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The Fermi contact term is closely related to the atomic hyperfine constants. It often dominates the hyperfine splittings. The quality of the wave function near the origin and the correlation effect between electrons are two main factors which affect the numerical accuracy of the Fermi contact term. It is not an easy task to compute the Fermi contact term with high precision for a general atom. In the present paper, the Schrödinger equations of the
$ {\text{1s2s3s}}{\;^{\text{4}}}{\text{S}} $ ,$ {\text{1s2s4s}}{\;^{\text{4}}}{\text{S}} $ and$ {\text{1s2s2p}}{\;^{\text{4}}}{\text{P}} $ states of lithium atom and lithium-like ions (Z = 4–10) are solved by using Rayleigh-Ritz variational method in Hylleraas coordinates. The variational energiesenergy converges to an accuracy of 10–13. Then the Fermi contact terms for these states are calculated based on the high precision variation wave functions. In particular, the Drachman global method are adopted in order to improve the convergence of the Fermi contact term. The effect of finite nuclear mass on Fermi contact term, i.e. the first-order mass polarization coefficient is also calculated. The Fermi contact term converges to an accuracy of 10–10, which is the most accurate result at present. Our results can be used as a reference for other theoretical methods or relevant experimental studies.-
Keywords:
- lithium-like ions /
- Fermi contact term /
- Hylleraas coordinates /
- variational method
[1] Lamb W E, Retherford R C 1947 Phys. Rev. 72 241
Google Scholar
[2] Tiesinga E, Mohr P J, Newell D B, Taylor B N 2021 Rev. Mod. Phys. 93 025010
Google Scholar
[3] Pachucki K, Yerokhin V A 2010 Phys. Rev. Lett. 104 070403
Google Scholar
[4] Lu Z T, Mueller P, Drake G W F, Nörtershäuser W, Pieper S C, Yan Z C 2013 Rev. Mod. Phys. 85 1383
Google Scholar
[5] Yan Z C, Nortershauser W, Drake G W F 2008 Phys. Rev. Lett. 100 243002
Google Scholar
[6] Puchalski M, Pachucki K 2013 Phys. Rev. Lett. 111 243001
Google Scholar
[7] Qi X Q, Zhang P P, Yan Z C, Drake G W F, Zhong Z X, Shi T Y, Chen S L, Huang Y, Guan H, Gao K L 2020 Phys. Rev. Lett. 125 183002
Google Scholar
[8] Hiller J, Sucher J, Feinberg G 1978 Phys. Rev. A 18 2399 (Errata 1980 Phys. Rev. A 22 2293; 1979 Phys. Rev. A 20 378)
[9] Drachman R 1981 J. Phys. B:At. Mol. Phys. 14 2733
Google Scholar
[10] Yan Z C, McKenzie D, Drake G W F 1996 Phys. Rev. A 54 1322
Google Scholar
[11] Wang L M, Yan Z C, Qiao H X, Drake G W F 2012 Phys. Rev. A 85 052513
Google Scholar
[12] Yan Z C 2001 J. Phys. B: At. Mol. Opt. Phys. 34 3569
Google Scholar
[13] King F W 2013 Int. J. Quantum Chem. 113 2534
Google Scholar
[14] Zhuo L, Gou B C, Zhu J J 2009 Eur. Phys. J. D. 54 1
Google Scholar
[15] Zhuo L, Chen F, Gou B C 2010 Int. J. Quantum Chem. 110 1108
Google Scholar
[16] Drake G W F 2006 Springer Handbook of Atomic, Molecular, and Optical Physics (New York: Springer Science & Business Media) pp199–219
[17] 王黎明, 周挽平, 严宗朝 2021 中国科学: 物理学 力学 天文学 51 074203
Google Scholar
