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g-boson excitations in the interacting boson model
Nuclear 0
Physics
A398 (1983) 235252
North-Holland
Publishing
g-BOSON
Company
EXCITATIONS
K. HEYDE,
IN THE INTERACTING
P. VAN ISACKER’,
M. WAROQUIER,
BOSON
G. WENES
MODEL
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 (Revised
22 September 1982 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 whereas, by explicitly taking into account both proton and neutron freedom, the more microscopically founded IBA-2 approach results ‘). Recently,
in many
transitional
nuclei
near
closed
is obtained degrees of
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. 235
Postal,
20-364,
01000 Mexico,
DF.
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. 13q14), we will concentrate
on a detailed
study
framework
of the
Recently,
particular
problem
IBM. using
of the hexadecapole
the
some
studies
degree have
group-theoretical
been
reduction
of freedom devoted of
the
in the on
this
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+ where H,, describes expansion notation
the standard
.,4++i”,.(Cl-),
IBA-1 hamiltonian
(2.1) ‘). Here, we use the multipole
H,, = &,n,+~Q.Q+dL.L+K”P+.P+q3T3.T3+q4T4.T4, using
the
multipole
operators
as
defined
in
the
Dronten
(2.2) lectures
notes
of
K. Heyde ef al. j g-boson excitations
L Fig. 1. Schematic
237
SN
representation of the I(sd)N) and the I(sd)N-’ @g) configurations (anharmonic quadrupole vibrational spectra).
in the SU(5) limit
Iachello 24), i.e. Q = (d+s+s+d)‘2’-~(d+d)(2), L = --JG(,+d)(‘), P s
$(d. d) -$s.
T3 = (cI+~)‘~‘,
s), 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 one obtains
the general
(sd)-g boson
interaction
(2.2)
hamiltonian
Hi”, (CT) = ci@+a+d+B)(2)+5@+~)(2)}.
Q@).
(2.3)
K. Heyle et al.
238
After recoupling
(2.3) one finally
: CJ-hoson excitations
obtains: (2.4)
with H,i,(CT)
= @‘s+)‘~‘.
(~~)‘4’ + h.C.
+(cJ+~+)‘~‘.
(&)“‘+h.c.
(2.5)
(~+d+)‘L’(d”a)‘L’+h.c.+....
+Xx5 L
The first term in eq. (2.4)
= (,y+ s + )‘4’ . (&I)‘“‘.
here, SU(5) and O(6) nuclei,
In the nuclei we study its main structure
IsN), IsN- ‘d), IsN-‘d2), and the g-boson
coupled
(2.6)
the ground-state
. . .,
band
has as
(2.7)
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 lsNm2dg) 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’, T(E4) = e~(d+d)‘4’+e~‘@+s+s+9)‘4’.
(2.9) (2.10)
K. Heyde et al. / g-boson excitations
At this moment
the extra
effective
boson
charges
e$,
239
ez’, e$
parameters. Therefore, we have carried out most calculations E2 and E4 decay rates in a phenomenological way. 2.2.1. Perturbation can calculate,
theory.
Starting
from the mixing
in the SU(5) limit, the wave function 14:)
and
ez’ are new
of energy spectra
hamiltonian
and
of eq. (2.6), we
for the lowest J” = 4+ level as
= IsN-2d2;4+)-~~Ij~~sN-1g;4+),
(2.11)
with AE = 2(q, -E,)-Q the energy denominator which for the 2 _Y50 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 which for a general (J-2
yrast E2 transition,
= 2n,-2llT(E2)11J
1-
1Js1,
$$ i ldEl
can be generalized
= 2n,)
= &Wr,+
9
(2.12)
to the expression
l)n,(4n,+
1) (2.13)
For the hexadecapole
and the general
(J-4
transitions,
one obtains
(0W(E4)114:)
= -e$i
yrast matrix
element
~Jziqcij, IAEl
(2.14)
equals
= 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 fcontributions 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, interference will result in eq. implies that for negative values of efdi constructive (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
K. Heyde et al. / g-boson excitations
240
from their
C-values
by a number
proportional
to [‘ebb’. Therefore,
positive effective boson charges egg (2) in order to make the quadrupole negative compared with their C-values.
one will need moments
less
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 (i)
SU(5):
(ii)
O(6):
parameters
cd = 0.4 MeV;
c0 = 0.17 MeV; K” =
These parameters are IBA-2 parameters as as a standard set of For the unperturbed which is near to 24, in the 2 = 50 (A z eL<) 5 = 0 . 1 e.b,
for the hamiltonian
0.10 MeV;
of eq. (2.1): c2 = 0.10 MeV; c4 = 0.07 MeV; K’
.
