An interacting boson model description of octupole states in nucleic

ANNALS

182. 344-374

OF PHYSICS

(1988)

An Interacting Boson Model Description Octupole States in Nuclei

of

A. F. BARFIELD AND B. R. BARRETT Department

of Phyics.

University

of Arizona,

Tucson,

Arizona

85721

J. L. WOOD School

of Physics,

Georgia

Institute

of Technology,

Atlanta,

Georgia

30332

AND

0. SCHOLTEN * National

Superconducting

Cyclotron East Lansing,

Laboratory, Michigan

Michigan 48824

State

Universiiv.

Received October 26, 1987

The IBM-l +Jboson

model is described and systematically applied to the nuclei %m, 168&, 172yb, “sHf, and ‘*2W. Reasonable agreement with available energies and f?(m) transition rates is obtained in all cases except 16’Dy, which has the anomalous octupole band order K” = 2 -, O-, 1 -. This energy ordering of K- bands cannot be obtained within the present model. The parameter trends show considerable variation, indicating that the underlying fermionic subshell structure is very important in octupole states. CC) 1988 Academic Press, Inc. 156Gd

I%(jd,

l%DY,

162DY,

I. INTRODUCTION

Octupole states in nuclei have traditionally been associated with vibrational degrees of freedom [ 11. This leads to a simple picture, within the geometrical model, for the K” = O-, 1 -, 2 ~, and 3 ~ bands commonly seen in deformed even-even nuclei. Such states can also be described in an algebraic framework. Within the Interacting Boson Model (IBM) of Arima and Iachello [24], octupole states can be obtained in a simple manner by adding an f boson to the usual s-d boson space. This can be done either within the neutron-proton (IBM-2) framework, or within the original (IBM-l) framework in which neutrons and protons are not separately distinguished. Low-lying states appear to be equally well described by IBM-l and IBM-2 since these states are nearly symmetric to the interchange of proton and neutron degrees of freedom. For this reason, and because * Present address: Kernfysisch Versneller Instituut, 9747 AA Groningen, The Netherlands.

344 0003-4916188

$7.50

Copyright 0 1988 by Academic Press, Inc. AI1 rights of reproduction in any form reserved.

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of its relative simplicity, the IBM-l framework has been chosen for the present work. The aims of this study are: (1) to explore the IBM-l +f-boson model in order to gain insight into the Hamiltonian, transition operators, and wave functions, and (2) to apply the model systematically to a given mass region, in an attempt to obtain a global fit to energies and B(E3) values with smoothly varying parameters. Because of the availability of good data, the deformed rare-earth region is considered. The experimental signature for octupole collectivity is an enhanced B(E3) transition rate between the gs and the I” = 3- member of a band, as seen in inelastic scattering or Coulomb excitation. It is often found that a known octupole state is also strongly populated in a single-nucleon transfer reaction [S-S], indicating significant 2qp character. Such states are complex in nature. As employed here, the IBM is purely collective, with no microscopic content. It is of interest to see what kinds of problems arise in the use of this model in describing such complex “collective/2pq” states. Extensive microscopic studies of octupole states in deformed nuclei have been made by Neergdrd and Vogel [9, lo] and by Soloviev and co-workers [ 111 using RPA techniques. Such states have been studied macroscopically [ 123. Negativeparity states have been treated by boson methods other than the IBM. Boson expansion techniques [ 131, phononcore coupling [ 141, the Coherent State Model [15], self-consistent Hartree-Bose [16], and the Interacting Vector Boson Model (IVBM) [ 173 have been employed. The IVBM is a group-theoretical method based on Sp( 12, R) and is unrelated to the IBM. The question of possible static octupole deformation is also currently of considerable interest [IS]. Various versions of the IBM-1 +f-boson model have been applied to nuclei. Notably, the samarium isotopes [19,20], the N= 88 isotones [Zl], the barium isotopes [22], the krypton isotopes [23], ls6Gd [24], “‘Cd [25], and 168Er [26] have been considered. Octupole states have been treated within the U(5) limit of the IBM [3,27], and similarly, within the W(3) and O(6) limits [4,28]. Other IBM f-boson models exist in which the number off bosons is not restricted to one. An SU( 13) treatment [29] places f bosons on an equal footing with s and d bosons, while a U( 16) treatment includes dipole (p) bosons, as well [30]. These models include the possibility off bosons in the gs, corresponding to octupole deformation. Finally, a different kind of p boson is involved in the vibron model [31], which considers an alphaclustering (diple) mode and has nof-boson (octupole) content. The present work is the first systematic application of the IBM-l +f-boson model to an extended mass region. The IBM-l +f-boson model is described in Section II and its phenomenology is explored in Section III. The results of calculations for nine deformed rare-earth nuclei are given in Section IV and discussed in Section V. Finally, Section VI presents conclusions.

346

BARFIELD

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ET AL.

THEORY

A. Wave Functions In order to describe octupole states, the usual IBM-l (s-d) model space is extended to include a boson with angular momentum L = 3 and intrinsic negative parity (an f boson). Although, in principle, an arbitrary number off bosons can be included [3,4], only the simple case of one f boson is considered in the present model. This is justified in the rare-earth region where there is no evidence for two-phonon octupole states. It is assumed that the total number of bosons, N, is conserved, N=n,+n,+nf,

(1)

where n,, nd, and nf are the numbers of S, d. and f bosons, respectively, and nf= 0 or 1. It should be noted that the f boson is coupled to N- 1, rather than N, s and d bosons. For a given eveneven nucleus, the boson number N is determined by adding N, and N,, where N,(N,) is half the number of valence proton (neutron) particles or holes, counted from the nearest closed shell. The basis vectors can be written in terms of U(5), or seniority, basis states [3] as I~zIM)=lCNln,,v,n,,L,;n/;ZM) = CICN-n/l

nd,v, nd, L),O

In&/)X,,

(2)

where the subscript “c” denotes the positive-parity “core” of active s and d bosons, and the d-boson arguments are defined as in Ref. [3]. B. Hamiltonian The Hamiltonian

for the coupled system can be written

H = H,, + H,+ V+,

(3)

where Hsd is the usual IBM-l Hamiltonian, H/is the f-boson Hamiltonian, and Vsd, is the interaction between the f boson and’ the s and d bosons. The f-boson Hamiltonian is given by H/ = &

(4)

where E, is the energy associated with an f boson and $ is the number operator for f bosons. (The approximation of considering only nf= 0 or 1 is equivalent to the experimental constraint E, < 2~~, where E, is the excitation energy in any given nucleus.) The most general two-body interaction between an f boson and the s-d core can be written

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IN NUCLEI

vsq=c c, L[(d Xf+yL)X(aXf)‘L’]jjo’ L

+ u2df{ [(d+ ~f+)‘~’ x (s xf)“‘]&“‘+ + fi

hc.}

uzr[ (f+ x S+y3) x (7x s)“‘]jj”‘,

where “ x ” denotes a tensor product, i = Jm, the annihilation operator is defined as

(5)

and the spherical tensor form of

gyL(-l)(i.+qp,

(6)

so that aP = (- 1)’ L, and yjz = -( - l)“f-,. Using boson number conservation [Eq. (l)], the last term in Eq. (5) can be rewritten [(ffxs+)“‘X(JXS)(3)](0)=ri,~~~=[(N-l)~~-ri,~~]/~,

(7)

since n,s= n/ for nj= 0 or 1. If the first term on the right-hand side of Eq. (7) is included in Eq. (4), then the remainder of the sd-f interaction can be rewritten in equivalent multipole form, V~~“=Cojld~~+cILd.L/+c2Qd.Q,

. (f +d)(3) + cz,T, . (f+ XT)(~),

+ ~3 T ,

(8)

where

Ld=fi(dtxap, Qd=(StX a+ d X S)+ x(6tX a)"', (f +xf)"',

L,= 2 J;i Qf=

-2

fi(f+

@a) Pb) (9c)

xY)'~',

Pd)

T3 = (8

X

p3’,

(9e)

T4 = (6’

X

p4’.

