Influence of triaxial deformation on wobbling motion in even-even nuclei
IInfluence of triaxial deformation on wobbling motion in even-even nuclei
Bin Qi, ∗ Hui Zhang, Shou Yu Wang, and Qi Bo Chen † Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment,School of Space Science and Physics, Institute of Space Sciences,Shandong University, Weihai, 264209, People’s Republic of China Physik-Department, Technische Universit¨at M¨unchen, D-85747 Garching, Germany
The influence of triaxial deformation γ on the purely collective form of wobbling motion in even-even nuclei are discussed based on the triaxial rotor model. It is found that the harmonic approx-imation is realized well when γ = 30 ◦ for the properties of energy spectra and electric quadrupoletransition probabilities, while this approximation gets bad when γ deviates from 30 ◦ . A recent datafrom Coulomb excitation experiment, namely 3 +1 and 2 +2 for the Ru are studied and might besuggested as the bandhead of the wobbling bands. In addition, two types of angular momentumgeometries for wobbling motion, stemming from different γ values, are exhibited by azimuthal plots. I. INTRODUCTION
Two unambiguous fingerprints of the stable triaxialityof nuclei are chirality [1] and wobbling [2], which havebeen studied actively over the past two decades. Wob-bling motion was introduced by Bohr and Mottelson in1970s [2]. It is described as small amplitude oscillation ofthe total angular momentum vector with respect to theprincipal axis with the largest moment of inertia. Since2001, wobbling experimental evidence was first reportedin
Lu [3, 4], and later in
Lu,
Lu,
Lu,
Tanuclei [5–8]. In recent years, wobbling was reported inother regions as well:
Pr,
La in the A ∼
130 re-gion [9–11],
Pd in the A ∼
100 region [12],
Au and
Au in the A ∼
190 region [13, 14]. It is interesting tonote that all of the aforementioned wobbling motions arein odd- A nuclei.For odd- A nuclei, Frauendorf and D¨onau showed twodifferent possibilities of wobbling modes: longitudinalcase and transverse case [15]. The theoretical descrip-tions of wobbling motion of odd- A nucleus have beenattracted great attention, and extensively studied withthe triaxial particle rotor model (PRM) [15–19] andits approximation solutions [20–22], the random phaseapproximation [23–30], the angular momentum projec-tion (AMP) methods [10, 31], or the collective Hamilto-nian method [32, 33]. There are also some debates onthe interpretations [15] for the wobbling in odd- A nu-cleus [20, 34–36].Meanwhile, wobbling modes in even-even nuclei [2] hasbeen studied continuously in theory, e.g., see Refs. [37–44]. Recently, two new bands built on the two-quasiparticle π ( h / ) configuration were reported ineven-even nuclei Ba [45], which were lately interpretedas the transverse wobbling bands by PRM [46] and AMPmethod [47]. However, one notes that the experimentalevidence for the wobbling motion based on even-even nu- ∗ Electronic address: [email protected] † Electronic address: [email protected] cleus with zero quasi-particle configuration, namely theoriginally predicted purely collective form [2], is fragmen-tary yet. For instance, the possible evidence was pointedto the γ -band in Ru [48]. Unfortunately, there werenot interband γ rays connecting between the candidatesof wobbling band.The recent advent of new-generation detectors hasbeen opening a great possibility to explore a new areaof the collective rotation physics, in which one interest-ing exploration is searching for the wobbling mode withpurely collective form. Prior to this, the investigation forthe variation of the wobbling excited bands with respectto the triaxial parameter γ could be helpful for the exper-imental exploration. In addition, a clear picture of theangular momentum geometry and its evolution for thewobbling excitation with purely collective form will alsobe helpful to better understand the wobbling phenomenain odd- A nuclei. Motivated by the above considerations,in this paper we discuss