Nuclear Physics A398 (1983) 235252

0 North-Holland Publishing Company

g-BOSON EXCITATIONS IN THE INTERACTING BOSON MODEL

K. HEYDE, P. VAN ISACKER’, M. WAROQUIER, G. WENES and Y. GIGASE

Institute for Nuclear Physics, Proeftuinstraat, 13 B-9000 Gent, Belgium

and

.I. STACHEL

Insritut ftir Kernchemie der Universitiit, Maim, D-6500 Maim, Germany

Received 22 September 1982 (Revised 8 November 1982)

Abstract: We have extended the interacting boson model (IBM) by including the g-boson degree of freedom. Schematic model calculations have been carried out in the two different limits: SU(5)

and O(6). Particular applications have been carried out for i@‘Ru a nucleus intermediate between , SU(5) and O(6). In all cases, energy spectra, E2 and E4 transition rates have been studied in detail

and compared with the most recent experimental data for lo4Ru.

1. Introduction

The interacting boson model (IBM) has been developed during the last years to

a level where it is possible to give a unified description of low-lying quadrupole

collective excitations, using an interacting system of s- and d-bosons ’ -“). Making

no distinction between protons and neutrons, the IBA-1 approximation is obtained

whereas, by explicitly taking into account both proton and neutron degrees of

freedom, the more microscopically founded IBA-2 approach results ‘).

Recently, in many transitional nuclei near closed shells as well as in strongly

deformed nuclei, low-lying extra states have been observed experimentally, that

cannot be accounted for within the (sd)N boson space. Especially near the 2 = 50

closed-shell mass region, i.e. in the Zr, MO, Ru, Pd, Cd, Sn,. . nuclei, extra

J” = O+, 2+ and 4+ levels occur below an excitation energy needed for creating a

two-quasiparticle configuration 4- 12). Moreover, some of these levels are

characterized by very peculiar E2 and EO decay modes 5, 9). Also, the Z = 82 closed

shell region is characterized by similar low-lying excitations, especially in the Hg

[refs. 13* ‘“)I and Pt [ref. “)I nuclei. In some well-studied strongly deformed nuclei,

such as the doubly even Gd nuclei, experimental evidence i6) for the importance of

the hexadecapole degree of freedom has resulted. Since the possible origin of some

of these low-lying excitations was attributed as being due to particle-hole

t Present address: Instituto de Fisica, UNAM, Apdo. Postal, 20-364, 01000 Mexico, DF.

235

236 K. Heyde et al. 1 pboson excitations

excitations across a closed shell and have already been discussed in the framework

of the particle-core coupling model 9, lo. “) and in the IBM 6, “3 ll. 13q 14), we will

concentrate on a detailed study of the hexadecapole degree of freedom in the

framework of the IBM. Recently, some studies have been devoted on this

particular problem using the group-theoretical reduction of the SU(15)

group 17- 19). Here, we shall carry out, in the framework of IBA-1, schematic model

calculations in which the coupling of a g-boson to an exact SU(5) or O(6) core is

studied.

Application of the more schematic calculation to a specific region of transitional

nuclei, i.e. the Ru nuclei [ lo4Ru 1, is carried out. The coupling of a g-boson to an

SU(3) core within a schematic model and applied to the even-even Gd isotopes,

was carried out by the authors ‘O). Finally, we point out the importance of

incorporating the g-boson degree of freedom in many nuclei near closed-shell

configurations and also in strongly deformed nuclei.

2. The SU(5) and O(6) limits

2.1. SCHEMATIC CALCULATIONS IN THE EXACT LIMITS

In nuclei with low-lying anharmonic quadrupole vibrational excitations, in

which the excitation energy of the two-phonon states is of the order of the energy

needed to break a pair of nucleons and create a (j)‘= J 4,6,8 ,,,, paired state, mixing

between the two-phonon and intrinsic excitations can result ‘l). These conditions

are mainly fulfilled when one type of nucleon (protons or neutrons) is away from

a closed-shell configuration by f4 nucleons at most, i.e. the Pd, Cd, Sn, Te, Xe

nuclei near 2 = 50. Concentrating on the J” = 4+ coupled two-nucleon

configurations, sometimes a collective enhancement can result so that one can

speak of a collective g-boson excitation. A more detailed study on the degree of

collectivity of such low-lying J” = 2+, 3-, 4+ excitations was carried out recently

by Akkermans et al. 22, 23) in the fram ework of the projected BCS formalism.

