Hyperfine structure and isotope shift measurements on spin-forbidden transitions of atomic gadolinium

Upendra M. Adhikari; Kevin D. Battles; Clayton E. Simien

doi:10.1364/OSAC.2.001548

1. Introduction

Gadolinium (Gd) is a rare-earth element with atomic number Z = 64, belonging to the lanthanide series on the periodic table. The electronic ground state is [Xe]4f⁷5d¹6s² configuration, which implies that all shells up to the 5p shell are fully occupied as in noble gases. Of the ten valence electrons, two of them fill the 6s shell, one is in the 5d shell, and the remaining seven are in the 4f shell, leaving it with an electron vacancy [1]. This submerged shell structure provides for several atomic physics applications: for precision measurements Gd has transitions with enhanced sensitivities to variation of fine structure constant [2], with increased suppressions of black-body radiation shifts which could enable a more accurate optical clock [3]. Moreover, for quantum gas investigations, Gd has a large ground state magnetic moment $7{\mu _B}$ in addition to six stable fermionic and bosonic isotopes: ¹⁵⁵Gd (15% abundance), ¹⁵⁷Gd (16% abundance), ¹⁵⁴Gd (2% abundance), ¹⁵⁶Gd (20% abundance), ¹⁵⁸Gd (25% abundance), and ¹⁶⁰Gd (22% abundance), which will allow for novel dipolar physics investigations.

Gadolinium ground state is odd parity and consist of five fine structure levels $^9{D_J}$ with total electronic angular momentum ${J_{g\; }} = 2 - 6$. The closest even parity states are [Xe]4f⁸6s² $^7{F_J}$ and [Xe]4f⁷5d¹6s 6p $^{11}{F_J}$, and electric-dipole transition (E1) from the ground state levels to these states are known as intercombination lines. These $\Delta S \ne 0$ spin forbidden transitions occur in relatively heavy elements since the spin-orbit and spin-spin interactions become large enough to create an appreciable admixture of states with different total spin quantum numbers. Spin forbidden transitions are important for next generation frequency standards, quantum computing [4], and implementing narrow line laser cooling for the production of dipolar quantum degenerate gases [5]. Furthermore, optical spectra of these lines are important sources of information; optical isotope shifts (IS) provide information about the change in nuclear charge distribution between the isotopes, and for more efficient isotope separation and enrichment techniques to be developed [6]. For gadolinium, the spin forbidden transitions between $^9{D_J} \to {}^{11}{F_J}$ manifolds are of special interest. Unfortunately, there is no available spectroscopic information about these lines. This makes it impossible to explore the aforementioned research avenues with atomic gadolinium. Moreover, Gd is unique among the other rare earths in that it has one 5d electron, which may lead to additional and surprising features as compared to other lanthanide elements.

In this Letter, we present for the first time isotope shift studies of spin-forbidden transitions in atomic gadolinium 0 – 15000 cm⁻¹ energy range. Figure 1 shows the partial energy-level diagram of the ground and excited state configurations with the spin-forbidden transitions labeled. Measurements were performed to investigate two intercombination lines in the visible region. In this study we performed optogalvanic (OG) absorption spectroscopy with a tunable Ti-Sapphire laser in a gadolinium gas discharge cell [7]. This sensitive technique proves useful for achieving good signal-to-noise when investigating the relatively weak line intensities of intercombination transitions.

Fig. 1. Energy level diagram of the ground (4f⁷5d6s²) and excited (4f⁷5d6s6p) states of transitions in this study.

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2. Experimental results

The Gd plasma discharge, which provides a source of atoms and ions in their ground and excited states, provides optical access, and is filled with neon as a buffer gas. A homemade high voltage power supply drives the discharge in series with a 20-kΩ resistor. The power supply is operated at 300 V, resulting in 10 mA current sustained through the discharge. Laser light for the experiments was produced using a commercial continuous-wave Ti-sapphire laser (M2 SOLSTIS). The laser was locked to an external high finesse cavity for improved linear frequency scanning, and the linewidth of the laser was less than 100 kHz. The absolute wavelength measurements of the laser at 725 nm and 745 nm were determined with a high precision wavemeter (Bristol 621A-VIS). The laser excitation power level was always set in order to avoid saturation effects to prevent nonlinearites in the line intensities. Moreover, a portion of the laser light was directed to a Fabry-Perot etalon (FPe) with a FSR of 1.5 GHz, which was used as a relative frequency reference to measure transition splittings and widths. The laser output was directed inside the discharge cell.