systematically the wobbling ex-citation in even-even nuclei using triaxial rotor model. II. DISCUSSIONA. Influence of γ value on the harmonicapproximation The moment of interia (MoI) is a key parameter to de-scribe the wobbling excitation. The hydrodynamical MoIis very reasonable for the triaxial deformed nuclei, and isconsistent with cranking shell model [34]. In Fig. 1(a),we present the hydrodynamical MoI of the three principalaxes [49], J k = J sin ( γ − πk ) , (1)with γ the triaxial deformation parameter and the unit of J . In the range of 0 ≤ γ ≤ π/ k = 1 , , m -), short ( s -) and long ( l -) axis,respectively. Obviously, J , i.e., the MoI of the m -axis,is the largest.The harmonic approximation (HA) for the wobblingexcitation and the theoretical framework of triaxial ro- a r X i v : . [ nu c l - t h ] S e p - 3 0 0 3 0 6 0 9 00 . 00 . 20 . 40 . 60 . 81 . 0 Moment of inertia g [ d e g ] ( a ) ( b ) I = 5 (cid:1)
I = 2 1 (cid:1)
I = 1 3 (cid:1)
H A T R M n = 1 g [ d e g ] (cid:1) w [keV] FIG. 1: (a) The hydrodynamical MoI of the three principalaxes (denoted by k = 1 , ,
3) as functions of γ . The unit istaken as J . (b) The wobbling frequency as functions of the γ calculated by HA formula Eq. (2) and TRM using Eq. (5)with n = 1. tor model (TRM) can be found in Ref. [2]. Using thehydrodynamical MoI, the wobbling frequency calculatedby HA formula (cid:126) ω = I (cid:20)(cid:18) J − J (cid:19) (cid:18) J − J (cid:19)(cid:21) / , (2)as functions of the γ for I = 5 , , (cid:126) are shown inFig. 1(b). Here, we take a value of J = 100 (cid:126) MeV − ,which is slightly larger than ∼ (cid:126) MeV − in Lu [15].The wobbling frequency is the smallest at γ = 30 ◦ , andincreases as the γ deviates from 30 ◦ . It increases dramat-ically for γ < ◦ or γ > ◦ . The (cid:126) ω value is direct pro-portion to 1 / J . Thus, if J takes value of 20 (cid:126) MeV − (suitable for Pr [15]), (cid:126) ω will be five times as large asthese values in Fig. 1(b).For comparison, the (cid:126) ω extracted from TRM are alsoshown, and the (cid:126) ω in HA becomes better in agreementwith the frequency extracted from TRM when γ is closerto 30 ◦ and spin is larger.As shown in Fig. 1(b), the γ degree of freedom is veryimportant in determining the properties of triaxial nuclei.To examine the quality of HA, we calculate the resultsof all γ values systematically in the TRM. As is known,the nucleus described as ( β , γ ) have the identical shapewith ( β , − γ ), ( β , ± γ ± ◦ ), where β is the quadrupledeformation. Thus only the results in the γ ranging from0 ◦ to 60 ◦ are sufficient for discussion. Moreover, due to the symmetry of J k with respect to γ = 30 ◦ as shown inFig. 1, the results for 30 ◦ + ∆ γ (55 ◦ to 35 ◦ ) are identicalto the corresponding results for 30 ◦ − ∆ γ (5 ◦ to 25 ◦ ).Thus we can only focus on the results for γ from 0 ◦ to30 ◦ .The energy spectra, wobbling energies, as well as thereduced electric quadrupole transition probabilities asfunctions of spin for the several lowest bands calculatedby TRM for γ changing from 5 ◦ to 25 ◦ are shown inFig. 2 and for γ = 30 ◦ in Fig. 3, in comparison withthose obtained by the HA formulas. The energy spectraare obtained by diagonalizing the TRM Hamiltonian [2],ˆ H = (cid:88) k =1 ˆ I k J k = A ˆ I + A ˆ I + A ˆ I , (3)with A k = 1 / (2 J k ). The wobbling energies, defined asthe energy differences between the excited states and theyrast state, are extracted as [43] E wob = E ( n, I ) − E (0 , I ) , (4)for even n bands, and E wob = E ( n, I ) −
12 [ E (0 , I −
