Longitudinal Wobbling in $^{133} $ La

Longitudinal Wobbling in

133

La

S. Biswas,1 R. Palit,1 U. Garg,2 G. H. Bhat,3 S. Frauendorf,2 W. Li,2 J. A. Sheikh,3 J. Sethi,1 S. Saha,1 Purnima Singh,1 D. Choudhury,1 J. T. Matta,2 A. D. Ayangeakaa,2 W. A. Dar,3 V. Singh,4 S. Sihotra4

arXiv:1608.07840v1 [nucl-ex] 28 Aug 2016

1

Department of Nuclear and Atomic Physics, Tata Institute of Fundamental Research, Mumbai 400005, India 2 University of Notre Dame, Indiana 46556, USA 3 Department of Physics, University of Kashmir, Srinagar 190006, Indiana 4 Department of Physics, Panjab University, Chandigarh 160014, India (Dated: August 30, 2016) Excited states of 133 La have been investigated to search for the wobbling excitation mode in the low-spin regime. Wobbling bands with nω = 0 and 1 are identified along with the interconnecting ∆I = 1, E2 transitions, which are regarded as fingerprints of the wobbling motion. An increase in wobbling frequency with spin implies longitudinal wobbling for 133 La, in contrast with the case of transverse wobbling observed in 135 Pr. This is the first observation of a longitudinal wobbling band in nuclei. The experimental observations are accounted for by calculations using the quasiparticle triaxial rotor with harmonic frozen approximation and the triaxial projected shell model approaches, which attribute the appearance of longitudinal wobbling to the early alignment of an h11/2 neutron pair.

2 The atomic nucleus is a fascinating mesoscopic system, which continues to reveal new collective excitation modes due to improved sensitivity in the experimental techniques [1–3]. For quadrupole deformed nuclei, loss of axial symmetry generates new types of collective excitations. The novel characteristic rotational features of triaxial nuclei are chirality and wobbling. The focus of the present communication is the wobbling mode, which emerges because a triaxial nucleus can carry collective angular momentum along all three principal axes. In contrast, an axial nucleus cannot rotate about its symmetry axis. The wobbling mode is well known for triaxial-rotor molecules [4]. It appears as a family of rotational bands based on excitations with increasing angular momentum components along the two axes with smaller moments of inertia. Its appearance in even-even triaxial nuclei was analyzed by Bohr and Mottelson [5]. It is characterized by an excitation energy, E(I) (= ~ω) increasing with the angular momentum I and enhanced collective ∆I = 1, E2 interconnecting transitions with reduced transition probabilities proportional to 1/I. The evidence for this mode is rather limited. In a survey [6] across the nuclear chart using the finite-range liquiddrop model (FRLDM), soft triaxial ground-state shapes are predicted for nuclides around Z = 62, N = 76 and Z = 44, N = 64 and Z = 78, N = 116. As reviewed by Ref. [7], most of these nuclides seem to only execute large-amplitude oscillations around the axial shape. Partial evidence for stabilization of ground state triaxiality in these regions has been the observation of odd-I-low staggering of the quasi γ band [8–11]. Ref. [12] discussed the best case 112 Ru: The quasi γ band splits at high spin into the one- and two- phonon wobbling bands with the expected ω ∝ I. However, the data in 112 Ru [9] do not contain the required information about the B(E2, I → I − 1) values to clearly establish the wobbling character. More compelling evidence for the wobbling phenomenon was observed in the A ∼160 mass region at high spin in the odd- A Lu and Ta isotopes [13–17] and at low spin in the A ∼130 mass region in 135 Pr [18]. The presence of the odd quasiparticle in these odd-A nuclei modifies the wobbling mode in substantial way, which provides additional evidence for the triaxial shape. Frauendorf and D¨onau have recently analyzed the modifications semiclassically [12]. They distinguish between two different kinds of wobbling modes for odd-A nuclei. For the “longitudinal” mode, the angular momentum of the odd particle is parallel to the axis with the largest MoI, whereas for the “transverse” mode, it is perpendicular to this axis. The two modes can be recognized by the I-dependence of the wobbling energy Ewob (I) (= ~ωwob ), which increases or decreases, respectively. It is given by Ewob (I) = E(I, nω = 1) − [E(I − 1, nω = 0) + E(I + 1, nω = 0)]/2.

