PH
YSICAL RKVIE%
C
VOLUME
Validity of the broken-pair
Department
16,
DECEMBER 1977
NUMBER 6
for N
approximation
= 50,
even-A nncleist
S. Haq and Y. K. Gambhir
of Physics, Indian Institute of Technology, Bombay-400 076, India
(Received 15 March 1977)
The validity
of the broken-pair approximation as an approximation to the seniority shell model is
The results of the broken-pair approximation and the seniority shell model, obtained by
investigated.
employing identical input information (single-particle levels and their energies, effective two-body matrix
elements, "Sr inert core) for X = 50, even-A nuclei are compared. A close agreement obtained between the
calculated broken-pair approximation and the seniority shell model energies for Zr, "Mo, "Ru, and "Pd
nuclei and large (95-100/e) overlaps between the broken-pair approximation and the senority shell model
wave functions for 'Mo, demonstrates
the validity of the broken-pair approximation in this region and in
general its usefulness as a good approximation to the seniority shell model.
NUCLEAR STRUCTURE Shell model, seniority shell model, broken-pair approximation, quasiparticle Tamm-Dancoff (QTD}, projected QTD; 90Zr, ~ Mo, 9 Ru,
96Pd
Recently, shell-model (SM) calculations have
been reported' for N = 50 nuclei, with "Sr as an inert core. The valence protons were restricted to
the 2py/g and lgg(2 shell-model orbitals. The single-particle (s.p. ) energies and all the nine protonproton matrix elements of the effective two-body
interaction were determined phenomenologically by
directly fitting the experimental data. It is expected that the phenomenologically determined input parameters would take into account, at least
partially, the effect of the neglected configurations.
Unfortunately, there is no direct way of verifying
this explicitly. The shell-model or even the seniority ( —4) shell-model (SSM) calculations in the
enlarged configuration space (i. e. , including 2P, &,
and lf, &, or M, &, and lg, &, s.p. states in addition
to the 2p, &, and lg, &, states) cannot be performed
due to prohibitively large dimensions of the configuration space. However, such enlarged configuration mixing calculations can be performed
within the framework of approximate methods.
The shell-model or SSM calculations are needed to
check the validity of approximate methods by comparing their results with the corresponding SM
(SSM) results. One such ayproximate method
which has been extensively studied in the past' is
the quasiparticle Tamm-Dancoff (QTD) method
(including its allied versions). The biggest drawback of the QTD method is that the quasiparticle
basis states are not exactly the eigenstates of the
particle number operator. Consequently, the calculated eigenstates are superpositions of states
with a different number of particles. It also leads
to spurious states, the removal of which is necessary before diagonalizing the Hamiltonian matrices. To remedy these defects, the use of num-
ber-projected quasiparticle basis states was proposed. ' It was shown that number projection is important and is essential, especially for the cases
where the number of valence nucleons is small.
in practice, these calculations are
Unfortunately,
quite involved. The broken-pair approximation
(BPA)'& is an imyrovement over the projected
quasiparticle Tamm-Dancoff method and is an approximation to the seniority shell model. The purpose of this note is to check the validity of the
BPA by comparing its results for X=50, even-A
nuclei with the corresponding SM results of the
Ref. 1.
%e sketch below only the relevant points of the
BPA (for details see Ref. 4). The BPA assumes,
as a first approximation, that the ground state for
2n (identical) nucleons is built up by the repeated
application (n times) of the pair distributed operator S, on the particle vacuum. Thus the approximate ground state is written as
s", io&,
where
i.)
S, =pl. k(&i. +1)"'~0,(i.
(2)
The operator Azt„(j, j~) creates an antisymmetric,
unnormalized, coupled two-particle (in s.p. states
and ) state with a total angular momentum Z
and projection M. The distribution coefficients
Q, in S, are determined by minimizing the expectation value of the Hamiltonian in the state (1).
The structure of state (1) is exactly the same as
that of the 2n-particle component of the BCS state
(quasiparticle vacuum). The only difference is
that inthe BPA the coefficients (p, = v, /
vu,+' = 1)
j,
j,
„u,'
S.