1) + E (0 , I + 1)] , (5)for odd n bands, in which E ( n, I ) denotes the energy ofspin I in the n -th excited band. The reduced electro-magnetic transition probabilities are calculated by theoperator [2] ˆ M ( E , µ ) = (cid:114) π ˆ Q µ , (6)with the obtained eigen TRM wave functions. Thequadrupole moments in the laboratory frame ( ˆ Q µ ) andthe intrinsic system ( ˆ Q (cid:48) µ ) are connected by the relationˆ Q µ = D ∗ µ ˆ Q (cid:48) + (cid:0) D ∗ µ + D ∗ µ − (cid:1) ˆ Q (cid:48) = D ∗ µ Q cos γ + (cid:0) D ∗ µ + D ∗ µ − (cid:1) √ Q sin γ. (7)For small triaxial deformation γ = 5 ◦ and 10 ◦ , thereis rather large difference between the HA and TRM re-sults. For γ = 15 ◦ , the wobbling energies for I > (cid:126) of n = 1 band in HA are in agreement with those inTRM, while for n > γ increasing, the quality ofagreement between TRM and HA becomes better. HAresults for both energy spectra and wobbling energies arein nice agreement with TRM over the whole spin rangefor small n =1, 2 phonon wobbling bands for γ = 25 ◦ .When γ = 30 ◦ , the HA formulas could give very gooddescriptions for the TRM results, which implies that therotational axis exhibit a very good harmonic oscillationswith respect to m -axis with the largest MoI. S p i n I [ (cid:1) ] Ewob [MeV]E [MeV]
H A T R M n = 0n = 1n = 2n = 3n = 4 g = 5 o H A T R M g = 1 0 o g = 1 5 o g = 2 0 o g = 2 5 o n = 4 n = 3 n = 2n = 1 n = 0 fi n = 0 n = 1 fi n = 1 n = 2 fi n = 2 H A
T R M
B(E2:I fi I-1) [e2b2] n = 0 fi n = 1 n = 1 fi n = 0 S p i n I [ (cid:1) ] B(E2:I fi I-2) [e2b2] g = 2 5 o g = 2 0 o g = 1 5 o g = 1 0 o g = 5 o n = 1 fi n = 2 n = 2 fi n = 1 FIG. 2: Upper panels: The energy spectra and wobbling energies for several lowest bands calculated by TRM (dots) comparewith those by HA formulas (lines) for γ changing from 5 ◦ to 25 ◦ . Lower panels: The intraband and interband B ( E
2) valuesfor n = 0, 1, and 2 bands calculated by TRM compare with those by HA formulas. The intraband and interband B ( E
2) in HA formulasare calculated as [2]: B ( E nI → n, I ± ≈ π e Q (8) B ( E nI → n − , I − π e nI (cid:16) √ Q x − √ Q y (cid:17) (9) B ( E nI → n + 1 , I − π e n + 1 I (cid:16) √ Q y − √ Q x (cid:17) , (10)where Q and Q are the quadrupole moments with re-spect to the m -axis, and x = (cid:112) [ α/ ( (cid:126) ω ) + 1] / y = (cid:112) [ α/ ( (cid:126) ω ) − / α ≡ ( A + A − A ) I . Thequadrupole moment Q = (cid:112) e Q + e Q takes values of √ π e b in the calculations, which is close to the valueof ∼ e b in Lu isotopes [3].For the intraband B ( E , I → I −
2) values, the HAresults given by Eq. (8) are constants, which are indepen-dent of spin I and wobbling phonon number. This equa-tion results from the approximation of (cid:104) I, K, , − | I − , K (cid:48) (cid:105) ≈ (cid:104) I, K, , − | I − , K (cid:48) (cid:105) , with K = I − n , K (cid:48) = I − − n . The val-ues of K and K (cid:48) are taken based on the wobbling pic-ture with γ = 30 ◦ . A similar recipe for n = 0, 1 bandswas already made in Ref. [18]. After such modifications,the HA formula could describe well the characteristics ofTRM results, which show the increasing trend of intra-band B ( E
2) as the increase of wobbling phonon number.For interband B ( E , I → I −
1) values, the HA resultsexhibit a decreasing trend with respect to spin, whichare determined by the factor 1 /I in the HA formulasEqs. (9) and (10). The B ( E n, I → n − , I −
1) arevery small over almost the whole spin region. For each γ under our discussion, the strength of the interband B ( E n, I → n + 1 , I −
1) is smaller than that of theintraband B ( E n, I → n, I −
2) in the high spin regionby a factor of order n/I [2]. Again, the agreement be-tween HA values and TRM results becomes better as theincrease of γ .From both the comparisons of the energy spectra andelectric quadrupole transition probabilities of the HA and H AT R M n = 2n = 1
Ewob [MeV] ( a ) ( b ) n = 0 fi n = 0 n = 1 fi n = 1 n = 2 fi n = 2 ( c ) n = 0 fi n = 1 n = 1 fi n = 0 S p i n I [ (cid:1) ] B(E2:I fi I) [e2b2] B(E2:I-1 fi I) [e2b2]B(E2:I fi I-1) [e2b2]B(E2:I fi I-1) [e2b2] B(E2:I fi I-2) [e2b2] ( d ) n = 1 fi n = 2 n = 2 fi n = 1 ( e ) n = 2 fi n = 0 ( f ) n = 1 fi n = 0 n = 2 fi n = 1 FIG. 3: Left panels: The wobbling energies, the intraband and interband B ( E
2) values for the ground band and n = 1 , γ = 30 ◦ compare with those in HA. Right panels: The energy level scheme calculatedby the TRM for the ground band and n = 1 , B ( E
2) values.
