aa r X i v : . [ nu c l - t h ] O c t Wobbling excitations at high spins in A ∼ J. Kvasil and R. G. Nazmitdinov
2, 3 Institute of Particle and Nuclear Physics, Charles University,V.Holeˇsoviˇck´ach 2, CZ-18000 Praha 8, Czech Republic Departament de F´ısica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia (Dated: October 31, 2018)We found that in
Dy and
Yb the lowest odd spin gamma-vibrational states transform to thewobbling excitations after the backbending, associated with the transition from axially-symmetricto nonaxial shapes. The analysis of quadrupole electric transitions determines uniquely the sign ofthe γ -deformation in both nuclei after the transition point. PACS numbers: 21.10.Re,21.60.Jz,27.70.+q
Thanks to novel experimental detectors, a new frontierof discrete-line γ -spectroscopy at very high spins has beenopened in the rare-earth nuclei (see, for example, [1]).These nuclei can accommodate the highest values of theangular momentum, providing one with various nuclearstructure phenomena. The quest for manifestations ofnonaxial deformation is one of the driving forces in highspin physics in past few years [2]. The identification ofwobbling excitations is recognized nowadays as a convinc-ing proof of the nonaxiality. Wobbling excitations weresuggested first by Bohr and Mottelson for rotating even-even nuclei [3] and studied soon within simplified micro-scopic models [4] (see also Ref.5 and references therein).According to the microscopic approach [6, 7], the wob-bling excitations are vibrational states of the negativesignature built on the positive signature yrast (vacuum)state. Their characteristic feature is collective E2 tran-sitions with ∆ I = ± h between these and yrast states.First experimental evidence of such states in odd Lu nu-clei was reported only recently [8].The first analysis of the properties of the second tri-axial superdeformed band in Lu was based upon phe-nomenological particle-rotor calculations [9]. The abso-lute values of the irrotational moments of inertia were fit-ted and so-called ” γ -reversed” dependence of these mo-ments was introduced in order to obtain a reasonableagreement with the experimental data. It was shownin Ref.10 that the microscopic approach [7] may gain abetter insight into the observed phenomena. In the anal-ysis of [10], however, the constant mean-field deforma-tion parameters are used, which is not always justified.Moreover, the authors admitted that the kinematic mo-ment of inertia ℑ x was not described properly due to thestrong velocity dependence of the Nilsson potential (seediscussion in Ref.10). We recall that wobbling excitationsdepend on all three moments of inertia that characterizethe nonaxial shape. Therefore, a self-consistent descrip-tion of moments of inertia is a prerequisite of the mi-croscopic analysis of the nuclear wobbling motion. Themain aim of this Letter is to analyze new data on highspin states in Dy and
Yb [11, 12] within a micro-scopic approach [13] based on the cranked Nilsson modelplus random phase approximation (CRPA). In our ap- proach mean field parameters are determined from theenergy-minimization procedure. The proper descriptionof the moment inertia ℑ x is achieved using the recipesuggested in Ref.14. Our calculations suggest that someexcited states at high spins may represent wobbling ex-citations.Our model Hamiltonian isˆ H Ω = ˆ H − X τ λ τ ˆ N τ − Ω ˆ J x + V (1)The term ˆ H = ˆ H N + ˆ H add contains the Nilsson Hamil-tonian ˆ H N and the additional term that restores the localGalilean invariance of the Nilsson potential, broken in therotating frame [14]. This term is essential to obtain a cor-rect description of ℑ x -moment of inertia [13]. Althoughthe additional term ˆ H add breaks the rotational symme-try in the sense of Eq.(3) (see below), this effect can benegligibly small in the RPA order. The chemical poten-tials λ τ ( τ =n or p) are determined so as to give correctaverage particle numbers h ˆ N τ i . Hereafter, h ... i meansthe averaging over the mean field vacuum (yrast) stateat a given rotational frequency Ω. The interaction V in-cludes separable monopole pairing, monopole-monopole,and quadrupole-quadrupole terms to describe the posi-tive parity states. All multipole and spin-multipole op-erators have a good isospin T and signature r = ± x i = ( ω i /ω ) x i , which ensure the self-consistent condi-tions at the equilibrium deformation. Details about themodel Hamiltonian (1) can be found in Refs.13.The Nilsson-Strutinsky analysis of experimental dataon high spins in Dy [12] indicates that the positive par-ity yrast sequence undergoes a transition from the prolatetowards the oblate rotation. In our calculations the de-formation parameters β and γ are defined by means of theoscillator frequencies ω i = ω h − β q π cos ( γ − π i ) i ( i = 1 , , x, y, z ). To compare our results with avail-able experimental data [12], we consider the mesh on the β, γ plane: from γ = 60 (an oblate rotation aroundthe y-axis) to γ = − (an oblate rotation aroundthe x-axis) and β = 0 − .
