2229 m Th isomer from a nuclear model perspective
Nikolay Minkov
1, 2, ∗ and Adriana P´alffy
2, 3, † Institute of Nuclear Research and Nuclear Energy,Bulgarian Academy of Sciences, Tzarigrad Road 72, BG-1784 Sofia, Bulgaria Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany Department of Physics, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, D-91058 Erlangen, Germany (Dated: February 5, 2021)The physical conditions for the emergence of the extremely low-lying nuclear isomer m Th atapproximately 8 eV are investigated in the framework of our recently proposed nuclear structuremodel. Our theoretical approach explains the m Th-isomer phenomenon as the result of a very fineinterplay between collective quadrupole-octupole and single-particle dynamics in the nucleus. Wefind that the isomeric state can only appear in a rather limited model space of quadrupole-octupoledeformations in the single-particle potential, with the octupole deformation being of a crucial im-portance for its formation. Within this deformation space the model-described quantities exhibit arather smooth behaviour close to the line of isomer-ground state quasi-degeneracy determined by thecrossing of the corresponding single-particle orbitals. Our comprehensive analysis confirms the pre-vious model predictions for reduced transition probabilities and the isomer magnetic moment, whileshowing a possibility for limited variation in the ground-state magnetic moment theoretical value.These findings prove the reliability of the model and suggest that the same dynamical mechanismcould manifest in other actinide nuclei giving a general prescription for the search and explorationof similar isomer phenomena.
I. INTRODUCTION
Well supporting the current strong emphasis on in-terdisciplinary research, a unique extremely low-lying m Th isomer at approximately 8 eV [1–4] obviouslydisregards the recognized low-energy border of nuclearphysics firmly stepping on atomic physics territory. Al-though another low-lying nuclear excitation in
U alsoapproaches this limit with an order of magnitude largerenergy of 76 eV [5], currently m Th attracts much moreinterest since its energy lies in the range of accessibilityof present vacuum ultraviolet (VUV) lasers capable tohandle the wavelength of 150 nm ( ≈ m Th-based applicationssuch as the precise determination of temporal variationsin fundamental constants [9–12], the development of nu-clear lasers in the VUV range [13], detection improve- ∗ Electronic address: [email protected] † Electronic address: Adriana.Palff[email protected] ments in satellite and deep space navigation, gravitationwaves, geodesy, precise analysis of chemical environmentand others.Towards the aforementioned applications, recent ex-periments have confirmed the existence of the isomer [14]and have determined the isomer mean half-life in neu-tral Th atoms [15]. Furthermore, the magnetic dipolemoment µ of the nuclear isomeric state (IS) was deter-mined for the first time through laser spectroscopy exper-iments [16, 17] providing the value of µ IS = − . µ N .Then, three very recent experiments proposed newly up-dated values for the isomer energy, E IS = 8 . E IS = 8 . E IS = 8 . γ -ray energy differences).These advances, although not yet reaching the accu-racy needed for a nuclear clock, pose new challenges andinspire new studies of the m Th problem from the nu-clear structure side.
Th belongs to the light actinidenuclear mass region known for the presence of enhancedcollectivity and shape dynamic properties suggesting acomplicated interaction between the collective motion ofthe even-even core and the individual motion of the sin-gle neutron. The single-particle (s.p.) states of the latterdetermine the
Th ground state (GS) with K π = 5 / + and the IS with K π = 3 / + based on the 5/2[633] and3/2[631] s.p. orbitals. Here, π denotes the parity and K refers to the projection of the total nuclear angularmomentum on the body-fixed principal symmetry axis ofthe system, respectively. We use the usual Nilsson nota-tion K [ N n z Λ] with N , n z and Λ being the asymptoticNilsson quantum numbers [19]. Although it is intuitivelyclear that the entire nuclear structure dynamics should a r X i v : . [ nu c l - t h ] F e b essentially influence the appearance and the properties ofthe isomer, only limited work has addressed this aspectin the past. Thus, predictions for the B ( M
1) and B ( E / + → / + transition probabilities have beenmade in Refs. [20, 21] using the quasiparticle-plus-phononmodel [22] without particular focus on the isomer proper-ties. Furthermore, in Refs. [23, 24] estimates were madefor the isomer B ( M
1) transition rate using the Alagabranching ratios [25], and for the IS magnetic moment µ IS based on the Nilsson model [26]. The obtained value µ IS = -0.076 µ N essentially differs from the recently avail-able experimental value of − . µ N [16].Understanding the physical mechanism behind the m Th phenomenon requires a thorough investigationof the interplay of all involved collective and s.p. de-grees of freedom, and identification of all structure ef-fects which could allow the appearance of an excitationin the eV energy scale. Since the latter is beyond reachfor the accuracy of nuclear models generally speaking,the implementation of such a task would require theapplication of a sophisticated theoretical method whichcan provide the necessary conclusion by juxtaposing re-sults and information gained from different perspectivesand observables such as energies, transition probabilitiesand magnetic moments. Motivated by the considerationsabove we have recently put forward a complete nuclear-structure model approach that takes into account the ax-ial quadrupole-octupole (QO) (pear-shape) deformationmodes typical for the nuclei in the actinide region both inthe collective and s.p. degrees of freedom of the nucleus[27]. The formalism involves in the even-even nuclearcore the so-called coherent QO model, describing collec-tive axial quadrupole and octupole vibrations with equaloscillation frequencies non-adiabatically coupled to therotation motion [28–31], while the odd-nucleon motion isdescribed within a deformed shell model (DSM) includingreflection-asymmetric Woods-Saxon potential [32] andpairing correlations of Bardeen-Cooper-Schrieffer (BCS)type with blocking of the unpaired nucleon orbital [33].The odd-nucleon motion is coupled to the collective mo-tion by a Coriolis interaction taken into account throughperturbation theory. The model spectrum has the formof quasi-parity-doublet bands built on the ground andexcited quasi-particle (q.p.) states. In this scheme theIS appears as a q.p. band head of an excited quasi-parity-doublet. The model framework allows a rathercomplete and intrinsically consistent spectroscopic treat-ment of the nucleus including its IS.Based on this model description we were able inRef. [27] to predict the B ( M
1) and B ( E
2) reduced prob-abilities for the IS 3 / + → / + transition. For B ( M . − .
008 Weisskopfunits (W.u.), well below the earlier deduced values of0.048 W.u. [23, 24] and 0.014 W.u. [21], corroborat-ing the experimental difficulties to observe radiative iso-mer decay [34–36]. For the electric quadrupole B ( E B ( E µ IS and of the GS, µ GS , by taking into accountattenuation effects in the spin and collective gyromag-netic factors, without changing the model parametersoriginally adjusted in Ref. [27]. The result for µ IS in therange from µ IS = − . µ N to − . µ N is in rather goodagreement with the recent experimental values ( − . − .
4) [17] and − . µ GS was obtained in the range µ GS = 0 . − . µ N , overesti-mating the latest reported and older experimental valuesof 0 . µ N [38] and µ GS = 0 . µ N [39], respectively,and being in agreement with an earlier theoretical predic-tion µ GS = 0 . µ N based on the modified Woods-Saxonpotential [40]. Our model analysis in Ref. [37] showedthat the Coriolis K -mixing interaction lowers µ GS push-ing it towards the experimental values, while its effecton µ IS is negligible due to the circumstance that the K π = 3 / + IS has no mixing partner with angular mo-mentum I π = 3 / + in the GS band.These results raise several important questions to ourunderstanding of the m Th problem from the nuclearstructure perspective, which we address in this work. (i)To which extent does the shape dynamics play a role forthe emergence of such a nuclear structure phenomenonas the tiny energy difference between the IS and the GS?(ii) What is the degree of arbitrariness in the choice ofparameters providing the model predictions? The basicinput in DSM are the quadrupole ( β ) and octupole ( β )deformations, which determine the s.p. orbitals on whichthe GS and IS are formed. It is, therefore, important toidentify the regions in the ( β , β ) deformation space ofDSM which provide a relevant model treatment of theisomer and the overall spectroscopic properties of the nu-cleus. To clarify this question, in this work we performDSM calculations on a grid in a wide range in the QOdeformation space covering the regions of physical rele-vance for a nucleus in the actinide mass region around Th. A next question that we address is (iii) whetherby including the experimental GS and IS magnetic mo-ment values in the model fits made for different pairsof DSM QO deformations, a better reproduction of µ GS could be achieved? How would the model predictionsfor the other spectroscopic quantities and in particularfor B ( M
1) and B ( E
2) change? Finally, (iv) is m Th aunique phenomenon appearing by chance, or the consid-ered dynamical mechanism could provide the presence ofsimilar not yet observed phenomena in other nuclei? Inthis work we aim to provide answers to these questions,prove the degree of reliability of the model predictions,and clarify details of the mechanism which governs theappearance of the IS.This work is structured as follows. Sec. II reviews themodel formalism in a self-contained form together withdetails on its application to the m Th problem. In Sec.III results from the calculations in the QO deformationspace of DSM with the corresponding behaviour of theIS energy, B ( M B ( E
2) transition rates and the ISand GS magnetic moments are presented and discussed.In Sec. IV we summarize our analysis and conclude onthe reliability of the suggested model mechanism. Wethereby provide our updated theoretical predictions forall discussed observables and answer the questions for-mulated above.