(W

and

The positive-parity

Hamiltonian,

Hsd, can also be written in a multipole

H,d=~rid+aOPt~P+a,L,~L,+a,Q,~Q,+a3T3~T3+a4T4.T4,

form [32], (loa)

where

P=g(a.a)-,.,I.

(lob)

It has been found that the octupole and hexadecapole terms in Eq. (10) are seldom necessary. The corresponding terms in Eq. (8) are, similarly, not ordinarily used. Instead, a so-called “exchange” term is included in the sd-f interaction, :E&. Edj = 5:(dt

Xf)"'

. (f’

X

a)(3’:.

(11)

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BARFIELD

ET AL.

This term is a particular linear combination of the live multipole terms in Eq. (8). It is added to the Hamiltonian for phenomenological reasons, since a spectrum with K” = l- or 2- as the lowest octupole band cannot be obtained without such a term (see Section IIIA and Ref. [4]). The exchange term can be placed on an equal footing with the monopole-monopole, dipole-dipole, and quadrupole-quadrupole terms in the sd-f interaction by considering a neutron-proton octupole-octupole interaction of the form

where

co,= s;x& + x3&$’

~7~)“’ + hc.,

p = lr, v.

(13)

As shown in Refs. [33, 343, the projection of this interaction onto the IBM-l model space results in a monopole-monopole term, a quadrupolequadrupole term, and the exchange expression of Eq. (1 l), complete with normal ordering. The dipole-dipole interaction in Eq. (8) changes the energies of odd-spin states relative to even-spin states within a negative-parity band, as discussed in Section IIIA. This term is of minor importance in the sd-f interaction, as is the monopole-monopole term. The average value of ri, is approximately the same for all low-lying states in well-deformed nuclei. In this case the monopole-monopole term in Eq. (8) is nearly constant and has little effect on the excitation spectra. Consequently, this term is neglected in the present work. The remainder of this paper is concerned primarily with applications to deformed nuclei [35]. The Consistent-Q Hamiltonian of Warner and Casten [36] has been shown to give a good description of positive-parity bands in deformed nuclei using only a few parameters and has been adopted for the core Hamiltonian H,y, in the present study. The complete Hamiltonian is then H=a,L,.L,+a,Q,.Qd+&~~~+A,Ld.L,+AZQd.Q, f A,: E&. Edf:.

(14)

It should be noted that most previous applications of the IBM-l +f-boson model have employed the SU(3) form of the quadrupole operator (with x= -fi/2) in the Qd. Q/ interaction. The more general form of the quadrupole operator, given in Eq. (9b), is employed here for both Qd. Qd and Qd. Q,.. C. Transition Operators The most general one-body E2 transition by the expression T’“’ where e is the boson (quadrupole)

operator within the s-d space is given

= eQd,

(15)

effective charge and Qd is defined in Eq. (9b).

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For systems containing f bosons, this operator could be expanded to include (ft xf)“‘. However, it is assumed that E2 collectivity is associated with d bosons, and thus the E2 transition operator given in Eq. (15) is utilized for both positive and negative parity states. The most general E3 transition operator with only onebody terms and one f boson is given by 7-‘E3’

where the parameter octupole operator

=

e30(3)

(16)

e3 represents a boson octupole CQc3) = S+xf+

effective charge; and the

x3(d+ x,?“)‘” + h.c.

(17)

has the same form as the IBM-l image of the neutron-proton operator of Eq. ( 13). (Thus it is possible to develop a consistent-octupole model [33,34] in much the same manner as the Consistent-Q model [36] is formulated for the E2 case.) An appropriate El transition operator includes both one- and two-body terms, T’E’)=a[(d+xf)“‘+(f+x~)(‘~]+(~/2lv)[(Q,xC0(3))(1)+(L0(3)~Qd)~’)],

(18)

where the quadrupole and octupole operators are defined in Eqs. (9b) and (17), respectively, and N is the boson number. Finally, reduced electric transition rates are defined: B(El;z,-,z~)=~(z~~lT’N’IIzi)/2/(2zi+ D. Quadrupole-Quadrupole

and Exchange

1).

Terms in the SU(3)

(19) Limit

It is often useful to consider Hamiltonian operators in the exact group limits of the IBM. In the present case, the s-d description given by the first two terms in Eq. (14) is close to the W(3) limit of the model [4]. Therefore, analytic expressions that are valid in the exact W(3) limit should be useful for qualitative estimates in deformed nuclei. Such expressions can be used to consider the effects of the quadrupolequadrupole and exchange terms, which dominate the f-sd interaction. SU(3) limit expressions are available [37] for the quadrupole-quadrupole and (fermion) exchange terms in the case of an odd nucleon coupled to an s-d core (in the Interacting Boson-Fermion Model, IBFM), and can be adapted to the octupole case, since the corresponding operators have the same form. (It should be noted that, although the octupole and IBFM exchange operators have the same form, they have different interpretations [34,38].) The octupole energy eigenfunctions can be written in an SU(3) basis, in the limit as N-+ co [37], as

I~~~M)=ICNl(5~),K,;f,K/;KZM), where I and p label the irreducible

representations

(20)

of SU(3) [4], and K,, Kf,and K

350

BARFIELD ETAL.

correspond to the angular momentum projections on the nuclear symmetry axis (in the geometrical picture). The allowed values for the K quantum number are K= IKckKfl.

(21)

The energy expressions in the SU(3) limit are given by [33] E, = A,d(4 - q)/d

(22)

and E,, = -A,1(4

- Kj)‘/36,

(23)

for the s&f quadrupole-quadrupole and exchange terms, respectively. Neglecting the f-sd dipole-dipole term, which is small, one can write the total energy for negative-parity states in the W(3) limit as E(K,,~,~)=E,+&,+E,+E,,, where E,. is the energy of the positive-parity gs energy,

(24)

core of s and d bosons, relative to the

E,=E([N-l](L,p),K,L)-E([N](d=2N,p=O),K=O,L=O).

The energy eigenvalue in the W(3)

(25)

limit is given by [4]

E([N](&~),K,L,M)=(;Ic--‘)L(L+l) - K(A2+ p2 + &l+ 3@+ p)),

Pa)

for the Hamiltonian H=

-~KQ.Q-K'L.L,

Wb)

where Q is given by Eq. (9b) with x = - & and L is given by Eq. (9a). For the simplest case in which the positive-parity core is described by the lowest SU(3) irreducible representation, (A, /A)= (2N- 2,0), KC= 0, and K= K,= 0, 1,2, or 3. The second-lowest irreducible representation has quantum numbers (A, p) = (2N - 6,2), with KC= 0 or 2, corresponding to the geometrical model beta and gamma bands, respectively. For KC= Ob the allowed K values are K = KY= 0, 1,2, 3 as before. However, K, = 2, results in 0 < K < 5, as given by Eq. (21). Such K bands can occur at relatively low energies, as discussed in the next section.