Resonant absorption of the laser beam lead to a change in the steady-state population of the atomic levels and in the impedance of the discharge, which was monitored as a change in the voltage drop across a ballast resistor as a function of laser frequency. The laser beam was amplitude modulated at a frequency of about 700 Hz. A capacitor in the discharge electrical circuit was used to output couple the optogalvanic induced voltage modulation, which was sent directly to a lock-in amplifier for phase sensitive detection. The lock-in time constant was set to 1s and the output filter was set to 6 dB per octave, and the spectrum as well as the transmission peaks through FP, were simultaneously recorded on a digital oscilloscope as the laser was tuned at a rate of $0.75\;GHz\; {s^{ - 1}}$, averaging over 32 sweeps for the final results. Figure 2 is a typical experimental spectrum which shows the single resonance from the three even isotopes ¹⁶⁰Gd, ¹⁵⁸Gd, and ¹⁵⁶Gd, and the hyperfine structure of the odd isotope ¹⁵⁵Gd.

Fig. 2. Optogalvanic spectrum of $^9{D_4} \to {}^{11}{F_3}$ transition for gadolinium isotopes. The experimental data is the solid thick dark line, a fit to Eq. (1) is the thin dashed line. The vertical lines indicate the relative intensities and positions of hyperfine components. The four intense hyperfine components of Gd 157 (Gd 155) isotope transitions in terms of F numbers are the following: a(a’):4.5,3.5; b(b’):5.5,4.5; c(c’):3.5,2.5; d(d’):2.5,1.5.

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The initial nonlinear regression procedure involved fitting the spectra to a sum of Voigt profiles. This step determined the Lorentzian and Gaussian contributions to the overall line shape of the observed transitions, and Doppler broadening was determined to be the dominant broadening mechanism in the discharge cell. The various isotope profiles of the intercombination transitions are resolved in this Doppler-limited study because the separations are larger than the broadening widths.

Based on the initial analysis the observed spectra can be accurately described by a sum of Gaussian functions given by the following [8]:

(1)$$I(\nu )= \mathop \sum \nolimits_n {I_n}exp\left[ {\frac{{ - {{({\nu - {\nu_n}} )}^2}}}{{0.36\Delta {\upsilon_i}^2}}} \right],$$

where ${I_n}$ is the intensity of the nth spectral component, ${\Delta }{\upsilon _i}$ is the half-width of the Gaussian profiles, and ${\nu _n}$ is the nth peak location. The excited state hyperfine constants and isotope shifts were determined by fitting Eq. (1) to the experimental data. This procedure utilized a non-linear least-square fitting routine which begins with a set of initial input variables which include Gaussian widths, magnetic dipole (${{\boldsymbol A}_{\boldsymbol i}}$) and electric-quadrupole hyperfine constants (${{\boldsymbol B}_{\boldsymbol i}}$), center-of-gravity energy shift, baseline offset, and line intensities.

In Fig. 2, the optogalvanic spectrum for of $^9{D_4} \to {}^{11}{F_3}$ transition is shown. The solid thick dark line represents the OG signal as a function of laser detuning, and the thin dashed line is a typical fit to experimental data using Eq. (1) given above. The black vertical markers indicate the fitted position of each transition component, with the size of the vertical marker representing the relative intensity of the transition. This graph illustrates the experimental data is well described by the sum of Gaussian profiles. The line intensities of the nth hyperfine transitions are given by

(2)$${I_{HYP,n}} = ({2{F_g} + 1} )({2{F_e} + 1} ){\left|{\left\{ {\begin{array}{{ccc}} {{J_g}}&{{F_e}}&I\\ {{F_g}}&{{J_e}}&1 \end{array}} \right\}} \right|^2}{|\langle{J_g}|| d ||{J_e}\rangle|^2},\; $$

where J, I, and F are the electronic, nuclear, and total angular momentum quantum numbers respectively, and the g and e denotes ground and excited states [9]. The $\langle{J_g}|| d ||{J_e}\rangle$ is the reduced dipole matrix element, and the $\left\{ {\begin{array}{{c}} {\ldots }\\ \ldots \end{array}} \right\}$ is a Wigner 6-j symbol. For fitting analysis the relative intensities of each hyperfine component was scaled relative to the most intense hyperfine component, thus these Rachah intensities were not free parameters in the nonlinear regression fitting routine [10].