II. QUADRUPOLE-OCTUPOLE CORE PLUSPARTICLE MODELA. Hamiltonian
The model Hamiltonian of axial QO vibrations androtations coupled to the s.p. motion with Coriolis inter-action and pairing correlations can be written in the form[27] H = H s.p. + H pair + H qo + H Coriol . (1)Here H s.p. is the single-particle (s.p.) DSM Hamil-tonian with the Woods-Saxon potential for fixed axialquadrupole, octupole and higher multipolarity deforma-tions ( β , β , β , β , β ) [32] providing the s.p. energies E K sp with given value of the projection K of the totaland s.p. angular momentum operators ˆ I and ˆ j , re-spectively on the intrinsic symmetry axis. H pair is thestandard BCS pairing Hamiltonian [33] which togetherwith H s.p. determines the quasi-particle (q.p.) spectrum (cid:15) K qp = (cid:113) ( E K sp − λ ) + ∆ , with the chemical potential λ and the pairing gap ∆ determined as shown in Ref. [41].Furthermore, H qo describes the oscillations of the even–even core with respect to the quadrupole ( ˜ β ) and oc-tupole ( ˜ β ) axial deformation variables mixed through acentrifugal (rotation-vibration) interaction [28, 29]. Itsspectrum is obtained in an analytical form by assumingequal frequencies for the quadrupole and octupole oscil-lations. The latter are known as the coherent QO mode(CQOM) and will be discussed in more detail in Sec. IIB.Hereafter, we distinguish the CQOM collective (dynami-cal) variables ˜ β and ˜ β from the fixed DSM deformationparameters β and β considered in this work (see Sec.IIB for clarification).Returning to the total Hamiltonian in Eq. (1), H Coriol involves the Coriolis interaction between the even-evencore and the unpaired nucleon [29]. It is treated as aperturbation with respect to the remaining part of Hamil-tonian (1) and then incorporated into the collective QOpotential of H qo defined for a given angular momentum I , parity π and s.p. bandhead projection K b leading toa joint effective term [42] H IK b qo = − (cid:126) B ∂ ∂ ˜ β − (cid:126) B ∂ ∂ ˜ β + 12 C ˜ β + 12 C ˜ β + (cid:101) X ( I π , K b ) d ˜ β + d ˜ β . (2) Here, B ( B ), C ( C ) and d ( d ) are quadrupole (oc-tupole) mass, stiffness and inertia parameters, respec-tively. The function (cid:101) X ( I π , K b ) determines the centrifu-gal term in which the Coriolis mixing is taken into ac-count and has the form: (cid:101) X ( I π , K b ) = 12 (cid:34) d + I ( I + 1) − K b +( − I + (cid:18) I + 12 (cid:19) a ( π,π b ) δ K b , − A (cid:88) ν (cid:54) = b ( K ν = ,K b ± (cid:104)(cid:101) a ( π,π b ) K ν K b ( I ) (cid:105) (cid:15) K ν qp − (cid:15) K b qp (cid:21) , (3)where d determines the collective QO potential origin, A is the Coriolis mixing strength defined in Ref. [42] and thesum is performed over q.p. states with energies (cid:15) K ν qp abovethe Fermi level. For the sum we consider in our numericalcalculations ten mixing orbitals. The quantity a ( π,π b )1 / = ππ b a ( π b ) − represents the decoupling factor for the case K b = 1 /
2, while the quantities (cid:101) a ( π,π b ) K ν K b ( I ) represent theCoriolis mixing factors given by (cid:101) a ( π,π b ) K ν K b = (cid:112) ( I − K b )( I + K b + 1) a ( π b ) K ν K b , K ν = K b + 1 (cid:112) ( I + K b )( I − K b + 1) a ( π b ) K b K ν , K ν = K b − ππ b ( − ( I + ) ( I + ) a ( π b ) − , K ν = K b = , (4)with a ( π b ) K ν K b = P bK ν K b N ( π b ) K ν N ( π b ) K b (cid:104)F ( π b ) K ν | ˆ j + |F ( π b ) K b (cid:105) = P bK b K ν N ( π b ) K b N ( π b ) K ν (cid:104)F ( π b ) K b | ˆ j − |F ( π b ) K ν (cid:105) . (5)The latter involve matrix elements of the s.p. operatorsˆ j ± = ˆ j x ± i ˆ j y between the parity-projected componentsof the s.p. wave functions F ( π b ) K b of the bandhead stateand the admixing state F ( π b ) K ν . Each s.p. wave functionis obtained in DSM [32] as an expansion in the axially-deformed harmonic-oscillator basis | N n z ΛΣ (cid:105) (with Λ +Σ = K ) F K = (cid:88) Nn z Λ C KNn z Λ | N n z ΛΣ (cid:105) . (6)In the case of reflection asymmetry ( β (cid:54) = 0) the wavefunction has a mixed parity and can be decomposed as F K = (cid:80) π sp= ± F ( π sp) K = F (+) K + F ( − ) K , with the s.p. par-ity given by π sp = ( − N = ±
1. The action of the s.p.parity operator ˆ π sp gives ˆ π sp F K = F (+) K − F ( − ) K , and forthe parity-projected parts one has ˆ π sp F ( ± ) K = ±F ( ± ) K . Inour approach the projection is made with respect to theexperimentally assigned good parity π b of the bandheads.p. state (see below). It is clear that in the presence ofoctupole deformation each s.p. orbital is characterized byan average (expectation) value of the parity determinedas [43] (cid:104) ˆ π sp (cid:105) = (cid:88) Nn z Λ ( − N | C KNn z Λ | , (7)with the expansion coefficients calculated in the DSM.The quantity (cid:104) ˆ π sp (cid:105) takes values in the interval − ≤(cid:104) ˆ π sp (cid:105) ≤ +1 in dependence on the octupole β andquadrupole β deformations entering the DSM.The quantity N ( π b ) K = (cid:104)(cid:68) F ( π b ) K (cid:12)(cid:12) F ( π b ) K (cid:69)(cid:105) in Eq. (5) is aparity-projected normalization factor, whereas P bK ν (cid:48) K ν = U bK ν (cid:48) U bK ν + V bK ν (cid:48) V bK ν involves the BCS occupation factors.The index b corresponds to the blocked s.p. orbital onwhich the collective spectrum is built. Since the BCS pro-cedure is performed separately for each (blocked) band-head orbital, the overlap integrals and the matrix ele-ments between states built on different bandhead orbitalsinvolve the average of both separate occupation factors P bb (cid:48) K ν (cid:48) K ν = (cid:16) P bK ν (cid:48) K ν + P b (cid:48) K ν K ν (cid:48) (cid:17) . The occupation fac-tors U and V and the q.p. energies (cid:15) K qp are obtained bysolving the BCS gap equation as done in Ref. [41] withthe pairing constant G = G N/P for neutron/proton sub-systems of N protons and Z neutrons determined as [19] G N/P = 1 N + Z (cid:18) g ∓ g N − ZN + Z (cid:19) . (8)Here the pairing parameter g is considered to vary be-tween the values g = 17 . g = 19 . g is considered to vary around the value g = 7 . B. Model solution, spectrum and wave functions
The spectrum which corresponds to the Hamiltonian(1) represents QO vibrations and rotations built on a q.p.state with K = K b and parity π b . It is obtained in twosteps. First, the s.p. and q.p. energy levels and wavefunctions are obtained through a DSM plus BCS calcu-lation performed for fixed β - and β - parameter valuesof the s.p. Woods-Saxon potential, providing the odd-nucleon energy contribution to the bandhead, (cid:15) K b qp , andthe Coriolis mixing factors (cid:101) a , Eq. (4), in the centrifu-gal part (cid:101) X , Eq. (3), of the QO Hamiltonian (2). In thesecond step the collective QO vibration-rotation energiesand wave functions are obtained through the solution ofthe Schr¨odinger equation for the two-dimensional poten-tial in the collective ˜ β - and ˜ β - variables of Hamiltonian (2). In the following we will address in more detail thissecond step.In the general case of arbitrary values of the Hamilto-nian parameters B , B , C , C and d , d , the solutionof the Schr¨odinger equation in ˜ β and ˜ β has to be ob-tained numerically. A transformation of variables intro-duces the ellipsoidal “radial” and “angular”coordinates,respectively, η = (cid:34) d ˜ β + d ˜ β ) d + d (cid:35) , φ = arctan (cid:32) ˜ β ˜ β (cid:114) d d (cid:33) , (9)such that ˜ β = pη cos φ, ˜ β = qη sin φ, (10)with p = (cid:112) d/d , q = (cid:112) d/d , d = 12 ( d + d ) . (11)An analytical solution for the spectrum of H qo can befound for a specific set of parameters, when assuming co-herent QO oscillations (the so-called coherent QO mode,CQOM) with a frequency ω = (cid:112) C /B = (cid:112) C /B ≡ (cid:112) C/B . Then, the two-dimensional potential in Hamilto-nian (2) obtains a shape with an ellipsoidal equipotentialbottom in the space of the collective deformation vari-ables ˜ β and ˜ β [28]. This allows a separation of theellipsoidal variables and reduction of the problem to theone-dimensional Schr¨odinger equation for an analyticallysolvable potential of Davidson type in the radial variable η . The motion with respect to this potential correspondsto a “soft” QO vibration mode without fixed minimain ˜ β and ˜ β . These should not be confused with thefixed β - and β - deformations in the s.p. Woods-Saxonpotential of the DSM. The CQOM approach has beensuccessfully applied to QO spectra of even-even and oddmass nuclei [28–31].The quadrupole and octupole semiaxes ˜ β sa2 and ˜ β sa3 ofthe ellipsoidal CQOM potential bottom are defined foreven-even nuclei as [30]˜ β sa λ ( I ) = [2 X ( I ) /d λ C λ ] / , λ = 2 , , (12)with the centrifugal factor X ( I ) = [ d + I ( I + 1)] /