(1)

The wobbling bands observed in the Lu and Ta isotopes are examples of transverse wobbling because the wobbling frequency decreases with increasing spin, which also holds for the very recently reported first observation of transverse wobbling at low spin in 135 Pr [18]. The lowering of wobbling frequency enhances the detection probability and made it possible to detect pattern of the enhanced reduced transition probabilities for interband E2 transitions, which is a crucial signal for wobbling. In all cases the high-j quasiproton has particle character. Its interaction with the triaxial core aligns its angular momentum with the short axis, and the medium axis has the maximal MoI, which results in transverse wobbling (see [12]). The present Letter reports on observation of longitudinal wobbling in the nucleus 133 La, which is an isotone of 135 Pr. The nω = 1 phonon band with band head at 13/2−, is found to decay to the nω = 0 phonon band by ∆I = 1, E2 transitions whose multipolarities have been determined on the basis of directional correlation from oriented states (DCO) and polarization measurements. The wobbling frequency is shown to increase with increasing spin indicating longitudinal wobbling. This is the first time that a longitudinal wobbling band has been observed in nuclei. Using the quasiparticle triaxial rotor (QTR) model with harmonic frozen approximation (HFA) and the triaxial projected shell model (TPSM), the transition from transverse to longitudinal wobbling mode is demonstrated to be caused by the early alignment of a pair of h11/2 with the short axis. A 52-MeV 11 B beam from the 14-UD Pelletron at Tata Institute of Fundamental Research (TIFR) was used to populate the excited states of 133 La via 126 Te(11 B, 4n) reaction. A layer of 1.1 mg/cm2 -thick enriched 126 Te evaporated on an Au backing of 9.9 mg/cm2 served as the target. The emitted γ rays were detected in Indian National Gamma Array (INGA), which consisted of 21 Compton suppressed clover HPGe detectors coupled with a digital data acquisition system [19]. A set of 3×108 two- and higher-fold events were collected during the experiment. The time stamped data were sorted in a γ-γ-γ cube and angle dependent γ-γ matrices, and the RADWARE software package [20] was used for further analysis of these matrices and cubes. A partial level scheme of 133 La containing the negative parity states studied in the present work, is shown in Fig. 1. This level scheme is based on detailed analysis of the γ-γ-γ coincidence relations, cross-over transitions, and relative intensities of the concerned γ rays. Spin and parity assignments to the states have been made on the basis of the measured DCO ratios (RDCO ) and polarization asymmetries of the transitions depopulating these states. The detectors at 90◦ and 157◦ were used to determine the DCO ratios [21]. The polarization of γ rays was extracted from the 90◦ detectors using the formula given in Refs. [22, 23].

3 43/2

* 1110 39/2 (39/2 ) 37/2

*

946

(643) 390

35/2

35/2

(824)

*

434 33/2

908

*474

972 31/2

29/2

31/2

922

* *

(1031)

934

*893

982

29/2

797

* 1104 25/2 307

843

448 349

27/2

27/2

25/2

152

168

1150

897

*

913 998

21/2

23/2

874

789

797 17/2

19/2 758

681

585 13/2

15/2 618

445 9/2 7/2 5/2

62 ns

11/2 346

58 477

404

131

FIG. 1. Partial level scheme of 133 La showing the previously known yrast band (nω = 0), the nω = 1 wobbling band, and the dipole band. The transitions marked with an asterisk are new. The intensities have been obtained from the 445-keV gated coincidence spectrum.