III.

EXPLORATION

OF MODEL

A. Dependenceof Octupole Bands on Negative-Parity Parameters

In this section, the effects of changes in various Hamiltonian parameters are explored. The Hamiltonian, Eq. (14), and the parameter ranges employed are par-

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351

a)

AJkeV)

b)

U(3)

Kc Kf -1

0,

Limit

.r t t

K”

(MBV I

3

27 3 0, 3

1P 0; oa I 2,

’

(79

:

22 2 ’

-20

0

20

40

A,(keV)

FIG. 1. Octupole bandhead dependence on the s&f quadrupole-quadrupole strength A,. Calculated IBM-l results (in (a)) are compared with SU(3) limit predictions (in (b)), where K= /Kc* K,/. In both cases, the exchange strength A, = 0. In (a), x = -0.63, a, = 2.5 keV, and aI = - 27 keV. In (b), the solid lines correspond to (A = 2N - 2, p =0) with E,.== 900 keV and the dashed lines correspond to (A = 2N - 6, p = 2) with E,. = 2000 keV. Other parameters are: N = 13, A, = 0, and E, = 1000 keV.

titularly appropriate for deformed nuclei. However, many of the results obtained are expected to be valid for non-deformed nuclei as well. The calculations were carried out using the computer codes PHINT and FBEM [39]. The relationship between the parameters of the present work and the input parameters for the computer codes is given in the Appendix. The dependence of octupole bandheads on the Qd. QY strength A, (with A, = A, = 0) is shown in Fig. la. The corresponding results in the large N SU(3) limit are shown in Fig. lb. The solid lines in Fig. la are smooth curves through the calculated results obtained for the lowest band of a given K-. The x’s indicate that one or more other unidentified bands are present (such as a second K” = 1 - band). The dashed lines are asymptotic extrapolations, for purposes of comparison with Fig. 1b. The solid lines in Fig. 1b correspond to an unexcited s-d core, (A= 2N - 2, p=O), K,=Og, where the subscript “g” denotes the gsb. The dashed lines in this

352

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figure result from the second-lowest irreducible representation of SU(3), (A= 2N - 6, p = 2), K, = 0, or 2,. An expected K" = 5 - band is not shown because the calculations were done for angular momenta up to I= 4 only. The W(3) limit predictions agree quite well with the calculations in this case. The structure seen in Fig. la at A, = 0 is due to the finite moment of inertia of the gsb. The lowest K” = l- band in Fig. la appears to be based on the gsb for A, 5 10 keV, and on the gamma band for A, 2 25 keV. The K" = 3 - band that is associated with K, = 0, appears about 400 keV above the corresponding 1~ band (K,.e2,Kf=3), for AZ-- 30 keV. This is consistent with the relative locations of the K" = 0: and 2: bands for this calculation (EB - E, = 350 keV). In the SU(3) limit, these bands are degenerate. Figure la shows a state labeled ZK" = OO- at about 3050 keV. This level has a rotational band built on it, with even spin only. Thus, it appears to be an even-spin branch of the P = 0 ~ band that is built on the gamma band. Since the core is not axially symmetric in this case, the O-~ band members need not be restricted to odd spins.

/oo-

a) K”c3--e

b1

--

-100

-

_

_

-50

50

0

m

SU(3)Limit

0;2- 3-f,53-Y. ' I-,3-,i-- . ' \ '\'o-9-,2-2-31 t-

E(MeV) 4f

I

,

2-

‘I 0;4-

2 0,;i
1;3F

2‘ & ,

k K, K”

3-. -----{$I; \-- \a_

t-51;

,,+j

0-

-100

-50

0

50

3-

A,(keV) FIG. 2. Octupole bandhead dependence on the s&f exchange strength results (in (a)) are compared with SU(3) limit predictions (in (b)), where the quadrupole strength A, = 0. Other parameters are given in the caption

A,.

The

calculated IBM-I In both cases,

K= IK, k K,I. to Fig.

1.

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353

t K” 0 0‘3 3i i2 2-1 oI 1-100

A, (keV) FIG. 3. Octupole bandhead dependence on the exchange strength for a constant value of Ax/AZ = 2. Other parameters are given in the caption to Fig. 1.

The dependence of octupole bandheads on the exchange strength A, is shown in Figs. 2a and 2b, with the energies resulting from the numerical calculations again being given in (a) and the SU(3) limit results in (b). As before, the second N(3) irreducible representation is important for understanding the calculated results. A critical concern is the energy ordering of the octupole bands: this depends interaction strongly on the parameters A, and A,. The quadrupoleequadrupole alone results in a K” = O- or 3 - octupole band being lowest in energy, depending on the sign of the parameter AZ. Similarly, the exchange term alone results in a 2or 3- band being lowest in energy, depending on the sign of the parameter A,. The presence of both interactions allows for the possibility of a K” = l- band being lowest in energy. When both A, and A, are negative, the model predicts an ordering of (O-, l-, 2-), (lV, O-, 2-), (l-,2-,0-) or (2-, l-,0-) for the three lowest Kp bands, depending on the relative values of A, and A, [33]. It is the ratio of these two strengths which is most important for the two lowest K- bands. This is seen in Fig. 3, where the splitting between the two lowest Kp bands is almost constant over a wide range of the parameter A, when the ratio A3/A2 is kept fixed. The splitting between the second and third K- bands depends upon the absolute value of A,. Evidently, the values of the parameters A,, A,, and cl (which is an additive constant) can be determined unambiguously only if three or more experimental octupole bands are known. The effect of the L,. L, (dipole-dipole) interaction on octupole states can be seen in Figs. 4a and 4b. The ordering of the K- bands is practically independent of the dipoledipole strength A,. This interaction, however, can have a pronounced effect on the order of levels within the Kp bands, as shown in Fig. 4b for the first lband. The odd-spin states are shifted, relative to the even-spin states, by mixing with the next-higher (K” = O- ) band, which has no even-spin states. The Ld. L, interaction thus has a “Coriolis-like” effect, changing the coupling between states with the same spin and K values differing by one [9]. Band distortion and K-mix-

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ET AL.

FIG. 4. Octupole bandhead dependence on the sd-SdipoleApole interaction strength A,. Bandhead behavior is shown in (a). Behavior of the lowest K” = 1~ band is shown in (b). For this calculation A, = -60 keV and A, = -120 keV. Other parameters are given in the caption to Fig. 1.

ing due to Coriolis coupling are common features of octupole bands [6,9, 121. A Coriolis term is added to the geometrical model Hamiltonian explicitly, producing substantial changes in the negative-parity results [9]. In the algebraic picture, a “Coriolis-like term,” -I. L,, arises naturally from the Ld. Ld interaction in ZZsd,Eq. (lo), since Ld = I- L,.. (The analogous situation in the IBFM is discussed in Ref. 40). The sd-f dipoledipole interaction, L,. Lf, also includes a Coriolis-like term. This latter interaction is not needed in most cases because the observed spectral structure is fairly well described without it. It is useful for fine-tuning in some cases. B. Negative-Parity Band Dependenceon Core Description Since the energies of excited K” = O+ and 2+ bandheads vary a great deal over the deformed region, it is of interest to consider the dependence of the negativeparity states on the calculated 0: and 2: energies. The Q,, . Qd strength, a,, is determined in the present study by fitting to the experimental gamma band. In principle, the beta band can also be considered in determining this parameter. One must be careful, however, because it is not always clear that excited O+ states in deformed nuclei belong to the s-d model space [41]. Octupole band sensitivity to changes in the strength of the Ld. L, interaction is not particularly relevant: the strength of this interaction, a,, is chosen to tit the moment of inertia of the gsb, which does not vary much for well-deformed nuclei. The positive-parity part of Eq. (14) is known to work well for the K” = O+ and 2+ bands in ‘68Er [36]. Parameter values appropriate for this nucleus are

