ЯДЕРНАЯ ФИЗИКА, 2007, том 70, № 8, с. 1480-1484
ЯДРА. Теория
ON THE Qn-Qp DEFORMATION ENERGY SURFACES OF NEUTRON-RICH OXYGEN ISOTOPES
© 2007 A. P. Severyukhin1)*, M. Bender2), H. Flocard3), P.-H. Heenen4)
Received October 31, 2006
We present an analysis based on the generator coordinate method that allows to disentangle the neutron and proton contributions to the properties of low-lying collective states in oxygen isotopes.
PACS:21.10.Dr, 21.10.Gv, 21.10.Re, 21.10.Ky, 21.60.-n, 27.30.+t
1. INTRODUCTION
Experimental progresses in the study of exotic nuclei stimulate new theoretical developments by opening the possibility of new collective modes. New features are expected in the low-energy spectra of nuclei away from the ^-stability line, among which new deformation modes or a change of the magic numbers known close to stability. Among the new tools developed for nuclear structure studies, the generator coordinate method (GCM) [1, 2] is particularly promising. It is based on the mean-field wave functions generated by constrained calculation, where an effective interaction of Skyrme or Gogny type or a relativistic Lagrangian are used.
The isoscalar quadrupole mode dominates the low-energy large-amplitude collective dynamics in nuclei close to the ^-stability line. In nuclei with a large neutron excess, one can wonder whether the isovector quadrupole mode will not become important, leading to nuclei with significantly different deformations for neutrons and protons. The strong proton shell structure at Z = 8 and the already good experimental knowledge of the complete oxygen isotope series from 16O up to the drip line at 24O [3—14] makes oxygen an attractive element to explore this mode. The experimental efforts on these isotopes have stimulated theoretical analysis based either on the mean-field method [7, 14—19] or on the shell model [11, 20, 21]. There is a rather complete list of references on that subject in [22]. An important result from both experimental and theoretical works
1)Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia.
2)CEA-Saclay DSM/DAPNIA/SPhN, Gif-sur-Yvette Cedex, France.
3)CNRS-IN2P3, Université Paris XI, CSNSM, Orsay Campus, France.
4)PNTPM, CP229, Université Libre de Bruxelles, Belgium. E-mail: sever@theor.jinr.ru
is the weakening of the N = 20 shell effect (both 26O and 28 O are unstable [4—6]) along with evidences for possible N = 14 [14] and N = 16 [18] shell closures.
The present study is performed within the projected GCM [23, 24] with a basis built from a set of mean-field N-body wave functions generated by constrained Hartree—Fock—Bogoliubov (CHFB) calculations. We have considered two kinds of constraints: (i) a single constraint on the axial mass quadrupole moment, which generates a one-dimensional deformation energy curve and (ii) two independent neutron and proton quadrupole constraints generating a deformation energy surface. Recently, we have applied our approach to calculate characteristics of the ground state and of the low-lying collective states in the nucleus 20O [22]. In the present work we discuss the motivation for this GCM analysis with the isovector collective variables, to investigate the evolution of collectivity along the oxygen isotope series.
2. THE METHOD
This beyond mean-field method has already been presented in details and applied to a large number of nuclei [16, 23—27]. Let us very briefly describe the different steps involved in this approach.
The starting point of the method is the construction of a set of N-body wave functions by the CHFB method [1, 2]. We shall first use as a reference the usual quadrupole constraint generated by the axial mass quadrupole moment Q = Qn + Qp, where Qn,QP are the axial neutron and proton quadrupole operators, respectively. Hereafter we describe such a constraint as "isoscalar" jn contrast with calculations in which both Qn and Qp are constrained independently. We use the qualifier "isovector" for such a double constraint. The Hamiltonian H is derived from the
ON THE Qn-Qp DEFORMATION ENERGY SURFACES
1481
Fig. 1. Contour lines of the particle-projected mean-field deformation energy surfaces of 18 24 O vs. the N5/3 and Z6/3 scaled expectation values of the neutron (Qri) and proton (Qp) moments, respectively. Only that part of the surface corresponding to prolate neutron and proton deformations is shown. The curves are labelled by the absolute energy in MeV. The dashed lines indicate the HFB paths obtained with an isoscalar constraint on Q.
Skyrme Sly4 interaction [28] together with a surface peaked density-dependent zero-range force acting in the particle—particle channel [29] with a strength of —1000 MeV fm3 and a pairing active space limited to an energy range of 5 MeV above and below the Fermi level. To avoid that pairing correlations collapse in the case of low density of single-particle levels around the Fermi surface, we use the approximate variation-after-projection Lipkin—Nogami method.