IO -
0.02 MeV ; q3 = 0.15 MeV.
obtained via a projection method 25) starting from the lo4Ru determined by Van Isacker and Puddu 5). They can be used parameters for the schematic SU(5) and O(6) calculations. g-boson energy, we take a typical value of &g= 2 MeV, the energy needed to create a two-quasiparticle excitation 100) mass region. The effective boson charges used are
e$’ = -0.1
e.b
and we vary ebi’ and e$) from 0 to +0.3
1 v=6,01
( v-8,0 I
. 1210-
=
.
q
2’
_
7’
-
6’
e. b in
1v=8,Obg *
a’6’_2
-
L’
-
12
-
IO
-
9
.
. a7’_-
6’
5’
6’_
l.’ -
i
2’
-
i
-6
.
.
Fig. 2. Pure W(5) limit for the specilic case of N = 8 (s,d) bosons. Also the lowest band for the l(sd)’ @g) configurations is given. Only the lowest bands are given. The quantum numbers between brackets are (u,nd).
K. Heyde et al. / y-boson excitations
steps of 0.1 e b. For the E4 operator, e. b2. In the latter
case, no experimental
and only an exploratory
calculation
below are done with a mixing
we use the boson hexadecapole parameter
charges
matrix
could be carried
strength
241
el;(d = eg) = 1.0
elements
out. All calculations
are known discussed
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;‘ 0.2UJ m
I 0.1-
0.1-
o.ok--e-%*
0.01
I
I
I
I
I
2
I
6
6
IO
I
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-
ls-
0 I61
\
\
\
\
3 \
\ \
2 \ I
\ \
\
\ \
\ I
Fig. 4. Same caption
I
I
I
t
\ I
IBA-I
0
’ IBAI
I
I
I
I
I
as 3, but for the yrast quadrupole moments. The label i on the different gives the effective boson charge e:’ = i(0. I ) e. b.
curves
(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
-RELATIVE
243
10
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.
only by comparing the calculated more phenomenological model. Therefore, 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 for coupling a g-boson to a SU(5) or O(6) core can be applied,
model calculations we should look for
K. Heyde et al.
244
i g-hosonexcitations
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)
[ =0 4: E, (MeV)
(cI+~)‘~’
4:
0.85
- 3.72
4;
1.45
[ = 0.4
0.2 (q +sp
E, (MeV) 0.79
(d+d)c4) - 3.70
[q +sp 1.50
1.37
E, (MeV) 0.64
(d+a)c4) - 3.70
[q +sp 2.43 1
1.21
1.67
-0.39
1.186
1.51
0.34
2.09
~ 0.41
- 3.409
1.87
- 0.008
2.20
2.09
0.65
5.266
2.06
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
0.084
0.266
2.12
-0.01
-0.013
4:3
3.05
2.92
0.0002
0.0003
2.97
-0.56
- 0.336
4L
3.10
2.92
- 0.096
3.12
-0.41
- 0.995
4:s
3.10
3.02
0.032
4:h
3.13
3.22
4:,
3.35
3.41
4:8
3.40
3.46
4:Cl
3.41
3.47
3.49
4;ll
3.45
3.57
3.52
4:
1.75
44’
2.00
4;
0.58 6.63
- 0.030
- 0.066
-0.853 0.053
2.18
0.0024
2.26
1.116
- 0.48 1 5.710
0.032
0.022
2.62 - 0.30 0.064
-0.151 0.104
2.63 2.68 2.70
-0.139 0.033
3.25
0.07
0.111
0.003
0.007
3.38 3.48 -0.019
- 0.008
3.48
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 transitions. 6: +4: (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 27,28) and the collective model studies of Gneus, Greiner and Hess model (AR) (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
explained
easily within
the standard
descriptions
245
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
E~ = 0.6 MeV K = -0.010
of lo4Ru [ref. “)I, and are given below:
F, = 2.0 MeV; 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 yband 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
U
EXI?
