-
CIIINFSE .JOUItNAL O F PflYSlCS
APRIL 1997
VOL. 35, NO. 2
Study of N = 90 Isotones in the
Interacting-Boson-Plus-One-Fermion-Pair
Model
L. M. Chen
Department of Physzcs, National Sun Yat-sen University,
Kaohsiung, Taiwan 804, R. 0. C.
(Received October 9, 1996)
The low-spin states and intermediately high spin states below the second backbendings of N = 90 even-mass isotones were studied in the interacting-boson-plusfermion-pair model. Energy spectra, effective moments of inertia and B(E2) value are
calculated. It was found that the energy spectra and the backbending behavior can be
reproduced quite well.
PACS. 21.60.-n - Nuclear-structure models and methods.
PACS. 21.60.E~ - Collective models.
I. Introduction
In recent years quite abundant high-spin states were discovered for the nuclei in
the transitional region. It is especially interesting to see that the nuclear spectroscopy of
N = 90 isotones, 154Gd, r5’jDy, 158Er and rsoYb shows very similar features [l-4]. The
low-lying energy levels have collective’ rotational spectra. The rotaitonal behavior is, of
course, more prominent for well deformed nuclei. As one goes to higher angular momentum
states 1 M 14, the reduction of collectivity is revealed by the lower values of quadrupole
moment and show “b ackbending ” if the conventional 2 J/hi2 vs (LJ)~ curves are plotted.
It is generally believed that this phenomenon can be explained in the collective core-plusquasiparticle alignment picture [5-91. The backbendings are considered as the band crossing
of the ground and quasiparticle bands.
The IBA (interacting boson approximation) plus-fermion-pair model is very suitable
for the calculations covering a string of isotopes or isotones [lo-131, since it is believed that
the nuclear shapes evolve at the nuclear-mass change.
In this work, we study the positive parity states of four N = 90 isotones ‘s*Gd,
156Dy, ls8Er and lsoYb in the IBA-I-plus-one-fermion-pair model. In the calculation only
one single-particle orbit i1si2 is considered since the Coriolis decoupling effect is the most
prominent for this orbit [14]. Taking 2 = N = 82 as the core, pure IBA-I assumes a valence
boson number Ng = 11,12,11,10 for the four nucleides, respectively.
II. The model
The model space contains two kinds of basis states:
156
..,
@ 1997 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
i.
L. M. Chen
VOL. 35
157
Where n, and nd mean the numbers of s and d bosons, respectively. Hence, ns + nd =
NB,nk + n& = NE - 1. Where NB means the total number of bosons. The U, Y(v’,Y’) are
the additional quantum numbers which are needed to specify the boson states. The symbol
j = 13/2 means fermion orbit angular momentum; J means the total angular momentum
for the one-fermion pair and assumes the values 4,6,. ..2j--1. TheJ=Oand2statesare
removed to avoid double counting.
The model Hamiltonian consists of three parts:
where
HE
z&dd+
.d+ UlP +.
HF =~?(2j + 1)‘j2[af
P + a2L. L + asQ. Q
x a$‘) + ; cVJ(2J + l)““[($
HBF = Q&B . I (CL’ X Zij)(‘) f /JQB[(cz~~x CX’)(~)
and
J5
3
X
2 - df X
x a:)cJ) x (5, x ?#)](‘I,
(6j X
Cj)(4)](2)
QB = [d+ x S f sf x i- T(d+ x i)]t21
Here Hg is the Hamiltonian of IBA-I in the multiple expansion form. HF is the fermion
Hamiltonian which includes the single-fermion energy &j and the twobody interaction parts.
HBF is the boson-fermion interaction Hamiltonian which is of quadrupole-quadrupole interaction form with Q and 0 as coupling strength parameters. The VJ’s, which are the
fermion-fermion interaction strengths, are calculated from the Yukawa potential with the
Rosenfeld mixture. The oscillator constant 2, is chosen equal to 0.96 A-Ii3 with A = 160.
The overall normalization of VJ’s is determined by requiring
The whole Hamiltonian is then diagonalized in the whole model space, and the
interaction parameters are determined by least-squares fittings to the energy spectra. Since
the interaction parameters are model space dependent, we actually should start with a
different set of interaction parameter for zero and one fermion-pair model space. However,
it is known that the IBA-I works very well for low-lying states, and for low-lying staes
the contribution from fermion alignment should be small. But for high energy states, the
major effect is come from boson-fermion interaction. Therefore, we use the low-lying states
to determine &d,ur, ~22, us, and adopt this set of value of &d, al, ~2,~s for all two kinds of
model space. Physically this means we adopt a unified core for the nucleus in the low energy
region and high energy region.
