Transverse wobbling

F. Donau1, ∗ and S. Frauendorf2, †

1Institut fur Strahlenphysik, Helmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Germany2Department of Physics, University of Notre Dame, South Bend, Indiana 46556

PACS numbers: 21.10.Re, 23.20.Lv, 27.70.+q

I. INTRODUCTION

II. TRANSVERSE AND LONGITUDINALWOBBLING

Although the first description of the collective wob-bling mode was based on even-even nuclei, all exam-ples to date have been observed in odd-A nuclei. Infact the unpaired, highly-aligned i13/2 quasiproton hasproven to play a pivotal role for the observation of wob-bling in lutetium and tantalum nuclei. As first suggestedin Ref. [7], the i13/2 quasiproton drives the nuclear shapetoward a larger deformation than the other quasiparticlestates located near the Fermi surface. At these larger de-formations, a significant gap in the quasineutron energylevels opens at N = 94 and stabilizes an asymmetricshape with γ ≈ 20◦.

Following the initial discovery of the first wobblingstructure in 163Lu [8], particle-rotor model calculationswere primarily used to describe features of the exoticwobbling mode, see Refs. [9–12]. Random-phase approx-imation calculations were also able to reproduce experi-mental results of the wobbling bands, see Refs. [13–16].In particular, the unusually large transition strength ra-tios B(E2)out/B(E2)in could be described in both mod-els. However, the previous particle-rotor model phonondescription of the wobbling energy has not been able toreproduce a systematic feature of the known wobblingbands. Figure 1 displays the energy splitting betweenthe wobbling band and the πi13/2 sequence upon whichthe wobbling structure is based. In all four cases wherethe wobbling energies are known, it is found that the en-ergy splitting reduces with spin. That is, the wobblingband is becoming lower in energy, with respect to theπi13/2 band, as spin increases. As outlined below, withthe previous assumptions in the particle-rotor calcula-tions, the wobbling sequence should actually increase inenergy with spin. However, we propose a different set ofassumptions that can not only reproduce the same large

FIG. 1: Energy splitting between the n = 1 wobbling bandand the πi13/2 n = 0 sequence in 163,165,167Lu and 167Ta.

∗Electronic address: doenau@hzdr.de†Electronic address: sfrauend@nd.edu

B(E2)out/B(E2)in ratios, but also describe the reduc-tion in energy splitting between the wobbling and πi13/2

sequences.

A. Simple, transverse, and longitudinal Wobblers

Bohr and Mottelson [1] described the motion of a tri-axial nucleus and pointed out the presence of wobblingexcitations for an even-even system. Starting from theHamiltonian of a rigid triaxial rotor they derive the ener-gies for the wobbling excitations as the small amplitudelimit of the nutation of the angular momentum vector

H = A3I(I + 1) +(

n +12

)~ωw, (1)

where the wobbling phonon energy ~ωw is equal to

~ωw = 2I[(A3 − A1)(A2 − A3)]1/2, (2)

and the Ak parameters are related to the moments ofinertia (Jk) about the three principle axes as

Ak =~2

2Jk. (3)

Here, it is assumed that the 3-axis is the axis withthe maximal moment of inertia. However, to account forthe presence of a highly-aligned unpaired quasiparticle,the triaxial rotor Hamiltonian must be replaced by thetriaxial particle rotor Hamiltionian

H = A3(J3 − j1)2 + A1(J1 − j1)2 + A2(J2 − j2)2, (4)

XXXXX ist das richtig mit j1, j2 ???????where jk is the angular momentum associated with theunpaired quasiparticle and Jk is the total angular mo-mentum about the respective axis.

1. Semiclassical analysis

The consequences of the presence of the unpairedquasiparticle can be best understood in terms of the clas-sical motion of the angular momentum vector J . Its or-bits are determined by the conservation of angular mo-mentum

J2 = J21 + J2

2 + J23 = I(I + 1) (5)

2

and energy

E = A3(J3 − j)2 + A1J21 + A2J

22 , (6)

where for simplification we assume that the angular mo-mentum of the unpaired quasiparticle is rigidly alignedwith axis 3. The classical orbit of J is the intersection ofthe angular momentum sphere (5) with the energy ellip-soid (6).

Let us first consider the case of a simple triaxial ro-tor without the additional quasiparticle, i. e. j = 0 inequation (6). Fig. 2 illustrates three types of orbits fora given angular momentum value J , which is the radiusof the angular momentum sphere (5)). We assume forthe rotational inertial parameter the ratios A1 = 6A3

and A2 = 3A3. The size of the energy ellipsoid increaseswith the energy E. The yrast line corresponds to uniformrotation about the 3-axis with the maximal moment ofinertia. The upper panel shows an orbit just above theyrast line, which represents the harmonic wobbling mo-tion as discussed by Bohr and Mottelson. The middlepanel shows the orbit called separatrix. It has the en-ergy of the unstable uniform rotation about the 2-axiswith the intermediate moment of inertia. The frequencyof this orbit is zero, because it takes infinite long timeto get to or to depart from the point of the labile equi-librium (uniform rotation about the 2-axis). The orbitswith larger energy than the separatrix revolve the 1-axis.The lower panel shows one example.

Quantum mechanically one has to take into accountthe invariance of the rotor with respect to rotations byπ/2 about its principal axes. It has the consequence thatthe states corresponding to revolution about the 3-axishave a signature quantum number α = I+even number,which is fixed by the quantum number I of the angularmomentum. The signature alternates between 0 and 1,starting with 0 for the yrast line. Above the separatrix,there are two classical orbits with the same energy, whichrevolve the positive and negative 1-half axes. Quantummechanically they make an even and odd linear combina-tion, which have signature 0 and 1. The rotational bandsappear as ∆I = 1 sequences above the separatrix and as∆I = 2 sequences below.

