Calculation of Hyperfine Structure in Tm ii

Abstract

The first measurements of the magnetic dipole hyperfine structure constants A in singly ionized thulium revealed substantial discrepancies with the corresponding theoretical calculations. Subsequent measurements expanded the very limited available dataset and demonstrated that two of the previously reported experimental A values were incorrect, thereby motivating new theoretical calculations. In this work, we employ the configuration interaction method to calculate the A constants for several low-lying levels in Tm ii, with the random-phase-approximation corrections also taken into account. Our results show good agreement with the new experimental data and provide reliable predictions for additional states where measurements are not yet available.

1. Introduction

Thulium is located near the end of the lanthanide series in the periodic table of the elements and has only one stable isotope, ${}_{69}{}^{169}{}_{\mathrm{}}\mathrm{Tm}$, with nuclear spin $I=\frac{1}{2}$. Thus, the hyperfine interaction of electrons in its ions with the nuclear moments is determined solely by the magnetic dipole contribution, while the electric quadrupole contribution is absent. The resulting hyperfine structure (HFS) of the electronic levels is therefore characterized exclusively by the magnetic dipole HFS constant A.

Singly ionized thulium, Tm ii, as a system with an open f shell, has a complex and dense spectrum, which was studied experimentally [1,2] and theoretically [3,4]. Although the experimental research on transition rates and lifetimes is sufficiently extensive [5,6,7,8,9,10,11,12] (see Refs. [3,4] for theoretical predictions), only two papers report measurements of the HFS, and results of only one corresponding calculation have been published. Mansour et al. [13] used collinear fast-ion-beam laser spectroscopy to determine HFS constants A for eight levels. Their measurements of the A constants for the four lowest states were compared with the results of the multiconfiguration Dirac-Hartree-Fock (MCDHF) calculations (see Ref. [14] for calculation details). They found a clear discrepancy between the experimental data and theoretical predictions for the ${\left(4f^{13}\left({}^{2}F_{5/2}\right)6s_{1/2}\right)}_2^o$ level and concluded that configurations containing an open s shell are in substantial disagreement with the calculations. Recently, Kebapcı et al. [15] reported A constants extracted from the hollow-cathode-discharge-lamp emission spectra recorded by a Fourier transform spectrometer. Parts of the spectra in other wavelength ranges had previously been analyzed to study HFS and transition rates in neutral thulium [16,17,18]. They determined 27 additional A constants beyond those reported by Mansour et al. [13] and showed that two of the values in that work, including the one for the ${\left(4f^{13}{(}^2{F}_{5/2})6s_{1/2}\right)}_2^o$ level, were incorrect. Their revised value for this level partially resolves the discrepancy with the MCDHF result reported in Ref. [13] and motivates new calculations.

Meanwhile, an experimental campaign at CERN aimed to measure nuclear moments and changes in root-mean-square radii in radioactive thulium isotopes is ongoing [19,20,21]. As a groundwork for these experiments, the hyperfine structure and transition rates in the stable isotope are also being studied at COALA, an offline collinear laser spectroscopy setup for high-precision measurements at TU Darmstadt (Germany) [22]. This technique enables the determination of the A constants with significantly higher accuracy than the approaches mentioned above. Furthermore, a spectroscopic investigation of Tm ii focused on establishing its use as a platform for advanced quantum applications is currently underway at the University of California, Los Angeles [23].

In this study, we calculate the magnetic dipole HFS constants A and Landé g factors for the low-lying levels of the $4f^{13}6s$ and $4f^{13}5d$ configurations in Tm ii using a configuration interaction (CI) method with the random-phase approximation (RPA) included. Our results show good agreement with the available measurements and provide predictions for states where experimental data are not yet available. Atomic units $\hslash =e=m_{\mathrm{e}}=4\pi {\epsilon }_0=1$ are used throughout unless otherwise stated.