6. At each rotational fre-quency, we have determined the equilibrium deformationparameters ( β, γ ) by minimizing the mean-field energy E MF = h ˆ H Ω i on the mesh. In the vicinity of the back-bending this procedure becomes highly unstable. In or-der to avoid unwanted singularities for certain values ofΩ, we followed the phenomenological prescription [16] forthe definition of the pairing gap parameter (see details inRefs.13). Parameters of the Nilsson potential were takenfrom Ref.[17]. In our calculations we include all shellsup to N = 9. Near the transition point we extendedour configuration space up to N = 10 shells. The dif-ference between results from the former and the lattercases was small and all presented results are obtainedwith N = 0 − ls and l potentials, taking into completeaccount ∆ N = 2 mixing produced by them. This im-proves the accuracy of the mean field calculations, sincethe ”single stretched” ls and l potentials break the ro-tational symmetry. FIG. 1: (Color online) Equilibrium deformations in β - γ planeas a function of the angular momentum I = h ˆ J x i − / h ). The equilibrium deformations for Dy providethe lower mean field energies in the region − π/ < γ < γ -valuesdoes not exceed ∼ Dy.
Our results conform to the results of the Nils-son+Strutinsky shell correction method (compare ourFig.1 with Fig.3c in Ref.12), although we obtain slightlydifferent values for the equilibrium deformations. In theanalysis of Ref.12 the pairing correlations are missing,while the hexadecapole deformation is not included in thepresent calculations. The triaxiality of the mean field setsin at the critical rotational frequency ¯ h Ω c which triggersthe backbending in the considered nuclei due to differentmechanisms. We obtain ¯ h Ω c ≈ . h → h )and ¯ h Ω c ≈ . h → h ) for Yb and
Dy,respectively. The contribution of the additional term wascrucial to achieve a good correspondence between the calculated and experimental values of the crossing fre-quency in each nucleus. In
Dy we obtain that the γ -vibrational excitation ( K = 2) of the positive signa-ture tends to a zero in the rotating frame at the tran-sition point, in close agreement with experimental data.At the transition point there are two indistinguishablemean-field energy minima with different shapes: axiallysymmetric and strongly nonaxial. The increase of therotational frequency changes the axial shape to the non-axial one with a negative γ -deformation ( γ ∼ − o ). Incontrast, the axially symmetric configuration in Yb isreplaced by the two-quasiparticle one with a small neg-ative γ -deformation. There, the backbending occurs dueto the rotational alignment of a neutron i / quasipar-ticle pair. The nonaxiality evolves quite smoothly.In the CRPA approach the positive ( r = +1) and neg-ative ( r = −
1) signature boson spaces are not mixed,since the corresponding operators commute and H Ω = H Ω ( r = +1)+ H Ω ( r = − h ˆ H Ω ( r = +) , ˆ N τ i = 0. Theother one is related to the spherical symmetries of themean field h ˆ H Ω ( r = +) , ˆ J x i = 0. While the positive sig-nature excitations are analyzed in Ref.13, the main focusof this Letter is wobbling excitations that belong to thenegative signature sector. The negative signature RPAHamiltonian has the formˆ H Ω [ r = −
