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Ab initio calculation of hyperfine-structure constant A 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

Ab initio calculation of hyperfine-structure constant A 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
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  • 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 S1/2, P1/2,3/2 and nD3/2 (n = 7-11) states are to from the direct effect; however, the dominant contributions for the 6D3/2, and nD5/2 (n = 6-11) states are due to the pair-correlation and the core-polarization, respectively. For D5/2 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 207−213,220−228Fr 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}$.
      Corresponding author: Tang Yong-Bo, ybtang@whu.edu.cn
    [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

    [4]

    Jönsson P, He X, Fishcher C F, Grant I 2007 Computer Physics Communications. 177 597

    [5]

    Dzuba V A, Flambaum V V, Kozlov M G 1996 Phys. Rev. A 54 3948

    [6]

    Dzuba V A, Johnson W R 1998 Phys. Rev. A 57 2459

    [7]

    Angstmann E J, Dzuba V A, Flambaum V V 2004 Phys. Rev. A 70 014102

    [8]

    Dinh T H, Dzuba V A, Flambaum V V, Ginges J S M 2008 Phys. Rev. A 78 054501

    [9]

    Kozlov M G, Porsev S G, Johnson W R 2001 Phys. Rev. A 64 052107

    [10]

    Pal R, Safronova M S, Johnson W R, Derevianko A, Porsev S G 2007 Phys. Rev. A 75 042515

    [11]

    Blundell S A, Johnson W R, Liu Z W, Sapirstein 1989 Phys. Rev. A 40 2233

    [12]

    Eliav E, Vikas M J, Ishikawa Y, Kaldor U 2005 Chem. Phys. 311 163

    [13]

    Mani B K, Angom D 2011 Phys. Rev. A 83 012501

    [14]

    Kallay M, Nataraj H S, Sahoo B K, Das B P, Visscher L 2011 Phys. Rev. A 83 030503

    [15]

    Nandy D K, Singh Y, Sahoo B K 2014 Phys. Rev. A 89 062509

    [16]

    Borschevsky A, Eliav E, Vilkas M J, Ishikawa Y, Kaldor U 2007 Phys. Rev. A 75 042514

    [17]

    Eliav E, Kaldor U, Ishikawa Y 1996 Phys. Rev. A 53 3050

    [18]

    Chaudhuri R K, Chattopadhyay S, Mahapatra U S 2013 J. Phys. Chem. A 117 12616

    [19]

    Tang Y B, Lou B Q, Shi T Y 2017 Phys. Rev. A 96 022513

    [20]

    Tang Y B, Gao N N, Lou B Q, Shi T Y 2018 Phys. Rev. A 98 062511

    [21]

    Byrnes T M R, Dzuba V A, Flambaum F F, Murray D W 1999 Phys. Rev. A 59 3082

    [22]

    Mukherjee D, Sahoo B K, Nataraj H S, Das B P 2009 J. Phys. Chem. A 113 12549

    [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

    [24]

    Sahoo B K, Aoki T, Das B P, Sakemi Y 2016 Phys. Rev. A 93 032520

    [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

    [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

    [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

    [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

    [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

    [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

    [32]

    Grossman J S, Orozco L A, Simsarian J E, Sprouse G D, Zhao W Z 1999 Phys. Rev. Lett. 83 935

    [33]

    Sansonetti J E 2007 J. Phys. Chem. Ref. Data 36 497

    [34]

    Gomez E, Aubin S, Orozco L A, Sprouse G D, Iskrenova-Tchoukova E, Safronova M S 2008 Phys. Rev. Lett. 100 172502

    [35]

    Dzuba V A, Flambaum V V, Sushkov O P 1984 J. Phys. B: At. Mol. Phys. 17 1953

    [36]

    Owusu A, Dougherty R W, Gowri G, Das T P 1997 Phys. Rev. A 56 305

    [37]

    Safronova M S, Johnson W R, Derevianko A 1999 Phys. Rev. A 60 4476

    [38]

    Sahoo B K, Nandy D K, Das B P, Sakemi Y 2015 Phys. Rev. A 91 042507

    [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

    [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

    [41]

    Kien F L, Balykin V I, Hakuta K 2005 J. Phys. Soc. Jpn. 74 910

    [42]

    Ingvar L 1978 Int. J. Quantum Chem. 12 33

    [43]

    Sinha D, Mukhopadhyay S, Mukherjee D 1986 Chem. Phys. Lett. 129 369

    [44]

    Blundell S A, Johnson W R, Sapiratein J 1991 Phys. Rev. A 43 3407

    [45]

    Porsev S G, Beloy K, Derevianko A 2010 Phys. Rev. D 82 036008

    [46]

    Sahoo B K, Sur C, Beier T, Das B P, Chaudhuri R K, Mukherjee D 2007 Phys. Rev. A 75 042504

    [47]

    Safronova M S, Safronova U I 2011 Phys. Rev. A 83 052508

  • 图 1  212Fr原子S1/2, P1/2, P3/2, D3/2和D5/2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 S1/2, P1/2, P3/2, D3/2 and D5/2 states of 212Fr.

