google Scholar
[8]	Dinh T H, Dzuba V A, Flambaum V V, Ginges J S M 2008 Phys. Rev. A 78 054501 Google Scholar
[9]	Kozlov M G, Porsev S G, Johnson W R 2001 Phys. Rev. A 64 052107 Google Scholar
[10]	Pal R, Safronova M S, Johnson W R, Derevianko A, Porsev S G 2007 Phys. Rev. A 75 042515 Google Scholar
[11]	Blundell S A, Johnson W R, Liu Z W, Sapirstein 1989 Phys. Rev. A 40 2233 Google Scholar
[12]	Eliav E, Vikas M J, Ishikawa Y, Kaldor U 2005 Chem. Phys. 311 163 Google Scholar
[13]	Mani B K, Angom D 2011 Phys. Rev. A 83 012501 Google Scholar
[14]	Kallay M, Nataraj H S, Sahoo B K, Das B P, Visscher L 2011 Phys. Rev. A 83 030503 Google Scholar
[15]	Nandy D K, Singh Y, Sahoo B K 2014 Phys. Rev. A 89 062509 Google Scholar
[16]	Borschevsky A, Eliav E, Vilkas M J, Ishikawa Y, Kaldor U 2007 Phys. Rev. A 75 042514 Google Scholar
[17]	Eliav E, Kaldor U, Ishikawa Y 1996 Phys. Rev. A 53 3050 Google Scholar
[18]	Chaudhuri R K, Chattopadhyay S, Mahapatra U S 2013 J. Phys. Chem. A 117 12616 Google Scholar
[19]	Tang Y B, Lou B Q, Shi T Y 2017 Phys. Rev. A 96 022513 Google Scholar
[20]	Tang Y B, Gao N N, Lou B Q, Shi T Y 2018 Phys. Rev. A 98 062511 Google Scholar
[21]	Byrnes T M R, Dzuba V A, Flambaum F F, Murray D W 1999 Phys. Rev. A 59 3082 Google Scholar
[22]	Mukherjee D, Sahoo B K, Nataraj H S, Das B P 2009 J. Phys. Chem. A 113 12549 Google Scholar
[23]	Sakemi Y, Harada K, Hayamizu T, Itoh M, Kawamura H, Liu S, Nataraj H S, Oikawa A, Saito M, Sato T 2011 J. Phys. Conf. Ser. 302 012051 Google Scholar
[24]	Sahoo B K, Aoki T, Das B P, Sakemi Y 2016 Phys. Rev. A 93 032520 Google Scholar
[25]	Atutov S N, Calabrese R, Corradi L, Dainelli A, Mauro C D, Khanbekyan A, Mariotti E, Minguzzi P, Moi L, Sanguinetti S, Stancari G, Tomassetti L 2008 Proc. SPIE 7027 70270C
[26]	Ekström C, Ingelman S, Wannberg G, Skarestad M 1978 Physica Scripta 18 51 Google Scholar
[27]	Coc A, Thibault C, Touchard F, Duong H T, Juncar P, Liberman S, Pinard J, Lermé J, Vialle J L, Büttgenbach S, Mueller A C, Pesnelle A, the ISOLDE Collaboration 1985 Phys. Lett. B 163 66 Google Scholar
[28]	Coc A, Thibault C, Touchard F, Duong H T, Juncar P, Liberman S, Pinard J, Carre M, Lermé J, Vialle J L, Büttgenbach S, Mueller A C, Pesnelle A, the ISOLDE Collaboration 1987 Nucl. Phys. A 468 1 Google Scholar
[29]	Arnold E, Borchers W, Duong H T, Juncar P, Lermé J, Lievens P, Neu W, Neugart R, Pellerin M, Pinard J, Vialle J L, Wendt K, the ISOLDE Collaboration 1990 J. Phys. B 23 3511 Google Scholar
[30]	Arnold E, Borchers W, Carré M, Duong H T, Juncar P, Lermé J, Liberman S, Neu W, Neugart R, Otten W, Pellerin M, Pinard J, Pesnelle A, Vialle J L, Wendt K, the ISOLDE Collaboration 1989 J. Phys. B 22 L391 Google Scholar
[31]	Bauche J, Duong H T, Juncar P, Liberman S, Pinard J, Coc A, Thibault C, Touchard F, Lermé J, Vialle J L, Büttgenbach S, Mueller A C, Pesnelle A, the ISOLDE Collaboration 1986 J. Phys. B 19 L593 Google Scholar
[32]	Grossman J S, Orozco L A, Simsarian J E, Sprouse G D, Zhao W Z 1999 Phys. Rev. Lett. 83 935 Google Scholar
[33]	Sansonetti J E 2007 J. Phys. Chem. Ref. Data 36 497 Google Scholar
[34]	Gomez E, Aubin S, Orozco L A, Sprouse G D, Iskrenova-Tchoukova E, Safronova M S 2008 Phys. Rev. Lett. 100 172502 Google Scholar
[35]	Dzuba V A, Flambaum V V, Sushkov O P 1984 J. Phys. B: At. Mol. Phys. 17 1953 Google Scholar
[36]	Owusu A, Dougherty R W, Gowri G, Das T P 1997 Phys. Rev. A 56 305 Google Scholar
[37]	Safronova M S, Johnson W R, Derevianko A 1999 Phys. Rev. A 60 4476 Google Scholar
[38]	Sahoo B K, Nandy D K, Das B P, Sakemi Y 2015 Phys. Rev. A 91 042507 Google Scholar
[39]	Duong H T, Juncar P, Liberman S, Mueller A C, Neugart R, Otten E W, Peuse B, Pinard J, Stoke H H, Thibault C, Touchard F, Vialle J L, Wendt K, the ISOLDE Collaboration 1987 Europhys. Lett. 3 175 Google Scholar
[40]	Barber Z W, Stalnaker J E, Lemke N D, Poli N, Oates C W, Fortier T M, Diddams S A, Hollberg L, Hoyt C W, Taichenachev A V, Yudin V I 2008 Phys. Rev. Lett. 100 103002 Google Scholar
[41]	Kien F L, Balykin V I, Hakuta K 2005 J. Phys. Soc. Jpn. 74 910 Google Scholar
[42]	Ingvar L 1978 Int. J. Quantum Chem. 12 33
[43]	Sinha D, Mukhopadhyay S, Mukherjee D 1986 Chem. Phys. Lett. 129 369 Google Scholar
[44]	Blundell S A, Johnson W R, Sapiratein J 1991 Phys. Rev. A 43 3407 Google Scholar
[45]	Porsev S G, Beloy K, Derevianko A 2010 Phys. Rev. D 82 036008 Google Scholar
[46]	Sahoo B K, Sur C, Beier T, Das B P, Chaudhuri R K, Mukherjee D 2007 Phys. Rev. A 75 042504 Google Scholar
[47]	Safronova M S, Safronova U I 2011 Phys. Rev. A 83 052508 Google Scholar

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Ab initio calculation of hyperfine-structure constant A of Fr and evaluation of magnetic dipole moments of Fr isotopes

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Ab initio calculation of hyperfine-structure constant A of Fr and evaluation of magnetic dipole moments of Fr isotopes

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Corresponding author: Tang Yong-Bo, ybtang@whu.edu.cn

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