2. Foran odd- A nucleus the expression for the semixes takesthe form˜ β sa λ ( I π , K b ) = [2 (cid:101) X ( I π , K b ) /d λ C λ ] / , λ = 2 , , (13)with (cid:101) X ( I π , K b ) determined in Eq. (3). Comparing theexpressions of X ( I ) and (cid:101) X ( I π , K b ) it becomes clearthat for the odd-nucleus the semiaxes ˜ β sa λ =2 , ( I π , K b ) inEq. (13) differ from the semiaxes ˜ β sa λ ( I ) (12) of the orig-inal even-even core CQOM potential because here the (cid:101) X ( I π , K b ) factor includes the additional term ( − K b ) aswell as the Coriolis mixing and decoupling contributionsfrom the single nucleon. The CQOM potential semiaxesobey the relation [30]˜ β sa3 ˜ β sa2 = 1 (cid:112) p − , (14)with p defined in Eq. (11) determining the relative contri-bution of the quadrupole and octupole collective modesin the coherent QO motion. We note that the value p = 1corresponds to equal values of both semiaxes, i.e., to acircle form of the CQOM potential bottom. In terms ofthe coherence assumption concept this means that boththe quadrupole ˜ β and octupole ˜ β deformation modesenter the collective CQOM motion with the same weight.This case will be discussed later in the paper and is ex-emplified in Figs. 4 and 5.Despite the missing single ( ˜ β , ˜ β )- minimum in theCQOM potential, the collective QO states of the systemare still characterized by the so-called dynamical defor-mations determined by the density maxima of the QOvibration wave function. Explicitly, the CQOM QO vi-bration wave function is given by [28, 30]Φ π qo nkI ( η, φ ) = ψ Ink ( η ) ϕ π qo k ( φ ) , (15)where the radial part ψ Ink ( η ) = (cid:115) c Γ( n + 1)Γ( n + 2 s + 1) e − cη / ( cη ) s L sn ( cη ) (16)involves generalized Laguerre polynomials in the variable η , with s = (1 / (cid:113) k + b (cid:101) X ( I, K ) and c = √ BC/ (cid:126) , thelatter having the meaning of a reduced QO oscillator fre-quency, and Γ( z ) denotes the Gamma function. The an-gular part in the variable φ appears with a positive ornegative parity π qo of the collective QO mode as follows ϕ + k ( φ ) = (cid:112) /π cos( kφ ) , k = 1 , , , ... ; (17) ϕ − k ( φ ) = (cid:112) /π sin( kφ ) , k = 2 , , , ... . (18)The maxima of the density | Φ π qo nkI | → | Φ π qo nkI ( ˜ β , ˜ β ) | calculated in the ( ˜ β , ˜ β ) space pin down the dynami-cal deformation values [30]. Strictly speaking, the dy-namical deformation is defined by the expectation valueof the square of the corresponding multipole (deforma-tion) operator in the CQOM state, but considering thedensity maximum is enough to locate its position in the( ˜ β , ˜ β ) space. The positions of these maxima are sit-uated outside of the potential bottom ellipse and movefurther out with increasing angular momentum. They es-sentially characterize the collective dynamical behaviourof the nucleus in the presence of a coherent mode. Thiswill be illustrated in Sec. III B for the present model ap-plication in Th. It will be seen that the CQOM dy-namical deformations appearing in the overall collectivespectrum of the nucleus are reasonably correlated withthe intrinsic Woods-Saxon DSM QO deformations. We should stress here, however, that the dynamicalQO deformations in CQOM do not need to ultimatelycoincide or even to be close to the fixed Woods-Saxondeformations β and β of the DSM. Imposing artificiallysuch a constraint would deprive the overall algorithm ofthe capability to incorporate the individual (separate)dynamic properties of the collective and s.p. degrees offreedom (carried by the available data) and, therefore, ofthe possibility to plausibly reproduce the interaction be-tween them. The present model formalism does not puta constrain on both potentials but rather leaves themto independently feel, as much as possible, the corre-sponding physical conditions which govern the nuclearcollective and intrinsic motions and their very fine in-terplay. As it will be seen in the following Sec. II E, inthe case of m Th, the DSM deformations β and β determine the hyperfine (from the nuclear point of view)conditions for the appearance of the K π = 3 / + isomerwhile the dynamical CQOM deformations in the ( ˜ β , ˜ β )-space reflect the conditions imposed by the overall collec-tive spectrum which complement the microscopic isomer-formation mechanism.By taking the analytical CQOM solution together withthe result of the DSM plus BCS calculation the QO coreplus particle spectrum built on the given q.p. bandheadstate is obtained in the form [27] E tot nk ( I π , K b ) = (cid:15) K b qp + (cid:126) ω (cid:20) n + 1 + (cid:113) k + b (cid:101) X ( I π , K b ) (cid:21) . (19)Here b = 2 B/ ( (cid:126) d ) has the meaning of a reduced inertiaparameter, while n = 0 , , , ... and k = 1 , , , ... standfor the radial and angular QO oscillation quantum num-bers, respectively, with k odd (even) for the even-parity(odd-parity) states of the core [28, 29]. The levels of thetotal QO core plus particle system, determined by thegiven n and pair of k (+) and k ( − ) values for the stateswith I π =+ and I π = − , respectively, form a split doubletwith respect to the parity, called a quasi-parity-doublet[29, 31].The corresponding wave functions can be constructedin three steps. First, the quadrupole-octupole vibrationwave function of the CQOM is calculated according toEq. (15). Second, we can construct the unperturbed QOcore plus particle wave function [31, 42]:Ψ π,π b nkIMK ( η, φ, θ ) = 1 N ( π b ) K (cid:114) I + 116 π Φ ππ b nkI ( η, φ ) × (cid:104) D IM K ( θ ) F ( π b ) K + ππ b ( − I + K D IM − K ( θ ) F ( π b ) − K (cid:105) , (20)where D IM K ( θ ) are the rotation (Wigner) functions andΦ ππ b nkI ( η, φ ) are the QO vibration functions (15) with π qo = ππ b . In Eq. (20) the relevant part F ( π b ) K = F (+) K or F ( − ) K of the s.p. wave function F K given in Eq. (6) istaken by projecting the latter with respect to the exper-imentally assigned bandhead parity π b = + or − , thusproviding a good parity of the total core-plus-particlewave function.Finally, the Coriolis perturbed wave function (cid:101) Ψ ≡ (cid:101) Ψ π,π b nkIMK b corresponding to Hamiltonian (1) with thespectrum (19) is obtained in the first order of pertur-bation theory and has the form (cid:101) Ψ = 1 (cid:101) N IπK b Ψ π,π b nkIMK b + A (cid:88) ν (cid:54) = b C IπK ν K b Ψ π,π b nkIMK ν , (21)where K ν = K b ± , , the expansion coefficients read C IπK ν K b = (cid:101) a ( π,π b ) K ν K b ( I ) (cid:15) K ν qp − (cid:15) K b qp , (22)while the normalization factor is given by (cid:101) N IπK b = (cid:68) (cid:101) Ψ π,π b nkIMK b (cid:12)(cid:12) (cid:101) Ψ π,π b nkIMK b (cid:69) = 1 + 2 A (cid:88) ν (cid:54) = bK ν = K b = C IπK ν K b δ K ν K b P bK ν K b N ( π b ) K ν N ( π b ) K b (cid:68) F K ν ( π b ) (cid:12)(cid:12) F K b ( π b ) (cid:69) + A (cid:88) ν , (cid:54) = bK ν ,ν = K b ± , C IπK ν K b C IπK ν K b δ K ν K ν × P bK ν K ν N ( π b ) K ν N ( π b ) K ν (cid:68) F K ν ( π b ) (cid:12)(cid:12) F K ν ( π b ) (cid:69) . (23) C. Electric and magnetic transition rates
Expressions for the reduced B ( E B ( E B ( E Q µ ( Eλ ) = (cid:115) λ + 14 π (4 − δ λ, ) ˆ Q λ (cid:88) ν D λµν ,λ = 1 , , , µ = 0 , ± , ..., ± λ, (24)with the explicit form of the operators ˆ Q λ given by Eqs.(31)–(33) in [30].The expression for the B ( M
1) reduced transition prob-ability was obtained by using the standard core plus par-ticle magnetic dipole ( M
1) operator (e.g. see Eq. (3.61)in Ref. [33]) written asˆ M (cid:114) π µ N (cid:104) g R ( ˆ I − ˆ j ) + g s ˆ s + g l ˆ l (cid:105) , (25)after taking it in the intrinsic frame. The operators ˆ s andˆ l in Eq. (25) correspond to the s.p. spin and orbital mo-menta and ˆ j = ˆ l + ˆ s . The quantities g s and g l are the spin and orbital gyromagnetic factors, respectively, and g R isthe collective gyromagnetic factor. The orbital factor is g l = 0 (1) for neutrons (protons), while the spin factoris taken as g s = q s g free s , with g free s = − .