In order to describe the motion of J we introduce thecanonical variables J3 and ϕ,

J1 = J⊥cosϕ, J2 = J⊥sinϕ, J⊥ =√

J2 − J23 , (7)

where ϕ is the angle of the projection of J onto the 1-2plane. The phase space for the one-dimensional motionon the angular momentum sphere is −π ≤ ϕ < π and−J ≤ J3 ≤ J . Fig. 3 shows several orbits in phasespace. Orbit 1 corresponds to wobbling about the 3-axis.Orbit 4 is the separatrix. Orbit 7 corresponds to wob-bling about the 1-axis. According to classical mechanics,the period of the orbit T = 2πdS/dE, where S is thephase space area enclosed by the orbit. The orbits in Fig3 are calculated for energies that increase by the same

FIG. 2: (Color online) Angular momentum sphere and en-ergy ellipsoid for a simple triaxial rotor with the rotationalconstants A1 = 6A3 and A2 = 3A3. The intersection line isthe classical orbit of the angular momentum vector relativeto the body fixed frame. The three panels correspond to theorbits 1, 4, and 6 in Fig. 3.

3

-Π -Π

20 Π

2Π

-2-1

012-Π -

Π

2 0Π

2 Π

-2-1012

Φ

J 31

23

45

67

FIG. 3: Classical orbits of the angular momentum vector for asimple triaxial rotor with the rotational constants A1 = 6A3

and A2 = 3A3. The unit of angular momentum is j. Theangular momentum is J = 2j. The energy difference betweenthe orbits is 0.4, where the energy unit is A3j

2

.

amount. As seen, the difference ∆S is maximal near theseparatrix, which means the period T has a maximumand the frequency ω = 2π/T has a minimum. In classi-cal mechanics the energy increases continouusly, and thefrequency of the separatrix goes to zero, as mentionedabove. In quantum mechanics the increase of the energyis discrete, such that ∆S = h, i. e. the energy distancebetween adjacent levels has a minimum at the separatrix.A stable minimum is defined at the point where J3 = Jabout which small oscillations describe the wobbling mo-tion. The derivation of the energy of the wobbling mode(2) is given in [1]. Figs. 4.33 and XXX in Bohr and Mot-telson [1] show the quantal spectrum of a triaxial rotor.Below the separatrix the one sees the ∆I = 2 wobblingbands, which come together the when approaching theseperatrix and merge into ∆I = 1 sequences above theseparatrix.

In the presence of the odd quasiparticle one mustdistinguish between the case of its angular momentumj = (j1, j2, j3) being aligned with the axis of the largestmoment of inertia, which we refer to as ”longitudinal”,and the case of j being perpendicular to this axis, whichwe refer to as ”transverse”. The longitudinal case issimilar to the simple rotor, only that the energy ellipsoidis shifted by j upwards. The yrast line corresponds touniform rotation about the 3-axis and the lowest excitedstates to wobbling about this axis, similar to the upperpanel of Fig. 2. The orbits are shown in Fig. 4. Thewobbling bands have alternating signature ±j, startingwith J at the yrast line. For the transverse transverserotor we assume that the 2-axis has the largest momentof inertia, and the 1-axis the smallest. We use thesame ratios as for the simple rotor, i. e. the rotationalconstants A1 = 6A2 and A3 = 3A2.

-Π -Π

20 Π

2Π

-2-1

012-Π -

Π

2 0Π

2 Π

-2-1012

Φ

J 3

FIG. 4: Classical orbits of the angular momentum vector for alongitudinal triaxial rotor with the rotational constants A1 =6A3 and A2 = 3A3. The angular momentum **?? I = 2j.The unit of angular momentum is j. The energy differencebetween the orbits is 0.3, where the energy unit is A3j

2

.

**** Note the particle angular momentum j is alignedalong the 3-axis like before.*******

As illustrated in Fig. 5, the yrast line consists of twopieces. At low angular momentum it corresponds to ro-tation about the 3-axis. The energy ellipsoid touches theangular momentum sphere at the point J3 = J on the3-axis. The yrast energy is E = A3(J − j)2. The lowenergy orbits represent wobbling about the 3-axis. Fig.7 displays the intersection of the lowest orbit in Fig. 8,which shows the orbits in phase space. At the criticalangular momentum Jc = jA3/(A3 − A2) the rotationalaxis of the yrast line flips to the direction of the pointJ1 = 0, J2 =

√J2 − J2

c , J3 = Jc, where the energy ellip-soid touches the angular momentum sphere from inside.The upper panel of Fig. 9 displays the intersection ofthe energy ellipsoid with angular momentum sphere fora slightly higher energy. In Fig. 10, it is the first orbitenclosing the touching point. At E = A3(J − j)2, thereis the separatrix, which is illustrated in the middle panelof Fig. 9. Above the separatrix the orbits revolve aboutthe 3-axis as shown in the lower panel of Fig. 9.

For J < Jc, the yrast line is E = A3(J − j)2. It con-tinues as separatrix for J > Jc. As discussed above forthe simple rotor, the classical frequency of the separatrixis zero. This means that the frequency of the wobblingmode with infinitesimal amplitude goes to zero at J = Jc,where uniform rotation about the 3-axis becomes unsta-ble, and the new branch of the yrast line starts. Thatis, the wobbling frequency decreases with J such thatit reaches zero at Jc. Quantum mechanically, the yraststaes have signature j for J < Jc and the first wobblingstate has signature −j. It encloses the fixed area h inphase space, which means its energy decreases with J . It

4

1.0 1.5 2.0 2.5 3.002468

J

Ene

rgy

S

FIG. 5: Energy of the yrast line and the separatrix (S) forthe transverse rotor with the rotational constants A1 = 6A2

and A3 = 3A2. The unit of angular momentum is j, and theenergy unit is A3j

2.

1.0 1.5 2.0 2.5 3.002468

J

Ene

rgy

S

FIG. 6: Energy of the yrast line and the separatrix (S) forthe transverse rotor with the rotational constants A1 = 6A2

and A3 = 3A2. The unit of angular momentum is j, and theenergy unit is A3j

2.

merges with the yrast with yrast line, which becomes a∆I = 1 sequence for J > Jc. In case of the longitudinalrotor, there is no bifurcation of the yrast line, which isreflected by an increase of the wobbling frequency withJ .