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

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

    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
    7S1/26001.769403.5630.939124(94)9064.2(2)[27]
    9064.4(1.5)[39]
    8S1/21538.032014.1017.371986(19)
    9S1/2631.98792.1913.66784(9)
    10S1/2321.24396.8611.95419(9)401(5)[29]
    11S1/2185.41225.7111.10212(9)225(3)[29]
    12S1/2116.42141.2510.80
    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
    7P1/2642.481198.1041.961181(9)1189.1(4.6)[28]
    1187.1(6.8)[39]
    1192.0(2)[32]
    8P1/2228.04372.0433.66371(5)373.0(1)[39]
    9P1/2106.78167.2130.88
    10P1/258.3589.5329.46
    11P1/235.2653.3828.51
    12P1/222.8834.2427.68
    7P3/251.0597.8843.5596(3)97.2(1)[27]
    97.2(1)[39]
    8P3/218.6732.5137.8232(3)32.8(1)[39]
    9P3/28.8915.0035.83
    10P3/24.918.1534.75
    11P3/23.004.9234.01
    12P3/21.973.2033.33
    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
    6D3/233.2592.9161.2779(5)
    7D3/216.8230.1739.6529(3)
    8D3/28.6513.8132.2013(1)13.0(6)[29]
    9D3/24.937.5028.757(1)7.1(7)[29]
    10D3/23.064.5226.64
    11D3/22.032.9324.97
    6D5/213.14–53.92126.38–54(5)
    7D5/26.32–13.64150.21–15(3)
    8D5/23.20–5.67161.12–6(1)–7.1(6)[29]
    9D5/21.81–2.96166.96–3.3(6)–3.6(4)[29]
    10D5/21.12–1.72170.59
    11D5/20.74–1.10173.16
    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7P1/27P3/2${\mu}$
    ${A_{{\rm{expt}}.}}$[33]${{\mu} _{1/2}}$${A_{{\rm{expt}}.}}$[33]${{\mu} _{3/2}}$${{\mu} _{{\rm{present}}}}$${{\mu} _{{\rm{expt}}{\rm{.}}}}$[33]
    207Fr9/290.7(6)3.85(3)3.85(3)3.89(9)
    208Fr7874.8(3)4.723(2)72.4(5)4.784(33)4.753(33)4.75(10)
    209Fr9/21127.9(2)3.914(1)93.3(5)3.963(21)3.939(22)3.95(8)
    210Fr6946.3(3)4.379(1)78.0(2)4.418(11)4.399(20)4.40(9)
    211Fr9/21142.1(2)3.964(1)94.9(3)4.031(13)3.998(34)4.00(8)
    212Fr51187(7)4.577(26)97.2(1)4.588(5)4.583(30)4.62(9)
    213Fr9/21150(8)3.991(28)95.3(3)4.047(13)4.019(30)4.02(8)
    220Fr1–73.2(5)–0.691(5)–0.691(5)–0.67(1)
    221Fr5/2808(12)1.558(23)65.5(6)1.545(14)1.552(25)1.58(3)
    222Fr233(1)0.623(19)0.623(19)0.63(1)
    223Fr3/283.3(9)1.179(13)1.179(13)1.17(2)
    224Fr142.1(7)0.397(7)0.397(7)0.40(1)
    225Fr3/277(3)1.090(42)1.090(42)1.07(2)
    226Fr17(1)0.066(9)0.066(9)0.071(2)
    227Fr1/2316(2)1.491(9)1.491(9)1.50(3)
    228Fr2–41(2)–0.77(4)–0.77(4)–0.76(2)
    DownLoad: CSV
  • [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

    [4]

    Jönsson P, He X, Fishcher C F, Grant I 2007 Computer Physics Communications. 177 597

    [5]

    Dzuba V A, Flambaum V V, Kozlov M G 1996 Phys. Rev. A 54 3948

    [6]

    Dzuba V A, Johnson W R 1998 Phys. Rev. A 57 2459

    [7]

    Angstmann E J, Dzuba V A, Flambaum V V 2004 Phys. Rev. A 70 014102

    [8]

    Dinh T H, Dzuba V A, Flambaum V V, Ginges J S M 2008 Phys. Rev. A 78 054501

    [9]