Here, we start from the situation drawn schematically in fig. 1, which can be

used to visualize in the SU(5) and in the O(6) limit, the actual level ordering.

Hence, for a hamiltonian describing a l(sd)N) system in interaction with

I(sd)N-l @ g) configurations, we can write

H = H,,+sgg+ .,4++i”,.(Cl-), (2.1)

where H,, describes the standard IBA-1 hamiltonian ‘). Here, we use the multipole

expansion notation

H,, = &,n,+~Q.Q+dL.L+K”P+.P+q3T3.T3+q4T4.T4, (2.2)

using the multipole operators as defined in the Dronten lectures notes of

K. Heyde ef al. j g-boson excitations 237

L SN

Fig. 1. Schematic representation of the I(sd)N) and the I(sd)N-’ @g) configurations in the SU(5) limit

(anharmonic quadrupole vibrational spectra).

Iachello 24), i.e.

Q = (d+s+s+d)‘2’-~(d+d)(2),

L = --JG(,+d)(‘),

P s $(d . d) -$s. s),

T3 = (cI+~)‘~‘, T4 = (LI+~)‘~‘,

Q, = @+J)“‘. (2.3)

Assuming that the interaction hamiltonian between the (sd)- and g-bosons,

originates in the quadrupole-quadrupole interaction and using as the quadrupole

operator, including contributions from the g-boson,

Q’” E (d+s+s+d)‘2’+x(d+a)‘2’+~@+a+d+B)(2)+5@+S)(2)r (2.2)

one obtains the general (sd)-g boson interaction hamiltonian

Hi”, (CT) = ci@+a+d+B)(2)+5@+~)(2)}. Q@). (2.3)

238 K. Heyle et al. : CJ-hoson excitations

After recoupling (2.3) one finally obtains:

with

H,i,(CT) = @‘s+)‘~‘. (~~)‘4’ + h.C.

+(cJ+~+)‘~‘. (&)“‘+h.c.

+Xx5 (~+d+)‘L’(d”a)‘L’+h.c.+.... L

(2.4)

(2.5)

The first term in eq. (2.4) <Q. Q, splits the states originating from the same l(sd)N)

core state, coupled to a g-boson but does not mix configurations. Since the nuclei

we study are SU(5) and O(6) nuclei in which the quadrupole moment Q(2:) is

generally small (but non-zero), the energy shifts (<Q. Q,) will also be small. This is

in contrast to SU(3) nuclei in which the 5Q. Q, term is of major influence in

determining the splitting between different K-bands 20).

The general mixing hamiltonian H,,,(Cr) (see eq. (2.5)) has been restricted to

only the first term in the present calculations, i.e.

In the nuclei we study

its main structure

H,,,(Cr) = (,y + s + )‘4’ . (&I)‘“‘. (2.6)

here, SU(5) and O(6) nuclei, the ground-state band has as

IsN), IsN- ‘d), IsN-‘d2), . . ., (2.7)

and the g-boson coupled band, with the lowest unperturbed energy, has as its main

structure

Is ’ - ‘g), IsN - 2dg), (9 - ‘d2g), . . (2.8)

Therefore, the first term from eq. (2.5) induces the most important coupling

between these bands. The other contributions in the mixing hamiltonian of eq. (2.5)

cause a mixing of the ground-state band 4+ level with 4+ levels from the JsNP3d2g)

and lsNm2d g) configurations, respectively. Without a microscopic theory at present

explaining the different interaction terms in eqs. (2.4) and (2.5), we have carried out

the mixing calculations in a highly schematic way, using the most effective coupling

hamiltonian corresponding with the SU(5) limit. In order to study the particular

coupling mechanisms in the SU(5) and O(6) limit for all terms of eq. (2.4), a

detailed study will be carried out [see also ref. 20) for a similar study but in the

SU(3) limit]. In further discussions, we restrict ourselves to the inclusion of only

one g-boson excitations.

In the same spirit, the electromagnetic transition operators are modified:

T(E2) = e~‘(d+s+sfd)‘2’+e~~(d+d)‘2)+e~)~q+d+d+~)’2’+e~’@+~)‘2’, (2.9)

T(E4) = e~(d+d)‘4’+e~‘@+s+s+9)‘4’. (2.10)

K. Heyde et al. / g-boson excitations 239

At this moment the extra effective boson charges e$, ez’, e$ and ez’ are new

parameters. Therefore, we have carried out most calculations of energy spectra and

E2 and E4 decay rates in a phenomenological way.

