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The wave functions, energy levels, and oscillator strengths of Be+ ions and Li atoms are calculated by using a relativistic potential model, which is named the relativistic configuration interaction plus core polarization method (RCICP). The calculated energy levels in this work are in good agreement with experimental levels tabulated in NIST Atomic Spectra Database, and the difference appears in the sixth digit after the decimal point. The present oscillator strengths are in good agreement with the existing theoretical and experimental results. By means of these energy levels and oscillator strengths, the electric-dipole static polarizabilities and hyperpolarizabilities of the ground states are determined. The contributions of different intermediate states to the hyperpolarizabilities of the ground state are further discussed. For Be+ ions, the present electric-dipole polarizability and hyperpolarizability are in good agreement with the results calculated by Hartree-Fock plus core polarization method, the finite field method and relativistic many-body method. The largest contribution to the hyperpolarizability is the term of
$\alpha _{\text{0}}^{\text{1}}{\beta _0}$ . For Li atoms, the present electric-dipole polarizability is in good agreement with the available theoretical and experimental results. However, the present hyperpolarizability is different from the other theoretical results significantly. Moreover, the hyperpolarizabilities calculated by different theoretical methods are quite different. The biggest difference is more than one order of magnitude. In order to explain the reason for these differences, we analyze the contributions of different intermediate states to the hyperpolarizability in detail. It is found that the sum of the contributions of the 2s→npj$\left( {n \geqslant 3} \right)$ and npj→ndj$\left( {n \geqslant 3} \right)$ to hyperpolarizability is approximately equal to that term of$\alpha _{\text{0}}^{\text{1}}{\beta _0}$ . The total hyperpolarizability, which is the difference between the sum of the contributions of the 2s→npj$\left( {n \geqslant 3} \right)$ and npj→ndj$\left( {n \geqslant 3} \right)$ to hyperpolarizability and$\alpha _{\text{0}}^{\text{1}}{\beta _0}$ , is relatively small. Consequently, this difference magnifies the calculated error. If the uncertainties of the transition matrix elements are less than 0.1%, the uncertainty of hyperpolarizability is more than 100%. Therefore, the differences of hyperpolarizabilities for the ground state of Li atoms, calculated by various theoretical methods, are more than 100% or one order of magnitude.-
Keywords:
- electric-dipole polarizability /
- hyperpolarizability /
- Be+ ions /
- Li atoms /
- the relativistic model potential method
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表 1 Be+离子和Li原子的截断参数
${\rho _{l, j}}$ (单位: a.u.)Table 1. Cut-off parameters
${\rho _{l, j}}$ of Be+ ions and Li atoms (in a.u.).State j ρl, j Be+ Li 2s 1/2 0.9552 1.40880 2p 1/2 0.8789 1.28466 3/2 0.8775 1.28396 3d 3/2 0.1287 2.324 5/2 0.1284 2.330 表 2 Be+离子和Li原子基态和部分低激发态相对于原子实的能级, 实验值(Exp.)是来自于NIST的数据(单位: a.u.)
Table 2. Energy levels of the ground state and some low-lying states of Be+ ions and Li atoms relative to atomic core. Experimental values (Exp.) are from the NIST data (in a.u.).
State j Be+ Li RCICP Expt.[42] RCICP Expt.[42] 2s 1/2 –0.66924767 –0.66924755 –0.1981419 –0.1981419 2p 1/2 –0.52376962 –0.52376949 –0.1302358 –0.1302358 3/2 –0.52373967 –0.52373953 –0.1302343 –0.1302343 3s 1/2 –0.26719384 –0.26723337 –0.0741684 –0.0741817 3p 1/2 –0.22954214 –0.22958234 –0.0572264 –0.0572354 3/2 –0.22953331 –0.22957356 –0.0572260 –0.0572354 3d 3/2 –0.22247809 –0.22247805 –0.0556055 –0.0556057 5/2 –0.22247565 –0.22247565 –0.0556051 –0.0556055 4s 1/2 –0.14313397 –0.14315285 –0.0386096 –0.0386151 4p 1/2 –0.12811380 –0.12813485 –0.0319693 –0.0319744 3/2 –0.12811009 –0.12813115 –0.0319691 –0.0319744 4d 3/2 –0.12512357 –0.12512455 –0.0308153 –0.0312735 5/2 –0.12512257 –0.12512345 –0.0308152 –0.0312734 5s 1/2 –0.08905659 –0.08906605 –0.0236202 –0.0236365 5p 1/2 –0.08159826 –0.08160960 –0.0203583 –0.0203739 3/2 –0.08159637 –0.08160765 –0.0203583 –0.0203739 5d 3/2 –0.08006698 –0.08006725 –0.0124153 –0.0200122 5/2 –0.08006648 –0.08006670 –0.0124152 –0.0200122 表 3 Be+离子基态和部分低激发态之间跃迁的振子强度, “Diff.”表示用RCICP方法计算的结果与NIST结果之差的百分比
Table 3. Oscillator strengths of transitions between the ground state and some low-lying states of Be+ ions. “Diff.” represents the difference in percentage form calculated by RCICP method and NIST results.