1] = 12 X µ E µ b + µ b µ − χ X µ =1 , ˜ Q ( − )2 µ , (2)where E µ = ε i + ε j ( E ¯ i ¯ j = ε ¯ i + ε ¯ j ) are two-quasiparticle energies and b + µ ( b µ ) is a quasi-boson cre-ation (annihilation) operator [13]. Hereafter, the in-dex µ runs over ij , ¯ i ¯ j and the index µ is a projec-tion on the quantization axis z. The double stretchedquadrupole operators ˜ Q ( − )1 = ξ ˆ Q ( − )1 ( ξ = ω x ω z /ω ),˜ Q ( − )2 = η ˆ Q ( − )2 ( η = ω x ω y /ω ) are defined by means ofthe quadrupole operators ˆ Q ( r ) m = i m +( r +3) / ( ˆ Q m +( − ( r +3) / ˆ Q − m ) / p δ m ), where ˆ Q λm = ˆ r λ Y λm (m=0,1,2). The symmetry broken by the external rota-tional field (the cranking term) implies[ H Ω , ˆ J y ∓ i ˆ J z ] = ± Ω( ˆ J y ∓ i ˆ J z ) (3)(hereafter, we use in all equations ¯ h = 1). This con-dition is equivalent to the condition of the existence ofthe negative signature solution ω ν = Ω created by theoperator ˆΓ † = ( ˆ J z + i ˆ J y ) / q h ˆ J x i [19]. We recall thatˆ H add in ˆ H (Eq.(1)) breaks Eq.(3) in general. However,to meet the condition (3) we determine the strength con-stant from the requirement of the existence of the RPAsolution ω ν = Ω. As a result, the violation is unessential(see below).We solve the RPA equations of motion for normalmodes [ ˆ H Ω , ˆ O † ν ] = ω ν ˆ O † ν with ˆ O † ν = P µ ( ψ ( ν ) µ b + µ − φ ( ν ) µ b µ )(cf [13]). The solution leads to a couple of equations forunknown coefficients˜ R ν = − √ h ˆ O ν , ˜ Q ( − )1 i , ˜ R ν = i √ h ˆ O ν , ˜ Q ( − )2 i (4)Resolving these equations one obtains the secular equa-tion F ( ω ν ) = det ( D − χ ) = 0 (5)that determines all negative signature RPA solu-tions ω ν . The matrix elements D km ( ω ν ) = P µ ˜ f k,µ ˜ f m,µ C kmµ / ( E µ − ω ν ) involve the coefficients C kmµ = ω ν for k = m and E µ otherwise; ˜ f m,µ are two-quasiparticle matrix elements of operators ˜ Q ( − ) m . Amongcollective solutions there are solutions that correspond tothe shape fluctuations of the system and the rotationalmode ω ν = Ω. With aid of Eq.(3) the system for theunknown coefficients ˜ R ν , can be cast in the form similarto the classical expression for the wobbling mode ω ν = w = Ω s [ ℑ x − ℑ eff ][ ℑ x − ℑ eff ] ℑ eff ℑ eff (6)with microscopic effective moments of inertia [7] ℑ eff , = ℑ y,z + Ω S ℑ x − ℑ y,z − ω ν S/ Ω ℑ z,y + Ω S (7)that depend on the RPA frequency. Here, ℑ x = h ˆ J x i / Ω, S = P µ J yµ J zµ / ( E µ − ω ν ) and ℑ y,z = P µ E µ ( J y,zµ ) / ( E µ − ω ν ). Equation (6) does not containthe solution ω ν = Ω.We obtain quite a remarkable correspondence betweenthe experimental and calculated values for the kine-matic moment of inertia for both nuclei (see top pan-els in Fig.2). The irrotational fluid moment of inertia ℑ ( irr )1 does not reproduce neither the rotational depen-dence nor the absolute values of the experimental oneas a function of equilibrium deformations (see Fig.2).The rigid body values provide the asymptotic limit offast rotation without pairing, if shell effects are smearedout (see discussion on shell effects at fast rotation inRef.20). Evidently, the difference between the rigid bodyand the calculated kinematic moments of inertia in bothnuclei decreases with the increase of the rotational fre-quency, although it remains visible at high spins. Atvery fast rotation ¯ h Ω > . x the wobbling excitationswith different collectivity could be found from Eq.(6), if TT x i [ M e V S [ M e V S - - ]] S W [ MeV ] Yb Dy FIG. 2: (Color online) Top panels: the kinematic ℑ x = h ˆ J x i / Ω (solid line), the rigid body ℑ ( rig )1 = mAR (cid:16) − p π β cos( γ − π ) (cid:17) (dashed line) and the hy-drodinamical ℑ ( irr )1 = π mAR β sin (cid:0) γ − π (cid:1) (dash-dottedline) moments of inertia are compared with the experimentalvalues (filled squares). Experimental values ℑ x = I/ Ω areconnected by dashed line to guide eyes (¯ h Ω = E γ / ℑ eff (dashed line) and ℑ eff (dash-dotted line) for the first RPAsolution ν = 1 obtained from Eq.