2. Harmonic wobbling

Now we consider small amplitude wobbling vibrationsabout the 3-axis. The angular momentum of the odd par-ticle is assumed to be firmly aligned with the 3-axis andcan be considered as a number. Then the Hamiltonianbecomes

H = A3(J3 − j)2 + A1J21 + A2J

22 , (8)

FIG. 7: (Color online) Angular momentum sphere and energyellipsoid for a transverse triaxial rotor with the rotational con-stants A1 = 6A2 and A3 = 3A2. The intersection line is theclassical orbit of the angular momentum vector relative to thebody fixed frame. The figure correspond to the lowest energyorbit in Fig. 8.

-Π -Π

20 Π

2Π

-1

0

1-Π -

Π

2 0Π

2 Π

-1

0

1

Φ

J 3

FIG. 8: Classical orbits of the angular momentum vector fora transverse triaxial rotor with the rotational constants A1 =6A2 and A3 = 3A2. The angular momentum J = 1.25j. Theunit of angular momentum is j. The energy increases from thetop to the bottom. The energy difference between the orbitsis 0.3, where the energy unit is A3jI

2****??. The separatrixin the lower part of the figure is additionally placed at itsenergy.

5

where j is a number. Using

J3 =√

J2 − J21 − J2

2 ≈ J − 12

(J2

1

J+

J22

J

), (9)

with J2 = I(I + 1), the Hamiltonian becomes

H = A3(J − j)2 +

(A1 − A3(J))J21 + (A2 − A3(J))J2

2 , (10)

A3(J) = A3

(1 − j

J

)(11)

The Hamiltonian has the form of the simple triaxialrotor Hamiltonian, except that A3 is replaced by theJ-dependent rotational constant A3(J) = A3(1 − j/J).Therefore, one has only to replace A3 by A3(J) in theexpressions derived by Bohr and Mottelson [1]. (The 1-axis has been exchanged by the 3-axis.) The wobblingfrequency becomes

~ωw = 2J [(A1 − A3(J))(A2 − A3(J))]1/2,

A3(J) = A3

(1 − j

J

), J =

√J(J + 1). (12)

The E2-transition probabilities are

B(E2, n, I → n, I ± 2) =5

16πe2Q2

2, (13)

B(E2, n, I → n − 1, I − 1) =5

16πe2 n

J(√

3Q0x −√

2Q2y)2, (14)

B(E2, n, I → n + 1, I + 1) =5

16πe2 n + 1

J(√

3Q0x −√

2Q2y)2, (15)

where [x

y

]=[12

(α

(α2 − β2)1/2± 1)]1/2

(16)

α =(A1 + A2 − 2A3(J)

)J, β = (A1 − A2) J, (17)

and Q0 and Q2 are the quadrupole moments of the tri-axial density relative to the 3-axis.

The harmonic approximation is only valid as long asJ1 << J and J2 << J . The J2 component has thelargest amplitude, which is reached at its classical turningpoint where J1 = 0. With the energy of the first wobblingexcitation of 3/2~ωw one finds for the turning point

J22 = 3J

[A1 − A3

A2 − A3

]1/2

<< J2, (18)

Eqn. (12) can be rewritten as

~ωw =j

J3

[(1 +

J

j

(J3

J1− 1))(

1 +J

j

(J3

J2− 1))]1/2

.(19)

FIG. 9: (Color online) Angular momentum sphere and energyellipsoid for a transverse triaxial rotor with the rotational con-stants A1 = 6A2 and A3 = 3A2. The intersection line is theclassical orbit of the angular momentum vector relative to thebody fixed frame. The three panels correspond to the orbitwith the smallest energy, the separatrix, and the next orbitin Fig. 10.

6

-Π -Π

20 Π

2Π

-3-2-1

0123-Π -

Π

2 0Π

2 Π

-3-2-10123

Φ

J 3

FIG. 10: Classical orbits of the angular momentum vectorfor a transverse triaxial rotor with the rotational constantsA1 = 6A2 and A3 = 3A2. The angular momentum I = 3j.The unit of angular momentum is j. The energy differencebetween the orbits is 0.4, where the energy unit is A3I

2. Thedot indicates the yrast ”orbit”, from where energy increases.The separatrix in the lower part of the figure is additionallyplaced at its energy.

Within the previous particle-rotor model studies [10] theassumption of A3 < A1, A2 (J3 > J1,J2) was made,which is the longitudinal wobbler in our terminology. Itis apparent that with this assumption both parenthesisby which J in Eqn. (19) is multiplied are always positive,such that the wobbling energy will increase with spin.Thus, the wobbling band must increase in energy withspin relative to the πi13/2 band; exactly the opposite ofthe trend observed in Fig. 1. This is the reason why pre-vious studies could not properly reproduce the reductionin energy splitting between the two bands.

We suggest that the observed wobbling excitations areof the transverse type, i. e. A2 < A3, A3 < A1

(J2 > J3, J3 > J1). Within Eqn. (19), the factor1 + J(J3/J2 − 1)/j decreases with J . The wobbling en-ergy will decrease for sufficiently large J , because it willbe zero for Jc = jJ2/(J1 − J2),+++**muss heissen Jc = jJ2/(J2 −J3),*************which is the previously discussed critical angular momen-tum where the separatrix bifurcates from the yrast line.There the yrast and the wobbling bands will merge into asingle ∆I = 1 sequence, which is based based upon tiltedrotational axis of the upper part of the yrast line of thetransverse rotor. Clearly with ~ωw reducing with spin,the energy splitting between the wobbling and πi13/2

bands will decrease, as is experimentally found as seenin Fig. 1.

Tanabe and Sugawara-Tanabe [11, 12] considered theless restrictive approximation that the odd particle is notrigidly fixed to the core. Its angular momentum may ex-ecute small amplitude oscillations. They find a moderatecoupling between the two oscillators, such that the lowest

states may be classified as being predominantly a wob-bling mode of the core or an excitation of the odd par-ticle. Our discussion above concerns only the first type,the wobbling modes.

