Role of static and dynamic pairing correlations in the superdeformed band of 152Dy

Volume 198, number 1

PHYSICS LETTERS B

12 November 1987

R O L E O F STATIC A N D D Y N A M I C P A I R I N G C O R R E L A T I O N S

IN THE SUPERDEFORMED BAND OF lS2Dy Y.R. S H I M I Z U a,~, E. V I G E Z Z I a,2 and R.A. B R O G L I A a,b a The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen 0, Denmark b Dipartimento di Fisica, Universitgl di Milano, and INFN Sezione di Milano, Via Celoria 16, 1-20133 Milan, Italy

Received 18 December 1986; revised manuscript received 19 August 1987

The value of the pairing gap of both protons and neutrons is zero essentially over the whole range of the rotational frequency where the ( + ,0)-superdeformed configuration of ~S2Dyhas been observed. Pairing fluctuations give rise to a dynamic dealignment by about 5h, and alters the frequency dependence of the angular momentum to make jr2)> .¢o) at low frequencies, the difference becoming smaller as the angular momentum increases. These results indicate the important role played by dynamical pairing correlations in the superdeformed rotational band of 152Dy.

Recently, a superdeformed band in ~52Dy has been identified, which extends from spin 60h down to spin 22h [ 1 ]. The observed dynamical m o m e n t o f inertia j ( 2 ) ~ 84h 2 MeV-~ is consistent with the rigid rotation of a prolate nucleus displaying a ratio 2:1 between larger and smaller axis, throughout the whole range of the observed rotational frequencies, 0.25 < hOgro~< 0.7 MeV. This result poses a challenge to reconcile the "rigid rotation" of a configuration with parity and signature ( + , 0 ) at frequencies as low as 0.25 MeV, within the existing framework of pairing theory [2-5 ]. On the other hand, studies of pairing correlations in rapidly rotating nuclei [5,6] show that pairing fluctuations [ 7 ] lead to important renormalizations o f the nuclear motion after the pairing collapse, that is, at rotational frequencies where the normal (unpaired) phase has been realized. Because the large shell gap associated with the superdeformed configuration strongly hinders the scattering of pairs o f particles across the Fermi surface, the role of the static pair-field is, in this configuration, reduced with respect to standard deformed systems. U n d e r these On leave from Department of Physics, Kyushu University 33, Fukuoka 812, Japan. 2 On leave from INFN, Sezione di Milano, Via Celoria 16, 120133 Milan, Italy.

circumstances, the effects o f pairing fluctuations are expected to be specially important. In the present paper we investigate the properties o f static and dynamic pairing correlations as a function of the rotational frequency for the superdeformed configuration of ~52Dy. Because o f the simplicity o f the model, we do not intend to carry out a detailed comparison with the experimental data but merely to assess the consequences implied by allowing the pairing field to fluctuate. This is done in the framework of the cranked BCS theory with fixed deformation parameters and taking into account the effects o f the fluctuations induced by the pairing correlations in the (harmonic) random phase approximation (RPA) [ 5,6 ]. The Strutinsky renormalization procedure, which is important for quantitative discussions, is not taken into account. The deformed single-particle potential used in the calculations is the Nilsson hamiltonian formulated in ref. [ 8 ]. Namely, the non-stretched 1.s and/2-terms are used with their parameters taken from ref. [ 9 ], and the major shell mixing induced by the cranking term is neglected. The resulting Nilsson diagram is similar to that of ref. [ 9], although in the present case the Z - - 6 6 shell gap at e2 ~ 0.6 is larger than the corresponding shell gap for Z = 64. The model space used contains the deformed harmonic oscillator shells with principal q u a n t u m numbers N - - 3 , 4, 5, 6, 7 and 2,