2456
HAQ AHO
are determined by minimizing the expectation
value of the Hamiltonian after number projection,
rather than before. The basis vectors, in addition
to state (1), defining the first BPA are constructed
by replacing one of the 8, operators in Eq. (1), by
an arbitrary two-particle creation operator, i.e. ,
they are the states
0
(j,j ~2M) = A ~~„(j,j,)S~ '
~
0)
.
(3)
j,j,
For 8= 0 the states q (
00) have seniority zero.
These are not orthogonal, and, furthermore the
state 8,"(0) is just a linear combination of the
states q'(j, 00). Therefore, for J=0, and orthonormal basis set is constructed from the states (3)
procedure. For J W 0
by Schmidt orthogonalization
the states 4 (j,j~ZM) are orthogonal and are seniority-two states, in which the distribution of
(n-1) zero-coupled pairs is restricted by the BPA
assumption of the approximate ground state [Le. ,
Eq. (1)]. Clearly, for two particles the BPA coincides with the seniority (exact) shell model. The
space spanned by the first BPA is therefore the
same as that spanned by seniority zero for the
J =0 and seniority two for the J w0 shell model.
Further, the maximum dimension of the first BPA
configuration space is the same as that of the twoparticle space, in contrast to large dimensions of
the seniority-zero and -two shell-model spaces.
This great reduction in the dimensions stems from
the particular structure of the BPA basis states.
It was shown' that the first BPA reduces to the projected two-quasiparticle Tamm-Dancoff theory if
one replaces the distribution coefficients Q,
(= v, /u, ) by the corresponding coefficients of the
quasiparticle theory.
The next step is the evaluation of the matrix elements of the Hamiltonian between the BPA states
(3). The explicit expressions, for these are given
in Ref. 4. Finally, the diagonalization of the
energy matrices determines the energy spectra
j,
Y. K. GAMBHIR
l6
value of the Hamiltonian in the state (1), because
of the inclusion of only two, 2P», (degeneracy 2)
and Ig, &, (degeneracy 10), s.p. states in the calculation. The energy matrices are generated and
diagonalized. The calculated BPA and the twoquasiparticle Tamm-Dancoff (2QTD) spectra together with the experimental data are plotted in
Figs.
' Zr,
for
1 and 2
"Mo, "Ru,
and
"Pd nuclei.
The analogous SM results, obtained by using a set
of parameters identical to Ref. 1, are also shown
(labeled EXACT) in the figures. The number of
calculated BPA and 2QTD energy levels is the
same for all four nuclei and corresponds to the independent number of two-particle configurations,
while in the SM the number of these states increases with the addition of valence protons.
%'e lay main emphasis on the detailed comparison of the BPA and the exact results. For the
two-particle case, i.e. , ~Zr, the BPA results are
identical to the SM results, as expected. It is gratifying to note from the Figs. 1 and 2 the excellent
agreement between the EXACT and the BPA spectra. This establishes the applicability of the BPA
and its validity as a good approximation to the SSM.
The 2QTD results are consistently bad for almost
all the cases. This is to be expected since the
number of valence nucleons (protons) considered
is small. For the detailed comparison of the wave
4.5—
4.0+
6+
3.5—
3.0-
7
4+
3
—
4+
2+
4
3
8+
+
2. 5-
5
and the wave functions.
0+
0+
4+
4+
%e present here the BPA results for N = 50, evenA ( 90Zr,
"Mo, "Ru,
6
action matrix elements. Kith this input information, the distribution coefficients fIt) are obtained by solving the gap and number equations of
the quasiparticle theory. These P, are expected to
be very close to the corresponding distribution
coefficients obtained by minimizing the expectation
—
—
4+
"Pd) nuclei,
obtained by
confining (as in Ref. 1) protons outside the "Sr inert core to 2P&)2 and 1geg 2 s.p. orbitals. The s.p.
energies and the proton-proton effective matrix
elements are taken from Ref. 1. This pheset of parameters (labeled
nomenological
SENIORITY in Ref. 1) was obtained by the least
squares fit to the energy levels, with an additional
constraint of seniority conservation on the interand
~4
g 8 \+
5
2+
1.5—
2
—
5
0+
2~
2
1.0—
0+
O+
EXPT
EXACT
BPA)
0+
2 QTD
0+
0+
EXPT
EXACT
0+
BPA
2QTD
(=
90Z
40
92
42
FIG. 1. The experimental (EXPT) and the calculated
broken-pair approximation (SPA) and the two-quasiparticle Tamm-Dancoff (2 QTD) spectra of SOZr and Mo.