M
EXP
U
M
U
EXP
t-4
EXI?
Llo'-
I 5 0
I
\ \ \ \'
. 8-
I
7*\:,
I__)
“F,
3-
‘\ ‘-
\
.
6.5-;\
-
‘_,._ \
. ‘.j-:
‘-;
\
;:-__
-
f-
_-_
2'O+---
2*----,_
($= iOi. O-
-I
010+__-.--
Fig. 6. Comparison (U: unmixed) and
of the experimental lo4Ru spectrum ‘) with IBA-1 calculations without when coupling a g-boson to the same IBA-1 I(sd)N) core, interacting hamiltonian of eq. (2.6) (mixed: M).
g-bosons via the
K. Heyde et 01. / g-bo.m
246
IBA-2 -
EXP
excitations
EXP
IBA-2
IBA-2
EXP
. 12
l
-
-
IO
-
\
‘I-IlO’
10hR
9’
~1 \ -
8’
i* -. =A0:..
‘\
l
‘( -17* ‘\
l
-8
-‘\
\
-6+
6
l
-. -
1
--
-j-
7;:
-
L’
d .
2’-
2
0’
3’
-*
-
d
‘3’
7
\ -L’
‘-
I I
5’
--
1’5-1’
=IL’) 4’
Yf.
\ \-
---
U
---2+
o-o
l
t
1
1
__-2.
0
Fig.
7. Comparison
---
d
of the experimental
1
lo4Ru spectrum’) with parameters of ref. 5).
the IBA-2
calculations,
using
the
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, approach to describe which is typical of an asymmetric rotor 32). An alternative 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
K. Heyde
et al. / y-boson
excitations
247
0.5 -
i 0.L -
t
5
m
w; ;; 0.3-1
? ,” ,IBA-I
z m 0.2-
0 IBA-2
I
0.1 -
o.oIL 0
2
4 ___
6
8 I,-
IO
12
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 ec2) = e$) = 0 +O.l,
eit’ = 0.1 e. b, e$ = -0.05 e. b and the g-boson f 0.2 and kO.3 e. b. In each case, we also compare
charges 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 ‘).
K. Heyde et al. / g-boson excitations
248
O’r------
0.1.-
t “0 N 0
0.3 -
l-4 3
-1
t zo.2 .. N w
-
lBA0
7
1
1
0.1 -
IBA-2 2 3
4
6 __
Fig. 9. Same caption
a
10
‘Y--
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 0.010
I
I
I
249
I
IBA- 1 t No N 0 N
I
fjoo5
I-
-
.i.l m
1 \\ 1 \
\ \
I
‘1 \\ \\ \’
0.001
Fig. 10. Same caption
ly-
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, to the configurations
in a schematic approach, the influence of coupling a g-boson I(sd)N) on the description of energy spectra and E2 and E4
K. Heyde el al.
250 I
I
: g-hosonexcitations I
I
I
I
IBA-I
1
T
T
IBA-1 0 -1 -2
-3
IBA-2
O
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 labeled with a parameter i, indicating the effective boson
in the yrast band. The curves charge ez’ = i(O.l) e’ b.
are
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
IWONL
(Instituut
Nijverheid
Fonds
voor
Wetenschappelijk
voor
aanmoediging
van
en Landbouw)
for constant
support.
Onderzoek)
het Wetenschappelijk
and
G.W.