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33

Volume 198, n u m b e r 1

PHYSICS LETTERS B

3, 4, 5, 6 for neutrons and protons, respectively. The pairing force strengths G, = 16/A, Gp = 20/A ( M e V ) were used. They reproduce the experimental even-odd mass difference in the normal deformed band within the same model space. In all calculations shown below, we have used the deformation parameters e2=0.58 and 64=7=0, independent of the rotational frequency. A more realistic Nilsson-Strutinsky type calculation (cf. ref. [ 10] and references therein) led to similar values of ~2 and 7 for the superdeformed minimum, which are rather constant as a function of the rotational frequency. However, the hexadecapole deformation (E4~0.02 at hO)ro,=0) is found to increase as the spin increases, and to slightly change the single-particle spectra [ l 0]. We have checked the dependence of the results presented below on the single-particle spectrum by carrying out the calculations for two values ofe2 (0.56 and 0.60) and found essentially no changes. It should also be mentioned that increasing the deformation, some orbits from the proton N = 7 shell come down to an energy region rather near the Fermi-surface. We have repeated the calculations including the N = 7 proton shell and found that the effect of the associated orbitals on the pairing fluctuations is small, although in this case the proton static pair gap Zip survives to slightly larger rotational frequency (fia~rot~ 0.28 MeV, see below). In fig. l we show the self-consistent values of the pairing gap and the correlation energy induced in the superdeformed configuration by the pairing fluctuations. Also shown are the values of the angular momentum as a function of the rotational frequency calculated both with and without the contributions arising from dynamic pairing correlations. The static pairing fields of both neutrons and protons are completely quenched at horror~ 0.25 MeV. Actually the neutron pairing gap is zero at all frequencies ~', while the proton pairing gap at fitO~o,= 0 amounts only to 60% of the odd-even mass difference. This result was expected. In fact, because of the large single-particle energy gap (1.5-2 MeV) at N = 86, Z = 66, the dimensionless order parameter ~2 ~ If the value E2= 0.605 is used, the neutron pairing gap is z/, ~ 0.3 MeV at htoro~-- 0 and vanishes at ha) ro, ~ 0.1 MeV. ~2 Here ~2 is the effective shell-degeneracy and D is the singleparticle gap between occupied and unoccupied orbits.

34

12 November 1987

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Fig. I. (a) Correlation energy associated with the ( + , 0 ) superdeformed configuration of ~52Dy induced by the pairing fluctuations for both neutrons and protons. In the inset the static pairing gaps are shown. (b) The continuous curve represents the total angular m o m e n t u m , plotted as a function of the rotational frequency. The dashed and dash-dotted lines show the neutron and proton contributions, respectively. The experimental data taken from ref. [ 1 ] is also shown. (c) Same as (b) but including the effect of pairing fluctuation (cf. eq. ( I )). The singular behaviour caused by the cusp of the proton contribution to Ecorr at the critical frequency was smoothened out as discussed in ref. [6]. The overestimation of the calculated results is due to the use of the Nilsson potential without the Strutinsky renormalization.

Volume 198, n u m b e r 1

PHYSICS LETTERS B

which measures the competition between pairing and single-particle shell effects [ 11 ] is of the order of unity, indicating that the system is close to the phase transition between the normal and superfluid phases already at ho9~o,= 0. Although the static pairing gaps collapse at rather low frequencies, the dynamic pairing fluctuations play an important role on the values of a variety of observables like the angular momentum, up to rotational frequencies as high as 1 MeV. In particular, the results displayed in figs. lb and lc indicate that they give rise to a dynamic dealignment of the system measured by

x= G/D

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and amounting to 5 h - 3 h at hogrot=0.3-0.8 MeV, respectively. In this context, it should be noticed that the absolute value of the calculated angular momentum displayed in fig. 1 overestimates the experimental value by 10-15%. This is due to the fact that the Strutinsky renormalization of the angular momentum has not been performed. Therefore, no absolute comparison with the data is possible. However, from this figure we can see that the dealignment (eq. (1)) is of the same order of magnitude as the Strutinsky shell corrected contribution to the angular momentum. Pairing fluctuations also affect the rotational frequency dependence of the angular momentum. This is seen in the difference between the kinematic and dynamic moments of inertia J<~)=I~/Ogro, and j ~ 2 ) =OIJOogrot, (cf. fig. 2). If the effect of pairing fluctuations is neglected (fig. 2a) j ~ ) > j < 2 ) , and the difference becomes larger as hto rot increases. The inclusion of the pairing fluctuation reverses the picture, in qualitative agreement with the data. Because of the ubiquitous role pairing fluctuations have as a source of renormalization of the properties of strongly rotating nuclei, it seems important to get direct information on these modes of excitation. This could be accomplished through two-nucleon transfer reactions (cf. refs. [12,13]). The superdeformed ~52Dy configuration seems to be an ideal case to carry out these studies. This is because the large single-particle shell gap, caused by the superdeformation, makes the ( + , 0 ) configuration undergo a phase transition into the normal phase at quite low rotational frequencies. The resulting system resembles,

12 November 1987

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Fig. 2. The kinetic and the dynamic m o m e n t s of inertia, J < ~ and j<2) as a function of the rotational frequency: (a) results without taking into account pairing fluctuations, (b) results including pairing fluctuations. The absolute value of J is overestimated because the Strutinsky renormalization of the angular m o m e n t u m was left out.

to some extent, the one known from (t,p) reactions on 2°8Pb at ho)ro~=0 (cf. ref. [14] and references therein). This is shown in fig. 3, where the strength functions for two-neutron transfer are shown in the range hogrot=0.3-0.7 MeV. At hO)rot=0.3 MeV a pronounced peak at hoJRvg= 1.5 MeV is observed, which resembles that associated with the collective pairing vibrations based on the ground state of 2°Spb. The situation will be even more pronounced at h O J r o t ~ 0 . Increasing the rotational frequency, the collectivity of the vibrations is reduced and the pairing strength spreads over many non-collective roots (Landau damping) resembling the behaviour observed in strongly rotating normal "deformed" nuclei [ 5,15 ]. We conclude that the zero-point motion induced by pairing vibrations play a crucial role in the value 35

