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利用相对论组态相互作用模型势方法计算了B2+和B+离子的波函数、能级和振子强度, 进一步得到B2+离子2s1/2, 2p1/2, 2p3/2, 3s1/2态的电偶极极化率和基态2s1/2的超极化率, 以及B+离子2s2 1S0与2s2p 3P0态的电偶极极化率. B2+离子2p1/2和2p3/2的偶极极化率为负值, 基态2s1/2的超极化率的贡献主要来自于与极化率相关的
$ {{\rm{\alpha }}}^{1}{{\rm{\beta }}}_{0} $ 项. 对于B+离子, 钟跃迁2s2p 3P0 → 2s2 1S0在室温下的黑体辐射频移是0.01605 Hz, 该黑体辐射频移比碱土金属原子的钟跃迁黑体辐射频移小1—2个数量级.The wave functions, energy levels, and oscillator strengths of B2+ ions and B+ ions are calculated by using a relativistic potential model, which is named the relativistic configuration interaction plus core polarization (RCICP) method.The presently calculated energy levels are in very good agreement with experimental energy levels tabulated in NIST Atomic Spectra Database, with difference no more than 0.05%.The presently calculated oscillator strengths agree very well with NIST and some available theoretical results. The difference is no more than 0.6%. By using these energy levels and oscillator strengths, the electric-dipole static polarizability of the 2s1/2, 2p1/2, 2p3/2, and 3s1/2 state and static hyperpolarizability of the ground state 2s1/2 for B2+ ion, as well as electric-dipole static polarizability of the 2s2 1S0 state and 2s2p 3P0 state for B+ ion are determined, respectively. The polarizability of the 2p1/2 state and 2p3/2 state of B2+ ion are negative. The main reason is that the absorption energy of the 2p1/2,3/2 → 2s1/2 resonance transition is negative. The contribution to the polarizability of the 2p1/2 state and 2p3/2 state are both negative. For the tensor polarizability of the 2p3/2 state, the main contribution from the 2p3/2 → 2s1/2 transition and 2p3/2 → 3d5/2 transition are 2.4963 a.u. and –0.2537 a.u., respectively, and the present RCICP result is 2.1683 a.u. The largest contribution to the hyperpolarizability of the ground state 2s1/2 originates from the term of$ {\alpha }^{1}{\beta }_{0} $ . The electric-dipole static polarizability of the 2s2 1S0 state and 2s2p 3P0 state of B+ ion are 9.6220 a.u. and 7.7594 a.u., respectively. The presently calculated blackbody radiation (BBR) shift of the 2s2p 3P0 → 2s2 1S0 clock transition is 0.01605 Hz. This BBR shift is one or two orders of magnitude smaller than that for alkaline-earth-metal atom.-
Keywords:
- electric-dipole polarizability /
- hyperpolarizability /
- B2+ ions /
- B+ ions
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表 1 B2+离子的截断参数
$ {\rho }_{l, j} $ (单位: a.u.)Table 1. Cut-off parameters
$ {\rho }_{l, j} $ of B2+ ions (in a.u.).State j $ {\rho }_{l, j} $ 2s 1/2 0.72951 2p 1/2 0.67398 3/2 0.67164 3d 3/2 0.91441 5/2 0.91355 
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表 2 B2+离子的基态和部分低激发态相对于原子实的能级, 实验值(Expt.) [33]是来自于NIST的数据(单位: a.u.), “Diff.”表示用RCICP方法计算的结果与NIST结果之差的百分比
Table 2. Energy levels of the ground state and some low-lying states of B2+ ions relative to atomic core. Experimental values (Expt.) [33] are from the NIST data (in a.u.). “Diff.” denotes the difference in percentage from calculated by RCICP method and NIST results.
State j RCICP Expt.[33] Diff./% 2s 1/2 –1.3939235 –1.3939235 0 2p 1/2 –1.1735867 –1.1735867 0 3/2 –1.1734313 –1.1734313 0 3s 1/2 –0.5728008 –0.5728632 0.01 3p 1/2 –0.5146980 –0.5147730 0.01 3/2 –0.5146520 –0.5147274 0.01 3d 3/2 –0.5005686 –0.5005686 0 5/2 –0.5005553 –0.5005553 0 4s 1/2 –0.3108609 –0.3108905 0.01 4p 1/2 –0.2874707 –0.2875098 0.01 3/2 –0.2874514 –0.2874920 0.01 4d 3/2 –0.2815308 –0.2815324 0 5/2 –0.2815252 –0.2815268 0 4f 5/2 –0.2812848 –0.2812705 0.01 7/2 –0.2812820 –0.2812676 0.01 5s 1/2 –0.1948639 –0.1948793 0.01 5p 1/2 –0.1831864 –0.1832067 0.01 3/2 –0.1831765 –0.1831970 0.01 5d 3/2 –0.1801535 –0.1801552 0 5/2 –0.1801507 –0.1801523 0 5f 5/2 –0.1800204 –0.1800138 0 7/2 –0.1800190 –0.1800124 0
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表 3 B2+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[41]之差的百分比
Table 3. Oscillator strengths of transitions between the ground state and some low-lying states of B2+ ions. “Diff.” represents the difference in percentage form calculated by RCICP method and NIST results.
