Theoretical investigation on hyperfine structure and isotope shift for 5d106s 2S1/2→5d96s2 2D5/2 clock transition in Hg+

Zhang Xiang; Lu Ben-Quan; Li Ji-Guang; Zou Hong-Xin

doi:10.7498/aps.68.20182136

Abstract

The Dirac-Hartree-Fock approximation is adopted to calculate the mass shift and the field shift for the 5d¹⁰6s ²S_1/2→5d⁹6s^{2 2}D_5/2 clock transition in Hg⁺. It is found that the field shift is much larger than the mass shift so that the latter can be neglected in the isotope shift. In addition, we estimate that the isotope shifts of the levels related to the 5d¹⁰6s ²S_1/2→5d⁹6s^{2 2}D_5/2 clock transition of Hg⁺ is on the order of about 10⁴ GHz, while the hyperfine structure splitting is in a range of 1−10 GHz. However, the isotope shift of the 5d¹⁰6s ²S_1/2→5d⁹6s^{2 2}D_5/2 clock transition is on the same order of magnitude as the hyperfine structure splitting. Therefore, the hyperfine structure splitting must be taken into account for predicting the frequency shifts of the clock transition between different isotopes. On the basis of these results, we perform a multi-configuration Dirac-Hartree-Fock calculation on the field shift of the 5d¹⁰6s ²S_1/2→5d⁹6s^{2 2}D_5/2 clock transition in Hg⁺ and the hyperfine interaction constants of the upper and the lower levels involved. In order to give accurate theoretical results of these physical quantities, we systematically consider the main electron correlations in the atomic system by using the active space method. The restricted single and double (SrD) excitation method is used to capture the correlation between the 5d and the 6s valence electrons, and the correlation between the 3s, 3p, 3d, 4s, 4p, 4d, 5s, 5p, and 5d core and the valence electrons. The isotope shifts and hyperfine structure splitting for this transition of several stable mercury isotopes are given. In particular, the uncertainty of the calculated isotope shift between ¹⁹⁹Hg⁺ and ¹⁹⁸Hg⁺ is about 2%, compared with the experimental measurement available. Using these results, we predict the absolute frequency values of this transition for seven mercury isotopes, which provides theoretical reference data for experiments. Moreover, the calculated isotope shifts and hyperfine structures are also useful for studying the structure, property and nucleon interaction of mercury nucleus.

Keywords:

hyperfine structure /
isotope shift /
multi-configuration Dirac-Hartree-Fock method /
mercury ion clock

Authors and contacts

1.
College of Liberal Arts and Science, National University of Defense Technology, Changsha 410073, China
2.
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
3.
National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, China

Corresponding author: Li Ji-Guang, li_jiguang@iapcm.ac.cn ; Zou Hong-Xin, hxzou@nudt.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11604385, 91536106, 11204374, 11874090) and the Research Project of National University of Defense Technology, China (Grant No.ZK17-03-11).

