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在星载原子钟中, 传统原子钟(铷钟、铯钟和氢钟)会因原子之间和原子与器壁的碰撞产生多普勒频移, 使跃迁谱线比较宽, 而利用电磁场约束粒子的离子钟则避免了这些因素, 使其具有较高的频率准确度和稳定度. 在离子钟中, 汞离子具有最高的基态超精细能级跃迁频率, 并对温度和空间磁场的敏感度较低, 稳定度指标与主动型氢钟相当, 且重量轻(约3 kg)、体积小[1,2], 所以是理想的星载原子钟, 可用于导航二期后续、深空导航等项目[3]. 精确的汞离子钟还可以作为精密测量平台探测精细结构常数
$\alpha $ 是否随时间变化[4]或轻夸克质量与量子色动力学标度因子比率[5], 在低能区检验标准模型, 探索新物理.自然界中稳定存在的汞天然同位素共有7种, 分别为196Hg, 198Hg, 199Hg, 200Hg, 201Hg, 202Hg和204Hg. 表1给出了它们的相关参数, 包括同位素质量数、相对原子质量、元素丰度、均方根核电荷半径、核自旋、核磁偶极矩和核电四极矩. 其中, 199Hg和201Hg的核自旋不为零而使其能级具有超精细结构.
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+ – – 表 1 7种天然汞同位素及相关参数
Table 1. Related parameters of seven natural mercury isotopes.
199Hg+是光钟的重要候选离子[10,11], 对于199Hg+离子5d106s 2S1/2(F = 0)→5d96s2 2D5/2(F = 2) 态跃迁谱线的实验测量, Rafac等[12]、Bergquist等[13]和Oskay等[14]已经将测量精度提高到了10–17量级. 其中, Oskay等[14]给出绝对频率精度最高, 为1064721609899144.94 (97) Hz, 可以作为光学频率基准, 199Hg+也被列入新的秒定义候选离子. 198Hg+离子5d106s 2S1/2→5d96s2 2D5/2态跃迁谱线也有实验精确测量, 其值为35514.304(6) cm–1[15]. 对于无超精细结构的202Hg+和198Hg+离子, 它们约194 nm的深紫外光可被用来对四极阱中的199Hg+和201Hg+离子进行光抽运[1,5]. 另外, 在激光电离分离汞同位素离子技术中[16,17], 需要通过参考具体同位素频移来调谐用于光电离的激光, 选择性地电离制备特定的汞同位素离子. 但是对于汞同位素离子5d106s 2S1/2和5d96s2 2D5/2等能级及能级之间跃迁同位素位移(isotope shift, IS)的研究则缺少相关理论和高精度的实验结果.
本文中我们利用相对论多组态Dirac-Hartree-Fock(MCDHF)方法计算了钟跃迁谱线的IS以及相关两个能级的超精细结构常数, 与已有实验值比较, 我们的IS计算精度已经达到2%左右. 在此基础上, 获得了其他6种汞同位素离子相对于199Hg+钟跃迁频率的偏移量, 并理论预言了各种汞同位素可能的钟跃迁绝对频率值. 该理论计算结果可以为实验测量其他汞同位素离子跃迁谱线提供参考, 大幅减小实验扫谱范围, 降低实验难度, 为实现不同汞同位素离子的光学频率标准提供帮助. 另外, 汞离子IS为研究汞原子核的结构、性质及核子相互作用规律提供了必要的理论参数[18].
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在MCDHF方法下, 原子态波函数
$\varPsi$ 是由线性的组态波函数$\varPhi$ 展开的[19], 即$\varPsi \left( {\gamma PJ{M_J}} \right) = \sum\limits_i {{c_i}\varPhi \left( {{\alpha _i}PJ{M_J}} \right)} .$ 其中: ci是混合系数;
${\alpha _i}$ 是附加量子数, 它与宇称P、总角动量J及其z方向的分量MJ一起使描述的组态波函数具有唯一性. 理论上, 用无限个组态波函数去展开这个原子态波函数就能得到体系真实的原子态波函数. 但是, 实际计算中只能选择有限个组态波函数, 其数目多少和选择方式决定了对电子关联的描述程度.在自洽场(self-consistent field, SCF)过程中, 基于变分原理, 混合系数和单电子相对论轨道通过求解MCDHF方程来同时优化. 在得到单电子轨道基后, 进一步扩大组态空间, 利用相对论组态相互作用(relativistic configuration interaction, RCI)方法可以包含更多电子关联效应. 另外, 在RCI计算中还能包括Breit相互作用[20]和量子电动力学(quantum electrodynamics, QED)修正[21,22]. 本文的计算利用了基于MCDHF方法发展的GRASP程序包[23].
