Theoretical predictions for the magnetic dipole moment of $^{229m}$Th

TL;DR

This paper presents a new nuclear model prediction for the magnetic dipole moment of $^{229m}$Th, aligning closely with recent experimental measurements and offering insights into nuclear structure and transition probabilities.

Contribution

The authors develop a coupled collective quadrupole-octupole and single-particle nuclear model that accurately predicts the isomer's magnetic dipole moment, improving upon previous Nilsson model predictions.

Findings

Predicted isomer magnetic dipole moment of -0.35μ_N matches experimental -0.37(6)μ_N.

Model overestimates ground state dipole moment by 40%.

Provides insights into parity mixing, Coriolis effects, and gyromagnetic ratio g_R.

Abstract

A recent laser spectroscopy experiment [J. Thielking et al., Nature (London) 556, 321 (2018)] has determined for the first time the magnetic dipole moment of the 7.8 eV isomeric state $^{229m}$Th. The measured value differs by a factor of approximately 5 from previous nuclear theory predictions based on the Nilsson model, raising questions about our understanding of the underlying nuclear structure. Here, we present a new theoretical prediction based on a nuclear model with coupled collective quadrupole-octupole and single-particle motions. Our calculations yield an isomer magnetic dipole moment of $\mu_{ IS}= -0.35\mu_N$ in surprisingly good agreement with the experimentally determined value of $-0.37(6)\mu_N$, while overestimating the ground state dipole moment by a factor 1.4. The model provides further information on the states' parity mixing, the role and strength of the Coriolis mixing and the most probable value of the gyromagnetic ratio $g_R$ and its consequences for the transition probability $B(M1)$.

Tables

Table 1. Table 1: Calculated ground state and isomer MDM μ GS subscript 𝜇 GS \mu_{\mbox{\scriptsize GS}} and μ IS subscript 𝜇 IS \mu_{\mbox{\scriptsize IS}} , respectively, obtained for 229 Th for several q R subscript 𝑞 𝑅 q_{R} ( g R subscript 𝑔 𝑅 g_{R} quenching) values in comparison with previous nuclear theory predictions and experimentally deduced MDM values.

$\mu \left({\mu }_N\right)$	$q_R$ (this work)					nuclear theory			laser spectroscopy
	$1.0$	$0.8$	$0.7$	$0.6$		Ref. Chasman et al. (1977)	Ref. Dykhne and Tkalya (1998)		Ref. Gerstenkorn et al. (1974)	Ref. Safronova et al. (2013)	Ref. Müller et al. (2018)	Ref. Thielking et al. (2018)
${\mu }_{\text{GS}}$	0.654	0.591	0.559	0.528		0.54	–		0.46(4)	0.360(7)	–	–
${\mu }_{\text{IS}}$	$-0.253$	$-0.300$	$-0.323$	$-0.347$		–	$-0.076$		–	–	$\left(-0.3\right)$–$\left(-0.4\right)$	$-0.37\left(6\right)$

Table 2. Table 2: Predicted B ​ ( M ​ 1 ) 𝐵 𝑀 1 B(M1) values (in W.u.) for several transitions involving yrast (yr) and excited (ex) QPD states of 229 Th obtained for four q R subscript 𝑞 𝑅 q_{R} values in comparison with available experimental data.

Decay	$q_R$				Exp. nnd
	$1.0$	$0.8$	$0.7$	$0.6$
${\frac{3}{2}}_{\text{ex}}^{+}\to {\frac{5}{2}}_{\text{yr}}^{+}$	0.0081	0.0068	0.0062	0.0056	–
${\frac{7}{2}}_{\text{yr}}^{+}\to {\frac{5}{2}}_{\text{yr}}^{+}$	0.0096	0.0043	0.0025	0.0011	0.0110 (40)
${\frac{9}{2}}_{\text{yr}}^{+}\to {\frac{7}{2}}_{\text{yr}}^{+}$	0.0185	0.0097	0.0065	0.0038	0.0076 (12)
${\frac{9}{2}}_{\text{yr}}^{+}\to {\frac{7}{2}}_{\text{ex}}^{+}$	0.0144	0.0147	0.0149	0.0151	0.0117 (14)

