A self-consistent study of triaxial deformations in heavy nuclei

Volume 18 I, number 3,4

PHYSICS LETTERS B

4 December 1986

A SELF-CONSISTENT STUDY OF TRIAXIAL DEFORMATIONS IN HEAVY NUCLEI N. R E D O N , J. MEYER, M. M E Y E R lnstitut de Physique Nuclbaire I, Universitb Claude Bernard Lyon L 43, Boulevard du 11 Novembre 1918, 1=-69622 Villeurbanne Cedex, France

P. Q U E N T I N Laboratoire de Physique Thborique e, Universitb de Bordeaux L Chemin du Solarium, F-33170 Gradignan, France

M.S. WEISS Lawrence Livermore National Laboratory, Livermore, CA 94550, USA

P. B O N C H E Service de Physique Thborique, DPhG, CEN Saclay, F-91191 Gif sur Yvette Cedex, France

H. F L O C A R D Division de Physique Thborique e, Institut de Physique Nucl~aire, F- 91406 Orsay Cedex, France

and P.-H. H E E N E N 3 D~partement de Physique Nucl~aire Th~orique, Universit~ Libre de Bruxelles, CP 229, B-I 050 Brussels, Belgium

Received 15 July 1986; revised manuscript received 18 September 1986

Lattice Hartree-Fock + BCS calculations for axially asymmetrical solutions have been extended to heavy nuclei. The deformation energy surfaces in a (fl, 7) sextant for the ~aSm and ~9:Osnuclei exhibit a shallow triaxial minimum, while a valley connecting smoothly the oblate and prolate minima is found in the ~86Ptnucleus.

Self-consistent calculations o f q u a d r u p o l e deform a t i o n energy surfaces have recently been performed [1] for a n u m b e r o f exotic nuclei with A = 100. The axially a s y m m e t r i c solutions o f the constrained H a r t r e e - F o c k ( H F ) variational equations have been obtained through lattice calculations resulting from a discretization o f the configuration space on a three-dimensional rectangular mesh. In this letter, we report on the application o f this numerical procedure to heavier nuclei where the evidence o f various structural effects related to triaxial J And IN2P3. 2 Formation associre au CNRS. 3 Maitre de Recherche FNRS. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )

q u a d r u p o l e d e f o r m a t i o n s has been brought up. The recent d a t a on neutron-deficient isotopes are relevant for such a subject. The only alternative to these calculations that carries the same a m o u n t o f physical input is the H a r t r e e - F o c k Bogoliubov a p p r o a c h o f G i r o d and G r a m m a t i c o s [ 2 ] which uses a projection o f the wave-functions on h a r m o n i c oscillator basis states. The decay o f 13SEu into 138Sm studied in the Hej e t coupled to a mass-separator experiments performed at SARA in Grenoble [ 3 ], whose results have been confirmed by (H.I., xnylry ) reaction studies [4], yields the following results: (2) the second 2 + level lies below the first 4 + level, (ii) the sum o f the ener-

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gies of the two first 2 + levels is equal within less than 1% to the first 3 + level energy. Both features can be explained with a Bohr collective model based on the rotation oftriaxial intrinsic states [5 ]. However, one notes that the second property can be understood as well in terms o f ),-vibrations around an axially symmetrical equilibrium shape. In this region, the existence of triaxial deformations is also suggested by the observed bunching of the levels 1 3 / 2 - , 1 5 / 2 - and 1 7 / 2 - , 19/2 - for the N = 77 isotopes of neodymium and cerium. On the other hand these states are ordered as regular 1 1 / 2 - coupled band for the N = 75 isotopes. In phenomenological rotor plus quasi-particle approaches [6] this feature is explained as the signature of a transition towards triaxial even-even cores. For a very long time, the isotopes with 76 ~
4 December 1986

Table 1 Constant pairing matrix elements (in MeV) for protons (Gp) and neutrons (Gn) for the nuclei considered. Nucleus

Gp (MeV)

Gn (MeV)

t3SSm

0.218 0.159 0.160

0.195 0.134 0.126

tS6pt J°2Os

dramatic increases in the number of real variational quantities whose order o f magnitude roughly goes as A 2, where A is the nucleon number. Typically a nucleus like ~86pt at moderate deformations (fl ~<0.3) is well represented within a box of 2 8 × 2 4 × 2 4 fm 3 with a mesh size o f 1 fm. This corresponds to a calculation with about 1.5 × 106 variational quantities whereas one needed about a third o f it for A ~ 100 nuclei. On a CRAY-I computer the time per Hartree-Fock iteration is about 8 s for ~86pt and slightly less than 100 iterations were needed to reach a convergence. Computations have been carried out at five values of the asymmetry angle (? = 0 ° , 15 ° , 30 °, 45 °, 60 ° ) paving the relevant part o f the sextant with about 50 different H F solutions. Pairing correlations have been included in the standard constant Gapproximation, as sketched in ref. [ 1 ]. The values o f the matrix element G are listed in table 1, which reproduces the pairing properties o f the nuclei considered. The deformation energy surfaces are displayed in figs. 1-3. They are plotted in terms of two parame-