IBM

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I

STATES I

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IN NUCLEI

I

I I 1

K* I-

-

1 o2

1.5

2-

1 it

“’

I

I

I

I

I

0.8

i.0

i.2

1.4

EIO:)

MeV

FIG. 5. Octupole bandhead dependence on the beta-bandhead energy. The changes in the energy of the 0: level were obtained by adding a pairing term to the Hamiltonian. Other parameter values are: N=.16, a,=6SkeV, a2= -17.5keV, I= -0.49, &,=1240keV. A,=O, A,= -37keV, and A, = -120 keV.

employed for the calculations in this section. The calculated 02 energy is most easily studied by adding a pairing term to the s-d Hamiltonian, giving H,,=a,P+.P+a,L,.L,+a,Q,.Q,;

(27)

cf. Eqs. (9) and (10). The parameter a,, can be varied to change the 02 energy. The results of such a calculation are shown in Fig. 5 for ten Kp bands. The strength of the quadrupole term a2 has been adjusted as necessary in order to keep the gammaband energies approximately constant. An W(3) limit analysis indicates that the lowest three octupole bands in Fig. 5 are based on the gsb. The second 1~ band and the lowest 3 - band are built on the gamma band (K, = 2, K,= l), for the values of the parameters A, and A, employed. The second O- band and the 4band are similarly based on the gamma band (K, = 2, K,.= 2), while the K” = 2; and 1; bands are built on the beta band. The energy of the gamma bandhead is easily changed by varying the quadrupole strength a,. This is seen, for example, in the W(3) limit from Eq. (26a), E(2j+)=E([N]

(J.=2N-4,p=2),

= 6a, - (6N- a) a,,

where a, = rc’, a2 = --2x, and Figure 6 shows the octupole The corresponding quadrupole strength a, has been adjusted used in this calculation. Thus

L=2)-E([N]

(1=2N,,u=O),

L=O) (28)

the energy eigenvalue for the gs has been subtracted. band dependence on the gamma-bandhead energy. strength a, is shown along the top axis. The dipole to keep the 2: energy constant. No pairing term is all the octupole energies, including those for the three

356

BARFIELD

0

10 1

'

ET AL.

o,(keV) 20 1

I 05 El2;)

FIG. 6. quadrupole

Octupole strength

bandhead dependence a2 is also shown. Other

30 1

-

I 1.0 MN

,'K.

I 1.5

on the gamma-bandhead energy. The corresponding parameters are given in the caption to Fig. 5.

s-d

lowest K- (which are built on the gsb) increase and decrease with the gammabandhead energy. The second K” = 3 - band is included in Fig. 6, along with the ten K bands shown in the previous figure. This band is built on the s-d core corresponding to the (A = 2N - 10, p = 4) W(3) irreducible representation. There are three K- bands (not shown) between the 1; and 3, bands. Figure 7 shows the dependence of the B(E3; 3- -+ 0:) values on the s-d

0

0,

2

I t.0

I 0.5

1.5

E(2;)MeV

FIG. 7. Dependence of E(E3) K= 3 band is built on the gamma a2 is also shown. The Hamiltonian are: e3 = 0.054 eb’12 and x3 = 4.5.

values on the gamma-bandhead energy for four octupole bands. The band and all others are built on the gsb. The s-d quadrupole strength parameters are given in the caption to Fig. 5. The F”’ parameters

357

IBM OCTUPOLESTATESINNUCLEI

quadrupole strength a, and, consequently, on the gamma-bandhead energy. The lowest K- band is shown for K= 0, 1,2, 3. This figure shows that the transition rates are more sensitive to the core description than are the energies, for low-lying states. It is interesting that the B(E3) to the gs for the K” = 3 ~ band is substantial, even though this band is built on the gamma band, in this case. Finally, one can consider alternatives to the s-d core description given by the first two terms in Eq. (14) [33]. Outside of the deformed region it is advantageous to include an Endterm in H,, [42]. It appears that the low-lying negative-parity spectrum is essentially independent of the core description, so long as the gsb and gamma bandhead are well-fitted. The negative-parity parameters corresponding to different core descriptions are, of course, not the same. In particular, the f-boson energy sf must be adjusted to compensate for changes in binding energy, among other things. The interested reader is referred to Ref. [33] for details. IV. STUDY OF DEFORMED RARE-EARTH

NUCLEI

A. Approach The octupole model described in Section II is now applied to nine deformed rareearth nuclei, chosen primarily on the basis of available octupole data. The selection of octupole bands is made on the basis of known B(E3) transition rates. No attempt is made to fit negative-parity bands which lack B(E3; 0: + 3 -) strength. In some of these nuclei there are low-lying 2qp bands with K < 3 and negative parity. These are manifestly outside of the present model space. Band assignments for calculated negative-parity bands are made primarily on the basis of (calculated) B(E2) transition rates, since it is assumed that matrix elements for intraband transitions are larger than those for interband transitions. Some caution is needed because of K mixing due to Coriolis coupling. It is not always clear, even experimentally, which states belong to which K bands. The B(E2)-based band assignments for states below about 2 MeV are usually unambiguous. B. Determination of Parameters The positive-parity parameters a, and a2 are chosen to give the best possible lit to the gsb and the gamma bands of the nucleus in question, without regard to any excited K” = Of states or to the parameter values employed for neighboring nuclei. The parameter x in the quadrupole operator Qd is determined by fitting the experimental ratio R=B(E2;2;

-O:)/B(E2;2:

40:)

(29)

in the usual consistent-Q manner [36]. The positive-parity parameters are given in Table I. The negative-parity parameters are given in Table II, along with the experimental ordering of the K- bands and the boson numbers. The parameter .sl is determined 595/182/2-12

TABLE Positive-Parity

Parameters

(kae;r,

’ 54Sm ‘=Gd “‘Gd lssDy 16*Dy ‘68Er ‘72Yb

Note. The numbers “R=B(E2;2:~O:)/B(~;2:~0:). h Ref. [43]. ’ Ref. [44]. d Refs. [44, 121. y Ref. [45]. 1 Refs. [46, 473. g Refs. [48, 491. h Refs. [SO, 511. ’ Ref. [52].

in the Study R”

(kaet,

0.0 2.0 3.0 5.5 6.0 6.5 4.5 5.0 2.5

“‘Hf ISZW

Employed

F”’