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References

[1]	Prestage J D, Weaver G L 2007 Proc. IEEE 95 2235 Google Scholar
[2]	Tjoelker R L, Prestage J D, Burt E A, Chen P, Chong Y J, Chung S K, Diener W, Ely T, Enzer D G, Mojaradi H, Okino C, Pauken M, Robison D, Swenson B L, Tucker B, Wang R 2016 IEEE Trans. Ultrason. Ferroelectr. Freq. Control 63 1034 Google Scholar
[3]	Prestage J D, Chung S K, Thompson R J, Neal P M 2009 IEEE International Frequency Control Symposium Joint with the 22nd European Frequency and Time forum Besancon, France, April 20-24, 2009 p54–7
[4]	Rosenband T, Hume D B, Schmidt P O, Chou C W, Brusch A, Lorini L, Oskay W H, Drullinger R E, Fortier T M, Stalnaker J E, Diddams S A, Swann W C, Newbury N R, Itano W M, Wineland D J, Bergquist J C 2008 Science 319 1808 Google Scholar
[5]	Larigani S T, Burt E A, Lea S N, Prestage J D, Tjoelker R L 2009 International Frequency Control Symposium Joint with the 22nd European Frequency and Time forum Besancon, France, April 20-24, 2009 pp774–777
[6]	Coursey J S, Schwab D J, Tsai J J, Dragoset R A http://physics.nist.gov/Comp [2018-10-27]
[7]	Zucker M A, Kishore A R, Sukumar R, Dragoset R A http://physics.nist.gov/EDI [2018-10-27]
[8]	Angeli I, Marinova K P 2013 At. Data Nucl. Data Tables 99 69 Google Scholar
[9]	Stone N J 2005 At. Data Nucl. Data Tables 90 75 Google Scholar
[10]	Prestage J D, Janik G R, Dick G J, Maleki L 1991 Conference on Precision Electromagnetic Measurements Ottawa, Ontario, Canada, Canada, June 11-14, 1990 pp270–271
[11]	Tjoelker R L, Prestage J D, Maleki L 1996 Telecommun. Data Acquis. Prog. Rep. 126 1
[12]	Rafac R J, Young B C, Beall J A, Itano W M, Wineland D J, Bergquist J C 2000 Phys. Rev. Lett. 85 2462 Google Scholar
[13]	Bergquist J C, Rafac R J, Young B, Beall J A, Itano W M, Wineland D J 2001 Proc. SPIE 4269 1 Google Scholar
[14]	Oskay W H, Diddams S A, Donley E A, Fortier T M, Heavner T P, Hollberg L, Itano W M, Jefferts S R, Delaney M J, Kim K, Levi F, Parker T E, Bergquist J C 2006 Phys. Rev. Lett. 97 020801 Google Scholar
[15]	Bergquist J C, Wineland D J, Itano W M, Hemmati H, Daniel H U, Leuchs G 1985 Phys. Rev. Lett. 55 1567 Google Scholar
[16]	Matveev O I, Smith B W, Winefordner J D 1998 Opt. Lett. 23 304 Google Scholar
[17]	Zou H X, Wu Y, Chen G Z, Shen Y, Liu Q 2015 Chinese Phys. Lett. 32 054207 Google Scholar
[18]	Cheal B, Cocolios T E, Fritzsche S 2012 Phys. Rev. A 86 042501 Google Scholar
[19]	Grant I P 2007 Relativistic Quantum Theory of Atoms and Molecules (New York: Springer) pp259-388
[20]	Li J G, Jönsson P, Godefroid M, Dong C Z, Gaigalas G 2012 Phys. Rev. A 86 052523 Google Scholar
[21]	Fullerton L W, Rinker G A 1976 Phys. Rev. A 13 1283 Google Scholar
[22]	Dyall K G, Grant I P, Johnson C T, Parpia F A, Plummer E P 1989 Comput. Phys. Commun. 55 425 Google Scholar
[23]	Jönsson P, Gaigalas G, Bieroń J, Fischer C F, Grant I P 2013 Comput. Phys. Commun. 184 2197 Google Scholar
[24]	McDaniel E W, McDowell M R C 1975 Case Studies in Atomic Physics Ⅳ (Amsterdam: North-Holland) pp197–298
[25]	Jönsson P, Parpia F A, Fischer C F 1996 Comput. Phys. Commun. 96 301 Google Scholar
[26]	Tupitsyn I I, Shabaev V M, Crespo López-Urrutia J R, Draganić I, Orts R S, Ullrich J 2003 Phys. Rev. A 68 022511 Google Scholar
[27]	Filippin L, Beerwerth R, Ekman J, Fritzsche S, Godefroid M, Jönsson P 2016 Phys. Rev. A 94 062508 Google Scholar
[28]	Shabaev V M 1985 Theor. Math. Phys. 63 588 Google Scholar
[29]	Palmer C W P 1987 J. Phys. B At. Mol. Phys. 20 5987 Google Scholar
[30]	Shabaev V M, Artemyev A N 1994 J. Phys. B At. Mol. Opt. Phys. 27 1307 Google Scholar
[31]	Jönsson P, Froese C F 1997 Comput. Phys. Commun. 100 81 Google Scholar
[32]	Nazé C, Gaidamauskas E, Gaigalas G, Godefroid M, Jönsson P 2013 Comput. Phys. Commun. 184 2187 Google Scholar
[33]	Blundell S A, Baird P E G, Palmer C W P, Stacey D N, Woodgate G K 1987 J. Phys. B: At. Mol. Phys. 20 3663 Google Scholar
[34]	Fischer C F, Brage T, Jönsson P 1997 Computational Atomic Structure - An MCHF Approach (London: Institute of Physics Publishing) pp67-86
[35]	Brage T, Proffitt C, Leckrone D S 1999 Astrophys. J. 513 524 Google Scholar
[36]	Simmons M, Safronova U I, Safronova M S 2011 Phys. Rev. A 84 052510 Google Scholar
[37]	Guern Y, Méhu A B, Abjean R, Gilles A J 1976 Phys. Scr. 14 273 Google Scholar
[38]	Itano W M 2006 Phys. Rev. A 73 022510 Google Scholar