In the later part of fitting procedures the parameters Ed, al, ~22, us are kept essentially
at constant values except some fine tunings which can improve the overall quality of fitting.
It is expected that the interaction parameters will very smoothy versus mass numbers in
general.
VOL.35
STIJDYOF N = 90 ISOTONES IN THE ."
158
TABLE I. Adopted interaction parameters and the overall root-mean-square deviations i n
MeV.
Nuclides
&d
a1
a2
a3
CY
P
El
ls4GD
lssDy
1ssEr
rsoYb
0.544
0.453
0.564
0.846
0.032
0.010
0.013
0.074
0.0023
0.0023
0.0023
0.0023
-0.0065
-0.0065
-0.0065
-0.0056
0.136
0.177
0.174
0.068
0.006
0.006
0.006
0.006
1.455
1.580
1.570
1.537
RMSD
0.080
0.081
0.072
0.053
Energy Spectra of
Energy Spectra of
‘-
‘* Gd
Dy
7.0
6.0
5.0
40
-
-48. -
_,r
_-i“
- 1.5.
>
J
-
---I,’
-
- I,’
-
--It*
$
P
-In.
I”
20
00
-
I,
--II.--
--10*1
-
-r _ _,.
-
_o.
-
-4’
_*’
-
-
-_6’
_*,
5.0
0’
-
-
(.V
-4’ 2’ -0’
zz
-
3.0
7,-M
M.-d
-
-r -
-
=
’
M
2: =
=I.--f,*
4.
=;;
=;:
-I’
1.0
mea
m.0
E.O
m.0
iho
-0.
FIG. 1. Calculated and experimental energy
levels for 154Gd. The experimental
data are adopted from Ref. [l].
FIG 2. Calculated and experimental energy
levels for lssDy. The experimental
data are adopted from Ref. [2].
III. R e s u l t s
The adopted interaction parameters are shown in Table I. In general, the parameters
vary smoothly versus the boson numbers. It was found that the L.L term and the parameter
,B can be unified for all isotopes. The value of p is small. Therefore, the mixings between the
NE pure boson configurations and (NB - l)-boson-plus-one-fermion-pair configurations are
small in general. The weak mixings between NB and (LYE - l)-boson-plus-one-fermion-pair
configurations seem to be consistent with the general analysis in Ref. [5]. The d-boson
L. M. Chcn
VOL. 35
159
Energy Spectra of
“‘Er
7.0
I
Energy Spectra of
I10 Yb
x.4
--IF
100
2.’
*a.%.
- 21
5.0
80
--m
-
_.
-21’
-
-
-
-3
-
--12’
-
-20.
-
-
-I8
-
-
--16’
_--m
---I,.
_-Is.
-
i
6.0
-lb
26
6.
3.0
- -,I.
--
5
I
I.’
.,-
-
II‘
---II’
- #,U
---Iv
-1.0
IX
- I’
- ,- -
I.
_
‘
_-
1.-
-
1’
- ;:
_
-r
- --:I
--
-44.:
20
2’
--c
0.0
---o
n”
_
:
6.
- - 2.
I.,
40
:
-,6’
_ ].
FE-z.
_
0’
l.+IO’
-,8’
c.,
rh..
E.,
n-
E.,
Tk.
CA*
ib
I.,
n..
FIG. 3. Calculated and experimental energ) i
levels for 15”Er. The experimental
data are adopted from Ref. [3].
FIG. 4. Calculated and experimental energy
levels for “‘Yb. The experimental
data are adopted from Ref. [4].
single particle energy Ed increaes as the boson number is decreased. This is because we
go to a less deformed region as the mass number is increased. As is well know in IBA
calculations the value of &d increases when one goes to a less deformed region. The value
of the mixing parameter Q is closely related to the values of ~j. Except 154Gd, for much
higher values of Ej higher values of a are needed in general to fit the data.
The calculated and experimental energy spectra for 154Gd, rs6Dy, 15’Er, and 160Yb
are shown in Figs. (l)-(4). In general, the fits are quite well. All of the states are
considered up to I” = 2N$, going to higher states I” > 2fVg, the fits are not good. This is
probably because only one fermion orbital is introduced in the model and that these higher
spin states are outside the model space of this work. The fits for the p and y bands are not
as good as those for the ground band. These are quite accepted because the p and y bands
are not taken into accound in the weaks coupling IBA type calcutaionls [13-171 due to the
truncation of the model space. The backbending properties of the moments of inertia are
usually displayed in the conventional 2,7/h2 vs (~LJ)~ plot. In Figs. (S)-(8) the backbending
plots for 154Gd, 156Dy, “‘Er, and lsoYb are shown. In general, the calculated curves show
backbends similar to the experimental data.