Equations

$$H=H_{\text{mbox}s.p.}+H_{\text{mbox}pair}+H_{\text{mbox}qo}+H_{\text{mbox}Coriol}.$$

$$\tilde{\Psi }=\frac{1}{{\tilde{N}}_{I\pi K_b}}\left[{\Psi }_{nkIM{K}_b}^{\pi ,{\pi }^b}+A\sum _{\nu \ne b}{C}_{K_{\nu }{K}_b}^{I\pi }{\Psi }_{nkIM{K}_{\nu }}^{\pi ,{\pi }^b}\right],$$

$${\Psi }_{nkIMK}^{\pi ,{\pi }^b}(\eta ,\varphi ,\theta )=\frac{1}{N_K^{({\pi }^b)}}\sqrt{\frac{2I+1}{16{\pi }^2}}{\Phi }_{nkI}^{\pi \cdot {\pi }^b}(\eta ,\varphi )$$

$$\times$$

$$\displaystyle\mu=\mu_{N}g_{R}I+\frac{1}{I+1}\frac{1}{\widetilde{N}_{I\pi K_{b}}^{2}}\Bigg{[}K_{b}\frac{M^{\pi^{b}}_{K_{b}K_{b}}}{N^{(\pi^{b})}_{K_{b}}}$$

$$+2A{K}_b\sum _{\begin{array}{c}\nu \ne b\\ K_{\nu }=K_b=\frac{1}{2}\end{array}}{\delta }_{K_{\nu }{K}_b}{C}_{K_{\nu }{K}_b}^{I\pi }\frac{P_{K_{\nu }{K}_b}^b{M}_{K_{\nu }{K}_b}^{{\pi }^b}}{N_{K_{\nu }}^{({\pi }^b)}{N}_{K_b}^{({\pi }^b)}}$$

$$\displaystyle+A^{2}\mkern-18.0mu\!\!\sum_{\begin{subarray}{c}\nu_{1,2}\neq b\\ K_{\nu_{1},\nu_{2}}=K_{\nu}\\ =K_{b}\pm 1,\frac{1}{2}\end{subarray}}\!\!\mkern-18.0mu\delta_{K_{\nu_{1}}\!K_{\nu_{2}}}\!K_{\nu}C^{I\pi}_{K_{\nu_{1}}\!K_{b}}C^{I\pi}_{K_{\nu_{2}}\!K_{b}}\frac{P^{b}_{K_{\nu_{1}}\!K_{\nu_{2}}}M^{\pi^{b}}_{K_{\nu_{2}}\!K_{\nu_{1}}}}{N^{(\pi^{b})}_{K_{\nu_{1}}}N^{(\pi^{b})}_{K_{\nu_{2}}}}\Bigg{]}.$$

Full text

Introduction. The persistent interest of the metrology and atomic and nuclear physics communities in the spectroscopic properties of the actinide nucleus 229Th is related to the exceptionally low-energy $7.8$ eV isomeric state 229mTh Beck et al. (2007); Bec (2009). Vacuum ultraviolet (VUV) laser access to this nuclear state is believed to allow a number of applications such as a new “nuclear clock” frequency standard E. Peik and Chr. Tamm (2003); Peik and Okhapkin (2015); Campbell et al. (2012), the development of nuclear lasers in the optical range Tkalya (2011) or a more precise determination of the temporal variation of fundamental constants Flambaum (2006); Berengut et al. (2009); Rellergert et al. (2010). While so far direct laser excitation of the isomer remains illusive due to the poor knowledge of the exact transition frequency, recent experiments could demonstrate the existence of the isomer von der Wense et al. (2016), and measure the isomer mean half-life in neutral Th atoms Seiferle et al. (2017). In 2018, laser spectroscopy experiments determined for the first time the magnetic dipole moment (MDM) of the nuclear isomeric state as $-0.37(6)\mu_{N}$ Thielking et al. (2018); Müller et al. (2018), where $\mu_{N}$ stands for the nuclear magneton. This breakthrough is particularly important because it opens the possibility to experimentally probe the nuclear isomeric state via optical spectroscopy on the electronic hyperfine structure Beloy (2014). From the nuclear structure point of view, however, the measured MDM value poses a riddle: the only theoretical MDM prediction so far was providing the value $\mu_{\mbox{\scriptsize IS}}=-0.076\mu_{N}$ Dykhne and Tkalya (1998), which is about five times smaller than the measured value. This calls for new theoretical efforts to understand the physical mechanism behind the Th isomer.