~ 0

0o

1000

2000 Q ( f r n 2)

Fig. I. Potential energy surface of '3SSm in the (Qo, Y) representation. The mass quadrupole moment Qo (in fm~) and the angle 7 (in degree) have been defined in the text. The dot locates the static equilibrium solution. The energy difference between two contour lines is 0.5 MeV.

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0°

0

~000

2000

Q (fro 2 ) Fig.2. Same as fig. 1 for ~"6pt.

ters: Qo (in barn) and 7 (in degrees), which results from a parametrisation of expectation values of the three operators Q,= 3x2, - r 2 through Q,=Qo × cos(7 +i2n/3). In agreement with experimental data, we find a potential energy surface for the ~38Sm nucleus with a triaxial minimum around Qo ~ 8 b and 7 ~ 2 5 °. This minimum is 0.6-0.7 MeV below the two axial local minima which happen to be almost degenerate in energy. The spherical barrier height is usual to 2.5 MeV. For Qo above the equilibrium value, the nucleus is much softer on the prolate edge as compared to the oblate edge. The existence of such asymmetric equilibrium solutions had already been suggested for

60 °

0

1000

2000 O (fro 2 }

Fig.3. Same as fig. 1 for '92Os.

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N = 7 6 isotones (with 58~
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The main feature of the ~86pt deformation energy surface is a rather shallow valley connecting the absolute prolate minimum to a plateau extending from 7 = 30 ° to 7 = 60 ° at an energy located ~ 1.6 MeV above the equilibrium point. For increasing values of y, this valley bends slightly in the direction of lower Qo values (from ~ 18 b to ~ 13 b). This somewhat soft character is consistent with the data in neighbouring o d d - N platinum isotopes. The spherical barrier Bs is found to be 5.6 MeV high. The energy difference Vpo is consistent with the findings of the macroscopic-microscopic approach of ref. [ 14] while their B~ value is ~ 2 MeV lower. The corresponding deformation parameter values were found to be f12= 0.22 and f14= - 0.06 which is roughly consistent with our findings (fl2=0.26, f14= - 0 . 0 4 6 ) . Our results are also in fair agreement with the Woods-Saxon calculations of G r t z et al. [18] (fl2=0.24, f14= - 0 . 0 4 ) as well as with the pairing plus quadrupole force calculation of Kumar and Baranger [19] (fl2=0.188) and of Ansari [20] (f12=0.19, f14= - 0.03). The authors ofref. [ 18] have emphasized the role of the hexadecapole deformation in stabilizing the prolate shape, whereas Ansari [20] has illustrated the importance of including all the components Y4/,of the hexadecapole tensor. Both effects are of course self-consistently included in our calculations. The absence of hexadecapole terms in the parametrised potential energy surfaces of ref. [ 21 ] may explain why an oblate instead o f a prolate equilibrium solution is found in this approach for this nucleus. The '92Os nucleus exhibits a shallow triaxial minimum in a prolate-oblate valley for Qo~ 12 b and 7 ~ 30°, located at 350 keV below the local minimum on the prolate edge. The energy difference between the absolute minimum and the local oblate minimum is ~ 2 MeV. The existence of such a rather flat valley has been obtained for this region in the calculations of Kumar and Baranger [ 19], Grtz et al. [ 18] and the data parametrisation of ref. [21]. Now, whereas the pairing plus quadrupole approaches of refs. [ 19,20 ] give equilibrium solutions at y = 60 ~"and 46 °, respectively, the Strutinsky-type approaches of refs.[ 14,15 ] leads to prolate axial solutions with fl2=0.12, fl4=-0.05 and fl2=0.16, f l 4 = - 0 . 0 5 , respectively. This discrepancy should of course not be stressed too much in view of the very weak depth 226

4 December 1986

Table 2 Comparison of experimental [22] and calculated binding energies (in MeV) of the considered nuclei. For 13SSmthe energy is deduced from the systematics of neighbouring nuclei. Nucleus