Volume 198, number 1

PHYSICS LETTERS B '

I

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12 November 1987

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Fig. 3. The two-neutron transfer strength functions for hcorot=0.3, 0.4, 0.5, 0.6, 0.7 MeV. The continuous (dashed) curve represents the pair addition (pair removal) modes. An energy averaging parameter of 200 keV is used (cf. ref. [6]). a n d r o t a t i o n a l f r e q u e n c y d e p e n d e n c e o f the a n g u l a r m o m e n t u m a s s o c i a t e d w i t h the ( + , 0 ) s u p e r d e f o r m e d c o n f i g u r a t i o n o f ~52Dy. T h e study o f the s u p e r d e f o r m e d b a n d t h r o u g h the t r a n s f e r o f t w o n u c l e o n s i n d u c e d by h e a v y i o n s can p r o v i d e d i r e c t i n f o r m a t i o n o n the p a i r a d d i t i o n a n d p a i r r e m o v a l modes. O n e o f the a u t h o r s ( Y R S ) is i n d e b t e d to the N i s h ina M e m o r i a l F o u n d a t i o n for financial support. T h i s p r o j e c t has b e e n e x e c u t e d w i t h a g r a n t o f the C o m m e m o r a t i v e A s s o c i a t i o n for t h e J a p a n W o r l d Exposition.

References [ 1] P.J. Twin, B.M. Nyako, A.H. Nelson, J. Simpson, M.A. Bentley, H.W. Cranmer-Gordon, P.D. Forsyth, D. Howe, A.R. Mokhtar, J.D. Morrison, J.F. Sharpey-Schafer and G. Sletten, Phys. Rev. Lett. 57 (1986) 811. [2] J.L. Egido and P. Ring, Nucl. Phys. A 383 (1982) 189; A 388 (1982) 19; Phys. Lett. B 95 (1980) 331; J.L Egido, HJ. Mang and P. Ring, Nucl. Phys. A 334 (1980) 1;A 341 (1980) 229. [3] U. Mutz and P. Ring, J. Phys. G 10 (1984) L39; S.Y. Chu, E.R. Marshalek, P. Ring, J. Krumlinde and J.O. Rasmussen, Phys. Rev. C 12 (1975) 1017; L.F. Canto, P. Ring and J.O. Rasmussen, Phys. Lett. B 161 (1985) 21; 36

S. Bose, J. Krumlinde and E.R. Marshalek, Phys. Lett. B 53 (1974) 136. [4] J. Krumlinde and Z. Szymanski, Phys. Lett. B 36 (1971) 157; W. Nazarewicz, J. Dudek and Z. Szymanski, Nucl. Phys. A 436 (1985) 139. [5] M. Gallardo, M. Diebel, S. Frauendorf and R.A. Broglia, Phys. Lett. B 166 (1986) 252. [6] Y.R. Shimizu, J.D. Garrett, R.A. Broglia, M. Gallardo and E. Vigezzi, to be published. [7] D.R. Bes and R.A. Broglia, Nucl. Phys. 80 (1966) 289. [8] A. Bohr and B.R. Mottelson, Nuclear structure, Vol. 2 (Benjamin, Reading, MA, 1975). [9] T. Bengtsson and I. Ragnarsson, Nucl. Phys. A 436 (1985) 14. [ 10] I. Ragnarsson and S. Aberg, Phys. Lett. B 180 (1986) 191. [ 11 ] R.A. Broglia, C. Riedel and B. Sorensen, Nucl. Phys. A 107 (1968) 1. [ 12 ] M. Guidry, Proc. Conf. Nuclear structure with heavy ions, eds. R.A. Ricci and C. Villi (Editrice Compositori, Bologna, 1975) p. 99. [ 13 ] R.A. Broglia, Theory of nuclear structure and reactions, eds. M. Lozano and G. Madurga (World Scientific, Singapore, 1985) p. 133. [ 14] R.A. Broglia, O. Hansen and C. Riedel, Adv. Nucl. Phys. 6 (1973) 287. [ 15 ] R.A. Broglia, M. Diebel, F. Barranco and S. Frauendorf, XXIII Intern. Meeting (Borrnio), ed. 1. Iori, Ricerca Scientifica ed Educazione Permanente, Universith di Milano, Suppl. No. 47 (1985) 1; R.A. Broglia and M. Gallardo, Nucl. Phys. A 447 (1985) 484.