    Kozlov M G, Porsev S G, Johnson W R 2001 Phys. Rev. A 64 052107

    [10]

    Pal R, Safronova M S, Johnson W R, Derevianko A, Porsev S G 2007 Phys. Rev. A 75 042515

    [11]

    Blundell S A, Johnson W R, Liu Z W, Sapirstein 1989 Phys. Rev. A 40 2233

    [12]

    Eliav E, Vikas M J, Ishikawa Y, Kaldor U 2005 Chem. Phys. 311 163

    [13]

    Mani B K, Angom D 2011 Phys. Rev. A 83 012501

    [14]

    Kallay M, Nataraj H S, Sahoo B K, Das B P, Visscher L 2011 Phys. Rev. A 83 030503

    [15]

    Nandy D K, Singh Y, Sahoo B K 2014 Phys. Rev. A 89 062509

    [16]

    Borschevsky A, Eliav E, Vilkas M J, Ishikawa Y, Kaldor U 2007 Phys. Rev. A 75 042514

    [17]

    Eliav E, Kaldor U, Ishikawa Y 1996 Phys. Rev. A 53 3050

    [18]

    Chaudhuri R K, Chattopadhyay S, Mahapatra U S 2013 J. Phys. Chem. A 117 12616

    [19]

    Tang Y B, Lou B Q, Shi T Y 2017 Phys. Rev. A 96 022513

    [20]

    Tang Y B, Gao N N, Lou B Q, Shi T Y 2018 Phys. Rev. A 98 062511

    [21]

    Byrnes T M R, Dzuba V A, Flambaum F F, Murray D W 1999 Phys. Rev. A 59 3082

    [22]

    Mukherjee D, Sahoo B K, Nataraj H S, Das B P 2009 J. Phys. Chem. A 113 12549

    [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

    [24]

    Sahoo B K, Aoki T, Das B P, Sakemi Y 2016 Phys. Rev. A 93 032520

    [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

    [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

    [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

    [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

    [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

    [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

    [32]

    Grossman J S, Orozco L A, Simsarian J E, Sprouse G D, Zhao W Z 1999 Phys. Rev. Lett. 83 935

    [33]

    Sansonetti J E 2007 J. Phys. Chem. Ref. Data 36 497

    [34]

    Gomez E, Aubin S, Orozco L A, Sprouse G D, Iskrenova-Tchoukova E, Safronova M S 2008 Phys. Rev. Lett. 100 172502

    [35]

    Dzuba V A, Flambaum V V, Sushkov O P 1984 J. Phys. B: At. Mol. Phys. 17 1953

    [36]

    Owusu A, Dougherty R W, Gowri G, Das T P 1997 Phys. Rev. A 56 305

    [37]

    Safronova M S, Johnson W R, Derevianko A 1999 Phys. Rev. A 60 4476

    [38]

    Sahoo B K, Nandy D K, Das B P, Sakemi Y 2015 Phys. Rev. A 91 042507

    [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

    [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

    [41]

    Kien F L, Balykin V I, Hakuta K 2005 J. Phys. Soc. Jpn. 74 910

    [42]

    Ingvar L 1978 Int. J. Quantum Chem. 12 33

    [43]

    Sinha D, Mukhopadhyay S, Mukherjee D 1986 Chem. Phys. Lett. 129 369

    [44]

    Blundell S A, Johnson W R, Sapiratein J 1991 Phys. Rev. A 43 3407

    [45]

    Porsev S G, Beloy K, Derevianko A 2010 Phys. Rev. D 82 036008

    [46]

    Sahoo B K, Sur C, Beier T, Das B P, Chaudhuri R K, Mukherjee D 2007 Phys. Rev. A 75 042504

    [47]

    Safronova M S, Safronova U I 2011 Phys. Rev. A 83 052508

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  • Received Date:  21 January 2019
  • Accepted Date:  09 March 2019
  • Available Online:  06 June 2019
  • Published Online:  01 May 2019

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

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

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 S1/2, P1/2,3/2 and nD3/2 (n = 7-11) states are to from the direct effect; however, the dominant contributions for the 6D3/2, and nD5/2 (n = 6-11) states are due to the pair-correlation and the core-polarization, respectively. For D5/2 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 207−213,220−228Fr 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}$.