To evaluate the hyperfine coefficients for ¹⁵⁵Gd and ¹⁵⁷Gd we incorporated into Eq. (1) the energy of the hyperfine structure component given by

(3)$${E_{i{\;\ }}}({{F_i},{J_i},{\;\ }I} ) = \frac{1}{2}{A_i}{C_i} + \frac{{3{C_i}({{C_i} + 1} )- 4I({I + 1} ){J_i}({{J_i} + 1} )}}{{8I({2I - 1} ){J_i}({2{J_i} - 1} )}}{B_i},$$

where ${C_i} = \; {F_i}({{F_i} + 1} )- {J_i}({{J_i} + 1} )- I({I + 1} )$ [11]. This calculated spectra was compared to the experimental data with adjustment of the free input parameters for χ² -minimization. To ensure the fitting routine finds a global chi-squared minimum the excited state hyperfine coefficients for the given transitions of different Gd fermionic isotopes are linked using a known scaling law which is described by the following equation [12,13]:

(4)$$\; {A_{155}} = \left( {\frac{{{I_{155}}}}{{{I_{157}}}}} \right)\left( {\frac{{{\mu_{155}}}}{{{\mu_{157}}}}} \right){A_{157}},$$

where ${\mu _{155,157}}$ and ${I_{155,157}}$ are the nuclear magnetic moments and spins respectively for the two fermionic isotopes. The lower hyperfine parameters are known and taken from magnetic resonance experiments, thus are held fixed during the analysis [14]. The cumulative best fit results for the excited state hyperfine coefficients (A_e ,B_e) are shown in Table 1.

Table 1. Hyperfine coefficients A_e and B_e of excited states.

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The isotope shifts are calculated with respect to ¹⁶⁰Gd as the reference isotope. The primary source of systematic uncertainty is due to resolution of the FP-cavity, which we take as one-half of the full-width-half-maximum. Thus, the systematic uncertainties are added in quadrature with statistical error obtained from a χ² analysis of the average of 32 measurements, and are given as the combined standard error. Although the hyperfine structure of the ¹⁵⁷Gd isotope is not fully resolved in this study, we were able to determine isotope shift with respect to center of gravity using the sophisticated analysis described above. The cumulative results for all measurements are given in Table 2.

Table 2. Gd I measured isotope shifts (ν^A – ν¹⁶⁰) of $^9{D_J} \to {}^{11}{F_J}$ transitions. All values in GHz.

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The measured isotope shift $\delta \upsilon _i^{AB}$ between two isotopes with mass numbers A and B for a given transition can be described as the sum of the nuclear dependent field shift (FS) and mass shift (MS) [15]:

(5)$$\delta \upsilon _i^{AB} = {F_i}{\lambda ^{AB}} + \delta {\nu _M}\frac{{A - B}}{{AB}}$$

where ${F_i}$ is the electronic factor which is proportional to the change of electron charge density at the nucleus in transition under investigation. The coefficient ${\lambda ^{AB}}$ describes the change of the radial nuclear charge parameters. The mass shift term ($\delta {\nu _M}$) is the direct sum of two contributions: the normal mass shift ($\delta {\nu _{NM}}$) and specific mass shift ($\delta {\nu _{SM}}$). The normal mass shift arises from the isotopic dependence of the electronic reduced mass. The specific mass shift is due to momenta correlation between the electrons orbiting the nucleus. These mass dependent terms overall affect the kinetic energy of the nucleus.

The normal mass shift is simply calculated using the following equation

(6)$$\; \delta {\nu _{NM,i}} = {M_{NM,i}} = \frac{{{m_e}}}{{{m_n}}}\frac{{A - B}}{{AB}}{\nu _i}, $$

where ${\nu _i}\; \; $is the transition frequency, and ${m_e}$ and ${m_n}$ are the electron and neutron mass respectively. The $\delta {\nu _{SM}}\; $has a similar functional form ($\; \delta {\nu _{SM,i}} = \; {M_{SM,i}}\; $), however it is difficult to evaluate this term theoretically. The field shift component is written as

(7)$$\; \; \delta {\nu _{FS,i}} = \; {F_i}{\lambda ^{AB}}\; , $$

where ${\lambda ^{AB}}$ is the nuclear charge parameter for the isotope pair.

When $\delta {\nu _{NM}}$ is subtracted from the measured isotope shift, Eq. (5) can be rewritten as the following:

(8)$$\frac{{AB}}{{A - B}}{\;\ }\delta \upsilon _i^{AB} = {F_i}\frac{{AB}}{{A - B}}{\lambda ^{AB}} + \delta {\nu _{SM,i}}.\; $$

The left-hand side of Eq. (8) is referred to as the modified residual isotope shift, and the term $(AB/A - B)\; $is the mass factor (${M_{AB}}$) for a pair of isotopes A and B. The relationship between $\frac{{AB}}{{A - B}}{\;\ }\delta \upsilon _i^{AB}$ and $\frac{{AB}}{{A - B}}{\lambda ^{AB}}$ is linear and can be used to determine ${M_{SM}}$ and ${F_i}$ or the nuclear charge parameter ${\lambda ^{AB}}$ in a King plot [16].

