Analysis of octupole instability in medium-mass and heavy nuclei

Nuclear Physics A429 (f984) 269-295 @?Nosh-HoIland Publishing Company

ANALYSIS OF OCIWPULE INSTABILITY IN TEDIUM-MASS AND HEAVY NUCLEI W. N~REWICZ$, ~~r~e~r

P. OLANDERS

and I. RAGNARSSON

of Maikematicai Physics,Lund Instituteof Technology,S-.22@07 Lund, Sweden J. DUDEK Centre de Recherches Nucizhires, F-67037 Strasbourg Cedex, France and nte Niels Bohr Instifute, DK-2100 Copenhagen 0, Denmark G. A. LEANDER ~~~S~~

Oak Ridge Assoc~ted animosities, Oak Ridge, ?%messee, USA

P. MdLLER** Lawrence Berkeley National Luborutory,Berkeley, Califontia, USA and E, RUCHOWS~ KVI, ~ni~~~ty

of @oningen,~~n~ngen~ TRe Netherlan~ and

Instituteof ExperimentalPhysics, Universityof Warsaw, Warsaw,Poland Received 8 May 1984 Abstract. S~tinsk~ty~ potential-energy ~lculations are avowed using various models For the microscopic as well as macroscopic energy terms. The following conclusions are nearly modelinde~ndent: Ra-Th nuclei around A==225 have ground-state equilibrium shapes with both quadrupole and octupole deformation; nuclei in the heavy barium region are just barely unstable with respect to octupole distortion of the quadrupole shape, while lighter nuclei are stable with respect to octupole deformation, albeit softness occurs e.g. in the region of the very light and the very heavy 2 =n36 isotopes. Tables of ground-state equilibrium shapes, calculated with the WoodsSaxon potential, are presented for doubly-even nuclei in the Ba and Ra-Th regions of octupoledeformed nuclei.

1. Introduction When the concept of spontaneous intrinsic symmetry-breaking in many-body systems was introduced ‘) to nuclear physics around 1952, it raised the long-standing question of whether intrinsic pasty-violating collective asymmet~ may occur in the * On leave from Institute of physics, Technical University, Pl-00-662 Warsaw, Poland. ** On leave from the Depa~ment Sweden.

of Mathematical 269

Physics, Lund Institute of Technology, Lund,

270

W. Nazarewicz et al. f Octupole instability

ground states of some nuclei. The first investigations within the deformed shell model did not lend major support to this idea [e.g* refs.“‘)], and unnatural-parity excited states were described in a basis of few-quasip~icle excitations relative to a reflection symmetric state [e.g. refs. “*‘)I. The advent of the St~tinsky method 8, led to a new series of investigations after 1970 [e.g. refs. ““)], Also WartreeFock calculations with defo~ation constraints were carried out for very light nuclei [e.g. ref. ‘“)I. ~though in some nuclei the intrinsic energy was lowered slightly by the octupole degree of freedom, the corresponding very-shallow potential minima were not considered significantly different from the soft potentials obtained previously. Since 1981, however, several studies have indicated that the Strutinsky method does in fact lead to significant stabilization of oetupole deformation in the Ra-Th region’3-‘6). At the same time other studies presented both experimental and theoretical arguments for collective dipole and octupole deformation [e.g. refs. i7-3’)]. The purpose of the present investigation is: (a) to elarify the developments in the theory responsible for the current rest&s, and generally to investigate the model sensitivity of predictions about octupole deformation in nuclear ground states ; (b) to search for octupole equilibrium deformations in regions of nuclei other than the Ra-Th region; (e) to present tables ‘of octupole equilib~um shapes, calculated with the version of the Woods-Saxon model developed by the Warsaw group 32-34).

The approaches applied here are all based on the St~tinsky method *> in which the bulk-energy contribution is obtained from a macroscopic liquid-drop model formula, whereas the microscopic shell correction is derived from the single-partide energy spectrum [for details see e.g. ref. 35), and references therein]. 2.1. THE MACROSCOPIC

ENERGY

TERM

Early St~tinsky calculations used the sharp-surface liquid-drop formula with the parameters of Myers and Swiatecki 36): E(defomted)-E(sphere)=(BG-l)aCZ2fA”3+(Bs-l)a~A2’3,

(11

where the shape dependence is contained in the normalized Coulomb and surface area integrals SC and B,,while ac and ai are the Coulomb and surface energy constants. In recent years there has been increasing use of a modified surface term 37). In this term the surface diffuseness and the finite range of the nuclear interaction are simulated by a folding integral over the equivalent sharp surface: E~~shape~ = --a,”

(2)

U? Nazarewicz

271

et al. / Octupole instability

I--

I

I 1.5 -

/ _ -__

MOD. L.D. STD. L.D.