III. MICROSCOPIC CONSIDERATIONS

A. Cranking moments of inertia and particle-rotorcalculations

In the considered case, the i13/2 quasiproton is a nearlyperfect particle (chemical potential in the low part of theintruder shell), which aligns with the short axis. Thisorientation minimizes the energy of the attractive short-range interaction between the orbital and the triaxialcore, because it correspond to maximal overlap. In thecase of a quasiparticle that is a nearly perfect hole (chem-ical potential in the high part of the intruder shell) thepreferred orientation is the long axis. Qualitatively, it isexpected that the medium axis has the largest momentof inertia, because the deviation of the density distribu-tion from rotational symmetry is for this axis larger thanfor the other axes. Hence, the transverse wobbler is thetypical case. This is also the case for the odd quasipar-ticle being of hole type. Previous studies [9–12] assumedA3 < A2, A1, i. e. the case of the longitudinal wobbler.Accordingly, they found that the wobbling frequency in-creased with I, in contrast to experiment. Ref. [12] useda common scaling factor for all moments of inertia whichincreased the moment of inertia of the yrast line. Thecorresponding decrease of the scale of Ak results in aproportional decrease of ~ωw.

We studied the transverse wobbling scenario for 163Lu.This nucleus was chosen as it is was the first case ofwobbling ever identified and it is perhaps the best ex-ample of this phenomenon. The ratios between the mo-ments of inertia for classical irrotational or rotational flowcan not be considered as quantitatively acceptable. Forthis reason we calculated them microscopically by meansof cranking the Z = 70, N = 92 core about its threeprincipal axes. Using the TAC code [21] with a typicalTSD of ε = 0.39 γ = 17o and zero pairing, we foundJ3/~2=70 MeV, J2/~2=77 MeV, and J1/~2=13 MeVfor the short, medium, and long axis, respectively Theratios differ qualitatively from the ones assumed in theprevious work [10, 10–12]. As expected, the moment ofinertia is largest for the medium axis, i. e. the the wob-bler is transverse. We multiplied the calculated momentsof inertia by a common I-depend factor, which was cho-sen such that************ ????????? *************???We diagonalized the particle-rotor Hamiltonian (4) nu-merically using the code by [? ] assuming the samedeformation for the potential and no pairing as usedfor the TAC calculations. A scaling factor was multi-plied to all moments of inertia. Fig. ?? shows the ex-perimental wobbling frequency determined as ~ωw(I) =

7

E1(I) − (E0(I − 1) + E0(I + 1))/2, where E0(I) are theyrast energies and E1(I) the energies of the one phononwobbling band. For the two phonon band the energies2~ωw = E2(I)−E0(I) are shown. The particle-rotor val-ues were calculated in the same way. The small differencebetween J3 and J3 ?????????? ....

In addition, the calculated B(E2)out/B(E2)in ratiosfor the n = 1 band are shown in Fig. ?? along withthe experimental values from Ref. [8]. One may seea satisfactory agreement between the two values whichis lending further credence to the tilted wobbling de-scription. We mention that a decline in the ratiosB(E2)out/B(E2)in with spin is predicted by both thetilted *********?????********** wobbling assumptionand the previous assumptions made in the particle-rotorcalculations (see Fig. 5 in Ref. [8]). However, experimen-tal values measured in 163Lu [8], 165Lu [18], 167Lu [19],and 167Ta [? ] remain constant with spin in each case.Constant quadrupole deformation was assumed in bothcases, and as measured in 163Lu, there appears to be adecrease in the quadrupole moment with spin [20]. TheB(E2)in would reduce in this case, which may accountfor the discrepancy observed between the models and theexperiment.

B. Quasiparticle random phase approximation(QRPA)

The QRPA is the adequate microscopic approach fordescribing the wobbling motion in terms of correlatedtwo-quasiparticle excitations. This method has beenalready applied by several authors [13, 23–25] to anal-yse measured energies and electromagnetic transitionrates of wobbling bands. Our calculations concentratespecifically on the case of the i13/2 TSD bands in 163Luwhich has the most complete set of data for the nuclearwobbling motion. Our RPA calculations are based ona quadrupole-quadrupole (QQ) interaction the averagepart of it which is producing the deformed quadrupolepotential and the residual part is driving the wobblingexcitations.Firstly, we start with a fixed strength of the QQ inter-action and calculate the shape parameters (ε, γ) of thequadrupole potential selfconsistently for each frequencyand perform the QRPA with these parameters.Secondly, the reversed case is studied, i.e. the deforma-tion parameters (ε, γ) are chosen arbitrary, for instance,from Nilsson-Strutinsky minimization as in Ref.[20] andderive afterwards the strength of residual QQ interactionby restoring the rotational invariance of the resultingHamiltonian. This was applied e.g. in Ref.([25]) for aQRPA with a deformed Woods-Saxon potential.Thirdly, we investigate how additional residual inter-action terms in the QRPA can influence the wobblingbands. This is because a more quantitative descriptionof both the experimental wobbling frequencies and themeasured E2/M1 properties turned out to be difficult.

In particular, the fact that the measured ∆I=1 cross-over transitions show predominantly E2 character witha rather weak M1 admixture is not obtained in theframework of the residual interactions used so far.

A. Selfconsistent QRPA (sc QRPA)

The theoretical framework of the QRPA calcula-tions is formally similar to the one in a recent study ofchiral vibrations [27]. The Hamiltonian is

H =∑

τ=π,ν

[ h◦τ − ∆τ (P †

τ + Pτ ) − λτ Nτ ] − ωJ1

− κ0

2

∑m=−2,2

(−1)mQmQ−m + Vadd

. (20)

The operator h◦τ is the spherical part of the Nilsson

Hamiltonian. The isospin index τ = π, ν distinguishesthe neutron and proton contributions, respectively. Theterm ∆τ (P †