Hrd Nucleus

I

- 36.0 -32.5 - 27.5 - 27.0 - 20.0 -17.5 -22.5 -21.5 -36.0 in parentheses

IBM

x 0.15 0.14 0.14 0.14 0.13 0.13 0.12 0.12 0.13

-0.72 -0.54 -0.63 -0.45 -0.54 -0.49 -0.80 -0.49 -0.45

are the experimental

TABLE Negative-Parity

Parameters

Exp

0.015 0.025 0.017 0.03 1 0.021 0.023 0.007 0.024 0.03 1

uncertainties

0.016(2)h 0.026( 1)’ O.O17(l)d 0.032(2)’ 0.022( 1 y 0.022( 1 )R O.O07(3)h 0.024( 1)’ 0.030( 1)’

in the last digit,

II Employed

in the Study

Boson numbers

Nucleus

(,b%‘)

x3

0

0.0

0.07

-0.50

-75

1.7

0.07

-0.35

-120

2.0

0.07

-0.30

-20

-90

4.5

0.07

-0.30

0 0

-42 -57

-135 -145

3.2 2.3

0.07 0.07

4.1 3.5

-37

-150

4.0

N

N,

N,

‘%m

o-

l-

11

5

6

1430

10

-50

‘56Gd

l-O-2-

12

5

7

1100

10

-45

ls8Gd

l-O-2-

13

6

7

1260

10

-60

“*Dy

2-l-O-

13

5

8

650

0

‘62Dy

2-o-

15

7

8

u

168Er ‘72Yb

1-2-Ol-O-2-

16 16

9 10’

7 6

1340 1250

lo

I-

‘78Hf

2- I-

15

182W

2-

13

9

(kk)

AJA,

K”order

(keV)

s

750

0

4

480*

0

Note. The experimental ordering of included. Boson numbers with bars above “This ordering of K- bands cannot be D The experimental data are insufficient

(keV)

o*

(keV)

-loo*

the octupole bands and the relevant boson them indicate hole rather than particle degrees obtained within this model. to determine this parameter with confidence.

0.07

1.7

0.07

0.3

numbers are of freedom.

IBM

OCTUPOLE

STATES

359

IN NUCLEI

so as to make the average energy of the calculated bandheads the same as the corresponding experimental average. The L,. L,. strength A r is chosen, when necessary, to give the correct level ordering within the lowest KP band. The parameters A, and A, depend on the relative positions of the K- bandheads. As discussed in Section IIIA, the absolute values of these two parameters are less important than the ratio A,/A, for the lowest K- bands. A unique determination of the negative-parity parameters is impossible for those cases in which only one or two octupole bands are known. In cases for which it is possible to describe the known octupole band(s) with only the Qd. Q/or only the exchange interaction, the strength of the remaining interaction is arbitrarily taken to be zero. The parameter sets for the nuclei ls4Sm and r8*W, for which this was done, are not unique. The octupole effective charge, e3, is taken constant for all the nuclei studied. Therefore, the parameter x3, alone, is responsible for differences in B(E3) patterns for different nuclei. 2500-

6+-

500

6-

t 0t Experiment

t

,

, Thearv

,

25002.---

,(“22-

2000;: 2

-3.or4’

5.-

3.1.-

i5005.-

w lOOO-

4-3;y

3.I--K= :,--

3.-

5---1-pYy. 31-K”zO-

5OcFIG. 8. Energy spectra for 15”Sm The first and third experimental negative-parity bands have been identified as octupole. The level at 1811 keV is collective, either octupole or hexadecapole (i.e., 3m or 4+ ). The data are from Refs. [43, 533.

500 t

1000

t500-

-

Experiment

5--

, ,

Theory

K”

I

5--

,

473-1 475)G (4*,3-l

K":4-

KR=2-

$--...4-3-2.-

-3.

3-2--

4-3- K”:2.:r ,-‘,.E K”:oK”=l-

5-l-.4--

- IL-l

r

uI*4.3.3.gE 1-y t-z K=O~

I

[24, 54, 551.

FIG. 9. Energy spectra for ls6Gd The K” = 1~ and 2 experimental bands have been identified as octupole. The data are from Refs.

w

5 9

2000

2500 -

3000-

o-

Expernent

,

Theory

6+-

5-- K”22-

5-p 4.3.2--

,

I

FIG. 10. Energy spectra for lisGd The lowest K" = 1 -, 0 -, and 2 experimental bands have been identified as octupole. The data are from Refs. [ 12, 561.

I

6’-

-

2000

FIG.

identiCed

I

Experiment

, 8

5.-

4'-

4*-

51 4--2z 23-& p=2-

Theory

1-z

3-

z:y

2’K”.2’

1

6'cj+-

K” ~0’

6*-

K”c2’

*,-;4t

6+-

8'-

6*5*4*;:&

158 66DY92

,

-

-

-

has been

1.K":(j-

3--

5.-

2'o+y, K :O

11. Energy spectra for “*Dy. The K” = 2- band as octupole. The data are from Refs. [56, 573.

500 t

5: EOQ2 w lOoQ-

-

OL

r

2500

2500

-

2000

-

-

,

-(3-j

K=

Experiment

Experiment

1

L

4*-

6+-

,

,,

,

Set 1

Theory

4'-

6+-

8+-

,,

6+-22' ,j+4;;*= K”Z2’

Set2

3.I.--K"zO-

o+K”=O’

,

, J

!

FIG. 12. Energy spectra for ‘6ZDy. The K” = 2 -, O-, and 1 bands have been identified as octupole, as well as the 3- state at 2318 keV. The calculated negative-parity spectra are discussed in the text. The data are from Refs. [58, 591. The negative-parity parameters are (c,, AZ)= (950, -52) keV for set I and (1020, -25) keV for set 2. In both cases A, = 0 and A 1 = - 155 keV.

5ooL

lOOO-

lAi 1500-

5: 2

2500

O-

5cQ-

> 15000) 5 w 1000 -

2000

2500-

15CQ-

-

-

-

state

FIG. 14. experimental

3-

-

-o+

=;*

X8+ 1

+

3’ 0’

Other bonds K”

,--

Experiment

4’-.--

6+-

8’-

w-K”:2-

5’;:=

,

K”

5: 4.p T2;

K” :O’

,

-

-I-

-3.

, Theory

3--

5.-

n/

;*+=I K=q,’

4+-

6+-

a+-

K”z2’

:*r 2’-

‘7720Yb102 1

O-, and 2Energy spectra for 172Yb. The K"= l-, bands have been identiiied as octupole, as well as the at 2032 keV. The data are from Refs. [SO, 621.

500-

% f L&J 15Oc-

2000

2500

500-

I-Kc ,-

,

lOOO-

,Theory

$= K=.O+

o-

500-

2 1 w lOOO-

1500-

2000-

2500~

1000 -

1500-

I

Experiment

4’2’o*--K” zO*

6’-‘=+

&-4c

6+5’7

‘66eBErt00

FIG. 13. Energy spectra for 16sEr. The K”=l(1431 keV), 2-, and O- experimental bands have been identified as octupole, as well as the 3- state at 2257 keV. The data are from Refs. [S, 60, 613.

s ,” =

2OQo-

2500-

o-

500-

Id IOOO-

5: 2

2000

2500-

I

-8.

-8-

=5;-

Other bands K” -0’

2-E

f-x

K”z2’

2.- TK?T:,,(==2-

‘y

5.

5.-

Experiment

6*-

8*-

6'5'4*3'2'-

,

z:=. 3-3-y 2.2-p 1-y K~=2- K =1-

, Theory

6'-

4+-

g-0'5p.

‘:;Hf,06

5--

K”=O’

2'-

FIG. 15. Energy spectra for “sHf. The K" = 2 experimental las been identilied as octupole. The data are from Refs. [63-65,

25CW

2500-

,

band 773.