Cited By

图 1 汞同位素离子5d¹⁰6s ²S_1/2→5d⁹6s^{2 2}D_5/2跃迁相对于¹⁹⁹Hg⁺离子的FS和MS随质量数变化的趋势

Figure 1. Trends of field shift and mass shift for the 5d¹⁰6s ²S_1/2→5d⁹6s^{2 2}D_5/2 transition in mercury isotope ions with respect to ¹⁹⁹Hg⁺ as the increase of mass number.

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图 2 ¹⁹⁹Hg⁺和¹⁹⁸Hg⁺离子的超精细能级结构图

Figure 2. Hyperfine level structure diagram of ¹⁹⁹Hg⁺ and ¹⁹⁸Hg⁺.

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表 1 7种天然汞同位素及相关参数

Table 1. Related parameters of seven natural mercury isotopes.

Isotopes’ mass number	Relative atomic mass^[6]	Abundance^[7]	R/fm^[8]	I/$\hbar$	$\mu $/nm^[9]	Q/barn^[9]
196	195.9658326 (32)	0.15%	5.4385	0+	–	–
198	197.96676860 (52)	10.04%	5.4463	0+	–	–
199	198.96828064 (46)	16.94%	5.4474	1/2–	+0.5058855(9)	–
200	199.96832659 (47)	23.14%	5.4551	0+	–	–
201	200.97030284 (69)	13.17%	5.4581	3/2–	–0.5602257(14)	+0.387(6)
202	201.97064340 (69)	29.74%	5.4648	0+	–	–
204	203.97349398 (53)	6.82%	5.4744	0+	–	–

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表 2 电子关联对能量本征值的影响

Table 2. Effect of electron correlations on energy eigenvalues.

n	Active orbitals	Virtual orbitals	NCF	Energy eigenvalue/10⁴ Hartrees
DF			1/1	–1.964857825739/–1.964840329639
7	5d6s	7s, 6p, 6d, 5f, 5g	310/1631	–1.964887721767/–1.964870006459
8	5spd6s	8s, 7p, 7d, 6f, 6g	4047/19457	–1.964907829871/–1.964890991924
9	4spdf5spd6s	9s, 8p, 8d, 7f, 7g	29884/151235	–1.964927346267/–1.964910124355
10	3spd4spdf5spd6s	10s, 9p, 9d, 8f, 7g	69579/334460	–1.964929839430/–1.964912598231
11	3spd4spdf5spd6s	11s, 10p, 10d, 9f, 7g	103101/480763	–1.964930723063/–1.964913507368

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表 3 汞同位素离子相对¹⁹⁹Hg⁺离子5d¹⁰6s ²S_1/2→5d⁹6s^{2 2}D_5/2钟跃迁的场位移 (单位: GHz)受电子关联的影响

Table 3. Effect of electron correlations on the FS (in GHz) of the 5d¹⁰6s ²S_1/2→5d⁹6s^{2 2}D_5/2 transition in mercury isotope ions (relative to ¹⁹⁹Hg⁺).

n	¹⁹⁶Hg⁺	¹⁹⁸Hg⁺	²⁰⁰Hg⁺	²⁰¹Hg⁺	²⁰²Hg⁺	²⁰⁴Hg⁺
DF	–9.01296	–1.11476	7.80962	10.8553	17.6634	27.4329
7	–9.20985	–1.13911	7.98023	11.0925	18.0493	28.0321
8	–8.81504	–1.09028	7.63813	10.6169	17.2755	26.8305
9	–9.11351	–1.12720	7.89674	10.9764	17.8605	27.7389
10	–9.12483	–1.12860	7.90656	10.9901	17.8827	27.7734
11	–9.14646	–1.13127	7.92530	11.0161	17.9250	27.8392

