(Revised 12 October 1992). Abstract: The coupling of pairing vibrations with
nuclear-shape oscillations has been investigated. The collective hamiltonians.
NUCLEAR PHYSICS A
Nuclear Physics A554 ( 1993 ) 4 13-420 North-Holland
Coupling
of nuclear shape oscillations pairing vibrations*
Stanislaw Zaktad
Pilat and Krzysztof
Fizyki Teoretwznej
University
M.C.S.
with
Pomorski PL-20031 Lubtin, Poland
Received 23 July 1992 (Revised 12 October 1992)
The coupling of pairing vibrations with nuclear-shape oscillations has been investigated. The collective hamiltonians were obtained within the generator-coordinate method using the gaussian overlap approximation, as well as in the cranking model. The projection onto proper particle number was taken into account, A Nilsson single-particle hamiltonian with pairing forces and long-range residual interaction in the local approximation was used. For the axial quadrupole oscillations the lowest-lying Oi energy levels were estimated for Ba (N = 68-78) and Xe ( N = 64-76) isotopes. The coupling of the pairing vibrations and quadrupole oscillations significantly lowers the collective energy levels, and show a much closer agreement to the experimental data than in the one-dimensional calculations.
Abstract:
1. Introduction
Pairing degrees of freedom are of great importance in the description of collective phenomena in atomic nuclei. The influence of pairing vibrations in spontaneous fission was studied with use of the concept of the collective hamiltonian within the cranking model ‘,I) and in the generator coordinate method using the gaussian overlap approximation (GCM +GOA) 3). A strong coupling between the pairing vibrations and the fission mode was found. This effect decreases the estimates of the spontaneous fission lifetimes by several orders of magnitude [cf. also ref. “)I. In this work we present some qualitative results on the coupling between the pairing vibrations and the nuclear surface oscillations. We have used both the models mentioned to estimate the lowest energy levels of the collective quadrupole oscillations including the influence of the pairing degrees of freedom. The theoretical estimates of the collective vibrations energy levels give spectra which are systematically spread out in energy by a factor almost 2 when compared with the existing experimental values [cf. e.g. ref. ‘)I. Correspondence to: Dr. K. Pomorski, Technische Universitat Miinchen, Physik Department T30, James Franckstr. 1, W-8046 Garching, Germany. l This work is supported by the Polish Committee of Scientific Research under MEN contract No. 203119101. 0375.9474/93/$06.00
@ 1993 - Elsevier
Science
Publishers
B.V. All rights reserved
414
Fig. I. The mass parameter
B,,
calculated for ““Ba in the GCM+GOA the cranking method (solid line).
model
(dashed
line) and in
On the other hand calculations done in ref. “) show that the maximum of the ground-state wave function for the collective pairing vibrations is systematically shifted from the minimum of the collective potential towards smaller values of the pairing gap A. This effect originates from the fact that the mass tensor components increase rapidly as A decreases [a well-known result “‘)I. The results of similar calculations performed in both models (GCM and cranking) 14) may be seen in figs. l-3 for “‘Ba for example. From these considerations lower values of the collective energy levels can be expected due to the coupling between the pairing and shape vibrations. We adapted here our model developed in ref. ‘) within the GCM + GOA formalism to describe the coupling of the pairing and axial quadrupole oscillations. To check the dependence of this effect on the model we have also used the cranking
Fig. 2. The collective
potential
obtained for ““Ba in the GCM +GOA cranking method (solid line).
model
(dashed
line) and in the
S. Piiat,
0.0
Fig. 3. The normalized
K. Pomwski
0.2
/ Nuclear
0. 1
0.6
415
shape
0.H
ground-state wave function obtained for ““Ba in the GCM +GOA line) and in the cranking method (solid line).
model (dashed
method described in ref. ‘), in which projection onto good particle number has been taken into account. From the collective hamiltonian so obtained the values of the lowest 0’ levels were estimated for Ba (N = 68-78) and Xe (N = 64-76) isotopes.