1500

2000

2500

o-

t

500 -

lOcc-

K”

2'o'K"zO'

4'-

6'-

8+-

5--G-.K”z5-

Experiment

K”=4-

KKz2-

4--45--5+ 3-p 2.-

I

-0'

-6'

Xher bands

Krzfj-

,(“:2’

4'7' 2'-

,,

182W 74 108

5-L-4-3.-

K”z2-

4-3-2-y

Theory

FIG. 16. Energy spectra for IxzW. The K" = 2- experimental has been identified as octupole. The data are from Ref. 1661.

2 5 w

F 2

1500-

2000-

2500-

,

band

364

BARFIELD

ET AL.

C. Results of Calculations Spectra for the nine nuclei studied are given in Figs. 8-16. The positive-parity spectra include the gsb and gamma band, up to angular momentum Z= 8 and 6, respectively, for both theory and experiment. The first two levels for the second calculated K” = Of band are also shown. Experimental excited Of bandheads are shown in an inset, along with other low-lying bandheads. All experimentally known negative-parity bandheads (including 2qp bands) up to about 2 MeV are shown. Levels up to I” = 5-- are given for the lowest negative-parity bands. Those experimental states with known octupole collectivity are specified in the figure captions. The energy spectra for ls4Sm and 156Gdare given in Figs. 8 and 9, respectively. Both of these nuclei have one or more states with I” assignment (3-, 4’) that are collective (seen in inelastic scattering experiments). Spectra for the nuclei “‘Gd and “‘Dy are presented in Figs. 10 and 11, respectively. The K” = O- band for “*Dy is not well reproduced. The octupole spectrum of 162Dy cannot be described at all, within this model. Figure 12 shows two attempts to lit this nucleus, Set 1 fits the k? = 2 - and O- bands, resulting in a l- band that is below, rather than above, the other two bands. Set 2, which reproduces the 2- and 1 - bands, results in a very high O- band. It appears that the ‘62Dy K- band order (22, O-, l-) is not possible in the present model. The spectra for 16*Er are shown in Fig. 13. The calculated K” = 3 - bandhead is in good agreement with the collective experimental state at 2257 keV. All other levels shown in the experimental negative-parity inset are 2qp bandheads; see, e.g., Section VC. The lowest calculated K” = 4- band is shown in the theory inset. The decay modes of the ZK” = 44- states have not been investigated. The lowest even-spin O- bandhead is also shown. There are no known experimental O- states in even-even deformed nuclei, to the authors’ knowledge. The spectra for ‘72Yb are shown in Fig. 14 and those for “‘Hf and “‘W are given in Figs. 15 and 16, respectively. The calculated K” = O- band for ‘78Hf and the I.band for ls2W are speculative, since they result from parameters that are not well determined by the data. The calculated B(E3) results, presented in Table III, are compared with experiment and also with the RPA results of Neerglrd and Vogel [9]. The energies of the 3levels are listed also. The lowest RPA K” = 3 ~ bands are non-collective in nature- and have been omitted from Table III, except for the nucleus 16’Er, for which there are data for comparison. (The 2qp states are within the domain of the RPA calculations, but outside of the IBM model space.) The K values for the highest experimental levels for 16*Er are not known and may not, in fact, correspond to the calculated levels shown. The 3 ~ levels in the predicted K” = 1, bands have relatively small B(E3) values with respect to the gs. This is to be expected since they are built on the gamma band. Second K” = 1 - bands are known experimentally for “‘Gd and 168Er. The predicted K” = 3- bands, also built on the gamma band, have significant B(E3; 3- -0:) for ‘68Er and ‘72Yb. The experimental E3 strength is fragmented over many states for these two nuclei. A large value of the parameter x3, given in Table II, is needed in order to reproduce this fragmentation.

IBM

OCTUPOLE

STATES

TABLE Comparison

of IBM

B(E3;

3

+ 0:)

IBM K” iS4Sm

lsbGd

15*Gd

“*DY

E(3-) 1.09 1.57 2.41 1.26 1.58 1.91 2.41 2.56 1.12 1.39 1.81 2.34 2.46 1.29 1.50 1.97 2.33 2.39

124 51 8 176 6: 0 1 164 7: i 219 29 15

‘72Y b

“*Hf

with

Experiment

and RPA

Results

Experiment

E(33)

NE31

E(33)

B( E3)”

1.7 1.3 2.0 1.3 1.8 1.7

0 183 49 220 19 19

1.01 1.58

129(29)d

1lOd 77d

1.28 1.54 1.85

244( IO)’ < 19’

166’

1.2 1.7 1.6 1.98

113 49 19

1.04 1.40 1.86

177( 10)’ 33(4)’

1.3 1.5 2.0

223

1.40 1.54 1.67

329(71)8

134 26

149( 10)’

23

1.21 1.36 1.74 2.32 1.43 1.54 1.63 1.83 1.91 2.00 2.02 2.27 2.32

:

B(a)’

63’

9Y :Y

234”

(:

‘62Dy

““Er

III

Results RPA”

B(E3)

365

IN NUCLEI

1.:3

65

1.27

94

1.3 1.6 1.9 2.0 1.6 1.4 1.7

1.91 k 2.40 2.30 2.65 2.72 2.78 2.86 3.06 1.23 1.74 1.84 2.74 2.79 1.33 1.47 2.09 1.40 2.05

54

2.2

: 59 7;

13 :r; 4

61(9)’ 71(14)’

2.49 :: 2 :: 47 4: 76 ; 109 57

1.4 1.9 1.7

86

1.4 1.6 2.2 1.5 1.7

139

2.03 1.32 1.43

i 199 0

1.37

Note. Energies are given in MeV and transition ’ Ref. [9]; results after Coriolis coupling. ’ Coulomb excitation. ’ Inelastic scattering. d Ref. [43]. ’ Ref. [46]. ’ Ref. [67]. c Ref. [68]. h Ref. [45].

1:

1.22 1.71 1.82

;y:: 74” 32”. ’ 76( 14)”

rates in 10m4 e2b3. ’ Ref. [49]. ’ Ref. [26]. k 2qp band, outside IBM model space. ’ K value not experimentally determined. m Ref. [SO]. “Ref. [62]. “Ref. [44]. JJRef. [52].

lll(17)P

366

BARFIELD

V. A. Comparison

ET AL.