τ + Pτ ) accounts for the pair field where P †τ

and Pτ mean the familiar monopole pair operators. Thegap parameters ∆τ of these fastly rotating states areconsidered to be reduced: below the cranking frequencyω=0.45 MeV we take ∆π=0.45 MeV for the proton gapand ∆ν=0.35 MeV for the neutron gap, and ∆τ=π,ν = 0above. The term λτ Nτ implying the particle numberoperators Nτ is introduced as usual to accomplish thatwe get for the average particle numbers ⟨Nπ⟩ = Z and⟨Nν⟩ = N , respectively by appropriate choice of theFermi energy λτ . For describing the rotational proper-ties of the TSD band we add in Eq.(20) a one-dimensionalcranking term −ω J1 . This corresponds to the assump-tion that the mean direction of the angular momentum isnot tilted, i.e. is aligned along a principal axis which wehave chosen to be the 1-axis. Because of technical rea-sons we shall nevertheless use the 3D-TAC model even ifthe cranking axis is not tilted in this case. The next termis an isoscalar quadrupole-quadrupole (ISQQ) interactiondefined by the mass quadrupole operators Qm = Qm(π)+Qm(ν) with Qm(τ) =

√4π/5 (r/b◦)2Y2m(τ) implying

the oscillator length b◦ = 1.01A1/3. When requiring self-consistency then the strength constant κ0 determines theshape (ε, γ) of the deformed potential v that is generatedthrough the mean field contribution of the ISQQ inter-action

v = v(ε, γ) = κ0⟨Q0⟩Q0 + κ0⟨Q2⟩(Q2 + Q−2). (21)

The reference state ⟩ is the short hand notation for theTAC state |ω, ε, γ⟩ of the πi13/2 TSD band refering to thedeformation parameters (ε, γ) of the modified oscillatormodel [28]. Denoting the c-numbers ⟨Q0,2⟩ as q0,2(ε, γ)the selfconsistency conditions demand searching the de-formation parameters which for a given frequency ω sat-isfy the relations

κ0⟨Q0⟩ ≡ κ0q0(ε, γ) = ~ω0 2/3 ε cos γ

κ0⟨Q2⟩ ≡ κ0q2(ε, γ) = −~ω0 2/3 ε sin γ/√

2. (22)

8

TABLE I: Equilibrium values of the deformation parameters(ε, γ) in the frequency region ω = 0.15-0.50 MeV/~. Thestrength parameter of the ISQQ interaction is κ0= 0.0196MeV

ω(MeV/~) ε γ(deg)

0.15 0.398839 9.248

0.20 0.397926 9.362

0.25 0.396632 9.486

0.30 0.394788 9.631

0.35 0.392064 9.798

0.40 0.387658 9.977

0.45 0.381236 11.575

0.50 0.377065 11.619

Note that the above conditions lead to a stable equi-librium shape only if one renders the volume conserva-tion by taking the frequency factor as constant ~ω0 =41A−1/3 . Collecting the mean field part from the Hamil-tonian H, Eq.(20), one obtains the one of the 1D-TACmodel h′ = h − ωJ1 [21] where h is given by

h = h◦ − ∆τ (P † + P ) − λN −

− ~ω0

23

ε ( cos γQ0 −sin γ√

2(Q2 + Q−2) ). (23)

The last term Vadd

in Eq.(20) stands for additional

0,37 0,38 0,39 0,40 0,41 0,426

8

10

12

14

(de

g)

0,01,0E-030,0100,0250,0500,100,150,200,250,300,36

E (MeV)

FIG. 11: (Color online) Total routhian surface for the TSDconfiguration in 163Lu at ω = 0.45 MeV/~.

residual interaction terms to be studied below.The diagonalization of the TAC hamiltonian h is done inan oscillator basis with the quantum numbers {n, l, j, m}including the orbits of the three main shells n = 4 − 6.The search for the equilibrium needs to be performedwith diabatic tracing (c.f. [21]) the selected (πi13/2, νg)configuration of the TSD band. The strength of theISQQ interaction κ0= 0.01960 MeV has been adaptedonce at ω = 0.15 MeV/~ such that the triaxial equilib-rium shape comes close to the suggested values (ε, γ) ≈

(0.4,+20◦ ) of the experimental TSD band [20]. The

FIG. 12: (Color online) Experimental and calculated mo-ments of inertia of the TSD band in 163Lu. The calculatedmoment of inertia is J1 = ⟨J1⟩/ω.

resulting shapes for the interval ω=0.15-0.50 of thecranking frequency are presented in table I. It is seenthat for the ISQQ interaction the actual triaxiality γ ≈9-12◦ is lower than in Ref. [20] but close to the valuesfound by Shoji and Shimizu [25] with Nilsson-Strutinskyminimization (see discussion in Ref.[25]). The relativechange of the deformation (ε, γ) to higher rotationalfrequencies is small. Nevertheless the precision of theselfconsistency is required in the subsequent QRPAcalculation in order to derive numerically safely theexcitation energies and E2/M1 properties of the wob-bling band. Again, as pointed out already by the aboveauthors, one has to select the sector of positive γ-valuesin order to generate wobbling excitations. Above thefrequency ω=0.5 MeV/~ the s.c. solutions change tothe tilted axis regime which is not pursued in thispaper. In Fig. 11 we show the total routhian surface

0,20 0,25 0,30 0,35 0,40 0,45 0,500

1

2

3

4

exp. TAC calc.

B(

E22)

(e2 b2 )

(MeV)

163Lu

FIG. 13: (Color online) Experimental and calculatedB(E2,I→I-2) values of the TSD band.

for ω=0.45 MeV/~ as obtained by diabatic tracing theTSD configuration with the TAC code. The ISQQinteraction gives a relatively shallow minimum withinthe deformation surface. In Fig. 12 the experimentaland calculated moments of inertia J1 are compared.The calculated moments of inertia come out to be 10-15% larger than the experimental ones. The frequenciesof the TSD bands are derived by using the standarddefinition ω(I) = (E(I) − E(I − 2))/2 where I = I − 1and E(I) denote the transitional spin and the levelenergies, correspondingly. In Fig. 13 we present theexperimental B(E2) values of the I→I-2 transitions ofthe TSD g-band [20] and the ones calculated with thes.c. TAC model. The polarisation charges ep=(1+Z/A)eand en=Z/A e were adopted for representing the protons

9

and neutron parts of the electric quadrupole operator.

