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本文首先在Dirac-Hartree-Fock近似下理论评估了Hg+离子5d106s 2S1/2→5d96s2 2D5/2钟跃迁的质量位移(mass shift, MS)和场位移(field shift, FS)在其同位素位移(isotope shift, IS)中的相对贡献, 发现MS远小于FS而可以被忽略. 在此基础上, 通过系统地考虑该原子体系中主要的电子关联效应, 计算了这条钟跃迁FS的精确值以及涉及到的上下两个能级的超精细结构常数, 并得到了几种稳定汞同位素离子该跃迁的IS和超精细结构分裂. 其中, 计算的199Hg+和198Hg+离子之间的钟跃迁频率偏移与已有实验测量值相比误差为2%左右. 最终, 本文给出了汞离子7种常见同位素该谱线的绝对频率值, 为实验上的谱线测量提供了有效的理论依据.
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关键词:
- 超精细结构 /
- 同位素位移 /
- 多组态Dirac-Hartree-Fock方法 /
- 汞离子光钟
The Dirac-Hartree-Fock approximation is adopted to calculate the mass shift and the field shift for the 5d106s 2S1/2→5d96s2 2D5/2 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 2S1/2→5d96s2 2D5/2 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 2S1/2→5d96s2 2D5/2 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 2S1/2→5d96s2 2D5/2 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
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Google Scholar
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图 1 汞同位素离子5d106s 2S1/2→5d96s2 2D5/2跃迁相对于199Hg+离子的FS和MS随质量数变化的趋势
Fig. 1. Trends of field shift and mass shift for the 5d106s 2S1/2→5d96s2 2D5/2 transition in mercury isotope ions with respect to 199Hg+ as the increase of mass number.
图 2 199Hg+和198Hg+离子的超精细能级结构图
Fig. 2. Hyperfine level structure diagram of 199Hg+ and 198Hg+.
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 2S1/2→5d96s2 2D5/2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 2S1/2→5d96s2 2D5/2 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 2S1/2和5d96s2 2D5/2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 2S1/2 and 5d96s2 2D5/2 states of 199Hg+ and 201Hg+.
n 199A1/2 199A5/2 201A1/2 201A5/2 201B5/2 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 2S1/2→5d96s2 2D5/2跃迁谱线的绝对频率值
Table 5. Absolute frequency values of the 5d106s 2S1/2→5d96s2 2D5/2 transition in mercury isotope ions.
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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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