PHYSICAL REVIE%
Quadrupole
21,
VOLUME
C
moments and
NUMBER
MARCH
8
1.
980
E2 transition rates
in the Zr region with wave functions of the
broken pair approximation
Y. K. Gambhir, S. Haq, and J. K. Suri
Indian Institute
of Technology, Bombay-400 076, India
(Received 10 May 1979)
=
The E2 transition rates and the quadrupole moment of the first 2+ state for X 50 even A nuclei and for
even Zr isotopes are calculated using the structure wave functions of the broken pair approximation,
'Sr as an inert core. The broken pair approximation results are in excellent agreement with the
assuming
corresponding shell model results obtained by using identical input information. This fully demonstrates the
usefulness of the broken pair approximation as a valid, reliable, and practical approximation to the shell
model.
NUCLEAR STRUCTURE Shell model, broken pair approximation (BPA), generalized BPA, E2 transition &ates, quadrupole moment, M=50 evenA nuclei,
even Zr isotopes.
"
the brokenIn our previous communications,
pair approximation (BPA)' ' has been demonstrated
to be a valid and reliable approximation to the
shell model (SM) in the Zr region for identical as
well as nonidentical valence nucleons. This conclusion has been based on an excellent agreement
obtained between the BPA and SM energy spectra
for the nuclei in this region. It is known, however,
that the electromagnetic moments and transition
rates are sensitive to the detailed structure of the
wave functions, and, therefore, these quantities
provide a more faithful test of the reliability of
the BPA. With this motivation and for completeness, we here report the results of the BPA calculations for the quadrupole moments of the first
2' states [Q(2;)] and the B(E 2) values for various
transitions between the levels of these nuclei.
Here again, the BPA values for Q(2;) and B(E2)
transition rates are found to be very close to the
corresponding SM values and, at the same time,
reproduce equally well the corresponding experimental results. This observation allows for complete faith in BPA and helps in applying BPA with
confidence, especially to cases where SM calculations cannot be performed.
In BPA, the approximate ground state P, ) for
p pairs of identical particles is assumed to be
particle creation operator with total angular momentum J' and projection M. The state $0) reduces
to the number projected BCS state (corresponding
to 2P particles) if one replaces the BPA ground
state parameters v (u) by the corresponding BCS
values of occupation (nonoccupation) probabilities.
In the next approximation, the BPA basis states
used to diagonalize the Hamiltonian are constructed by replacing one S„ the pair distributed operator, in Eq. (I) by an arbitrary two particle creation operator A~. These states are
~
(ab))=A~
jab)tt,
~O).
The motivation and justification for choosing these
basis states are discussed in detail in Ref. 3.
For nonidentical nucleons, the BPA basis states
can be obtained through a straightforward extension' of Eq. (2). These can be written as
~
l.
..
e„(P,P,) e, „( . )]:&.
Here the subscripts p and n refer to protons and
neutrons, respectively.
The resulting BPA eigenvectors representing
the nuclear state P«) can be expressed in the
following form:
~
~
(plt2)
J
r JpJ (P1P2
ulu2)
~
i%I,(P1P
)2
(n, n2) Z„
x 8y~ (u, n, )]„).
u, "~
Qa ~
(n, /2)' 'At, (aa)
~O),
a
with v, '+u, '=
', and
I; 0, = j, + —
A~z„(ab)
is a two
The electromagnetic reduced transition probabilities and static moments are simply related to
the quantity
in which
~
21
(4)
1124
J&M &) and ~Z,.M,.)
are the pertinent nu-
1980 The American Physical Society
QUADRUPOLE
MOMENTS
E2 TRANSITION
AND
For N = 50 nuclei shell model (SM) calculations
Sr core have already been reported by
Gloeckner et al. ' In their calculation the valence
protons were confined to 2pzy2 and 1g, &, shell
model orbitals, and the input parameters, namely,
the single particle (sp) energies and the p-p ef-
clear states which, in BPA, are of the form given
Eq. (4). The single particle tensor operator
Q~ of rank A. corresponds to the specific electromagnetic multipole transition and its second
quantized version is
with
by
Q~=
Q Q~ Q
Q
goi
fective matrix elements (me), were determined
~
OO
where at(a) is the particle creation (annihilation)
operator.