Wang L M, Zhou W P, Yan Z C 2021 Sci. Sin-Phys. Mech. Astron., 51 074203
Google Scholar
[18] Barrois R, Lüchow A, Kleindienst H 1996 Chem. Phys. Lett. 249 249
Google Scholar
[19] Yan Z C 2003 J. Phys. B:At. Mol. Opt. Phys. 36 2093
Google Scholar
[20] Barrois R, Bekavac S, Kleindienst H 1997 Chem. Phys. Lett. 268 531
Google Scholar
[21] Glass R 1978 J. Phys. B:At. Mol. Phys. 11 3469
Google Scholar
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表 1 锂原子
${\text{1s2s3s}}{\;^{\text{4}}}{\text{S}}$ 态的非相对论变分能量的收敛情况, 核质量取为无穷大Table 1. Study of the convergence of nonrelativistic variational energies of the
${\text{1s2s3s}}{\;^{\text{4}}}{\text{S}}$ state of lithium atom, suppose the mass of the nucleus is infinite.$\varOmega$ Number of terms $E(\varOmega)$/a.u. $R(\varOmega)$ 4 210 –5.212 747 426 455 68 5 462 –5.212 748 209 017 03 6 924 –5.212 748 244 064 12 22.33 7 1716 –5.212 748 246 979 61 12.02 8 3003 –5.212 748 247 204 70 12.95 9 5000 –5.212 748 247 216 73 18.71 10 8000 –5.212 748 247 223 54 1.77 11 12370 –5.212 748 247 224 70 5.87 12 18560 –5.212 748 247 224 71 89.09 13 27130 –5.212 748 247 224 84 0.10 $\infty $ –5.212 748 247 224 8(2) Yan[19] $\infty $ –5.212 748 247 225(5) King[13] 1904 –5.212 748 246 6 
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表 2 类锂离子体系
${\text{1s2s3s}}{\;^{\text{4}}}{\text{S}}$ ,${\text{1s2s4s}}{\;^{\text{4}}}{\text{S}}$ 和${\text{1s2s2p}}{\;^{\text{4}}}{\text{P}}$ 态的非相对论能量, 核质量取为无穷大 (原子单位)Table 2. Nonrelativistic energies of the
${\text{1s2s3s}}{\;^{\text{4}}}{\text{S}}$ ,${\text{1s2s4s}}{\;^{\text{4}}}{\text{S}}$ and${\text{1s2s2p}}{\;^{\text{4}}}{\text{P}}$ states of lithium-like ions, suppose the mass of the nucleus is infinite (in a.u.).Ion ${\text{1s2s3s} }{\;^{\text{4} } }{\text{S} }$ ${\text{1s2s4s} }{\;^{\text{4} } }{\text{S} }$ ${\text{1s2s2p} }{\;^{\text{4} } }{\text{P} }$ ${\text{Li}}$ –5.212 748 247 224 8(2)
–5.212 748 247 225(5)a–5.158 393 473 137 2(4)
–5.158 393 473 2(2)a–5.368 010 154 030 5(2)
–5.368 010 153 9(2)a${\text{B} }{ {\text{e} }^{\text{+} } }$ –9.619 844 613 890 47(2)
–9.619 844 58(3)b–9.462 507 112 198 5(2)
–9.462 507 0(2)b–10.066 652 477 404 7(4)
–10.066 652(4)c${{\text{B}}^{2 + }}$ –15.389 482 739 000 99(1)
–15.389 482 7(0)b–15.136 079 140 469(2)
–15.136 078(8)b–16.267 610 175 163 7(4)
–16.267 610(1)c${{\text{C}}^{3 + }}$ –22.520 800 619 349 34(1)
–22.520 800 5(8)b–22.194 572 767 231(2)
–22.194 572(4)b–23.969 555 014 323 5(3)
–23.969 555(0)c${{\text{N}}^{4 + }}$ –31.013 515 458 020 68(2)
–31.013 515 4(3)b–30.615 962 427 558(2)
–30.615 962(2)b–33.172 011 265 984 3(3)
–33.172 011(2)c${{\text{O}}^{5 + }}$ –40.867 505 629 631 74(1)
–40.867 505 5(9)b–40.399 023 507 082(4)
–40.399 023(2)b–43.874 766 296 947 8(4)
–43.874 766(2)c${{\text{F}}^{6 + }}$ –52.082 709 993 193 28(2)
–52.082 709 9(6)b–51.543 484 487 834(1)
–51.543 484(2)b–56.077 710 886 696 3(4)
–56.077 710(8)c${\text{N}}{{\text{e}}^{7 + }}$ –64.659 094 417 708 25(1)
–64.659 094 3(8)b–64.049 233 634 687(2)
–64.049 233(3)b–69.780 783 217 935 2(2)
–69.780 783(2)c注: a, 文献[19]; b, 文献[18]; c, 文献[20].