/

O-

‘““a”/ / /

1

If

._;;

t I.5 -

l-

I 0

I 0.05

I 0.10

43 fop&) Fig. I. Comparison of octupote-deformation-energy curves obtained with the standard liquid-drop formula ‘) (dashed curves) and the modified formula with a finite-range interaction in the surface energy term 13) (solid curves). The shell correction was obtained from the WS potential.

where CT= ]r - r’] and a is the range of Yukawa-plus-exponential folding function while a,” is the generalized surface energy constant. As a result of this modifi~tion the surface becomes softer with respect to small wiggles, so that for a given stiffness against quadrupole deformation the stiffness against higher multipole deformations is reduced. A second modification concerns the Coulomb energy term where the surface diffuseness correction is now evaluated exactly ‘3*38). The importance of these modifications for ground-state octupole deformations is illustrated in fig. 1, which shows the total energy for two selected nuclei (‘&Ba and =*Ra) plotted as a function of octupole deformation (p3) at the quadrupole and hexadecapole deformations (& and &) which minimize the total energy. The solid curves were obtained with the modification mentioned above using the parameters of ref. 38). The dashed curves were obtained with the sharp-surface area term and the standard parameters of Myers and Swiatecki 36). The same Woods-Saxon singleparticle potential was used in both cases to calculate the microscopic energies. It is seen that the supposedly more realistic modified surface energy term does play some role in the development of octupole-deformed minima, and that it actually tips the scale in soft nuclei like the neutron-rich ones around ‘46Ba. 2.2. THE MICROSCOPIC

ENERGY TERM

Three single-particle potentials are considered: the modified oscillator (MO), folded Yukawa (FY) and Woods-Saxon (WS) potentials with parameters from

272

W. Nazarewicz et al. / Octupoie ~~tab~fity

refs. 39*‘3)and ref. 34), respectively. We shall conclude that it’is probably more the different fits of the single-particle levels than the models themselves that are tested when different potentials are compared. On the basis of the general condition for appearance of shell structure, Bohr and Mottelson 40) anticipated a greater propensity for octupole deformation in flatbottomed potentials (like FY and WS) as compared with the harmonic oscillator. The argument in its simplest form is that classical closed trajectories may be triangular in Bat-bottomed potentials whereas in an oscillator potential the closed orbits are elliptic. It then follows that in realistic potentials the orbitals with AZ = 3 may come close together in energy. In the harmonic oscillator where all degenerate orbitals have Al = 2 (or 4,6 . .>, such orbitals stay relatively far apart. The shell structure in the single-particle spectra for a harmonic oscillator and for a more realistic nuclear potential is illustrated in fig. 2. car small I-values the oscillator gives an appropriate level order, but the Al = 3 orbitals come closer together with increasing I only in a realistic potential. The coupling between these orbitals due to an octupole potential term, which microscopically generates octupole deformation, has therefore the strongest effect in heavy nuclei. The tendency towards octupole deformation occurs just above the closed shells, i.e. for N or 2 = 34, 56, 90, 134. In the MO model some of the effects on the shell spacing generated in more realistic potentials by a flat bottom is brought about by the t2 term. Therefore, the Al = 3 subshells come close in the MO potential. From this point of view, all models that give a reasonable level order for spherical shape are expected to have a similar tendency towards octupole deformation. This tendency in different models can be tested by comparing the magnitude of the octupole coupling between the spherical 2fy12 and li13,* shells. These two shells lie close together in the proton single-particle spectrum, so the octupole interaction matrix elements between suborbitals can be extracted using a two-level approximation. It has been shown previously “) that the FY potential does in fact lead to an octupole coupling which is about 1.6 times stronger than obtained with the MO potential. We have now verified that the WS potential gives results almost identical with the FY model for these couplings. However, variations between models of the energy separation of such closely spaced subshells may override this difference between the MO potential and the flat-bottomed potentials (see fig. 7 below). Since all three models should reflect the structure presented in fig. 2, similarities are expected between the shell corrections obtained with the three models. For the MO potential, the shell correction has been plotted 14)as a function of both octupole deformation and neutron number. Corresponding results for the WS model are displayed in figs. 3-6. It can be seen that the shell correction drives towards octupole deformations near the same particle numbers in both models, namely 56, 88 and 134. In addition, the surface in fig. 3 has a valley extending towards p3 distortion at particle number 34. These results are not very sensitive to small variations in the ground state quadrupole deformation, so constant values of & (and p4) have been

W. Nazarewicz et al. / Octupole instability

&g:_ \ \-

N:l

-\

/ lj-&-

@

/ =;,s=,\

-./A,-----

,--29-c \

N=5

e’

\

//

\

\

<”

1112 97/z ds/2 ,::;; 1

.---2f

--,’

h/2 f512 b/z '1312 1

‘,Y_

p;;

@

__&)A-

~

.

__J30

d3/z h/z 5112 1 y30

97/z J d5/z

-lg--( 99/Z

Nz3 -c---if__:

Y30

i

/\-

--Ih-/= \

/;-2d-: ‘.

9/z

99/z

__3s

N.4

,-

\ /’

213

/--2P-y_

p1/2 '512

x5-

1

p3/2

f7/2

harm.osc.

flat bottom i MO : I’- term1

spm-orbit

Fig. 2. The splitting of the harmonic-oscillator energy levels caused by a more realistic the 1’ term in the MO potential) and by the spin-orbit potential, respectively. The couplings of the octupole Y30 term are also indicated. The splitting, to the lowest order, according to the formula given at the bottom of the figure. Note the general tendency the energy

denominator

to increase

with decreasing

radial shape (or most important can be obtained for the value of

mass number.

used in figs. 3-6. Empirically “) it is specifically for the particle numbers listed above that negative-parity states are observed at relatively low energies in doublyeven nuclei.