Transitions RCICP NIST[42] Theor.[25] Diff./% 2s1/2→2p1/2 0.16624 0.16596 0.1661 0.17 2s1/2→2p3/2 0.33258 0.33198 0.3322 0.18 2s1/2→3p1/2 0.02760 0.02768 0.0277 0.29 2s1/2→3p3/2 0.05517 0.05540 0.0553 0.42 2p1/2→3s1/2 0.06434 0.06438 0.0644 0.06 2p3/2→3s1/2 0.06436 0.06438 0.0644 0.03 2p1/2→4s1/2 0.01022 0.01039 0.0102 1.64 2p3/2→4s1/2 0.01022 0.01039 0.0102 1.64 2p1/2→3d3/2 0.6320 0.6320 0.6319 0.00 2p3/2→3d3/2 0.0632 0.0632 0.0632 0.00 2p3/2→3d5/2 0.5689 0.5689 0.5688 0.00 3s1/2→3p1/2 0.2768 0.2767 0.2767 0.04 3s1/2→3p3/2 0.5538 0.5535 0.5535 0.05 3p1/2→3d3/2 0.08069 0.08113 0.0811 0.54 3p3/2→3d3/2 0.08059 0.08103 0.081 0.54 3p3/2→3d5/2 0.07256 0.07294 0.073 0.52 3p1/2→4s1/2 0.1346 0.1347 0.1346 0.07 3p3/2→4s1/2 0.1346 0.1347 0.1346 0.07 表 4 Li原子基态和部分低激发态之间跃迁的振子强度, “Diff.”表示用RCICP方法计算的结果与NIST结果之间差别的百分比
Table 4. Oscillator strengths of transitions between the ground state and some low-lying states of Li atoms. “Diff.” represents the difference in percentage form calculated by RCICP method and NIST results.
Transitions RCICP NIST[42] Theor.[29] Diff./% 2s1/2→2p1/2 0.24915 0.24899 0.2490 0.06 2s1/2→2p3/2 0.49832 0.49797 0.4981 0.07 2s1/2→3p1/2 0.00157 0.00157 0.0016 0.00 2s1/2→3p3/2 0.00313 0.00314 0.0032 0.32 2p1/2→3s1/2 0.11058 0.11050 0.1106 0.07 2p3/2→3s1/2 0.11059 0.11050 0.1106 0.08 2p1/2→4s1/2 0.01285 0.01283 0.0128 0.16 2p3/2→4s1/2 0.01285 0.01283 0.0128 0.16 2p1/2→3d3/2 0.63876 0.63858 0.6386 0.03 2p3/2→3d3/2 0.06388 0.06386 0.0639 0.03 2p3/2→3d5/2 0.57489 0.57472 0.5747 0.03 3s1/2→3p1/2 0.40512 0.4051 0.405 0.00 3s1/2→3p3/2 0.81027 0.8100 0.810 0.03 3p1/2→3d3/2 0.07397 0.0733 0.0744 0.91 3p3/2→3d3/2 0.00740 0.00736 0.0074 0.54 3p3/2→3d5/2 0.06657 0.0663 0.0669 0.41 3p1/2→4s1/2 0.22325 0.2230 0.2232 0.11 3p3/2→4s1/2 0.22325 0.2230 0.2232 0.11 3d3/2→4p1/2 0.01453 0.01497 0.015 2.94 3d3/2→4p3/2 0.00290 0.00299 0.003 3.01 3d5/2→4p3/2 0.01743 0.01796 0.018 2.95 表 5 Be+离子基态的电偶极极化率
$\alpha _{\rm{0}}^{\rm{1}}$ 和超极化率${\gamma _{\rm{0}}}$ , “Diff.”表示用RCICP方法计算的γ0与其它理论数据之间差别的百分比, 括号内的值表示不确定度Table 5. Electric-dipole polarizability and hyperpolarizability of the ground state of Be+ ions. “Diff.” represents the difference of γ0 in percentage form calculated by RCICP and other theoretical method. The values in parentheses indicate the uncertainties.