    • 超精细相互作用描述的是核外电子与核的电磁多极矩之间的相互作用. 由其引起的能级劈裂称为超精细结构. 准确了解原子体系能级的超精细结构对于高精度光谱测量实验, 原子钟, 量子信息, 核电磁多极极矩的确定, 以及与激光冷却和激光囚禁相关的多个领域都是非常重要的. 超精细相互作用是一种短程力, 对相对论效应和电子关联效应非常敏感. 精确计算超精细结构常数需要同时考虑这两种效应. 相对论效应通过解Dirac-Fock (DF)方程很容易被考虑进去, 因此决定计算精度的是电子关联效应. 目前在原子物理领域, 相对论框架下的多体方法比如: 相对论组态和多组态相互作用方法[1-4]、相对论多体微扰理论方法[5-9]、相对论耦合簇理论方法[10-12], 以及这些方法的其他改进版本[13-18]已广泛用于处理电子关联效应. 最近我们基于B样条基, 发展了一套用于原子结构性质计算研究的相对论耦合簇理论方法, 并开发了相应的程序包. 应用这种方法计算研究了重原子Tl和Fr的能级、跃迁矩阵元、态寿命和极化率等性 质[19,20]. 已证实这套方法能精确地计算出重原子体系的长程性质. 本文尝试应用这套理论方法计算Fr原子的磁偶极超精细结构常数, 用以检测本方法应用到短程性质时的性能和精度. Fr是研究宇称不守恒效应(parity non-conservation, PNC)和测量电子永久电偶极矩(electric dipole moment, EDM)的重要候选体系[21-25]. 理论计算Fr原子的超精细结构常数是非常有意义的. 比如PNC振幅是一个不可观测量, 需要理论提供这个参数, 其理论计算精度就是通过对比磁偶极超精细结构常数A的精度来评估. 另一方面, 结合超精细结构常数的精确测量值便可导出核的电磁多极矩. 这是目前确定核电磁多极矩精确的方法之一, 因其并不依赖核模型.

      在过去的几十年中, 已有不少实验和理论工作报道了Fr原子超精细结构常数和核结构参数. 实验测量方面: 1978年, Ekström等[26]利用原子-光束磁共振技术测量208−213,220−222Fr同位素的核自旋和磁偶极矩; 1985—1990年, ISOLDE合作组[27-31]先后测量了212Fr原子多个态的超精细结构常数. 1999年, Grossman等[32]测量了208−212Fr的7P1/2态的超精细结构. 2007年, Sansonetti[33]综述了Fr原子跃迁能、核结构参数、同位素位移和超精细结构常数等性质. 2008年, Gomez等[34]测量了210Fr原子9S态的磁偶极超精细常数, 并结合精确的理论值定出210Fr的磁偶极矩. 理论方面: 1984年, Dzuba等[35]应用相对论多体微扰方法计算了211Fr原子nS1/2 (n = 7—8)和nP1/2,3/2 (n = 7—8)态的磁偶极超精细结构常数; 1997年, Owusu等[36]应用相对论连接图多体微扰理论方法研究了212Fr原子S态磁偶极超精细相互作用中的原子实极化效应. 由于多体微扰理论仅考虑部分主要的电子关联效应, 因此这些理论结果精度并不高. 1999年, Safronova等[37]应用相对论全阶理论方法(等价于相对论线性耦合簇理论方法)计算了211Fr原子nS1/2 (n = 7—8)和nP1/2,3/2 (n = 7—8)态的磁偶极超精细结构常数, 计算精度相比之前的多体微扰理论结果有了较大的提高. 最近, Sahoo等[38]采用基于高斯基的多种相对论多体方法计算了210,212Fr原子17个态的超精细结构常数, 并系统地研究了电子关联效应. 这17个态不仅包括了之前计算过的S和P态, 还首次报道了多个D态的值. 除了高激发态10S和11S的磁偶极超精细结构常数与测量值有4%—5%的差异外, Sahoo等[38]的理论结果均与测量值[27-29,32,39]符合. 大部分理论计算和实验测量都是针对Fr原子低能态超精细结构常数, 目前可参考的高激发态超精细常数非常少. 而同族的碱金属元素Rb和Cs原子, 已有很多高激发态都有精确的测量值和计算值. Sahoo等[38]采用的是基于高斯基组的理论方法, 由于大数目的高斯基组可能发生线性相关等数值问题, 优化基底中的非线性参数需要特别处理, 这个优化参数也是非常耗时的, 这使得他们的方法很难应用到高激发态性质的研究. 与高斯函数不同, B样条是定义在有限区间的精确的分段多项式[40], 由B样条构成的基组是近似完备的, 并不存在线性相关问题, 这类基不仅可以描述低能束缚态, 还能很好描述高激发态、里德伯态和连续态[41].

      本文首先应用基于B样条基的相对论簇耦合理论方法计算了212Fr原子S, P和D态的磁偶极超精细结构常数, 一方面通过与可参考的实验值对比检测理论方法, 另一方面通过研究这些计算结果调查5个序列态中的电子关联趋势. 最后, 通过结合7P态磁偶极超精细结构常数测量值, 进一步定出了208−228Fr核磁偶极矩$\mu$. 本文采用原子单位制.