2.2.1. Perturbation theory. Starting from the mixing hamiltonian of eq. (2.6), we

can calculate, in the SU(5) limit, the wave function for the lowest J” = 4+ level as

14:) = Is N-2d2;4+)-~~Ij~~sN-1g;4+), (2.11)

with AE = 2(q, -E,)-Q the energy denominator which for the 2 _Y 50 mass region

has negative value of the order of 2 1 MeV. Subsequently, we derive the E2

reduced matrix element as

(2: 11~(E2)114: > = @3J20 1- $$ i ldEl [ 1 Js 9 1 , (2.12)

which for a general yrast E2 transition, can be generalized to the expression

(J-2 = 2n,-2llT(E2)11J = 2n,) = &Wr,+ l)n,(4n,+ 1)

For the hexadecapole transitions, one obtains

(0W(E4)114:) = -e$i IAEl ~Jziqcij,

(2.13)

(2.14)

and the general yrast matrix element equals

(J-4 = 2nd-4llT(E4)llJ = %‘I,) = -eg)i&

‘$(25+1) n,(n,-l)(N-n,+l)(N-n,+2). (2.15)

From the expressions (2.12) and (2.13) one can notice that the Z- and f-

contributions to the yrast B(E2) values will result into positive interference if the

product (e$)/ei:))c becomes a negative quantity. Taking eit) as a positive effective

charge (N 0.1 e. b in the schematic calculations, this is also a typical value

obtained from IBA calculations in this particular mass region 5), and having [ > 0,

implies that for negative values of efdi constructive interference will result in eq.

(2.13) and for positive values of egd , (*) destruction interference. Already from the

lowest order in perturbation theory one notices that the yrast B(E2) values can

become larger than the standard IBA-1 and IBA-2 results. Hexadecapole strength

obtained via particle inelastic scattering will determine further information about

the effective charge es’, using eqs. (2.14) and (2.15).

The quadrupole moment for the yrast band members however will only deviate

240 K. Heyde et al. / g-boson excitations

from their C-values by a number proportional to [‘ebb’. Therefore, one will need

positive effective boson charges egg (2) in order to make the quadrupole moments less

negative compared with their C-values.

2.1.2. Schematic calculations. Starting from the knowledge gained from the

analysis of subsect. 2.1.1, we carry out more detailed but still schematic

calculations, coupling a g-boson to an exact SU(5) and O(6) core. Thereby, we use

the following parameters for the hamiltonian of eq. (2.1):

(i) SU(5): cd = 0.4 MeV; c0 = 0.17 MeV; c2 = 0.10 MeV; c4 = 0.07 MeV;

(ii) O(6): K” = 0.10 MeV; K’ = 0.02 MeV ; q3 = 0.15 MeV.

These parameters are obtained via a projection method 25) starting from the lo4Ru

IBA-2 parameters as determined by Van Isacker and Puddu 5). They can be used

as a standard set of parameters for the schematic SU(5) and O(6) calculations.

For the unperturbed g-boson energy, we take a typical value of &g = 2 MeV,

which is near to 24, the energy needed to create a two-quasiparticle excitation

in the 2 = 50 (A z 100) mass region. The effective boson charges used are

eL<) = 0 1 e.b, e$’ = -0.1 e.b and we vary ebi’ 5 . and e$) from 0 to +0.3 e. b in

( v-8,0 I

7’ 6’ . .

q 2’ _ - 1210-

. . a7’_- 6’ 5’ IO -

1 v=6,01

* L’ a’6’_2 -

i i 6’_ -

l.’ - 2’

Fig. 2. Pure W(5) limit for the specilic case of N = 8 (s,d) l(sd)’ @g) configurations is given. Only the lowest bands are

brackets are (u,nd).

1 v=8,Obg

- 12

. - IO

- 9 .

. -6

bosons. Also the lowest band for the given. The quantum numbers between

K. Heyde et al. / y-boson excitations 241

steps of 0.1 e b. For the E4 operator, we use the boson charges el;(d = eg) = 1.0

e. b2. In the latter case, no experimental hexadecapole matrix elements are known

and only an exploratory calculation could be carried out. All calculations discussed

below are done with a mixing strength parameter of [ = 0.2.

