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## 留言板

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

## 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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• #### 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 ${\mu}$.

#### 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 [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

#### Cited By

• 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.

•  [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 ${\mu}$.

Reference (47)

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