the
Onderzoek
in
References 1) 2) 3) 4) 5) 6) 7) 8)
A. Arima and F. Iachello, Ann. Rev. Nucl. Sci. 31 (1981) 75 Interacting bosom in nuclear physics, ed. F. Iachello (Plenum, New York, 1979) Interacting Bose-Fermi systems in nuclei, ed. F. Iachello (Plenum, New York, 1981) P. Federman and S. Pittel, Phys. Rev. C20 (1979) 820 P. van Isacker and G. Puddu, Nucl. Phys. A348 (1980) 125 M. Sambataro and G. Molnar, Nucl. Phys. A376 (1982) 201 J. Stachel, P. van Isacker and K. Heyde, Phys. Rev. C25 (1982) 650 J. Stachel, N. Kaffrell, E. Grosse, H. Emling, H. Folger, R. Kulessa and D. Schwalm, Nucl. Phys. A383 (I 982) 429 9) K. Schreckenbach, A. Mheemed. G. Barreau. T. von Egidy. H. R. Faust, H. G. Borner, R. Brissot, M. L. Stehs, K. Heyde, P. van Isacker, M. Waroquier and G. Wenes, Phys. Lett. 1lOB (1982) 364 IO) K. Heyde, P. van Isacker, M. Waroquier. G. Wenes and M. Sambataro, Phys. Rev. C25 (1982) 3160 I I) M. Sambataro, Nucl. Phys. A380 (1982) 365 12) G. Wenes, P. van Isacker. M. Waroquier. K. Heyde and J. van Maldeghem. Phys. Lett. 98B (1981) 398; Phys. Rev. C23 (1981) 2291 13) P. Duval and B. R. Barrett, Phys. Lett. 1OOB(1981) 223 14) P. Duval and B. R. Barrett, Nucl. Phys. A376 (1982) 213 15) J. Wood, in Frontier research in nuclear physics, ed. D. H. Feng el al. (Plenum, New York, 1981) p. 123 16) P. B. Goldhoorn, M. N. Harakeh, Y. Iwasaki. L. W. Put, F. Zwarts and P. van Isacker, Phys. Lett. 103B (1981) 291 17) L. J. B. Goldfarb, Phys. Lett. 104B (1981) 103 18) Hua-Chuan Wu, Phys. Lett. 1lOB (1982) 1 19) R. P. Ratna Raju, Phys. Rev. C23 (1981) 518 20) P. van Isacker, K. Heyde, M. Waroquier and G. Wenes. Phys. Lett. 104B (1981) 5; Nucl. Phys. AjSO (1982) 383 21) V. Paar in Heavy ion, high-spin states and nuclear structure, vol. 2 (IAEA, Vienna, 1975) p. 179 22) J. N. L. Akkermans and K. Allaart, Z. Phys. A304 (1982) 245, and to be published 23) J. N. L. Akkermans, Ph. D. thesis, Vrije Universiteit Amsterdam, 1981, unpublsihed 24) F. Iachello, in Dronten nuclear structure summer school, ed. C. Abrahams, K. Allaart and A. E. L. Dieperink (Plenum, New York, 1981) p. 53 25) 0. Scholten, Ph. D. thesis, University of Groningen, 1980, unpublished 26) T. Otsuka, A. Arima and F. Iachello, Nucl. Phys. A309 (1978) I 27) A. S. Davydov and G. F. Fillippov, Nucl. Phys. 8 (1958) 237 28) H. Toki and A. Faessler. Z. Phys. A276 (1976) 35 29) G. Gneuss and W. Greiner, Nucl. Phys. A171 (1971) 449 30) P.O. Hess, M. Seiwert, J. Maruhn and W. Greiner, Z. Phys. A296 (1980) 147 31) P. van Isacker and Jin-Quan Chen, Phys. Rev. C24 (1981) 684 32) K. Heyde, P. van Isacker, M. Waroquier and G. Wenes, to be published 33) A. E. L. Dieperink and R. Bijker, Phys. Lett. 116B (1982) 77
Physics
A398 (1983) 235252
North-Holland
Publishing
g-BOSON
Company
EXCITATIONS
K. HEYDE,
IN THE INTERACTING
P. VAN ISACKER’,
M. WAROQUIER,
BOSON
G. WENES
MODEL
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 (Revised
22 September 1982 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 whereas, by explicitly taking into account both proton and neutron freedom, the more microscopically founded IBA-2 approach results ‘). Recently,
in many
transitional
nuclei
near
closed
is obtained degrees of
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. 235
Postal,
20-364,
01000 Mexico,
DF.