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表 3 类锂离子体系
${\text{1s2s3s}}{\;^{\text{4}}}{\text{S}}$ ,${\text{1s2s4s}}{\;^{\text{4}}}{\text{S}}$ 和${\text{1s2s2p}}{\;^{\text{4}}}{\text{P}}$ 态的质量极化系数(原子单位)Table 3. Mass polarization coefficients of the
${\text{1s2s3s}}{\;^{\text{4}}}{\text{S}}$ ,${\text{1s2s4s}}{\;^{\text{4}}}{\text{S}}$ and${\text{1s2s2p}}{\;^{\text{4}}}{\text{P}}$ states of lithium-like ions (in a.u.)Ion ${\varepsilon _1}$ ${\varepsilon _2}$ ${\varepsilon _3}$ ${\text{1s2s3s} }{\;^{\text{4} } }{\text{S} }$ ${\text{Li}}$ –0.019 098 689 45(3) –0.346 258 57(4) 0.174 909 1(5) ${\text{B} }{ {\text{e} }^{\text{+} } }$ –0.028 190 435 40(2) –0.865 722 62(2) 0.897 689 5(2) ${{\text{B}}^{2 + }}$ –0.033 290 376 39(3) –1.642 090 43(4) 2.614 098 9(2) ${{\text{C}}^{3 + }}$ –0.034 289 481 61(2) –2.698 194 80(6) 5.879 648 9(1) ${{\text{N}}^{4 + }}$ –0.031 162 328 61(2) –4.056 337 23(2) 11.361 993 82(3) ${{\text{O}}^{5 + }}$ –0.023 902 016 05(6) –5.738 638 37(2) 19.842 156(2) ${{\text{F}}^{6 + }}$ –0.012 506 894 13(5) –7.767 142 43(4) 32.215 000(3) ${\text{N}}{{\text{e}}^{7 + }}$ 0.003 023 055 68(3) –10.163 856 84(2) 49.489 432 4(2) ${\text{1s2s4s} }{\;^{\text{4} } }{\text{S} }$ ${\text{Li}}$ –0.018 619 468 74(3) –0.272 708 10(3) 0.061 978 4(3) ${\text{B} }{ {\text{e} }^{\text{+} } }$ 0.035 099 392 18(3) –2.306 028 72(2) 35.296 396(2) ${{\text{B}}^{2 + }}$ 1.242 706 153 8(3) –5.493 869 41(3) –41.733 915(2) ${{\text{C}}^{3 + }}$ 2.125 081 744 6(3) –5.285 653 44(3) –13.261 767 4(3) ${{\text{N}}^{4 + }}$ 3.138 461 490 6(2) –6.795 076 43(3) –15.010 263 8(2) ${{\text{O}}^{5 + }}$ 4.330 570 648 1(3) –8.699 185 86(2) –22.191 657 9(2) ${{\text{F}}^{6 + }}$ 5.707 148 244 9(2) –10.854 895 13(2) –33.822 354 0(1) ${\text{N}}{{\text{e}}^{7 + }}$ 7.269 691 423 1(3) –13.212 273 43(5) –50.532 032(2) ${\text{1s2s2p} }{\;^{\text{4} } }{\text{P} }$ ${\text{Li}}$ 0.197 556 864 869(4) –0.743 722 190(3) 0.462 807 90(2) ${\text{B} }{ {\text{e} }^{\text{+} } }$ 0.532 973 840 148(7) –1.776 435 078(2) 1.072 866 24(5) ${{\text{B}}^{2 + }}$ 1.026 077 002 714(1) –3.219 212 106(3) 1.903 522 30(1) ${{\text{C}}^{3 + }}$ 1.675 951 877 937(6) –5.070 384 099(3) 2.956 052 91(3) ${{\text{N}}^{4 + }}$ 2.482 252 239 327(2) –7.329 612 081(3) 4.231 693 02(2) ${{\text{O}}^{5 + }}$ 3.444 825 571 973(3) –9.996 821 053(2) 5.731 157 12(2) ${{\text{F}}^{6 + }}$ 4.563 595 540 943(6) –13.071 999 313(5) 7.454 857 77(5) ${\text{N}}{{\text{e}}^{7 + }}$ 5.838 520 069 099(5) –16.555 151 141(2) 9.403 043 87(2)