2.3. STRUTINSKY

ENERGY

The dependence of the potential energy on the octupole deformation the three models is illustrated in fig. 7. From different combinations

within of the

214

W. Nazarewicr

SHELL

zz

et al. / Octupole instability

ENERGY

29

CBETA2=0.30

BETA4=0.0)

36 PROTON NUMBER

43

50

Fig. 3. WS proton-shell energy landscape (not including pairing energy), for a typical ground-state deformation (& = 0.3, p4 = 0), plotted versus proton number, Z = 22 - 50, and octupole deformation ps. The thick lines separate regions of positive and negative shell energy. The most interesting quantity is the slope in the ps direction. The calculations were carried out for protons in %r and scaled with A-‘13 for different proton numbers.

SHELL

0.24

ENERGY I

CBET

BETA4=0.0>

1.30,

1.5 0. 16 I 0.12

0.08

0.04

1

/

1 1

1

I

I 34

41

48 NEUTRON NUMBER

Fig. 4. Same as fig. 3, but for neutron

numbers

I 55 N = 34 - 62.

62

W Nazarewicz SHELL

0.24

ENERGY

et al. / Octupole instability CBETA2=0.17,

215

BETA4=I

0.20

0.16 2 0.12

0.08 1 0.04

0.00

! 74

81

86 NEUTRON NUMBER

95

Fig. 5. Same as fig. 3, but for neutron numbers N = 74- 102 at & = 0.17, & = 0.04 and with parameters from ‘&Ba.

0.08

-’

0.04

-

I

0.00

/

’ 120

127

134 NEUTRON NUMBER

141

148

Fig. 6. Same as fig. 3, but for neutron numbers N = 12G148 at pz = 0.15, p4 = 0.08 and with parameters from 222Ra.

216

W

Nazarewicz

et al. / Octupole instability

“octupole-driving” particle numbers (cf. figs. 3-61, four regions of likely candidates for octupole deformed equilibrium shapes emerge: neutron-de~cient isotopes with 2 = 90 and N = 134, neutron-rich isotopes with Z = 34, N = 56 and 2 = 56, N = 88, and neutron-deficient nuclei with Z = N- 34. One nucleus from each of these

2

I

5

I

I

I

I

> 0.4 -

3460,

!,

-

224Th

0.2 -

-*-*-

F.Y.

-----

M.Q.

-

W.S.

0

0.1 fi3

Fig. 7. Potential energy of tictupole deformation, calculated by the Strutinsky method, relative to the reflection-symmetric (Pa = 0) constrained minimum. Results are shown for the FY, MO and WS singleparticle potentials specified in the text, and for four nuclei in different mass regions where the microscopic shell-energy term favours octupole deformation.

regions is presented in fig. 7. The two nuclei 2$$h,s4 and ‘:zBa9,, represent the most favoured candidates for octupole deformation in their respective regions. In the lighter mass regions, represented by zzKrs6 and $Ge38, the nuclei which combine the most favoured neutron and proton numbers are “too far from P-stability”. For the WS and FY results in fig. 7, the & and & values are obtained from the equilibria

W. Nazarewicz et ai. / Octupoie instability

277

determined under the constraint of reflection symmetry (& = 0). For the MO model, which gives somewhat different equilib~um deformations e.g. in the light Ra-Th region, the same p2 and p4 values as for the WS model were used. The macroscopic contribution to the deformation energy in fig. 7 is calculated using the modified surface and Coulomb diffuseness terms of refs.37*38).Qualitatively, the results do not depend on the choice of single-particle model. The nuclei ‘*Se and 92Kr are soft but stable with respect to octupole defo~ation (see discussion below). A shallow octupole-deformed minimum develops for 146Baand a deeper minimum occurs for 224Th. The differences between the models in the three lighter nuclei are to some extent caused by different values of the ground-state quadrupole deformation, which in turn affects the stiffness of the macroscopic energy term with respect to the octupole deformation. The largest differences occur in 224Th and will now be discussed. The MO potential gives only a shallow minimum. More complete calculations within the MO model were presented in refs. 4’*42),where, however, the results of the so-called simple consistency condition must be disregarded, see ref. 43). The FY and WS potentials give deeper minima than the MO potential, in qualitative agreement with the effect anticipated for a more realistic radial shape, cf. subsect. 2.2. The differences between the FY and WS potentials are, however, even larger than those between the WS and MO potential. This seems largely due to a difference in the single-particle energy separation between precisely the strongly interacting rrf,,2 and rri13,z subshells mentioned above. Their energy separation in a spherical **‘%I potential well is about 50 keV for the FY, about 280 keV for the MO and about 600 keV for the WS potential. The energy separation between the strongly interacting neutron orbitals vggi2 and vjls12 varies much less: 1520 keV, 1400 keV and 1370 keV for the FY, MO and WS potentials, respectively. The difference between the FY and WS potential-energy curves for 22&Ih in fig. 7 can be reduced to roughly half by using a different set of FY proton single-particle potential parameters - the FY(1) set of ref. “) -with a large rrf,,* - 7ri13,2separation similar to the corresponding separation of the WS orbitals. A special feature of the MO model is the large N = 136 spherical gap in the single-particle spectrum which prevents many of the transitional nuclei from becoming deformed in this model. Still, the spacing between the suborbitals which couple via the octupole component in the potential is quite similar in both the FY and MO models. Therefore, the different depths in the ‘*‘?h energy curves in fig. 7 reflect the larger octupole coupling in the FY model. For the WS potential, on the other hand, the larger coupling compared to the MO model is compensated for by a larger vf7/2 - +,3,2 separation. Summa~sing the results for 224Th,the calculated stability of octupole deformation appears to be governed primarily by the spacing between strongly interacting subshells. Given the spacing, however, the exact form of the potential also seems to be of importance.