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. 13q14), we will concentrate
on a detailed
study
framework
of the
Recently,
particular
problem
IBM. using
of the hexadecapole
the
some
studies
degree have
group-theoretical
been
reduction
of freedom devoted of
the
in the on
this
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+ where H,, describes expansion notation
the standard
.,4++i”,.(Cl-),
IBA-1 hamiltonian
(2.1) ‘). Here, we use the multipole
H,, = &,n,+~Q.Q+dL.L+K”P+.P+q3T3.T3+q4T4.T4, using
the
multipole
operators
as
defined
in
the
Dronten
(2.2) lectures
notes
of
K. Heyde ef al. j g-boson excitations
L Fig. 1. Schematic
237
SN
representation of the I(sd)N) and the I(sd)N-’ @g) configurations (anharmonic quadrupole vibrational spectra).
in the SU(5) limit
Iachello 24), i.e. Q = (d+s+s+d)‘2’-~(d+d)(2), L = --JG(,+d)(‘), P s
$(d. d) -$s.
T3 = (cI+~)‘~‘,
s), 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 one obtains
the general
(sd)-g boson
interaction
(2.2)
hamiltonian
Hi”, (CT) = ci@+a+d+B)(2)+5@+~)(2)}.
Q@).
(2.3)
K. Heyle et al.
238
After recoupling
(2.3) one finally
: CJ-hoson excitations
obtains: (2.4)
with H,i,(CT)
= @‘s+)‘~‘.
(~~)‘4’ + h.C.
+(cJ+~+)‘~‘.
(&)“‘+h.c.
(2.5)
(~+d+)‘L’(d”a)‘L’+h.c.+....
+Xx5 L
The first term in eq. (2.4)
= (,y+ s + )‘4’ . (&I)‘“‘.
here, SU(5) and O(6) nuclei,
In the nuclei we study its main structure
IsN), IsN- ‘d), IsN-‘d2), and the g-boson
coupled
(2.6)
the ground-state
. . .,
band
has as
(2.7)
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 lsNm2dg) 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’, T(E4) = e~(d+d)‘4’+e~‘@+s+s+9)‘4’.
(2.9) (2.10)
K. Heyde et al. / g-boson excitations
At this moment
the extra
effective
boson
charges
e$,
239
ez’, e$
parameters. Therefore, we have carried out most calculations E2 and E4 decay rates in a phenomenological way. 2.2.1. Perturbation can calculate,
theory.
Starting
from the mixing
in the SU(5) limit, the wave function 14:)
and
ez’ are new
of energy spectra
hamiltonian
and
of eq. (2.6), we
for the lowest J” = 4+ level as
= IsN-2d2;4+)-~~Ij~~sN-1g;4+),
(2.11)
with AE = 2(q, -E,)-Q the energy denominator which for the 2 _Y50 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 which for a general (J-2
yrast E2 transition,
= 2n,-2llT(E2)11J
1-
1Js1,
$$ i ldEl
can be generalized
= 2n,)
= &Wr,+
9
(2.12)
to the expression
l)n,(4n,+
1) (2.13)
For the hexadecapole
and the general
(J-4
transitions,
one obtains
(0W(E4)114:)
= -e$i
yrast matrix
element
~Jziqcij, IAEl
(2.14)
equals
= 2nd-4llT(E4)llJ = %‘I,) =
-eg)i&
‘$(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 fcontributions to the yrast B(E2) values will result into positive interference if the product (e$)/ei:))c becomes a negative quantity. Taking eit) as a positive effective charge (N 0.1 e. b in the schematic calculations, this is also a typical value obtained from IBA calculations in this particular mass region 5), and having [ > 0, interference will result in eq. implies that for negative values of efdi constructive (2.13) and for positive values of egd (*), destruction interference. Already from the lowest order in perturbation theory one notices that the yrast B(E2) values can become larger than the standard IBA-1 and IBA-2 results. Hexadecapole strength obtained via particle inelastic scattering will determine further information about the effective charge es’, using eqs. (2.14) and (2.15). The quadrupole moment for the yrast band members however will only deviate
K. Heyde et al. / g-boson excitations
240
from their
C-values
by a number
proportional
to [‘ebb’. Therefore,
positive effective boson charges egg (2) in order to make the quadrupole negative compared with their C-values.
one will need moments
less
2.1.2. Schematic calculations. Starting from the knowledge gained from the analysis of subsect. 2.1.1, we carry out more detailed but still schematic calculations, coupling a g-boson to an exact SU(5) and O(6) core. Thereby, we use the following (i)
SU(5):
(ii)
O(6):
parameters
cd = 0.4 MeV;
c0 = 0.17 MeV; K” =
These parameters are IBA-2 parameters as as a standard set of For the unperturbed which is near to 24, in the 2 = 50 (A z eL<) 5 = 0 . 1 e.b,
for the hamiltonian
0.10 MeV;
of eq. (2.1): c2 = 0.10 MeV; c4 = 0.07 MeV; K’
.