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原子核的电磁多极矩与电子之间的超精细相互作用可以表示为
${H_{{\rm{hfs}}}} = \sum\limits_{k \geqslant 1} {{{{T}}^{\left( k \right)}} \cdot {{{M}}^{\left( k \right)}}} .$ 其中, T(k)和M(k)分别是电子和原子核空间的k阶球张量算符[24]. k = 1表示磁偶极超精细相互作用, k = 2表示电四极超精细相互作用. 进一步, 磁偶极和电四极超精细结构常数A, B可以分别表示为[25]
$ \begin{split} & {A_J} = \frac{{{\mu _I}}}{I}\frac{1}{{{{\left[ {J\left( {J + 1} \right)} \right]}^{{\rm{1/2}}}}}}\left\langle {\gamma J\left\| {{{{T}}^{\left( 1 \right)}}} \right\|\gamma J} \right\rangle , \\ & {B_J} = 2{Q_I}{\left[ {\frac{{J\left( {2J - 1} \right)}}{{\left( {J + 1} \right)\left( {2J + 3} \right)}}} \right]^{1/2}}\left\langle {\gamma J\left\| {{{{T}}^{\left( 2 \right)}}} \right\|\gamma J} \right\rangle. \end{split} $ 相应的磁偶极和电四极超精细相互作用的一阶能量修正可表示为
$ \begin{split} & {W_{M1}}\left( {J,J} \right) = \frac{1}{2}{A_J}C, \\ & {W_{E2}}\left( {J,J} \right) = {B_J}\frac{{\frac{3}{4}C\left( {C + 1} \right) - I\left( {I + 1} \right)J\left( {J + 1} \right)}}{{2I\left( {2I - 1} \right)J\left( {2J - 1} \right)}}, \end{split} $ 其中C = F(F + 1) – J(J + 1) – I(I + 1).
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早期处理原子结构时, 把原子核看成是质量无穷大的点电荷. 但实际上, 原子核是有限质量的质子和中子组成的量子体系, 其内部结构不同也影响着核外电子, 进而造成不同同位素之间原子能级存在不同程度的位移, 这一位移量被称为IS. IS分为质量位移(mass shift, MS)和场位移(field shift, FS)[20,26,27]. 对于较轻元素, 依赖于核有限质量的MS贡献较大. 对于较重元素, 依赖于核电荷分布的FS贡献较大, 后面的讨论将细致比较两者在汞离子IS中占的实际比重.
对于质量为M的原子核, 在(
$\alpha $ Z)4m2/M近似下[28-30](其中m是电子质量), MS对应的哈密顿量为$ \begin{split} {H_{{\rm{MS}}}} & = {H_{{\rm{NMS}}}} + {H_{{\rm{SMS}}}},\\ {H_{{\rm{NMS}}}}& = \frac{1}{{2M}}\sum\limits_{j = 1}^N \left( {{p}}_j^2 - \frac{{\alpha Z}}{{{r_j}}}{{{\alpha }}_j} \cdot {{{p}}_j}-\right.\\ &\quad \left. \frac{{\alpha Z}}{{{r_j}}}\left( {{{{\alpha }}_j} \cdot {{C}}_j^1} \right){{C}}_j^1 \cdot {{{p}}_j} \right), \\ {H_{{\rm{SMS}}}}&=\frac{1}{{2M}}\sum\limits_{j \ne k}^N \left( {{{p}}_j} \cdot {{{p}}_k} - \frac{{\alpha Z}}{{{r_j}}}{{{\alpha }}_j} \cdot {{{p}}_k}-\right. \\ &\quad \left. \frac{{\alpha Z}}{{{r_j}}}\left( {{{{\alpha }}_j} \cdot {{C}}_j^1} \right){{C}}_j^1 \cdot {{{p}}_k} \right). \end{split} $ 其中, Z为核电荷数;