278

W. Nazarewicz

3. Calculations

ef al. / Octupole instability

with the Woods-Saxon potential

This section describes the calculations of the shell corrections, potential-energy surfaces and equilibrium shapes obtained from systematic search throughout the regions where possible octupole deformation is indicated by figs 3-6. A WS singleparticle potential is used with a “universal” set of potential parameters 34), i.e. a set describing the single-particle states for all A 3 40 nuclei. The macroscopic energy term is the finite-range energy formula of ref. 13).

3.1. SHAPE

PARAMETRISATION

We use a /3-parametrization harmonics:

in which the nuclear shape is expanded in spherical

R(6) = C(P)&{1 +c &Ko(cos

6)) 3

(3)

where R(8) denotes the distance from the origin of the coordinate frame to the nuclear surface, and C(p), with p = {&, p3, p4, . . .}, is calculated in such a way that the volume enclosed by the surface of eq. (3) is constant. We restrict ourselves to axially symmetric shapes, which is phenomenologically justified in the heavy Ba and Ra-Th regions, where it is a K” -O- octupole band that is observed at low energies. Calculations with the present WS model in the 2 = 32 - 40 region 45) with the inclusion of a nonaxial deformation parameter y give axially deformed but often y-soft ground-state shapes. In the present work &, p3 and p4 are varied as independent shape coordinates. The numerical calculations were performed on a three-dimensional lattice of &, /13 and p4 mesh points in typical steps of 0.04, 0.03 and 0.02, respectively. In order to optimize the geometry of the liquid drop, non-zero values are also assigned to ps and &. The values of p5 and & for given &, p3 and p4 are defined as the ones which minimize the macroscopic energy of some arbitrarily chosen nucleus (here 222Ra). The sharp-surface formula of Myers and Swiatecki 36) is used in this context, rather than a finite-range formula, since it is assumed to be more effective for the purpose of smoothing out surface wiggles that arise from a pure p2, p3 and p4 parametrization. Fig. 8 illustrates the role of (p5, p6) at a large elongation (p2 = 1.0). Here it is seen that the optimal values of @Is,/16) lower the liquid-drop energy by about 0.5 MeV compared to that at (p5, p6) = (0,O). Thus it seems that the higher multipoles are less importnat in the p-parametrization than in the a-parametrization of the MO model, where s5 [ref. “)I and also e6 [ref. “‘)I have a larger effect. In order to find the optimal values of p5 and p6, the liquid-drop energy was minimized with respect to these parameters for a large number of realistic combinations of the deformation parameters p2, p3 and p4, and a x2 fit was performed to obtain the

219

0.06

0.04

0.02

0.00

-0.02

-0.04

0.00

0.02

0.04

0.06 BETA5

0.08

0.

IO

0.12

Fig. 8. The macroscopic energy as a function of & and & for given values of &, & and p4, using a standard liquid-drop formulax6). The contour lines, labeled in MeV, are 0.1 MeV apart. In this case about 0.5 MeV can be gained by non-zero & and &

coefficients of the following numerical approximation fis = &(0.177&+0.655/X,-

formulas:

0.0352fl; +0.0089),

~,=0.2215~:+0.1055~~+0.1476~,~,-0.0285&

(4) (5)

where -0.14 &d 1.O, 0 d /I3 c 0.3 and -0.1 c /?,~0.3. The terms included in these polynomials correspond to the lowest-order combinations of multipoles A = 2,3,4 which can couple to multipolarity 5 or 6. A linear term in & is iincluded in eq. (4) due to the c.m. shift. Note that & is highly sensitive to &.

3.2. THE

A= 80 AND

A= 100 MASS

REGIONS

It is convenient to divide the region of 32~ 2~40 into two groups with the characteristic mass numbers A = 80 (Ns 44) and A = 100 (N 2 56) (the nuclei around N = 50 are spherical). Empirically, the nuclei around 74Kr [ref. ““)I and “‘Sr [ref. “)I have pronounced prolate shapes. In the A = 80 region competition between different shapes leads to shape coexistence in the Ge, Se and Kr isotopes [see e.g. ref. ““)I. The present calculations are limited to near-spherical and prolate shapes. The calculated strongly-prolate structures in the A = 80 mass region are determined mainly by the 2 = N = 38,40 energy gaps in the single-particle level scheme. In the

280

W. Nazarewicz et al. / Octupole instability

A = 100 mass region prolate structure result from a combination of the proton 2 = 38, 40 and neutron N = 60,62 prolate gaps [see e.g. refs. 49*5”)]. In spherical nuclei with particle numbers between 28 and 50 the only valence orbitals which couple through octupole deformation are p3/2 and g9/z as illustrated in fig. 2. Similarly, in nuclei with particle numbers between 50 and 82, the dS12and h, ,/2 couple. The small number of active subshells in these nuclei makes the octupole effect more sensitive to quad~pole deformation than in heavier nuclei. Thus for

74

0.15

KRYPTON

-

SHELL

ENERGY

0 12 0.09 0.06 0.03 s;’

0.00 -0

iz m

03

-0.06 -0.09 -0

12

-0.

is 43

10

-0.00

0

10

0.20

0.30

0.40

BETA2 90

KRYPTON

-

SHELL

ENERGY

BETA2

Fig. 9. The microscopic (shell plus pairing) energy for ‘%r (top) and for %r (bottom), obtained from the WS potential as a function of & and &, The higher multiple parameters &, 8% and & have the same values as in fig. 10. The contour lines are labeled in MeV.