TRM, it is found that the agreement are very nice for γ changing from ∼ ◦ to ∼ ◦ over the whole spin rangefor n =0, 1, and 2 bands. B. K m of wobbling motion The above discussion to judge the quality of HA isbased on the observable of energy and electric quadrupoletransition probability. We further analyze the informa-tion of angular momentum to understand this question.For this purpose, the root mean square of projectionof total angular momentum along the m -axis, namely (cid:104) K m (cid:105) / are calculated in TRM as (cid:104) K m (cid:105) / = (cid:104) IM | ˆ I | IM (cid:105) / = (cid:104) IM | ( ˆ I + + ˆ I − ) / | IM (cid:105) / . (11)Here, the | IM (cid:105) is the eigen wave function of TRM, | IM (cid:105) = (cid:88) K ≥ C IK | IM K + (cid:105) , (12)expanded on the basis | IM K + (cid:105) = (cid:115) I + 116 π (1 + δ K ) (cid:2) D IMK + ( − I D IM − K (cid:3) . (13)The root mean square of K m with γ changing from 5 ◦ to 30 ◦ for I = 4, 5, 12, 13, 20, 21 (cid:126) are shown in Fig. 4. It is found that the relationship (cid:104) K m (cid:105) / = I − n with n = 1, ... , n max are satisfied strictly for γ = 30 ◦ . Thedifferences between (cid:104) K m (cid:105) / and I − n will increase if γ gradually deviates from 30 ◦ , or if the phonon number n gradually increases.The above variation as γ and n can be understood bythe K m structure, namely the probability distribution ofdifferent K m values. As examples, the K m structure forall states with different phonon number at spin 12 (cid:126) and13 (cid:126) are shown in Table I.Let us first investigate the case of γ = 30 ◦ . It is mucheasier to express the results from γ = 90 ◦ , which has theidentical shape with γ = 30 ◦ except the m -axis is chosenas the quantum 3-axis. Due to J = J , A = A = 4 A in Eq. (3) when γ = 90 ◦ , the Hamiltonian reads nowˆ H = 12 ( A + A )( ˆ I − ˆ I ) + A ˆ I . (14)Thus the projection K m is good quantum number. Fromthe calculated results the following relationship are sat-isfied strictly for n = 1,..., n max , K m = I − n. (15)When γ deviates from γ = 30 ◦ , the non-diagonal termin Eq. (3) will introduce the K -mixing, K m is not a goodquantum number. As shown in Table I, for n = 0 groundstate of 12 (cid:126) , the components of K m = 12 is over 90%when γ = 25 ◦ and 20 ◦ , and decreases to 76% when γ =15 ◦ . For n = 2 phonon state, the component of K m = I − γ = 25 ◦ . TABLE I: K m − structure for I = 12 and 13 (cid:126) I = 12 (cid:126) γ = 30 ◦ γ = 25 ◦ γ = 20 ◦ γ = 15 ◦ n=0 100% | (cid:105) | (cid:105) + 2% | (cid:105) | (cid:105) +8% | (cid:105) + 1% | (cid:105) | (cid:105) +15% | (cid:105) + 5% | (cid:105) n=2 100% | (cid:105) | (cid:105) +15% | (cid:105) + 2% | (cid:105) | (cid:105) + 29% | (cid:105) + 15% | (cid:105) | (cid:105) + 20% | (cid:105) + 19% | (cid:105) n=4 100% | (cid:105) | (cid:105) + 31% | (cid:105) + 13% | (cid:105) | (cid:105) + 26% | (cid:105) + 20% | (cid:105) | (cid:105) + 19% | (cid:105) + 19% | (cid:105) n=6 100% | (cid:105) | (cid:105) + 24% | (cid:105) + 20% | (cid:105) | (cid:105) + 23% | (cid:105) + 21% | (cid:105) | (cid:105) + 28% | (cid:105) + 13% | (cid:105) n=8 100% | (cid:105) | (cid:105) + 21% | (cid:105) + 20% | (cid:105) | (cid:105) + 23% | (cid:105) + 15% | (cid:105) | (cid:105) + 26% | (cid:105) + 14% | (cid:105) n=10 