The wave functions resulting from the minimization of the expectation value of H plus the constraint are denoted |$a), where a labels the collective variable. For an isoscalar constraint, a is identified with the expectation value {Q), generating a dynamics along a single collective dimension, the total deformation. In particular, the relative contributions of {Qn) and {Qp) to the total quadrupole moment {Q) are fixed by the CHFB minimization process. When Qn, Qp are taken as independent constraints, the collective variable a is two-dimensional and is identified with a = {{Qn), {Qp)}. Details of the calculations
with the isovector collective variables are discussed in [22].
Wave functions with good angular momentum and particle numbers are obtained by restorations of symmetry on |$a) [1,2, 24]:
\$,Ja) = N PoO PNPZ |$«), (1)
where N is a normalization coefficient, P00, PN, PZ are projectors onto the angular momentum J, neutron number N, and proton number Z, respectively.
In the next stage of our method, N-body collective wave functions (CWFs) are linear combinations of states |$, J a):
\*,J,k) = £ fia\$,Ja),
(2)
and the unknown weight functions fJa are determined by a minimization (f EJ = 0 ) of the total energy
j_ (*,J,k\H\*,J,k) (*,J,k\*,J,k) '
EJ =
(3)
Œ
-1.24 0
1.24 2.48
-130 -
-140 -
-150 -
-160 -
-200 0 200 400
(Q), fm2 P2
-1.24 0
1.24 2.48
-200
0 200
(Q), fm2
400
transformation of the weight functions fJa:
gj,k (a) = ^ lja?fk,a> >
Fig. 2. Nucleus 20O: (a) The particle-number-projected mean-field deformation energy curve (solid line), particle-number- and spin-projected mean-field deformation energy curve for the value J = 0 (dashed line) and the abscissa of the point indicates the mean deformation (P2) of the corresponding collective wave function of the ground state. (b) The collective wave function of the ground state.
This leads to the Griffin—Hill—Wheeler equations
EJ - EJkXia,J = 0. (4)
a'
Their solution requires the prior calculation of the off-diagonal matrix elements for the energy and norm GCM kernels:
J = {$,Ja\H\$,Ja>), (5)
J = {$,Ja\$,Jd).
For each J value, Eq. (4) gives a spectrum of correlated states corresponding the variable(s) a. A set of orthonormal CWFs are obtained by an integral
where
xj V2
represents the kernel whose square convolution gives 1J. The collective wave function gJ,k (a) provides an information of the spreading of the CWF over the collective space spanned by the variable(s) a.
3. RESULTS
Figure 1 shows the contour lines of the particle-number-projected mean-field deformation energy surfaces of 18-24O, the deformations being limited to prolate ones. For an easy comparison of different isotopes, the deformations have been expressed in terms of the quantities (Qn)/N5/3 and (Qp)/Z5/3. Such coordinates take into account the trivial scaling of the neutron and proton quadrupole moments associated with differing values of N and Z. One can see that for the heavier isotopes, the HFB ground state is spherical. The one-dimensional path (dashed line) defined by a constraint on the mass quadrupole moment deviates from the bisector, indicating a tendency along this optimal path of different deformations for neutrons and protons. At low deformations, this tendency increases with the neutron number, the neutron deformation being larger than the proton one. It remains to determine how this tendency is affected by the two-dimensional dynamics as a function of the neutron and proton deformations.
To illustrate this dynamics, we concentrate on the isotope 20O. In Fig. 2, we show results one-dimensional energy curve obtained as a function of a = (Q). The solid line in Fig. 2a corresponds the particle-projected mean-field deformation energy curve. It displays the usual pattern of an oxygen isotope: a well-defined spherical minimum and a steep rise of the energy versus deformation. For the J = 0 projected curve (dashed line in Fig. 2a) which according to Eq. (5) corresponds to H°,a, there are two minima for two deformed mean-field configurations, an oblate and a prolate one. This figure illustrates the energy gain due to projection and to the configuration mixing (the solution of Eq. (4)). The CWF of the ground state is plotted in Fig. 2b. As expected the CWF is localized in the vicinity of (Q) = 0. On the other hand, it displays an asymmetry favoring prolate deformation.
In Fig. 3, the results obtained when the neutron and proton quadrupole deformations, i.e., a = = {(Qn), (Qp)} are constrained separately are given.
a
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-100
-100
100
200
300 < Q n), fm2
0
-126
< Q n), fm 126
0 <Qp), fm2
252
Fig. 3. Nucleus 20O: (a) Contour lines of the particle-number- and spin-projected mean-field deformation energy surface for J =0 (the absolute energy in MeV) vs. expectation values of the neutron (Qn) and proton (Qp) moments, the dashed line is the deformation path followed
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