The model wave functions can be further tested against the experimental transition quadrupole moment Qu. In the IBA-plus-fermion-pair model the electric quadrupole
operator is given by
T(‘) = eBQ~ + oeF(a:
X zLj)c2) + /!I?eB[(a: X a:)(“)d-
d+ X (61 X ti,)(4)](2)
L
STUDYOF N =90 ISOTONES IN THE ...
160
r
Backbending
lYGd
VOL. 35
Backbending
‘Tly
M Exp.
D - 0 THeo.
FIG. 5. Calculated (dotted currve) and ex-
FIG. 6. Calculated (dotted currve) and ex-
perimental (solid curve) backbending
plot for ls4Gd.
perimental (solid curve) backbending
plot for ls6Dy.
The values of (Y and /3 are adopted from those in the Hamiltonian obtained in the
energy level fittings. The effective charges eB and eF are adopted to be 0.15 and 0.37 eb,
respectively. Those values are similar to those used in previous similar calculations [lo-131.
It was found the dominant contribution to B(E2) values comes from the boson part. The
comparisons between the calculated and experimental values are shown in Figs. (9)-(12).
For ls8Er the suddenly reduction of B(E2) values at I = 14 can be reproduced quite well.
The model fails to reproduce the reduction of B(E2) values at I = 10 and increase at 1 = 12
in lsoYb. This is because the B(E2) val ues depend on the wave functions quite sensitively
and the calculated wave functions of I = 12 and I = 10 states have too small overlaps, such
that the calculated value shows a very steep drop at I = 12. The calculated B(E2) values
of rs4Gd agree quite well with the experimental data and do not have any significant drop
versus variation of I 5 10.
-_-
VOL. 35
L. M. Chen
161
Backbending
Backbending
‘%
“‘ET
,
150.0
0
~
1
100.0
2.
>
-A
:
z2
%
“s
a
r.
100.0
E
0-S Exp.
D----C Theo.
50.0
50.0
0.0
0
)
0.10
0.0
Cl.O( I
-
0.20
0.10
-.
0.20
(Ao)‘( Mev)*
FIG. 8. Calculated (dotted currve) and experimental (solid curve) backbending
plot for 16’Yb
FIG. 7. Calculated (dotted currve) and experimental (solid curve) backbending
plot for 15sEr.
E24ransition
l”Gd
E2-transition
In Dy
0
2.0
FIG. 9. Calculated and experimental B(E2)
values of the yrast band for 154Gd.
The experimental data are adopted
from Ref. [l].
FIG. 10
Calculated and experimental B( E2)
values of the yrast band for “‘Dy.
The experimental data are adopted
from Ref. [2].
STUDYOF N=WJISOTONESINTHE~.~
162
IV.
VOL.35
Discussions
The even-mass N = 90 A = 1 5 4 N 160 isotones were studied in the boson-plusfermion-pair model. The model space in this work is small and the complexity of calculation
can be reduced considerably. It was found the interaction parameters L L term and
boson-fermion coupling strength can be unified in our work. The high -spin states above
backbends are usually dominated by the fermion-pair configurations. The low-lying states
of the ground-state band and the quasi-p and quasi-y bands are dominated by pure boson
configurations. In general, the model can reproduce the moments of inertia near backbends
only qualitatively. The calculation on the effective moments of inertia and B(E2) values,
however, suggest that the model used is an over-simplified one.
The model can be extended in several directions. Such as the decoupling in the horbitals or consider the contributions of a g boson into the model. However, when the model
are extended, the model space becomes too large and we have also too many interaction
parameters to be determined. This is why we stick to the simplified model including the
decoupling up to two fermions in the ils,‘L orbital. In this calculation, the shape change vs.
the mass number is taken care by changing the boson interaction parameters. The shape
change within one nucleus vs. spin is taken care implicitly by choosing different values of
&j and the sets of core-fermion coupling parameters
EZ-transition
IYE,
FIG.
1. Calculated and experimental B(E2)
values of the yrast band for lsaEr.
The experimental data are adopted
from Ref. [3].
E2-transition
‘-%I
FIG. 12. Calculated and experimental B(E2)
values of the yrast band for 16’Yb,
The experimental data are adopted
from Ref. [4].
VOL. 35
L. M. Chen
163
Acknowledgments
The author is grateful to Prof. S. T. Hsieh and H. C. Chiang. This work was
supported by the National Science Council, Republic of China, under Grant No. NSC-852112-MllO-016.
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