2. The model Following ref. ‘) we have taken a product of BCS type wave function and neutrons as the generator function jq), viz.
for protons
where E? is the elongation parameter, A,, A,, are the pairing gaps and (p,,, q,, are the gauge angles. Such a form for the generator function ensures an approximate particle-number projection ‘). The microscopic many-body hamiltonian was assumed to have the form ~i=ci,,(E~)--~j‘t~+tjp:l,r-h(~-(~)).
(2.2)
Here fiti,,is the single-particle Nilsson-type hamiltonian and the pairing hamiltonian tipair is a sum of the proton and neutron parts. The last term is the particle-number constraint. The two-body, long-range residual interaction is taken here in a local approximation ‘)),
(2.3) in which from the self-consistency
condition
we have the strength
constant (2.4)
416
S. Pifar, K. Pomorski
Both models (cranking the following form:
and GCM+GOA)
/ Nuclear shape
lead to the collective
hamiltonian
in
(2.5) The collective
inertia
tensor
in the GCM + GOA model
has the form
(2.6) where
yi, is the width tensor
of the overlap (2.7)
The mass parameters
obtained
in the cranking
model
fulfill the equation (2.8)
where at creates a quasiparticle is equal to
with energy
V(q)
E,, in state 1v). The collective
=mw
W4) 9
where 8, is the zero-point energy which represents the collective correlation in the ground-state energy. In the GCM + GOA model it is equal to
potential
(2.9) present
(2.10) whereas in the cranking method one assumes %‘.= 0. In the form of the collective hamiltonian (5) g represents the determinant of the metric tensor. In the GCM + GOA it is equal to the width tensor of the overlap (7) while in the cranking model it is identical to the mass tensor (8). The matrix elements in eqs. (6)-( 10) were calculated microscopically. Similarly to refs. 233)we have used the Strutinsky renormalization procedure “‘) calculating the (qlfilq) matrix element. The deformed Nilsson potential with the standard set of the parameters was used “). The pairing-force strength was adjusted to restore the experimental mass differences, according to refs. “,3). Considering the pairing gap A as dynamical satisfy the BCS number equation at each point N,,=2
parameter, A is usually in the A-mesh:
C u;(A,h). I’__0
chosen
to
(2.11)
It was demonstrated in ref. “) (within GCM) that such a procedure leads to the situation when pairing vibrational energy levels contain contributions from all 2p2h states even when the pairing strength G +O. It results in an overestimate of the energy-level values. The use of A as a dynamical parameter is suggested as one of the possibilities to overcome this problem.
S. Pi/at,
Following
this suggestion
only
K. Pomorski
partly
/ Nuclear
for numerical
ourselves to the F?, A, cpset of collective variables. number equation (11) only for Ami, corresponding potential. spectrum)
411
shape
reasons
we have
limited
However, A now fulfills the BCS to the minimum of the collective
In this case (with h depending only on the deformed single-particle all terms containing the derivatives ah/ad vanish. The projection onto
good particle an additional
number provided term 13):
by the form of the generator
.p=-g(N,,-Z (CIC
1 0;)’ V-,0
function
(1) then gives
(2.12)
which should be added to the collective potential energy (the so-called quasirotational term). At the minimum in the collective potential it vanishes identically.
3. Results
of the calculations
We performed the numerical calculations to estimate the values of the lowest collective quadrupole vibrational states for the Ba (N = 68-78) and Xe (64-76) isotopes. We find that the expected strong dependence for the mass parameters on the pairing gap A holds for B,, as well as for BFZJ. For the BeJtL component in the vicinity of the equilibrium deformation value this dependence is much weaker. This effect was obtained both in GCM + GOA as well as in the cranking model and is due to shell effects [cf. ref. 14) for details]. In figs. 4 and 5 the BI.>*.?components calculated for lzhBa within the GCM +GOA model and in the cranking method respectively are presented as functions of E? and A,,,,,, (the mass tensor does not depend on (P”, cp,). This result indicates that shell effects decrease the coupling between pairing and shape vibrations. It also means that all components of the mass tensor should be included
into the calculations.