826 (5 . q s is an attenua-tion factor usually supposed to be q s = 0 . − .
7, takinginto account spin-polarization effects [44]. The collectivegyromagnetic factor g R is often associated with the ratio g R = Z/ ( Z + N ), with Z and N being the proton andneutron numbers, respectively, adopted on the basis ofthe liquid-drop-model [45]. However, it is known that inmost deformed nuclei g R is lowered with respect to thisratio by 20%-30% or more [46, 47], with the attenuationbeing explained by the influence of the pairing interactionon the collective moment of inertia [48–50]. Therefore,in Ref. [37] we have introduced the relevant quenchingfactor q R such that g R = q R Z/ ( Z + N ), showing that onthe basis of several earlier theoretical and experimentalanalyses, it can be taken for Th as low as q R ∼ . q s and q R play an important role in the model prediction of the B ( M
1) transition rates and magnetic moments and theirconsideration with further slightly lower values may shedmore light on the
Th formation mechanism.The following common form of the expressions for bothtypes ( T ) of the electric ( T = E ) and magnetic ( T = M )transition with multipolarity λ between initial (i) andfinal (f) states was derived [27] B ( T λ ; π b i I i π i K i → π b f I f π f K f )= R T λ δ π bf π bi (cid:2) (1 + π f π i ( − λδ T,E ) / (cid:3) × (cid:101) N I f π f K f (cid:101) N I i π i K i (cid:34) δ K f K i C I f K f I i K i λ P b f b i K f K i M π bf π bi K f K i N ( π bf ) K f N ( π bi ) K i + A C I f K f I i K f λ (cid:88) ν (cid:54) = iK ν = K i ± , δ K f K ν C I i π i K ν K i P b f K f K ν M π bf π bi K f K ν N ( π bf ) K f N ( π bi ) K ν + A C I f K i I i K i λ (cid:88) ν (cid:54) = fK ν = K f ± , δ K ν K i C I f π f K ν K f P b i K ν K i M π bf π bi K ν K i N ( π bf ) K ν N ( π bi ) K i + A (cid:88) ν (cid:48)(cid:48) (cid:54) = fK ν (cid:48)(cid:48) = K f ± , (cid:88) ν (cid:48) (cid:54) = iK ν (cid:48) = K i ± , δ K ν (cid:48)(cid:48) K ν (cid:48) C I f K ν (cid:48)(cid:48) I i K ν (cid:48) λK ν (cid:48)(cid:48) − K ν (cid:48) × C I f π f K ν (cid:48)(cid:48) K f C I i π i K ν (cid:48) K i P b f b i K ν (cid:48)(cid:48) K ν (cid:48) M π bf π bi K ν (cid:48)(cid:48) K ν (cid:48) N ( π bf ) K ν (cid:48)(cid:48) N ( π bi ) K ν (cid:48) (cid:35) , (26)where the factor R T λ = Eλ = 2 λ + 14 π (4 − δ λ, ) R λ ( π b i n i k i I i → π b f n f k f I f )(27)involves integrals on the radial and angular variables inCQOM (see Eqs. (35)–(41) and Appendixes B and C inRef. [30]) and R T M = 34 π µ N (28)involves the nuclear magneton µ N . Also here M π bf π bi K f K i = (cid:104)F ( π bf ) K f |F ( π bi ) K i (cid:105) , for T = E (cid:104) ( g l − g R ) K i δ K f K i (cid:104)F ( π bf ) K f |F ( π bi ) K i (cid:105) +( g s − g l ) (cid:104)F ( π bf ) K f | ˆ s |F ( π bi ) K i (cid:105) (cid:105) , for T = M, (29)where ˆ s is the z component of the spin operator in spher-ical representation. The factors C I K I K λµ in Eq. (26) areClebsch-Gordan coefficients. The integrals in Eq. (27) de-pend on the model parameters c , defined below Eq. (16),and p , Eq. (11), both determining the electric transitionprobabilities [30].The reduced transition probability expression (26)contains first-order and second-order K -mixing effects.First-order mixing terms practically contribute withnonzero values only in the cases K i/f = K ν = 1 /
2, i.e.,when a K i/f = 1 / K ν = 1 / K = 1 , | K i − K f | ≤ K -values despite the axial symmetry assumedin both CQOM and DSM parts of Hamiltonian (1). Westress that although often disregarded in the literature, itis only through the Coriolis mixing that the M E m Th are rendered possiblewithin the model discussed here.
D. Magnetic moment
The described model formalism allows us to obtainthe magnetic-dipole moment in any state of the quasi-parity doublet spectrum characterized by the Corio-lis perturbed wave function (cid:101) Ψ IMK b (21). The mag-netic moment is determined by the matrix element µ = (cid:113) π (cid:104) (cid:101) Ψ IIK b | ˆ M | (cid:101) Ψ IIK b (cid:105) , where ˆ M is the zeroth spher-ical tensor component of the operator ˆ M
1, Eq. (25),taken after transformation into the intrinsic frame (seeChapter 9 of Ref. [47]). Thus we obtain the followingexpression for the magnetic moment in a state with col-lective angular momentum I and parity π built on a q.p. bandhead state with K = K b and π = π b : µ = µ N g R I + 1 I + 1 1 (cid:101) N IπK b (cid:34) K b M π b K b K b N ( π b ) K b + 2 A K b (cid:88) ν (cid:54) = bK ν = K b = δ K ν K b C IπK ν K b P bK ν K b M π b K ν K b N ( π b ) K ν N ( π b ) K b + A (cid:88) ν , (cid:54) = bK ν ,ν = K ν = K b ± , δ K ν K ν K ν C IπK ν K b C IπK ν K b P bK ν K ν M π b K ν K ν N ( π b ) K ν N ( π b ) K ν (cid:35) , (30)with M π b K µ K ν ≡ M π b ,π b K µ K ν being defined in Eq. (29) ( T = M ) and all other quantities being already defined above.We note that the complete expression would involvean additional decoupling term applying for the case of K b = 1 / M Th no K = 1 / K b = 1 /
2, but we keep itfor consistency with the B ( Eλ ) and B ( M
1) expressions(26). Thus, in the present application of the model onlythe third term is important for the Coriolis mixing in themagnetic moment.One can easily check that in the case of missing Coriolismixing Eq. (30) appears in the usual form of the particle-rotor expression, e.g. Eq. (3.62) in Ref. [33], in which theintrinsic gyromagnetic ratio g K is g K b = 1 K b N ( π b ) K b ] (cid:104)F ( π b ) K b | g s · Σ + g l · Λ |F ( π b ) K b (cid:105) . (31)Equation (31) still takes into account the circumstancethat in the case of nonzero octupole deformation we haveto apply the projected and renormalized s.p. wave func-tion as explained below Eq. (20). In the case of miss-ing octupole deformation (nonmixed s.p. wave function),Eq. (31) reduces to the standard “reflection-symmetric”expression (3.63) in Ref. [33]. In this way the presentmodel expression for the magnetic-dipole moment inEq. (30) is consistent with the relevant limiting cases. E. Model application in Th The CQOM plus DSM-BCS model framework de-scribed above contains a number of parameters that aredetermined according to the physical conditions whichgovern the structure and dynamics of the nucleus
Thand to the available experimental data. These param-eters are the two already discussed Woods-Saxon DSMQO deformations β and β , the five CQOM parameters,namely the QO oscillator frequency ω , the reduced inertiafactor b in Eq. (19), the parameter d in Eq. (3) and theparameters c and p from Eqs. (16) and (11), respectively,entering Eq. (27), the Coriolis mixing constant A , andthe two pairing parameters g and g entering Eq. (8).As it will be detailed below, the first two (Woods-SaxonDSM QO deformation) parameters are determined in aregion of the deformation space providing ultimate DSMconditions for the formation of the m Th isomer. Thepairing constants are fixed for the overall study to valuesin a range typical for the adjacent regions of nuclei. Fi-nally, the five CQOM parameters and the Coriolis mixingconstant are adjusted in a fitting procedure to quantita-tively reproduce the positive- and negative-parity levelsof
Th with energy below 400 keV as well as the avail-able experimental data on transition rates and magneticmoments at each particular Woods-Saxon DSM QO de-formation. As explained in Sec. II B, the five CQOMparameters also determine the shape of the collective po-tential and the corresponding dynamical deformations inthe ( ˜ β , ˜ β )-space. The rather fine parameter determina-tion procedure described here is based on the followingphysical assumptions:1. The considered part of the spectrum consists of twoquasi-parity-doublets: an yrast one, based on the K b = 5 / + GS corresponding to the 5/2[633] s.p.orbital and a nonyrast quasi-parity-doublet, builton the isomeric K b = 3 / + state corresponding tothe 3/2[631] orbital. Both orbitals are very closeto each other providing a quasidegeneracy of theGS and IS. This condition primary depends on thechoice of the quadrupole ( β ) and octupole ( β ) de-formation parameters in DSM and on the BCS pair-ing contribution in the q.p. energy of both states.2. Both quasi-parity-doublets correspond to coherentQO vibrations and rotations with the same radial-oscillation quantum number n = 0, the lowest pos-sible angular-oscillation number k (+) = 1 for thepositive-parity sequences and one of the few lowestpossible k ( − ) = 2 , , k ( − ) = 2 value inthe both quasi-parity-doublets. This suggests com-pletely identical QO vibration modes superposedon both GS and IS. The vibration modes aloneobviously do not cause any mutual displacementof the two quasi-parity-doublets, but the term K b in the centrifugal expression (cid:101) X ( I π , K b ) in Eq. (3)does. It directly mixes the collective energy withthe bandhead and down-shifts the K b = 5 / + levelsequences with respect to the K b = 3 / + ones.This term affects the mutual displacement of ISand GS and, therefore, plays a role in the finallyobserved quasidegeneracy effect.3. The Coriolis mixing affects the total spectrum andthe IS-GS displacement as well through the cor- responding perturbation sum in Eq. (3). As re-alized in Ref. [37], the mixing directly affects the I b , K b = 5 / + GS which gets an admixture fromthe I = 5 / + state of the IS-based band, whereasthe I b , K b = 3 / + IS remains unmixed due to themissing I = 3 / + counterpart in the yrast (GS)band. The corresponding effect of the Coriolis mix-ing in the GS is that it lowers the value of the GSmagnetic moment. On the other hand it raises the B ( M