In a typical King plot coordinates are given by the normalized reduced residual isotope shift which is defined by the following equation [17]:

(9)$$\delta \upsilon _{R,i}^{AB} = {M_{AB}} \times ({\delta \upsilon_i^{AB} - \delta {\nu_{NM,i}}} )\times \left[ {\frac{{{A_{std}} - A_{std}'}}{{{A_{std}}A_{std}'}}} \right],$$

for an arbitrary chosen pair of standard isotopes ${A_{std}}$ and $A_{std}^,$. This diagram is produced when Eq. (9) for a particular transition i of interest is plotted on the y-axis versus the reduced residual isotope shift of a reference line j plotted on the x-axis. As a result, the plotted points form a straight line. The straight line in the King plot is the following expression

(10)$$\delta \upsilon _{R,i}^{A,B} = \frac{{{F_i}}}{{{F_j}}}\delta \upsilon _{R,j}^{A,B} + \left( {\delta {\nu_{SM,i}} - \frac{{{F_i}}}{{{F_j}}}\delta {\nu_{SM,j}}} \right)\left( {\frac{{{A_{std}} - A_{std}'}}{{{A_{std}}A_{std}'}}} \right)$$

The slope of the straight line is an experimental quantity yielding the ratio ${F_i}/{F_j}$ in the field shift of the two lines. The y-intercept of the line is expressed as

(11)$$y = \left( {\delta {\nu_{SM,i}} - \frac{{{F_i}}}{{{F_j}}}\delta {\nu_{SM,j}}} \right)\left( {\frac{{{A_{std}} - A_{std}'}}{{{A_{std}}A_{std}'}}} \right)\; . $$

For this work the isotopes ¹⁶⁰Gd and ¹⁵⁸Gd are used as the standard pair. Thus, the normalized reduced isotope shift for each transition i in Table 2 is plotted against the reduced shifts of a reference line j. Figure 3 is a King plot of the 726 nm transition on the ordinate. The reference transition values on the absicissa are the [Xe]4f⁷5d¹6s² → [Xe]4f⁷5d¹6s6p transition at 422 nm [18,19]. This reference line is is nearly a pure (ns² → nsnp), thus the wavefunction should have a relatively small admixture of other configurations.

Fig. 3. King-plot for 726 nm line from Table 1. The normalized reduced isotope shifts are plotted against the normalized reduced isotope shift of a reference transition at 422 nm [15]. The straight line is a least-square fit, and the error bars represent standard error of regression.

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Thus, ${M_{SM,j}}$ for this transition can be assumed negligible according to the semi-empirical relation ${M_{SM}} \approx ({0 \pm 0.5} ){M_{NM}}$ [20]. However, for this particular reference line, isotope shift data suggests the relation is not valid; hence the mass shift factor ${M_{SM,j}}$ of the 422 nm line is used in the analysis of this work.

The straight lines in Figs. 3 and 4 are a result of linear regression analysis. The King plot is linear for the 726 nm transition. However, analysis of the 743 nm line suggests nonlinearity in the graph, which will be discussed further below. From the values of the y-intercept obtained using linear regression analysis, specific mass shift values of the reference lines, and slopes of the fitted lines in conjunction with isotope field shift constants of the reference lines; the field shift factor ${F_i}$ can be extracted.

Fig. 4. King-plot for 743 nm line from Table 1. The circled data point does not fall on the regression line, which indicates nonlinearity of the plot.

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Moreover, since the field shift factor is defined as ${F_i} = \; {E_i}f(Z )$ and ${E_i}$ is a coefficient related to the nonrelativistic electron density at the nucleus for the transition of interest, and $f(Z )$ is a correction factor for finite nuclear size and relativistic effects. The term ${E_i}$ is difficult to calculate, however from the King plots relative values can be inspected [21]. Table 3 gives the derived values of the field shift coefficient, specific mass shift, normal mass shift, and ${E_i}/{E_j}$ ratios (line slopes). The uncertainties specified with the final derived quantities are calculated using error propagation analysis [22].