For an E2 transition
2P'2
@@2 g
off y
and the operator for the quadrupole
(
16
moment is
)1/2gE2
Q~ being a one body operator, expression (5) will
require evaluation of the terms of the following
type only:
&P, „J~b) II fl'll
RATES IN THE. . .
g, ;,„&(«)&.
(9)
This expression (9) for the reduced matrix element
can easily be evaluated by expressing O~ in a form
in which the particle creation operator appears on
the left and the annihilation operator on the
right, so that the final expression reduces to an
overlap integral of the BPA basis states corresponding to p+1 (n+1} particles. The explicit expression for (9} is given in the appendix of Ref. 4.
Explicit numerical calculations have been carried out for N = 50 even A (identical valence nucleons) and for even Zr isotopes (nonidentical
valence nucleons) assuming "Sr as an inert core.
by directly fitting the observed spectra of N= 50
nuclei. We use, in our BPA calculation for this
case, the same input information to facilitate
direct comparison with the corresponding shell
model results. First, the BCS -gap and number
'
equations are solved to obtain the BPA ground
diastate parameters. The Hamiltonian is then
gonalized in the space spanned by the BPA basis
states (2). The resulting energy spectra for N=50 .
even A nuclei are remarkably close to the corresponding SM spectra as reported in a previous
communication. ' The BPA eigenvectors thus obtained are used to calculate the quadrupole moment of the first 2' state and E2 transition rates
for the various transitions. The results for transition rates are summarized in Table I. The effective charge for the proton (ef, f) has been chosen to
be 1.72e, which is the same as used in the SM
calculation by Gloeckner et al. 7 The oscillator
size parameter b =2.124 fm is used. This b value
reproduces the rms radius for the nuclei (A -9296) in this region. The calculated total mean lifetime includes the contribution due to internal conversion factor listed in the same table. It is amply clear from the table that the BPA and seniority
SM results are in excellent agreement, and both
reproduce, reasonably well, the corresponding ex-
TABLE I. E2 transition rates for %=50 even A. nuclei. Seniority SM results are taken from Gloeckner et aE. (Ref.
The references for corresponding experimental values, wherever available, are cited in Ref. 7.
S„(keV)
Nucleus
80Zr
Transition
2+
4+
0+
6'
4'
2
+
92Mo
6+
2+~ 0+
4+
2'
6+
4'
8+
6+
2+
0'
4+
2
6+~ 4+
8+ ~6+
0+
2+
4+
+
4+
2'~
8+~ 6+
Experimental
BPA
2182
2288
770
282
891
371
141.5
133
1509
773
330
1498
770
148
1428
755
133
1407
771
311
282
145
133
1395
282
770
282
133
Internal
conversion
Experimental
Mean lifetime (ns)
Seniority SM
(0.14 + 0.01) x 10
0.32 x 10+
~ ~ ~
0.32
0
0.017
0.29
~ ~
~ ~ ~
BPA
0.35 x 10~
0.89 x 10+
1.03
180
+ 9
(0.45 + 0.08) x 10+
2.22 ~ 0.07
275
+ 10
0
0
0.024
0.33
7).
107
(102
+ 10
+ 7) x 103
224
0.62 x 10+
2.29
242
0.64 x 10~
0.03
3.12
249
339
50
57
0.64 x 10
0.42
4.3 x 103
0.65 x 10
0.48
5.p x 103
0.77 x 10&
0.12
4p
2.2 x10'
Y. K. GAMBHIR, S. HA@,
1126
AND
"Pd
J.
K. SURI
21
Identical behavior is predicted
also. The corresponding experimental results are available only for "Zr and
"Mo, and these are in full conformity with the
calculated values.
For even A Zr isotopes, the valence protons'
have been confined to 2pz/2 and 1g, /, sp oribitals
as before, and valence neutrons have been restricted to 2d, /, and 3s]/2 sp states. The p-p
interaction me and b, q~ (the energy spacing between 2p», and lg, &, ) are the same as used for
N= 50 even A nuclei, while the phenomenological
set of the n-n and n pint-eraction me and &q„(the
sp energy spacing for neutrons) used here are taken directly from the earlier SM analysis. ' The
latter set of parameters has been determined' in
the shell model analysis by directly fitting the observed energy spectra of "'"Y, "Zr, and
"~'"Nb nuclei. The details of the fitting procedure are given in Ref. 8. The BCS gap and number
equations are solved for protons and neutrons separately using the above mentioned set of phenomenological interaction me and energy spacings.