    2.   理论与方法
    • Fr是目前元素周期表中最重的碱金属元素, 是一个单价原子系统. 体系处于$\upsilon $价电子轨道时的精确波函数可表示为

      其中, Ω表示波算符; $\left| {{\phi _\upsilon }} \right\rangle $是参考状态, 即零阶DF波函数.

      在耦合簇理论(coupled-cluster)框架[42,43]下, 波算符表示成簇算符S的指数形式:

      根据电子激发的数目, 簇算符S可展开成

      其中, N是体系的电子数, ${S_n}$表示电子的n激发. 在仅考虑电子单激发(single excitation)和双激发(double excitation)的耦合簇理论近似下, 系统的波函数被简化为

      仅考虑单双激发的簇耦合方法用CCSD表示. 若忽略式子中的非线性项, 波函数近似为

      这种近似称为线性簇耦合方法(用LCCSD表示), 等价于目前原子物理领域Blundell等[44]发展的全阶方法.

      根据相对参考态所激发的粒子和洞的数目, 簇算符S被分成

      其中${S^{\left( {0,0} \right)}}$${S^{\left( {0,1} \right)}}$分别表示原子实激发和价电子激发的簇算符. 前者是用于计算闭壳层体系波函数, 后者对应价电子态波函数. 将其进行二次量子化后, 原子实簇算符表示为

      而价电子簇算符表示为

      式中, 字母$a,b$代表原子实轨道; $r,s$为虚轨道; $\upsilon $表示价电子轨道; ${a^\dagger }\left( a \right)$是粒子产生(湮灭)算符, $S_{ \cdot \cdot \cdot }^{ \cdot \cdot \cdot }$是簇振幅. 利用广义Bloch方程即可导出簇算符的耦合方程

      式中$\chi = \varOmega - 1$, $W$称为折叠算符; $P$$Q$分表表示作用于模型空间及其正交互补空间的投影算子. 在实际计算中, 首先迭代求解${S^{\left( {0,0} \right)}}$的耦合方程, 直至收敛. 然后利用已获得的${S^{\left( {0,0} \right)}}$, 用类似的迭代方法求解${S^{\left( {0,1} \right)}}$的方程.

      在完成上面迭代求解之后, 单体算符$O$在态$\left| {{\psi _{\upsilon} }} \right\rangle $的期望值可用下列式子计算得到

      在我们最近的工作[19,20]中已对(12)式做了具体展开. 在本文中, $O$是磁偶极超细相互作用算符, 即

      其中${\mu}$是核磁矩, $F\left( r \right)$是一个用来考虑Bohr-Weisskopf效应的磁化分布函数. 本文采用均匀球分布模型

      式中${R_N}$表示磁化半径. 本文采用与核电荷分布半径一样的值.

      磁偶极超精细结构常数由下式定义:

      定义一个磁偶极超精细耦合常数a,

      通过结合实验值Aexpt.和理论计算值ather.即可导出核的磁偶极矩, 即

    3.   结果与讨论
    • 我们之前的工作已报道了Fr原子能量和跃迁性质[19]. 本文基于之前所得到的波函数计算了Fr原子的磁偶极超精细结构常数. 在Fr的众多同位素中, 212Fr多个态的磁偶极超精细结构常数具有精确的测量值. 212Fr原子的磁偶极矩和核自旋的实验推荐值分别为4.62和5.