With the results of the s.c. TAC calculation theQRPA is performed following the general formalism asoutlined in the textbooks (e.g. [29]). We mention onlythe important steps of the QRPA and refer for moredetails to our recent paper [27].First, the Hamiltonian (20) is rewritten in quasiparticle(qp) representation,

H = h′ + V4qp, (24)

where h′ is the diagonalized TAC Hamiltonian

h′ = E◦ +∑

i

eiα†i αi. (25)

The set {α†i , αi} denotes the qp operators and V4qp is

the residual qp interaction which contains the remain-ing 4qp terms of the ISQQ interaction. Then, the quasi-boson approximation α†

i α†j ⇒ b†ij is applied such that the

Hamiltonian, Eq. (24) is expressed in terms of bosons,H ⇒ HRPA, keeping only terms up to second order[29]. This Hamiltonian can be diagonalized by using theQRPA equation[

HQRPA, O†λ

]= Eλ

QRP AO†

λ (26)

which yields the phonon excitation energies EλQRP A

andthe phonon excitation operators O†

λ defined by

O†λ =

∑µ

(Xλµ b†µ − Y λ

µ bµ). (27)

The amplitudes Xλµ and Y λ

µ are found as a result of solv-ing the standard set of linear equations following fromEq.(26). The energetically lowest solution is identifiedwith the wobbling excitation where excluding the rota-tional excitation at EQRP A = ω that has to be excludedas spurious solution. Because the Hamiltonian H hasgood 1-signature symmetry we can restrict the RPA di-agonalization to the branch of negative 1-signature thatis the signature of the wobbling phonons.The E1/M1 cross-over transition amplitudes from theTSD wobbling state to the TSD g-band are calculatedin QRPA by

⟨w|Mm(E2/M1)|0⟩ = ⟨0|OwMm(E2/M1)|0⟩ (28)

where |w⟩ means the wobbling phonon state, |0⟩ denotesthe QRPA vacuum state at the cranking frequency ω andthe transition operators are

Mm(E2) = epr2pY2m(p) + enr2

nYm(n), (29)

Mm(M1) =34π

g(l)p l1m(p) + g(s)

p s1m(p) + g(s)n s1m(n).

The component m is assigned to the transition I → I −m. Further, the orbital g-factor for M1 is g

(l)p = 1 µN

for protons and 0 for neutrons. The spin g-factors g(s)p

and g(s)n are 0.7 times the values for the free proton or

neutron. The B-values are finally given as

B(E2/M1, I → I ∓ 1) =∣∣∣⟨w ∣∣∣M±1(E2/M1)

∣∣∣ 0⟩∣∣∣2 (30)

The ISQQ term in the Hamiltonian (20) is the most im-

0,20 0,25 0,30 0,35 0,40 0,45 0,50

0,1

0,2

0,3

0,4

exp. calc. ISQQ

Ewob

(MeV

)

(MeV/ )

163Lu

FIG. 14: (Color online) Excitation energy of the wobblingband in 163Lu as a function of the rotational frequency. Ex-perimental values (blue diamonds) are from [20]. QRPA cal-culation (solid line) with selfconsistent ISQQ interaction.

portant interaction term because it generates both thedeformed mean field and main part of the residual in-teraction in the QRPA. We remember that the strengthκ◦ of the ISQQ interaction is fixed by the selfconsistencyfor the whole frequency range ω =0.15-0.5 MeV/~. Thefactorized form of the ISQQ term allows us to reduce thesolution of the QRPA equation to searching the zerosof a corresponding dispersion relation which provides usthe QRPA energies E

QRP A. In Figs. (14 - 16) we present

the QRPA result for the wobbling energies and the cross-over B(E2) and B(M1) values I → I − 1 obtained withthe sc. ISQQ interaction. The B-values of the transi-tions I → I + 1 are in general at least one order smallerand therefore not displayed. The calculated wobblingenergies EQRP A(ω) follow the deceasing tendency of themeasured ones but are substantially lower approachingzero above ω=0.45 MeV which signalizes the change toa tilted rotational axis as mentioned above. The exper-imental wobbling energy decreases linearly yet continu-ing well above zero beyond ω=0.60 MeV. The calculatedB(E2) reach only the half of the measured value even ifthe latter show large error bars. The comparison of theexperimental and the calculated B(M1) values is shownin Fig. (BM11QQa) in logarithmic scale because the mea-sured one are very small. The calculated B(M1) are morethan a factor ten larger. In summary, the QRPA withthe sc ISQQ interaction reflects only partly the observedproperties of the TSD bands.

10

0,20 0,25 0,30 0,35 0,40 0,45 0,500,0

0,2

0,4

0,6

0,8

1,0

exp. calc. ISQQ

(MeV/ )

B(E2

1) (e

2 b2 )163Lu

FIG. 15: (Color online) B(E2) values of the cross-over (I →I − 1) transitions between the TSD wobbling band and theTSD ground band in 163Lu. Notations as in Fig. 14.

0,20 0,25 0,30 0,35 0,40 0,45 0,501E-3

0,01

0,1

1

exp. calc. ISQQ

(MeV/ )

log

B(M

11) (

)

163Lu

FIG. 16: (Color online) B(M1) values of the cross-over (I →I−1) transitions between the TSD wobbling band to the TSDground band in 163Lu. Notations as in Fig. 14.

B. QRPA study for arbitrary shape parameters

In this section the above selfconsistency betweenthe shape parameters and the common strength of theQQ interaction is given up. This means we use thesame deformed mean field Hamiltonian h as before inEq. (23) but with freely chosen values of (ε, γ). The lostconsistency is restored as in Ref. [25] by constructingthe residual interaction from the requirement that theresulting Hamiltonian H = h + V4qp becomes rotationalinvariant. Actually the Hamiltonian H is written as

H = h − 12

3∑m=1

κmF 2m (31)

where h is our deformed mean field Hamiltonian,Eq. (23). The residual interaction V4qp is built up fromthe operators Fm defined by the commutators with the

angular momentum operators Jm, i.e.

