Octupole deformations in the very extended nuclear shapes of the A ∼ 180 region

cs .-__

__ k!!B

ELSEVIER

2.5 May 1995

PHYSICS

LETTERS

B

Physics Letters B 351 (1995) 18-23

Octupole deformations in the very extended nuclear shapes of the A - 180 region R.R. Chasman a, L.M. Robledo b aPhysics Diuision, Argonne National Laboratory, Argonne, IL 60439-4843, USA b Dept. de Fisica Teorica C-Xl, Uniuersidad Autonoma de Madrid, E-28049 Madrid Spain Received 9 March 1995 Editor: G.F. Bertsch

Abstract

Energy surfaces in the A N 180 region have been investigated on an extended grid in a four-dimensional deformation space consisting of quadrupole, o&pole, hexadecapole and necking deformations. A lowering in the energy surface of several MeV caused by octupole deformation, is found for many nuclides in the vicinity of 176W at very large values of the elongation and necking. Octupole softness of the known extended minima is found to be quite similar in most instances.

In the past few years, calculations have shown that superdeformed nuclear shapes are accessible in the A = 150 [1,2] and the A = 190 [3] mass region. These predictions have been experimentally [4,5] verified. In recent studies, using both the cranked Strutinsky method [6] with a necking degree of freedom and using the Gogny interaction with the HFB method [7], we have found a minimum in the energy surfaces of nuclei in the A = 180 mass region characterized by very extended shapes with axis ratios of 2.2 : 1. Prior studies of the effects of octupole deformation on superdeformed shapes [8] were carried out on a grid in a &, &, two-dimensional deformation space with p4 chosen to minimize liquid drop energies. In these calculations, no static octupole minima were found. However, a softness to octupole vibrations was noted in the regions of 19’Hg at I = 40 and of rsoGd at I = 60. Because the liquid drop energy increases rapidly with & it was observed that values of & larger than 0.1 give shapes that are highly

excited relative to reflection symmetric shapes. A study of Hg isotopes at I = 0 shows [9] that the minimum at & _ 0.8 in laoHg is slightly deformed in an octupole sense. Calculations at I = 80 in the A = 150 mass region show [lo] that octupole deformation plays an important role in stabilizing the hyperdeformed minimum that has been reported [ill in “‘Dy. Some time ago, we noted that the inclusion of P&OS 0) deformations in the description of the light actinides accounts for much of the discrepancy between observed and calculated masses. When one includes P,(cos 0) deformation, octupole minima become considerably shallower [12] and octupole effects are best characterized as “strong correlations” or incipient deformation, in agreement with the many-body study of octupole correlations [13] in the actinides that predates Strutinsky calculations in this region. In our studies of very extended nuclear shapes in the mass region A = 170-200, on a large three-dimensional grid in a space consisting

0370-2693/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDZ 0370-2693(95)00327-4

R.R. Chasman, L.M. Robledo/Physics

of quadrupole, hexadecapole and necking deformations, we found that the inclusion of a necking degree of freedom is important for the accurate description of the 2.2 : 1 shapes that appear to be most accessible in the nuclei around ‘*‘Os. It gives shapes quite similar to those given by our HFB calculations in which only (Q,) is constrained (and varied). In leading order, the necking degree of freedom corresponds to a P,(cos 19) deformation, when v4 is non-zero. With this in mind, it seems worthwhile to carry out a large scale study of oc!upole effects in a four dimensional deformation space, consisting of quadrupole, octupole, hexadecapole and necking degrees of freedom. In addition, we include a cranking frequency in order to study the energy surfaces at non-zero spins. It is only at high spins, that the very extended minima become accessible. At Z = 0 these calculations are quite manageable, because states with different values of J, do not mix and the matrices to be diagonalized are comparatively small. The cranking term in the Hamiltonian mixes states of different J, and the matrices to be diagonalized include all values of J,. The inclusion of reflection asymmetric degrees of freedom doubles the size of the matrices to be diagonalized and quadruples the computation time, The recent availability of massively parallel processor systems makes it feasible to consider cranked Strutinsky calculations in a four dimensional deformation space. We examine here the 72 even-even nuclides going from Z = 70 to 84 and for N = 98 to 114. In this work, we emphasize superdeformed and hyperdeformed shapes. In the absence of octupole correlations, several extended minima have been found in calculations [3,6,7,9,14,15] covering this mass region; one that becomes yrast in the A = 170 region, near Z = 80 with an axis ratio of 2.5 : 1; three minima in the A = 180 region, one characterized by axis ratios of 1.5 : 1, that becomes yrast near Z = 40, a second with axis ratios of 2.2 : 1 that becomes yrast near Z = 60 and a third with axis ratios of 2.9 : 1 that becomes yrast about Z = 80. In addition there are the well known superdeformed minima around “‘Hg with axis ratios of N 1.6 : 1. We use a Woods-Saxon potential to calculate shell corrections and a liquid drop model with parameters that have been given [3,6] previously. Here,