      表1列出了212Fr原子主量子数$ n \leqslant 12$的S态的磁偶极超精细结构常数. $ {A_{{\rm{DF}}}}$$ {A_{{\rm{CCSD}}}}$分别表示用DF方法和CCSD方法计算得到的值. $ {A_{{\rm{DF}}}}$$ {A_{{\rm{CCSD}}}}$之间的差值表示电子关联效应的贡献. 表中第4列为电子关联效应的百分比$ \varDelta = \dfrac{{{A_{{\rm{CCSD}}}}-{A_{{\rm{DF}}}}}}{{{A_{{\rm{CCSD}}}}}}\times $100%. 最后两列分别列出了其他理论值[38]和实验[27,29,39]. 从表1可以观察到, 对于基态7s1/2的超精细结构常数A, 本文理论值ACCSD和测量值的差异在4%左右, 而Sahoo等[38]的结果在1%以内. 本文方法没有考虑三激发和量子电动力学效应(quantum electrodynamics, QED). 根据Porsev等[45]关于碱金属Cs的工作, 三激发态和QED对Cs原子基态超精细结构常数A的贡献在2%左右. Fr比Cs电子数多, 可以估算三激发态和QED对Fr原子基态的超精细结构常数的贡献在2%—3%, 若加上这部分贡献, 本文理论值与实验测量值的差异就降低到1%—2%. Sahoo等[38]推荐的值包括部分三激发态的贡献, 因此文献[38]结果比本文结果更接近实验值. 这也说明, 对于碱金属原子基态超精细结构常数而言, 若理论计算精度要控制在1%左右, 就须考虑三激发态和QED效应的贡献. 对于其他S态, 只有10S和11S具有实验测量值. 从表1还可以看出, Sahoo等[38]推荐的结果与测量值有4%—5%的差异. 本文计算值与实验值符合, 都在实验测量值所给不确定度范围内. 相比基态, 这类高激发S态的电子关联效应更弱, 因此更容易计算到高精度. 从整个序列态来看, nS (7—9)态, 本文结果均大于Sahoo等[38]的结果, 但是10S态的超精细结构常数A反而是小于他们的值. 为了找到这种反常差异的原因, 对比了两者的零阶DF值, 10S和11S的DF值分别是353.32和171.98 MHz, 与本文结果321.24和185.41 MHz有差异. 通过对比发现, 本文和文献[38]中电子关联效应的贡献是一样的, 因此可以判定文献[38]的结果是由于基组本身导致的.

      能级${A_{{\rm{DF}}}}$${A_{{\rm{CCSD}}}}$$\varDelta /$%Ref.[38]实验值
      7S1/26001.769403.5630.939124(94)9064.2(2)[27]
      9064.4(1.5)[39]
      8S1/21538.032014.1017.371986(19)
      9S1/2631.98792.1913.66784(9)
      10S1/2321.24396.8611.95419(9)401(5)[29]
      11S1/2185.41225.7111.10212(9)225(3)[29]
      12S1/2116.42141.2510.80

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

      表2列出了212Fr原子主量子数$n \leqslant 12$的P态的磁偶极超精细结构常数. 对于P态而言, 仅有7P和8P态有可参照的实验值[27,28,32,39]和理论值[38]. 本文ACCSD值与其他理论值以及实验值均符合, 差异在1%以内. 报道了nP (n = 9—12)态的磁偶极超精细结构常数. 采用CCSD方法计算得到的nP (n = 9—12)态的磁偶极超精细结构常数, 其精度在1%—2%. 这为将来实验测量和其他理论方法提供了一个参考.

      能级${A_{{\rm{DF}}}}$${A_{{\rm{CCSD}}}}$$\varDelta /$%Ref.[38]实验值
      7P1/2642.481198.1041.961181(9)1189.1(4.6)[28]
      1187.1(6.8)[39]
      1192.0(2)[32]
      8P1/2228.04372.0433.66371(5)373.0(1)[39]
      9P1/2106.78167.2130.88
      10P1/258.3589.5329.46
      11P1/235.2653.3828.51
      12P1/222.8834.2427.68
      7P3/251.0597.8843.5596(3)97.2(1)[27]
      97.2(1)[39]
      8P3/218.6732.5137.8232(3)32.8(1)[39]
      9P3/28.8915.0035.83
      10P3/24.918.1534.75
      11P3/23.004.9234.01
      12P3/21.973.2033.33

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

      表3列出了212Fr原子主量子数$ n \leqslant 11$的D态的磁偶极超精细结构常数A. 但D态与S态、P态不同, 具有测量值的态并不是最低的D态, 而是8D和9D态[29]. 对于8D3/2和9D3/2态, 利用CCSD方法计算的值与Sahoo等[38]的理论值以及实验测量值都是符合的. 对于8D5/2和9D5/2态, 本文CCSD值小于测量值, 不在测量值的不确定度范围内; 本文CCSD值与Sahoo等[38]的计算值是非常接近的, 在其不确定度范围内. 除6D3/2以外的其他D态, 本文CCSD值与Sahoo等[38]的理论值是符合一致的. 对于6D3/2态, 本文CCSD值比Sahoo等[38]的推荐值大18%左右. 从表3可以观察到6D3/2的电子关联效应非常强, 电子关联效应的贡献是零阶DF值的两倍左右, 这表明除单双激发近似外的高阶激发态也有比较明显的贡献. 换言之, 若要精确计算6D3/2态的超精细结构常数, 需要考虑三激发态甚至更高阶的激发贡献. 综合3个表格也可以看出, D态的电子关联效应趋势并不如S和P态那么规范.