Fm = [h, iJm]. (32)

where the index m = 1, 2, 3 denotes the principal axes ofthe deformed potential. Apparently, the above commuta-tors shall generate again quadrupole operators due to thequadrupole deformed field in h. The strength constantsκm are determined by demanding rotational invariance:

[H, iJm] = [h − 12

3∑n=1

κnF 2n , iJm] = 0. (33)

The solution of this equation gives

κ−1m = ⟨[[h, iJm], iJm]⟩. (34)

Below the commutators relations (32, 34) are explicitlyevaluated. The final results are written in terms of thecombined quadrupole operators Q1± and Q2± defined by

Q1+ =Q1 + Q−1

i√

2, Q1− =

Q1 − Q−1√2

,

Q2+ =Q2 + Q−2√

2, Q2− =

Q2 − Q−2

i√

2. (35)

Introducing the following expressions implying the ex-pectation values of the quadrupole operators

Q =√⟨Q0⟩2 + ⟨Q2+⟩2 ,

sin γ = −⟨Q2+⟩Q

, cos γ =⟨Q0⟩Q

. (36)

then the final Hamiltonian is found to be

H = h +13

~ω◦ε

Q[

sin γ

sin γQ 2

2−+

sin (γ + 2π/3)sin (γ + 2π/3)

Q 21+

+

+sin (γ − 2π/3)sin (γ − 2π/3)

Q 21−

]. (37)

Hence, the residual interaction of Hamiltonian H for theRPA is fully determined when the mean field part h isspecified, i.e. in the case of the above deformed potentialby choosing deformation the deformations (ε, γ). Thisincludes the selfconsistent treatment of the Hamiltonian(20) which reads in the notations (35)

H = h′ − κ◦

2

∑µ=0,1±,2±

Q2µ (38)

which in comparison to the one in Eq.(37) contains ad-ditionally the terms Q 2

0and Q 2

2+that are driving the

beta-gamma vibrations. In this case one has to searchfor deformations (ε, γ)s.c. which satisfy the selfconsistentconditions (cf. Eq. (22))

κ◦

2=

13

~ω◦ε

Q, sin γ = −⟨Q2+⟩

Q= sin γ. (39)

11

According to Eq. (26) one recognizes that for theselfconsistent deformation (ε, γ)s.c. the common pref-actor in the Hamiltonian (37) becomes equal to κ◦/2and the three ratios of the sinus terms get simply one.The Hamiltonian H, Eq. (20) is from the beginningrotational invariant, i.e. the commutator relation (33)is automatically satisfied. In the Hamiltonian Eq.(37)rotational invariance is forced for any choice of thedeformation parameters (ε, γ). Therein the relativestrengths of the three interaction terms Q2

k± are not anymore equal indicating the distortion in the correspond-ing system. However, with the special values of κ1,2,3

locally a rotational symmetry is restored by satisfying inaverage the commutator relations (33).We have studied firstly the case when inserting in theHamiltonian Eq.(37) the (ε, γ) deformations given intable II which were obtained by a Nilsson-Strutinskyminimization [20]. Notice that the triaxiality of these

TABLE II: Deformation parameters (ε, γ) of 163Lu in the fre-quency region ω = 0.15-0.55 MeV/~ obtained from a Nilsson-Strutinsky (NS) minimization [20]

ω(MeV/~) ε γ(deg)

0.15 0.3815 18.75

0.2 0.3892 19.2

0.25 0.3968 19.64

0.3 0.4044 20.12

0.35 0.408 20.41

0.4 0.3991 20.72

0.45 0.3908 21.3

0.5 0.3852 21.78

0.55 0.3812 22.34

deformations is an about 10 degrees larger than thecorresponding selfconsistent values.

In Fig.(17) the resulting wobbling frequencies areshown together with the experimental values. Comparedwith the wobbling frequencies obtained above for the scISQQ model (cf. Fig. (14)) the calculation with the NSdeformations gives a more flat tendency that is not seenin the experiment, however, the break down of the RPAis slightly retarded. Furthermore, a 20 percent increaseis found for the electric cross-over transition strengthwhen comparing the B(E21) values presented in Fig. 15and Fig. 18. Otherwise no decrease is obtained for thecorresponding magnetic cross-over transition strength asseen in Fig. (16) and Fig. (19) that display the B(M11)values. Hence the larger γ-values predicted by theNilsson-Strutinsky calculation leads not to a pronouncedimprovement description of the TSD bands of 163Lu asthe selfconsistent calculation.

In Ref. [25] is argued that a substantial increase ofthe triaxiality γ with the rotational frequency ω couldexplain the observed larger strength of the cross-overE2 transitions. We tried the case of maximal triaxiality

0,20 0,25 0,30 0,35 0,40 0,45 0,500,1

0,2

0,3

0,4 calc =30 calc NS exp

(MeV/ )

Ewob

(MeV

)

FIG. 17: (Color online) Wobbling frequencies in 163Lu asa function of the rotational frequency. Experimental values(blue diamonds) are from [20]. Calculated values (solid line)are obtained with the Nilsson-Strutinsky (NS) deformationsin table II and calculated values (red diamond) for the fixeddeformation point (ϵ = 0.4, γ = 30 deg).

0,20 0,25 0,30 0,35 0,40 0,45 0,500,0

0,2

0,4

0,6

0,8

1,0 calc = 30 calc NS exp

(MeV/ )

B(E2

1) (e

2 b2 )

FIG. 18: (Color online) B(E2) values of the cross-over (I →I − 1) transitions between the TSD wobbling band and theTSD ground band in 163Lu. Notations as in Fig. 14.

γ = 30 deg and ε = 0.4. The results are presented inFigs.(17-19). Indeed, wobbling frequencies obtained withthis γ−value are enlarged and exceed the experimentalfinding such that they could be adjusted by choosing anappropriate triaxiality between 22-30 degrees. However,concerning the above mentioned descrepancy of thecross-over transition strength, the increase of γ bringspractically no further improvements of the B(E2/M1)values. Hence, all in all the use of the isoscalar QQresidual interaction has given only partly an explanationof all experimental findings together. This is the reasonto implement additional residual interaction terms fora possible perfection of the description of the TSD bands.