100% | (cid:105) | (cid:105) + 29% | (cid:105) + 21% | (cid:105) | (cid:105) + 33% | (cid:105) + 18% | (cid:105) | (cid:105) + 28% | (cid:105) + 17% | (cid:105) n=12 100% | (cid:105) | (cid:105) + 38% | (cid:105) + 12% | (cid:105) | (cid:105) + 34% | (cid:105) + 16% | (cid:105) | (cid:105) + 33% | (cid:105) + 17% | (cid:105) I = 13 (cid:126) γ = 30 ◦ γ = 25 ◦ γ = 20 ◦ γ = 15 ◦ n=1 100% | (cid:105) | (cid:105) + 7% | (cid:105) | (cid:105) +21% | (cid:105) + 5% | (cid:105) | (cid:105) + 29% | (cid:105) + 15% | (cid:105) n=3 100% | (cid:105) | (cid:105) +23% | (cid:105) + 7% | (cid:105) | (cid:105) + 22% | (cid:105) + 20% | (cid:105) | (cid:105) + 25% | (cid:105) + 17% | (cid:105) n=5 100% | (cid:105) | (cid:105) + 24% | (cid:105) + 21% | (cid:105) | (cid:105) + 27% | (cid:105) + 16% | (cid:105) | (cid:105) + 23% | (cid:105) + 15% | (cid:105) n=7 100% | (cid:105) | (cid:105) + 35% | (cid:105) + 19% | (cid:105) | (cid:105) + 23% | (cid:105) + 22% | (cid:105) | (cid:105) + 23% | (cid:105) + 22% | (cid:105) n=9 100% | (cid:105) | (cid:105) + 36% | (cid:105) + 14% | (cid:105) | (cid:105) + 31% | (cid:105) + 28% | (cid:105) | (cid:105) + 31% | (cid:105) + 29% | (cid:105) n=11 100% | (cid:105) | (cid:105) + 42% | (cid:105) + 12% | (cid:105) | (cid:105) + 33% | (cid:105) + 18% | (cid:105) | (cid:105) + 30% | (cid:105) + 21% | (cid:105) I = 4 (cid:1) Æ Km æ / g = 5 o g = 1 0 o g = 1 5 o g = 2 0 o g = 2 5 o g = 3 0 o I = 5 (cid:1) Æ Km æ / I = 1 2 (cid:1)
I = 1 3 (cid:1) p h o n o n n u m b e r
I = 2 0 (cid:1) Æ Km æ / I = 2 1 (cid:1)
FIG. 4: The root mean square of the projections of totalangular momentum on the m -axis ( K m ) with γ changing from5 ◦ to 30 ◦ for I = 4 , , , , , (cid:126) . Here, we think the probability of the K m = I − n component larger than 50% might be chosen as a reason-able criteria to judge the quality of HA approximation.Based on this suggested criteria, wobbling bands are re- alized perfectly for γ = 30 ◦ . For spin 12 (cid:126) and 13 (cid:126) , nicewobbling occurs for n = 1 , , γ = 25 ◦ , and for n = 1 phonon excitation when γ = 20 ◦ .As the spin increasing, the HA wobbling approximationbecomes better. For the states of 20 (cid:126) and 21 (cid:126) , the prob-ability of K m = I − n larger than 50% is n = 1 , , , γ = 25 ◦ . The obtained conclu-sion to judge the quality of HA from the probability ofthe K m = I − n component is very consistent with thosejudgements from the energy and transition. C. Level scheme of wobbling band with γ = 30 ◦ According to the above discussion, the stable largetriaxial deformation is necessary for the realization ofthe wobbling excitation. For the realistic even-even nu-clei, the stable triaxial deformation is rare in the groundstate [50]. A stable rigid triaxial deformation with γ ∈ (25 ◦ , ◦ ) is indeed a relatively strict condition. It mightbe one reason why the purely collective form was difficultto be observed in experiment in the past decades.We would further explore level scheme for γ = 30 ◦ ,which shows a very good wobbling picture, with the hy-pothesis of a stable rigid triaxial deformation. Bohr andMottelson discussed the excited energies of γ = 30 ◦ verybriefly in appendix 6B of textbook [2]. In Fig. 3, thewobbling energies and the B ( E