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表 4 ¹⁹⁹Hg⁺和²⁰¹Hg⁺离子5d¹⁰6s ²S_1/2和5d⁹6s^{2 2}D_5/2态的磁偶极(A单位: MHz)和电四极(B单位: MHz)超精细结构常数

Table 4. Magnetic dipole A (in MHz) and electric quadrupole B (in MHz) hyperfine interaction constants for the 5d¹⁰6s ²S_1/2 and 5d⁹6s^{2 2}D_5/2 states of ¹⁹⁹Hg⁺ and ²⁰¹Hg⁺.

n	¹⁹⁹A_1/2	¹⁹⁹A_5/2	²⁰¹A_1/2	²⁰¹A_5/2	²⁰¹B_5/2
DF	36812.0	986.665	–13585.7	–364.216	796.132
7	39090.5	1263.67	–14426.7	–466.447	755.219
8	38761.2	795.021	–14305.1	–293.490	765.173
9	40556.1	951.973	–14967.5	–353.908	936.169
10	40967.0	951.669	–15119.2	–351.307	961.161
11	41133.9	963.552	–15180.8	–355.692	966.809
Ref. [38]		963.5		–355.7	839.4
Ref. [37]	40460		–14960
Ref. [35]	42366	1315	–15527	–482	859
Ref. [36]	41477		–15311

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表 5 汞同位素离子5d¹⁰6s ²S_1/2→5d⁹6s^{2 2}D_5/2跃迁谱线的绝对频率值

Table 5. Absolute frequency values of the 5d¹⁰6s ²S_1/2→5d⁹6s^{2 2}D_5/2 transition in mercury isotope ions.

Ions	¹⁹⁶Hg⁺	¹⁹⁸Hg⁺	¹⁹⁹Hg⁺	²⁰⁰Hg⁺	²⁰¹Hg⁺	²⁰²Hg⁺	²⁰⁴Hg⁺
ν/GHz	1064683.30	1064691.31*	1064721.61^[14]	1064700.37	1064719.89	1064710.37	1064720.28
*Experiment value is 1064691.95 GHz^[15].