(5) ℑ x > ℑ eff , ℑ eff ( or ℑ x < ℑ eff , ℑ eff ). The rotationalbehavior of the effective moments of inertia for the firstRPA solution of Eq.(5) (see Fig.2) suggests that this so-lution may be associated with a wobbling mode.To identify the wobbling mode among the solutions ofEq.(5) it is instructive to introduce new variables, simi-lar to ones in [5] : r ν = ˜ R ν / ( ξA ), r ν = ˜ R ν / ( ηB ), where A = h ˆ Q + √ Q i , B = 2 h ˆ Q i . By means of Eqs.(4)and ˆ O ν =Ω ≡ ˆΓ, we obtain exact definitions for the un-knowns r Ω1 , associated with the redundant mode ω ν = Ω: r Ω1 = − / q h ˆ J x i , r Ω2 = 1 / q h ˆ J x i . With aid of thesedefinitions, exploiting the fact that the components ofthe quadrupole tensor commute, one can define the un-knowns r w = 12 q h ˆ J x i (cid:18) W W (cid:19) / , r w = 12 q h ˆ J x i (cid:18) W W (cid:19) / (8)and show (cf Ref.5) that they are associated with thewobbling mode. Here, W = (1 / ℑ eff − / ℑ x ), W =(1 / ℑ eff − / ℑ x ). It is convenient to use the variables c ν =4 h ˆ J x i r ν r ν . From the definitions of r Ω1 , , r w , it follows that c ν =Ω ≡ − , c ν = w ≡ only the secular equation for the quadrupole op-erators, Eq.(5), the condition Eq.(9) enables us to iden-tify the redundant and the wobbling modes. Note thatthe variables r ν , (or c ν ) can be only defined for nonaxialshapes. S W [ MeV ] (+,1) bands Yb n = 0 B3yrast c n S w n [ M e V ] S W [ MeV ] (+,1) bands = 0 n B3yrastB10 Dy FIG. 3: (Color online) Top panels: rotational dependenceof the negative signature RPA solutions with odd spins ( π =+ , α = 1). The redundant mode ω ν = Ω is denoted as ”0” andis displayed by the dotted line. Number in a circle denotes theRPA solution number : 1 is the first ν = 1 RPA solution etc.Different symbols display the experimental data associatedwith B1,B2...bands (the band labels are taken in accordancewith the definitions given in Ref.11). Bottom panels: therotational dependence of the coefficients c ν ∼ r ν r ν (see text)that are determined by the solutions of Eq.(5) The experimental level sequences for all observed up-to-date rotational bands in
Yb and
Dy are takenfrom Ref.11. All rotational states are classified by quan-tum number α which is equivalent to our signature r .The negative signature states ( r = −
1) correspond to α = 1 and are associated with odd spin states in even-even nuclei. All considered bands are of the positive par-ity π = +. To elucidate the structure of observed states,we define the experimental excitation energy in the rotat-ing frame ¯ hω ν (Ω) exp = R ν (Ω) − R yr (Ω) as a function ofthe rotational frequency Ω [21]. Here, the Routhian func-tion R ν (Ω) = E ν (Ω) − ¯ h Ω I ν (Ω). The energy ¯ hω ν (Ω) exp can be compared with the RPA results, ¯ hω ν (Ω), calcu-lated at a given rotational frequency.Top panels of Fig.3 display the redundant mode andfour lowest RPA solutions of Eq.(5) as a function of therotational frequency. We recall that these solutions arefound at different equilibrium deformations (see Fig.1).Indeed, in both nuclei the criteria Eq.(9) uniquely deter-mines the redundant and the wobbling modes. In Fig.3the redundant mode is manifested as a straight line (seetop panels), while the corresponding coefficient c Ω = − Yb it is known only one negative signature γ -vibrational state. The first RPA solution ( ν = 1) is anegative signature gamma-vibrational mode (with oddspins) till ¯ h Ω ≈ . h Ω ≈ . ν = 1 solution may be used as a guidelinefor possible experiments on identification of the wobblingexcitations near the yrast line. The first negative signa-ture RPA solution in Dy can be associated with thenegative signature gamma-vibrational excitations withodd spins. After the transition from the axial to non-axial rotation, at ¯ h Ω ≈ . h Ω ≈ . , band ac-cording to Ref.12). On this basis we propose to considerthe B10 band as the wobbling band in the range of val-ues 0 . < ¯ h Ω < . h ≤ I ≤