1) and B ( E
2) transition probabilities.The assumptions above sketch the mechanism whichmay lead to the formation of a quasidegenerate pair of5 / + GS and 3 / + IS in
Th. We see that the very fineinterplay between the involved collective and s.p. degreesof freedom is directly governed by the Woods-Saxon DSMQO deformations β and β , the pairing strength deter-mined by the parameters g and g in Eq. (8) and theCoriolis mixing strength determined by the parameter A in (3). The remaining CQOM parameters ω , b , d , c and p influence the isomer energy through the overall fit ofthe energy spectrum, transition rates and magnetic mo-ments. Within the above physical mechanism the 3 / + IS of
Th appears as an essentially s.p., i.e., micro-scopic, effect, the energy and electromagnetic propertiesof which, however, are formed under the influence of thecollective dynamics of the nucleus.In Ref. [27] the above algorithm was applied throughseveral steps, including the choice of β and β in DSMbased on information available for neighbouring even-even nuclei (see the beginning of next section), tuningof the pairing constants in BCS to reach a rough prox-imity of GS and IS and subsequent fine adjustment ofthe collective CQOM parameters together with the K -mixing constant A to obtain overall model descriptionand predictions. It was demonstrated that at the ex-pense of a minor deterioration of the agreement betweenthe overall theoretical and experimental spectrum, onecan exactly reproduce the IS energy of about 8 eV. Ofcourse, such a refinement is of little practical significancesince it is beyond the genuine accuracy provided by anynuclear structure model.Few comments regarding the results in the next sec-tion should be given here in advance. We remark thatsome model parameters are not completely independentregarding particular physical observables. Thus, thechange in the IS-GS displacement due to variation in theDSM QO deformations could be compensated by varia-tions in the pairing constants or the K -mixing constant A . Therefore, one of the important issues to be clarifiedis the extent to which the different model parametersare correlated in the problem and how we can constrainthem to reach most unambiguously the correct solution.Our numerical study showed that if we fix the pairingparameters in Eq. (8) to the values of g = 18 .
805 MeVand g = 7 .
389 MeV, which were tuned in the modeldescription in Ref. [27], the further analysis and drawnconclusions also apply for the pairing strengths adoptedin Refs. [19] and [41]. Therefore, hereinafter we use
FIG. 1: K -values for the (a) GS and (b) IS s.p. orbitals and the respective average parities (c), (d) (cid:104) π sp (cid:105) appearing in theDSM within the space of quadrupole and octupole deformations. The regions of relevant deformations providing the correct K GS = 5 / K IS = 3 / (cid:104) π sp (cid:105) > K IS = 3 / the above fixed g and g parameter values while direct-ing our study to the examination of the QO deformationspace of the DSM. Another point is that in Ref. [37] the ISand GS magnetic moments were predicted without tak-ing their experimental values into the model adjustmentprocedure. In the present work we include the magneticmoments into the fitting procedure by considering all ob-servables in the fit analysis (energies, transition rates andmagnetic moments) on the same footing. We will alsoinvestigate to what extent the gyromagnetic quenchingfactors q s and q R can be reasonably varied for the re-production of the GS and IS magnetic moments. Thisanalysis aims to reduce the arbitrariness in the modelpredictions for the Th IS properties.
III. NUMERICAL RESULTS AND DISCUSSIONA. Determination of the deformed shell modeldeformation space
The Woods-Saxon DSM shape parameters β and β represent a basic input of our model and their valuesare decisive for the model predictions. Hereafter under“QO deformations and/or parameters” we will under-stand these two quantities unless otherwise specified. InRef. [27] the quadrupole-deformation parameter β waschosen by varying it between the experimental values0.230 and 0.244 available for the neighbouring even-evennuclei Th and
Th, respectively [51]. Simultaneouslythe octupole-deformation parameter β was varied to ob-tain the GS and IS orbitals very close to each other, withleading 5/2[633] and 3/2[631] components in the respec-0 FIG. 2: Average parity (cid:104) π sp (cid:105) in the (a) GS and (b) IS s.p. orbitals appearing in DSM within the model-defined QO deformationspace.FIG. 3: (a) S.p. and (b) q.p. energy (in keV) of the 3 / + isomer orbital with respect to the 5 / + GS orbital appearing inDSM within the model-defined space of QO deformations. The q.p. energy is obtained with pairing parameters g = 18 . g = 7 .
389 MeV used in Eq. (8). See text for further explanations. tive s.p. wave-function expansions given in Eq. (6), andwith positive average values of the parity (cid:104) π sp (cid:105) > β between 0.1and 0.2 in the neighboring even-even nuclei Th and
Th.In this study we identify the ( β , β ) deformationspace which could provide a relevant model descriptionof the m Th isomer similar to the one obtained in Refs. [27, 37] which had considered the values β = 0 . β = 0 . . ≤ β ≤ .
26 and0 ≤ β ≤ .
15 which are supposed to include the QO de-formations physically relevant for a nucleus in the massregion of
Th. At each point of the grid we obtain the K -value and the average parity (cid:104) π sp (cid:105) for the last occu-pied s.p. orbital, which is supposed to determine the GSand for the next (first) non-occupied orbital, candidatefor the IS. Note that the calculation does not involvethe collective (CQOM) part of the model and the onlyentering parameters are the two Woods-Saxon DSM de-formations.The result of this calculation is shown in Fig. 1. Figs. 1(a) and 1 (b) present in color coding the ( β , β ) areas1 FIG. 4: The CQOM QO wave-function density | Φ π qo nkI ( ˜ β , ˜ β ) | from Eq. (15) as a function of ˜ β and ˜ β for the I π = 5 / + GS(with k = 1) and 5 / − state (with k = 2) of the yrast quasi-parity-doublet in Th. We use here the DSM+CQOM fit with q s = q R = 0 . β , β )=(0 . , . β sa , are shown as red circles.FIG. 5: The same as Fig. 4 (lower panels), but for the I π = 15 / + and 15 / − yrast quasi-parity-doublet states of Th. Notethe considerable increase in the QO semiaxes and the corresponding expansion of the dynamical-deformation peak positions inthe space of the collective CQOM variables ˜ β and ˜ β . K values appear for the GS and ISorbitals, respectively. For the GS orbital the K = 5 / K = 3 / K = 5 / K = 3 / K = 3 / β , β )- region in which the DSM pro-vides the required 5/2[633] and 3/2[631] orbitals for theGS and IS, respectively. Furthermore, considering theinformation from Fig. 1 (c) and 1 (d), one can identifyin a similar way the regions with positive and negativeaverage values of the parity in the GS and IS orbitals, re-spectively. By retaining only the (cid:104) π sp (cid:105) > β , β )-regiongiven by the thick triangle contour in the four plots. Thisregion includes all QO deformations from the consideredspace which are relevant within the DSM regarding thecurrent experimental information and theoretical inter-pretation of the K π = 3 / + isomer in Th. Hereinafterwe call this region our “model deformation space”.Based on the above result, we can draw the follow-ing important conclusion. Considering the long-adopted K -value and parity of the m Th IS, our DSM pre-diction shows that this isomer can only exist at essen-tially nonzero octupole deformation of the s.p. potential.More precisely, one can say that the coexistence of the K π = 3 / + IS together with the K π = 5 / + GS requiresthe presence of nonzero octupole deformation, as seenfrom Fig. 1(a). In fact our more extended calculations inthe QO deformation grid show that for β < . K π = 5 / + GS and K π = 3 / + IS orbitals goes down and further reaches the β = 0 line.However, this occurs around β ∼ .