This procedure is justified because, for the cases
considered here, the valence protons and neutrons
fill different major shells and therefore the n. -p
interaction is expected to be much weaker. An
orthonormal set of BPA basis states is then constructed from (8) to diagonalize the total Hamiltonian to yield the energy spectra and wave functions. The energy spectra calculated by this generalized broken pair approximation (GBPA) for
Zr isotopes is found to be very close' to that obtained in the SM analysis by Gloeckner. ' These
GBPA (nonidentical particles case) nuclear wave
functions are then used to calculate the quadrupole moment of the first 2' state and the B(E2)
values for 0
transitions for Zr isotopes. The
effective charge for proton (e~«) is chosen to be
1.765e which is identical to that used by Gloeckner. ' In the SM analysis, ' the effective charge for
neutron (e",«) has been determined by fitting the
experimental B(E2) values for 0'- 2' transitions
separately for each isotope. We have chosen e",«
= 1.2e which is approximately the average of the
(8 protons).
by the seniority SM
t
Experimental:
io-
&
SenipritySM:
BPA
Ol
Vl
PV
I
CD
X
E
05-
0C
X
01I
I
I
Zr
I
94@
Mp
FIG. 1. The mean lifetime of the
N = 50 even A nuclei.
96pd
first 2' state for
"
perimental results. This demonstrates that the
BPA is equally capable of predicting the dynamic
nuclear properties which are sensitive to the detailed structure of the wave functions.
To test the effects of configuration mixing, the
BPA calculations have been repeated in the pure
proton g, &, space. The results thus obtained are
found to be significantly different from those in
the full g9/2 py/2 space. For example, the mean
lifetimes for the case of 2'-0' transition in the
pure (vg, &, )" basis are 0.12 x 10 ' ("Zr), 0.49
x 10 ' ("Mo), 0. 65 x 10 ' ('4Ru), and 0.11 x 10 '
("Pd) in contrast to the corresponding results
0.35x10', 0.64x10', 0.65x10',
and O. VVx10'
respectively (Table I) in full space. These differences arise mainly because of the different
amplitudes of the relevant components of the wave
functions and because of the dressing of (p, &, )'
pairs. This indicates, as expected, that E2 transition rates are sensitive to configuration mixing.
The variation of the mean lifetime for the 2'-0'
transition with the addition of protons is plotted in
Fig. 1. The figure reveals that the calculated
(BPA) mean lifetime is minimum for "Zr (2 protons) and is almost the same for "Mo and '4Ru
(4 and 6 protons), while it is slightly larger for
-2'
TABLE Q. B(E2) values for even Zr isotopes. The corresponding SM results have been calculated using the values
of the respective neutron and proton reduced matrix elements reported by Gloeckner {Ref. 8) (with e",ff = 1.2e, eef f
= 1.765e). Experimental values for 9 '9 Zr and 9 '94Zr have been taken from Hefs. 9 and 10, respectively.
(~, IIEzll~,
E„(keV)
Nucleus
Transition
0+
0+
0+
2+
~ 2+
2+
Experimental
934
918
1758
&
&&yll
&2 II &;&z
B(E2) e fm x10
GBPA
SM
6BPA
SM
GBPA
Experimental
878
1072
1927
3.42
3.37
2.92
2.25
2.91
16.23
17.14
17.14
15.95
17.68
16.80
0.79 + 0.,07
0.56 + 0,06
0.55 + 0.22
1.94
SM
GBPA
0.65
0.58
0.66
0.63
0.63
0.64
QUADRUPOLE
90' 92'
--~-- 90-zr
.
9'--
MOMENTS
9'
96Z
94
96
wo Ru
AND
E2 TRANSITION
RATES IN THE. . .
Experimental
Shell Model
0-
Pd
1127
~
GBPA
1.