TABLE I. The mixing ratios (δ), E2 fractions (=δ 2 /(1+δ 2 )) and the experimental and theoretical transition probability ratios B(M 1out ) out ) and B(E2 for the transitions from the nω = 1 to the nω = 0 band in 133 La. Experimental ratios have been obtained B(E2in ) B(E2in ) from measured intensities of the transitions. Iπi → Iπf Eγ δ E2 Fraction (%) 13/2− → 11/2− 618 -1.48+0.45 68.6+13.1 −0.32 −9.3

B(M 1out ) B(E2in )

(Expt.)

-

17/2− → 15/2− 758 -2.05+0.39 −0.30

80.8+5.9 −4.5

0.107+0.035 −0.028

-2.60+0.46 −0.47

87.1+4.0 −4.0

25/2− → 23/2− 982 -3.07+0.47 −0.65

21/2− → 19/2− 874

29/2 → 27/2 −

−

1031

-

B(M 1out ) B(E2in )

(TPSM) 0.231

B(E2out ) B(E2in )

(Expt.)

-

B(E2out ) B(E2in )

(TPSM/QTR+HFA) 0.341/0.299

0.210

1.127+0.140 −0.130

0.643/0.324

0.056+0.018 −0.019

0.142

0.716+0.079 −0.079

0.487/0.311

90.4+2.6 −3.7

0.039+0.011 −0.015

0.127

0.545+0.057 −0.059

0.437/ 0.269

-

-

0.033

-

0.212/0.221

Prior to the present work, the nucleus 133 La was studied through heavy-ion fusion evaporation reactions using a small detector array [24]. The present work confirms the previous results. In addition, we have observed several new γ-ray transitions; these have been marked with an asterisk in Fig. 1. The yrast and the yrare bands have been extended to I π = 43/2− and I π = 29/2− and identified as the nω = 0 and nω = 1 phonon wobbling bands, respectively. A dipole band has also been observed at high spin up to I π = 39/2−. The characteristic feature of wobbling bands is that the connecting transitions must be of ∆I = 1, E2 character [13]. The spectra which have been usd to extract the DCO Ratio and Polarization for the 874-keV transition are shown in Figs. 2(a) and (b) respectively. Fig. 2(c) shows the angular disrtibution plot for this transition. The RDCO and the polarization asymmetry for the 874-keV (21/2− → 19/2−) transition were calculated as a function of mixing ratio (δ), which is the ratio of the reduced matrix elements for the E2 and M 1 components of a parity non-changing ∆I = 1 transition. The contour plot of theoretical RDCO and polarization along with the experimental values is shown in Fig. 2(d). In the calculation of RDCO , the width of the sub-state population (σ/I) was assumed to be 0.3. This comparison gives a mixing ratio of −69◦ for the 874-keV transition and firmly confirms its ∆I = 1, E2 nature. In Fig. 2(e), the a2 − a4 contour plot along with the experimental data point has been shown. The mixing ratios obtained from both the DCO Ratio-Polarization and angular distribution method have been shown in Fig. 2(f) for the 758-, 874- and 982-keV connecting transitions.. However, the mixing ratios for the linking transitions mentioned in Table I

4 3000

1000

(a) I γ (measured at 157 °; gated by γ2 at 90 °)

150

(b) Perpendicular

(c)

Parallel

1

I γ (measured at 90 °; gated by γ2 at 157 °) 1

2000

Counts

100

Counts

γ1: 874 keV γ2: 445 keV

1000

Iγ

500

50

a2 = -0.68 ± 0.06 a4 = 0.21 ± 0.09

0 0 860

880

860

Energy (keV) (d)

-2

0.2

-2.5

-50

0.1

-4.5

-0.05

1.5

1

(f)

-3

-4 0

0

-0.8

0.5

-3.5

0.05

-0.6

1

0

δ 50

69

0.5

-0.5

-1.5

0.25

a4

Polarization

+0.62

0.15

0

-0.4

0

-1

cos(θ)

δ = -2.24 -0.95

0.3

0 -0.2

0

890

-1

-0.47

-69

880

(e)

+0.46

δ = -2.60

0.2

870

Energy (keV)