1, which is far be-yond the deformation limits typical for this mass region.Thus, we can conclude that the octupole deformation ap-pears to be of a crucial importance for the formation ofthe m Th isomer according to the present knowledgeon the corresponding GS and IS angular momenta andparities.Furthermore, we note that the deformation values( β = 0 . , β = 0 . Th and
Th obtainedin the relativistic Hartree-Bogoliubov model calculations[52]. Nevertheless, the precise determination and predic-tion of the m Th isomer properties as well as the deeperunderstanding of the mechanism governing its formationrequires a more detailed examination of the model de-scriptions obtained for various deformations fixed in theoutlined model space. In the following our study is fo-cused on this task.A direct consequence of the location of the model
FIG. 6: Energy RMS values in keV for the GS (yrast) andIS (excited) bands together obtained by the model fit on agrid within the model-defined QO deformation space. We use q s = q R = 0 .
6. The open star indicates the location of thedeformations ( β , β )=(0 . , . space at nonzero octupole deformation is that the GSand IS s.p. orbitals provided by the DSM always appearwith mixed parity which has to be projected in the to-tal core plus particle wave function, as seen in Eq. (20),and implemented in the model procedure applied in [27].The average parity (cid:104) π sp (cid:105) of both orbitals calculated fromEq. (7) as a function of the QO deformations within themodel space is illustrated in Fig. 2. The results showthat while in the GS the quantity (cid:104) π sp (cid:105) varies within thelimits 0 . − .
46, in the IS the parity mixing is evenmuch stronger with (cid:104) π sp (cid:105) varying between 0 and 0.14.The black side of the triangle in Fig. 2(b) corresponds tothe (left) border of the space where the average parity ofthe IS turns from positive to negative values. This re-sult shows that the mechanism governing the formationof the isomer is even more complicated due to the fineparity-mixed structure of the s.p. wave functions and theaccordingly applied projection procedure.Other important quantities delivered by the DSM arethe s.p. and q.p. energies for the IS orbital determinedwith respect to the corresponding energies in the GS or-bital, E sp3 / + = E / + sp − E / + sp and E qp3 / + = (cid:15) / + qp − (cid:15) / + qp .The lowering of the q.p. energy with respect to the s.p.energy can be controlled through additional tuning of thepairing constants as shown in Ref. [27]. However, as ar-gued at the end of Sec. II E, here we use the g and g values fixed in [27] focusing our analysis on the defor-mation dependencies. In Fig. 3 both E sp3 / + and E qp3 / + for the IS are plotted as functions of β and β . Asexpected, they show an identical dependence but withdifferent nominal values. In addition, along the rightside of the triangle the q.p. and s.p. content of the iso-mer energy goes to zero, i.e. the two orbitals, 5/2[633]and 3/2[631] mutually degenerate. This is an important3 FIG. 7: Isomer energy E IS obtained by the model fits on a model-defined QO deformation space grid for two differentcombinations of q s and q R . The open star indicates the location of the deformations ( β , β )=(0 . , . β , β )=(0 . , . limit of the model deformation space. In fact the blacklines in Fig. 3 correspond to the crossing of both orbitalswhen leaving the model space to enter the blue area with K = 3 / K = 5 / K π = 3 / + and the IS obtains K π = 5 / + . Theproximity to this line from the model space interior de-termines the degree of the q.p. quasidegeneracy effect.For the pair of QO deformations ( β , β )=(0 . , . / + q.p. energy yields E qp3 / + = 2 .
196 keV. We note that this is not the final ISenergy in which additional contributions take a part asexplained in Sec. II E. The upper side of the triangle cor-responds to the crossing of the K π = 3 / + orbital with a K π = 7 / − orbital with leading component 7/2[743] [seered area in Fig. 1(b)]. It is not of a particular interestfrom the isomer-formation point of view. B. Coherent quadrupole-octupole model fits in thedeformed shell model deformation space
At this point we are ready to examine the behaviourof the model description and prediction for the physi-cal observables of interest within the DSM deformationspace. We are especially interested in the correspondingbehaviour of the B ( M
1) and B ( E
2) IS transition ratesand of the IS and GS magnetic moments µ GS and µ IS .To this end we have performed full model fits by ad-justing the five CQOM parameters, ω , b , d , c , p andthe Coriolis mixing constant A with respect to the ex-perimental quasi-parity-doublet spectrum, the availabletransition rates and magnetic moments at each point ofthe deformation space grid with the pairing constantsfixed as described above. Thus, for each pair of Woods- Saxon DSM QO deformations we obtain the full spectro-scopic description of the nucleus storing the quantities ofinterest for our systematic analysis presented below.The five adjusted CQOM parameters show a smoothbehavior along the deformation space with values consis-tent with those obtained in Ref. [27]. We therefore refrainfrom addressing further numerical details here. We notehowever that the parameter p defined in Eq. (11) is closeto unity p ≈ β , β )-modelspace. This parameter indicates the relative contributionof the quadrupole and octupole modes in the CQOM.Thus, according to Eq. (14) all parameter fits lead to apractically circular bottom of the CQOM potential with˜ β sa2 ≈ ˜ β sa3 , showing that the model describes the collec-tive quasi-parity-doublet structure of the Th spectrumwith equal weights of the quadrupole ˜ β and octupole ˜ β deformation modes. Considering the DSM deformationparameters β = 0 .
240 and β = 0 .
115 used in Ref. [27],we obtain for the 5 / + GS and 3 / + IS states valuesbetween 0.12 and 0.17 for the two equal ˜ β sa2 , semiaxes(which define the circle radius of the CQOM potentialbottom).Inspection of the odd-nucleon contribution to˜ β sa2 , determined in Eq. (13) through Eq. (3) at( β , β )=(0 . , . Ththe Coriolis mixing causes a negligible decrease in ˜ β sa2 , compared with the core case, Eq. (12), while a consid-erable decrease is caused by the term ( − K b ). Thuswhile in the core+particle case the GS ˜ β sa2 , = 0 . − K b )] these valuesbecome 0.149. Similarly, in the isomeric state thecore+particle ˜ β sa2 , = 0 . | Φ π qo nkI ( ˜ β , ˜ β ) | fromEq. (15) for the I π = 5 / + GS and its negative-parity4
FIG. 8: B ( M
1) isomer transition values obtained by the model fits on a grid within the model-defined QO deformation spaceat four different combinations of q s and q R . The open star indicates the location of the deformations ( β , β )=(0 . , . β , β )=(0 . , . counterpart 5 / − obtained with the parameters of theCQOM fit at ( β , β )=(0 . , . − K b ) termin (cid:101) X ( I π , K b ) dropping the negligible Coriolis mixingterm. The wave function density was calculated forthe quenching parameter set ( q s , q R )=(0 . , . β , β ) parameters, confirming the discussion inSec. II B on the distinction between dynamical deforma-tion parameters in the CQOM and the DSM deformationparameters. We also note that the CQOM potentialbottom semiaxes ˜ β sa2 , (the red circles) do not changebetween the positive- and negative-parity counterpartsin the quasi-parity-doublet since the Coriolis mixingterm only mixes states with the same (bandhead) parity.This situation, however, would be different in a spectrum build on the K b = 1 / Th) where the decoupling term in (cid:101) X ( I π , K b ) in Eq. (3) would act in opposite directions onthe semiaxes lengths of the opposite-parity counterparts.We have checked that the density plots for the I π =3 / + IS and its 3 / − quasi-parity-doublet counterpart(not given here) look very similar to those in Fig. 4. Toinvestigate the effect of higher angular momentum val-ues we show the CQOM QO wave-function densities forthe I = 15 / ± states of the yrast quasi-parity-doubletin Fig. 5. The CQOM potential semiaxes ˜ β sa2 , and thecorresponding dynamical deformations considerably in-crease with angular momentum, reaching at I = 15 / ± values larger than 0 .
2. This shows that the dynamical de-formation is responsible for the higher-energy part of thespectrum which otherwise would not be felt by the s.p.potential. We may conclude that the model algorithmrather carefully takes into account also the influence ofthe collective dynamics at the higher angular momenta,which reflects on the overall deformation characteristics5
FIG. 9: B ( E
2) isomer transition values obtained by the model fits on the model-defined QO deformation space grid at fourdifferent combinations of q s and q R . The open star indicates the location of the deformations ( β , β )=(0 . , . β , β )=(0 . , . of the CQOM potential.Finally, special attention is given below to the be-haviour of the Coriolis mixing constant A which, as al-ready mentioned, plays an important role in the forma-tion of the IS energy and electromagnetic properties. Thecalculations were repeated for four pairs of ( q s , q R ) valuesof the quenching factors considered for the spin and col-lective gyromagnetic rates. As the analysis of magneticmoments made in Ref. [37] suggests the need of ratherstrong attenuation of the latter, here we consider q s and q R with slightly lower values compared with the lowestpair ( q s , q R )=(0 . , .