IO -
0.02 MeV ; q3 = 0.15 MeV.
obtained via a projection method 25) starting from the lo4Ru determined by Van Isacker and Puddu 5). They can be used parameters for the schematic SU(5) and O(6) calculations. g-boson energy, we take a typical value of &g= 2 MeV, the energy needed to create a two-quasiparticle excitation 100) mass region. The effective boson charges used are
e$’ = -0.1
e.b
and we vary ebi’ and e$) from 0 to +0.3
1 v=6,01
( v-8,0 I
. 1210-
=
.
q
2’
_
7’
-
6’
e. b in
1v=8,Obg *
a’6’_2
-
L’
-
12
-
IO
-
9
.
. a7’_-
6’
5’
6’_
l.’ -
i
2’
-
i
-6
.
.
Fig. 2. Pure W(5) limit for the specilic case of N = 8 (s,d) bosons. Also the lowest band for the l(sd)’ @g) configurations is given. Only the lowest bands are given. The quantum numbers between brackets are (u,nd).
K. Heyde et al. / y-boson excitations
steps of 0.1 e b. For the E4 operator, e. b2. In the latter
case, no experimental
and only an exploratory
calculation
below are done with a mixing
we use the boson hexadecapole parameter
charges
matrix
could be carried
strength
241
el;(d = eg) = 1.0
elements
out. All calculations
are known discussed
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;‘ 0.2UJ m
I 0.1-
0.1-
o.ok--e-%*
0.01
I
I
I
I
I
2
I
6
6
IO
I
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-
ls-
0 I61
\
\
\
\
3 \
\ \
2 \ I
\ \
\
\ \
\ I
Fig. 4. Same caption
I
I
I
t
\ I
IBA-I
0
’ IBAI
I
I
I
I
I
as 3, but for the yrast quadrupole moments. The label i on the different gives the effective boson charge e:’ = i(0. I ) e. b.
curves
(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
-RELATIVE
243
10
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.
only by comparing the calculated more phenomenological model. Therefore, 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 for coupling a g-boson to a SU(5) or O(6) core can be applied,
model calculations we should look for
K. Heyde et al.
244
i g-hosonexcitations
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)
[ =0 4: E, (MeV)
(cI+~)‘~’
4:
0.85
- 3.72
4;
1.45
[ = 0.4
0.2 (q +sp
E, (MeV) 0.79
(d+d)c4) - 3.70
[q +sp 1.50
1.37
E, (MeV) 0.64
(d+a)c4) - 3.70
[q +sp 2.43 1
1.21
1.67
-0.39
1.186
1.51
0.34
2.09
~ 0.41
- 3.409
1.87
- 0.008
2.20
2.09
0.65
5.266
2.06
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
0.084
0.266
2.12
-0.01
-0.013
4:3
3.05
2.92
0.0002
0.0003
2.97
-0.56
- 0.336
4L
3.10
2.92
- 0.096
3.12
-0.41
- 0.995
4:s
3.10
3.02
0.032
4:h
3.13
3.22
4:,
3.35
3.41
4:8
3.40
3.46
4:Cl
3.41
3.47
3.49
4;ll
3.45
3.57
3.52
4:
1.75
44’
2.00
4;
0.58 6.63
- 0.030
- 0.066
-0.853 0.053
2.18
0.0024
2.26
1.116
- 0.48 1 5.710
0.032
0.022
2.62 - 0.30 0.064
-0.151 0.104
2.63 2.68 2.70
-0.139 0.033
3.25
0.07
0.111
0.003
0.007
3.38 3.48 -0.019
- 0.008
3.48
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 transitions. 6: +4: (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 27,28) and the collective model studies of Gneus, Greiner and Hess model (AR) (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
explained
easily within
the standard
descriptions
245
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
E~ = 0.6 MeV K = -0.010
of lo4Ru [ref. “)I, and are given below:
F, = 2.0 MeV; 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 yband 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
U
EXI?