Table 3. Isotope shift constants values for M_NM , M_SM, F_i , and E_i / E_j

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The field shift factor obtained from the 726 nm King plot is negative, which illustrates that optical frequencies decrease when the rms nuclear charge radius increases in conjunction with increasing neutron numbers of Gd isotopes for this particular transition. In addition, ${M_{SM}}$ is greater than ${M_{NM}}$ by a least a factor of 10 for both lines, which is expected for transitions with changes in d electrons; moreover it suggest admixtures from other configurations in the excited states. Also, the positive ${E_i}/{E_j}$ ratio obtained for the 726 nm transition in this study, indicate the [Xe]4f⁷5d¹6s6p is the dominant contribution to the upper level configuration as confirmed by NIST atomic data base [23].

The 743 nm transition isotope shift data has an asterisk attached to it due to the fact the King plot data demonstrated nonlinearity. More specifically, the distance between the circled data point and the regression line is greater than the standard error of regression. This nonlinearity can be due to the nonfactorization of the electronic and nuclear parameters in the expression for the field isotope shift. This becomes possible when two or more nuclear parameters are not proportional to each other and appear in different combinations for different atomic transitions. In addition, it has been recently proposed that measurements of King plot nonlinearity can be used to search for hypothetical new light bosons [24]. This is a novel assessment, and worth investigating further in future experiments.

3. Summary and conclusions

In conclusion, we report the first isotope shift studies of spin forbidden transitions between the $^9{D_J} \to {}^{11}{F_J}$ manifolds at 726 nm and 743 nm wavelengths. Furthermore, we determined the corresponding magnetic dipole and electric quadrupole hyperfine coefficients for the upper level for these corresponding transitions. In addition, using King plot analysis the values of the isotope shift constants: field shift coefficients (${F_i}$), specific mass shifts (${M_{SM}}$), normal mass shifts (${M_{NM}}$) and slopes for these lines have been determined. Moreover, we noticed an anomaly in our study with respect to the 743 nm transition, which demonstrated King plot nonlinearity. The spectroscopic results presented on the spin forbidden transitions studied in this work are important for understanding nuclear charge distribution variations, implementing narrow band laser cooling for dipolar physics investigations, and will aid in the development of novel isotope separation and enrichment techniques for medical and nuclear technological applications.

Funding

Directorate for Mathematical and Physical Sciences (MPS) (PHY-1404496).

Acknowledgment

We thank J.C.E. Asher for support and guidance with this research.

References

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Upper Level	$A_{e}^{157}$ (GHz)	$B_{e}^{157}$ (GHz)
4f⁷5d¹6s6p¹¹F₃	−0.0766(48)	0.5895 (40)
4f⁷5d¹6s6p¹¹F₅	0.0839 (45)	0.9673 (41)

Line (nm)	Transition J_g → J_e	Lower Energy (cm⁻¹)	Upper Energy (cm⁻¹)	¹⁵⁸Gd	¹⁵⁷Gd(cg)	¹⁵⁶Gd	¹⁵⁵Gd(cg)
726.266	⁹D₄ → ¹¹F₃	532.977	14298.311	0.9640(35)	2.1538(202)	2.5629(36)	3.2427(119)
743.014	⁹D₆ → ¹¹F₅	1719.087	15174.000	1.0540(35)	0.4862(654)	2.0288(37)	2.4763(118)

Line (nm)	Transition	M_NM (GHz amu)	M_SM (THz amu)	F_i (GHz/fm²)	E_i / E_j
726.266	⁹D₄ → ¹¹F₃	226.4	40.6(8)	−27.1(16)	1.98(16)
743.014*	⁹D₆ → ¹¹F₅	221.3	−104(1.5)*	44.5(26)*	−3.26(27)*

Upper Level	$A_{e}^{157}$ (GHz)	$B_{e}^{157}$ (GHz)
4f⁷5d¹6s6p¹¹F₃	−0.0766(48)	0.5895 (40)
4f⁷5d¹6s6p¹¹F₅	0.0839 (45)	0.9673 (41)

Line (nm)	Transition J_g → J_e	Lower Energy (cm⁻¹)	Upper Energy (cm⁻¹)	¹⁵⁸Gd	¹⁵⁷Gd(cg)	¹⁵⁶Gd	¹⁵⁵Gd(cg)
726.266	⁹D₄ → ¹¹F₃	532.977	14298.311	0.9640(35)	2.1538(202)	2.5629(36)	3.2427(119)
743.014	⁹D₆ → ¹¹F₅	1719.087	15174.000	1.0540(35)	0.4862(654)	2.0288(37)	2.4763(118)

Hyperfine structure and isotope shift measurements on spin-forbidden transitions of atomic gadolinium

Abstract

1. Introduction

2. Experimental results

3. Summary and conclusions

Funding

Acknowledgment

References

Cited By

Figures (4)

Tables (3)

Equations (11)

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