-_
CHINESE JOURNAL OF PHYSICS
VOL. 32, NO. 6-I
DECEMBER 1994
Study of W Isotopes in the Interacting-Boson-Plus-Fermion-Pair Model
L. M. Chen
Department of Physics, National Sun Yat-Sen University,
Kaohsiung, Taiwan 804, R. 0. C.
(Received June 7, 1994)
Even-mass W isotopes are studied in the interacting boson-plus-fermion-pair-model.
The energy spectra of the ground-state bands and the side bands can be reproduced
quite well. The effective moments of inertia and B(E2) values can be reproduced
qualitatively.
PACS. 21.60.-n - Nuclear-structure models and methods.
.
I. INTRODUCTION
In recent years the phenomena of shape evolution and the moment of inertia anomaly
have created considerable interest in nuclear structure studies. The low-lying energy levels
usually show rotational behavior. The rotational behavior is, of course, more prominent
for well deformed nuclei. As one goes to higher angular momentum states, the collective
reduction is revealed by the lower values of the quadrupole moment. The moment of inertia
anomaly known as the first “backbending” usually shows up around I M 12 and the second
“b ackbending” around I x 28. It is generally believed that the complicated interplay
between the collective and single-particle degrees of freedom is responsible for the high-spin
anomalies. The single-particle degrees of freedom are induced by Coriolis decoupling of
the fermion pair in the high-j single-particle orbitals [l]. Therefore, high-spin anomalies
are generally analyzed in a core plus quasi-particle model [2-6] and attributed to particle
alignment and band crossings.
The interacting boson model has been applied extensively in order to correlate the
nuclear collective properties. The model was extended to include single-particle degrees of
freedom [7-121. This algebraic model has the advantage of including all kinds of collective
degrees of freedom on an equal footing. Since it is believed that the nuclear shapes evolve
at the nuclear-mass change, the IBA-plus-fermion-pair model is very suitable for calculations covering a string of isotopes. Also, it is well known that it is possible to calculate
809
@ 1994 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
810
STUDY OF W ISOTOPES IN THE INTERACTING-BOSON-PLUS-. . .
VOL. 32
the collective properties of nuclei with different deformations by a smooth variation of the
interaction parameters contained in the model [13-161. The light W isotopes were selected
since abundant high spin experiment data are available. Of particular interest in W nuclei
are the systematics of the first backbending attributed to the Coriolis decoupling effect of
the fermion pair in the ir3i2 orbit [17,18]. The high-spin states of several W isotopes were
analyzed theoretically and can be correlated qualitatively by using the cranked shell model
[19]. In this work, we apply the IBA-plus-fermion-pair model to perform an extensive calculation on the light even-mass 166’“176W isotopes. We want to make more quantitative
theory-experimental comparisons which cover the low-spin and high-spin regions simultaneously.
II. THE MODEL
The model space contains two kinds of basis states:
Here n,(n:), w(n&) are the numbers of s- and d-bosons, n, + nd = n: + n& + 1 = NB,
where NB is the total number of bosons. The j which denotes the fermion orbital angular
momentum can assume the value 13/2 (i,,,,), since it is believed to be the most important
one from an analysis of the Coriolis matrix elements [l]. J is the total angular momentum
for the one-fermion pair. The J = 0, 2 fermion-pair states are omitted to avoid double
counting [ll]. T he V, 7 (v’, 7’) stand for the additional quantum numbers which are needed
to specify a given boson state. The boson-fermion combined model space as described is
too massive. In order to make the calculation feasible we have to make some truncation of
the model space. Since for each value of L we need only calculate the first few low-energy
states, we couple the lowest energy states in both the fermion and boson subspaces to form
the total model space. The energies for yrast boson states increase quite rapidly as LB
increases. Also, the T = 1 two body matrix elements increase monotonically as J increases.