19

Letters B 351 (1995) 18-23

we extend our shape space to include octupole deformations, by setting {r2[cos20 exp( -+v,)

+ sin28 exp(tv,)

+ (3.5)“” v&(cos

0)

+ (4.5) 1’2 v‘&(cos

B)])

x (1 - vnk sin20)-r = R;.

(1)

The quadrupole deformation v2 is varied from 0 to 4.90 in steps of 0.05; the hexadecapole deformation, v, is varied from -0.12 to 1.8 in steps of 0.02 (however the range of v, is determined by v2 and there are never more than 65 values of v, used for any value of v,); vnk is varied in steps of 0.2 from -1. to 0.4 and in steps of 0.1 from 0.4 to 0.7. Finally, we let vg vary from 0 to 0.20 in steps of 0.05. We restrict the shapes to those having surface areas less than 1.27 in units of the surface area of the spherical nucleus. A value of 1.26 corresponds to the surface area of two spherical nuclei, each having half the mass of the original nucleus and we feel that the details of fission are more appropriately studied with Hartree-Fock calculations. We have done additional calculations over limited grids in v2, vq and vnbkfor vg values of 0.0, 0.3 and 0.4. In the calculations on the limited grid for vg = 0, we allowed surface areas of up to 1.32 in order to examine the effects of the 1.27 restriction. Also, the calculations for vs = 0.30 were carried out to study the most extended shapes and we allowed surface areas of up to 1.30. When one is dealing with large quadrupole deformations and high spins, it is not useful to consider energy surfaces in a space labeled by v2, v,. For large values of v2, one finds that the minimum energy becomes roughly constant no matter how large one makes the quadrupole deformation v2, because increases in v2 are offset by changes in v, such as to keep the nuclear elongation roughly constant. The value of the quadrupole deformation, v2, is near 4.0 at the 2.2 : 1 minimum in the OS region. In the absence of v, and vnk deformations, this would mean axis ratios of w 50 : 1 rather than 2.2 : 1. Similarly the actual necking in (or out) of a shape depends on v2, v, and vnk in a complicated way. It is useful to consider more physical characterizations

20

R.R. Chasman, L.M. Robledo/Physics

of the nuclear shape, such as elongation and necking as has been done [16] in fission studies. We make a two-dimensional projection of the energy surface in terms of the physical quantities v2 and qnk defined below. For pure va deformations, the maximum extension of a nucleus in the Z-direction is exp (iv,>; i.e. u2 = 1.5 l&Z,,,). We define a physically meaningful quantity, ~a, equal to v2 for purely quadrupole deformations, as r/2 = 1.5 ln(Z,,,),

Letters B 351 (1995) 18-23

F” 0.6

(2)

where Z,,, is i the length of the liquid drop shape. 72 and ZJ* do not usually coincide when other deformation parameters are non-zero. When v2 is the only non-zero deformation, we set v* to zero by defining vti =X,,,(e=