39 ¯ h for thisband). Note that the band B10 contains the states with31 ¯ h −
53 ¯ h . However, our conclusion is reliable only forthe states with I = 33¯ h −
39 ¯ h (or up to ¯ h Ω < . h Ω ≈ . h Ω > . I >
39 ¯ h for the B10 band) one may expect theonset of octupole deformation in the yrast states. Theoctupole deformation is beyond the scope of our analysisand will be discussed in forthcoming paper.In the microscopic approach [5] the electric transitionprobabilities from the wobbling states take the same formas in the macroscopic rotor model [3]. Indeed, for inter-band transitions (from one-phonon to yrast states) wehave (cf [13, 19]) B ( E I ν → I ± yr ) ≈ (10) (cid:12)(cid:12)(cid:12) i √ h ˜ O ( − )( E )2 , ˆ O † ν i /η ∓ √ h ˜ O ( − )( E )1 , ˆ O † ν i /ξ (cid:12)(cid:12)(cid:12) Here, ˆ M ( E ) = ( eZ/A ) ˆ M . In virtue of Eqs.(4),(8), onecan obtain for the quadrupole transitions from the one-phonon wobbling state to the yrast states B ( E I w → I ± yr ) ≈ (11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) W W (cid:17) A ( E ) ∓ (cid:16) W W (cid:17) B ( E ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / (4 h ˆ J x i )For intraband transitions we have (see [19] and Eq.(43)in Ref.13) B ( E Iν → I − ν ) ≈ (cid:12)(cid:12)(cid:12) √ h ˆ Q ( E )0 i − h ˆ Q ( E )2 i (cid:12)(cid:12)(cid:12) (12)Expressions (10), (11), (12) are obtained in high spinlimit I ≫
1. To understand a major trend of thequadrupole transitions, we employ relations from thepairing-plus-quadrupole model : mω ǫ cosγ ′ = χ h Q i , mω ǫ cosγ ′ = − χ h Q i (cf Ref.2). By means of theserelations and a definition of the quadrupole isoscalarstrength χ = 4 πmω / h r i ≈ πmω / (3 AR ) ( R ≈ . A f m ) one obtains from Eq.(11) B ( E I n w = 1 → I ± yr ) ≈ (13) Θ ǫ " W W ! sin ( π − γ ′ ) ± W W ! sinγ ′ / h ˆ J x i , where Θ = (9 / π ) e Z R . Equation (13) yields thefollowing selection rules for the quadrupole transitionsfrom the one-phonon wobbling band to the yrast one (for W , > a ) − o < γ < B ( E I n w → I − yr ) > B ( E I n w → I + 1 yr ) (14) b ) 0 < γ < o : B ( E I n w → I + 1 yr ) > B ( E I n w → I − yr )For the intraband transitions we obtain B ( E I n w → I − n w ) ≈
12 Θ ǫ cos ( π − γ ′ ) (15)One observes from Eq.(15) that for the transitions alongthe yrast line ( n w = 0) the onset of the positive (nega-tive) values of γ -deformation leads to the increase (de-crease) of the transition probability along the yrast line.Moreover, the decay from one-phonon wobbling statesto the yrast line R ( ± ) = B ( E I n w = 1 → I ± yr ) /B ( E I n w → I − n w ) ∼ /I ( h ˆ J x i ≈ I ≫ γ . However, the rotational evolu-tion of the nonaxiality may affect this tendency. We pre-dict almost a constant behaviour for the ratio R ( − ) ≈ . h Ω > . h Ω ∼ . ∼ e f m ( ∼ e f m ) in Yb(
Dy). We obtain a good correspondence between theshape evolution and the selection rules (14) for both nu-clei (see top panels of Fig.4 and Fig.1). The transitionprobabilities, Eq.(10), are calculated by means of the ψ ( ν ) µ and φ ( ν ) µ phonon amplitudes. The results for thefirst negative signature RPA solution (which is associ-ated with a wobbling mode) are compared with thoseobtained with the aid of the effective moments of inertia(see Eqs.(7),(11)). Evidently, if the ”spurious” solution(the redundant mode) would be not removed from Eq.(5),two estimations (10) and (11) (based on different secu-lar equations (5) and (6), respectively) would producedifferent numerical values. A good agreement betweenboth results (see Fig.4) is the most valuable proof of theself-consistency of our calculations. The observed negli-gible differences are due to the approximate fulfillmentof the conservation laws (3), caused by the additionalterm. In Yb, starting from ¯ h Ω ∼ . I' = I - 1I' = I + 1 Dy B ( M ); II ' y r n = [ m N B ( E ); II ' y r n = [ e ]f m I' = I - 1I' = I + 1 r = -1 Yb I' = I - 1I' = I + 1 r = -1