To be more specific, for a given value of L, we select all possible values of Lb up to L or 2n&
(if L > 2n&) to couple with the lowest two values of J that are compatible with the angular
momentum coupling rules. For example, for L = 12 the boson-fermion pair model includes
all pure boson states with L = 12, all one-fermion-pair states with Lb = 0, J = 12; Lb = 2,
J = 10, 12; L:, = 4, J = 8, lo,.. - Lh = 12, J = 6, 4. Since the excitation energies for the
boson state increase quite rapidly as LB goes to higher values, so the low-LB, high-J basis
states are, in fact, the most important ones. The truncated high-LB, high-J basis states
will usually produce very high excitation energy states and can be omitted. This justifies
the choice of the truncation scheme.
The model Hamiltonian adopted is of the standard form 191:
L.M.CHEN
VOL.32
811
H=HB+HF+HBF,
where
HB
HF
Edd+‘~+ulP+.P+u22.~+u3~.g,
= &j(2j + l)“*[af X iij](O)
=
+$ CVJ(2J + 1)‘12[(UJf
X .f)(J) X (Cj X iij)(J)](o) ,
.I
HBF
=
Q
=
B
GB . {a(aT X 6j)(2) + @[(a: X af)t4) X J- d+ X (&j X cj)(4)](2)},
d+xs+sxd-
(y2 p+
x
J)]
(2) .
[
.
Here HB is the Hamiltonian of IBA-1 in the multiple expansion form. HF is the fermion
Hamiltonian which includes the single-fermion energy and the fermion-fermion interaction
terms. HBF is the boson-fermion interaction Hamiltonian which is of the quadrupolequadrupole interaction form with a and p as coupling strength parameters. The VJ’s, which
are the fermion-fermion interaction strengths, are calculated from a Yukawa potential with
the Rosenfeld mixture. Harmonic-oscillator wave functions with the oscillation constant
v = 0.96A-‘/3fm-2 with A = 160 are assumed. The overall strength normalization is fixed
by requiring
(jjlVljj)~=2 - (jjlVljj)~=~ = 2 MeV.
The whole Hamiltonian is then diagonalized in the selected model space. The interaction parameters are determined by least-squares fitting to the energy spectra of the W
isotopes. However, it is known that IBA-1 works quite well for low-lying states, and for
low-lying staes the contribution from fermion alignment should be small. Therefore, we
use the low-lying states to determine Ed, al, uz, as. The values of CY, ,Ll and Ej are then
determined by least-squares fitting with the energies of the high-spin states. Physically this
means we adopt a similar core for the nucleus in the low energy region and allow different
core-fermion interactions for the fermion-pair model spaces in the high energy region. In
the later part of the fitting procedures the parameters &d, al, ~3, and ,0 are kept at essentially constant values, ~2, cy and Ej are varied smoothly versus mass numbers in general.
The effective moments of inertia are then calculated and the resulting wave functions are
used to calculate B(E2) values.
812
STUDY OF W ISOTOPES IN THE INTERACTING-BOSON-PLUS-.
VOL. 32
III. RESULTS
.
III-l. Interaction Parameters
The values (all in MeV) &d = 0.543, al = 0.040, as = -0.0082, and p = 0.050
are adopted for all 16&176W isotopes. The other adopted mass dependent interaction
parameters and the overall root-mean-square deviation for each nucleus are shown in Table
I. In general, the parameters vary smoothly with the change of the masses. Several places,
however, show quite a quick change of the parameters. A gap exists in the parameters of the
17*W and 174W isotopes. The parameter a2 shows an anomalous change from 17*W to 174W.
This probably indicates a change in the collective shape from 17*W to 174W. The singlefermion energy Ej of 176W is considerably larger than that of 174W. This indicates that the
Coriolis decoupling of the il3/2 orbit of 176W is much harder to excite as compared with
lighter-mass W isotopes. It was found that the coupling parameter (Y is closely correlated
to the values of&j and we need a higher value of a to get good fittings, if&j is higher. This
is consistent with the experimental observation that the fermion-pair alignment is weaker
for 17*W and the moment of inertia does not show any backbend [20].