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[1]	Prestage J D, Weaver G L 2007 Proc. IEEE 95 2235 Google Scholar
[2]	Tjoelker R L, Prestage J D, Burt E A, Chen P, Chong Y J, Chung S K, Diener W, Ely T, Enzer D G, Mojaradi H, Okino C, Pauken M, Robison D, Swenson B L, Tucker B, Wang R 2016 IEEE Trans. Ultrason. Ferroelectr. Freq. Control 63 1034 Google Scholar
[3]	Prestage J D, Chung S K, Thompson R J, Neal P M 2009 IEEE International Frequency Control Symposium Joint with the 22nd European Frequency and Time forum Besancon, France, April 20-24, 2009 p54–7
[4]	Rosenband T, Hume D B, Schmidt P O, Chou C W, Brusch A, Lorini L, Oskay W H, Drullinger R E, Fortier T M, Stalnaker J E, Diddams S A, Swann W C, Newbury N R, Itano W M, Wineland D J, Bergquist J C 2008 Science 319 1808 Google Scholar
[5]	Larigani S T, Burt E A, Lea S N, Prestage J D, Tjoelker R L 2009 International Frequency Control Symposium Joint with the 22nd European Frequency and Time forum Besancon, France, April 20-24, 2009 pp774–777
[6]	Coursey J S, Schwab D J, Tsai J J, Dragoset R A http://physics.nist.gov/Comp [2018-10-27]
[7]	Zucker M A, Kishore A R, Sukumar R, Dragoset R A http://physics.nist.gov/EDI [2018-10-27]
[8]	Angeli I, Marinova K P 2013 At. Data Nucl. Data Tables 99 69 Google Scholar
[9]	Stone N J 2005 At. Data Nucl. Data Tables 90 75 Google Scholar
[10]	Prestage J D, Janik G R, Dick G J, Maleki L 1991 Conference on Precision Electromagnetic Measurements Ottawa, Ontario, Canada, Canada, June 11-14, 1990 pp270–271
[11]	Tjoelker R L, Prestage J D, Maleki L 1996 Telecommun. Data Acquis. Prog. Rep. 126 1
[12]	Rafac R J, Young B C, Beall J A, Itano W M, Wineland D J, Bergquist J C 2000 Phys. Rev. Lett. 85 2462 Google Scholar
[13]	Bergquist J C, Rafac R J, Young B, Beall J A, Itano W M, Wineland D J 2001 Proc. SPIE 4269 1 Google Scholar
[14]	Oskay W H, Diddams S A, Donley E A, Fortier T M, Heavner T P, Hollberg L, Itano W M, Jefferts S R, Delaney M J, Kim K, Levi F, Parker T E, Bergquist J C 2006 Phys. Rev. Lett. 97 020801 Google Scholar
[15]	Bergquist J C, Wineland D J, Itano W M, Hemmati H, Daniel H U, Leuchs G 1985 Phys. Rev. Lett. 55 1567 Google Scholar
[16]	Matveev O I, Smith B W, Winefordner J D 1998 Opt. Lett. 23 304 Google Scholar
[17]	Zou H X, Wu Y, Chen G Z, Shen Y, Liu Q 2015 Chinese Phys. Lett. 32 054207 Google Scholar
[18]	Cheal B, Cocolios T E, Fritzsche S 2012 Phys. Rev. A 86 042501 Google Scholar
[19]	Grant I P 2007 Relativistic Quantum Theory of Atoms and Molecules (New York: Springer) pp259-388
[20]	Li J G, Jönsson P, Godefroid M, Dong C Z, Gaigalas G 2012 Phys. Rev. A 86 052523 Google Scholar
[21]	Fullerton L W, Rinker G A 1976 Phys. Rev. A 13 1283 Google Scholar
[22]	Dyall K G, Grant I P, Johnson C T, Parpia F A, Plummer E P 1989 Comput. Phys. Commun. 55 425 Google Scholar
[23]	Jönsson P, Gaigalas G, Bieroń J, Fischer C F, Grant I P 2013 Comput. Phys. Commun. 184 2197 Google Scholar
[24]	McDaniel E W, McDowell M R C 1975 Case Studies in Atomic Physics Ⅳ (Amsterdam: North-Holland) pp197–298
[25]	Jönsson P, Parpia F A, Fischer C F 1996 Comput. Phys. Commun. 96 301 Google Scholar
[26]	Tupitsyn I I, Shabaev V M, Crespo López-Urrutia J R, Draganić I, Orts R S, Ullrich J 2003 Phys. Rev. A 68 022511 Google Scholar
[27]	Filippin L, Beerwerth R, Ekman J, Fritzsche S, Godefroid M, Jönsson P 2016 Phys. Rev. A 94 062508 Google Scholar
[28]	Shabaev V M 1985 Theor. Math. Phys. 63 588 Google Scholar
[29]	Palmer C W P 1987 J. Phys. B At. Mol. Phys. 20 5987 Google Scholar
[30]	Shabaev V M, Artemyev A N 1994 J. Phys. B At. Mol. Opt. Phys. 27 1307 Google Scholar
[31]	Jönsson P, Froese C F 1997 Comput. Phys. Commun. 100 81 Google Scholar
[32]	Nazé C, Gaidamauskas E, Gaigalas G, Godefroid M, Jönsson P 2013 Comput. Phys. Commun. 184 2187 Google Scholar
[33]	Blundell S A, Baird P E G, Palmer C W P, Stacey D N, Woodgate G K 1987 J. Phys. B: At. Mol. Phys. 20 3663 Google Scholar
[34]	Fischer C F, Brage T, Jönsson P 1997 Computational Atomic Structure - An MCHF Approach (London: Institute of Physics Publishing) pp67-86
[35]	Brage T, Proffitt C, Leckrone D S 1999 Astrophys. J. 513 524 Google Scholar
[36]	Simmons M, Safronova U I, Safronova M S 2011 Phys. Rev. A 84 052510 Google Scholar
[37]	Guern Y, Méhu A B, Abjean R, Gilles A J 1976 Phys. Scr. 14 273 Google Scholar
[38]	Itano W M 2006 Phys. Rev. A 73 022510 Google Scholar

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Theoretical investigation on hyperfine structure and isotope shift for 5d¹⁰6s ²S_1/2→5d⁹6s^{2 2}D_5/2 clock transition in Hg⁺

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Corresponding author: Li Ji-Guang, li_jiguang@iapcm.ac.cn ; Zou Hong-Xin, hxzou@nudt.edu.cn

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