Tilted axis rotating states in 182Os yrast

21 April 1994 PHYSICS LETTERS B

Physics LettersB 325 (1994) 283-288

ELSEVIER

Tilted axis rotating states in 182Os yrast Takatoshi Horibata ", Naoki Onishi b o Department of Information System Engineering, Aomori University, Aomori-city Aomori 030, Japan and Cyclotron Laboratory, The Institute of Physical and Chemical Research (RIKEN), Hirosawa 2-1, Wako-shi, Saitama 351-01, Japan o Institute of Physics, College of Arts and Sciences, University of Tokyo, Komaba Meguroku Tokyo 153, Japan

Received 19 November1993 Editor: G.F. Bertsch

Abstract

We find tilted axis rotating solutions in the yrast states of the ~82Osby self-consistent three dimensional cranking calculations. Since the method is free from the restrictions of the linear approximation, we can manipulate solutions far from the regions where the conventional cranking + RPA approach could reach. Present solutions suggest a possible new interpretation for the backbending phenomena.

The dynamics of nuclear rotation is still worth accepting challenges from new interpretations because of its unexpected comprehensive nature of collectivity. Chowdhury et al. [ 1 ] report an observation of high-K isomers which decay abnormally fast into the yrast states in the ~82Os nucleus. Since the yrast states are considered to be low-K states, these decay processes suggest an existence of new decay mechanisms because the processes violate the K-selection rule significantly. They claim that the large y-deformation mixes the Kquantum number and prompt the penetration between high-K isomeric deformation aligned prolate state and the well known rotation aligned state where the angular momentum is perpendicular to the symmetry axis of y = 0 prolate shape. Indeed, theoretical study [2] revealed that the K-isomerism is strongly affected by y-softness, i.e. fluctuation of y. It is also important to point out that K-mixing due to the Coriolis force involves the breaking of the isomerism. In this nucleus, the energy of odd spin members of the quasi-y-band is abnormally low. The wobbling motion associated with the y-deformation is predicted Elsevier ScienceB.V. SSD103 70-2693 ( 94 ) 00261-5

to appear as a part of high spin survivals of the yvibrational band in the nucleus [ 3 ]. Since this motion accompanies shape oscillations, y-degrees of freedom are also expected to play a crucial role in the motion. Therefore this nucleus may offer a nice testing ground for questioning the quality of models inquiring dynamical mechanisms inherent in such general rotational motions. An extensive microscopic calculation on this nucleus has been performed, in which the properties of the negative signature sy-band were discussed in connection with the K-mixing effect and the dynamical moment of inertia [4]. It was shown that the analysis suggests a character change in the band to the wobbling motion at the high spin part of it. Because the approaches used so far have been mainly based on linear theory [ 5 ] they would not be adequate in describing the dynamics which causes drastic character changes such as the violation of the K-selection rule or the abnormal spectrum found in the quasi-y-band. Kerman and one of the present authors [6] proposed a semi-classical theory based on the time dependent variational theory which can describe non-uniform

T. Horibata, N. Onishi/ Physics Letters B 325 (1994) 283-288

284

rotation including precession and wobbling, which which angular velocity is no longer time independent. The theory treats three components of the angular momentum, with respect to a certain intrinsic frame of coordinate system, as the dynamical variables together with the Euler angles. We employ this method to investigate the facets of rotational motion involved in this nucleus which may not be reached by the conventional linear theory such as cranking plus RPA. In this paper we focus our discussion on the character of potential surface in the three dimensional angular momentum space and report briefly very characteristic results from the self-consistent calculation. The previous work for application of the theory is limited to only Hartree-Fock space to describe high spin states such as from 30h to 40h. We extend the method to the Hartree-Fock-Bogoliubov space in order to take into account the pairing correlation, which plays an important role in the lower angular momentum states like J < 30h. Especially, interplay of the pairing correlation with the rotation alignment caused by Coriolis force is considered to be a dominant mechanism in this angular momentum region. The model Lagrangian for the general rotational motion including precession and wobbling is written

aH Ix=--~-

and R = c u r l S .