In fig. 2, we show part of the unperturbed (no mixing) SU5 and SU(5) @ CJ

spectra for a (sd)N=8 boson system. In figs. 3 and 4 we show the yrast B(E2) values

and the static quadrupole moments for the yrast band members respectively. In

each case, we also compare with the results obtained with the full IBA-1

hamiltonian [see eq. (2.6)], with the projected IBA-2 parameters of lo4Ru [ref. “)I.

Some remarkable features do result:

(i) In the yrast band, for negative values of egd, (2) for both the SU(5) and the O(6)

limit, the B(E2) cut-off is shifted to higher angular momenta (see fig. 3). Due to the

constructive interference between the C- and the f-amplitudes, B(E2) values result

which are doubled compared to the standard IBA-1 calculated values. We also

show the yrast B(E2) values for positive values of es) values, in which case,

destructive interference results. In the O(6) limit, the B(E2) enhancement in the

yrast region is even more pronounced compared with the SU(5) limit.

OL-

f c-4

" 03-

i;j

1

F;‘ 0.2- UJ

m

I

0.1 -

o.ok--e-%*

0.1 -

0.01 I I I I I I

2 I 6 6 IO 12

Fig. 3. Calculated B(E2) values in the yrast band in the exact SU(5) and O(6) limits, when coupling a g-boson to the I(sd)N) configurations. The label i on each curve indicates the effective boson charge

es) = i(O.1) e. b. The result called IBA-I corresponds to the pure limit with no g-boson mixing.

K. He_vde ct al. / pboson excitations

-Is-

\ \ IBA-I

I I I I t I

- ls-

0 I61

\ \

\

\

\ 3 \

\ \

2

\ \ I \ \ \ 0

’ IBA- I I I I I I

Fig. 4. Same caption as 3, but for the yrast quadrupole moments. The label i on the different curves gives the effective boson charge e:’ = i(0. I ) e. b.

(ii) Analoguous remarks hold for the E2 transitions within the y-band.

(iii) For the I, + I, E2 transitions, the IBA-1 results remain basically unaltered

when incorporating the g-boson admixtures in both the SU(5) and O(6) limits.

(iv) For positive values of the effective boson charge eki’ the yrast quadrupole

moments become less negative. This reduction becomes very pronounced in the

O(6) limit for values of eki’ = 0.3 e. b (see fig. 4).

We have also calculated in both the SU(5) and O(6) limits the hexadecapole

excitation matrix elements (4+))T(E4)110:) for the charges e?J = e:t’ = 1.0 e. b2. In

lig. 5, we show the SU(5) results. Here, only the (~+s+s+,c#~) operator contributes.

Thus only if < # 0, the lowest J” = 4: level which is mainly a C-state, can

become excited. For the O(6) limit, where both the (~y+~+s+jj)(~) and the (dfd)(4)

operators contribute, the separate contributions are given in table 1, as a function

of the mixing strength [ ([ = 0, 0.2 and 0.4). Within the calculations carried out

here, one observes that only two levels become strongly excited. Here, however,

one should keep in mind that we have taken equal boson charges for both

eg) and el;‘d. This simplification is by no means obvious and has to be tested via

hexadecapole excitation experiments.

The parameters eg’ and e$) used in the schematic calculations cannot easily be explained starting from the underlying shell structure. The Otsuka, Arima and

Iachello method 26) (OAI) of mapping shell-model matrix elements into boson

matrix elements is only possible in the framework of the proton-neutron version of

the interacting boson model (IBA-2). The IBA-1 model on the other hand is a

K. Heyde et al. / g-boson excitations 243

- 10 -RELATIVE UNITS -

Fig. 5. Hexadecapole excitation matrix elements (relative units) in the W(5) limit for different values of the coupling strength [ (see eq. (2.6)), when coupling a g-boson to the I(sd)N) configurations.

more phenomenological model. Therefore, only by comparing the calculated

electromagnetic E2 properties for a specific nucleus with the existing experimental

data we will be able to deduce reasonable values for the extra boson effective

charges ez’ and el,zd’, when including the g-boson degree of freedom in the present

description. Moreover, one should remind that the present schematic calculations

in both the SU(5) and O(6) limit have been carried out using the restricted mixing

hamiltonian of eq. (2.6). A careful study of the other possible mixing and splitting

terms of eqs. (2.4) and (2.5) is under way.