From the theoretical point of view, predictions on 229mTh have been provided based on three approaches. Dykhne and Tkalya Dykhne and Tkalya (1998) used the Nilsson model Nilsson (1995) to estimate the reduced transition probability $B(M1)$ and the isomer MDM, the former based on Alaga rules Alaga et al. (1995). Ruchowska and co-workers Ruchowska et al. (2006) used the quasiparticle-plus-phonon model to predict the reduced transition probabilities $B(M1)$ and $B(E2)$ for the isomeric transition. Finally, the present authors have put forward a nuclear model approach that takes into account the collective quadrupole-octupole vibration-rotation motion (typical for the nuclei in the actinide region) the motion of the unpaired nucleon within a reflection-asymmetric deformed potential with pairing correlations and the Coriolis interaction between the single nucleon and the core Minkov and Pálffy (2017). This model predicted the $B(M1)$ value for magnetic decay of the isomer in the limits of $0.006-0.008$ Weisskopf units (W.u.), well below earlier deduced values of 0.048 W.u. Dykhne and Tkalya (1998); Tkalya et al. (2015) and 0.014 W.u. Ruchowska et al. (2006), thus potentially offering an explanation for the recently reported experimental difficulties to observe the radiative decay of the isomer Jeet et al. (2015); Yamaguchi et al. (2015); von der Wense L. (2018).

It is the purpose of this Letter to provide new nuclear structure predictions for the MDM of the 229Th ground and isomeric states by extending the model approach in Ref. Minkov and Pálffy (2017). The model parameters are independent of MDM experimental data but rather rely on experimental energy levels and transition rates providing a further test of the model’s predictive capability. Based on different models for the collective gyromagnetic ratio $g_{R}$, we obtain the isomeric-state MDM in the range $\mu_{\mbox{\scriptsize IS}}=-0.25\mu_{N}$ to $-0.35\mu_{N}$, well in agreement with the experimental values. The ground-state (g.s.) MDM value on the other hand is obtained in the range $\mu_{\mbox{\scriptsize GS}}=(0.53-0.66)\mu_{N}$. This overestimates the latest reported value of $0.36\mu_{N}$ deduced from state-of-the-art atomic structure calculations for Th3+ ions in Ref. Safronova et al. (2013) on the basis of electronic hyperfine splitting data Campbell et al. (2011) or the older experimental result $\mu_{\mbox{\scriptsize GS}}=0.45\mu_{N}$ Gerstenkorn et al. (1974). Our values for the g.s. MDM are similar to the prediction $\mu_{\mbox{\scriptsize GS}}=0.54\mu_{N}$ based on a modified Wood-Saxon potential Chasman et al. (1977). Our calculations further show that as a peculiarity, the single particle (s.p.) orbital for 229mTh features very strong parity mixing. Furthermore, we find that whereas the effect of the Coriolis mixing is of crucial importance for the existence and strength of the isomeric transition, it is relatively weak in the g.s. MDM and negligible in the isomeric MDM. Finally, a comparison of our theoretical prediction with the experimental results supports the consideration of a strong quenching of the collective gyromagnetic ratio $g_{R}$ in 229Th and in consequence a lower reduced transition probability $B(M1)$ than so far assumed.