ISSSm ~s6pt 192Os

B theory

experiment

1132.24 1474.02 1522.55

1136.68 +0.45 1478.10 _+0.11 1 5 2 6 . 1 3 8+ 0.005

of the dip exhibited in our calculations around y = 3 0 °. Insofar as the equilibrium BCS wave-functions are representative of the ground state properties, we compare in table 2 the binding energies with the data [ 22 ]. The theoretical figures have been corrected for a 0.5% error due to the mesh size (whose amount has been checked locally with 0.1 fm mesh size calculations) [1]. The "experimental" 138Sm energy has been deduced from the binding energies for neighbouring nuclei [22]. In fig. 4 the evolution of neutron energy levels is displayed as a function of the axial quadrupole variable Qo for the lS6pt nucleus. (Note that in such a spectrum the hexadecapole moment is fixed by the

-10 ~

~ 27._-_~:_ -. . . .

-1000

7s1~

0

_---

1000

-"

X

Q (fro e )

Fig.4. Neutron single-particle energies c. (in MeV) as a function of the quadrupole moment Qo (in fro:) in ~e6pt. "['he evolution of positive (negative, respectively) parity levels is represented by solid (dashed, respectively) curves.

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between 7 = 0 and 7 = 30 °, is to be related with the calculated magicity of the N = 1 1 6 - 1 1 8 neutron numbers for these values of the constraints. This work extends the self-consistent approach to deformation energy surfaces for both axial and nonaxial quadrupole modes developed earlier [ 1] to heavy nuclei. It further demonstrates the capability of the present approach to perform systematic calculations. The relevance of any static calculation of this kind, however, will only be fully exploited when taking into account in a consistent microscopic way all the dynamical effects. For the rotational degrees of freedom alone, a projection of the intrinsic states on good angular momentum states may appear to be important for the existence oftriaxial shapes as demonstrated in ref. [25] by using the hamiltonian of Kumar and Baranger [ 19]. An alternative way, which further allows the coupling of rotational and vibra-

variational process.) These Nilsson spectra are very similar to those obtained with the A = 165 parametrisation of the modified harmonic oscillator [23]. The same global agreement is observed for the extrapolated parameters of ref. [24] apart from an inversion of the 2d3n and 3st/2 neutron levels. Finally the variation of the neutron energy levels of 186pt for Qo = 16 b as a function of 7 is shown in fig. 5. As clearly shown in these figures a rather large gap is observed for 108 neutrons at y = 0 ° which could be related to the observed stability of the axial prolate shape in '86pt. This minimum, already seen in fig. 4, disappears upon breaking the axial symmetry as exhibited in fig. 5. Similarly the gap oberved for 116 neutrons around ~,= 30 ° is consistent with the location in the (fl, 7) plane of the static equilibrium solution of '92Os. Moreover, the almost degenerate character of the variational solutions for (2o~ 12 b

~n(MeV

)

....

t

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

4 D e c e m b e r 1986

°

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

- ....

~

!

3

.:

-5

--

7

_

.

_ :'_

-10

7

........

.

. . . . . . . .

; . . . . . . . . .

-.

5e" . . . . . . . . . . . . .

e- ....

z

-

_ "''"

-

o- . . . . .

~

_ . . . . . .

+

• 9

-

--

-

~

-e3

• . . . . . . . . . . . . .

45

~--___.___.q~..

- -

-el

131 7e- . . . . . . . . . . . .

o- ~

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

o

~

. . e -

-

_ ~

.....

~

....

03 "411

- -

: -15 600

45*

30 °

15 o

0°

Fig.5. Neutron single-particle energies e, (in M e V ) as a function o f the asymmetry, angle y (in degrees) for Qo = 1600 fm 2 in 's6pt. Straight solid (dashed, respectively) lines connect levels o f positive (negative, respectively) parity calculated for y = 0 °, 15 ~, 30 o 45 ° and 60 °. The levels in the axial cases are labelled by twice the usual projection q u a n t u m n u m b e r Q value.

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tional degrees o f f r e e d o m , w o u l d m a k e use o f the B o h r collective h a m i l t o n i a n a p p r o a c h . In the adiabatic t i m e - d e p e n d e n t H a r t r e e - F o c k f r a m e w o r k o f ref. [ 26 ], o n e w o u l d be able to c o m p u t e c o n s i s t e n t l y all the ingredients o f the B o h r h a m i l t o n i a n . F o r the rather soft nuclei s t u d i e d here it is q u i t e clear t h a t the d e f o r m a t i o n d e p e n d e n c e o f mass p a r a m e t e r s and m o m e n t s o f inertia is o f crucial i m p o r t a n c e to determ i n e the a m o u n t o f s p r e a d i n g in the (fl, 7) sextant o f the associated c o l l e c t i v e w a v e - f u n c t i o n s . T h e s e calculations h a v e b e e n m a d e possible by c o m p u t i n g facilities p r o v i d e d by the C e n t r e de Caicul Vectoriel p o u r la R e c h e r c h e . P a r t o f this w o r k has b e e n p e r f o r m e d u n d e r the auspices o f the U S D O E by the L a w r e n c e L i v e r m o r e N a t i o n a l L a b o r a t o r y u n d e r contract N W 7 4 0 5 - E N G - 4 8 and also u n d e r the N A T O c o n t r a c t R G 8 5/0195.