2.2. APPLICATION TO THE Ru NUCLEI: ““Ru

In order to study a region of nuclei in which the schematic model calculations

for coupling a g-boson to a SU(5) or O(6) core can be applied, we should look for

244 K. Heyde et al. i g-hoson excitations

TABLE 1

The hexadecapole excitation matrix elements in the limit of coupling a g-boson to an O(6) core (the separate contributions of the operator T(E4) of eq. (2.10) are given)

4:

[ = 0 0.2 [ = 0.4

E, (MeV) (cI+~)‘~’ (q + sp E, (MeV) (d+d)c4) [q + sp E, (MeV) (d+a)c4) [q + sp

4: 0.85 - 3.72 0.79

4; 1.45 1.37

4: 1.75 0.58 1.67

44’ 2.00 6.63 2.09

4; 2.20 2.09

4: 2.33 2.34

4: 2.35 2.36

48’ 2.50 2.44

4; 2.78 2.84

4:0 2.80 2.86

4:, 2.85 2.89

4:, 2.95 2.90

4:3 3.05 2.92

4L 3.10 2.92

4:s 3.10 3.02

4:h 3.13 3.22

4:, 3.35 3.41

4:8 3.40 3.46

4:Cl 3.41 3.47

4;ll 3.45 3.57

- 3.70 1.50

-0.39 1.186

~ 0.41 - 3.409

0.65 5.266

- 0.030 - 0.066

- 0.30 -0.151

0.064 0.104

0.084 0.266

0.0002 0.0003

- 0.096 -0.139

0.032 0.033

-0.019 - 0.008

0.64

1.21

1.51

1.87

2.06

2.18

2.26

2.62

2.63

2.68

2.70

2.12

2.97

3.12

3.25

3.38

3.48

3.48

3.49

3.52

- 3.70 2.43 1

0.34 -0.853

- 0.008 0.053

0.0024 - 0.48 1

1.116 5.710

0.032 0.022

-0.01 -0.013

-0.56 - 0.336

-0.41 - 0.995

0.07 0.111

0.003 0.007

nuclei intermediate between both limits. It was shown recently by Stachel et al. ‘3 8,

that the Ru and Pd nuclei, in the mass A N 100 region, are probably good

transitional nuclei between SU(5) and O(6). Moreover, recent experiments “) on

lo4Ru point towards the necessity of incorporating extra degrees of freedom

besides the s- and d-bosons:

(i) There is a sudden increase in the B(E2) values between the 4: + 2: and the

6: +4: transitions.

(ii) There is no clear evidence for a reduction in the yrast B(E2) values, which,

conform with the IBA-1 predictions, should show a cut-off at spin I” = 16+.

Not only IBA-1, but also calculations in the framework of the asymmetric rotor

model (AR) 27,28) and the collective model studies of Gneus, Greiner and Hess

(GGH) 29.30) are unable to explain the feature (i).

The schematic model calculations of subsect. 2.1. give some hope for explaining

both features by including besides the J(sd)N) states, also I(sd)N-l @ g) states.

Although no unique evidence exists for the observation of a low-lying hecadecapole

state in lo4Ru, near E, 5 2 MeV, a J” = (4+) level has been observed that cannot be

K. Heyde et al. / g-boson excitations 245

explained easily within the standard descriptions of low-lying collective quadrupole

excitations.

Here, we have started from the full IBA-1 hamiltonian of eqs. (2.1)-(2.3) and

with a g-boson mixing hamiltonian as described by eq. (2.6). The IBA-1 parameters

are obtained from the IBA-2 parameters of lo4Ru [ref. “)I, and are given below:

E~ = 0.6 MeV F, = 2.0 MeV;

K = -0.010 MeV; K’ = 0.010 MeV; K” = 0.098 MeV; q3 = 0.050 MeV;

[ = 0.2.

The calculated spectrum is shown in fig. 6 where both the results for [ = 0

(unmixed calculation : U) and for [ = 0.2 (mixed calculation using the g-boson : M)

are indicated and compared with the recently obtained experimental data of ref. *).

In fig. 7, we show for comparison the results obtained using the proton-neutron

IBA-2 formalism for the same values of parameters in lo4Ru. Both calculations

give similar results.

The overall agreement of the IBA-1 calculation with experiment is good but for

the y-band, a very pronounced staggering effect (3+, 4+), (5+, 6+), . . . shows up

which is only weakly seen in the experimental data. This strong staggering in the y-

band is a consequence of the fact that the corresponding IBA-1 hamiltonian

potential energy surface (seen the large pairing part of the IBA-1 parameters given

5

loLRu U M EXI?