      能级${A_{{\rm{DF}}}}$${A_{{\rm{CCSD}}}}$$\varDelta / $%Ref.[38]实验值
      6D3/233.2592.9161.2779(5)
      7D3/216.8230.1739.6529(3)
      8D3/28.6513.8132.2013(1)13.0(6)[29]
      9D3/24.937.5028.757(1)7.1(7)[29]
      10D3/23.064.5226.64
      11D3/22.032.9324.97
      6D5/213.14–53.92126.38–54(5)
      7D5/26.32–13.64150.21–15(3)
      8D5/23.20–5.67161.12–6(1)–7.1(6)[29]
      9D5/21.81–2.96166.96–3.3(6)–3.6(4)[29]
      10D5/21.12–1.72170.59
      11D5/20.74–1.10173.16

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

    • 图1给出了212Fr原子S1/2, P1/2, P3/2, D3/2和D5/2态磁偶极超精细结构常数A中电子关联效应. 根据图1可以观察到前4个序列态的关联效应随着主量子数n的增加而减小, 而第5个序列D5/2态的电子关联效应随着主量子数的增加而增大. S1/2态电子关联效应最弱, 除了基态外, 其他S态的关联效应均小于20%. 然而, D5/2态电子关联效应最强, 均大于100%. 电子关联效应数值大于零阶DF值, 符号相反, 因此这两个数值间会存在抵消. 通常强的抵消会导致较差的结果. 正如表3显示, Sahoo等[38]针对D5/2序列态, 得到了比序列S和P态更大的不确定度. 对于D5/2序列态这类反常的电子关联趋势, 在其他原子体系也有观察到, 比如单价碱土离子体系[46]、碱金属原子Rb[47]和Fr[38].

      Figure 1.  Electron correlation effects in hyperfine-structure constant A for S1/2, P1/2, P3/2, D3/2 and D5/2 states of 212Fr.

      (12)式的展开式中有3项对超精细结构常数有显著贡献. 第一项是零阶的DF值, 由于这一项直接由DF波函数计算得到, 我们称之为直接效应, 用符号ADF表示; 第二项来自表示式$OS_2^{\left( {0,1} \right)}$及其复共轭; 第三项来自$OS_1^{\left( {0,1} \right)}$及其复共轭. 依据文献[38], 将这两项称为原子实极化效应和对关联效应, 其值分别用符号ACPAPC表示. 符号AT表示这三项的总和. 图2给出了这3项以及这3项的总和与总值A的比值. 从图2可以观察到: 1)对于S和P态, 最大的贡献项来自直接效应, 其次是原子实极化效应和对关联效应; 超过95%的贡献来自于这3项; 原子实极化效应贡献率随着主量子数n的增大逐渐趋于一常数; 2)对于6D3/2态, 最大贡献来自对关联效应, 其次是直接效应, 原子实极化效应非常小; 而对于其他D3/2态, 最大贡献仍然是来自直接效应, 其次是对关联效应; 主要3项的贡献占了总值的90%; 3)对于D5/2态, 不同项之间存在相互抵消现象; 除了这3项贡献外, 其他展开项的贡献也是非常大的; 同时也可以观察到在D5/2序列中不同的态其趋势也有差异. 这与S和P态有着截然不同的趋势. 这种反常的趋势也许就是D5/2态磁偶极超精细常数计算精度不如S和P态的原因.

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

      Fr原子磁偶极超精细结构常数中电子关联趋势的详细研究, 为后续应用这套理论方法研究Fr原子宇称反演不守恒以及电子的电偶极矩提供了一个重要参考.

    • 依据上述计算显示212Fr原子7P态磁偶极超精细结构常数的计算精度在1%以内, 因此可以结合7P态磁偶极超精细常数A的测量值得到磁偶极矩$\mu$. 原子核有一定的体积和电荷分布. 对于不同质量的同位素, 其原子核核电荷的空间分布不同, 因此不同同位素的能级之间是有差异的. 这种微小差异称为场移效应, 可通过两参数的 Fermi 核模型在自洽场计算中予以考虑. 对于不同质量的同位素, 磁化半径${R_N}$也有差异. 严格地讲, 计算任意一种质量数的Fr, 都要求考虑这两部的贡献. 换言之, 需要重新输入相应的参数并进行重新计算. 由于全电子处理Fr原子的相对论耦合簇计算量大, 若每一种同位素都要进行一次新的计算不太现实. 因此本文利用一个三阶微扰理论研究了这两部分效应对7P态磁偶极超精细结构常数的贡献. 本文计算结果显示这两部分效应对7P态的贡献是非常小的. 由于本文CCSD理论精度在1%左右, 因此可以忽略这些小的效应.