Method $\alpha _{\rm{0}}^{\rm{1}}$/a.u. γ0/a.u. Diff./% RCICP 24.504(32) –11529.971(84) Coulomb approximation[43] 24.77 Variation-perturbation Hylleraas CI[44] 24.5 Hylleraas[24] 24.489 Asymptotic correct wave function[45] 24.91 Variation-perturbation FCCI[46,47] 24.495 Hartree-Fock plus core polarization[22] 24.493 –11511 0.16 Hylleraas[22] 24.4966(1) –11521.30(3) 0.08 Relativistic many-body calculation[25] 24.483(4) –11496(6) 0.29 The finite field method[30] 24.5661 –11702.31 1.49 表 6 中间态对Be+离子基态超极化率的贡献, RCICPC表示2s→2pj, 2pj→3dj跃迁的约化矩阵元用NIST[42]结果替换之后计算的结果, 括号内的值表示RCICP相对于RCICPC的不确定度(单位: a.u.)
Table 6. Contributions to the hyperpolarizability of the ground state of Be+ ions. RCICPC represents that the reduced matrix elements of the 2s→2pj、2pj→3dj transitions are replaced by NIST[42] results. The values in parentheses indicate the uncertainties of RCICP relative to RCICPC (in a.u.).
Contr. RCICP RCICPC RMBT[25] $\tfrac{1}{18}$T (s, p1/2, s, p1/2) 34.34(2) 34.32 32.605(53) $-\tfrac{1}{18}$T (s, p1/2, s, p3/2) 68.68(5) 68.63 68.886(92) $-\tfrac{1}{18}$T (s, p3/2, s, p1/2) 68.68(5) 68.63 68.886(92) $\tfrac{1}{18}$T (s, p3/2, s, p3/2) 137.35(10) 137.25 137.669(109) $T({\rm{s, }}{{\rm{p}}_{j'}}, {\rm{ s}}, {\rm{ }}{{\rm{p}}_{j''}})$ 308.04(12) 308.83 308.046(178) $\tfrac{1}{18}$T (s, p1/2, d3/2, p1/2) 202.75(16) 202.59 202.031(121) $\tfrac{1}{18\sqrt{10} }$T (s, p1/2, d3/2, p3/2) 40.55(4) 40.51 40.403(18) $\tfrac{1}{18\sqrt{10} }$T (s, p3/2, d3/2, p1/2) 40.55(4) 40.51 40.403(18) $\tfrac{1}{180}$T (s, p3/2, d3/2, p3/2) 8.11(1) 8.10 8.080(3) $\tfrac{1}{30}$ T (s, p3/2, d5/2, p3/2) 437.85(40) 437.45 438.434(148) $T({\rm{s}}, {{\rm{p}}_{j'}}, {{\rm{d}}_j}, {{\rm{p}}_{j''}})$ 729.79(43) 729.17 729.351(192) $\alpha _{\rm{0}}^{\rm{1}}{\beta _0}$ 1999.67(6.95) 1992.72 1995.743(382) γ0(2 s) –11529(84) –11456 –11496(6) 表 7 Li原子基态的电偶极极化率
$\alpha _{\rm{0}}^{\rm{1}}$ 和超极化率${\gamma _{\rm{0}}}$ , 括号内的值表示不确定度(单位: a.u.)Table 7. Electric-dipole polarizability and hyperpolarizability of the ground state of Li atoms. The values in parentheses indicate the uncertainties (in a.u.).