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表 4 锂原子
${\text{1s2s3s}}{\;^{\text{4}}}{\text{S}}$ 态的费米接触项的收敛情况, 核质量取为无穷大Table 4. Study of the convergence of nonrelativistic variational energies of the
${\text{1s2s3s}}{\;^{\text{4}}}{\text{S}}$ state of lithium atom, suppose the mass of the nucleus is infinite.$\varOmega$ Number of terms ${f_{\text{c}}}$/a.u. $R\left( \varOmega \right)$ 4 210 114.945 400 79 5 462 114.945 834 87 6 924 114.945 803 15 –13.68 7 1716 114.945 820 05 –1.88 8 3003 114.945 821 97 8.78 9 5000 114.945 821 19 –2.45 10 8000 114.945 820 99 4.04 11 12370 114.945 821 02 –7.78 12 18560 114.945 821 02 8.02 13 27130 114.945 821 01 –0.26 $\infty $ 114.945 821 01(3) Yan[12] $\infty $ 114.945 823(2) King[13] 1904 114.945 79
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表 5 类锂离子体系
${\text{1s2s3s}}{\;^{\text{4}}}{\text{S}}$ ,${\text{1s2s4s}}{\;^{\text{4}}}{\text{S}}$ 和${\text{1s2s2p}}{\;^{\text{4}}}{\text{P}}$ 态的费米接触项, 核质量取为无穷大Table 5. Fermi contact terms of
${\text{1s2s3s}}{\;^{\text{4}}}{\text{S}}$ ,${\text{1s2s4s}}{\;^{\text{4}}}{\text{S}}$ and${\text{1s2s2p}}{\;^{\text{4}}}{\text{P}}$ states of lithium-like ions, suppose the mass of the nucleus is infinite.Ion ${f_{\text{c}}}$/a.u. ${\text{1s2s3s} }{\;^{\text{4} } }{\text{S} }$ ${\text{1s2s4s} }{\;^{\text{4} } }{\text{S} }$ ${\text{1s2s2p} }{\;^{\text{4} } }{\text{P} }$ ${\text{Li}}$ 114.945 821 01(3)
114.945 823(2)a114.756 210 17(5)
114.756 21(2)a112.298 053(6)
112.269 8b${\text{B} }{ {\text{e} }^{\text{+} } }$ 277.186 422 61(3) 273.548 196 54(2) 270.128 559(2) ${{\text{B}}^{2 + }}$ 547.507 013 93(5) 498.901 685 71(3) 533.170 916(3) ${{\text{C}}^{3 + }}$ 953.552 047 46(4) 859.720 632 46(5) 928.427 902(4) ${{\text{N}}^{4 + }}$ 1522.968 426 9(2) 1366.477 417 5(2) 1482.900 739(3)
1482.85c${{\text{O}}^{5 + }}$ 2283.403 624 4(2) 2042.089 145 6(3) 2223.589 964(6)
2223.711d${{\text{F}}^{6 + }}$ 3262.505 305 2(2) 2910.646 989 8(5) 3177.495 846(5)
3177.40c${\text{N}}{{\text{e}}^{7 + }}$ 4487.921 213 6(2) 3996.354 065 8(2) 4371.618 515(2)
4371.347d注: a, 文献[12]; b, 文献[21]; c, 文献[15]; d, 文献[14].