Model approach. The model Hamiltonian is taken in the form Minkov and Pálffy (2017)

[TABLE]

Here, $H_{\mbox{\scriptsize s.p.}}$ is the s.p. Hamiltonian of the deformed shell model (DSM) with a Woods-Saxon (WS) potential for axial quadrupole, octupole and higher-multipolarity deformations Cwiok et al. (1987) providing the s.p. energies $E^{K}_{\mbox{\scriptsize sp}}$ with given value of the projection $K$ of the total and s.p. angular momentum operators $\hat{I}$ and $\hat{j}$, respectively, on the intrinsic symmetry axis. $H_{\mbox{\scriptsize pair}}$ is the Bardeen-Cooper-Schrieffer (BCS) pairing Hamiltonian Ring and Schuck (1980). This DSM-BCS part provides the quasi-particle (q.p.) spectrum $\epsilon^{K}_{\mbox{\scriptsize qp}}$ as implemented in Ref. Walker and Minkov (2010). $H_{\mbox{\scriptsize qo}}$ represents a coherent QO motion (CQOM) of the even–even core as considered in Refs. Minkov et al. (2006, 2007). $H_{\mbox{\scriptsize Coriol}}$ involves the Coriolis interaction between the core and the unpaired nucleon (see Eq. (3) in Minkov et al. (2007)). It is treated as a perturbation with respect to the remaining part of (1) and then incorporated into the QO potential of $H_{\mbox{\scriptsize qo}}$ Minkov (2013); Minkov and Pálffy (2017). The spectrum of (1) is then obtained through the solution of the CQOM problem Minkov et al. (2006, 2007, 2012) superposed to the q.p. spectrum obtained in the DSM-BCS problem. It has the form of quasi-parity-doublets (QPD) ensuing from the QO vibrations and rotations Minkov et al. (2013) built on q.p. bandhead (b.h.) states with given $K=K_{b}$ and parity $\pi^{b}$ Minkov (2013); Minkov and Pálffy (2017).

The Coriolis perturbed wave function $\widetilde{\Psi}\equiv\widetilde{\Psi}^{\pi,\pi^{b}}_{nkIMK_{b}}$ corresponding to the Hamiltonian (1) reads

[TABLE]

where $K_{\nu}=K_{b}\pm 1,\frac{1}{2}$; $C^{I\pi}_{K_{\nu}K_{b}}$ are $K$-mixing coefficients, $\widetilde{N}_{I\pi K_{b}}$ is a normalization factor and $A$ is the Coriolis mixing constant Minkov and Pálffy (2017). The unperturbed QO core-plus-particle wave function in Eq. (2) has the form Minkov (2013)

[TABLE]

where $D^{I}_{M\,K}(\theta)$ are the rotation functions of the three Euler angles, $\Phi^{\pi\cdot\pi^{b}}_{nkI}(\eta,\phi)$ are the QO vibration functions in radial ($\eta$) and angular ($\phi$) coordinates with corresponding quantum numbers $n$ and $k$ (see Minkov et al. (2012, 2013) for details) and $\mathcal{F}_{K_{b}}^{(\pi^{b})}$ is the parity-projected component of the s.p. wave function of the b.h. state determined by DSM Cwiok et al. (1987). The quantity N_{K}^{(\pi^{b})}=\left[\left\langle\mathcal{F}_{K}^{(\pi^{b})}\big{|}\mathcal{F}_{K}^{(\pi^{b})}\right\rangle\right]^{\frac{1}{2}} is the corresponding parity-projected normalization factor. Note that the projection is compulsory since for nonzero octupole deformation the s.p. wave function obtained in DSM is parity mixed, i.e., it contains components possessing both parities, $\mathcal{F}_{K}=\mathcal{F}_{K}^{(+)}+\mathcal{F}_{K}^{(-)}$. The corresponding expectation value of the s.p. parity operator is then $-1\leq\langle\hat{\pi}_{\mbox{\scriptsize sp}}\rangle=\langle\mathcal{F}_{K}|\hat{\pi}_{\mbox{\scriptsize sp}}|\mathcal{F}_{K}\rangle\leq 1$. The projection $\mathcal{F}_{K_{b}}^{(\pi^{b})}$ is taken with respect to the experimentally confirmed parity which is for the ground and isomeric states in 229Th $\pi^{b}=+1$. In our case, the average $\langle\hat{\pi}_{\mbox{\scriptsize sp}}\rangle$ is 0.4014 for the ground state and 0.0101 for the isomeric state, respectively, showing that the parity mixing is very strong in the DSM solution for 229mTh. We note that the parity-projection plus renormalization procedure plays a considerable role in the resulting MDM values.