Method $\alpha _{\rm{0}}^{\rm{1}}$ γ0 RCICP 164.05(8) 1920(3264) The coupled cluster (all single, double and triple substitution)[1] 164.19 2880 Finite-field quadratic configuration interaction[1] 164.32 1020 Hylleraas[31] 164.112(1) 3060(40) The relativistic coupled-cluster method[48] 164.23 Relativistic variation perturbation[49] 164.084 Relativistic all-order methods[29] 164.16(5) Variation perturbation[33] 164.10 3000 Semiempirical pseudopotentials[26] 164.08 65000 Frozen core Hamiltonian with a semiempirical polarization potential[50] 164.21 Finite-field fourth-order many-body perturbation theory[34] 164.5 4300 Configuration interaction[35] 164.9 37000 Relativistic ab initio methods[51] 164.0(1) The restricted Hartree-Fock[32] 170.1 –55000 The Rydberg-Klein-Rees inversion method with the quantum defect theory[52] 164.14 3390 Exp.[53] 164(3) Exp.[54] 164.2(11) 表 8 中间态对Li原子基态超极化率的贡献, RCICPC表示2s→2pj, 2pj→3dj跃迁的约化矩阵元用NIST[42]结果替换之后计算的结果, “Diff.”表示RCICP与RCICPC之间差别的百分比, 括号内的值表示RCICP相对于RCICPC的不确定度
Table 8. Contributions to the hyperpolarizability of the ground state of Li atoms. RCICPC represents that the reduced matrix elements of 2s→2pj, 2pj→3dj transitions are replaced by NIST[42] results. “Diff.” represents the difference in percentage form between RCICP method and RCICPC. The values in parentheses indicate the uncertainties of RCICP relative to RCICPC.
Contr. RCICP/a.u. RCICPC/a.u. Diff. /% $ \frac{1}{18} $T (s, p1/2, s, p1/2) 8314(2) 8312 0.03 $ -\frac{1}{18} $T (s, p1/2, s, p3/2) 16629(5) 16624 0.03 $ -\frac{1}{18} $T (s, p3/2, s, p1/2) 16629(5) 16624 0.03 $ \frac{1}{18} $T (s, p3/2, s, p3/2) 33259(11) 33248 0.03 $T({\rm{s}}, {{\rm{p}}_{j'}}, {\rm{s}}, {{\rm{p}}_{j''}})$ 74833(13) 74809 0.02 $ \frac{1}{18} $T (s, p1/2, d3/2, p1/2) 33812(13) 33799 0.04 $ \frac{1}{18\sqrt{10}} $T (s, p1/2, d3/2, p3/2) 6762(3) 6759 0.04 $ \frac{1}{18\sqrt{10}} $T (s, p3/2, d3/2, p1/2) 6762(3) 6759 0.04 $ \frac{1}{180} $T (s, p3/2, d3/2, p3/2) 1352(0) 1352 0.00 $ \frac{1}{30} $ T (s, p3/2, d5/2, p3/2) 73033(40) 72993 0.05 $T({\rm{s}}, {{\rm{p}}_{j'}}, {{\rm{d}}_j}, {{\rm{p}}_{j''}})$ 121723(42) 121661 0.03 $\alpha _{\rm{0}}^{\rm{1}}{\beta _0}$ 196396(268) 196128 0.14 γ0(2 s) 1920(3264) 4109 170 -
[1] Kassimi E B, Thakkar A J 1994 Phys. Rev. A 50 2948Google Scholar
[2] Dür W, Briegel H J 2002 Phys. Rev. Lett. 90 067901Google Scholar
[3] Childress L, Taylor J M, Sørensen A S, Lukin M D 2005 Phys. Rev. A 72 052330Google Scholar
[4] Jiang L, Taylor J M, Sørensen A S, Lukin M D 2007 Phys. Rev. A 76 062323Google Scholar
[5] Gorshkov A V, Rey A M, Daley A J, Boyd M M, Ye J, Zoller P, Lukin M D 2009 Phys. Rev. Lett. 102 110503Google Scholar
[6] Wineland D J, Drullinger R E, Walls F L 1978 Phys. Rev. Lett. 40 1639Google Scholar
[7] Neuhauser W, Hohenstatt M, Toschek P E, Dehmelt H 1978 Phys. Rev. Lett. 41 233Google Scholar
[8] Flury J 2016 J. Phys. Conf. Ser. 723 012051Google Scholar