2) values for the two low-est wobbling bands calculated by TRM with γ = 30 ◦ areshown in comparison with those from the HA formulas.The HA formulas in panel (e) and (f) are new deducedin this paper according to the method in the textbook [2]as B ( E n, I → n − , I ) ≈ π e Q (16) B ( E n, I − → n − , I ) ≈ π e nI (cid:16) √ Q y − √ Q x (cid:17) . (17)In the right panel, we show level scheme from the TRMresults of the ground band and the n = 1 and 2 wobblingbands. The values of transition energies are marked, andthe thickness of the transition is proportional to B ( E E (0 , I + 2) − E (0 , I ) = E (1 , I + 5) − E (0 , I + 6) = E (2 , I + 8) − E (1 , I + 9) , e.g., 60 keV is the tran-sition energy for (0 , → (0 , , → (0 ,
6) and(2 , → (1 , E (0 , I + 2) − E (0 , I ) = E (1 , I − − E (1 , I −
3) = E (2 , I − − E (2 , I − , → (0 , , → (1 ,
5) and(2 , → (2 , E ( n, I + 4) − E ( n, I + 2)] − [ E ( n, I + 2) − E ( n, I )] =40 keV.These relationships are understood as follows. Fromthe Hamiltonian of TRM in Eq. (14), one obtains E ( n, I ) = A I ( I + 1) + 6 A I ( n + 12 ) − A n . (18)Alternatively, from the HA formula ( A = A = 4 A )with (cid:126) ω = 2 I (cid:112) ( A − A )( A − A ) = 6 IA , one obtains E ( n, I ) = A I ( I + 1) + 6 A I ( n + 12 ) . (19)Therefore, from either Eq. (18) or Eq. (19), one gets E ( n, I + 2) − E ( n, I ) = 4 A ( I + 3 n ) + 12 A , (20)and thus the above relationships 2 and 3. Furthermore,from Eq. (18), one gets E ( n + 1 , I + 5) − E ( n, I + 6) = 4 A ( I − n ) + 12 A (21)and thus relationship 1. Note that the HA formulaEq. (19) can not derive the relationship 1 due to thelack of − A n term. In addition, each values of energyin the level scheme will change according to the rule of1 / J for different J .There are one interesting thing worthwhile to be notedfrom Fig. 2 and Fig. 3(c), (d), the interband B ( E , I → I −
1) for n = 1 → n = 0 and n = 2 → n = 1 arestrongly suppressed for both HA and TRM results. Sim-ilar conclusions were obtained in Refs. [42]. In the ob-served wobbling bands in odd- A nuclei, the interband B ( E , I → I −
1) exists and links the wobbling excitedband and yrast band, e.g. see Refs. [4, 9]. It could beinferred that the linking transitions between the wob-bling bands of even-even nuclei are different from thosein odd- A nuclei.As shown in the level scheme in Fig. 3, the B ( E , I → I −
1) from n = 0 to n = 1 wobbling band will not occurspontaneously due to it needs to absorb energy. Suchtransitions are suggested to be realized by the method ofCoulomb excitation, which has explored the transitionsat the lower spin region of triaxial or octuple deformednuclei recently [53, 54]. - 9 0 009 01 8 0 g = 3 0 o q (deg) - 9 0 0 g = 2 9 o - 9 0 0 9 0 g = 2 5 o n = 0 - 9 0 009 01 8 0 q (deg) - 9 0 0 - 9 0 0 9 0 n = 2 - 9 0 009 01 8 0 q (deg) - 9 0 0 - 9 0 0 9 0 n = 4 - 9 0 009 01 8 0 q (deg) - 9 0 0 - 9 0 0 9 0 n = 6 - 9 0 009 01 8 0 q (deg) - 9 0 0 - 9 0 0 9 0 n = 8 - 9 0 009 01 8 0 q (deg) - 9 0 0 - 9 0 0 9 0 n = 1 0 - 9 0 009 01 8 0 q (deg) - 9 0 0 j ( d e g ) - 9 0 0 9 0 n = 1 2 FIG. 5: Azimuthal plots for states with phonon number n = 0, 2, 4, 6, 8, 10, and 12 at I = 12 (cid:126) calculated by TRMwith γ = 30 ◦ , 29 ◦ , and 25 ◦ . D. Two types of angular momentum geometries