DISCUSSION

with the Data

For most of the nuclei studied, the energy spectra are in good agreement with experiment for the states fitted (from one to three octupole bands per nucleus). The Coriolis effects are well reproduced, as are bandhead energies. In addition, predicted higher-lying K” = 1~ and 3- bandheads are consistent with available data. The energies of these bandheads are often high with respect to the data, however. This is not surprising, because the Hsd core is too simple for a correct description of the higher-lying bands. For example, the nucleus 158Gd, shown in Fig. 10, has two K” = 0 + bands and a 4 + band below 1500 keV that are not considered in the core description. Another example concerns the nucleus 17*Yb, which has an unusually high gamma band. As seen in Fig. 14, there are two O+ bands and a 3+ band that are below the gamma band, but not included in the IBM core. Despite this, the three lowest octupole bands are reasonably well described for this nucleus. The B(E3) data from inelastic scattering experiments are often rather different from the Coulomb excitation results, as can be seen in Table III. The IBM results are generally in good agreement with one or both sets of data for low-lying 3states. The distribution of E3 strength among the experimental bands is well reproduced by the theory. B. Comparison

with other IBM

Calculations

Three of the nuclei considered have been studied by other workers within the IBM-l +f-boson framework. The energy spectra reported for 154Sm [19] and is6Gd [24] are consistent with the present results, as are the Hamiltonian parameters employed. The negative-parity results reported for ‘68Er by Govil et al. [26], however, are not consistent with our results. Using the parameters given in Ref. [26] and our method of assigning states to K bands, we do not reproduce their band structure, which includes a low-lying K” = 3 - band. It appears that the authors of Ref. [26] did not consider the intraband B(E2) values in making band assignments. In addition, the calculated B(E3) values of Ref. [26] are not reasonable and not representative of the IBM, since other choices for the E3 parameters give very different results. In this regard, we note that it is hazardous to try to establish parameters on the basis of only one nucleus and expect meaningful results. Also, one should always keep in mind that the IBM is a collective theory that should not be applied to 2qp states, unless 2qp degrees of freedom are explicitly added to the model space. C. Experimental

Data on Microscopic

Structure

in 168Er

The presence of both collective and 2qp bands within the same energy region complicates matters when it comes to deciding which data should be considered in the application of a model. In this regard, it is extremely useful to have spectroscopic data on the states involved. The nucleus 168Er has been particularly well

IBM

OCTUPOLE

STATES

IN NUCLEI

367

studied experimentally. Such studies include (d, d’) [61], (a, a’) Coulomb excitation C491, h Y) and he-,) C601, (AP’) C691, (n, n’r) C701, (d,p) and (t, d) C711, (4 a) C71, (hi) and (P, f) l?31, and (d, d’ ) and (a, a’) inelastic scattering [26]. We consider this nucleus as an example of how spectroscopic information is used to characterize states. Data on E2, E3, and E4 collectivity are obtained in inelastic scattering and Coulomb excitation experiments. Such data are given in Table III, for the octupole degree of freedom. Information about microscopic structure is obtained in singlenucleon transfer reactions. For 16*Er, the Nilsson qp levels just below and just above the neutron Fermi surface are 3’ [633t] and i- [521j]. These qv levels constitute the gs for the nuclei 16’Er and ‘69Er, respectively [72]. The corresponding F and F+ 1 q7-clevels are $- [5231] and $’ [4111], which constitute the gs for 16’Ho [73] and 169Tm [72], respectively. The ‘69Tm(t, a)‘68Er reaction populates states with 2qx components in which one of the Nilsson levels is i.+ [411J]. Similarly, states populated in the 167Er(d,P)‘68Er reaction have components of 2qv configurations containing $’ [633f]. Data from such experiments, as reported, for example, in Refs. [7, 8, 711, have led to 2qp assignments for the K” = 4 -, 3 -, and 6- bands between 1 and 2 MeV in 16*Er. The lowest negative-parity band in 16*Er, K” = 4-, is strongly populated in both neutron and proton-transfer reactions. It is identified with the two (F, F+ 1) configurations, vv(~‘[633f]@t-[5211]) and nn(%-[523t]0ft[4111]). The 4band at 1905 keV is dominated by the former configuration and the 4- band at 2059 keV is dominated by the latter configuration. It is obvious that these three K” =4bands are mixed by the proton-neutron interaction, accounting for the appearance of a “non-collective” band at such low energy (K” = 4- at 1094 keV). The same qp levels, when combined in the opposite way (spin unfavored configurations), result in K= 3 and are identified with the K” = 3;(vv) and K” = 3; (7~7~)bands. The K” = 3; and 6; bands are known to have 2qn and 2qv character, respectively. The K” = 1; band is weakly populated in single-nucleon transfer. Although it is clearly collective, the gamma band, K” = 2:) in 16*Er is strongly populated in (t, a); it has a large component of the configuration 7rn(~+[411~]@f+[411~]) in its wave function. Similarly, the lowest octupole band, K” = 1 -, is strongly populated in (d, p), being dominated by the 2qv configuration {i’ [633T] Q 2 ~ [ 51211). Such states can be collective, despite large 2qp components, if the other configurations in the wave function combine constructively. D. Discussion of Parameter Trends The negative-parity parameters Ed, A,, A,, and x3 are given in Table II. It can be seen that the parameters for the nucleus lssDy deviate from the general trends. The relatively small values of q and IA, 1 result from the K”= 2-, I-, OP band ordering, which is anomalous with respect to the other nuclei. There is a discontinuity in the A, trend for the nucleus “‘Yb, also. Once again, the band ordering

368

BARFIELD

ET AL.

K” = l-, OP, 2 ~ is anomalous, with respect to the general trend. The ratio As/A, does not vary smoothly with mass number. In fact, it cannot, since it depends on the experimental octupole band ordering, given also in Table II. The K- order, for a given nucleus, clearly depends on which Nilsson qn and qv levels are near the Fermi surface. The ratio A,/A, increases as the lowest K- band goes from O- to 1~ to 2-. For K” = 3 ~ to be lowest, A, and/or A, must be positive. There are at present no known deformed nuclei, for which the lowest octupole band is K” = 3 -, with sufficient .octupole data to determine the parameters .sr, A,, and A, with any degree of confidence. It should be noted that K- band energies go down as A, is decreased. The f-boson energy Q. compensates, bringing the calculated energies back up, as necessary. The low value of s,for the nucleus ls2W is related to the arbitrary choice of AZ = 0, for this nucleus. The calculated B(E3)‘s are essentially determined by the parameter x,, since the boson octupole effective charge, e3, is constant for all the nuclei studied. The parameter x3 has values that fall into two different categories, -0.5 < 1(360.5, and x3 z 3 f 1 (cf. Table II). Small values of x3 describe nuclei for which the E3 strength is concentrated in one or two low-lying states. The larger values of x3 are necessary, as already noted, when the E3 strength is fragmented over many states. We note that the two nuclei for which this is the case, 16’Er and 172Yb, have neutron boson particles and proton boson holes. All other nuclei studied have all-particle or all-hole degrees of freedom. E. The 16*Dy Failure and the K- order 2 -, O-, 1 Given the strong role played by 2qp admixtures in octupole states and the absence of separate neutron and proton degrees of freedom in the IBM-l model, it is not surprising that the model should fail in some cases. It is of interest to consider the expected K- band ordering resulting from a single octupole degree of freedom. The degeneracy of the 3 ~ octupole state seen in spherical nuclei is broken by the quadrupole deformation, j12. For prolate deformation this would be expected to result in the set of K bands shown schematically in Fig. 17a. The energies of the observed K- bands in a given nucleus depend on the location of the Fermi level. Near the beginning of a well-deformed region the Fermi level is closest to the Oenergy, leading to an expected ordering of (O-, 1 -, 2-, 3- ). Near the end of the deformed region the Fermi level is closest to the 3- energy and the ordering is then (3-, 2-, 1 -, OP). The assumption of a smooth transition between these two situations leads to the schematic picture shown in Fig. 17b. The dashed lines separate the regions of different Kp band ordering: (O-, l-, 2-, 33), (1-,0P,2P, 33), (l-,2-,O-,3-), (2-, ll,3-,0-), (2-,3-, l-,0-), and (3-, 2-, l-,0-). In this simple picture, the band order K”=2-,O-, l-, seen in the nucleus 162Dy, does not arise. The 2qn configuration {q-[5231] @ $+[411f]} 2- is expected to be low in energy for all the deformed dysprosium isotopes, since the two Nilsson components

