Shape and electromagnetic properties of the 229mTh isomer

We examine the physical conditions, and specifically the role of the quadrupole-octupole deformation, for the emergence of the 8 eV “clock” isomer 229mTh. Our nuclear structure model suggests that such an extremely low-energy state can be the result of a very fine interplay between the shape and single-particle (s.p.) dynamics in the nucleus. We find that the isomer can only appear in a rather limited region of quadrupoleoctupole deformation space close to a line along which the ground-state and isomer s.p. orbitals 5/2[633] and 3/2[631], respectively, cross each other providing the isomer-formation quasi-degeneracy condition. The crucial role of the octupole deformation in the formation mechanism is pointed out. Our calculations within the outlined deformation region show a smooth behaviour of the 229Th electromagnetic properties, including the isomer decay rate, allowing for their more precise theoretical determination.


Introduction
The 229m Th nuclear isomer attracts much interest related to its exceptionally low energy of approximately 8 eV which appears to be accessible for the contemporary vacuum ultraviolet (VUV) lasers [1] and thus opens the possibility for a number of valuable applications such as the establishing of a new frequency standard, "nuclear clock" [2][3][4], development of nuclear lasers in the optical range [5] and others [6,7]. The achieving of these goals, however, requires very precise knowledge of the isomer energy, lifetime and the related nuclear electromagnetic characteristics. Following this motivation several recent experimental studies have been focused on the isomer decay modes and provided estimates about its life time in charged [8] and neutral [9] electronic states. Further, the magnetic dipole moment µ in the isomeric state (IS) was determined for the first time in laser spectroscopy experiments providing the value of µ IS = −0.37 (6)µ N [10,11]. Then, three very recent experiments provided newly updated values for the isomer energy, E IS = 8.28 (17) eV [12] from internal conversion electron spectroscopy, E IS = 8.30(92) eV [13] by determining the transition rates and energies from the above level at 29.2 keV and E IS = 8.10(17) eV [14] from a micro-calorimetric determination of absolute γ-ray energy differences.
Yet the direct nuclear decay transition of the 229m Th isomer is not observed experimentally. This imposes the need of detailed theoretical examination and prediction of its nuclear electromagnetic characteristics. The problem is additionally complicated due to the rich deformation properties of the actinide nuclei to which the nucleus 229 Th belongs. Trying to address the above aspects recently we ini- * e-mail: nminkov@inrne.bas.bg tiated a theoretical study [15], exploring a nuclear model approach which incorporates the shape-dynamic properties together with the intrinsic structure characteristics typical for the actinide region where the quadrupole-octupole (QO) deformation manifests. The formalism includes a description of the collective QO vibration-rotation motion coupled to the motion of the single (odd) nucleon within a reflection-asymmetric deformed potential with pairing correlations and fully microscopic treatment of the Coriolis interaction. The approach allows one to determine the energy and radiative decay properties of the 229m Th isomer as a part of the entire low-lying positiveand negative-parity spectrum and transition probabilities observed in the nucleus 229 Th. On this basis, we have shown that the extremely small isomer energy can be explained as the consequence of a very fine interplay between the rotation-vibration degrees of freedom and the motion of the unpaired neutron. The model calculations predict for the reduced probability B(M1) for magnetic decay of the isomer a value between 0.006 and 0.008 W.u. which is considerably smaller than earlier deduced values of 0.048 W.u. [16,17] and 0.014 W.u. [18]. This result provides the currently adopted estimate for the 229m Th nuclear life-time of about 10 4 sec explaining the experimental difficulties to observe the direct radiative decay of the isomer [19,20] and suggests a new finer accuracy target for further measurements. In a subsequent work [21] we have calculated the magnetic moment of the isomer, µ IS and of the ground state (GS), µ GS , by taking into account attenuation effects in the spin and collective gyromagnetic factors, without changing the model parameters originally adjusted in Ref. [15]. The result for µ IS in the range from µ IS = −0.25µ N to −0.35µ N is in rather good agreement with the recent experimental values −0.37 (6) [10] and (−0.3)-(−0.4) [11]. On the other hand, µ GS was obtained in the range µ GS = 0.53 − 0.66 µ N , overestimating the latest reported and older experimental values of 0.360(7) µ N [22] and µ GS = 0.45µ N [23], respectively, and being in agreement with an earlier theoretical prediction µ GS = 0.54µ N based on the modified Woods-Saxon potential [24].
The aforementioned theoretical studies pointed out the complexity of the 229m Th problem emphasizing on the need of consistent description of all related observables, detailed examination of the physical conditions for the forming of the isomer and further constraining of the arbitrariness in the model description to ensure maximal reliability of the obtained predictions as a tool in the establishing of the new nuclear clock standard. To this end in our very recent work [25] we investigated the shape deformation conditions under which such a low-energy isomer can appear in 229m Th and examined the behaviour of the model solution in the relevant space of quadrupole and octupole deformations clarifying the limits of reliability of our model predictions. In this work we colligate the results of our approach obtained so far trying to outline the frames of the general prescription for the search and investigation of nuclear excitations on the border between nuclear and atomic physics.
In Sec. 2 we briefly present the model formalism. In Sec. 3 we give numerical results and illustrate the main points in the consistent study of 229m Th shape characteristics and their influence on the isomer electromagnetic properties. In Sec. 4 concluding remarks are given.

