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Quasiparticle RPA with finite rank approximation for Skyrme interactions

A finite rank separable approximation for the particle-hole RPA calculations with Skyrme interactions is extended to take into account the pairing. As an illustration of the method energies and transition probabilities for the quadrupole and octupole excitations in some O, Ar, Sn and Pb isotopes are calculated. The values obtained within our approach are very close to those that were calculated within QRPA with the full Skyrme interaction. They are in reasonable agreement with experimental data.


I. INTRODUCTION
Many properties of the nuclear states can be described within the random phase approximation (RPA) [1][2][3][4]. Among many microscopic nuclear models aiming at a description of the properties of nuclear excitations the most consistent model employs an effective interaction which can describe, throughout the periodic table, the ground states in the framework of the Hartree-Fock (HF) approximation and the excited states in the small amplitude limit of the time-dependent HF, or the random phase approximation (RPA). The Gogny's interaction [5] and the Skyrme-type interactions [6] are very popular now. Such models are quite successful not only for predicting accurately nuclear ground state properties [7,8] but also for calculating the main features of giant resonances in closed-shell nuclei [9,10] and singleparticle strengths near closed shells [11]. Taking into account the pairing effects enables one to reproduce also many properties of collective states in open-shell nuclei [12][13][14][15].
It is well known that due to the anharmonicity of vibrations there is a coupling between one-phonon and more complex states [2,4]. The main difficulty is that the complexity of calculations beyond standard RPA (e.g., for studying damping mechanisms of collective excitations) increases rapidly with the size of the configuration space and one has to work within limited spaces. From another point of view more phenomenological models that assume some simple separable form for the residual nucleon-nucleon interaction while the mean field is modelized by an empirical potential well allow one to calculate nuclear excitations in very large configuration spaces since there is no need to diagonalize matrices whose dimensions grow with the size of configuration space. The well-known quasiparticle-phonon model (QPM) of Soloviev et al. [4] belongs to such a model. Very detailed predictions can be made by QPM for nuclei away from closed shells [16].
The possibility to solve easily the RPA problem in a large configuration space when the residual particle-hole (p-h) interaction is separable was the motivation for proposing in our previous work [17] a finite rank approximation for the p-h interaction resulting from Skyrme-type forces. Thus, the self-consistent mean field can be calculated in the standard way with the original Skyrme interaction whereas the RPA solutions would be obtained with the finite rank approximation to the p-h matrix elements. It was found that the finite rank approximation reproduces reasonably well the dipole and quadrupole strength distributions in Ar isotopes.
In the present work, we extend the finite rank RPA calculations to take into account pairing effects. As an application we present results of calculations for low-lying 2 + and 3 − states in some O, Ar, Sn and Pb isotopes. This paper is organized as follows: in Section II we sketch our method for constructing a finite rank interaction for the quasiparticle RPA (QRPA) case. In Section III we discuss details of calculations and show how this approach can be applied to treat different multipole states in wide excitation energy regions. Results of calculations for characteristics of the quadrupole and octupole states in some nuclei are given in Section IV. Conclusions are drawn in Section V.