2. Theory

The interaction of the electrons of an ion with the nuclear magnetic moment is described by the operator

$$H_{\mathrm{hfs}}={\displaystyle \frac{1}{c}}\mathsf{\mu }·{\mathit{T}}^{\left(1\right)}.$$

(1)

Here, c is the speed of light, $\mathsf{\mu }$ is the operator of the nuclear magnetic moment, acting in the nuclear subspace, and electronic operator ${\mathit{T}}^{\left(1\right)}$ is the sum of the one-electron operators ${\mathit{t}}^{\left(1\right)}\left(i\right)$,

$${\mathit{T}}^{\left(1\right)}=\sum _i{\mathit{t}}^{\left(1\right)}\left(i\right)=\sum _i{\displaystyle \frac{\left[{\mathit{r}}_i\times {\mathsf{\alpha }}_i\right]}{r_i^3}},$$

(2)

where $\mathsf{\alpha }$ are the Dirac matrices, and the point-dipole approximation is employed. We do not take into account the finite-nucleus magnetization distribution, since the corresponding correction (the Bohr-Weisskopf effect [24]) is smaller than the uncertainty associated with electron-correlations effects. It is worth noting that for heavy atoms and ions with fewer valence electrons, the theoretical accuracy is high enough that both the Bohr-Weisskopf and quantum electrodynamical (QED) corrections to the HFS become important [25,26]. The magnetic dipole HFS splitting of the state with total angular momentum J can be expressed in terms of the HFS constant A,

$$A={\displaystyle \frac{1}{c}}{\displaystyle \frac{\mu }{I}}{\displaystyle \frac{⟨J\left|\right|{\mathit{T}}^{\left(1\right)}\left|\right|J⟩}{\sqrt{J(J+1)(2J+1)}}},$$

(3)

where $\mu =⟨II\left|{\mu }_z\right|II⟩$ is the magnetic moment of the nucleus. According to Ref. [27], the value $\mu =-0.2310\left(15\right){\mu }_N$, where ${\mu }_N$ is the nuclear magneton, is used in the present calculations for ¹⁶⁹Tm. The reduced matrix element of the operator ${\mathit{T}}^{\left(1\right)}$ between the many-electron states in Equation (3) is evaluated under the density matrix formalism, which allows us to express it via one-electron matrix elements [28].

The Landé g factor of a fine-structure level, $g_J$, defined by

$$\Delta E^{\left(1\right)}=g_J{\mu }_{\mathrm{B}}{M}_{J}B,$$

(4)

where $\Delta E^{\left(1\right)}$ is the linear energy-level shift induced by the interaction with an external homogeneous magnetic field $\mathit{B}$, which is assumed to be aligned with the z axis, $M_J$ is the z projection of the total angular momentum, and ${\mu }_{\mathrm{B}}$ is the Bohr magneton, may serve as an estimate of the accuracy of the wave function of a state and, consequently, of its A constant. The g factors are calculated from the relation

$$\Delta E^{\left(1\right)}=⟨J{M}_J|V_{\mathrm{m}}|J{M}_J⟩,$$

(5)

where the operator

$$V_{\mathrm{m}}={\displaystyle \frac{1}{2}}\sum _i[{\mathit{r}}_i\times {\mathsf{\alpha }}_i]·\mathit{B},$$

(6)

represents the interaction with the magnetic field and i enumerates the electrons of the ion. A more comprehensive calculation, including QED and other corrections (see, e.g., Ref. [29]), is beyond the scope of the present paper.

To calculate the A constants and Landé g factors in Tm ii, we use the 14-electron CI method while keeping the 54 electrons of the Xe-like core frozen. We use a basis set of one-electron functions with orbital angular momentum $l⩽4$ and principal quantum number $n⩽10$, designated as $10spdfg$, and constructed as follows. First, we solve the Dirac-Hartree-Fock equations for the $[1s^2,\dots ,4f^{13},6s]$ electrons. Then, all electrons are frozen and the electron from the $6s$ shell is moved to the $5d$ shell, and the relativistic $5d$ orbitals are constructed in the frozen core potential. The same procedure is applied to form the $6p$ orbitals, while the remaining virtual orbitals are constructed following the procedure described in Refs. [30,31]. The configuration space is formed by allowing single and double excitations from the $4f^{13}6s$ and $4f^{13}5d$ configurations to the whole $10spdfg$ basis set. The calculations with the smaller $9spdfg$ and larger $11spdfg$ basis sets demonstrate convergence with respect to the basis-set size. Higher-rank excitations can not be included due to computational limitations. However, it was recently found that they do not substantially affect the A constants in neutral Tm [32], which presumably also holds for the ion.

The interaction with the frozen core for the A constants is partly accounted for by the RPA [33]. The RPA corrections are calculated using an extended basis set, $35spdfg$, and added to the radial one-electron matrix elements of the operator ${\mathit{T}}^{\left(1\right)}$ in the calculation of A according to Equation (3). A complementary approach to account for core-valence as well as for valence-valence correlations using many-body perturbation theory was described in Ref. [34] and employed in the Tm i calculations [18].