-1

-0.5

0

0.5

a2

DCO Ratio

-5 700

DCO Ratio-Polarization Angular Distribution

800

900

1000

Energy (keV)

FIG. 2. (color online). (a) The spectra for the DCO Ratio (b) The perpendicular and parallel spectra for polarization (c) The angular distribution plot (the red points are the experimental points, the black solid curve shows the best fit to the data and the blue dashed curve shows the curve for a pure ∆I = 1, M1 transition) for the 874-keV transition with a gate on 445-keV transition. (d) Contour plot for DCO Ratio vs. Polarization as a function of the mixing ratio (δ) (the red point on the ellipse is the experimental data point and the blue points represent the δ values) for 874-keV transition with a gate on 445-keV transition. (e) a2 -a4 contour plot for the 874-keV transition and (f) Comparison of mixing ratios extracted from DCO Ratio-Polarization method (black) and Angular Distribution Method (red) for 758, 874 and 982 -keV transitions.

are from the DCO-Polarization method. The E2 fraction increases for the connecting transitions with increasing spin, indicating enhancement of wobbling with increase in angular momentum. The experimental wobbling energies, Ewob (see Eq.(1)) of the isotones 133 La and 135 Pr are plotted in Fig. 3 as a function of spin. The wobbling frequency for 133 La is increasing with angular momentum, which along with the mixing ratios of the interconnecting transitions suggests longitudinal wobbling in 133 La [12], while it is decreasing in 135 Pr, for which transverse wobbling has been established [18]. The change from longitudinal to transverse in the neighboring isotones is a surprising result and warrants a detailed investigation. For both isotopes, the h11/2 quasiproton has particle character, i. e., it is expected to align with the short axis, which is confirmed by cranking calculations [24, 25]. According to the general arguments of Ref. [12], both isotones should have been transverse. 0.8 135

Pr (Exp.)

0.7

133

La (Exp.)

135

Pr (TPSM)

0.6

135

Ewob (MeV)

Pr (HFA)

133

0.5

La (TPSM)

133

La(HFA)

0.4 0.3 0.2 0.1 0

6

8

10

12

_

14

16

18

Spin, I (h )

FIG. 3. (color online). Comparison between experiment (solid lines) and theory (TPSM (dashed lines), and QTR+HFA (dash-dotted lines)) of the variation of wobbling frequency for nω = 1 band in 135 Pr (Black),and 133 La (Red), respectively.

To identify the origin of this abrupt change from transverse to longitudinal wobbling motion, plots of the spin of the yrast band as function of the rotational frequency, I(ω), for the two nuclei are displayed in Fig. 4. As discussed in the previous studies [24, 25], there is an alignment of a pair of h11/2 quasineutrons with the short axis, which occurs early and in a gradual manner in 133 La (see Fig. 7(b) of Ref. [24]), but late and rather rapidly in 135 Pr (where it combines

5 40

(a)

(b)

133

La (Exp.)

35

135

Pr (Exp.)

132

134

133

135

133

135

La (Exp.)

Pr (Exp.)

La (QTR+HFA)

_

Spin, I ( h )

30

Pr (QTR+HFA)

La (TPSM)

Pr (TPSM)

25 20 15 10 5 0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

_

Rotational Frequency, ω (MeV/ h )

0.5

0.6

FIG. 4. (color online). Spin (I) vs. Rotational frequency (ω) plots for (a) 133 La (experimental (black circles connected by black solid line), QTR+HFA (red solid line), TPSM (blue dashed line)) and 132 La (experimental (green circles connected by green solid line)). (b)135 Pr (experimental (black squares connected by black solid line), QTR+HFA (red solid line), TPSM (blue dashed line)) and 134 Pr (experimental (green squares connected by green solid line)).