W. Nazarewicz

et al. / Octupole instability

281

typical deformed ground-state shapes, the octupole driving forces are weak and confined to nuclei with particle numbers 2, N = 34 and 2, N = 58 (figs. 3 and 4). For spherical shapes these effects are more pronounced and extend also to particle numbers 36,38 and 60,62. The (&, /&) deformation dependence of the shell energies for two krypton isotopes is illustrated in fig. 9; the corresponding total energy landscapes are shown in fig. 10. Neither case has an octupole equilibrium deformation, although there is softness with respect to octupole distortions. 74

0. I5

KRYPTON

-

TOTAL

ENERGY

0.12

0.06

-0.06

BETA2

90

0. I5

KRYPTON

-

TOTAL

ENERGY

0.12 0.09 0.06 0.03 0.00 -0.03 -0.06 -0.09 -0.12

-0.

IS -0.

IU

-O.OU

0.

IO

0.20

0.30

0.40

BETA2

Fig. 10. The Strutinsky energy for ‘%r and %r as a function of p2 and & obtained by adding WS-shell-plus-pairing corrections to the macroscopic deformation energies of a finite-range model. The energy is minimized with respect to p4 in each point, while & and ps are chosen according to eqs. (4) and (5). The contour lines, labeled in MeV, are 0.1 MeV apart.

282

W. Nazarewicr et al. / Octupoie instability

stiffness in &for light and heavy nuclei in the 2 = 32 - 38 region is illustrated in fig. 11. A stiffness parameter C, is defined from the equation The

E = E. +;C&,

(6)

where the constant C, is extracted from the minimum energy at & = 0 (denoted by Eo) and p3 = 0.03. If the C, value of the Strutinsky energy is smaller than Cy (the stiffness of the macroscopic part of the energy) the nucleus is said to be ‘soft’ towards octupole deformation. As an inset in fig. 1I the values of CT for 78Kr are plotted versus quadrupole deformation & {with & = 0 in this special case). It can be seen that the variations of c;lac with p2 are small, of the order of 15% when ,S2

I

I

I

1

I

I

I

‘“I

I

I

I

I

I

I

1:75 -

l!

% z

II

u”

850

-

O-

\ 0

0.1

0.2

0.3

\

Stable

32 L

\

‘,

\

\

\

0.4

I

I

I

I

I

I

I

32

34

36

38

40

42

44 Neutron

”

Octupole

I

I

I

I

I

I

62

64

56

66

60

62

1 64

Number

Fig. 11. CaIEulated stiffness ~e~cients towards p3 defo~ations, Cr, for neutron-deficient and neutronrich isotopesof s&e, s.$e, s&r and $3r. For comparison, the corresponding vaIues calculated with the liquid-drop model are given for some different mass numbers at & = 0.20 (dashed lines) and, in addition, as a function of deformation & for constant mass in the inset. The liquid-drop vafue is defined along paths of constant & while the microscopic value is calculated for a path where the energy for constant ps is minimized with respect to &(&). As a consequence of the strong octupole driving force for N = 56, the largest &-softness is obtained in the neutron-excess region. Note the rather strong correlation between quadrupole and octupole degrees of freedom, with the deformed nuclei being stiffer in the & direction. In ?3r the lowest minimum is spherical (indicated by s) but there also exists a secondary deformed minimum having a much larger C, value.

W Nararewicz

et al. / Octupole instability

283

changes from &= 0 to 0.4. To show the mass dependence of the macroscopic stiffness and to mark the different mass areas horizontal lines are drawn for the C y values corresponding to A = 70, 80 and 90 (where Pt = 0.2 and /3_,= 0.0 was assumed). According to the results presented in fig. 11 only the very neutron-deficient Ge and Se nuclei of the A = 80mass region show significant softness towards octupole deformation. In fact, the C, coefficients are strongly correlated to quadrupole deformation with large values for the most well-deformed nuclei of Kr and Sr. This is an example of a general observation: where there is a pronounced ground-state minimum, for example at a large p2, it is not expected that the breaking of some other symmetry lowers the energy significantly. Therefore octupole softness or octupole deformation should be enhanced in the transitional nuclei, where there are no pronounced quadrupole minima at p3 = 0. The same observation holds for the non-axial degrees of freedom. The transitional nuclei are generally y-soft, and therefore the coupling between the octupole and non-axial degrees of freedom should in principle not be excluded. In the A = 100 mass region of fig. 11 there are more nuclei with octupole softness. Again, the stiffness is correlated with the quadrupole shape: the Sr isotopes, which have larger deformations, also have larger C, values. Note the low values of stiffness parameter for the N = 52 Kr and Sr isotopes, which are both spherical, and the much larger value for the secondary deformed minimum in ztSrs2. 3.3. THE