II. HAMILTONIAN OF THE MODEL AND QRPA
We start from the effective Skyrme interaction [6] and use the notation of Ref. [18] containing explicit density dependence and all spin-exchange terms rather than the original form of Ref [6] where density dependence at the HF level was introduced by a three-body contact force and where some spin-exchange terms were dropped. The exact p-h residual interactionṼ res corresponding to the Skyrme force and including both direct and exchange terms can be obtained as the second derivative of the energy density functional with respect to the density [19]. Following our previous paper [17] we simplifyṼ res by approximating it by its Landau-Migdal form in the momentum space: where k i , σ i and τ i are the nucleon momentum, spin and isospin operators, and N 0 = 2k F m * /π 2h2 with k F and m * standing for the Fermi momentum and nucleon effective mass. For Skyrme interactions all Landau parameters with l > 1 are zero. Here, we keep only the l = 0 terms in V res and in the coordinate representation one can write it in the following form: The expressions for F 0 , G 0 , F ′ 0 , G ′ 0 in terms of the Skyrme force parameters can be found in Ref. [18]. Because of the density dependence of the interaction the Landau parameters of Eq.(2) are functions of the coordinate r. In what follows we use the second quantized representation and V res can be written as: where a + 1 (a 1 ) is the particle creation (annihilation) operator and 1 denotes the quantum numbers (n 1 l 1 j 1 m 1 ), where In the above equation, j 1 ||Y J ||j 3 is the reduced matrix element of the spherical harmonics are the radial integrals: where the radial wave functions u(r) are related to the HF single-particle wave functions: As it is shown in [17] the radial integrals can be calculated accurately by choosing a large enough cutoff radius R and using a N-point integration Gauss formula with abscissas and weights r k , w k . Thus, the residual interaction can be presented as a sum of N separable terms.
So we employ the hamiltonian including an average nuclear HF field, pairing interactions, the isoscalar and isovector particle-hole (p-h) residual forces in a finite rank N: We sum over the proton(p) and neutron(n) indexes and the notation {τ = (n, p)} is used.
A change τ ↔ −τ means a change p ↔ n. The single-particle states are specified by the quantum numbers (jm), E j are the single-particle energies, λ τ the chemical potentials. V (0) τ is the interaction strength in the particle-particle channel, κ (M k) (κ (Sk) ) are the multipole (spin-multipole) interaction strengths in the p-h channel and they can be expressed via the Landau parameters as: The monopole pair creation, the multipole and spin-multipole operators entering the normal products in Eq.(10) are defined as follows: where f j ′ j are the single particle radial matrix elements of the multipole and spinmultipole operators: One can see that the hamiltonian (10)  In what follows we work in the quasiparticle representation defined by the canonical Bogoliubov transformation: The hamiltonian (10) can be represented in terms of bifermion quasiparticle operators and their conjugates [4]: We introduce the phonon creation operators where the index λ denotes total angular momentum and µ is its z-projection in the laboratory system. One assumes that the ground state is the QRPA phonon vacuum | 0 , i.e. Q λµi | 0 = 0. We define the excited states for this approximation by Q + λµi | 0 . For the QRPA the following relation is valid: The quasiparticle energies (ε j ), the chemical potentials (λ τ ), the energy gap and the coefficients u,v of the Bogoliubov transformations (17) are determined from the BCS equations with the single-particle spectrum that is calculated within the HF method with the effective Skyrme interaction. Making use of the linearized equation-of-motion approach [1]: with the condition: one can derive the QRPA equations [3,4]: In QRPA problems there appear two types of interaction matrix elements, the 2 )qτ matrix related to forward-going graphs and the B (λ) (j 1 j ′ 1 )τ (j 2 j ′ 2 )qτ matrix related to backward-going graphs. For our case we get the following expressions: where ε jj ′ = ε j + ε j ′ and u (±) jj ′ = u j v j ′ ± v j u j ′ . One can find a prescription how to solve this system and to find the eigen-energies and phonon amplitudes in Appendix A (see also [17]). The matrix dimensions never exceed 4N × 4N independently of the configuration space size. The derived equations have the same form as the QRPA equations in the QPM [4,20], but the single-particle spectrum and parameters of the p-h residual interaction are calculated making use of the Skyrme forces.