90’) - exp( -3r12),

0.8

1.0

(3) -

This definition remains useful when we extend the deformation space to include va as X,,(0 = 90”) does not change with vs. We define the term axis ratio as [(Z2)/(X2>]1/2. As is the case with v2, the value of v3 is not always a useful gauge for octupole shapes. For example, we found that the energy of the 2.2: 1 minimum in the A = 180 region hardly changes for values of va between 0 and 0.2. Such extreme octupole softness would be a very exciting result! However, an examination of the liquid drop shape shows essentially no change for this variation of u3. To get a true picture of octupole effects, it is more useful to consider a coordinate that is proportional to (Qs), the octupole moment, rather than to the deformation parameter vs. We define v3, bearing close resemblance to v3, by dividing (Qa ) by R3 and choosing a proportionality factor that makes q3 the same as y3 for the superdeformed minimum in lg2Hg at vg = 0.1. We have extended calculations to v3 = 0.4 for grid points in the vicinity of the 2.2 : 1 minimum to get a better idea of the response of the nucleus to octupole deformation for such shapes. The most striking effect arising from the inclusion of octupole deformation is a - 7 MeV lowering of some points on the energy surface for large deformations and large values of the necking. The effects are particularly large for neutron numbers from 100 to 108 and become much smaller by N = 112. This is not a liquid drop effect, as the liquid drop energies

1.2

-0.30

-0.20

-0.10

0

q nk Fig. 1. Energy surface of 17~W at Z = 70. Only reflection symmetric shapes are included. The numbers on the contours are excitation energy in MeV. The coordinates qz and Q are defined in the text.

F”

0.6

-0.80

-0.20

-0.10

0

“rlnk Fig. 2. Energy surface of 176W at Z = 70. Reflection asymmetric shapes are also included in this figure. See caption for Fig. 1.

R.R. Chasman, L.M. Robledo /Physics Letters B 351 (199)

do not vary substantially as a function of vg. Actually, there are two slightly different regions where one gets lowering of energy with octupole deformation. The first is for q2 of N 0.9 (axis ratios of - 3.1: 1) and the second for q2 of N 1.1. (axis ratios of N 3.7 : 1). In Fig. 1, we plot contours of the energy surface of 176W at I = 70 for reflection symmetric shapes, where the VEM (very extended minimum) is almost yrast. In Fig. 2 we show the energy surface with octupole deformation included. The first thing to note is that the minima are not shifted when octupole deformations are introduced. The second is the subtantial changes in the lower left of the two figures. For q2 = 1.1, at the point labeled A in Fig. 2, there is a * 7 MeV decrease in the energy and for qz = 0.9, at the point labeled B, the decrease is N 2 MeV. The shaded areas in the figures show regions not covered in the calculations. A detailed examination of shell corrections shows that there are large positive shell corrections for the reflection symmetric shapes for both neutrons and protons; and moderate negative shell corrections for the reflection asymmetric shapes in the region of point A. For larger neutron numbers, the neutron shell corrections are reduced considerably for reflection symmetric shapes and the differences become small. The gains in energy associated with octupole deformation in the vicinity of point B decrease with increasing 2. For many points, the greatest lowering of the energy in the vicinity of point A occurs for the maximum value of va that we have considered ( vg = 0.2). Therefore we calculated a limited number of grid points for vg = 0.30 for shapes in this region with the values of v2, v4 and r~,,~chosen to cover a moderate range in the vicinity of the v3 = 0.2 mini-

aq,, -2.0

,, , ,, ,, , -1.6 -1.2

-0.6 -0.4

(, 0

, I, 0.4

, , , ,I

(, 0.6

1.2

1.6

2.0

Fig. 3. Liquid drop shapes for reflection symmetric and reflection asymmetric shapes for r/s = 1.1 (point A in Fig. 2).