MASS

REGION

AROUND

‘46Ba

The combination of proton number Z = 56 and neutron number N = 90 leads to octupole-soft or octupole-deformed nuclei around ‘$Ba9,,. These nuclei are rather far from stability, but the low-energy properties up to N = 90 have been studied experimentally for the Ba and Cs isotopes [e.g. refs. s’-54)]. In a similar way as for Z = 60 - 70 nuclei stable deformations are reached for N = 86 - 90. A small number of nuclei around the neutron-rich isotope ‘&Ba are predicted to have octupole deformed equilibrium shapes. The calculated deformations are listed in table 1, according to which octupole deformation occurs in ‘$Xe, in the Ba isotopes with N = 88 - 92 and in the Ce isotopes with N = 86, 88. The present calculations indicate in addition that none of the Nd isotopes (Z = 60) are octupole deformed, a result which differs from previous calculations ‘*) where a modified oscillator potential was used. This difference may be due to the fact that minimization with respect to hexadecapole deformation was not performed in ref. il). The shell-correction energy for “‘Ba is plotted in fig. 12 versus & and &. It is seen to be octupole driving at all p2 values for 0 < & 6 0.28. This situation can occur when high-j spherical subshells which couple by the octupole component in an average field are relatively close to each other, which is the case for rrhll,2 - rTTdsf2 and vi,3,2 - vf?,* orbitals. Quadrupole deformation splits apart the suborbitals, but

284

W Nuzarewicz et al. / Octupole instability TABLE

1

Equilibrium shapes for selected even-even nuclei in the mass region around I”Ba obtained from the minimization of the potential energy N

A

St

B3

84

86 88 90 92 94

140 142 144 146 148

0.092 0.122 0.146 0.169 0.192

0.000 0.000

0.017 0.000 0.000

0.0536 0.063 1 0.068 1 0.0755 0.0806

86 88 90 92 94

142 144 146 148 150

0.114 0.149 0.172 0.203 0.227

0.001 0.068 0.079 0.044 0.000

0.0489 0.0615 0.069 1 0.0879 0.0901

84 86 88 90 92 94

142 144 146 148 150 152

0.006 0.130 0.170 0.212 0.231 0.251

0.000 0.017 0.064 0.000 0.000 0.000

0.0006 0.0451 0.0618 0.0872 0.0903 0.0893

86 88 90 92 94

146 148 150 152 154

0.155 0.216 0.248 0.260 0.269

0.000 0.000 0.000 0.000 0.000

0.0480 0.0795 0.0903 0.0860 0.0893

The Strntinsky method with a deformed WS single-particle potential and a Yukawa-plus-exponential model for the macroscopic energy were used. Values for & and & are givenby eqs. (4), (5).

tends to conserve the distance between the AK = 0 pairs of suborbitals which interact via the Y3,,field. The shell effects are strong enough to give an octupole deformed minimum at /I3 =L;: 0.08 (pz = 0.17) which in this nucleus lies about 200 keV below the lowest energy for symmetric shape, cf. fig. 13. The minimum valley in the & direction corresponds to a fairly constant &. The total potential energy along the minimum valley is plotted in fig. 14 for all even Ba isotopes with N = 86 - 94. The very shallow minima for ‘&Ba are in fact the best separated ones in this region of nuclei. This result may be correlated to the experimental energy of the lowest negative-parity states 53*54),plotted versus neutron number in the inset of fig. 14. The other nuclei in the region surrounding 2 = 56, N =90 are soft towards octupole deformation, see fig. 15. The ones with octupole deformed ground states are indicated in fig. 15 by C, < 0. The general trend here is similar to the trend in the lighter mass regions: the stiffness C, is larger for nuclei with well-deformed ground states.

W Nazarewicz 146

0. I8

285

et al. / Octupole instability

BARIUM

-

SHELL

ENERGY

0.15 0.12 0.09 0.06 0.03 0.00 -0.03 -0.06 -0.09 -0.12 -0.15 -0.

I8 -0.08-0.04-0.00

0.04

0.08 0.12 BETA2

0.16

0.20

0.24

0.28

Fig. 12. Same as fig. 9, but for ‘46Ba. At a fixed value of p2, the shell energy is octupole deformation driving everywhere (with 1.5-2 MeV lower values for & = 0.18 than for & = 0).

0.

I8

0.

I5

0.12 0.09 0.06 0.03 0.00 -0.03 -0.06 -0.09 -0.12 -0.15 -0.18

Fig. 13. Same

-0.08-0.04-0.00

0.04

0.08 0.12 BETA2

0.16

0.20

as fig. 10, but for ‘46Ba. Note the shallow minimum at &=0.08 extended region of almost constant energy up to & = 0.12.

0.24

0.28

(&=0.17)

and the

286

W. Nazarewiczet al. / Octupoleinstability

Fig. 14. Octupole deformation energy curves for neutron-rich Ba isotopes, after minimi~tion with respect to & and & The inset shows the energy of the lowest negative-parity states that are observed experimentally in these isotopes. The experimental data were taken from refs. 53*54).

3.4. THE NUCLEI IN THE THORIUh4 REGION

The nuclei with 2 = 90 and IV = 134 are susceptible with respect to octupole deformations, but it has been only during the last few years that stronger indications for octupole softness has obtained. For 2 = 90, N = 134 nuclei the MO model gives shallow minima g*4’,42 ) with .Q # 0 and E ---0. The FY model gives deeper octupole deformed minima 14) around E = 0.13. The equilibrium shapes calculated with the present WS potential are in between (see table 2). The discrepancies between the different models are discussed in more detail in sect. 2 above. The calculated WS potential-energy surfaces are shown in figs. 16a-c for the even sequence 220-224Ra,where a transition from spherical shape to quadrupole deformation takes place. The shell correction for 224Ra plotted in fig. 17 is similar to that in the co~espondi~g plot for 14%a, but with the region of lowest energy shifted towards slightly smaller quadrupole deformation.