111-2. Energy Levels
The calculated and experimental energy spectra of lssW, 17’W, 17*W and 174W
[17,18,21-251 are shown in Figs. l-4 for comparison. The energy spectra for other isotopes
in the isotope string are not shown since they are similar to the ones of adjacent mass
number. From the figures we find that the excitation energies of high-spin states as well
as those of low-spin states can be reproduced quite well. The states with spins I >_ 12 are
usually dominated by the boson-plus-fermion-pair configuration. Near the transition point
I = 12, states are usually mixtures of the two corresponding kinds of configurations.
TABLE I. Adopted interaction parameters and the overall root-mean-square deviations in
MeV
a2
a
sj
RMSD
lssW
lssW
17ow
172~
174w
176~
0.0066
0.058
1.407
0.024
0.0055
0.058
1.234
0.040
0.0043
0.058
1.183
0.046
0.0029
0.270
1.600
0.061
0.0036
0.289
1.685
0.040
0.0044
0.450
2.220
0.014
VOL. 32
L.M.CHEN
813
-(28’)
6.0
7.0
-24
6.0
"cv
-22
5.0
-26
>^
9
iiT
-Ia+
-16
-14
-12
4.0
3.0
-lb
20
(lo')
--c
1.0
d
0.0
4+
T
-O+
&P.
Theo.
FIG. 1. Calculated and experimental energy levels for IsaW.
11.c
(32'
9s
a.c
-8’
36
5
6.C
$
iii
5.0
-f
72.
36
4.0
-,r-1s
-16
-14 -14
-17
-1u
-c
F
d
r
&P.
T?lea.
FIG. 2. Calculated and experimental energy levels for 170W.
814
STUDY OF W ISOTOPES IN THE INTERACTING-BOSON-PLUS-. . .
6.0-
37’
5.0_
-6
4.0
-16.
'"W
-18
S
%
3
3.0
-14
-
2.0
1
2
-6+
1.0
-c
-e
T
0.0
”
4.
Theo.
FIG. 3. Calculated and experimental energy levels for 17?-W.
6.0
5.0
-*
4.0
1,‘
W
-10’
-16.
S
f
cc
3.0
-14
-
2.0
1
1
_10+
-e'
1.0
-6
0.0
Exp.
-5
-_uTheo.
FIG. 4. Calculated and experimental energy levels for 174W.
__.___._ _.
VOL.32
L.M.CHEN
VOL.32
815
111-3. Effective Moments of Inertia
The backbending properties of the moments of inertia are usually displayed in the
conventional 2J/ti2 (defined as (41 - 2)/(E1 - EI_2)) versus (tiw)’ (defined as (EI plot which is more sensitive to the moment
Ez-2)2/{1(1 + 1y - [(I - 2)(1- 1)]‘/2}2)
of inertia anomalies. In Figs. 5-8 the backbending plots of 1ss~170*172~176W are shown. In
general, the calculated results show backbending behaviours very similar to the experimental data. Comparing the figures we can see that the property of backbending tends to
become less prominent as the mass number increases.
111-4. Electromagnetic E2 Transition Rates
In the IBA-plus-fermion-pair model the electric quadrupole operator is given by [9]
Tf21 = PQB + oZ( UT
X tij)t2)
+/3eB[(af X (If)(*) X if- d+ X
.
(2ij X iij)(4)](2).
The values of a and /3 are adopted from those in the Hamiltonian obtained in the energy
level fittings. The values of the effective charges eB and eF are taken to be 0.15 and
0.37, respectively. Those values are similar to those used in previous similar calculations
[ll-13,15,16].
150.0
-7
100.0
>
&
“c
:
N
50.0
0.0
0.00
0.04
0.00
0.12
(hwfphNZ)
0.16
0.20
FIG. 5. Calculated (dotted curve) and experimental (solid curve) backbending plot for “‘W.
816
STUDY OF W ISOTOPES IN THE INTERACTING-BOSON-PLUS-. .
0.0 ’
0.00
0.04
0.08
0.12
0.16
0.20
VOL. 32
I
0.24
tFiw)'vJeV')
FIG. 6. Calculated (dotted curve) and experimental (solid curve) backbending plot for l”W.
0.0 OSJU
0.04
0.08
(llw)’ (MeV2 )
0.12
FIG. 7. Calculated (dotted curve) and experimental (solid curve) backbending plot for 172W.