      应用7P1/2和7P3/2态的超精细耦合常数a, 结合实验测量值[27-29,32,39], 导出了Fr同位素核的磁偶极矩, 相应结果在表4中列出. 计算中磁偶极超精细耦合常数分别是a(7P1/2) = 1296.64 MHz和a(7P3/2) = 105.94 MHz. ${{\mu} _{1/2}}$${{\mu} _{3/2}}$分别表示通过7P1/2和7P3/2态磁偶极超精细结构常数得到的磁偶极矩. ${{\mu} _{{\rm{present}}}}$表示两者的平均值, 即${{\mu} _{{\rm{present}}}} = \dfrac{{{{\mu} _{1/2}} + {{\mu} _{3/2}}}}{2}$. ${{\mu} _{{\rm{expt}}{\rm{.}}}}$是来自文献[33]. ${{\mu} _{1/2}}$${{\mu} _{3/2}}$的不确定度是依据测量值的不确定度给出. ${{\mu} _{{\rm{present}}}}$的不确定度依据$ {{\mu} _{\text{present}}} $${{\mu} _{1/2}}$${{\mu} _{3/2}}$之间的最大差值给出. 通常情况下, 这种方式所给出的不确定度都要大于由实验测量值导致的不确定度. 从表4可以看出, 对于同位素208−213Fr, ${{\mu} _{1/2}}$小于${{\mu} _{{\rm{expt}}{\rm{.}}}}$, 而${{\mu} _{3/2}}$大于${{\mu} _{{\rm{expt}}{\rm{.}}}}$, 二者的平均值${{\mu} _{{\rm{present}}}}$${{\mu} _{{\rm{expt}}{\rm{.}}}}$几乎完全一致.实验不确定度比要本文不确定度大约两倍. 其他同位素220−228Fr的${{\mu} _{{\rm{present}}}}$与实验值${{\mu} _{{\rm{expt}}{\rm{.}}}}$也符合得非常好. 由于这些同位素只有一个态具有测量值并有大的不确定度, 因此相比较208−213Fr, 220−228Fr的${{\mu} _{{\rm{present}}}}$具有更大的不确定度.

      同位素核自旋7P1/27P3/2${\mu}$
      ${A_{{\rm{expt}}.}}$[33]${{\mu} _{1/2}}$${A_{{\rm{expt}}.}}$[33]${{\mu} _{3/2}}$${{\mu} _{{\rm{present}}}}$${{\mu} _{{\rm{expt}}{\rm{.}}}}$[33]
      207Fr9/290.7(6)3.85(3)3.85(3)3.89(9)
      208Fr7874.8(3)4.723(2)72.4(5)4.784(33)4.753(33)4.75(10)
      209Fr9/21127.9(2)3.914(1)93.3(5)3.963(21)3.939(22)3.95(8)
      210Fr6946.3(3)4.379(1)78.0(2)4.418(11)4.399(20)4.40(9)
      211Fr9/21142.1(2)3.964(1)94.9(3)4.031(13)3.998(34)4.00(8)
      212Fr51187(7)4.577(26)97.2(1)4.588(5)4.583(30)4.62(9)
      213Fr9/21150(8)3.991(28)95.3(3)4.047(13)4.019(30)4.02(8)
      220Fr1–73.2(5)–0.691(5)–0.691(5)–0.67(1)
      221Fr5/2808(12)1.558(23)65.5(6)1.545(14)1.552(25)1.58(3)
      222Fr233(1)0.623(19)0.623(19)0.63(1)
      223Fr3/283.3(9)1.179(13)1.179(13)1.17(2)
      224Fr142.1(7)0.397(7)0.397(7)0.40(1)
      225Fr3/277(3)1.090(42)1.090(42)1.07(2)
      226Fr17(1)0.066(9)0.066(9)0.071(2)
      227Fr1/2316(2)1.491(9)1.491(9)1.50(3)
      228Fr2–41(2)–0.77(4)–0.77(4)–0.76(2)

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

    4.   结 论
    • 本文应用基于B样条基的相对论耦合簇理论方法计算了212Fr原子S, P和D态的磁偶极超精细结构常数A, 其中nP (n = 9—12)和nD (n = 10—11)态的结果是首次报道. 与其他理论方法计算值及测量值比较显示, 本文方法能精确计算出磁偶极超精细结构常数这类短程性质. 因此这套方法也能很好用于原子宇称不守恒和探索电子EDM的相关研究中. 同时, 也系统地研究了这5个序列态中电子关联趋势, 观察到D态具有一些反常的电子关联特征. 此外, 结合7P态磁偶极超精细结构常数A的测量值, 得到了207−213,220−228Fr的磁偶极矩$\mu$. 本文值与之前的测量值符合得非常好, 其中207−213Fr磁偶极矩$\mu$的精度提高了两倍左右.

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