L-

I

lo'- \ \ \ I__)

5 \'

0 3- I

U M EXP

. 8- 7*\:,

“F, ‘\ ‘-

\ . 6.- - 5 -;\

‘_,._ \ ‘- .

‘.- j-: -; -

\

2*----,_

U M EXP U t-4

f- _ -_ ;:-__ ($=

2'-

iOi- O+---

. O-

EXI?

I

010+__-.-- -I Fig. 6. Comparison of the experimental lo4Ru spectrum ‘) with IBA-1 calculations without g-bosons (U: unmixed) and when coupling a g-boson to the same IBA-1 I(sd)N) core, interacting via the

hamiltonian of eq. (2.6) (mixed: M).

246 K. Heyde et 01. / g-bo.m excitations

IBA-2 EXP

- 12 .

IBA-2 EXP IBA-2 EXP

- - 9’ \

‘I -IlO’ ~1 \ - 8’

‘\ ‘( -17*

‘\ l

-8 - -‘\ -6+

\ \ \- 5’

l

--- 6 -. - -L’ \

l

- IO 10hR

l i* =A 0: -. . .

U

--

- 1’5-1 ’ =IL’)

Yf. d - 4’ ‘3’ -j-

7;: - d 7

2’- 2 .

0’ I I ‘- 3’

1

t

-* l

-- - L’ ---2+ o-o

__-2.

0 1 1

--- d 1 Fig. 7. Comparison of the experimental lo4Ru spectrum’) with the IBA-2 calculations, using the

parameters of ref. 5).

above, being close to the O(6) limit) shows the typical y-soft (or even y-unstable in

the extreme O(6) limit) potential in the classical (/?, y) plane. We have studied this

staggering effect in the IBA-1 hamiltonian by including cubic terms in the d-boson

operators 3’s 32). Also, by studying the potential surface of cubic d-boson terms in

the IBA-1 hamiltonian via the classical limit expressions 31), one can obtain stable

triaxial shapes starting from a y-unstable O(6) potential. Thereby, the (3+,4+),

(5+,6+) staggering gradually changes into the (2+, 3+), (4’, 5+), . . . staggering,

which is typical of an asymmetric rotor 32). An alternative approach to describe

non-axial symmetric features in terms of the neutron-proton IBA-2 has been

proposed recently by Dieperink and Bijker 33). A possible connection between the

effect of introducing a g-boson in the interacting boson model (IBA-l), the

addition of cubic terms in the interacting boson model (IBA-1) and the approach

of Dieperink and Bijker needs to be clarified since the three models induce particular

triaxial deformation effects.

We have also calculated the B(E2) values, static quadrupole moments and a

number of branching ratios. Extensive comparison with the experimental data of

ref. 8, is carried out (see figs. 8-12). We have used the E2 operator of eq. (2.9) with

247

0.5 -

t 0.L -

5 m

w; ;; 0.3-

? ,” ,-

z m 0.2-

I

0.1 -

K. Heyde et al. / y-boson excitations

-1

IBA-I

0

IBA-2

o.oIL 0 2 4 6 8 IO 12

___ I,-

i

Fig. 8. Yrast E(E2) values, calculated using the IBA-I hamiltonian (see subsect. 2.2) coupled with one g-boson excitations. The IBA-2 calculations are also given5). The curves correspond to di&rent effective boson charges e$ and are labeled with a parameter i (egd w = i(O.l) e. b). The experimental data

are for lo4Ru [ref. “)I.

the effective charges eit’ = 0.1 e. b, e$ = -0.05 e. b and the g-boson charges

ec2) = e$) = 0 +O.l, f 0.2 and kO.3 e. b. In each case, we also compare with the

IgA-1 ([ = 0; aid IBA-2 [ref. “)I calculations. Here, one notices that for a value of

e!$r = -0.2 e. b, both the order of magnitude of the yrast B(E2) values and the

shift of the B(E2) cut-off to much larger angular momenta are observed (see fig. 8).

The same features are also observed and qualitatively reproduced for the y-band

B(E2) values (see fig. 9). For the I, + I, - 2 B(E2) values, which normally are very

small, the general trend again is well described for a value of es) = -0.2 e. b (fig.