M
EXP
U
M
U
EXP
t-4
EXI?
Llo'-
I 5 0
I
\ \ \ \'
. 8-
I
7*\:,
I__)
“F,
3-
‘\ ‘-
\
.
6.5-;\
-
‘_,._ \
. ‘.j-:
‘-;
\
;:-__
-
f-
_-_
2'O+---
2*----,_
($= iOi. O-
-I
010+__-.--
Fig. 6. Comparison (U: unmixed) and
of the experimental lo4Ru spectrum ‘) with IBA-1 calculations without when coupling a g-boson to the same IBA-1 I(sd)N) core, interacting hamiltonian of eq. (2.6) (mixed: M).
g-bosons via the
K. Heyde et 01. / g-bo.m
246
IBA-2 -
EXP
excitations
EXP
IBA-2
IBA-2
EXP
. 12
l
-
-
IO
-
\
‘I-IlO’
10hR
9’
~1 \ -
8’
i* -. =A0:..
‘\
l
‘( -17* ‘\
l
-8
-‘\
\
-6+
6
l
-. -
1
--
-j-
7;:
-
L’
d .
2’-
2
0’
3’
-*
-
d
‘3’
7
\ -L’
‘-
I I
5’
--
1’5-1’
=IL’) 4’
Yf.
\ \-
---
U
---2+
o-o
l
t
1
1
__-2.
0
Fig.
7. Comparison
---
d
of the experimental
1
lo4Ru spectrum’) with parameters of ref. 5).
the IBA-2
calculations,
using
the
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, approach to describe which is typical of an asymmetric rotor 32). An alternative 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
K. Heyde
et al. / y-boson
excitations
247
0.5 -
i 0.L -
t
5
m
w; ;; 0.3-1
? ,” ,IBA-I
z m 0.2-
0 IBA-2
I
0.1 -
o.oIL 0
2
4 ___
6
8 I,-
IO
12
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 ec2) = e$) = 0 +O.l,
eit’ = 0.1 e. b, e$ = -0.05 e. b and the g-boson f 0.2 and kO.3 e. b. In each case, we also compare
charges with the
IgA-1 ([ = 0; aid IBA-2 [ref. “)I calculations. Here, one notices that for a value of e!$r = -0.2 e. b, both the order of magnitude of the yrast B(E2) values and the shift of the B(E2) cut-off to much larger angular momenta are observed (see fig. 8). The same features are also observed and qualitatively reproduced for the y-band B(E2) values (see fig. 9). For the I, + I, - 2 B(E2) values, which normally are very small, the general trend again is well described for a value of es) = -0.2 e. b (fig. 10). Concerning the static quadrupole moments in the yrast band (fig. ll), even for the larger values of eBB (2), the experimental data of ref. 8, are approached. These results are also compatible with triaxial shapes, since in the framework of the asymmetric rotor 27, 28) the values and the systematics in the yrast band are well described ‘).
K. Heyde et al. / g-boson excitations
248
O’r------
0.1.-
t “0 N 0
0.3 -
l-4 3
-1
t zo.2 .. N w
-
lBA0
7
1
1
0.1 -
IBA-2 2 3
4
6 __
Fig. 9. Same caption
a
10
‘Y--
as 8, but for the y-band.
Finally we have compared some measured branching ratios with the SU(5) @ g, O(6) 0 g, the IBA-1 @ g boson calculations described above and also with the IBA-2 calculations of ref. 5). Here again, one observes an overall good agreement for the IBA-1 @ g-boson calculations, using the effective charge e’$) = -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 0.010
I
I
I
249
I
IBA- 1 t No N 0 N
I
fjoo5
I-
-
.i.l m
1 \\ 1 \
\ \
I
‘1 \\ \\ \’
0.001
Fig. 10. Same caption
ly-
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, to the configurations
in a schematic approach, the influence of coupling a g-boson I(sd)N) on the description of energy spectra and E2 and E4
K. Heyde el al.
250 I
I
: g-hosonexcitations I
I
I
I
IBA-I
1
T
T
IBA-1 0 -1 -2
-3
IBA-2
O
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 labeled with a parameter i, indicating the effective boson
in the yrast band. The curves charge ez’ = i(O.l) e’ b.
are
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
IWONL
(Instituut
Nijverheid
Fonds
voor
Wetenschappelijk
voor
aanmoediging
van
en Landbouw)
for constant
support.