The above CQOM-DSM-BCS formalism allows us to calculate the MDM in any state of the spectrum by using the complete Coriolis perturbed wave function (2). We consider the standard core plus particle magnetic dipole ($M1$) operator $\hat{M}1=\sqrt{3/4\pi}\mu_{N}\left[g_{R}(\hat{I}-\hat{j})+g_{s}\,\hat{s}+g_{l}\,\hat{l}\right]$, with $\hat{s}$ and $\hat{l}$ being the operators of the s.p. spin and orbital momenta ($\hat{l}+\hat{s}=\hat{j}$), respectively, and $g_{s}$ the spin gyromagnetic factor. The MDM is determined by the matrix element $\mu=\sqrt{\frac{4\pi}{3}}\langle\widetilde{\Psi}_{IIK}|\hat{M}1_{0}|\widetilde{\Psi}_{IIK}\rangle$ where $\hat{M}1_{0}$ is the zeroth spherical tensor component of $\hat{M}1$ taken after transformation to the intrinsic frame (see Chapter 9 of Ref. Eisenberg and Greiner (1970)). Thus we obtain the following MDM expression for a state with collective angular momentum $I$ and parity $\pi$ built on a q.p. b.h. state with $K=K_{b}$ and $\pi=\pi^{b}$:

[TABLE]

The complete expression would involve an additional so-called decoupling term applying for the case $K_{b}=1/2$ Eisenberg and Greiner (1970), that we have disregarded in Eq. (4), since it is not applicable for the states under consideration here. Similarly, the second term in the brackets of Eq. (4) is also not relevant for our problem. We use $M^{\pi^{b}}_{K_{\mu}K_{\nu}}=(g_{l}-g_{R})K_{\nu}\delta_{K_{\mu}K_{\nu}}\langle\mathcal{F}_{K_{\mu}}^{(\pi^{b})}|\mathcal{F}_{K_{\nu}}^{(\pi^{b})}\rangle+(g_{s}-g_{l})\langle\mathcal{F}_{K_{\mu}}^{(\pi^{b})}|\hat{s}_{0}|\mathcal{F}_{K_{\nu}}^{(\pi^{b})}\rangle$, where $g_{l}=0\,(1)$ is the orbital gyromagnetic factor for neutrons (protons), $g_{s}=0.6\ g_{s}^{\mbox{\scriptsize free}}$ is the attenuated spin gyromagnetic factor with $g_{s}^{\mbox{\scriptsize free}}=-3.826\,(5.586)$ for neutrons (protons) Ring and Schuck (1980) and $g_{R}$ is the collective gyromagnetic factor which will be discussed below. The factors $P^{b}_{K_{\nu_{1}}K_{\nu_{2}}}$ involve the BCS occupation probabilities as shown in Ref. Minkov and Pálffy (2017). The third term in the brackets of Eq. (4) is important for the Coriolis mixing. One can easily check that in the case of missing Coriolis mixing, Eq. (4) appears in the usual form of the particle-rotor expression Ring and Schuck (1980) and is consistent with the relevant limiting case.

Numerical results. We calculate the 229Th ground and isomeric state MDM in Eq. (4) using the wave function (2). Following the procedure described in Ref. Minkov and Pálffy (2017), we have chosen the quadrupole ($\beta_{2}$) and octupole ($\beta_{3}$) deformations entering DSM, the BCS pairing parameters, the collective CQOM parameters and the Coriolis mixing strength such that both states $5/2^{+}$ g.s. and the isomeric $3/2^{+}$ form a quasi-degenerate pair and the low-lying part of the 229Th spectrum ens is well reproduced. This set of parameters is the same as the one in Ref. Minkov and Pálffy (2017) except for the Coriolis mixing constant $A$ which was slightly shifted from 0.158 to 0.184 keV. The latter is due to fixing a minor numerical inaccuracy found in the Coriolis mixing coefficients. This leads to a negligible change in the other energies and slight change in the transition rates discussed below.