L. M. CHEN
VOL. 32
817
100.0
i
>
%
z
:
CI
50.0
0.04
VW’ WV’ )
FIG. 8. Calculated (dotted curve) and experimental (solid curve) backbending,plot for 176W.
.
The experimental data for the B(E2) values of the W isotopes are quite meager. In
Figs. 9-11 the calculated B(E2) values of the yrast bands o f 168v’70y172W are compared with
the available experimental data. For rssW the reduction of B(E2) values after I = 12 c a n
p--__*__--4,
I
-.
,’
,’
A’
-.
‘*.
‘.
7
I
I
7
‘\
,h--,’
‘a’
,I
/
d
0.0’0.0
’
,
2.0
‘- W
I
B(E2.6*+4*)
t
4.0
_d
(
6.0
Em.
>56
c6.0
n
10.0
’
120
’
14.0
t
16.0
’
16.0
I
FIG. 9. Calculated and experimental B(E2) values of the yrast band for lsaW. The experimental
data are adopted from Ref. [22].
818
STUDY OF W ISOTOPES IN THE INTERACTING-BOSON-PLUS-.
VOL. 32
200.0
s
Gw,
m
100.0
o.oo;
2.0
4.0
I
6.0
6.0
FIG. 10. Calculated and experimental B(E2) values of the yrast band for 17’W. The experimental
data are adopted from Ref. [23].
600.0
500.0
\
\
lRW
\
s
400.0
s
m
w
m 300.0
200.0
100.0 I
0.0
2.0
4.0
6.0
6.0
10.0
12.0 14.0
16.0
16.0
20.0
I
FIG. 11. Calculated and experimental B(E2) values of the yrast band for 172W. The experimental
data are adopted from Ref. [24].
VOL.32
L.M.CHEN
819
be reproduced qualitatively. However, the calculated values decrease much more rapidly
than the experimental data. This indicates that the model used in this work is oversimplified
and the overlap between the states near the backbends is too small. The experimental
B(E2,6+ + 4+) (> 56) and B(E2,8+ + 6+) (> 93) values are not available, here the
calculated values are 173 and 170 W.u respectively. The calculated B(E2) values of 17’W
agree quite well with the experimental data. The calculated B(E2) values of 172W can be
reproduced qualitatively but are generally small compared with the experimental data.
IV. DISCUSSION
.
The even-mass W isotopes are studied in the boson-plus-fermion-pair model. The
high-spin anomalies can be attributed to band crossing of the fermion-pair band and the
ground-state band. The states above the first backbend (28 2 I 2 12) are usually dominated by the one-fermion-pair configurations. The low-lying states are dominated by pure
boson configurations. For neutron rich isotopes the corresponding i1si2 single-particle energies become quite high and thus reduce the probability of Coriolis decoupling. So the
change in the single-fermion energy suggests that the fermion-pair decoupling excitation
is much harder for W isotopes with mass number 2 176. The calculation of the effective
moment of inertia and B(E2) values, however, indicates that the model is an oversimplified
one. The real situation may be more complicated. Although the model in this calculation already includes most important configurations, other orbitals such as the h1,12-orbit
may also contribute to the particle alignments. They probably will improve the theoryexperiment agreements in B(E2) val ues and backbending plots. However, in introducing
more orbitals in the calculation, the model space becomes too large and we also have too
many interaction parameters to be determined. This is why we stick to the simplified model
including decoupling of up to two fermions in the i,si2 orbit. Another possible extension is
to include the shape change in a more direct manner. In this calculation, the shape change
vs. the mass number is taken care of by changing the boson i - ,f interaction parameter,
core-fermion coupling parameter cr and single fermion energy Ej.
ACKNOWLEDGMENTS
In particular, the author wishes to thank Prof. S. T. Hsieh and Prof. H. C. Chiang.
This work was supported by the National Science Council, Republic of China, under Grant
No. NSC-83-0208-MllO-016.
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STUDY OF W ISOTOPES IN THE INTERACTING-BOSON-PLUS-. . .
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