IBM

E

OCTUPOLE

STATES

IN NUCLEI

369

3--

b) 0I23-

EF-

FIG. 17. A simple schematic picture of octupole band ordering. Dependence on the quadrupole deformation /I2 is shown in (a). Dependence on the Fermi level E, is given in (b), assuming a smooth transition between a well-deformed region (on the left) and a spherical region (on the right). Regions with different Km band ordering are separated by dashed lines.

are adjacent to each other, and the Fermi level for protons lies between them. The K” = 2 - octupole band in 162Dy96 is dominated by this coniiguration [SS]. For this nucleus, the Nilsson levels $‘[642f] and f-[523J] are on either side of the neutron Fermi level. Thus, a large 2qv admixture is expected (and has been predicted by Soloviev [74]) for the K” = 0 - octupole band. The octupole band order for this nucleus is, then, understandable from a microscopic point of view. It is quite possible that an adequate description of this nucleus would be obtained by adding a second, distinguishable, f boson (an f’ boson) to the IBM-l model space. Alternatively, an IBM-2 treatment could be employed. F. Strengths and Weaknesses of the Model The IBM-l +f-boson model presents a simple picture for octupole bands in nuclei. The Coriolis coupling effects are inherent in the model. The interplay between quadrupole and octupole degrees of freedom also occurs naturally in this model. In comparison with the one set of K” = 0 -, 1~) 2 -, 3 -- expected from a simple vibrational picture, many “octupole” bands, including K” > 4, arise naturally in the algebraic picture. Predictions of the model provide insight into higher-lying band structure. Only three or four parameters are needed in order to describe negative-parity states. Two parameters suffice for the determination of B(E3) value. In the present study, one E3 parameter is constant. Results compare well with the data, in most cases. On the other hand, states built primarily on single-particle degrees of freedom manifestly cannot be treated within the model. One must, in fact, consider the data carefully in order to identify those states to which fitting should be attempted.

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Because of the lack of microscopic content, parameter trends are not straightforward. Predictions cannot be made for nuclei for which no data are available. In some cases the model fails outright. Such failures, as well as the appearance of deviations in parameter trends, can serve as an indication that specific single-particle degrees of freedom are complicating the collective picture. The determination of parameters with any degree of confidence is, at present, a nontrivial task. This is particularly true where the data base is not large. Thus, a fit to a single nucleus is of questionable value. G. Suggestions for Future

Work

One would like to extend the scope of the study, eventually, to include nuclei outside of the deformed region, in a systematic way. In such an undertaking, it would be particularly useful to have more data for transitional nuclei. Before considering fits to specific nuclei, it would be instructive to consider the behavior of the d-f quadrupole-quadrupole and exchange terms in the O(6) and U(5) limits of the IBM-l. The subject of electric dipole transitions has not been dealt with in the present study. El transitions occurring in the energy region under consideration are not well characterized experimentally, and have not been systematically treated in the IBM. Such a treatment is under way. Microscopic calculations for the negative-parity parameters would be most interesting. This is not an easy task, particularly for deformed nuclei. The situation with respect to octupole states in deformed actinide nuclei is more complicated, in general, than the rare-earth case because of the presence of lowlying O+ states which appear [41] to be outside of the IBM model space. An adequate treatment would include configuration mixing, necessitating the introduction of more parameters into the Hamiltonian. It is doubtful that enough pieces of data exist, with which to determine these extra parameters. The actinide picture is further complicated by the likely presence of collective negative-parity states based on dipole degrees of freedom [30, 311. An IBM-2 +f-boson treatment is feasible, as is the extension of the IBM-l model space to include an f’ boson. Such treatments would, however, involve the addition of extra terms in the Hamiltonian and, hence, extra parameters. VI.

cO~~LusI0Ns

The IBM-l + f-boson model has been described and its phenomenology explored. The results of a systematic study of deformed rare-earth nuclei show that the model works reasonably well, in spite of the complex collective-plus-2qp nature of many octupole states. Nucleus by nucleus, with the exception of i6*Dy, the model gives a good description of the states fitted. The model predicts higher-lying band structure, thus providing some simple insight into excitations at higher energy. The reason for the failure of the model to describe 16*Dy may be due to the fact that it does not include separate neutron and proton degrees of freedom.

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On a global scale, the parameter sets are similar, but the trends are not simple. Although this is disappointing, it is not surprising and indicates that it is dangerous to draw conclusions about parameter values from a fit to a single nucleus. Our results would suggest that parameter values depend on the details of the underlying fermionic subshell structure. Often, octupole bands and 2qp negative-parity bands are found in the same energy region. The experimental evidence needs to be considered carefully since the IBM cannot be used to describe 2qp effects, unless such degrees of freedom are explicitly added to the model space, e.g., as fermion pairs in an IBFFM description [2]. In some cases, the data base is not sufficient to determine the IBM parameters with confidence. More experimental data are needed, both to test the predictions of the model and to extend the scope of the calculations. The present study illustrates both the limitations of the fBM and its power in correlating data and providing a simple view of nuclear structure. APPENDIX:

IBM- 1

PARAMETERS

Various definitions of the IBM operators can be found in the literature and it is not always clear how the different parameter sets are related. In the interest of clarity, we list in Table IV the input parameters for the IBM-l computer codes PHINT and FBEM [39] and their relationships to our parameters, as defined in Eqs. (9), (lo), and (14)-(18). Some particular sources of confusion are worth specific mention. The s-d quadrupole strength, which we call ul, is often called K [36]. This K is not the same as that of Eqs. (26a) and (26b) because of the factor of two in the latter equation. Although the definition of the pairing operator, P, given in Eq. (lob) is a comTABLE Relationship

Hamiltonian EPS=E PAIR = ad2 ELL=2a,

QQ =

2a2 OCT = a,/5 HEX = aJ5 CHQ = fix

IV

between the Parameters of the Present Work and the Input Parameters for the IBM-l computer codes PHINT and FBEM [39]

parameters HBAR3 = E, EPSD = A, FELL= A,

FQQ=Az FEX=

-A,

Transition operator parameters E2SD=e EtDD=,,beX

E3=e, E3DF=

fi

ezXs

EIDF= & El QE3 = fi

p/N

372

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ET AL.

mon one [32,75, 761, it differs from that given in the PI-IINT manual [39], resulting in an apparent inconsistency. The operator L,[Eq. (9a)] is sometimes defined with a minus sign, leading to possible confusion regarding the sign of the interaction L,. Ly Finally, in the computer code FBEM the coefficients of all terms of the form where bl and b2 are d- orf-boson operators, include the numerical fac(6, x bp tor (l/,/G). These numerical factors are sometimes missing in the literature. Thus, for example, the coefficient of the (8 ~7)~~) term is E3DF/+‘?, rather than E3DF. ACKNOWLEDGMENTS The authors are indebted to G. Leander and F. Iachello for many helpful discussions. On of us (A.F.B.) would like to thank UNISOR and the National Superconducting Cyclotron Laboratory for their hospitality at Oak Ridge National Laboratory and Michigan State University, respectively, where many of the calculations were done. This work was supported in part by the National Science Foundation under Grants PHY-84-05172 and PHY-8407858, and by the Department of Energy under Contracts DE-AS05-80ER10599 and DE-AC05-760ROOO33.

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