$\alpha$ 为精细结构常数;${{{\alpha }}_j}$ 为4 × 4的Dirac矩阵; pj为四维形式的电子动量算符. MS算符可以分为单体部分, 即正规质量偏移(normal mass shift, NMS)和两体部分, 即特殊质量偏移(specific mass shift, SMS), 分别对应HNMS和HSMS. NMS和SMS因子的形式分别为$ \begin{split} & {K_{{\rm{NMS}}}} = M\left\langle {\varPsi \left\| {{H_{{\rm{NMS}}}}} \right\|\varPsi } \right\rangle , \\ & {K_{{\rm{SMS}}}} = M\left\langle {\varPsi \left\| {{H_{{\rm{SMS}}}}} \right\|\varPsi } \right\rangle. \end{split} $ 对于FS, 不同同位素之间FS造成的能级频率位移
$\delta \nu $ 可表示为[31,32]$\delta \nu = \frac{{2{\text{π}}}}{{3h}}Z\left( {\frac{{{e^2}}}{{4{\text{π}}{\varepsilon _0}}}} \right)\Delta {\left| {\varPsi \left( 0 \right)} \right|^2}\delta \left\langle {{r^2}} \right\rangle. $ 其中,
${\left| {\varPsi \left( 0 \right)} \right|^2}$ 为电子在原子核内的概率;$\delta \left\langle {{r^2}} \right\rangle $ 为两同位素原子核电荷半径的均方根平方之差[33]. -
首先, 在单组态近似下计算了汞的7种同位素相对199Hg+离子的5d106s 2S1/2→5d96s2 2D5/2跃迁的MS和FS, 结果如图1所示. 由于实验测量的汞不同同位素的原子核均方根半径的平方随中子数增加而线性变化[8], 从图1中可以看出其同位素的FS也与同位素的质量数大致呈现线性关系. 另外, 也观察到汞不同同位素的MS基本相同. 值得注意的是, 对于Hg+离子的这条钟跃迁谱线, MS远小于FS, 其对IS的贡献远小于1%. 因此, 在IS的计算中可以忽略MS.
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
Figure 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.
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为了获得Hg+离子5d106s 2S1/2→5d96s2 2D5/2钟跃迁FS的准确结果, 需要考虑体系中的电子关联效应. 本文采用活动空间方法[34]产生组态空间来系统地捕获电子间的关联. 选取5d106s和5d96s2为参考组态, 把5d和6s电子看成价电子(V), 其他电子为芯电子(C). 组态空间利用限制性单双激发(SrD)方式扩展, 即活动的原子芯轨道中最多只允许一个电子激发出去, 而价电子可以被同时激发两个. 这样, 包括了芯电子与价电子(CV)和价电子之间的关联(VV). 为了使能量本征值和IS收敛, 关联轨道(virtual orbitals)是逐层加入的, 而且在SCF计算中每次只优化最新加入的轨道. 另外, 为了全面考虑原子芯电子(C)和价电子(V)之间的关联效应, 在逐层加入关联轨道的同时, 也逐层打开参考组态中占据的原子芯轨道(active orbitals), 即打开的原子芯轨道上的电子也可以被激发到价轨道或虚轨道上. 为了表述计算得到的原子态波函数的品质, 在表2中给出了具体的计算模型以及能量本征值随组态空间扩大的变化. n代表虚轨道最大的主量子数, DF代表单组态近似模型, NCF列两组数字分别代表参考组态5d106s 2S1/2和5d96s2 2D5/2在每个计算模型下组态的数目, Energy eigenvalue列下的两组值分别对应这两个组态的能量本征值.
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 表 2 电子关联对能量本征值的影响
Table 2. Effect of electron correlations on energy eigenvalues.
如表2所示, 计算是从DF近似(单组态近似)出发, 然后产生了5层虚轨道, 其中前3层轨道角动量包括spdfg, 最后两层轨道角动量为spdf. 对于主量子数大于8且角动量为g的轨道以及角动量全部为h的轨道, 计算结果显示它们的影响较小, 所以在虚轨道中不用添加. 在扩充虚轨道的同时, 我们还同时逐层打开了原子芯轨道直到n = 3.
表3给出了汞同位素离子相对199Hg+离子的FS收敛情况，第一列与表2的计算模型对应. 可以大致看出，随着越来越多的电子关联被考虑，汞同位素离子的FS趋向收敛. 比如，对于196Hg+离子，n = 8时的FS与n = 9相比相差3.28%，而n = 9时的FS与n = 10相比仅相差0.12%.
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 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+).
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使用上述计算FS的模型, 也计算了Hg+离子5d106s 2S1/2和5d96s2 2D5/2能级的磁偶极(A)和电四极(B)超精细结构常数, 其随电子关联的变化趋势展示在表4中. 为了反映同位素效应, 同位素的质量数标记在超精细结构常数的左上角. 另外, 原子态的角动量也标记在右下角. 可以看出, 随着组态空间的扩大, 磁偶极(A)和电四极(B)超精细结构常数均趋于收敛. 例如, 199Hg+ 5d106s 2S1/2态在n = 8时的磁偶极常数A与n = 9时的差别为4.43%, 而n = 9时的A值比n = 10的结果只大1%. 表4中最后4行给出了文献中已知的理论和实验值, 与文献[35]和[36]中理论结果相比, 在n = 11模型下的计算结果更接近文献[37]中的实验值, 其中, 计算的199Hg+和201Hg+离子5d106s 2S1/2态磁偶极超精细结构常数A与实验值的误差分别为1.67%和1.48%. 另外, 还得到了199Hg+和201Hg+离子5d96s2 2D3/2态的磁偶极超精细结构常数分别为2735.73 MHz和–1009.86 MHz, 201Hg+离子电四极超精细结构常数为841.17 MHz.
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 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+.
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图2描述了198Hg+和199Hg+离子超精细结构下5d106s 2S1/2→5d96s2 2D5/2钟跃迁的IS. 其中, ISS代表精细结构下能级的IS, HFSS代表超精细结构能级劈裂. 利用上面n = 11时计算的FS和超精细结构常数, 得到了199Hg+与198Hg+离子跃迁谱线的ISS之差为1.13 GHz, 199Hg+离子F = 2到F = 0超精细分裂(HFSS之差)为29.16 GHz, 因此这两个同位素钟跃迁的偏移(ν1与ν2之差)为这两项值之和, 即30.30 GHz, 该值即为199Hg+与198Hg+离子在超精细结构下的IS. 已知实验测量的199Hg+离子光钟钟跃迁频率为1064721609899144.94(97) Hz[14], 以及198Hg+离子光钟钟跃迁频率为1064.6919524 THz[15], 可以得到199Hg+与198Hg+ 离子5d106s 2S1/2→5d96s2 2D5/2钟跃迁的频率IS实验值为29.66 GHz, 我们的理论结果与其符合的很好, 两者差别仅为2.16%.
图 2 199Hg+和198Hg+离子的超精细能级结构图
Figure 2. Hyperfine level structure diagram of 199Hg+ and 198Hg+.
从计算中发现, Hg+离子能级的IS在104 GHz量级, 而超精细结构分裂在1～10 GHz量级. 但是对于跃迁IS, 其值是上能级5d106s 2S1/2和下能级5d96s2 2D5/2的IS之差, 该值与超精细结构分裂在相同的数量级. 结合表3中FS的计算结果也可以看出, 对于汞元素的几种同位素, 它们的FS都小于29.16 GHz的超精细结构分裂, 这导致计算IS时, 超精细结构的影响需要重点考虑.
在同时考虑了5d106s 2S1/2→5d96s2 2D5/2跃迁的IS和超精细结构, 给出了汞7种同位素离子该谱线的绝对频率值, 如表5所示. 其中, 201Hg+离子对应的跃迁是5d106s 2S1/2(F = 1)→5d96s2 2D5/2(F = 1), 199Hg+离子的是5d106s 2S1/2(F = 0)→5d96s2 2D5/2(F = 2). 除了199Hg+和198Hg+离子外, 其他汞同位素离子目前均没有相应实验测量值, 也没有其他理论计算结果来进行比较, 有待将来实验验证我们的理论计算结果.
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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本文首先在Dirac-Hartree-Fock近似下, 发现汞离子IS中FS要远大于MS. 进一步, 忽略MS的影响, 利用MCDHF方法, 我们考虑了原子体系中主要的电子关联效应, 获得了汞同位素离子5d106s 2S1/2→5d96s2 2D5/2钟跃迁在超精细结构下的IS. 其中, 计算的199Hg+和198Hg+离子的IS与实验值仅相差2.16%. 最后, 我们也理论预言了汞7种同位素5d106s 2S1/2→5d96s2 2D5/2钟跃迁谱线的绝对频率值. 本文的计算结果对进一步的实验测量工作提供了有价值的参考数据, 同时也对后续汞原子核相关性质的研究提供了理论依据.
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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
[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
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[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
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[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]
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[9] Stone N J 2005 At. Data Nucl. Data Tables 90 75
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[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
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[19] Grant I P 2007 Relativistic Quantum Theory of Atoms and Molecules (New York: Springer) pp259-388
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[24] McDaniel E W, McDowell M R C 1975 Case Studies in Atomic Physics Ⅳ (Amsterdam: North-Holland) pp197–298
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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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