0.5
0403-
rPV
02
yah
C)
0. )
lv
00
-O.l
Ill
UJ
A
I
90
05
0.2
c3
03
I
"Zr
'
FIG. 2. The quadrupole moment of the first 2 state
[Q(2 ~)] in units of e b for N =50 even & nuclei and even
Zr isotopes with Sr inert core.
'
For the oscillator size
SM values of Gloeckner.
parameter b, we have used the same value (2. 124}
as that used for K= 50 even-A nuclei. The B(E2}
values for the Zr isotopes are summarized in Table II. It is heartening to find an excellent agreement between the GBPA and the SM results. Further, both the GBPA and SM reproduce, remarkably well, the experimental B(E2) values. It is to
be noted that, in the limit of two (identical) andfour
(nonidentical) particles, the BPA formalism reduces to the exact shell model. The slight departure between the present and corresponding
SM results for two (Table I) and four (Table II)
particles is perhaps due to a slight difference in
b values used in the respective calculations.
The quadrupole moment Q(2,') of the first 2'
states of N = 50 even A nuclei and for even Zr isotopes is plotted in Fig. 2. The figure depicts the
variation of Q(2;) with the addition of neutrons and/
or protons. It is revealed that, for N = 50 even-A.
nuclei, Q(2;) decreases with the addition of protons and finally changes its sign (i.e. , becomes
negative) for "Pd which lies a little beyond the
~S. Haq and
Y. K. Gambhir,
(1977).
2Y. K. Gambhir, S. Haq, and
C 20, 381 (1979).
Y. K.
188,
Y. K.
C 3,
Gambhir,
Phys. Rev. C 16, 2455
J. K. Suri,
Phys. Rev.
A. Rimini, and T. Weber
Phys. Rev.
1573 (1969).
Gambhir, A. Rimini, and T. Weber, Phys. Rev.
1965 (1971).
SM. H. Macfarlane, in Lectures in Theoretical Physics,
edited by P. D. Kunz and W. E. Brittin (University of
Colorado Press, Boulder, Colorado, 1966), Vol. VIII C,
p. 583.
SP. L. Ottaviani and M. Savoia, Phys. Rev. 178, 1594
I
92Z r
I
"Zr
96Zr
FIG. 3. The B (E2: 0~8-2~) values in units of
e210 cm for even Zr isotopes.
Q(2,') for even Zr
of neutrons,
addition
the
isotopes decreases with
passes through a minimum (negative value for
~Zr), and again increases (i.e. , becomes positive} for "Zr. Such behavior is also expected
from the shell model analysis.
The B(E2) values are plotted in Fig. 3. For the
transition for even Zr isotopes, the calculated B(E2) values (GBPA} reproduce, quantitatively, the trend of the variation of B(E2) with the
addition of neutrons. The same remark holds
true for the corresponding SM results.
In conclusion, we remark that in the Zr region,
the BPA (GBPA) is equally successful in reproducing the observed dynamic nuclear properties such
as quadrupole moment and the E2 transition rates
which are sensitive to the detailed structure of the
wave functions. This confirms the faith in BPA
(GBPA) as a valid, reliable, and useful approximation of the shell model. Therefore, BPA(GBPA}
holds good promise for the majority of cases where
shell model calculations simply cannot be perhalf filled shell.
Similarly,
0'-2'
formed.
(1969); 187, 130 (1969); Nuovo Cimento 67, 630 (1970);
B. Lorazo, Phys. Lett. 29B, 150 (1969); Nucl. Phys.
A153, 255 (1970); K. Allart and E. Boeker, ibid. A168,
630 (1971); A198, 33 (1972); W. F. Van Gunsteren,
K. Allart, and E. Boeker, ibid. A266, 365 (1976).
~D. H. Gloeckner and F. J. D. Serduke, Nucl. Phys.
A220, 477 (1974).
D. H. Gloeckner, Nucl. Phys. A253, 301 (1975).
Yu. P. Gangrskii and I. Kh. Lemberg, Yad. Fiz. 1, 1025
(1965) [Sov. J. Nucl. Phys. 1, 731 (1965)].
L. N. Galperin, A. Z. Ilyasov, I. Kh. Lemberg, and
G. A. Firsonov, Yad. Fiz. 9, 225 (1969) [Sov. J. Nucl.
Phys. 9, 133 (1969)j.