18-23

21

Table 1 Energy lowering in MeV with octupole deformation at Z = 70 Yb

Hf

W

OS

Pt

Hg

Pb

PO

Z= 70

72

74

76

78

80

82

84

98

2.1 0.2

3.9 1.0

6.1 1.3

6.2 -

5.4 0.5

5.2 0.4

2.6 -

1.3 -

100

3.1 1.2

4.8 0.6

6.6 1.8

6.7 0.2

6.0 0.4

5.5 0.4

3.2 -

2.0 -

102

2.9 0.2

4.9 1.1

6.9 2.0

6.8 0.6

6.1 0.5

5.6 0.4

3.1 0.1

1.9 -

104

2.3 0.5

4.3 1.9

5.6 2.9

5.5 1.7

5.0 1.5

4.4 1.3

2.1 0.9

0.9 -

106

1.2 0.7

3.2 2.1

4.6 3.3

4.5 1.6

4.0 1.4

3.9 1.4

1.6 1.1

0.7 0.5

108

0.8 0.8

1.9 2.4

3.3 3.5

3.0 1.8

3.1 1.5

2.6 1.6

0.8 1.1

0.4

110

0.4 0.9

1.0 2.5

1.3 3.3

1.5 2.0

1.6 1.8

1.1 1.8

1.6

0.9

112

0.2 0.3

0.4 1.8

2.5

0.2 1.2

0.3 1.0

1.1

0.9

0.2

114

0.1

0.2

-

-

-

-

-

-

0.3

1.3

2.0

0.7

-

0.6

0.4

-

N

The upper entry for each value of Z and N is the energy lowering at point A of Fig. 2. The lower entry is for point B.

mum. In general, the vs = 0.30 points do not lower the energy further, but in a few instances there is a lowering of N 200 keV relative to the va = 0.20 points. In Fig. 3 we show the shapes that give the reflection symmetric minimum energy and the reflection asymmetric minimum energy at point A. The top half of the figure is the reflection symmetric shape and the bottom half is the reflection asymmetric shape. It is much easier to visualize each of the shapes by covering half of the figure. Fig. 3 also illustrates the utility of defining the energy surface in terms of r/_, and Q~. The shapes that are being compared in Fig. 3 are very similar, differing mostly in the position of the neck. The situation is much the same for point B. In Table 1 we show the energy lowering arising from reflection asymmetric shapes at points A and B at I = 70, for the nuclides considered in this study. These values are not very sensitive to angular momentum, between I = 60 and I = 80.

22

R.R. Chasman, L.M. Robledo/Physics

Looking at these results more closely suggests a qualitative explanation for much of what we see in terms of shell corrections for incipient fission [17] fragments. Such an approach has recently been sugested [18] to explain the hyperdeformed shapes in $!Q Th. Such arguments are at best suggestive, because the Coulomb field of the other incipient fragment is an important feature. Deformed shapes are much more energetically favored in the incipient fragments than they are in isolated fragments. There are two relevant fragment masses that pertain to this region. One is a spherical shape in the vicinity of 9oZr that arises from the magic neutron number 50 and the semi-magic proton number 40. The second is a deformed shape that is favored [19] for proton numbers from Z = 36 to Z = 44 and neutron numbers above N = 60, centered at ro4Zr. We can associate the Q = 0.90 shapes with incipient spherical fragments and the r/a = 1.1 region with incipt deformed fragments. For the nuclides around l8 Hg, one can have a symmetric nuclear shape with both incipient fragments corresponding to a spherical 9oZr, favoring symmetric shapes at r/a = 0.9 in this region. For the more elongated shapes corresponding to

I-

,

I

/

_.x' '0 i-r I I I I I II 4'@OS s3_ I=60 m 2.2:1 E Lu 2P , l/ /

/' / //

2l-

/'

-- *' I 1 7" -- 192 _. I =% 1.6:1

.d' .' IIfI(

11

0 II

I

0 I"1

/

/

/

/I' , / / . 7'1 I I II 1'1 0.05 0

I'

_HH O-T-t3 II ,I( 0.05 0

,

'@OS __ I=60 2.9:1p , ,

P 1.6:1

7"

1 I

_H

/

1 II 0.10

/

P

-

I 0.10

q3

Fig. 4. Energy as a function of q3 for characteristic extended minima in the A - 180 region. The lines are drawn to guide the eye.