6) considered in Ref. [37]. Thus, inthe present calculations the gyromagnetic quenching fac-tors were allowed to be as small as ( q s , q R )=(0 . , . E (3 / + ) IS in order tosee how the model fits “feel” its tiny energy scale aswell as to assess accordingly the relevance of the overall model description for the different deformations withinthe model space. C. Energy description
The primary quantity providing overall informationabout the relevance of the model descriptions in thedifferent points of the deformation space is the root-mean-square (RMS) deviation between the theoreticaland experimental energy levels. Its behaviour as a func-tion of the DSM QO deformation for calculations madewith quenching factors ( q s , q R )=(0 . , .
6) is illustratedin Fig. 6. We indicate with an open star the locationof the deformations ( β , β )=(0 . , . µ GS FIG. 10: GS magnetic moment values obtained by the model fits on a grid within the model-defined QO deformation spaceat four different combinations of q s and q R . The open star and the circle indicate the same sets of deformations as shown inFigs. 8 and 9. and µ IS are included in the fits. We note that the RMSfactor is close to this value over a larger area of the defor-mation model space, demonstrating the stability of themodel solutions with the variation of QO deformations.The upper part of the space with large β and β values,however, appears unfavoured. We have verified that inall regions of the space with the RMS close to 34 keVthe description of the overall energy spectrum and theavailable B ( M
1) and B ( E
2) transition rates is of simi-lar accuracy as the one reported in Ref. [27], with theobtained CQOM parameter values being close to thosein Ref. [27] (see Fig. 1 and Table 1 therein). We noticethat in the upper-left parts of the plot some lower RMSdeviations are obtained as low as ≈
30 keV, however, forthese deformations the model predictions for the isomerenergy are less favourable as analyzed below. In addition,we found (barely visible in Fig. 6) that towards the line ofthe 5/2[633]–3/2[631] degeneracy the RMS factor sharplyincreases. As discussed below in relation to the particu-lar observables, this is the result of the strong increase of the Coriolis K -mixing interaction which largely exceedsthe perturbation theory limitation and puts a constrainton the model description valid close to the 5 / + –3 / + orbital crossing.In Fig. 7 the theoretical isomer energy values E IS ob-tained by the model fits on the DSM QO space grid arepresented for two sets of gyromagnetic quenching values( q s , q R ) = (0 . , .
6) and (0 . , . β , β )=(0 . , . E IS ≈ E IS ≈ . β , β )=(0 . , . . , . E IS ≈ .
040 keV already approachingthe scale of the experimental value. We note that forthis pair of deformations the 3 / + q.p. energy yields E qp3 / + = 0 .
188 keV.All E IS values shown in Fig. 7 are obtained in the7fitting procedure on the same footing without particularrefinement. As already mentioned, one can easily achievethe exact experimental value of 0.008 keV through a veryfine tuning of model parameters (e.g. the K -mixing A ),with a minimal deterioration of the description in the re-maining energy levels (see also Ref. [27]). The plots inFig. 7 show that in the large areas of the model spacethe fits provide reasonable values of E IS which could berenormalized to the experiment in this manner. However,we also see that in the upper-left parts of the plots the E IS considerably increases up to 7-8 keV giving an indi-cation that at these deformations the remoteness of the5/2[633] and 3/2[631] orbitals (see Fig. 3) already makesit difficult for the model mechanism to achieve quaside-generacy. Also, we notice a thin stripe with large E IS values along the line of degeneracy which obviously indi-cates the limitation of the perturbation theory. Conclud-ing this part, our analysis of the RMS factor and isomerenergy values outlines certain limits of reliability of thepresent model application and favours the lower vertexof the model space around the deformation set used inRef. [27] in reasonable proximity to the 5 / + –3 / + or-bitals crossing line. D. B ( M and B ( E isomer transition rates FIG. 11: The “bare” s.p. GS magnetic moment values ob-tained without Coriolis mixing ( A = 0) on a grid within themodel-defined QO deformation space with q s = q R = 0 . β , β )=(0 . , . µ GS =0.55 µ N . The results obtained for the isomeric B ( M
1; 3 / +IS → / +GS ) and B ( E
2; 3 / +IS → / +GS ) transition rates areillustrated in Figs. 8 and 9, respectively, for the foursets of quenching factors. As in the energy analy-sis above, here we mark with an open star the value obtained by the model at the pair of QO deforma-tions ( β , β )=(0 . , . / + –3 / + orbitals degeneracy by considering in thegraphs of ( q s , q R )=(0 . , .
45) the pair of deformations( β , β )=(0 . , . B ( M
1; 3 / +IS → / +GS ) transition value showsan increase with the approaching of the degeneracy line.This is due to the circumstance that with the decreasingdistance between both orbitals 5 / + and 3 / + , the K -mixing effect generated by the matrix element in Eq. (5)sharply increases and this leads to the increase in theconnecting transition rates. It should be noted, how-ever, that in the model procedure this increase is coun-terbalanced by the adjustable parameter A which dropsaccordingly thus preventing a deterioration of the modeldescription due to the excessive mixing force. This will bediscussed in more detail in the following (see Fig. 13 andrelated text below). Keeping in mind this clarification,we notice in Fig. 8(d) that while the quenching of thegyromagnetic factors leads to a reduction of B ( M IS to0.005 W.u. at ( β , β )=(0 . , . B ( E
2; 3 / +IS → / +GS )transition rate in the model deformation space is ob-served in Fig. 9. We note that here all obtained B ( E IS values exceed the previous prediction [27] but stay in therange of the correction suggested in Ref. [37]. E. Ground-state and isomer magnetic moments
The calculated GS magnetic moment µ GS for thefour sets of quenching factors ( q s , q R ) is illustrated inFig. 10. The overall model behaviour of this quantityis such that it decreases both with the attenuation ofthe gyromagnetic factors and with the approaching ofthe 5 / + –3 / + orbital-degeneracy line. We see that for( β , β )=(0 . , . µ N , while withthe shift of the deformations to the point (0 . , . µ N . The latter result is obvi-ously due to the increasing Coriolis mixing which reducesthe value of µ GS as has been shown already in Ref. [37].However, it appears that for model conditions consid-ered physically reasonable, this is still not sufficient toreproduce the newer experimental value of 0.360(7) µ N [38], although the model reproduces fairly well the earliermeasured value of 0.46(4) µ N [39].It is instructive to check here also the “bare” val-ues of µ GS , i.e. those obtained by the pure s.p. wavefunction without including the Coriolis mixing. There-fore, in Fig. 11 we show the analog of Fig. 10(a) (with8 FIG. 12: IS magnetic moment values obtained by the model fits on the model-defined QO deformation space grid at fourdifferent combinations of q s and q R . The open star and the circle indicate the same sets of deformations as shown in Fig. 10. q s = q R = 0 .
6) in which µ GS is calculated in the absenceof Coriolis mixing with the K -mixing constant A = 0.In this case Eq. (30) reduces to the terms in its first linewith the second term involving the expression of Eq. (31).Here we first see that µ GS appears with considerablylarger values in the limits µ GS =0.55–0.60 µ N which alsoshow different behaviour in the DSM QO space comparedwith the Coriolis-mixing case. This result does not de-pend on the model-parameters fit and illustrates the gen-uine contribution of the QO deformation for the forma-tion of the GS magnetic moment of Th. Comparingboth plots we see that in the pure s.p. case without Cori-olis mixing, the lowest µ GS = 0 . µ N value appears inthe left-upper vertex of the triangle model space, whereasin the Coriolis-mixing case the low values (lower than thepure s.p. ones), appear in the lower vertex of the space.The above result leads us to the following conclu-sions. The Coriolis effect causes a decrease of the nu-clear magnetic moment in the Th GS throughoutthe model deformation space. It plays a considerablerole in our approach for fixing the GS magnetic mo- ment through the overall model fits, although this isstill not enough to reproduce the latest adopted experi-mental value. The appearance of essentially lower µ GS -values obtained through the adjusted K -mixing constant A compared with the corresponding pure s.p. µ GS -valuesshows that the increase of the model-controlled Coriolis-mixing towards the 5 / + –3 / + orbitals-degeneracy lineessentially determines the behaviour of the GS magneticmoment and dominates over the corresponding effect ofchanging QO deformation on the pure s.p. µ GS -values.This conclusion suggests that no considerably differentresult can be reached through further variation of defor-mations parameters in the DSM QO space.Figure 12 shows the calculated IS magnetic moment µ IS for the four sets of gyromagnetic quenching factorsconsidered ( q s , q R ). We note its relatively flat behaviouras a function of the QO deformation, with a slight in-crease towards the degeneracy line. Since µ IS is practi-cally not affected by the K -mixing effect, we may claimthat this dependence can be considered as the bare ef-fect of the changing structure of the s.p. wave functions9 FIG. 13: Values of the K -mixing constant A obtained by the model fits on the model-defined QO deformation space grid at twodifferent combinations of q s and q R . The open star and the circle indicate the same sets of deformations as shown in Fig. 10. along the deformation space. We see that in all plots ofFig. 12 the lowest value of µ IS appears in the left up-per corner of the model space similarly to the “bare”(s.p.) µ GS case (Fig. 11). For example in the case of q s = q R = 0 .