Transitions RCICP RMBPT[42] HR[43] NIST[41] Diff./% 2s1/2→2p1/2 0.121251 0.121101 0.121076 0.12099 0.22 2s1/2→2p3/2 0.242723 0.242501 0.242399 0.24215 0.24 2s1/2→3p1/2 0.051084 0.05108 0.01 2s1/2→3p3/2 0.102061 0.10240 0.33 2p1/2→3s1/2 0.046308 0.046288 0.04636 0.11 2p1/2→3d3/2 0.637937 0.63800 0.01 2p1/2→4s1/2 0.008193 0.008233 0.49 2p1/2→4d3/2 0.122573 0.12280 0.19 2p3/2→3s1/2 0.046346 0.046338 0.04636 0.03 2p3/2→3d3/2 0.063806 0.06381 0.01 2p3/2→3d5/2 0.574284 0.57430 0 2p3/2→4s1/2 0.008198 0.008236 0.46 2p3/2→4d3/2 0.012256 0.01228 0.20 2p3/2→4d5/2 0.110323 0.11050 0.16 3s1/2→3p1/2 0.203293 0.20310 0.10 3s1/2→3p3/2 0.406942 0.4068 0.04 3s1/2→4p1/2 0.048745 0.04850 0.51 3s1/2→4p3/2 0.097357 0.09700 0.37
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表 4 B2+离子基态与部分低激发态的静态电偶极标量极化率与张量极化率以及主要跃迁的贡献(单位: a.u.)
Table 4. Static electric-dipole scalar and tensor polarizability of the ground state and some low-lying state of B2+ ions and breakdowns of the contributions of individual transitions (in a.u.).
2s1/2 2p1/2 2p3/2 3s1/2 Contr. $ {\alpha }_{}^{{\rm{S}}} $ FCPC [44] Contr. $ {\alpha }_{}^{{\rm{S}}} $ Contr. $ {\alpha }_{}^{{\rm{S}}} $ $ {\alpha }^{{\rm{T}}} $ Contr. $ {\alpha }_{}^{{\rm{S}}} $ 2p1/2 2.4975 2.4953[44] 2s1/2 –2.4975 2s1/2 –2.4963 2.4963 3p1/2 60.218 2p3/2 4.9926 4.9872[44] 3d3/2 1.4084 3d5/2 1.2684 –0.2537 3p3/2 120.35 Remains 0.3433 0.3453[44] Remains 0.4959 Remains 0.6371 –0.0743 Remains 2.3125 Core[30] 0.0195 0.0195[44] Core 0.0195 Core 0.0195 Core 0.0195 Total 7.8529 7.8473[44] Total –0.5737 Total –0.5713 2.1683 Total 182.90 CICP[45] 7.8460 –0.56938 182.94 SCC[46] 7.85 FCG[47] 7.8591
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表 5 B2+离子基态的超极化率及其中间态对超极化率的贡献(单位: a.u.)
Table 5. Hyperpolarizability of the ground state of B2+ ion and the contributions to the hyperpolarizability (in a.u.).
Contributions $ {\gamma }_{0}\left(2{\rm{s}}\right) $ $ {\gamma }_{0}^{{\rm{C}}}\left(2{\rm{s}}\right) $ $ \dfrac{1}{18}T({\rm{s}}, {{\rm{p}}}_{1/2}, {\rm{s}}, {{\rm{p}}}_{1/2}) $ 1.251(1) 1.250 $ -\dfrac{1}{18}T({\rm{s}}, {{\rm{p}}}_{1/2}, {\rm{s}}, {{\rm{p}}}_{3/2}) $ 2.501(1) 2.500 $ -\dfrac{1}{18}T({\rm{s}}, {{\rm{p}}}_{3/2}, {\rm{s}}, {{\rm{p}}}_{1/2}) $ 2.501(1) 2.500 $ \dfrac{1}{18}T({\rm{s}}, {{\rm{p}}}_{3/2}, {\rm{s}}, {{\rm{p}}}_{3/2}) $ 5.001(1) 5.000 $T({\rm{s} }, { {\rm{p} } }_{ {j}^{'} }, {\rm{s} }, { {\rm{p} } }_{ {j}^{''} })$ 11.255(5) 11.250 $\dfrac{1}{18}T({\rm{s} }, { {\rm{p} } }_{1/2}{, {\rm{d} } }_{3/2}, { {\rm{p} } }_{1/2})$ 9.588(8) 9.580 $\dfrac{1}{18\sqrt{10} }T({\rm{s} }, { {\rm{p} } }_{1/2}{, {\rm{d} } }_{3/2}, { {\rm{p} } }_{3/2})$ 1.917(2) 1.915 $\dfrac{1}{18\sqrt{10} }T({\rm{s} }, { {\rm{p} } }_{3/2}{, {\rm{d} } }_{3/2}, { {\rm{p} } }_{1/2})$ 1.917(2) 1.915 $\dfrac{1}{180}T({\rm{s} }, { {\rm{p} } }_{3/2}{, {\rm{d} } }_{3/2}, { {\rm{p} } }_{3/2})$ 0.383(1) 0.382 $\dfrac{1}{30}T({\rm{s} }, { {\rm{p} } }_{3/2}{, {\rm{d} } }_{5/2}, { {\rm{p} } }_{3/2})$ 20.692(16) 20.676 $T({\rm{s} }, { {\rm{p} } }_{ {j}^{'} }{, {\rm{d} } }_{j}, { {\rm{p} } }_{ {j}^{''} })$ 34.497(28) 34.469 $ {\alpha }^{1}{\beta }_{0} $ 134.364(586) 133.778 RCICP –1063.346(6.645) –1056.701 UCHF[50] –1160 CHF[49] –1120
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表 6 B+基态和部分低激发态相对于原子实的能级值, 实验值(Expt.) [51]是来自于NIST的数据(单位: a.u.), “Diff.”表示用RCICP方法计算的结果与NIST结果之差的百分比
Table 6. Energy levels of the ground state and some low-lying states of B+ ions relative to atomic core. Experimental values (Expt.) are from the NIST data (in a.u.). “Diff.” denotes the difference in percentage from calculated by RCICP method and NIST results.
State RCICP Expt.[51] Diff./% 2$ {{\rm{s}}}^{2}{{}_{}{}^{1}{\rm{S}}}_{0} $ –2.318347 –2.318347 0 2s2p$ {{}_{}{}^{3}{\rm{P}}}_{0} $ –2.148235 –2.148233 0 2s2p$ {{}_{}{}^{3}{\rm{P}}}_{1} $ –2.148205 –2.148205 0 2s2p$ {{}_{}{}^{3}{\rm{P}}}_{2} $ –2.148178 –2.148132 0 2s2p$ {{}_{}{}^{1}{\rm{P}}}_{1} $ –1.9832 –1.983927 0.03 2$ {{\rm{p}}}^{2}{{}_{}{}^{3}{\rm{P}}}_{0} $ –1.867621 –1.867673 0 2$ {{\rm{p}}}^{2}{{}_{}{}^{3}{\rm{P}}}_{1} $ –1.867634 –1.867634 0 2$ {{\rm{p}}}^{2}{{}_{}{}^{3}{\rm{P}}}_{2} $ –1.867565 –1.867573 0 2$ {{\rm{p}}}^{2}{{}_{}{}^{1}{\rm{D}}}_{2} $ –1.852917 –1.851947 0.05 2$ {{\rm{p}}}^{2}{{}_{}{}^{1}{\rm{S}}}_{0} $ –1.736452 –1.736679 0.01 2s3s $ {{}_{}{}^{3}{\rm{S}}}_{1} $ –1.727042 –1.727053 0 2s3s $ {{}_{}{}^{1}{\rm{S}}}_{0} $ –1.690800 –1.691293 0.03 2s3p $ {{}_{}{}^{3}{\rm{P}}}_{0} $ –1.662206 –1.662280 0 2s3p $ {{}_{}{}^{3}{\rm{P}}}_{1} $ –1.662167 –1.662277 0.01 2s3p $ {{}_{}{}^{3}{\rm{P}}}_{2} $ –1.662006 –1.662261 0.02 2s3p $ {{}_{}{}^{1}{\rm{P}}}_{1} $ –1.661601 –1.661765 0.01 2s3d $ {{}_{}{}^{3}{\rm{D}}}_{1} $ –1.631934 –1.631936 0 2s3d $ {{}_{}{}^{3}{\rm{D}}}_{2} $ –1.631720 –1.631936 0.01 2s3d $ {{}_{}{}^{1}{\rm{D}}}_{2} $ –1.613116 –1.613545 0.03 $ 2{\rm{s}}4{\rm{s}}{{}_{}{}^{3}{\rm{S}}}_{1} $ –1.560411 –1.560423 0 $ 2{\rm{s}}4{\rm{s}}{{}_{}{}^{1}{\rm{S}}}_{0} $ –1.552914 –1.553177 0.02 $ 2{\rm{s}}4{\rm{p}} $ $ {{}_{}{}^{1}{\rm{P}}}_{1} $ –1.540973 –1.541075 0.01 $ 2{\rm{s}}4{\rm{p}} $ $ {{}_{}{}^{3}{\rm{P}}}_{1} $ –1.5366 –1.5367 0.01 $ 2{\rm{s}}4{\rm{p}} $ $ {{}_{}{}^{3}{\rm{P}}}_{2} $ –1.536439 –1.536726 0.02 $ 2{\rm{s}}4{\rm{p}} $ $ {{}_{}{}^{3}{\rm{P}}}_{0} $ –1.536693 –1.536726 0 $ 2{\rm{s}}4{\rm{d}} $ $ {{}_{}{}^{3}{\rm{D}}}_{2} $ –1.524938 –1.525210 0.02 $ 2{\rm{s}}4{\rm{d}} $ $ {{}_{}{}^{3}{\rm{D}}}_{1} $ –1.525198 –1.525210 0
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表 7 B+离子基态和部分低激发态之间电偶极跃迁的振子强度(单位: a.u.)
Table 7. Oscillator strengths of electric-dipole transitions between the ground state and some low-lying states of B+ ions (in a.u.).
Transition RCICP CICP[45] BCICP[52] MCHF-BP[53] MCHF[54] NIST.[41] 2s2 1S0 →2s2p 1P1 1.00092 0.99907 1.002 1.001 0.9976(22) 0.9990 2s2 1S0→2s3p 1P1 0.10829 0.10959 0.108 0.1087 0.1093(3) 0.1090 2s2 1S0→2s4p 1P1 0.05331 0.0530 0.0514 2s2 1S0→2s5p 1P1 0.02244 0.0230 0.0241 2s2p 3P0→2p2 3P1 0.34113 0.34298 0.365 0.3430 0.3427(2) 0.3400 2s2p 3P0 →2s3s 3S1 0.06437 0.06377 0.06397 0.0640 2s2p 3P0→2s3d 3D1 0.47657 0.47627 0.473 0.4759 0.4750 2s2p 3P0→2s4s 3S1 0.01170 0.0115 2s2p 3P0→2s4d 3D1 0.12480 0.125 0.1260
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表 8 B+离子2s2 1S0 和2s2p 3P0的电偶极极化率
Table 8. Electric-dipole polarizability of 2s2 1S0 and 2s2p 3P0 states of B+ ions
2s2 1S0 2s2p 3P0 Contributions polarizability/a.u. Contributions polarizability/a.u. 2s2 1S0→2s2p 1P1 8.9149 2s2p 3P0→2p2p 3P1 4.3326 2s2 1S0→2s3p 1P1 0.2511 2s2p 3P0→2s3d 3D1 1.7878 Remains 0.4365 Remains 1.6195 Core 0.0195 Core 0.0195 RCICP 9.6220 RCICP 7.7594 CI[55] 9.5750 CI[55] 7.7790 CI+MBPT[55] 9.6130 CI+MBPT[55] 7.7690 CI+all-orders[55] 9.6240 CI+all-order[55] 7.7720 CCD+ST [56] 9.5660 CICP[45] 9.6441 CICP[45] 7.7798 PRCC[29] 9.4130 CCSDpT[57] 10.395(22) RRV[58] 9.6210
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[1] 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 033201
Google Scholar
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Google Scholar
[3] Chou C W, Hume D B, Koelemeij J C J, Wineland D J, Rosenband T 2010 Phys. Rev. Lett. 104 070802
Google Scholar
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Google Scholar
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Google Scholar
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[7] McGrew W F, Zhang X, Fasano R J, Schäffer S A, Beloy K, Nicolodi D, Brown R C, Hinkley N, Milani G, Schioppo M, Yoon T H, Ludlow A D 2018 Nature 564 7734
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