3

(1)

k=l

The gyroscopic nature of the equation brings two constants of motion, the energy and the modulus of angular momentum vector. Therefore the angular momentum is confined to a sphere. The mass quadrupole moment is the most sensible order parameter for breaking rotational symmetry. Therefore the intrinsic frame of coordinate system is determined such that the principal axes (PA) of the mass quadrupole moment become axes of coordinate system,

<~l~kl q~>= o , Bk = ½(Qij +Qji)

where the gauge gradient is defined as (2)

and the energy is given by

H(jl) = (l~(j/) I/~l 4'(/3 >.

(3)

We truncate a model subspace from the full HFB space, in which wave functions are labeled by the three components of angular momentum with respect to an intrinsic frame, ji = (~(J,)

I?, I q,(j,) >.

(4)

The equation of the rotational motion is derived from Euler-Lagrange variational principle and then is written as follows,

dj j X I~ at = 1 + R . j '

( i j k cyclic).

(7)

By adding the particle number constraints, we obtain the variational equation for the constrained HFB,

=0.

(8)

/4=27-½K ~' ( - ) ~ ' Q ~ , Q _ ~ - y" g~Gt, G~. /z

Sk(j,) = (qb(jt)i ajO-~k[q~(js) >,

(6)

The model Hamiltonian employed in the present calculations is the pairing plus Q--Q force given by

as

Lw(OiO,,jj,) = ~ {oJkjk+jkS~(j~)}-H(j,),

where

(5)

(9)

T

While this Hamiltonian is not proper enough for a quantitative study, e.g., a precise comparison with experimental data, this is simple and contains the essential characteristics of deformed nucleus such as Nilsson's single particle spectrum. We solve the variational equation by the steepest descent method with the eight constraints. The single particle states available for active particles consist of [0i13/2, 2sl/2, ld3/2, ld5/2, 0g7/2, 0g9/2, lf7/2, 0h9/2, 0 h l l / ~ ] for proton and [0i11/2, lg9/2, 0i13/2, 2pl/2, lf5/2, 2p3/2, 0h9/2, 1t7/2, 0hl 1/2] for neutron states, respectively. While 74 states for proton and 78 states for neutron are open, 40 protons and 70 neutrons are frozen as the core nucleus. In the calculation for lSZOsthe proton valence particles and the neutron valence particles are taken to be the same; those are 36 particles in the respective model spaces. Although the size of the space may be small for quantitative discussions, this model space

T. Horibata, N. Onishi/ Physics Letters B 325 (1994) 283-288

covers Nilsson's single particle states in the vicinity of the Fermi surface. First we perform the calculation for one dimensional rotation, i.e. stationary rotation along the principal axis of the mass quadrupole moment (PAR), up to the state of angular momentum 30h. Since our discussion is concentrated mainly on studying the essential features of the possible mechanism of the rotational motion in this nucleus, we simply fix the force parameters so as to reproduce the excitation energies of the first few levels in the g-band. That is, the coupling constant of the Q-Q force takes on the value 1.523 × 10 -4 MeV/ fm -4, and the pairing force for proton g,~ and for neutron g~ are taken to be 0.155 MeV and 0.158 MeV, respectively. We observe abrupt phase change on the PAR-yrast line in the present numerical calculations with above set of parameters. This phase transition is of the first kind and therefore hysteresis takes place in the transitional region; namely overheating and supercooling. Indeed, we confirm them for both directions by performing cranking down calculation after cranking up Energy

9 r .....................