The parameter that needs special consideration is the collective gyromagnetic factor $g_{R}$. In Ref. Minkov and Pálffy (2017) the $B(M1)$ values were obtained by using the phenomenological expression $g_{R}=Z/(Z+N)$ with $Z$ and $N$ the proton and neutron numbers, respectively, adopted on the basis of the liquid-drop-model Way (1939). However, it is well known that similarly to $g_{s}$ (which is typically attenuated by the quenching factor 0.6 taking into account spin polarization effects Mottelson (1960)) in most deformed nuclei $g_{R}$ is also lowered by 20-30% or more Eisenberg and Greiner (1970). This effect has been proven both experimentally Bodenstedt (1962) and theoretically Nilsson and Prior (1961); Prior et al. (1968); Greiner (1965). We note that the experimental determination of $g_{R}$ is model dependent and in fact usually involves knowledge of MDM. In theory the lowering of $g_{R}$ is explained through a suppressed relative contribution of the proton system to the total moment of inertia due to the pairing interaction Nilsson and Prior (1961); Prior et al. (1968). Thus in odd-mass nuclei the quenching of $g_{R}$ is stronger when the odd particle is a neutron and weaker when it is a proton compared to the quenching in the adjacent even-even nuclei. In this basic approach $g_{R}$ is usually calculated through the Inglis-Belyaev cranking procedure Inglis (1954, 1955) using Nilsson and Prior (1961); Prior et al. (1968) the Nilsson deformed shell model Nilsson (1995) or applying a density-dependent Hartree-Fock plus BCS (HFBCS) model Sprung et al. (1979).

We introduce the quenching factor $q_{R}<1$, such that $g_{R}=q_{R}Z/(Z+N)$ with $Z/(Z+N)=0.393$ for 229Th. We have used the following $g_{R}$ reference values: $g_{R}=0.28$ and $0.27$ for 228Th and 230Th, respectively, obtained in the early cranked Nilsson approach Nilsson and Prior (1961); $0.24$ obtained for 230Th in the cranked HFBCS calculation Sprung et al. (1979); and $0.31\pm 0.03$ deduced in Ref. Ton et al. (1970) on the basis of experimental measurements for the ground state MDM and $M1/E2$ mixing ratios for two ground state intraband transitions in 229Th. Keeping in mind that for the odd-neutron nucleus $g_{R}$ is lower compared to the neighboring even-even cases we consider the quenching factors $q_{R}=0.7$ and 0.6 based on the cranked Nilsson and HFBCS calculations, respectively, and $q_{R}=0.8$ given by the experimental estimate Ton et al. (1970).

The MDM results are shown in Table 1 and compared with existing theoretical and experimental values. The previous nuclear theory approaches were based on the Nilsson model Dykhne and Tkalya (1998) and modified Woods-Saxon potential Chasman et al. (1977). The experimental data on the MDM stems from laser spectroscopy measurements of the electronic hyperfine splitting of Th ions. The early work in Ref. Gerstenkorn et al. (1974) extracted the hyperfine constants $A$ and $B$ from Th+ spectra and deduced based on rather simplified atomic structure matrix elements the ground state MDM of $\mu_{\mbox{\scriptsize GS}}=0.45(4)\,\mu_{N}$. This value was corrected in Ref. Safronova et al. (2013) to $\mu_{\mbox{\scriptsize GS}}=0.360(7)\,\mu_{N}$ based on a more recent measurement of the hyperfine structure of 229Th3+ ions Campbell et al. (2011) and state-of-the-art atomic structure calculations based on the coupled-cluster model. The first experimental observation of the isomer electronic hyperfine splitting in 229Th2+ was reported only recently Thielking et al. (2018). Based on this measurement, an isomer MDM value of $\mu_{\mbox{\scriptsize IS}}=-0.37(6)\,\mu_{N}$ Thielking et al. (2018) or in the range of between $-0.30$ and $-0.40\,\mu_{N}$ Müller et al. (2018) was extracted. Note that Ref. Thielking et al. (2018) makes use of the ground state MDM value $\mu_{\mbox{\scriptsize GS}}=0.360(7)\,\mu_{N}$ from Ref. Safronova et al. (2013) to extract the experimental value for $\mu_{\mbox{\scriptsize IS}}$.

Regarding our model predictions, an overall observation is the decrease of both MDMs $\mu_{\mbox{\scriptsize GS}}$ and $\mu_{\mbox{\scriptsize IS}}$ (the latter going towards more negative values) with the decrease of $g_{R}$. As seen from Table 1 we have obtained for the ground state MDM the model values in the range between $\mu_{\mbox{\scriptsize GS}}=0.655$ and $0.530\,\mu_{N}$ for $q_{R}$ taking the values 1.0, 0.8, 0.7 and 0.6. Comparing them to the experimental values in Refs. Gerstenkorn et al. (1974) and Safronova et al. (2013) we see that they overestimate the first one, $0.46(4)\,\mu_{N}$, by a factor between 1.42 and 1.15. The latter one is overestimated by a factor between 1.82 and 1.47. On the other hand our values for the isomer MDM which vary between $\mu_{\mbox{\scriptsize IS}}=-0.253$ and $-0.347\,\mu_{N}$ essentially corroborate the experimental values in Refs. Thielking et al. (2018) and Müller et al. (2018). We see that the three values obtained at $q_{R}=0.8$, 0.7 and 0.6 enter the error bar for the value of $\mu_{\mbox{\scriptsize IS}}=-0.37(6)\,\mu_{N}$ in Thielking et al. (2018). The value at $q_{R}=1$ underestimates (in absolute value) by a factor 0.84 the lower limit $-0.30\,\mu_{N}$ of the values in Ref. Müller et al. (2018). Comparison with previous theoretical results shows that our calculations corroborate the g.s. MDM values $\mu_{\mbox{\scriptsize GS}}=0.54\,\mu_{N}$ obtained by Chasman et al Chasman et al. (1977) and disagree with the isomer MDM value $\mu_{\mbox{\scriptsize IS}}=-0.076\,\mu_{N}$ obtained in Dykhne and Tkalya (1998). As a further check, the model predicts a spectroscopic electric quadrupole moment $Q_{GS}$=2.79 eb, close to the experimentally determined 3.11(16) eb Campbell et al. (2011) or 3.14(3) eb Bemis et al. (1988).

While the variation of $q_{R}$ ($g_{R}$) in Table I does not affect the energy levels and $B(E2)$ transition rates, it does change the predicted $B(M1)$ transition rates in Ref. Minkov and Pálffy (2017). This is illustrated in Table 2 where the model predictions for several $B(M1)$ values (including that of the isomer transition) obtained for the different $q_{R}$ ($g_{R}$-quenching) values are given together with the available experimental data. The transition rates in the first column (with $q_{R}=1.0$) are slightly different from the original calculation in Minkov and Pálffy (2017) due to the already mentioned correction in the numerical mixing coefficients. We see that the isomer-decay $B(M1)$ value (first row) gradually decreases with the decrease of $g_{R}$ in consistence with the MDM behavior. Faster decrease of $B(M1)$ with $g_{R}$ is observed for the yrast intraband transitions $7/2^{+}\rightarrow 5/2^{+}$ and $9/2^{+}\rightarrow 7/2^{+}$ whereas the B(M1) values for the yrast-excited interband transition $9/2^{+}_{\mbox{\scriptsize yr}}\rightarrow 7/2^{+}_{\mbox{\scriptsize ex}}$ practically remain unaffected. Comparison with the experimental data shows that the first intraband transition $7/2^{+}\rightarrow 5/2^{+}$ which is originally underestimated by the model goes even further down with $g_{R}$ by an order of magnitude. On the other hand the other intraband transition $9/2^{+}\rightarrow 7/2^{+}$ approaches rather well the corresponding experimental value for $q_{R}=0.7$.

The dependence in Tables 1 and 2 suggests that, since the quenching of $g_{R}$ is physically reasonable in the considered limits, we may state that the model predictions for $\mu_{\mbox{\scriptsize GS}}$ and $\mu_{\mbox{\scriptsize IS}}$ obtained with $g_{R}$ attenuated below the $Z/(Z+N)=0.393$ value better approach the corresponding experimental values with $\mu_{\mbox{\scriptsize IS}}$ firmly entering the uncertainty bars. In addition, this result suggests that a further slight decrease of the predicted isomer-decay $B(M1)$ value as compared to Ref. Minkov and Pálffy (2017) towards $B(M1)=0.005$ W.u. is probable.

A word is due on the role of the Coriolis mixing for the calculated MDM values. The attenuation of $g_{R}$ counteracts the mixing-effect on the connection between the two states. For example our numerical analysis shows that a slight raising of the Coriolis constant $A$ from 0.184 keV used in this work to the value of 0.230 keV leads to a slight decrease in $\mu_{\mbox{\scriptsize GS}}$ at $q_{R}=1.0$ from 0.654 to $0.643\,\mu_{N}$ with a negligible decrease in $\mu_{\mbox{\scriptsize IS}}$ in the fifth digit leaving the reported value of $-0.253\,\mu_{N}$ unaffected. At the same time the isomer $B(M1)$ transition rate raises from 0.0081 to 0.0121 W.u. The predictions for the $E2$ transitions are also affected with the isomer $B(E2)$ raising from 29 to 43 W.u. Conversely by setting the Coriolis mixing strength zero as a limiting case, $\mu_{\mbox{\scriptsize GS}}$ slightly increases to $0.677\,\mu_{N}$ whereas $\mu_{\mbox{\scriptsize IS}}$ remains again unaffected. This analysis reveals some peculiarities of the present model mechanism in 229Th. Whereas the Coriolis mixing strongly affects the isomer (interband) transitions being of crucial importance for their existence, it has only a small impact on the g.s. MDM and a completely negligible effect on the isomer MDM. The latter is explained by the circumstance, also checked numerically in wave function (2), that while the g.s. is mixed with the $I=5/2^{+}$ isomer-based state, the isomer state has no $I=3/2^{+}$ mixing-counterpart in the g.s. band due to the $K\leq I$ restriction. Thus, it appears that the connection between the ground and isomer state is mainly due to the admixture in the g.s. This effect enters the transition matrix elements and appears in the second power in the transition rates. We note that as the Coriolis mixing and the $g_{R}$ quenching counteract in the isomer transition rates and act in the same direction suppressing $\mu_{\mbox{\scriptsize GS}}$ it may be possible to bring the latter closer to the experimental value without drastic change in the predicted transition rates. However, this would cause an overall change in all model observables, energy levels, $B(E2)$ and $B(M1)$ values, and would rather call for a full readjustment of the model parameters. Further refinements such as more precise tuning of the QO deformation parameters entering DSM, possible involvement of hexadecapole deformation (suggested in Ref. Bemis et al. (1988)), or tuning of the spin-gyromagnetic quenching (similarly to $q_{R}$) as suggested in Ref. Ton et al. (1970) could provide an even more reliable prediction of 229mTh electromagnetic decay properties.

In conclusion, the CQOM-DSM-BCS model of the QPD spectrum and $B(M1)$ and $B(E2)$ transition rates in 229Th provides a good description of the MDM in the isomeric state. Our prediction $\mu_{\mbox{\scriptsize IS}}=-0.35\mu_{N}$ is in good agreement with the experimentally determined value and largely differs from the previous prediction in Ref. Dykhne and Tkalya (1998) based on the Nilsson model. In the same time, the ground state MDM is overestimated by a factor of approx. 1.4. We conclude that our results provide ground for further refined consideration of the interplay between the nuclear dynamic modes which determine the electromagnetic properties of the isomeric state as a part of the entire 229Th structure.

The authors would like to thank Christoph H. Keitel for fruitful discussions. This work is supported by the BNSF under contract DFNI-E02/6. AP gratefully acknowledges funding by the EU FET-Open project 664732.