The model approach
The model Hamiltonian is taken in the form [15] where H s.p. is the s.p. Hamiltonian of Deformed Shell Model (DSM) with a Woods-Saxon (WS) potential including axial quadrupole (β 2 ) and octupole (β 3 ) deformation parameters [26], which provides the s.p. energies E K sp with given value of the projection K of the total and s.p. angular momentum operatorsÎ andĵ, respectively on the intrinsic symmetry axis; H pair is the standard Bardeen-Cooper-Schrieffer (BCS) pairing Hamiltonian [27]. This DSM+BCS part provides the quasi-particle (q.p.) spectrum K qp as shown in Ref. [28]. H qo represents oscillations of the even-even core with respect to quadrupole and octupole axial deformation variables mixed through a centrifugal (rotation-vibration) interaction [29,30]; H Coriol involves the Coriolis interaction between the even-even core and the unpaired nucleon (see Eq. (3) in [30]). It is treated as a perturbation with respect to the remaining part of (1) and then incorporated into the QO potential of H qo defined for given angular momentum I, parity π and s.p. bandhead projection K b [15,31].
The spectrum of Hamiltonian (1) is obtained in the form of coherent (with the same frequency) QO vibrations and rotations built on a q.p. state with K = K b and parity where b is a model inertia parameter; n = 0, 1, 2, ... and k = 1, 2, 3, ... are radial and angular QO oscillation quantum numbers, respectively, with k odd (even) for the even (odd) parity states of the core [32]; ω is the frequency of the coherent QO mode (CQOM) [29,30] and X(I π , K b ) determines the centrifugal term in which the Coriolis mixing is taken into account [25]. The levels of the total QO core plus particle system, determined by a particular n and k (+) (k (−) ) for the states with given I π=+ (I π=− ) form a split (quasi) parity doublet [33]. The Coriolis perturbed wave function corresponding to Hamiltonian (1) with the spectrum (2) is obtained in the form in which the unperturbed core plus particle wave function Ψ π,π b nkI MK b involving the projection K b is mixed with the functions based on other s.p. states. In the case of nonzero octupole deformation β = 0 in the WS potential the s.p. states are characterized by average (mixed) parity and the corresponding s.p. wave functions are taken projected on the experimentally determined band-head parity π b (for details see Eqs. (20)-(23) in [25]).
Having the wave function (3) we obtain the reduced B(E1), B(E2), B(E3) and B(M1) transition probabilities for the energy spectrum (2) [15] as well as the magnetic moments in the ground and isomeric states, µ GS and µ IS , respectively [21] (for details see Ref. [25]). Because of the Coriolis mixing the states with different K b values appear connected through electromagnetic transitions which otherwise would be suppressed due to the axial symmetry of the system. This determines the model mechanism allowing for the M1 and E2 radiative decay of the K π = 3/2 + IS to the K π = 5/2 + GS.