::

'

to the maximum angular momentum state J - - 30h with a step of 0.5h. We obtain the PAR g-band from the cranking up calculation till J = 18h and the PAR s-band down to J = 10h by cranking down calculation. The PAR g-band crosses to the PAR s-band at J = 14h in energy. Fig. 1 shows the yrast line thus obtained from the one dimensional calculations. Although the calculated levels are located equally higher compared with the experimental data, the trend of the yrast line is well reproduced together with the band crossing point. Next, we show the feature of T-deformation in Fig. 2. The upper solid curve in the figure represents the angle of the ~/-deformation for the PAR g-band branch and the lower dotted curve represents the angle for the s-band branch. The deformation jumps by more than 13 degrees when two bands cross each other at J = 14h. This sudden change of the triaxiality is consistent with the analysis by Lieder et al. [7] and reflects a typical feature of the nuclear rotation; while the shape is likely to stretch along the rotating axis in the quantum liquid, the superfluidity, it is likely to shrink in the classical liquid. These characteristic features may manifest Curves of Yrast Line

I .........................

I .................................

!................................

I

::

i

i

i

i .............................

i ..................

..................... i..............................

285

.................................

............

J

', . . . . . . . . .

i

+

:;

6 k ...................... i ............................... i ............................... i ............................... i ...................... ,e.----! ............................. ':

i

;:

i

i

+

v

5 k ............................... i ............................... i ................................. i ........................... ~ 'i ................................. !............................... P A R g~ba

~'°'

4i

i

i

i

i

u.i

,

PAR siband

3 t- ................................................................

~

i

......

..-~

i . . . . . . . . . . .

+

ii

~, e x p .

i .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

i

.

.

.

.

.

.

.

ii

i

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

i

.

i

.- ................ ~ ..................................... i ................................ i ............................... fi

;-...............................i............................ ~.................................i.................................i...................

i

-

0 0

5

10 Angular

15 Momentum

20

25

30

Fig. I. The energies of PAR g-band (solid line) and PAR s-band (dotted line) are shown together with true yrast states including TAR (crosses). The PAR g-band and s-band cross at J = 14h and the hysteresis is illustrated by extending each line of the branches. The experimental energies are also indicated (diamonds) which are taken from Ref. [ 1 ].

286

T. Horibata, N. Onishi/ Physics Letters B 325 (1994) 283-288 Gamma 4

deformation

i

i

i

-2 -4

.

.

.

.

.

.

.

.

.

.

.

.

.

-6

.............................. ::..............................

i-..~... (

-8

.........................................................

i ......... ~ . . . . . . . . . . . . . . .

-10

I

..........................................................................................

'

> ..................... i

~. . . . . . . . . . . . . . . . . . . . . . . . .

......................... i . . . . . . . . . . . . . . . . . . . . . . .

,. . . . . . . . . . . . . . . . . . . . . . . . . .

,. . . . . . . . . . . . . . . . . . . . . .

~:1"i ........................................................................................................

-14 0

10 Angular

15 Momentum

20

25

30

Fig. 2. The angles of ~/-deformation are shown. The solid line and the dotted line indicate the angles for PAR g-band and s-band, respectively. The sign change takes place at J = 16h of the g-band.

themselves in the backbending phenomena in this nucleus. After constructing the PAR-yrast states we start from the PAR-states to explore a vast unknown three dimensional potential field where no self-consistent calculations have ever been performed for HFB-space. First we show a potential surface obtained by the present calculation at the middle of the phase transition, J = 15h. For illustration, we use geographical terms for the globe. The polar axis is taken as the symmetry axis of the deformation at J = Oh and the intersection of the prime meridian and the equator is chosen as PAR axis. Because of a symmetry of the space, the reflection symmetries with respect to the prime meridian and the equator, it is enough to show only a quadrant of the north--east hemisphere in Fig. 3. We find a local energy minimum at a distant point from the same angular momentum PAR-state which was obtained by the one dimensional rotation. The polar position of the point is (0, 27) in unit of degree, where the values in the parentheses represent the east-longitude and the north-latitude, respectively. The energy of this point is 0.3 MeV lower compared to that of the J - - 15h PAR-state. The