3. Results and Discussion

In Table 1, we compare the A constants calculated using the CI method, both without and with the RPA corrections (CI + RPA), with the experimental data from Refs. [13,15] and with the MCDHF calculations [13]. As can be seen from the table, our CI results are in good agreement with the previous MCDHF calculation [13]. They are slightly closer to experiment for the levels of the $4f^{13}6s$ configuration, but somewhat further from experiment for the levels of the $4f^{13}5d$ configuration. The overall consistency between the CI and MCDHF results reflects the similarity of the two methods. Furthermore, as the CI configuration space is reduced, the corresponding predictions tend to move even closer to those of the MCDHF calculation.

Table 1. A comparison of the magnetic dipole hyperfine structure constants A (in MHz) calculated using the CI and CI + RPA methods with the experimental results [13,15] for low-lying levels of odd parity in Tm ii. The results of MCDHF calculations [13] are also shown. The level energies and configuration assignments are taken from the NIST ASD [35].

				Experiment		Calculation
$\mathit{E}$ (cm−1)	$J$	Configuration	Term	Ref. [13]	Ref. [15]	CI	CI + RPA	MCDHF [13]
0	4	$4f^{13}{(}^2{F}_{7/2})6s_{1/2}$	${(7/2,1/2)}^o$		$-1038.3\left(16\right)$	$-945$	$-1040$
236.95	3	$4f^{13}{(}^2{F}_{7/2})6s_{1/2}$	${(7/2,1/2)}^o$		$293\left(6\right)$	164	253
8769.68	2	$4f^{13}{(}^2{F}_{5/2})6s_{1/2}$	${(5/2,1/2)}^o$	$914.9\left(10\right)$ *	$129\left(4\right)$	$-30$	120	$-67$
8957.47	3	$4f^{13}{(}^2{F}_{5/2})6s_{1/2}$	${(5/2,1/2)}^o$	$-1537.7\left(8\right)$		$-1401$	$-1506$	$-1326$
17,624.65	2	$4f^{13}{(}^2{F}_{7/2})5d_{3/2}$	${(7/2,3/2)}^o$	$-341.8\left(13\right)$		$-463$	$-350$	$-434$
21,713.74	3	$4f^{13}{(}^2{F}_{7/2})5d_{3/2}$	${(7/2,3/2)}^o$	$-313.5\left(19\right)$		$-362$	$-275$	$-351$

* In Ref. [15], this value was found to be incorrect and reanalyzed yielding a revised value of 125.7 MHz.

For the first and four levels of the $4f^{13}6s$ configuration, the difference between the CI results and experiment is below 10%, for the second level it is about 45%, and for the third level the calculated A constant even has the opposite sign. It is therefore instructive to examine the origin of these discrepancies more closely. The two dominant contributions to the A constants of these levels arise from the diagonal density-matrix elements of the one-electron f and s functions. For the first and fourth levels, the total angular momentum of the f-hole (or 13 f-electrons) and s-electron are parallel when coupled to the total angular momentum J of the level. In these cases, the f and s contributions to A have the same sign, resulting in a large magnitude of A. In contrast, for the second and third levels the momenta are antiparallel, and the contributions have opposite signs, partially canceling each other and leading to a small magnitude of A. This cancellation makes the A constants of these levels more difficult to calculate than those of the first and fourth ones. Taking all that into consideration, the measurement outcome for the third level, $A=914.9\left(10\right)$ [13], is doubtful, given that the A constants are known for the other three levels of the configuration. However, in the work of Mansour et al. [13], the A constants of only two of the four $4f^{13}6s$ levels were measured and calculated, making such an analysis impossible. We note that the experimental results for the ground and first excited states obtained at COALA [22] agree with the data of Kebapcı et al. [15], but have uncertainties smaller by about two orders of magnitude [36]. The RPA corrections significantly reduce the discrepancy between the CI results and experiment for all levels of the $4f^{13}6s$ configuration. In particular, they yield the correct sign of A for the third level, mainly by increasing the dominant one-electron s contribution discussed above.