with the alignment of a pair of h11/2 quasiprotons). Fig. 4 substantiates this by a comparison with the alignment for the πh11/2 νh11/2 bands in the odd-odd neighbors 132 La [26] and 134 Pr [27]. As expected, the odd quasineutron adds a nearly constant additional alignment in case of Pr. However, 132 La does not replicate the upbend seen in 133 La. This is naturally understood as the odd quasineutron blocking the upbend that is caused by the gradual alignment of a pair of h11/2 quasineutrons. To reveal the consequences for the wobbling band, we modified the simple QTR+HFA model of Ref. [12] by introducing a spin-dependent MoI about the short axis, which turned out to be essential for understanding the difference between the 133 La and 135 Pr. We used the expression Js = Θ0 + Θ1 R, R = I − i, where Js is MoI about the short axis, I is the total angular momentum, i the odd proton angular momentum, and R the core angular momentum. The experimental alignments shown in Fig. 4 are well accounted for with the parameters Θ0 = 10, 13 ~2 /M eV, Θ1 = 0.8, 0.2 ~/M eV , and i = 4.5, 5 ~ for La and Pr, respectively. The other two MoI’s were fixed at Jm = 21 ~2 /M eV (Jm is MoI about the medium axis), and Jl = 4 ~2 /M eV (Jl is MoI about the long axis) for both nuclides, as used in Ref. [12]. As seen in Fig. 3, the QTR+HFA calculation reproduces the change from transverse wobbling in 135 Pr to longitudinal in 133 La. The reason is the early increase of the MoI in 133 La compared to 135 Pr seen in Fig. 4. The ratios Jm /Js /Jl for 135 Pr and 133 La at I = 33/2 are, respectively, 21/15.4/4 and 21/19/4, which correspond to transverse and longitudinal wobbling. The QTR+HFA values of B(E2out )/B(E2in ) also account for the data reasonably well. The transition is understood as follows. Just above the band head both nuclides rotate about the short axis to which the h11/2 proton is aligned. For 135 Pr, the MoI of the short axis is smaller than the MoI of the medium axis (Js < Jm ). With increasing spin it becomes favorable to put more and more collective angular momentum on the medium axis, which has the larger MoI. This leads to the decrease of the wobbling frequency with spin, the hallmark of transverse wobbling. For 133 La, the MoI of the two axes are nearly the same (Js ∼ Jm ). The medium axis is no longer preferred by the collective angular momentum, which is now added to the short axis. This results in the increasing wobbling frequency with spin, which is the hallmark of longitudnal wobbling. To understand the results on a more microscopic level, we have carried out calculations in the TPSM approach. The details of the model can be found in Ref. [28]. The deformation parameters ǫ = 0.15, 0.16 and ǫ′ = 0.11 have been used for 133 La and 135 Pr, respectively, which are consistent with total routhian surface (TRS) calculations and FRLDM [24, 29]. The basis space consists of one-proton and one-proton plus two-neutron quasiparticle states below 2.5 MeV. The resulting energies for the lowest three bands are shown in Fig. 5, which are in good agreement with experiment. The second α = 1/2 sequence, which is interpreted as the signature-partner band of the α = −1/2 yrast band, is predicted at about 600 keV above the wobbling band. The value agrees with the signature splitting found by means of the TRS calculations of Ref. [24] (Fig. 5). The signature partner band in 135 Pr was predicted at similar energy by the TPSM calculations, but found at lower energy, somewhat above the wobbling band. The obvious question is: Where is the signature-partner to the yrast band in 133 La? We do not have a clear answer to this question. Our extensive efforts at identifying this band have borne no fruit and we must conclude that it is significantly non-yrast. This situation is not dissimilar to that in the isotone 135 Pr where it was clearly established that the main band sequence, previously assumed to be the signature-partner band, is in fact the wobbling band and the signature-partner band appears as a weak and short non-yrast sequence. The calculated branching for the decays from the first α = 1/2 band to the yrast band in Table I confirm the E2 dominance and thus the assignment as the wobbling band. As found in microscopic QTR [12] and quasipartical random phase approximation (QRPA) [30] calculations, the calculated ratios B(E2out )/B(E2in ) somewhat underestimate the data, whereas the calculated ratios B(M 1out )/B(E2in ) overestimate the data. The latter discrepancy could be removed by a coupling of the wobbling mode to the scissors mode, which reduces the B(M 1out ) [30]. The TPSM does not include such a coupling.