The label EXACT shows the corresponding shell-model
results of Ref. 1. (The first 4 state is a seniorityfour state in the EXACT spectra of Mo. )
VA I
IOIT Y OF
BROKEN-PAIR APPRO XI NATION
'FHK
FOR. . .
45~
N, =
40
35-
0
7+
?
0
25"
0+
/) g
8»
'lq+
$ 2.0-
4+
5
8+
8+
b
(8
4+
0+
0
-5
5
g+
5
6+
8+
4+
5
E6
LLI
5-
1.
2
+
2
1.0-
2+
0+
EXPT
0
EXACT
0
BPA
0
2QTD
EXACT
2+
0+
0
BPA
0
2QTO
S4
Ru
44
FIG. 2. The experimerital and the calculated spectra
of 94Ru and Pd (see caption of Fig. 1). EXACT* for
+Pd shows the result of the Total energy fit of Ref. 1.
functions, we rewrite the BPA states (3) in terms
of the shell-model states with seniority (&)
quantum numbers using the relation
I
(i)" »& =& "'t( 'fi )"'-~' (u)l'I
(s)"»&
where
report of this work was presented at
the Nuclear Physics and Solid State Physics Symposium (1976, Ahmedabad) heM by the Department of
Atomic Energy {DAE), Government of India.
/Work supported in part by the DAE (Govt. of India)
project No. BRNS/Physics/15/74.
~D. H. Gloeckner-and F.
D. Serduke, Nucl. Phys.
A220, 477 (1974).
2An extensive li. st of papers on the guasiparticle approx'
i,mation is given in M. Gmitro, A. Bimini,
Sawicki,
*A preliminary
J.
T. Weber,
J.
Phys. Rev. 173, 964 (1968).
M. H. Macfarlane, in Lectures in Theoreticu/
and
~t
for q~A&- v,
= 0 otherwise.
4
3.0"
q((A,' —v) I
Qg —Q' —&
Physics,
%e have calculated the overlaps between the BPA
and the exact wave functions for the "Mo nucleus.
In this case the exact wave functions have practichl?y no seniority mixing. For the negative
parity 4, 5" states, there appears only a single
BPA [Eq. (3)] configuration having seniority two.
Therefore these BPA states have 100% overlay
with the corresponding exact wave functions. For
the remaining states of the 'Mo nucleus, the overlays obtained with the yroyer phases of I(z)'J=O,
are 100%
u=O&, I(z)4/=2, 4, v=2& configurations,
for 0;, 0; states and =95% for 4=2;, 4;, 6;, 8; states,
respectively. Although we did not have the exact
wave functions for '4Ru and "Pd nuclei, similar
overlaps should be expected also in these cases.
In conclusion we remark that a close agreement
between the BPA and SM spectra and large overlaps obtained for N= 50, even-A nuclei, clearly establishes the validity of the BPA as a good approximation to the SSM in this region. This lends
confidence to the SPA and demonstrates its usefulness for the cases where the SSM calculations
cannot be performed.
Vife are thaxQrful to Dr. V. Potbhare for running a
shell-model code for the "Mo nucleus.
P. D. Kunz and %. E, Brittin (Univ. of ColorPress, Boulder, Colorado, 1966), Vol. VIIIC,
edited by
ado
p. 583; P. L. Ottaviani and M. Savoia, Phys. Bev. 178,
1594 (1969); 187, 130 (1969); Nuovo Cimento 67, 630
(1970) .
A. Bimini, and T. Weber, Phys. Rev.
188, 1573 (1969); Phys. Rev. C 3, 1965 (1971).
B. Lorazo, 3e cycle thesis, Universitk de Paris, Orsay,
1968 (unpublished); Phys. Lett. 298, 150 (1969); Nucl.
Phys. A153, 255 (1970); K. Allart and E. Seeker, Nucl.
Phys. A168, 630 (1971); A198, 33 (1972).
4Y. K. Gankhir,