result suggests the existence of tilted axis rotating (TAR-) states and the states possibly join the members of the new yrast states. The TAR is a stationary rotation since time derivative of angular momentum with respect to the intrinsic frame is proportional to the vector product of the angular momentum j and the gradient i~ (see Eq. ( 5 ) ) . At the minimum point of the potential the gradient is perpendicular to the sphere and eventually parallel to the angular momentum. This statement has been confirmed by Nazarewicz and Szymanski in terms of a schematic model [ 8 ]. Furthermore the strength sciof the constraint /~i to keep the principal axis turns out to be also vanishing. The existence of this kind of motion has already been predicted but the discussion was restricted to excited states or odd-particle nucleus [9]. The tilting of the axis starts from the states above J = 12h and the latitude of the tilted axis becomes the highest a t J = 16h. After the maximum tilted state has been identified the magnitude becomes small but the tilted axis never returns to merge with the PAR at higher angular momentum states. True yrast states obtained by the present calculation are plotted in Fig. 1. As shown in

T. Horibata, N. Onishi/ Physics Letters B 325 (1994) 283-288

287

Potential Surface Around Tilted Axis at 0=15

Energy (MeV) 0.4 0.2 0.0

20 50

North-latitude

0 Fig. 3. The energy surface of J = 15h sphere. The direction of angular momentum vector is represented by east-longitude and north-latitude. The energy contour lines are indicated in the figure. The energy minimum is found at (0, 27) in unit of degree.

the figure all yrast states above J = 12h are TAR states. The standard interpretation of the backbending phenomena is the band-crossing of the ground state band and the rotation aligned (RA) band of the high-j orbitals [ 10]. A typical microscopic theory is the PAcranking model based on HFB-space [ 11 ]. Because of the decoupled pair due to RA, the gap energy is reduced significantly in the RA-band. In the PA-cranking model, the symmetry of signature is maintained so that the quasi-particle state having negative energy possesses always the opposite signature to the dual state of positive energy. Hence the two energies are necessarily degenerate at zero energy and give rise to the level crossing. In the TAR-state, the symmetry of signature is broken down. Therefore odd angular momentum states are included in the states. From the symmetry properties of the potential surface, there is always found another minimum at the mirror image point with respect to the equator on the prime meridian. The tunneling back and forth motion recovers the signature (even-odd angular momentum) and gives rise to the splitting of the energies. In order words, the rarer the tunneling occurs the smaller the splitting is. This feature is favorable to the

explanation of the abnormally low energy of the odd angular momentum yrast states. The energy spectrum of quasi-particle would be interesting to look at. It is expected that the level crossing may not occur because of the noncrossing theorem in case of a coupled system. The energy surface in the transitional spin states looks very shallow and a very large fluctuation of the direction of the angular momentum, the wobbling motion, is expected. Evidently, this fact implies that any linear theories seem not applicable. Further investigation would be necessary and detailed studies on the wave function and the quasi-particle energy spectrum will be reported somewhere else as a full paper.

References [ 1 ] P. Chowdhury et al., Nucl. Phys. A 485 (1988) 136. [2] N. Tajima and N. Onishi, Nucl. Phys. A 491 (1989) 179. [3] A. Bohr and B. Mottelson, Nuclear Structure Theory (Benjamin, 1975) Vol. 2. [4] M. Matsuzaki, Nucl. Phys. A 509 (1990) 269. [5] I.N. Mikhailov and D. Jansssen, Phys. Lett. B 72 (1978) 303; E.R. Marshalek, Nucl. Phys. A 331 (1979) 429.

288

T. Horibata, N. Onishi/Physics Letters B 325 (1994) 283-288

[6] A.K. Kerman and N. Onishi, Nucl. Phys. A 361 (1981) 179; N. Onishi, Nucl. Phys. A 456 (1986) 279. [7] R.M. Lieder et al., Nucl. Phys. A 476 (1988) 545. [8] W. Nazarewicz and Z. Szymanski, Phys. Rev. C 45 (1992) 2771.

[9] S. Frauendorf, Nucl. Phys. A 557 (1993) 259; R. Bengtsson, Nucl. Phys. A 557 (1993) 277. [ 10] F.S. Stephens and R.S. Simon, Nucl. Phys. A 183 (1972) 257. [ 11 ] P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer-Vedag, 1980).