6 As seen in Fig. 3, the TPSM calculations are in fair agreement with the experimental wobbling frequencies for both La and 135 Pr. The change from transverse to longitudinal wobbling is caused by the earlier h11/2 two-quasineutron alignment in La as compared to Pr, which is illustrated in Fig. 4. In repeating the TPSM calculation with the same deformation for both isotones, we found that the change of the deformation modifies the alignment of the neutrons, which is known to be sensitive to the exact location of the chemical potential relative to the high-j intruder levels. As seen in Fig. 5, the dipole band lying at an excitation energy of about 3 MeV is also reproduced reasonably well by the TPSM approach. TAC calculations along the lines of Ref. [18] reproduced it with an accuracy comparable to that of its analog 135 Pr [18]. It has the same composition: The combination of the particle-like h11/2 proton with the pair of hole-like h11/2 neutrons results in a tilt of the rotational axis into the short-long plane, which is reflected in its dipole nature. 133

133

La

TPSM

Exp. -

43/2

-

-

43/2

39/2

5727

5890 -

-

37/2

6720

5683

39/2

5610

-

39/2 37/2

-

-

39/2

35/2 5013 33/2 4746

4651 -

33/2 -

35/2

-

29/2 -

31/2

2983 -

25/2 -

27/2

-

21/2

17/2

-

-

15/2

-

11/2

1922

1684 1020

13/2 426 0

-

21/2

-

19/2

-

25/2

2563

2299 -

-

23/2

3258

1189 634

-

17/2

-

13/2

3585 3077

2340

35/2

-

35/2

4664

-

-

31/2

-

3794 29/2

-

4305

-

29/2

33/2 4149 3790

-

31/2

-

3692

29/2

-

3790

-

-

29/2

27/2 3400 25/2 3167

Dipole Band

-

-

-

27/2

-

1821 1258

Signature Partner

Wobbling Band

Yrast Band

31/2

23/2

-

19/2

2758

1916

1126

25/2

15/2

-

11/2

445 0

27/2

-

25/2 21/2

17/2

-

-

-

-

2897

13/2

2000

5413 5160 4770 4336 3862 3414 3065 2913

Dipole Band

1203 618

Wobbling Band

Yrast Band

FIG. 5. (color online). Comparison of experimental levels with TPSM calculations for

133

La.

In summary, the nature of the wobbling mode in 133 La has been investigated. Mixing ratios of interband transitions obtained from the angular distribution and DCO-polarization measurements indicate their strong E2 nature. Surprisingly, 133 La shows longitudinal wobbling as the wobbling frequency is increasing with spin. This is the first observation of longitudinal wobbling band in nuclei. The quasiparticle plus triaxial rotor model with harmonic frozen approximation, as well as the triaxial projected shell model, were able to reproduce the experimental energies and transition rates, and clearly point to a transition from transverse wobbling in 135 Pr to longitudinal wobbling in 133 La. The change from transverse wobbling to longitudinal wobbling is understood as follows: In 135 Pr only the h11/2 quasiproton is aligned with the short axis of the triaxial density distribution and the medium axis has the largest MoI. This arrangement gives transverse wobbling. In 133 La an additional pair of h11/2 quasineutrons aligns early and gradually with the short axis. The additional alignment increases the effective MoI of the short axis, which becomes larger than the MoI of the medium axis. This arrangement gives longitudinal wobbling. In 135 Pr the h11/2 neutron alignment occurs later as a sharp band crossing, above which the longitudinal wobbling begins to appear [18]. We acknowledge the TIFR-BARC Pelletron Linac Facility for providing good quality beam. The help and cooperation of the INGA collaboration in setting up the array is also acknowledged. This work has been supported in part by the Department of Science and Technology, Government of India (No. IR/S2/PF-03/2003-I), the U.S. National Foundation (Grant No. PHY-1419765), and the US Department of Energy (Grant DE-FG02-95ER40934).

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