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表 6 类锂离子
${\text{1s2s3s}}{\;^{\text{4}}}{\text{S}}$ ,${\text{1s2s4s}}{\;^{\text{4}}}{\text{S}}$ 和${\text{1s2s2p}}{\;^{\text{4}}}{\text{P}}$ 态的费米接触项的一阶质量极化系数Table 6. First-order mass polarization coefficients of Fermi contact terms of
${\text{1s2s3s}}{\;^{\text{4}}}{\text{S}}$ ,${\text{1s2s4s}}{\;^{\text{4}}}{\text{S}}$ and${\text{1s2s2p}}{\;^{\text{4}}}{\text{P}}$ states of lithium-like ions.Ion 一阶质量极化系数$ f_{\text{c}}^1 $/a.u. ${\text{1s2s3s} }{\;^{\text{4} } }{\text{S} }$ ${\text{1s2s4s} }{\;^{\text{4} } }{\text{S} }$ ${\text{1s2s2p} }{\;^{\text{4} } }{\text{P} }$ ${\text{Li}}$ 1.895 4(2) 0.790 2(5) 4.832 8(3) ${\text{B} }{ {\text{e} }^{\text{+} } }$ 9.284 0(3) 103.777(2) 13.732(1) ${{\text{B}}^{2 + }}$ 27.430 7(2) 154.971(2) 28.485(4) ${{\text{C}}^{3 + }}$ 63.325 0(3) 41.512(2) 50.36(3) ${{\text{N}}^{4 + }}$ 125.663(3) –0.590 2(3) 80.68(2) ${{\text{O}}^{5 + }}$ 224.854(2) –60.458(4) 120.73(3) ${{\text{F}}^{6 + }}$ 373.014(2) –156.223(3) 171.84(2) ${\text{N}}{{\text{e}}^{7 + }}$ 583.972(2) –302.256(5) 235.29(2)
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[1] Lamb W E, Retherford R C 1947 Phys. Rev. 72 241
Google Scholar
[2] Tiesinga E, Mohr P J, Newell D B, Taylor B N 2021 Rev. Mod. Phys. 93 025010
Google Scholar
[3] Pachucki K, Yerokhin V A 2010 Phys. Rev. Lett. 104 070403
Google Scholar
[4] Lu Z T, Mueller P, Drake G W F, Nörtershäuser W, Pieper S C, Yan Z C 2013 Rev. Mod. Phys. 85 1383
Google Scholar
[5] Yan Z C, Nortershauser W, Drake G W F 2008 Phys. Rev. Lett. 100 243002
Google Scholar
[6] Puchalski M, Pachucki K 2013 Phys. Rev. Lett. 111 243001
Google Scholar
[7] Qi X Q, Zhang P P, Yan Z C, Drake G W F, Zhong Z X, Shi T Y, Chen S L, Huang Y, Guan H, Gao K L 2020 Phys. Rev. Lett. 125 183002
Google Scholar
[8] Hiller J, Sucher J, Feinberg G 1978 Phys. Rev. A 18 2399 (Errata 1980 Phys. Rev. A 22 2293; 1979 Phys. Rev. A 20 378)
[9] Drachman R 1981 J. Phys. B:At. Mol. Phys. 14 2733
Google Scholar
[10] Yan Z C, McKenzie D, Drake G W F 1996 Phys. Rev. A 54 1322
Google Scholar
[11] Wang L M, Yan Z C, Qiao H X, Drake G W F 2012 Phys. Rev. A 85 052513
Google Scholar
[12] Yan Z C 2001 J. Phys. B: At. Mol. Opt. Phys. 34 3569
Google Scholar
[13] King F W 2013 Int. J. Quantum Chem. 113 2534
Google Scholar
[14] Zhuo L, Gou B C, Zhu J J 2009 Eur. Phys. J. D. 54 1
Google Scholar
[15] Zhuo L, Chen F, Gou B C 2010 Int. J. Quantum Chem. 110 1108
Google Scholar
[16] Drake G W F 2006 Springer Handbook of Atomic, Molecular, and Optical Physics (New York: Springer Science & Business Media) pp199–219
[17] 王黎明, 周挽平, 严宗朝 2021 中国科学: 物理学 力学 天文学 51 074203
Google Scholar
Wang L M, Zhou W P, Yan Z C 2021 Sci. Sin-Phys. Mech. Astron., 51 074203
Google Scholar
[18] Barrois R, Lüchow A, Kleindienst H 1996 Chem. Phys. Lett. 249 249
Google Scholar
[19] Yan Z C 2003 J. Phys. B:At. Mol. Opt. Phys. 36 2093
Google Scholar
[20] Barrois R, Bekavac S, Kleindienst H 1997 Chem. Phys. Lett. 268 531
Google Scholar
[21] Glass R 1978 J. Phys. B:At. Mol. Phys. 11 3469
Google Scholar
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