图 1  由中间态(Xe)$ 5{\rm d}6{\rm d} \; ^3{\rm F}_2 $激发的一价镧离子光谱的能量标定: 以文献[1]的89690.4—91639.8 cm^–1光谱(下图)为基准, 对89872.8—91783.2 cm^–1能区光谱(上图)^[19]重新标定, 平移了–10.3 cm^–1. 横轴能量以一价镧离子基态能量为零点

Fig. 1.  The energy calibration of the excited La⁺ spectrum via intermediate state (Xe)$ 5{\rm d}6{\rm d} \; ^3{\rm F}_2 $. We recalibrate the spectrum^[19] in the energy region 89872.8—91783.2 cm^–1 (upper figure) according to the spectrum^[1] in the energy region 89690.4—91639.8 cm^–1(lower figure), and the offset of the recalibration is –10.3 cm^–1. The zero point of energy is taken the energy of the ground state of La⁺.

图 2  考虑了对实验谱有主要贡献的本征通道之间相互作用的理论光谱(中间谱)与实验谱(上图)和本征通道谱(下图)$ 5{\rm d}\epsilon {\rm p}\left(\dfrac{5}{2}, \dfrac{1}{2}\right)_3, J^\pi=3^- $的比较. 中间的理论谱是根据(1)式, 考虑了$ J^\pi=3^- $的这些本征通道的相互作用: $ 5{\rm d}_{3/2}\epsilon {\rm f}_{5/2}$, $5d_{5/2}\epsilon f_{5/2}$, $5{\rm d}_{5/2}\epsilon {\rm p}_{1/2}$, $4{\rm f}_{5/2}\epsilon {\rm s}_{1/2}$, $4{\rm f}_{5/2}\epsilon {\rm d}_{3/2}$, $4{\rm f}_{7/2}\epsilon {\rm s}_{1/2}$, $4{\rm f}_{7/2}\epsilon {\rm d}_{3/2}$, $4{\rm f}_{7/2}\epsilon {\rm d}_{5/2} $和$ 6{\rm p}_{3/2}\epsilon {\rm d}_{5/2} $

Fig. 2.  Comparison of the experimental spectrum(upper figure), the theoretical spectrum(middle figure) considering interaction among these eigenchannels that give primary contribution to the experimental spectra and the eigenchannel spetrum(lower figure) for $ 5{\rm d}\epsilon {\rm p}\left(\dfrac{5}{2}, \dfrac{1}{2}\right)_3, J^\pi=3^- $ The theoretical spectrum shown in middle figure is obtained based on the Eq. (1), and included these eigenchannels with $ J^\pi=3^- $: $ 5{\rm d}_{3/2}\epsilon {\rm f}_{5/2}$, $5{\rm d}_{5/2}\epsilon {\rm f}_{5/2}$, $5{\rm d}_{5/2}\epsilon {\rm p}_{1/2}$, $4{\rm f}_{5/2}\epsilon {\rm s}_{1/2}$, $4{\rm f}_{5/2}\epsilon {\rm d}_{3/2}$, $4{\rm f}_{7/2}\epsilon {\rm s}_{1/2}$, $4{\rm f}_{7/2}\epsilon {\rm d}_{3/2}$, $4{\rm f}_{7/2}\epsilon {\rm d}_{5/2} $ and $ 6{\rm p_{3/2}\epsilon d}_{5/2} $.

图 3  一价镧离子的里德伯系列(下图)和自电离里德伯系列(上图)对应的模为1的量子数亏损随激发能量的变化关系. 实验值用实心圆点表示, 理论计算给出了$ J^\pi=3^- $的所有可能的束缚态能级和本征通道$ 5{\rm d}\epsilon {\rm p}\left(\dfrac{5}{2}, \dfrac{1}{2}\right)_3 $的谱峰能级对应的模为1的量子数亏损, 用空心圆点表示. 激发能量用有效量子数$ n^\star $表征, 利用了里德伯关系, $ E=E_\infty-\dfrac{Z^2 Ry.}{(n^\star)^2}, n^\star=n-\mu_{n, l, j} $, 这里$ E $是激发能量, $ n^\star $为有效量子数, $ \mu $为量子数亏损, $ E_\infty $ 是相应的电离阈值, $ Z $为离子实的有效正电荷数, $ Ry. $为里德伯常数

Fig. 3.  Quantum defect $ \mu $ mod 1 v.s. excited energy for Rydberg series(lower figure) and autoionization Rydberg series(upper figure) of La⁺. $ \bullet $: Experimental data. $ \circ $: theoretcal quantum defect mod 1 for all possible bound state energy levels with $ J^\pi=3^- $ symmetry and peak positions of eigenchannel $ 5{\rm d}\epsilon {\rm p}\left(\dfrac{5}{2}, \dfrac{1}{2}\right)_3 $. Excitation energy is represented by the effective quantum number $ n^\star $, according to Rydberg formula $ E=E_\infty-\dfrac{Z^2 Ry.}{(n^\star)^2}, n^\star=n-\mu $. Here, $ E $ is excitation energy, $ n^\star $ is effective quantum number, $ \mu $ is quantum defect, $ E_\infty $ is the ionization threshold, $ Z $ is the charge of the ionic core, and $ Ry. $ is Rydberg constant.

图 4  不同电离阈值得到的一价镧离子里德伯系列(有效量子数$ n^\star>67 $时, 对应激发能量范围为90120 cm^–1—90175 cm^–1)量子数亏损随激发能量的变化关系. 采用文献[1]给出的一价镧离子第一电离阈值90212.8 cm^–1, 得出的量子数亏损用实心圆点表示; 采用根据里德伯系列量子数亏损变化光滑性拟合的电离阈值90212.5 cm^–1, 得出的量子数亏损用空心圆点表示

Fig. 4.  Quantum defect $ \mu $ v.s. excited energy for the Rydberg series ($ n^\star>67 $, in the energy region 90120 cm^–1—90175 cm^–1 ) converging to the different ionization thresholds. $ \bullet $: quantum defects obtained by the ionization threshold 90212.8 cm^–1 from Ref. [1]. $ \circ $: quantum defects obtained by the ionization threshold 90212.5 cm^–1, which is fitted based on the quantum defect regular behavior for a Rydberg series.