References [ 1] P. Bonche, H. Flocard, P.-H. Heenen, S.J. Krieger and M.S. Weiss, Nucl. Phys. A 443 (1985) 39. [2] M. Girod and B. Grammaticos, Phys. Rev. Lett. 40 (1978) 361; Phys. Rev. C 27 (1983) 2317. [3] A. Charvet, T. Ollivier, R. B6raud, R. Duffait, A. Emsallem, N. Idrissi, J. Genevey and A. Gizon, Z. Phys. A 321 (1985) 697; N. Redon, T. Ollivier, R. B6raud, A. Charvet, R. Duffait, A. Emsallem, J. Honkanen, M. Meyer, J. Genevey, A. Gizon and N. Idrissi, Z. Phys. A, to be published. [4] S. Lunardi, F. Scarlassara, F. Soramel, S. Beghini, M. Morando, C. Signorini, W. Meczynski, W. Starzecki, G. Fortuna and A.M. Stefanini, Z. Phys. A 321 (1985) 177. [5] A.S. Davydov and G.F. Filippov, Nucl. Phys. 8 (1958) 237.

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[ 6 ] J. Meyer Ter Vehn, F.S. Stephcns and R.M. Diamond, Phys. Rev. Lett. 32 (1974) 1383. [7] J.H. Hamilton, P.G. Hansen and E.F. Zganjar, Rep. Prog. Phys. 48 (1985) 631. 18] J. Sauvage-Letessier, P. Quentin and H. Flocard, Nucl. Phys. A 370 (1981) 231. [9] B. Roussiere, C. Bourgeois, P. Kilcher, J. Sauvage, M.-G. Porquet and the ISOCELE Collab., Nucl. Phys. A 438 (1985) 93, and references quoted therein. [10l C.D. Lederer and V.S. Shirley, Table of isotopes, 7th ed.(Wiley, New York, 1978). [11]I. Berkes, At. Data Nucl. Data Tables (1986), to be published. [ 12] M. Beiner, H. Flocard, N. Van Giai and P. Quentin. Nucl. Phys. A 238 (1975) 29. [13] K.T.R. Davies, H. Flocard, S.J. Krieger and M.S. Weiss, Nucl. Phys. A 342 (1980) 11 I. [ 14] I. Ragnarsson, A. Sobiczewski, R.K. Sheline, S.E. Larsson and B. Nerlo-Pomorska. Nucl. Phys. A 233 (1974) 329. [ 15 ] K. Bencheikh, P. Quentin, J. Libert, M. Brack, J. Meyer and M. Meyer, to be published. [ 16] G.A. Leander and P. M611er, Phys. Lett. B 110 (1982) 17. [ 17] H. Flocard, P. Quentin and D. Vautherin, Phys. Lett. B 46 (1973) 304. [ 18] V. G~)tz, H.C. Pauli, K. Alder and K. Junker, Nucl. Phys. A 192 (1972) 1. [ 19] K. Kumar and M. Baranger, Nucl. Phys. A 110 (1968) 529. [201A. Ansari, Phys. Rev. C 33 (1986) 321. [21 ] P.O. Hess, J. Maruhn and W. Greiner, J. Phys. G 7 (1981) 737. [22] A.H. Wapstra and G. Audi, Nucl. Phys. A 432 (1985) 1. [23] S.G. Nilsson, C.F. Tsang, A. Sobiczewski, Z. Szymanski, S. Wycech, C. Gustafson, I.-L. Lamm, P. M611er and B. Nilsson, Nucl. Phys. A 131 (1969) I. [24] C. Ekstr6m, S. Ingelman, G. Wannberg and M. Skarestad, Nucl. Phys. A 292 (1977) 144. [ 25] A. Hayashi, K. Hara and P. Ring, Phys. Rev. Lett. 53 (1984) 337. [ 26] M. Baranger and M. V6n6roni, Ann. Phys. (NY) 114 (1978) 123.