Onderzoek)
het Wetenschappelijk
and
G.W.
the
Onderzoek
in
References 1) 2) 3) 4) 5) 6) 7) 8)
A. Arima and F. Iachello, Ann. Rev. Nucl. Sci. 31 (1981) 75 Interacting bosom in nuclear physics, ed. F. Iachello (Plenum, New York, 1979) Interacting Bose-Fermi systems in nuclei, ed. F. Iachello (Plenum, New York, 1981) P. Federman and S. Pittel, Phys. Rev. C20 (1979) 820 P. van Isacker and G. Puddu, Nucl. Phys. A348 (1980) 125 M. Sambataro and G. Molnar, Nucl. Phys. A376 (1982) 201 J. Stachel, P. van Isacker and K. Heyde, Phys. Rev. C25 (1982) 650 J. Stachel, N. Kaffrell, E. Grosse, H. Emling, H. Folger, R. Kulessa and D. Schwalm, Nucl. Phys. A383 (I 982) 429 9) K. Schreckenbach, A. Mheemed. G. Barreau. T. von Egidy. H. R. Faust, H. G. Borner, R. Brissot, M. L. Stehs, K. Heyde, P. van Isacker, M. Waroquier and G. Wenes, Phys. Lett. 1lOB (1982) 364 IO) K. Heyde, P. van Isacker, M. Waroquier. G. Wenes and M. Sambataro, Phys. Rev. C25 (1982) 3160 I I) M. Sambataro, Nucl. Phys. A380 (1982) 365 12) G. Wenes, P. van Isacker. M. Waroquier. K. Heyde and J. van Maldeghem. Phys. Lett. 98B (1981) 398; Phys. Rev. C23 (1981) 2291 13) P. Duval and B. R. Barrett, Phys. Lett. 1OOB(1981) 223 14) P. Duval and B. R. Barrett, Nucl. Phys. A376 (1982) 213 15) J. Wood, in Frontier research in nuclear physics, ed. D. H. Feng el al. (Plenum, New York, 1981) p. 123 16) P. B. Goldhoorn, M. N. Harakeh, Y. Iwasaki. L. W. Put, F. Zwarts and P. van Isacker, Phys. Lett. 103B (1981) 291 17) L. J. B. Goldfarb, Phys. Lett. 104B (1981) 103 18) Hua-Chuan Wu, Phys. Lett. 1lOB (1982) 1 19) R. P. Ratna Raju, Phys. Rev. C23 (1981) 518 20) P. van Isacker, K. Heyde, M. Waroquier and G. Wenes. Phys. Lett. 104B (1981) 5; Nucl. Phys. AjSO (1982) 383 21) V. Paar in Heavy ion, high-spin states and nuclear structure, vol. 2 (IAEA, Vienna, 1975) p. 179 22) J. N. L. Akkermans and K. Allaart, Z. Phys. A304 (1982) 245, and to be published 23) J. N. L. Akkermans, Ph. D. thesis, Vrije Universiteit Amsterdam, 1981, unpublsihed 24) F. Iachello, in Dronten nuclear structure summer school, ed. C. Abrahams, K. Allaart and A. E. L. Dieperink (Plenum, New York, 1981) p. 53 25) 0. Scholten, Ph. D. thesis, University of Groningen, 1980, unpublished 26) T. Otsuka, A. Arima and F. Iachello, Nucl. Phys. A309 (1978) I 27) A. S. Davydov and G. F. Fillippov, Nucl. Phys. 8 (1958) 237 28) H. Toki and A. Faessler. Z. Phys. A276 (1976) 35 29) G. Gneuss and W. Greiner, Nucl. Phys. A171 (1971) 449 30) P.O. Hess, M. Seiwert, J. Maruhn and W. Greiner, Z. Phys. A296 (1980) 147 31) P. van Isacker and Jin-Quan Chen, Phys. Rev. C24 (1981) 684 32) K. Heyde, P. van Isacker, M. Waroquier and G. Wenes, to be published 33) A. E. L. Dieperink and R. Bijker, Phys. Lett. 116B (1982) 77