III. DETAILS OF CALCULATIONS
In this work we use generally the standard parametrization SIII [21] of the Skyrme force.
Some examples of calculations with other parameter sets are presented in the next section.
Spherical symmetry is assumed for the HF ground states. The pairing constants V 0 τ are fixed to reproduce the odd-even mass difference of neighboring nuclei. As a result constant pairing gaps have values that are very close to ∆ = 12.0A −1/2 besides a case of semimagic nuclei.
It is well known [12,14] that the constant gap approximation leads to an overestimating of occupation probabilities for subshells that are far from the Fermi level and it is necessary to introduce a cut-off in the single-particle space. Above this cut-off subshells don't participate in the pairing effect. In our calculations we choose the BCS subspace to include all subshells lying below 5 MeV. In order to perform RPA calculations, the single-particle continuum is discretized [22] by diagonalizing the HF hamiltonian on a basis of twelve harmonic oscillator shells and cutting off the single-particle spectra at the energy of 190 MeV. This is sufficient to exhaust practically all the energy-weighted sum rule. As it was shown in our previous calculations [17] we have adopted the value N=24 for the finite rank approximation for the dipole and quadrupole excitations in Ar isotopes. Increasing the mass number and the multipolarity of excitations demands an increase of the rank to keep the calculations accurate. Our investigations enable us to conclude that N=45 is enough for multipolarities λ ≤ 3 in nuclei with A ≤ 208. Increasing N, for example, up to N=60 in 208 Pb changes results for energies and transition probabilities not more than by 1%, so all calculations in what follows have been done with N=45. Our calculations show that, for the normal parity states one can neglect the spin-multipole interactions as a rule and this reduces by a factor 2 the total matrix dimension. For example, for the octupole excitations in 206 Pb we need to invert a matrix having a dimension 2N=90 instead of diagonalizing a 1376 × 1376 matrix as it would be without the finite rank approximation. For light nuclei the reduction of matrix dimensions due to the finite rank approximation is 3 or 4. So, for heavy nuclei our approach gives a large gain in comparison with an exact diagonalization.
The Landau parameters F 0 , G 0 , F ′ 0 , G ′ 0 expressed in terms of the Skyrme force parameters [18] depend on k F . As it is pointed out in our previous work [17] one needs to adopt some effective value for k F to give an accurate representation of the original p-h Skyrme interaction. To fix the effective values of k F for the Landau parameters we use the selfconsistency relation [23]. From the following set of equations: one can get the average field potential corresponding to such a choice of the residual interaction: This potential can be compared to (m * /m)U HF (r) which is the leader term of a local equivalent potential in Skyrme-HF approach. It is possible to evaluate the effective value k F for every nucleus. One can show that this value is larger than the nuclear matter value in order to compensate for the effects of the neglected terms F 1 and G 1 . To calculate the dipole strength distributions we choose k F so that the spurious isoscalar dipole mode appears at zero excitation energy. The strongest renormalization of the k F values in comparison with the nuclear matter value takes place in light nuclei. For 208 Pb the effective value k F becomes rather close to the nuclear matter one.

IV. RESULTS OF CALCULATIONS
As a first example we examine the multipole strength distributions in 36 Ar. The calculated strength distributions are displayed in Fig.1. For the giant dipole resonance (GDR) and giant quadrupole resonance (GQR) QRPA gives results that are very similar to our previous calculations with the particle-hole RPA [17] because the influence of pairing on the giant resonance properties is weak. This is not the case for the first 2 + and 3 − states that will be discussed later. For the GDR energy centroid we get E c = 19.9 MeV and this value is rather close to the empirical systematics [24] The isoscalar GQR energy centroid is equal to E c = 18. in this case. The study of the influence of two-phonon configurations on properties of the low-lying states within our approach is in progress now.
Results of our calculations for the 3 − 1 energies and the transition probabilities B(E3) are compared with experimental data [29] in Table 2. Generally there is a better agreement between theory and experiment than in the case of quadrupole excitations. Other choices of the Skyrme forces do not improve the agreement obtained with SIII.
An additional information about the structure of the first 2 + , 3 − states can be extracted by looking at the ratio of the multipole transition matrix elements M n /M p that depend on the relative contributions of the proton and neutron configurations. In the framework of the collective model for isoscalar excitations this ratio is equal to M n /M p = N/Z and any deviation from this value can indicate an isovector character of the state. The M n /M p ratio can be determined experimentally by using different external probes [30][31][32]. Recently [13,26], QRPA calculations of the M n /M p ratios for the 2 + 1 states in some O and Ar isotopes have been done. The predicted results are in good agreement with experimental data [26].
Our calculated values of the M n /M p ratios for the 2 + 1 and 3 − 1 states are shown in Tables 3   and 4, respectively. Our results support the conclusions of Refs. [13,26] about the isovector character of the 2 + 1 states in 18,20 O and 38 Ar. As one can see from Table 4 our calculations predict that the M n /M p ratios for the 3 − 1 states are rather close to N/Z, thus indicating their isoscalar character. This conclusion remains valid for the SGII and SkI4 parameter sets.

V. CONCLUSION
A finite rank separable approximation for the particle-hole RPA calculations with Skyrme interactions that was proposed in our previous work is extended to take into account the pairing correlations. The QRPA equations are derived for this case. These equations are used to study the evolution of quadrupole and octupole excitations in nuclei away from stability. It is shown that the suggested approach enables one to reduce remarkably the dimensions of the matrices that must be inverted to perform structure calculations in very large configuration spaces.
As an illustration of the method we have used the finite rank p-h interaction derived from the Skyrme force SIII to calculate the energies and transition probabilities of the 2 + 1 and 3 − The index k run over the N-dimensional space (k=1,2,...,N). Solving the system of equations (24) one can get the following expressions for the phonon amplitudes: