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作为典型的单中子晕核, 11Be在原子及核物理研究中具有独特的意义. 本文针对类氦11Be2+离子, 采用相对论组态相互作用方法, 高精度计算了主量子数最高达$n = 8$的$n^{3}{\mathrm{S}}_1$和$n^{3}{\mathrm{P}}_{0,1,2}$态的能量与波函数. 通过将有限核质量修正算符直接引入Dirac-Coulomb-Breit哈密顿量, 使计算能够同时考虑相对论效应和质量相关修正. 基于计算的高精度能量与波函数, 本文进一步确定了$k^3{\mathrm{S}}_1 \rightarrow m^3{\mathrm{P}}_{0,1,2}$ ($k \leqslant 5$, $m \leqslant 8$)电偶极跃迁的振子强度, 精度达3到6位有效数字. 此外, 利用态求和法计算了$n'^3{\mathrm{S}}_1$ ($n' \leqslant 5$)态在宽光子频率范围内的动力学电偶极极化率, 在远离共振位置处结果最高可达10–6精度水平. 上述高精度计算结果为11Be2+离子在高精度测量中涉及的斯塔克频移评估以及光与物质相互作用的模拟等方面提供了重要的理论依据和关键输入参数.
11Be, as a typical one-neutron halo nucleus, is of unique significance in studying atomic and nuclear physics. The nucleus comprises a tightly bound 10Be core and a loosely bound valence neutron, forming an exotic nuclear configuration that is significantly different from traditional nuclear configuration in both magnetic and charge radii, thereby establishing a unique platform for investigating nuclear-electron interactions. In this study, we focus on the helium-like 11Be2+ ion and systematically calculate the energies and wavefunctions of the $n^{3}S_1$ and $n^{3}P_{0,1,2}$ states up to principal quantum number $n=8$ by employing the relativistic configuration interaction (RCI) method combined with high-order B-spline basis functions. By directly incorporating the nuclear mass shift operator $H_M$ into the Dirac-Coulomb-Breit (DCB) Hamiltonian, we comprehensively investigate the relativistic effects, Breit interactions, and nuclear mass corrections for 11Be2+. The results demonstrate that the energies of states with $n\leqslant 5$ converge to eight significant digits, showing excellent agreement with existing NRQED values, such as $-9.29871191(5)$ a.u. for the $^{3}S_1$ state. The nuclear mass corrections are on the order of 10–4 a.u. and decrease with principal quantum number increasing. By using the high-precision wavefunctions, the electric dipole oscillator strengths for $k^3S_1 \rightarrow m^3P_{0,1,2}$ transitions ($k \leqslant 5$, $m \leqslant 8$) are determined, resulting in low-lying excited states ($m\leqslant4$) accurate to six significant digits, thereby providing reliable data for evaluating transition probabilities and radiative lifetimes. Furthermore, the dynamic electric dipole polarizabilities of the $n'^3S_1$ ($n' \leqslant 5$) states are calculated using the sum-over-states method. The static polarizabilities exhibit a significant increase with principal quantum number increasing. For the $J=1$ state, the difference in polarizability between the magnetic sublevels $M_J=0$ and $M_J=\pm1$ is three times the tensor polarizability. In the calculation of dynamic polarizabilities, the precision reaches 10–6 in non-resonant regions, whereas achieving the same accuracy near resonance requires higher energy precision. These high-precision computational results provide crucial theoretical foundations and key input parameters for evaluating Stark shifts in high-precision measurements, simulating light-matter interactions, and investigating single-neutron halo nuclear structures. 
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图 1 11Be2+离子$2^3 S_1$和$3^3 S_1$态$|M_J|=1$磁子能级的动力学电偶极极化率, 垂直虚线表示共振位置. 橘黄色和蓝色数字分别表示$2^3 S_1(|M_J|=1)$和$3^3 S_1(|M_J|=1)$态的幻零波长, 玫红色数字表示使$2^3 S_1(|M_J|=1)$和$3^3 S_1 $$ (|M_J|=1)$态极化率相等的魔幻波长.
Fig. 1. Dynamic electric dipole polarizabilities of the 11Be2+ ion $2^3 S_1(|M_J|=1)$ and $3^3 S_1(|M_J|=1)$ states, with vertical dashed lines indicating the resonance positions. The orange and blue numbers represent the tune-out wavelengths for the $2^3 S_1 |M_J|=1$ and $3^3 S_1 |M_J|=1$ states, respectively, and the magenta numbers indicate the magic wavelengths at which the polarizabilities of the $2^3 S_1(|M_J|=1)$ and $3^3 S_1(|M_J|=1)$ states are equal.
图 2 11Be2+离子$4^3 S_1$和$5^3 S_1$态$|M_J|=1$磁子能级的动力学电偶极极化率, 垂直虚线表示共振位置. 紫色和绿色数字分别表示$4^3 S_1(|M_J|=1)$和$5^3 S_1(|M_J|=1)$态的幻零波长, 玫红色数字表示使$4^3 S_1(|M_J|=1)$和$5^3 S_1(|M_J|= $$ 1)$态极化率相等的魔幻波长.
Fig. 2. Dynamic electric dipole polarizabilities of the 11Be2+ ion $4^3 S_1(|M_J|=1)$ and $5^3 S_1(|M_J|=1)$ states, with vertical dashed lines indicating the resonance positions. The purple and green numbers represent the tunw-out wavelengths for the $4^3 S_1(|M_J|=1)$ and $5^3 S_1(|M_J|=1)$ states, respectively, and the magenta numbers indicate the magic wavelengths at which the polarizabilities of the $4^3 S_1(|M_J|=1)$ and $5^3 S_1(|M_J|=1)$ states are equal.
表 1 11Be2+离子$ n ^3{\mathrm{S}}_1(n\leqslant 8) $态能量的收敛性检验, 以及∞Be2+离子$ n ^3{\mathrm{S}}_1(6\leqslant n\leqslant 8) $态的能量. 小括号内的数字是计算不确定度
Table 1. Convergence test of energy (in a.u.) for the $ n ^3{\mathrm{S}}_1(n\leqslant 8) $ states of 11Be2+ ion. And the energy for the $ n ^3{\mathrm{S}}_1(6\leqslant n\leqslant 8) $ states of ∞Be2+ ion. The numbers in parentheses are computational uncertainties.
(N, $ \ell_m $) $ 2 ^3\mathrm{S}_1 $ $ 3 ^3\mathrm{S}_1 $ $ 4 ^3\mathrm{S}_1 $ $ 5 ^3\mathrm{S}_1 $ (40, 8) –9.298 711 8781 –8.548 347 5380 –8.301 788 8508 –8.190 993 6393 (40, 9) –9.298 711 9119 –8.548 347 5470 –8.301 788 8543 –8.190 993 6410 (40, 10) –9.298 711 8673 –8.548 347 5442 –8.301 788 8537 –8.190 993 6408 (45, 10) –9.298 711 9028 –8.548 347 5516 –8.301 788 8542 –8.190 993 6238 (50, 10) –9.298 711 8649 –8.548 347 5498 –8.301 788 8539 –8.190 993 6224 Extrap. –9.298 711 91(5) –8.548 347 55(2) –8.301 788 85(1) –8.190 993 62(3) –9.298 711 181[21] ∞Be2+[29] –9.299 176 21(4) –8.548 773 43(4) –8.302 202 22(4) –8.191 401 39(4) (N, $ \ell_m $) $ 6 ^3\mathrm{S}_1 $ $ 7 ^3\mathrm{S}_1 $ $ 8 ^3\mathrm{S}_1 $ (40, 8) –8.131 856 6822 –8.096 615 3793 –8.073 936 7761 (40, 9) –8.131 856 6832 –8.096 615 3799 –8.073 936 7765 (40, 10) –8.131 856 6831 –8.096 615 3798 –8.073 936 7764 (45, 10) –8.131 856 5642 –8.096 614 7583 –8.073 933 5599 (50, 10) –8.131 856 5546 –8.096 614 7052 –8.073 933 2679 Extrap. –8.131 856 6(1) –8.096 614 7(4) –8.073 933(4) ∞Be2+ –8.132 261 3(2) –8.097 017 8(6) –8.074 334(5) 
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表 2 11Be2+离子$ n ^3{\mathrm{P}}_{0, 1, 2} (n\leqslant 8) $态的能量, 以及∞Be2+离子$ n ^3{\mathrm{P}}_{0, 1, 2} (6\leqslant n\leqslant 8) $态的能量. 小括号内的数字是计算不确定度
Table 2. Energy (in a.u.) for the $ n ^3{\mathrm{P}}_{0, 1, 2} (n\leqslant 8) $ states of 11Be2+ ion, and for the $ n ^3{\mathrm{P}}_{0, 1, 2} (6\leqslant n\leqslant 8) $ states of ∞Be2+ ion. The numbers in parentheses are computational uncertainties.
n $ ^3{\mathrm{P}}_0 $(11Be2+) $ ^3{\mathrm{P}}_0 $(∞Be2+) $ ^3{\mathrm{P}}_1 $(11Be2+) $ ^3{\mathrm{P}}_1 $(∞Be2+) $ ^3{\mathrm{P}}_2 $(11Be2+) $ ^3{\mathrm{P}}_2 $(∞Be2+) 2 –9.176 279 04(4) –9.176 700 64(4)[29] –9.176 331 62(4) –9.176 753 22(4)[29] –9.176 264 02(4) –9.176 685 61(4)[29] –9.176 278 322[21] –9.176 330 730[21] –9.176 263 355[21] 3 –8.515 916 23(4) –8.516 331 41(4)[29] –8.515 929 14(4) –8.516 344 33(4)[29] –8.515 909 08(4) –8.516 324 31(4)[29] 4 –8.288 671 51(4) –8.289 080 63(4)[29] –8.288 676 58(4) –8.289 085 70(4)[29] –8.288 668 14(4) –8.289 077 27(4)[29] 5 –8.184 422 45(4) –8.184 828 10(4)[29] –8.184 424 95(4) –8.184 830 61(4)[29] –8.184 420 64(4) –8.184 826 30(4)[29] 6 –8.128 103 85(8) –8.128 507 44(8) –8.128 105 27(8) –8.128 508 86(8) –8.128 102 78(8) –8.128 506 37(8) 7 –8.094 272 36(8) –8.094 674 69(8) –8.094 273 24(8) –8.094 675 56(8) –8.094 271 7(1) –8.094 674 0(1) 8 –8.072 374 1(4) –8.072 775 7(4) –8.072 374 5(4) –8.072 776 2(4) –8.072 373(4) –8.072 775 2(4)
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表 3 11Be2+离子$ n ^3{\mathrm{S}}_{1}\rightarrow m ^3{\mathrm{P}}_{0, 1, 2} $跃迁的振子强度. 小括号中的数字是计算不确定度, 中括号中的数字表示10的幂次
Table 3. The oscillator strengths (in a.u.) for $ n ^3{\mathrm{S}}_{1}\rightarrow m ^3{\mathrm{P}}_{0, 1, 2} $ transitions of 11Be2+ ion. Numbers in parentheses are computational uncertainties. Numbers in square brackets represent the power of 10.
$ 2 ^3{\mathrm{S}}_1 $ $ 3 ^3{\mathrm{S}}_1 $ $ 4 ^3{\mathrm{S}}_1 $ $ 5 ^3{\mathrm{S}}_1 $ $ 2^3{\mathrm{P}}_0 $ 2.372 207(2)[–2] 9.872 733(2)[–3] 1.928 282(2)[–3] 7.371 365(4)[–4] $ 2^3{\mathrm{P}}_1 $ 7.113 520(4)[–2] 2.959 444(1)[–2] 5.780 477(2)[–3] 2.209 758(2)[–3] $ 2^3{\mathrm{P}}_2 $ 1.186 353(6)[–1] 4.935 354(6)[–2] 9.638 898(6)[–3] 3.684 637(4)[–3] $ 3^3{\mathrm{P}}_0 $ 2.803 438 7(2)[–2] 3.959 550 0(4)[–2] 2.196 932 9(1)[–2] 4.408 759(2)[–3] $ 3^3{\mathrm{P}}_1 $ 8.412 570(1)[–2] 1.187 268 3(2)[–1] 6.586 619 7(8)[–2] 1.321 851 6(4)[–2] $ 3^3{\mathrm{P}}_2 $ 1.401 611 4(8)[–1] 1.980 174(5)[–1] 1.098 388 7(8)[–1] 2.204 119(1)[–2] $ 4^3{\mathrm{P}}_0 $ 7.939 441 8(4)[–3] 2.930 796 5(4)[–2] 5.442 867(2)[–2] 3.485 147(2)[–2] $ 4^3{\mathrm{P}}_1 $ 2.382 271 5(1)[–2] 8.794 741(2)[–2] 1.632 008 6(4)[–2] 1.044 959 8(8)[–1] $ 4^3{\mathrm{P}}_2 $ 3.969 574(2)[–2] 1.465 153(2)[–1] 2.721 986(4)[–1] 1.742 527(2)[–1] $ 5^3{\mathrm{P}}_0 $ 3.436 979(4)[–3] 8.804 208(4)[–3] 3.165 094(4)[–2] 6.891 32(2)[–2] $ 5^3{\mathrm{P}}_1 $ 1.031 254(1)[–2] 2.641 763(1)[–2] 9.497 75(1)[–2] 2.066 303(6)[–1] $ 5^3{\mathrm{P}}_2 $ 1.718 454(2)[–2] 4.401 593(4)[–2] 1.582 203(2)[–1] 3.446 360(4)[–1] $ 6^3{\mathrm{P}}_0 $ 1.822 257(8)[–3] 3.988 31(2)[–3] 9.679 22(2)[–3] 3.443 62(4)[–2] $ 6^3{\mathrm{P}}_1 $ 5.467 55(4)[–3] 1.196 685(8)[–2] 2.904 307(4)[–2] 1.033 36(2)[–1] $ 6^3{\mathrm{P}}_2 $ 9.111 17(6)[–3] 1.993 96(1)[–2] 4.838 841(4)[–2] 1.721 39(1)[–1] $ 7^3{\mathrm{P}}_0 $ 1.089 63(8)[–3] 2.192 5(2)[–3] 4.470 8(2)[–3] 1.057 500(8)[–2] $ 7^3{\mathrm{P}}_1 $ 3.269 3(2)[–3] 6.578 4(6)[–3] 1.341 47(8)[–2] 3.173 09(6)[–2] $ 7^3{\mathrm{P}}_2 $ 5.448 1(6)[–3] 1.096 1(1)[–2] 2.235 1(1)[–2] 5.286 6(2)[–2] $ 8^3{\mathrm{P}}_0 $ 7.067(8)[–4] 1.350(1)[–3] 2.503(4)[–3] 4.926(4)[–3] $ 8^3P_1 $ 2.118 2(4)[–3] 4.051(4)[–3] 7.510(4)[–3] 1.479(2)[–3] $ 8^3{\mathrm{P}}_2 $ 3.530(2)[–3] 6.750(2)[–3] 1.252(2)[–2] 2.464(2)[–2]
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表 4 11Be2+离子$ n ^3{\mathrm{S}}_1(n\leqslant 5) $态静态电偶极极化率的收敛性检验. 小括号中的数字是计算不确定度
Table 4. Convergence test of static dipole electric polarizability (in a.u.) for the $ n ^3{\mathrm{S}}_1(n\leqslant 5) $ states of 11Be2+ ion. The numbers in parentheses are computational uncertainties
(N, $ \ell_m $) $ 2\, ^3 S_1(M_J=0/\pm 1) $ $ 3\, ^3 S_1(M_J=0/\pm 1) $ $ 4\, ^3 S_1(M_J=0/\pm 1) $ $ 5\, ^3 S_1(M_J=0/\pm 1) $ (40, 8) 14.888 529/14.891 730 343.889 786/343.954 302 2868.6928/2869.2072 14424.502/14427.048 (40, 9) 14.888 533/14.891 735 343.889 940/343.954 462 2868.6941/2869.2085 14424.508/14427.054 (40, 10) 14.888 538/14.891 742 343.890 034/343.954 574 2868.6946/2869.2092 14424.510/14427.058 (45, 10) 14.888 561/14.891 758 343.890 263/343.954 742 2868.6970/2869.2111 14424.544/14427.088 (50, 10) 14.888 528/14.891 735 343.889 933/343.954 502 2868.6944/2869.2092 14424.531/14427.080 Extrap. 14.888 58(6)/14.891 77(4) 343.890 4(7)/343.954 8(5) 2868.697(5)/2869.211(4) 14424.54(4)/14427.08(4)
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表 5 11Be2+离子$ n ^3{\mathrm{S}}_{1} (\leqslant 5) $态的动力学电偶极极化率及其计算不确定度, ω为外场频率, 原子单位
Table 5. The dynamic electric dipole polarizabilities and computational uncertainties for $ n ^3{\mathrm{S}}_{1} (\leqslant 5) $ states of 11Be2+ ion, where ω is the frequency of external field, in a.u.
ω (a.u.) $ 2 ^3\mathrm{S}_1(M_J=0/\pm 1) $ $ 3 ^3\mathrm{S}_1(M_J=0/\pm 1) $ $ 4 ^3\mathrm{S}_1(M_J=0/\pm 1) $ $ 5 ^3\mathrm{S}_1(M_J=0/\pm 1) $ 0.02 15.27929(3)/15.28277(2) 551.7125(9)/551.9742(7) –2126.974(5)/–2125.537(4) –1666.090(2)/–1665.446(2) 0.03 15.79888(3)/15.80274(3) 2348.47(3)/2355.50(2) –649.2535(8)/–648.9762(6) –638.422(2)/–638.155(2) 0.04 16.59145(4)/16.59592(3) –645.258(3)/–644.484(2) –317.9701(4)/–317.8436(3) –284.578(3)/–284.410(3) 0.045 17.11436(4)/17.11926(3) –361.0677(9)/–360.7746(7) –238.3984(3)/–238.3025(2) –171.451(4)/–171.301(4) 0.05 17.74088(4)/17.74631(3) –240.8547(5)/–240.6914(4) –183.3957(2)/–183.3195(2) –60.173(6)/–60.025(7) 0.055 18.49116(5)/18.49728(4) –175.3147(3)/–175.2190(3) –143.3993(2)/–143.3365(2) 102.35(2)/102.53(2) 0.06 19.39221(5)/19.39919(4) –134.5050(2)/–134.4378(2) –113.0578(2)/–113.00454(9) 672.43(7)/672.93(7) 0.065 20.48070(6)/20.48879(5) –106.9053(2)/–106.8553(2) –89.1419(1)/–89.09557(8) –1326.21(8)/–1325.85(8) 0.07 21.80763(7)/21.81719(6) –87.1505(2)/–87.11157(9) –69.56520(9)/–69.52388(6) –490.156(5)/–490.103(5) 0.075 23.44577(9)/23.45730(7) –72.41078(9)/–72.37938(7) –52.87530(8)/–52.83766(5) –338.839(2)/–338.785(2) 0.08 25.5025(2)/25.51673(8) –61.05605(8)/–61.03007(6) –37.95614(6)/–37.92106(5) –278.694(3)/–278.627(3) 0.085 28.1427(2)/28.1609(1) –52.08358(7)/–52.06161(5) –23.81351(7)/–23.77994(6) –257.546(6)/–257.452(6) 0.09 31.6335(2)/31.6576(2) –44.84405(6)/–44.82516(4) –9.35342(8)/–9.32025(7) –277.90(2)/–277.68(2) 0.095 36.4367(3)/36.4702(2) –38.89943(5)/–38.88295(4) 6.9806(1)/7.01487(9) –432.7(2)/–431.7(2) 0.10 43.4261(4)/43.4760(3) –33.94404(5)/–33.92949(4) 28.0790(2)/28.1170(2) 441.99(6)/442.72(6) 0.11 74.483(2)/74.6458(9) –26.18144(4)/–26.16973(3) 131.7548(7)/131.8358(8) 32.52(3)/32.53(3) 0.12 361.19(4)/365.51(3) –20.39655(3)/–20.38682(2) –486.980(5)/–486.775(5) –146(1)/–146(1) 0.13 –111.268(4)/–110.830(3) –15.91658(3)/–15.90826(2) –116.4122(2)/–116.4040(2) –8.5(2)/–8.4(2) 0.14 –45.6790(6)/–45.5965(5) –12.32375(2)/–12.31647(2) –68.80539(6)/–68.79816(6) 0.15 –27.7762(3)/–27.7422(2) –9.34226(2)/–9.33576(2) –46.6878(2)/–46.6794(2) 0.16 –19.4618(2)/–19.4433(1) –6.77800(2)/–6.77207(2) –28.4859(4)/–28.4748(4) 0.17 –14.68262(9)/–14.67079(7) –4.48302(2)/–4.47750(2) 27.568(5)/27.602(5) 0.18 –11.59257(6)/–11.58420(5) –2.33146(2)/–2.32622(1) –76.562(2)/–76.550(2) 0.19 –9.43904(5)/–9.43342(4) –0.19850(2)/–0.193392(9) –51.2307(7)/–51.2156(7) 0.20 –7.85783(4)/–7.85328(3) 2.06578(2)/2.070910(9) –47.611(3)/–47.577(3) 0.22 –5.70307(3)/–5.70005(2) 8.05211(2)/8.058053(9) 56.557(6)/56.635(6) 0.24 –4.31412(2)/–4.31196(2) 22.40003(2)/22.41095(2) 2.70(5)/2.70(5) 0.26 –3.35158(2)/–3.349930(9) –1580.80(4)/–1566.25(4) –5.1(6)/–5.0(6) 0.28 –2.64914(1)/–2.647830(7) –26.249939(7)/–26.248577(8) 0.30 –2.115953(8)/–2.114877(6) –12.800496(3)/–12.799903(3) 0.32 –1.698284(7)/–1.697376(5) –7.278323(3)/–7.277525(3) 0.34 –1.362367(6)/–1.361585(4) –2.571976(7)/–2.570642(7) 0.36 –1.085927(5)/–1.085241(4) 22.4313(3)/22.4461(3) 0.38 –0.853663(4)/–0.853051(3) –10.49399(3)/–10.49349(3) 0.40 –0.654682(4)/–0.654128(3) –4.67869(3)/–4.67806(3)
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