I' = I - 1I' = I + 1 S W [ MeV ] FIG. 4: (Color online) B(E2)- (top) and B(M1)- (bottom) re-duced transition probabilities from the one-phonon bands tothe yrast band. The negative signature phonon band is de-scribed by the first RPA solution ( r = − ψ ( ν =1) µ and φ ( ν =1) µ phonon ampli-tudes, are connected by solid lines. The results obtained bymeans of Eqs.(11), (17) (with the aid of the variables W , )are connected by thin lines, starting from the rotational fre-quency ¯ h Ω ∼ . r = −
1) with spin I to the yrast stateswith spin I ′ = I − h Ω ≥ . transition point), the negative signature phonon bandchanges the decay properties. The interband quadrupoletransitions from the one-phonon state to the yrast oneswith a lower spin dominate in the decay (∆ I = 1, thecase Eq.(14), a)). Similar results for the first negativesignature one-phonon band are obtained in Dy. Atlow angular momenta (¯ h Ω ≤ . I ′ = I ± I is the angular momentum ofthe excited state). At ¯ h Ω ∼ . γ -deformation. In turn, the phonon band de-cays stronger on the yrast states with angular momenta I ′ = I − I = 1, the case Eq.(14), a)), starting from¯ h Ω ≥ . B ( M I ν → I ± yr ) ≈ (cid:12)(cid:12)(cid:12) i h ˆ M ( M )1 µ =1 , ˆ O † ν i ∓ h ˆ M ( M )1 µ =0 , ˆ O † ν i(cid:12)(cid:12)(cid:12) (16)Here, ˆ M ( M )1 µ = µ N √ P Ai =1 ( g ( i,eff ) s [ σ ⊗ Y l =0 ] µ + g ( i,eff ) l [ l ⊗ Y l =0 ] µ ) is a magnetic dipole operator; µ N is the nucleon magnetons, g ( eff ) s , g ( eff ) l are the spinand orbital effective gyromagnetic ratios, respectively.Our results evidently demonstrate the dominance of B ( M I n W → I − yr ) (see bottom panels in Fig.4)in both nuclei. In the rigid rotor model, one can obtainfor the magnetic transitions from the wobbling to yraststates B ( M I ν → I ± yr ) ≈ h ˆ J x i ( √ W ∓ √ W ) √ W W ×× (cid:12)(cid:12)(cid:12)(cid:12) h ˆ M ( M )1 ν =1 [ r = +1] i (cid:12)(cid:12)(cid:12)(cid:12) (17)The full derivation will be presented elsewhere. Note thatthe dipole magnetic moment h ˆ M ( M )1 ν =1 [+] i increases quitedrastically, if a nucleus is undergoing the backbending[22]. For the wobbling states with W , >
0, Eq.(17)yields B ( M I n W → I − yr ) > B ( M I n W → I +1 yr ) (18)At high spin limit I ≫
1, the microscopic and rigid bodyvalues of the variables W , are very close. Thus, the macroscopic model supports the results of microscopiccalculations for the magnetic transitions. It appears thatthe magnetic transitions with ∆ I = 1¯ h always dominatefrom the wobbling to the yrast states, independently fromthe sign of the γ -deformation of rotating nonaxial nuclei.In summary, we predict that the lowest excited nega-tive signature and positive parity band in Yb trans-forms to the wobbling band at ¯ h Ω ∼ . h Ω > . B ( E I w → I − yr ) /B ( E I w → I + 1 yr ) > Dy after the backbend-ing as well, at ¯ h Ω > . , band in Dy [12], are wobbling excitations at therotational frequency values 0 . < ¯ h Ω < . γ -deformation. It is quite desirable, however, tomeasure the interband B ( E I = 1¯ h magnetic transitionsfrom the wobbling to the yrast states, independently fromthe sign of the γ -deformation. Acknowledgments
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