Fig. 4. The Ll,,, ,,, dependence
of the mass parameter B,, obtained within the GCM + GOA model. thick lines correspond to the ( F~),,~,,~.
The
418
S. Piiat,
K. Pomorrki
/ Nuclear
Fig. 5. The same as in fig. 4 but obtained
shape
in the cranking
method.
The quasirotational term (12) was found to be rather small, e.g. for “hBa its maximum value is around 0.04 MeV. Nevertheless, as the dA/dA term vanishes in the formulae for the mass parameters (A is independent on A ) the resulting estimates of the collective vibration levels are lowered. In table 1 we present the values of the lowest 0’ levels determined by numerical diagonalization of the collective hamiltonians obtained in both models for Ba (N = 68-78) and Xe (N = 64-76) isotopes. Two cases were considered: (i) onedimensional (F?) with BCS approximation of the pairing forces and (ii) with the coupling between pairing and shape vibrations taken into account (F~, A). We recall here also the results of the ref. ‘) where quadrupole oscillations were investigated with the use of the Bohr hamiltonian. As we mentioned above the calculations with the standard value of the pairing strength constant G gave the energy levels almost a factor 2 too large. Similar calculations performed with G decreased by 20% resulted in spectra which agree very well with the experimental values. In table 1 the lowest O+ states obtained in ref. ‘) (B.H.) for standard pairing strength G (s.P.s.) and weak pairing strength G (w.P.s.) are presented as well as the existing experimental data I’). The rather strong coupling effect lowering collective-vibration energy levels can be seen. For the lowest 0’ states presented here this effect is around 40% for the GCM and almost 50% for the cranking case. The latter are closer to the experimental ones. These results convince us that the coupling between the pairing and quadrupole oscillations, although weakened by the shell effects, plays an important role. It significantly lowers the value of the collective vibrational states which are now closer to the experimental values. The comparison with the results of ref. ‘) suggests that the weakening of the pairing constant strength simulates in’some way the dynamical treatment of the pairing vibrations and the coupling effect. We are of course aware of the simplicity of our model and do not counter these estimates as final results. This is also the reason why we have here only presented
419 1
TAHLI.
energy levels estimated
Lowest-lying
0’
the cranking
method.
Here
(F) denotes
within the GC’M+GOA
while (F, _t) denote estimates with the pairing vibrations denoted as B.H. Calculations (s.p.s.1 as well
as with
model as well as in
the results of the one-dimensional included.
calculations
Results of ref. ‘) are
done with the standard value of the pairing forces strength
the value
decreased
experimental
by 20%
(w.p.s.)
are shown.
Existing
data are also presented
GCM
B.H.
Cranking
Exp. F
F, j
P
F, A
s.p.5.
2.40
1.45
2.31
0.97
1.96
0.74
2.67
1.61
2.35
1.17
2.05
I .OO
2.82
1.95
2.56
1.22
2.25
1.01
2.97
2.55
2.74
1.34
9.43
1.49
3.15
2.73
2.91
1.65
2.63
1.70
3.45
2.02
3.18
1.98
2.84
I.54
1.76
2.21
1.23
1.95
0.93
1.x0
0.89
0.83
2.15
I .20
I.‘)1
0.82
1.77
0.54
0.01
2.38
1.43
2.15
1.0’)
2.05
1.04
W.p.S.
1.50
2.64
1.76
2.43
I .43
2.33
1.24
1.27
2.82
1.Y3
2.12
I .47
2.52
1.44
1.31
3.13
2.27
2.94
1.65
2.81
1.x0
1.58
3.45
2.53
3.34
7.08
3 .OY
2.27
1.7Y
estimates for the lowest-lying 0’ states. Nevertheless these studies show strong evidence that coupling between the pairing and the shape oscillations should be taken into account in theoretical calculations of the collective vibration spectra. The authors are very grateful to Dr. Roger careful reading of the manuscript.
Hilton
for helpful
discussion
and a
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