W. Nazarewicz et al. / Oetupole instubilify

I

I

88

88

I

I

SO 92 Neutron Number

287

I 94

Fig. IS. Same as fig. 11, but for neutron-rich isotopes of ,,Xe, 56Ba, &e and &4d. Negative values of C, indicate that the iowest energy is calculated for j3.,# 0. Tbe liquid drop values, C3maf’(fig. 11) lie around 200 MeV, so all nuclei represented in this figure are thus relatively soft.

The total energies for the deformed nuclei have the minimum valley in the p3 direction at a fairly constant &, cf. figs. l&-c. For *%J, the octupole driving force at spherical shape is strong enough to make the calculated equilibrium shape weakly octupole deformed even in the absence of quadrupole deformation (fig. 18). Actually, systematic deviations of the shape from the static equilibrium point may occur in soft nuclei like 220Ra, 220Th and 222U. In particular, collective rotational excitations could enhance and stabilize octupole as well as quadrupole deformation “). Potential-energy curves along the valley in & are plotted in fig. 19 for all the nuclei discussed here. In cases where p2 varies, the minimum valley is defined as illustrated by the dashed lines in figs. I6a and 18. The most favourable cases for stable octupole defo~ation appear to be 224Th and 224U,where about 0.5 MeV of potential energy is gained relative to symmetric shape, The experimental I” = l- excitation energies shown in the insets in fig. 19 are lowest in both Ra and Th for the isotope with neutron number just above the one

288

W. Nazurewicz

et al. / Octiipole inslability TABLE

Same as table 2

1 but for selected

2

even-even

nuclei from the Ra-Th region

N

A

82

B3

84

134

218

0.000

0.000

0.0002

132 134 136 138 140

218 220 222 224 226

0.003 0.004 0.118 0.144 0.166

0.000 0.000 0.078 0.060 0.000

0.0002 0.0011 0.0716 0.0824 0.0886

130 132 134 136 138 140 142

218 220 222 224 226 228 230

0.000 0.004 0.119 0.138 0.159 0.178 0.195

0.000 0.000 0.092 0.099 0.083 0.020 0.000

0.0000 0.0008 0.0739 0.0803 0.0872 0.0953 0.0922

130 132 134 136 138 140

220 222 224 226 228 230

0.001 0.114 0.134 0.150 0.177 0.192

0.000 0.096 0.107 0.108 0.050 0.000

0.0004 0.0678 0.0776 0.0819 0.1051 0.1029

130 132 134 136 138

222 224 226 228 230

-0.001 0.126 0.151 0.184 0.193

0.035 0.107 0.108

-0.0122 0.0672 0.0788 0.1100 0.1107

with the most stable octupole deformation in the calculations. The potential-energy surfaces suggest that this could be due to a secondary equilibrium configuration at spherical shape (figs. l&-c). In 222Raand 222Th,where the spherical configuration comes low in energy, it could interact with the deformed ground-state 0’ level and lower its energy “). This should effectively lead to a larger energy spacing between the Of and l- levels than in 224Ra and 224Th, since the l- level does not have a spherical analogue to interact with. An interpretation along these lines is supported by the experimental systematics of the rotational moment of inertia 18) and the product B(E2)E(2f) [ref. 27)], and by the spin dependence of the parity splitting 28). 4. Conclusions Potential-energy surfaces have been calculated for axially symmetric and reflection asymmetric deformations, using the Strutinsky method with a deformed WoodsSaxon potential and a finite-range macroscopic energy formula. A full minimization

0

20

0

16

0

12

0

08

0

04

0

00

-0

04

-0.08 -0.12 -0.16 -0.2%

-v.o4

-0.00

0.04

0.08 BETA2

0

12

-0.00

0.04

0.06 BETA2

0.12

0.

16

0.20

0.20 0.16 0

12

0

08

0

04

0

00

-0.04 -0.08 -0.12 -0.16 -0

70 -0.04

224

0.20

RADIUM

-

TOTAL

0.16

0.20

ENERGY

0.16

-0 -0.20

I6

-0.04

-0.00

0.04

0 08 BETA2

0.

I2

0.

16

0.20

Fig. 16. Same as fig. IO, but for the radium isotopes in the spherical-to-deformed transitional region, “ORa. 222Ra and 224Ra. Note that the combined (&, &) softness of ““Ra evolves into minima with &#Oand&#Oin 222Ra and z24Ra. The dashed line in the “‘Ra surface indicates the path which was used to construct the energy curve of fig. 19. For the other two isotopes, this path follows the valley of minimal energy at approximately constant &.

290

W

224

0.20 0.

Nazarewicz

et al. / Ociupole

RADIUM

-

instability

SHELL

ENERGY

16

0.12 0.08 0.04 0.00 -0.04 -0.08 -0.12 -0.16 -0.20

-0

.04

-0.00

0.04

0.08 BETA2

Fig. 17. Same as fig. 9, but for z24Ra. Note the similarity fig. 12.

0.12

0.

I6

with the ‘&Ba microscopic

0.20 energy

shown

in

0.20 0.

I6

0.12 0.08 0.04 0.00 -0.04 -0.08 -0.12 -0.16 -0.20

-0.04

-0.00

0.04

0.08 BETA2

0.

I2

0.16

0.20

Fig. 18. Same as fig. 10 but for the transitional nucleus 222U. The dashed line is used as for 220Ra in fig. 16. The crosses mark the minimum at & = 0 and & = 0.04 while the dots indicate a developing secondary minimum at pz = 0.08 and & = 0.07.