12

0,20 0,25 0,30 0,35 0,40 0,45 0,501E-3

0,01

0,1

1

calc = 30 calc NS exp

log

B(M

11) (

)

(MeV/ )

FIG. 19: (Color online) B(M1) values of the cross-over (I →I−1) transitions between the TSD wobbling band to the TSDground band in 163Lu. Notations as in Fig. 14.

0,20 0,25 0,30 0,35 0,40 0,45 0,500,0

0,1

0,2

0,3

0,4QQ

sc+ll

QQsc

Ew

ob (M

eV)

(MeV)

163Lu

exp

FIG. 20: (Color online) Wobbling frequencies in 163Lu asa function of the rotational frequency. Experimental values(blue diamonds) are from [20]. Calculated values are obtainedwith QRPA using (solid line) selfconsistent ISQQ interactionand (green line) additionally LL interaction (see Eq. (41))

C. Additional residual interactions

Before presenting below results of our QRPA cal-culations with additional interaction terms a note aboutthe removal the spurious rotational modes is in order.When adding interaction terms the strength constantsof which are not fixed by selfconsistency or rotationalinvariance the rotational modes become shifted awayfrom true energies E

QRP A= 0, ~ω and become mixed

with the wobbling mode such that the results mayimply spurious effects. Therefore we apply the methodproposed in Ref. [31] to eliminate rigorously the spuriousmodes. Actually, the QRPA Hamiltonian is comple-mented by the term κjJ ·J which acts like a spring forcefor the unwanted angle vibrations of the total angularmomentum J in the body-fixed system. Choosing the

stiffness parameter large as κj ≥ 102 the excitationenergies for the rotational spurious states become shiftedhigh outside the considered energy range.Let us consider firstly the spin-spin (SS) interactionwhich has been successfully applied in connection withthe scissors mode to explain the systematic accumulationof 1+ states between 3-5 MeV with considerable M1decay strengths. It is suggestive to investigate how extraSS interaction terms can influence the properties of thewobbling mode, in particular, whether the M1 strengthfound in Secs.A,B can be quenched. We shall includeboth the IS and the isovector (IV) spin-spin interactionsdefined by

V(is,iv)LL =

∑m=−1,1

(−1)m S(is,iv)m

S(is,iv)−m

,

S(is,iv)m

= Sm(τ = +1) ± Sm(τ = −1). (40)

Concerning the SS strength parameters we extrapolated

0,20 0,25 0,30 0,35 0,40 0,45 0,500,0

0,2

0,4

0,6

0,8

1,0

QQsc+ll QQsc

exp

B(E

21) (

e2 b2 )

(MeV)

163Lu

FIG. 21: (Color online) B(E2) values of the cross-over (I →I − 1) transitions between the TSD wobbling band and theTSD ground band in 163Lu. Calculated values are obtainedwith QRPA using (solid line) selfconsistent ISQQ interactionand (green line) additionally LL interaction (see Eq. (41)).

the A-dependent strength parameters given in the workof De Coster and Heyde [30] used there for QRPA cal-culations of the 1+ to our case. The SS interactions arethen added to the selfconsistent ISQQ Hamiltonian de-scribed in Sec. A.The result of the QRPA calculation done for the fre-quency ω = 0.3 MeV/~ can be summarized as follows:The IS and IV SS terms have only a negligible effect toboth the wobbling energy and the B(E2/M1) transitionproperties. The lowering of the B(M1) value is small i.e.the expected shift of the M1 strength into the scissor re-gion does not appear.A similar attempt was made with an additional isovec-tor QQ interaction term built up of the operators Qiv

m =Qm(π) - Qm(ν). Knowing the selfconsistent strength κ◦from table I we set the isovector strength κiv

◦ = r κ◦

13

0,20 0,25 0,30 0,35 0,40 0,45 0,501E-3

0,01

0,1

1

QQsc+ll

exp

lo

g B

(M11

) ()

(MeV)

163Lu QQsc

FIG. 22: (Color online) B(M1) values of the cross-over (I →I−1) transitions between the TSD wobbling band to the TSDground band in 163Lu. Calculated values are obtained withQRPA using (solid line) selfconsistent ISQQ interaction and(green line) additionally LL interaction (see Eq. (41)).

where the value of the ratio r was varied in the range -1.5 to - 3.5 [1]. This addition affects only a minor changeof the B(E2/M1) transition properties. However, onefinds an increase of the wobbling energy which can beused to fit the experimental wobbling energy by choosinga proper value r.The more appropriate contribution is an interaction termwhich is composed of the isovector orbital angular mo-menta:

V(iv)LL =

∑m=−1,1

(−1)m L(iv)m

L(iv)−m

,

L(iv)m

= Lm(τ = +1) − Lm(τ = −1). (41)

Such an interaction term is suggests the picture of ascissor-like vibration of the proton angular momentumagainst the neutron angular momentum. The micro-scopical origin of such a time-odd term could stem fromcurrent-current contributions of more sophisticated in-teractions. The actual presence of this interaction termwould signal that the wobbling motion is not pure an-gle vibration between the total spin vector and thequadrupole mass tensor but implies an interaction be-tween the proton and neutron currents.The effects of adding the LL interaction are shown inFigs.(20- 22). The calculated wobbling energy increasesrepelling LL term. We find a good match to the ex-perimental curve when choosing the strength constantκLL = −0.5 MeV/~2. The cross-over E2 transitionsstay almost unchanged which is expected from a current-current interaction. The above fixed LL interaction givesthe desired quenching of the B(M1) transition strengthwhich comes close to the measured values. We mentionthat an analogous isovector JJ interaction has qualita-tively the same effects as the LL interaction.In summary gives the QRPA with additional LL interac-

tion a satisfactory description of the wobbling frequen-cies and of the magnetic properties, yet a microscopicexplanation for observation of the unusually strong elec-tric cross-over transition in the TSD bands has still to befound.

14

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