Letters B 351(1995)

18-23

~a = 1.1, we expect the symmetric shapes to come closer in energy to the reflection asymmetric shapes as both of the incipient fragments have neutron numbers approaching N = 60. The Q = 0.9 energy differences are dominated by a large negative proton shell correction for the reflection asymmetric shape centered at Z = 74. As Z approaches 80, the proton shell corrections are quite similar for reflection symmetric and reflection asymmetric shapes. We next turn to the change in energy with octupole moment for typical examples of the extended minima.in this region. We emphasize that all deformation parameters are varied I freely on the grid, when the energy is calculated as function of Q. In several cases, the minimum washes out completely with increasing octupole deformation and we terminate the curve. For the 2.2: 1 minimum, we performed a calculation over a small grid, fixing ~a at 0.40, in order to get a better idea about the softness with regard to octupole deformation for this minimum. The final point for the 2.2 : 1 shape in 1820s should be considered an upper limit. To emphasize the disparity between ZJ~and r/a, we note that ua = 0.40 translates to 7j3_ 0.06 for this point. In Fig. 4 we plot the energies of typical minima as a function of r/a. The interesting thing to note here is that five of the six cases are quite similar; the excitation energy is _ 1 MeV for r), = 0.05. It is slightly higher for lg2Hg and a little lower for the 1.5 : 1 shape in rs20s. Only for the 2.9 : 1 minimum in rs20s, is the nucleus much stiffer with respect to octupole deformation. We plan to look at the question of octupole softness more closely by calculating the excitation energies of the negative parity states using the generator coordinate method. Generator coordinate method calculations of octupole softness in the superdeformed minimum of lg4Pb 1201 have been carried out at Z = 0. In this work, we have carried out a survey of nuclides in the A = 180 region on a extended fourdimensional grid in deformation space. Octupole deformation plays an important role in determining nuclear energy surfaces in many of the nuclides of the A = 180 mass region, particularly at the largest deformations, characterized by ~a = 1.1 (axis ratios of _ 3.7 : 1) and by r12= 0.9 (axis ratios of N 3.1: 1). We also find that the energy surface is more or less equally stiff to octupole deformation for

R.R. Chasman, L.M. Robledo /Physics

most of the superdeformed, very extended and hyperdeformed minima in this mass region. We thank J.L. Egido and R. Janssens for useful discussions on this work. The calculations reported here were carried out on the SP computer of the MCS Division of Argonne National Laboratory and at the NERSC facility at Livermore. The research of R.C. is supported by the US Department of Energy, Nuclear Physics Division under contract W-31-109ENG-38. The research of L.M.R. is supported in part by DGICyT, Spain under project PB91-0006. Our collaboration has been greatly facilitated by a NATO Collaborative Research Grant 921182.

References [l] K. Neergaard, V. Paskevitch A 262 (1976) 61.

and S. Frauendorf,

Nucl. Phys.

Letters B 351 (1995) 18-23

121J.

23

Dudek and W. Nazarewicz, Phys. Rev. C 31 (1985) 298. [31 R.R. Chasman, Phys. Lett. B 219 (1989) 227. [41 P.J. Twin et al., Phys. Rev. Lett. 57 (1986) 811. 151 E.F. Moore et al., Phys. Rev. Lett. 63 (1989) 360. [61 R.R. Chasman, Phys. Lett. B 302 (1993) 134. [71 J.L. Egido, L.M. Robledo and R.R. Chasman, Phys. Lett. B 322 (1994) 22. [81 J. Dudek, T.R. Werner and Z. Szymanski, Phys. Lett. B 248 (1990) 235. 191 W. Nazarewicz, Phys. L&t. B 305 (1993) 195. m S. Aberg, Lund Institute of Technology Report Lund-Mph92/09. et al., Phys. Rev. Lett. 71 (1993) 231. 1111 A. Gahndo-Uribarri ml R.R. Chasman, Phys. Len. B 175 (1986) 254. D31 R.R. Chasman Phys. Lett. B 96 (1980) 7. D41 J. Dudek, T. Werner and L.L. R&linger, Phys. Lett. B 211 (1988) 252. 1151 T.R. Werner and J. Dudek, At. Data Nucl. Data Tables 50 (1992) 179. b51 J.R. Nix, Ann. Rev. Nucl. Sci. 22 (1972) 65. [171 B.D. Wilkins et al., Phys. Rev. C 14 (1976) 1832. ml S. Cwiok et al., Phys. Lett. B 322 (1994) 304. WI R.R. Chasman, Z. Phys. A 339 (1991) 111. DO1 P. Bonche et al., Phys. Rev. Lett. 66 (1991) 876.