For the levels of the $4f^{13}5d$ configuration, for which only two measurements of the A constants exist, the RPA corrections are also important and reduce the discrepancy for the level at 17,624.65 cm⁻¹. For the level at 21,713.74 cm⁻¹, they as for the one at 17,624.65 cm⁻¹ decrease the magnitude of A obtained in the CI calculation; however, this makes it noticeably smaller than the experimental value. A possible explanation of the poorer agreement with experiment for this level, as for the one at 17,624.65 cm⁻¹, is discussed below. To this end, Table 2 presents the calculated A constants for all 11 experimentally known levels of the $4f^{13}5d$ configuration listed in the NIST ASD [35]. The comparison between calculated and experimental values of the Landé g factors is also shown in the table. Our calculated g factors are generally in very good agreement with the experimental data [35]. The only exception is the level at 21,713.74 cm⁻¹, where the calculated value disagrees with experiment. We attribute this discrepancy to the interaction of this level with other levels with $J=3$, in particular to the one at 23,934.73 cm⁻¹, which is not fully captured by our CI calculation. This, in turn, leads to a less accurate wave function and A constant for that level. To examine this hypothesis, g factor measurements for the other levels in Table 2 would be valuable. An experimental study of the corresponding A constants would also be of interest, in order to verify the CI + RPA predictions, which show a large variation of the A constants among the levels of the $4f^{13}5d$ configuration. Overall, the accuracy of the calculated A constants for the levels with good agreement in the g factors is expected to be similar to that for the level at 17,624.65 cm⁻¹.

Table 2. Magnetic dipole hyperfine structure constants A (in MHz) calculated using the CI and CI + RPA methods for the levels of the $4f^{13}5d$ configuration in Tm ii. A comparison of the calculated and measured Landé g factors is also given. The level energies, Landé g factors, and configuration assignments are taken from the NIST ASD [35].

				Landé g Factor		A (MHz)
$\mathit{E}$ (cm−1)	$J$	Configuration	Term	Experiment [35]	CI	Experiment [13]	CI	CI + RPA
17,624.65	2	$4f^{13}{(}^2{F}_{7/2})5d_{3/2}$	${(7/2,3/2)}^o$	1.48	1.469	$-341.8\left(13\right)$	$-463$	$-350$
20,228.75	5	$4f^{13}{(}^2{F}_{7/2})5d_{3/2}$	${(7/2,3/2)}^o$	1.020	1.018		$-312$	$-330$
21,133.68	6	$4f^{13}{(}^2{F}_{7/2})5d_{5/2}$	${(7/2,5/2)}^o$	1.167	1.166		$-249$	$-190$
21,713.74	3	$4f^{13}{(}^2{F}_{7/2})5d_{3/2}$	${(7/2,3/2)}^o$	1.22	1.273	$-313.5\left(19\right)$	$-362$	$-275$
21,978.77	2	$4f^{13}{(}^2{F}_{7/2})5d_{5/2}$	${(7/2,5/2)}^o$	1.02	1.017		$-442$	$-574$
22,141.96	1	$4f^{13}{(}^2{F}_{7/2})5d_{5/2}$	${(7/2,5/2)}^o$	1.325	1.324		$-757$	$-813$
22,457.51	4	$4f^{13}{(}^2{F}_{7/2})5d_{3/2}$	${(7/2,3/2)}^o$		1.184		$-318$	$-246$
23,524.09	4	$4f^{13}{(}^2{F}_{7/2})5d_{5/2}$	${(7/2,5/2)}^o$		1.033		$-303$	$-327$
23,934.73	3	$4f^{13}{(}^2{F}_{7/2})5d_{5/2}$	${(7/2,5/2)}^o$		1.043		$-332$	$-395$
24,273.20	5	$4f^{13}{(}^2{F}_{7/2})5d_{5/2}$	${(7/2,5/2)}^o$	1.18	1.172		$-263$	$-193$
28,874.14	4	$4f^{13}{(}^2{F}_{5/2})5d_{3/2}$	${(5/2,3/2)}^o$		0.833		$-492$	$-500$

4. Conclusions

In this work, we used the CI method to calculate the magnetic dipole HFS constants A and Landé g factors for the levels of the ground $4f^{13}6s$ and first excited $4f^{13}5d$ odd configurations in Tm ii. We found that the RPA corrections to the A constants are essential for achieving good agreement between theory and experiment for the levels of the ground configuration, for which, owing to its relative simplicity, a more detailed analysis has been presented. The obtained results provide predictions for states where experimental data are not yet available. Our approach can also be used to calculate the electric field gradient at the site of the nucleus, which gives rise to the electric quadrupole interaction in unstable isotopes with nuclear spin $I⩾1$. This information is important for planning experiments with radioactive isotopes; however, in contrast to transition rates and A constants, it can not be obtained in ground-step experiments using the stable isotope ¹⁶⁹Tm.