[9] Bregolin F, Milani G, Pizzocaro M, Rauf B, Thoumany P, Levi F, Calonico D 2017 J. Phys. Conf. Ser. 841 012015Google Scholar
[10] Pihan-Le Bars H, Guerlin C, Bailey Q G, Bize S, Wolf P 2017 arXiv: 1701.06902[gr-qc]
[11] Roberts B M, Blewitt G, Dailey C, Murphy M, Pospelov M, Rollings A, Sherman J, Williams W, Derevianko A 2017 Nat. Commun. 8 1195Google Scholar
[12] Bloom B J, Nicholson T L, Williams J R, Campbell S L, Bishof M, Zhang X, Zhang W, Bromley S L, Ye J 2014 Nature 506 71Google Scholar
[13] Chou C W, Hume D B, Koelemeij J C J, Wineland D J, Rosenband T 2010 Phys. Rev. Lett. 104 070802Google Scholar
[14] Brewer S M, Chen J S, Hankin A M, Clements E R, Chou C W, Wineland D J, Hume D B, Leibrandt D R 2019 Phys. Rev. Lett. 123 033201Google Scholar
[15] Hinkley N, Sherman J A, Phillips N B, Schioppo M, Lemke N D, Beloy K, Pizzocaro M, Oates C W, Ludlow A D 2013 Science 341 1215Google Scholar
[16] Brusch A, Le T R, Baillard X, Fouché M, Lemonde P 2006 Phys. Rev. Lett. 96 103003Google Scholar
[17] Barbe Z W, Lemke J E, Polt N D 2008 Phys. Rev. Lett. 100 103002Google Scholar
[18] Westergaard P G, Lodewyck J, Lorini L, Lecallier A, Burt E A, Zawada M 2011 Phys. Rev. Lett. 106 210801Google Scholar
[19] Derevianko A, Katori H 2011 Rev. Mod. Phys. 83 331Google Scholar
[20] Katori H, Takamoto M, Pal'Chikov V G, Ovsiannikov V D 2003 Phys. Rev. Lett. 91 173005Google Scholar
[21] Porsev S G, Safronova M S, Safronova U I, Kozlov M G 2018 Phys. Rev. Lett. 120 063204Google Scholar
[22] Tang L Y, Zhang J Y, Yan Z C, Shi T Y, Babb J F, Mitroy J 2009 Phys. Rev. A 80 042511Google Scholar
[23] Tang L Y, Yan Z C, Shi T Y, Babb J F 2014 Phys. Rev. A 90 012524Google Scholar
[24] Tang L Y, Zhang J Y, Yan Z C, Shi T Y, Mitroy J 2010 Phys. Rev. A 81 042521Google Scholar
[25] Safronova U I, Safronova M S 2013 Phys. Rev. A 87 032502Google Scholar
[26] Fuentealba P, Reyes O 1993 J. Phys. B: At. Mol. Opt. Phys. 26 2245Google Scholar
[27] Jiang J, Mitroy J, Cheng Y J, Bromley M W J 2016 Phys. Rev. A 94 062514Google Scholar
[28] Mitroy J, Safronova M S, Clark C W 2010 J. Phys. B: At. Mol. Opt. Phys. 43 202001Google Scholar
[29] Safronova M S, Safronova U I, Clark C W 2012 Phys. Rev. A 86 042505Google Scholar
[30] Yin D, Zhang Y H, Li C B, Gao K L, Shi T Y 2016 Sci. China Phys. Mech. 59 690011Google Scholar
[31] Tang L Y, Yan Z C, Shi T Y, Babb J F 2009 Phys. Rev. A 79 062712Google Scholar
[32] Stiehler J, Hinze J 1995 J. Phys. B: At. Mol. Opt. Phys. 28 4055Google Scholar
[33] Pipin J, Bishop D M 1992 Phys. Rev. A 45 2736Google Scholar
[34] Maroulis G, Thakkar A J 1989 J. Phys. B: At. Mol. Opt. Phys. 22 2439Google Scholar
[35] Nicolaides C A, Themelis S I 1993 J. Phys. B: At. Mol. Opt. Phys. 26 2217Google Scholar
[36] Kaneko S 1977 J. Phys. B: At. Mol. Opt. Phys. 10 3347Google Scholar
[37] Mitroy J, Zhang J Y, Bromley M W J 2008 Phys. Rev. A 77 032512Google Scholar
[38] Bhatia A K, Drachman R J 1997 Can. J. Phys. 75 11Google Scholar
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