In this work, we want to illustrate the angular momen-tum geometry of the wobbling motion by a probabilitydensity profile on the ( θ, ϕ ) unit sphere, called azimuthalplot [19, 51, 52]. Here, ( θ, ϕ ) are the orientation angles ofthe angular momentum vector I (expectation value with M = I ) with respect to the intrinsic frame. The polar FIG. 6: Schematic illustration of angular momentum geome-try at spin I = 12 (cid:126) for γ = 30 ◦ and γ = 25 ◦ . The orientationof arrows refer to the maxima in the azimuthal plots in Fig. 5. angle θ is the angle between I and the l -axis, whereas theazimuthal angle ϕ is the angle between the projection of I on the m - s plane and the m -axis. The profile can beobtained by relating the orientation angles ( θ, ϕ ) to theEuler angles ( ψ, θ, π − ϕ ), where the z axis in the labo-ratory frame is chosen along I . The profile is calculatedas [19, 52] P ( ν ) ( θ, ϕ )= (cid:104) I, θϕ | IIν (cid:105) = (cid:88) KK (cid:48) D I ∗ KI ( θ, ϕ, C ( ν ) IK C ( ν ) IK (cid:48) D IK (cid:48) I ( θ, ϕ, , (22)where C νIK are the expansion coefficients in Eq. (12).In Fig. 5, the obtained profiles P ( θ, ϕ ) are shown forthe ground state and all of the wobbling excited states atspin 12 (cid:126) . To visualize the results of Fig. 5, we show theschematic of angular momenta geometry in Fig. 6. Notethat the orientation of the angular momentum vector inthis figure just corresponds to the position of the maximaof P ( θ, ϕ ). We choose three results for γ = 30 ◦ , ◦ and25 ◦ , whose ratio of J m : J s : J l are 4 : 1 : 1, 4 . . . . P ( θ, ϕ ) is alwayslocated at θ = 90 ◦ , ϕ = 0 ◦ for the ground state, whichmeans along the m -axis. The excited states exhibit dif-ferent features for different γ values.For γ = 25 ◦ in the right panels of Fig. 5, P ( θ, ϕ ) showa very clear evolution of the angular momentum as theincrease in the phonon number: m -axis ( n = 0) −→ m - s plane ( n = 2 , −→ s -axis ( n = 6) −→ s - l plane ( n =8 , −→ l -axis ( n = 12). This process are also shown inFig. 6. In some previous discussions, e.g., Ref. [15, 43],similar picture was mentioned based on the case of J m : J s : J l = 6 : 2 : 1.The picture of γ = 30 ◦ is different from that of 25 ◦ .The P ( θ, ϕ ) is cylindrical symmetry with ( θ = 90 ◦ , ϕ =0 ◦ ) since J s = J l . One notes that since the length of s - and l - axis are different, such precessional motion of rotational axis with respect to m -axis still makes sense.For n = 2 , , ◦ , 45 ◦ and60 ◦ , respectively, indicating the amplitude of fluctuationof the rotation axis is getting larger. The orientation ofangular momentum with the largest probability are alsoshown in dotted line in Fig. 6.The results of γ = 29 ◦ is mixture of the characterbetween γ = 30 ◦ and 25 ◦ cases. n = 2 state of 29 ◦ isclose to the case of 30 ◦ , while n = 12 state of 29 ◦ is closeto the case of 25 ◦ .Furthermore, the azimuthal plots and the schematic ofangular momenta for the band of n = 0 , , m -axis decreasesas the spin increasing, for both n = 1 and n = 2 wobblingbands, which is consistent with the HA formula with theprecession amplitude (cid:112) n/I . E. Comparison with the recent data of Ru Recently, a multi-step Coulomb excitation measure-ment was carried out for
Ru isotope [54]. The exper-imental data of