Transition rates and magnetic moments in 229 Th
In Ref. [15] we applied the above DSM-CQOM approach to the low-lying part of the experimental 229 Th spectrum [34], as shown in Fig. 1 of [15]. The theoretical spectrum was obtained in the form of an yrast quasi paritydoublet (QPD) built on the 5/2[633] ground-state s.p. orbital and non-yrast QPD built on the 3/2[631] orbital corresponding to the 3/2 + isomeric state. The model parameters, β 2 and β 3 entering DSM, the BCS pairing parameters, the collective CQOM parameters and the Coriolis mixing strength were determined so that both states 5/2 + GS and the isomeric 3/2 + were obtained as a quasi-degenerate pair, showing that the latter can be easily set in the range  Table I of [15]. In Ref. [21] we calculated the magnetic moments µ GS and µ IS of 229 Th corresponding to the parameters of the model description in [15] without further adjustment by only taking the commonly used quenching q s = 0.6 of the spin gyromagnetic factor g s and by considering several values for the quenching q R = 1, 0.8, 0.7, 0.6 of the collective gyromagnetic factor g R based on earlier studies of 229 Th [36,37]. Here we illustrate this result in Fig. 1. We see that for µ IS (upper left plot) the theoretical predictions well approach the experimental value of −0.37 (6) µ N [10] firmly staying inside the uncertainty bars for q R = 0.7 and 0.6. For µ GS we see (upper right plot) that although the theoretical values obtained with decreasing q R approach the experimental values 0.360(7) µ N [22] and µ GS = 0.45µ N [23] they still remain outside of the uncertainty bars. At the same time the lower plot of Fig. 1 illustrates that the isomer B(M1) transition rate decreases with q R . All these results show that more detailed consideration of the magnetic moments together with other observables in 229 Th needs to be done in order to assess the limits of reliability of our approach in determining the electromagnetic properties of the 8 eV isomer. Thus, in Ref. [25] we implemented such a complex study of the deformation and electromagnetic properties of the 229m Th isomer. First we implemented DSM calculations over a net in the β 2 and β 3 deformation space outlining the region where the GS and IS s.p. orbitals appear with the correct K values and average parity. This region is illustrated in Fig. 2 where the q.p. energy corresponding to the 3/2 + isomer orbital with respect to the 5/2 + GS orbital is given as a function of β 2 and β 3 . We can point out two important observations: i) the QO region allowing the correct orbitals configuration essentially includes non-zero octupole deformation; ii) there is a well determined line of IS-GS orbital degeneracy along which one may expect the isomer formation. This result already reveals the crucial role of the deformation for the appearance and the overall properties of the 3/2 + isomer. Then, as a second step, we performed massive model DSM-CQOM fits at each point in the deformation grid considering energies, transition rates and magnetic moments all together. This allowed us to examine the behaviour of the obtained model description for all observables of interest within the QO deformation space. In Fig. 3 we illustrate the result obtained for the isomeric B(M1) and B(E2) transition rates and the magnetic moments µ GS and µ IS with the quenching factors q s = 0.55 and q R = 0.45. We see that all these quantities exhibit a smooth behaviour in dependence on β 2 and β 3 with the physically interesting theoretical val-ues appearing towards the IS-GS degeneracy line. We also remark that towards this line the model fairly well approaches the experimental isomer energy of about 8 eV. Considering the magnetic moments, now we find a region where µ GS (lower left plot) already reaches one of the experimental values µ GS = 0.45µ N [23], while for µ IS (lower right plot) the theoretical predictions always appear close to the experimental value of −0.37 (6) µ N [10]. Also, we see from the upper plots that the behaviour of the isomer transition rates within the QO space suggests a bit smaller values for the B(M1) and somehow larger values for the B(E2) transition compared to those predicted in Ref. [15].

Conclusion
Our study and all results presented in this work lead us to the important conclusion that the electromagnetic properties and subsequently the decay and life-time characteristics of the 229m Th isomer are essentially determined by the collective and intrinsic (s.p.) manifestation of nuclear shape deformation. Therefore, its detailed analysis in the aspect of any model description aiming to reach precise determination of the isomer energy and decay properties appears to be of a crucial importance. In a more general aspect the study suggests that the same dynamical mechanism may govern also in other nuclei excitations close to the border of atomic physics energy scale, thus paving the way for the systematic search for similar low-energy isomers in adjacent and remote nuclear regions.