6, Fig. 12(a), the corresponding lowest value µ IS = − µ N comes closer to the experimental result.However, the model fits have shown that the lower cornerprovides better predictions for µ GS , and we keep our at-tention on this region. Besides, for all considered quench-ing factors ( q s , q R ), the theoretical µ IS values appearingin Fig. 12 enter the error bars of the recent experimentalvalue of − µ N reported in Ref. [16, 17].The results presented so far already reveal importantdetails and relations characterizing the model mechanismupon which the m Th isomer is formed and its spectro-scopic properties develop. Obviously the proximity ofthe 5/2[633] and 3/2[631] s.p. orbitals plays a major roleproviding the overall condition for the appearance of alow-lying excitation. Now this is clearly quantified by allabove plots. On the other hand, it is also clear that theappearance of the isomer can not be due only to the or-bital quasidegeneracy. The reason is that at the distanceof few eV the mixing between the two orbitals becomesvery large and pushes all related observables in unphys-ical regions of magnitude. In this case the perturbationterms in the centrifugal part, Eq. (3), of the Hamilto-nian as well as in the Coriolis perturbed wave function,Eq. (21), collapse in a singularity. The model preventsthis situation mostly through the K term and K -mixingconstant A in Eq. (3), which allow us to properly situ-ate both GS and IS, i.e., to obtain the IS energy valueas small as necessary, by keeping the 3/2[631]–5/2[633]orbital distance aside from the degeneracy line. In thisrespect one can say that the physically adequate QO de-formations are slightly aside from this line.The model mechanism feature described above can beseen by following the behaviour of the Coriolis mixing constant A adjusted at each grid point in the deforma-tion space. We investigate this in Fig. 13 for two sets ofquenching factors, ( q s , q R ) = (0 . , .
6) and (0 . , . A range from zero to approxi-mately 0.5 keV. Towards the degeneracy line, where the K -mixing matrix element in Eq. (5) connecting the twoorbitals sharply increases as they approach each other,the adjustment algorithm strongly reduces the value of A .In this way the model “feels” the growing magnitude ofthe Coriolis mixing and tries to compensate its excessiveeffect on the considered observables through the parame-ter A keeping them in physically meaningful ranges. Pro-viding this balancing role of the parameter A and havingin mind all so far obtained model patterns for the spec-troscopic observables in Th we can be rather confidentin the consistency of the analysis made and the reliabilityof the QO deformation region outlined.Finally, it is interesting to identify the degree of spinand collective gyromagnetic factor attenuations requiredto reproduce in the present model both experimental µ GS and µ IS values. This is shown in Fig. 14, where the valuesof each of these quantities obtained in the model fits at( β , β )=(0 . , . µ GS andFig. 14(b) for µ IS ] as functions of the quenching factors q s and q R . The black lines denote the pairs of ( q s , q R )values which provide the corresponding µ GS and µ IS ex-perimental values. The crossing of both lines shows thepoint at which both magnetic moments are reproducedtogether. We see that this occurs at q s ≈ .
52 and arather low value of q R ≈ .
22, which corresponds to aquite strong attenuation of the collective gyromagneticfactor. Because this quenching magnitude is hard to jus-tify, we conclude that the agreement between the presenttheoretical model and the currently adopted experimen-tal value of the GS magnetic moment remains an openissue.0
FIG. 14: (a) GS and (b) IS magnetic moment values obtained by the model fits on a grid for the spin-gyromagnetic, q s , androtation-gyromagnetic, q R , attenuation factors. The black lines show the values that would reproduce the corresponding GSand IS experimental values. IV. SUMMARY AND CONCLUSION
In this work we have thoroughly examined the phys-ical conditions for the formation of the 8 eV isomer of
Th according to the model mechanism suggested byour QO vibration-rotation core plus particle approach.First, we have determined the model deformation spaceencompassing the Woods-Saxon DSM quadrupole andoctupole deformations which allow the appearance of theGS and IS with the experimentally adopted K -values andparities. We were able to clearly identify its borders con-strained by the average parity of the isomer and the cross-ings of the 3/2[631] orbital on which the IS is built withthe 5/2[633] orbital of the GS as well as with a 7/2[743]orbital. This space is rather limited and essentially con-strains the QO deformation in the s.p. potential withinthe ranges 0 . ≤ β ≤ .
255 and 0 . ≤ β ≤ . K π = 3 / + IS through this mecha-nism is only possible in the presence of essentially nonzerooctupole deformation in the s.p. potential.Furthermore, we have examined the dependence of theoverall DSM+CQOM model description on the QO de-formations within the DSM model space. Our analy-sis of the CQOM fits made in Sec. III B, including theCQOM potential-bottom semiaxes and dynamical defor-mations, showed that the collective CQOM conditionsunder which the m Th isomer is formed consistentlyinterrelate with the relevant conditions provided by theodd-nucleon degrees of freedom. Under these overall con-ditions we obtain for all of the considered observables, B ( M
1; 3 / +IS → / +GS ), B ( E
2; 3 / +IS → / +GS ), µ GS , µ IS and E (3 / + ) IS a smooth behaviour of the modelpredictions and descriptions compared with the resultsobtained for the fixed pair of Woods-Saxon DSM QO de- formations considered in our previous works [27, 37], withthe only peculiarity appearing close to the line of 5 / + –3 / + degeneracy where the mixing of both orbitals ex-ceeds the perturbation theory limits. On the other handthe corresponding behaviour of the model energy RMSfactors and Coriolis mixing constant shows that descrip-tions obtained with values of the above observables es-sentially deviating from those obtained in [27, 37] areof lower quality and/or violate the perturbation limit.For the remaining descriptions we have verified that inthe limits of moderate deviations of the considered ob-servables from the original values in [27, 37], the modelis renormalizable, so that through a small variation ofmodel parameters and on the expense of small deteriora-tions of the RMS factor, we can get very similar modelpredictions for the different pairs of QO deformations.Using this model feature as well as assuming possiblestronger attenuation of the spin and collective gyromag-netic factors, we have outlined rather narrow limits ofarbitrariness in which the model values of each of theabove quantities can vary by keeping its reasonable phys-ical meaning and predictability.Our main conclusion is that within the obtained modeldeformation space the applied QO core plus particle ap-proach provides a rather constrained prediction for themost important Th energy and electromagnetic char-acteristics related to the formation and manifestation ofthe 8 eV isomer. This allows us to generally reconfirmthe predictions initially made in Ref. [27, 37] and tosummarize them with a slight update: The B ( M
1) IStransition remains in the limits 0.006-0.008 W.u. withan open possibility towards lower values such as 0.005W.u.; the B ( E
2) IS transition may be considered withslightly higher values between 30 and 50 W.u., comparedwith those in Ref. [27]; the GS magnetic moment allows1a limited possibility for variation and remains with amodel value around 0.50 µ N possibly getting values assmaller as 0.43-0.48 µ N under a stronger assumption forthe gyromagnetic factors attenuation, thus covering theold experimental value of Ref [39], but still overestimat-ing the newer one of Ref. [38]; the theoretical IS magneticmoment firmly reproduces the recent experimental valuewithin the uncertainty limits reported in [16, 17] and thisis obtained under all considered model conditions; and fi-nally the model values for the isomer energy typically ob-tained around 1 keV and below suggest that with a smallvariation of parameters, and with the expense of a minordeterioration of the other energy levels, the model caneasily reproduce the experimental value, although this isof little importance due to the overall limitation of themodel accuracy in the energy-spectrum description.We note that for all of the above quoted values theother model observables (energy levels and transitionrates), for which experimental data are available, remaindescribed within the accuracy limits reported in Ref. [27].Thus our analysis suggests that within the above out-lined limits of arbitrariness the model predictions couldprovide reliable estimates for the m Th spectroscopiccharacteristics which could serve to the experiment infurther efforts to observe and control the yet elusive ra-diative isomer transition.Finally, the results obtained confirm the relevance ofthe model mechanism emphasizing on the role of the fine interplay between nuclear collective and intrinsic degreesof freedom as a plausible reason for the isomer formation.On this basis we conclude that the same dynamical mech-anism may govern also in other nuclei the formation ofexcitations close to the border of atomic physics energyscale. Such states may exist being not yet observed dueto experimental difficulties similar to those encounteredin m Th. As in this work we give a detailed prescriptionabout the examination and constraining of the physicalconditions under which such a phenomenon may emerge,it appears promising to extend the study to other nucleiin the same or other mass regions. In this aspect theclose neighbour
Th as well as the m U isomer wouldbe natural candidates for such a study. This could be asubject of future work.
ACKNOWLEDGMENTS
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