10). Concerning the static quadrupole moments in the yrast band (fig. ll), even for

the larger values of eBB , (2) the experimental data of ref. 8, are approached. These results

are also compatible with triaxial shapes, since in the framework of the asymmetric

rotor 27, 28) the values and the systematics in the yrast band are well described ‘).

248 K. Heyde et al. / g-boson excitations

O’r------

0.1. -

t 0.3 -

“0 N

0

l-4

3 t zo.2 - . . N w

7

0.1 -

-1

lBA- 1 0

1

IBA-2

2

3

4 6 a 10

__ ‘Y--

Fig. 9. Same caption as 8, but for the y-band.

Finally we have compared some measured branching ratios with the SU(5) @ g,

O(6) 0 g, the IBA-1 @ g boson calculations described above and also with the

IBA-2 calculations of ref. 5). Here again, one observes an overall good agreement

for the IBA-1 @ g-boson calculations, using the effective charge e’$) = -0.2 e. b

(see figs. 12a-e).

Concluding, we can say that incorporating a g-boson degree of freedom in the

IBA-1 description of collective excitations, extends the scope of this model.

Moreover in the case of lo4Ru, detailed agreement in both the energy spectra and

the B(E2) can be obtained.

K. Heyde et al. / y-boson excitations 249

0.010

t

No N

0

N I

fjoo5 I-

.- i.l m

I

0.001

Fig. 10. Same caption as 8, but for the I, -+ I,-2 transitions.

3. SU(3) nuclei

After the study of sect. 2, where a g-boson was coupled to an SU(5) and O(6)

core, the coupling of a g-boson to a SU(3) core, also in a schematic model, was

studied by the authors in sect. 3 of ref. 20). Application to the even-even Gd nuclei

was carried out in order to possible hexadecapole excitation.

In order not to duplicate this work here but still keep a coherent treatment of

the g-boson coupled to the three different limits of the interacting boson model, we

refer to ref. 20) for detailed calculations.

4. Conclusions

We have studied, in a schematic approach, the influence of coupling a g-boson

to the configurations I(sd)N) on the description of energy spectra and E2 and E4

I I I I

- 1 \ \ \ \ \ ‘1 1 \ \ \ \ \ ’

IBA- 1

- ly-

250 K. Heyde el al. : g-hoson excitations

O

I I I I I I

IBA-I

1 T T

IBA-1 0

-1 -2 -3 IBA-2

I I I I I I

2 L 6 8 10 12 - I,-

Fig. 11. Same caption as 8, but for the static quadrupole moments in the yrast band. The curves are labeled with a parameter i, indicating the effective boson charge ez’ = i(O.l) e’ b.

transitions. We have carried out detailed model studies in the SU(5), O(6) and

SU(3) limits as well as in more detailed situations [lo4Ru, even-even Gd nuclei, see

ref. ‘“)I.

The most pronounced influence of the g-boson configuration shows up in the

yrast B(E2) values. Much larger values (N factor of 2) can result as compared with

the IBA-1 calculations. Moreover, the yrast cut-off is shifted to higher angular

momenta. For deformed nuclei, the authors have indicated [see ref. ‘“)I in an

almost parameter free study how extra K” = 4+, 3’, 2+, . ., Of low-lying

hexadecapole bands can result. This particular result can also shed new light on

the observation in the Er nuclei of bands that could not be described within the

IBA-1 model only taking into account s- and d-bosons.

Finally, we remark that we only took into account one-g boson excitations. The

most general situation can only be studied seriously by reducing the SU( 15) group

of an interacting s-, d- and g-boson system.

The authors are most grateful to Prof. F. Iachello for stimulating this research

during the last year in the light of a systematic study of “intruder” excitations in

the IBA-1 model. They are indebted to Dr. J. Wood, Dr. R. Casten, Dr. D. Warner

and Dr. A. E. L. Dieperink for general discussions about the IBM approach. The

authors are grateful to the IIKW (Interuniversitair Instituut voor

Kernwetenschappen) for financial support. Moreover M.W. acknowledges the

K. Heyde et al. / g-boson excitations 251

252 K. Hey& et al. ! g-Boson excitations

NFWO (Nat’ionaal Fonds voor Wetenschappelijk Onderzoek) and G.W. the

IWONL (Instituut voor aanmoediging van het Wetenschappelijk Onderzoek in

Nijverheid en Landbouw) for constant support.

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