291

W. Nazarewicz et al. / Cktupole instability

1.0

-0.55:

1.0

P 7 ;

$ 5

0.5

c

5 -1.0 s.

220 I

224

1

I

9oTh

P

I

0.5 0

220

I

/ 220

?

i

\ i

i

230

$

E.

1.0

El_

c 0.5 -1.0 -

224 I -0.1

224 228

I

I

1‘

-1-

230

228

234

I

1

,

I

I

0

0.1

-0.1

0

0.1

J

Fig. 19. Same as fig. 14, but for s&n, ss Ra, ,711 and & isotopes. The behavior of the *%J curve can be understood from the dashed line in fig. 18. The experimental data are from refs. 27~5S-58).

on a (&, &.p4) lattice has been performed. Those regions of medium-mass and heavy nuclei have been considered, where a global survey of shell corrections suggests that reflection asymmetric ground-state equilibria may occur, namely for realistic proton-neutron combinations of particle numbers around 34,56,90 and 134. The main results can be summarized as follows: (i) Explicit formulas are given for the higher-order deformation parameters, pS and & (in the usual PA paramet~zation of the shape of a nucleus), which generate a family of as smooth shapes as possible. These formulas are valid for a wide range of nuclear shapes defined by &, p3 and &,, and they may be useful in various applications, e.g. in calculations of the properties of fissioning nuclei.

292

W. Nazarewicz

et al. / Octupole ins~ab~li~

(ii) The best prospects for ground-state octupole deformation in mass regions below A = 200 are found in the nucleus ‘&Ba. Further spectroscopic studies of this nucleus and its neighbours are called for. (iii) A broader range of octupole deformed nuclei is found in the light thorium region, in qualitative agreement with previous calculations with a FY potential 14). In the present work, the reflection asymmetry is most pronounced for 22?h and 224U, where there is about 0.5 MeV potential-energy gain relative to the reflectionsymmetric shape. (iv) No strong octupole effects were found either in the neutron-deficient A 12:80 mass region or the neutron-rich A- 100 mass region. The Ge and Se isotopes are calculated to be the most octupole-soft nuclei in these mass regions. (v) There is a correlation between the energies of the lowest negative-parity states [displayed in fig. 15a-c of ref. ‘“)I and the octupole softness or octupole deformation of nuclei in all the mass regions considered. (vi) Comparisons have been made with the results of other macroscopic and microscopic models. Within all the models there are tendencies towards octupole softness or octupole deformation in the ground states in the same mass regions. There are, however, some quantitative differences between the models: (a) The sharp-surface liquid drop is stiffer with respect to octupole distortions, as compared with the results of the finite-range macroscopic formula which makes a difference of a couple of hundred keV at realistic octupole deformations (fig. 1). (b) The standard modified oscillator potential gives usually smaller quadrupole defo~ations in the transitional regions 59) and weak octupole minima. The present FY(I1) potential gives deep octupole minima (the largest barrier at /I3 = 0 is about 1.5 MeV) at medium quadrupole deformations (E = 0.1) in the thorium region. The use of the present similarly-shaped WS potential leads to octupole barriers of at most 0.5 MeV in the thorium region. The differences (b) are mainly due to the differences in the relative positions of the single-particle levels in the vicinity of the Fermi level. Some of the differences between the MO model and the flat-bottomed models could also be explained as a general consequence of different radial shapes. As the radial shapes are similar in FY and WS potentials, the different results for the nuclei from the light thorium region seems to be due to the level spacing in the models. The nfT12- 7t& level spacing is especially important. It is large when considering 208Pb-the present WS model, the FY(1) model “) and experiment in 209Bi[ref. 55)]-but seems to become smaher in fits to deformed actinide levels - the present FY and MO models and the WS model of Chasman 16). Further and more detailed studies of the single-particle structure throughout the region of nuclei above 208Pbare clearly motivated. Notably the good overall agreement between MO, FY and WS in this work probably results only because the single-particle levels in all three models have been closely fitted to experimental data.

W. Nazarewicz et al. / Octupole instability

293

(vii) All the minima calculated for & # 0 are soft towards p3 = 0. Therefore collective zero-point motion is expected to spread the deformation considerably, and .it becomes a matter of preference whether the term ‘octupole deformation’ or ‘octupole instability’ is more appropriate. This semantic issue is familiar from the case of quadrupole transitional nuclei. There the rigid triaxial rotor is known to provide an acceptable paramet~zation of the quadrupole field because the only significant differences compared to dynamical models lie in higher states and matrix elements. These quantities do not much affect the primary observables and are generally not described realistically by any collective model [e.g. refs. 60-62)].Just as in the quadrupole case, the octupole field can be effectively similar for soft and stable deformation 30). The rigid approximation does not severely reduce the range of spectroscopic phenomena that may occur because quasiparticles may ‘parity decouple’ from the octupole deformation, just as they sometimes ‘rotationally decouple’ from quadrupole deformation, regardless of whether the deformation is soft or stable. Finally, let us briefly mention preliminary theoretical indications that the octupole deformation can be stabilized at high spin because it leads to an increase of the collective moment of inertia [cf. ref. ‘“)I. It may therefore happen that nuclei which are soft towards octupole deformation but without octupole minima can move to & f 0 with increasing rotational frequency. This work was partly supported by the Swedish Natural Science Research Council and the Office of Energy Research of the US Department of Energy under contract no. DE-AC05760R00033 with Oak Ridge Associated Universities. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15)

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