Ru are shown in Fig. 8(b), where theexcitation energies (in keV) and spin-parity values aregiven above the states. The widths and labels of the ar-rows represent the measured reduced E +2 and 3 +1 areconsidered as one band in the Ref. [54], while we separatethem in the present level schemes. Ref. [54] pointed outthat the data provides direct evidence of relatively rigidtriaxial deformation near the ground state.In Fig. 8(a) and (c), we show the results calculated byTRM with γ = 30 ◦ and 25 ◦ . The adopted parameterof MoI J ( ∼ (cid:126) MeV − ) and quadruple moment Q ( ∼ . e b) are adjusted for the energy and B ( E
2) valueof 2 +1 state.The calculated results are in agreement with the ex-perimental data qualitatively. As mentioned and empha-sized in Ref. [54], the relatively large 2 +2 → +1 and small2 +2 → +1 matrix elements, are strong indications of triax-ial deformation. These experimental characteristics arereproduced by the present calculations. In addition, thelarge 3 +1 → +2 and small 3 +1 → +1 matrix elements inexperiment are reproduced by TRM. Based on this, 3 +1 and 2 +2 for the Ru might be suggested as the bandheadof the one- and two-phonon wobbling bands.
III. SUMMARY
The influence of triaxial parameter γ on the wobblingexcitation in even-even nuclei are investigated using theTRM with the hydrodynamical MoIs. We suggest thatthe probability of the K m = I − n component larger than50% might be a reasonable criteria to judge the qualityHA. Based on this criteria and the characteristic of the
09 01 8 0 ( 0 , 4 ) ( 1 , 5 ) ( 2 , 4 ) ( 0 , 1 2 ) ( 1 , 1 3 ) ( 2 , 1 2 ) j ( d e g ) q (deg) ( 0 , 2 0 ) - 9 0 0 ( 1 , 2 1 ) - 9 0 0 9 0 ( 2 , 2 0 )
09 01 8 0 ( 0 , 4 ) ( 1 , 5 ) ( 2 , 4 ) ( 0 , 1 2 ) ( 1 , 1 3 ) ( 2 , 1 2 ) ( 0 , 2 0 ) - 9 0 0 ( 1 , 2 1 ) - 9 0 0 9 0 j ( d e g ) q (deg) ( 2 , 2 0 ) FIG. 7: Azimuthal plots for states ( n, I ), in which I = 4, 5, 12, 13, 20, 21 (cid:126) with n = 0, 1, 2, and the schematic illustration ofthe evolution of angular momenta for the wobbling band. Results for γ = 30 ◦ and 25 ◦ are shown in upper and lower panels,respectively. n=2n=1n=0 (a) TRM with γ = 30 ◦ (b) Experiment of Ru n=0 n=1 n=2 (c) TRM with γ = 25 ◦ FIG. 8: Comparisons between experimental level scheme of
Ru [54] and TRM results with γ = 30 ◦ and 25 ◦ . The excitationenergies (in keV) and spin-parity values are given above the states. The widths and labels of the arrows represent the reduced E energy spectra and electric quadrupole transition prob- abilities, wobbling motion in even-even nuclei could berealized well for the states with small n phonon numberwhen γ changing from ∼ ◦ to ∼ ◦ .The above condition for the restriction of γ value isa relatively strict condition and might be difficult toachieve in realistic nuclei, which might be one of reasonsfor wobbling bands of purely collective were difficult tobe observed in experiment. A recent data from coulombexcitation experiment, namely 3 +1 and 2 +2 for the Ruare studied and might be suggested as the bandhead ofthe candidate one- and two- phonon wobbling bands.From azimuthal plot, the angular momentum geome-try in the wobbling excitation has two types due to thedifferent MoI: one is exhibited in the case of γ ∼ ◦ and the other one in γ deviating from 30 ◦ . In a wobblingband with certain phonon number, the angle between an- gular momentum and m -axis exhibits a decreasing trendwith respect to spin. Acknowledgments