Effective restoration of dipole sum rules within the renormalized random-phase approximation
The dipole excitations for calcium and zirconium isotopes are studied within the fully self-consistent Hartree-Fock mean field incorporated with the renormalized random-phase approximation (RRPA) using the Skyrme interaction SLy5. The RRPA takes into account the effect of ground-state correlations beyond RPA owing to the Pauli principle between the particle-hole pairs that form the RPA excitations as well as the correlations due to the particle-particle and hole-hole transitions, whose effects are treated here in an effective way. By comparing the RPA results with the RRPA ones, which are obtained for isoscalar (IS) and isovector (IV) dipole excitations in Ca and Zr, it is shown that ground-state correlations beyond the RPA reduce the IS transition strengths. They also shift up the energy of the lowest IV dipole state and slightly push down the peak energy of the IV giant dipole resonance. As the result, the energy-weighted sums of strengths of both IS and IV modes decrease, causing the violation of the corresponding energy-weight sum rules (EWSR). It is shown that this sum rule violation can be eliminated by taking into account the contribution of the particle-particle and hole-hole excitations together with the particle-hole ones in a simple and perturbative way. Consequently, the ratio of the energy-weighted sum of strengths of the pygmy dipole resonance to that of the giant dipole resonance increases.
pacs:21.60.Jz, 21.10.Pc, 24.30.Cz, 24.30.Gd
The random-phase approximation (RPA) is one of the most frequently used methods in the theoretical study of vibrational collective excitations in low-energy physics. These collective modes include the low-lying vibrations and high-lying giant resonances. Within the RPA, these vibrational excitations are generated by the coherent superpositions of elementary excitations. For instance, the electric multipole excitations (, etc.), including the isoscalar (IS) and isovector (IV) giant resonances, consist of many particle-hole (ph) components.
The basic assumption of the RPA is the quasi-boson approximation (QBA), which implies that the operators of ph pairs behave like ideal bosons, ignoring the Pauli principle owing to their fermion structure. Algebraically, the QBA is the replacement of the exact commutation relation between the ph pair operators with its expectation value in the Hartree-Fock (HF) ground state, where with and being the particle (p) creation and hole (h) annihilation operators, respectively Ring (). This assumption perfectly holds when the RPA vibrations consist of many coherent ph excitations, for instance in the case of high-lying collective excitations such as the giant resonances or low-lying ones such as the first or states in medium and heavy nuclei, where the concept of a nuclear mean field works very well. However, in light nuclei the validity of the mean-field concept and the QBA itself become questionable. Hence it becomes necessary to take into account properly the fermion structure of the ph pairs, removing in this way the violation of the Pauli principle.
There have been several methods to preserve the Pauli principle between the ph pairs by including the correlations in the ground state, which are neglected within the QBA. These correlations are referred to as the ground-state correlations (GSC) beyond RPA. Among the earliest approaches to include GSC beyond RPA, the one proposed by Hara Hara () is rather transparent and simple because it evaluates the expectation value of the commutation relation in the correlated RPA ground state by using the diagonal approximation, resulting in the particle and hole occupation numbers, (). The latter are approximately expressed in terms of the RPA backward-going amplitudes . This leads to the renormalized RPA (RRPA) equation, nonlinear with respect to the amplitudes , which can be solved by iteration. In the RRPA equation the residual interactions are renormalized by the so-called GSC factor , which is a function of .
In a later development, Catara, Dang and Sambataro derived in Ref. CaDaSa () an equation for the GSC factor , which recovers the expression for , which was obtained by Hara in the limit of small . It has been shown by Dukelsky and Schuck that this equation produces the exact expression for the GSC factor in the equidistant multilevel pairing model Dukelsky (). However, the application of the RRPA using this GSC factor has been limited so far only to the energy and value of the lowest state in Gd and Pb CaDaSa (). The RRPA was also studied by using the equations of motion method in Ref. Rowe () and the Green’s functions techniques in Ref. Schuck (). An alternative derivation of the occupation numbers by using the number-operator method was carried out in Refs. Rowe (); Lenske (). This method was employed in Ref. Catara () to obtain an expression for the GSC factor within the improved RPA up to the order of , which was applied to the calculations in metallic clusters.
The results of Ref. CaDaSa () show that, due to the GSC beyond the RPA, the energy the state in these two nuclei increases upward, leading to the significant decrease of around 40 and 20 in the associate values in Gd and Pb, respectively. These effects are more pronounced than those predicted for the low-lying states Hara (). No investigation was carried out for the states, whose first moment , known as the energy-weighted sum rule (EWSR), is the most important. For the transitions, this EWSR is related to the model-independent Thomas-Reich-Kuhn sum rule TRK = 60 NZ/A (MeV mb) for the giant dipole resonance (GDR) in a nucleus with A nucleons (N neutrons and Z protons). When a velocity-dependent effective interaction, such as the Skyrme one, or the meson-exchange interaction is used, the velocity-dependent and/or exchange terms contribute to the IV multipole sum rules. The IV EWSR then exceeds the TRK as TRK with 0 1 Liu (). The RPA (with the velocity-independent interaction) fulfills the GDR sum rule. However, being multiplied by the factor , the RRPA matrix elements of the ph interactions are reduced. This leads not only to a shift down of the GDR energy but also to a decrease in the values, causing the violation of the GDR sum rule.
The problem of model-independent sum rule violation within the RRPA was first pointed out in Ref. Hirsch (), where the authors attributed the origin of the problem to the adopted structure of both the Hamiltonian and the transition operators, which neglect the scattering terms coming from the particle-particle (pp) and hole-hole (hh) transitions. In Ref. Catara (), it has been shown that the improved RPA does not fulfill the EWSR as well. A practical idea to restore the EWSR within the RRPA was suggested in the same paper in the same spirit of Ref. Hirsch () by noticing that the transition operator, in general has not only a ph part but also the particle-particle (pp) and hole-hole (hh) ones. Therefore, in order to restore the EWSR, the RRPA transition amplitudes should contain also the contributions from the pp and hh transitions.
As a matter of fact, the importance of the pp and hh transitions were evident already in the studies of the finite-temperature RPA (FT-RPA) in the 1970s - 1980s, where the FT-RPA equations were derived and studied in details Ignatyuk (); Sommer (); Vautherin (); DangJP (). The FT-RPA equations, which were derived in these papers for the separable interactions, are formally identical to the RRPA ones if the ph transitions are considered. The only difference is that the single occupation numbers within the FT-RPA are described by the Fermi-Dirac distributions for non-interacting fermions at finite temperature . It has been shown in Ref. Sommer (), that the GDR energy decreases with increasing for the ph transitions within the FT-RPA, because becomes smaller than 1 and decreases with increasing . However, after the pp and hh transitions are included together with the ph ones, the GDR energy remains nearly the same as its value when 0 at varies up to 4 MeV. The and ESWRs also remain essentially constant up to 2 MeV. As for the IV excitations with thin the FT-RPA, the results of calculations obtained for Ca, including ph as well as pp and hh transitions by using a zero-range force in Ref. Vautherin () have also shown that the EWSR remains nearly constant up to temperature 6 MeV. A similar conclusion was obtained in Ref. Lacroix (), where the Skyrme force SGII was used in calculations.
With increasing the number of neutrons, a phenomenon called pygmy dipole resonance (PDR) is predicted in neutron-rich nuclei and a number of theoretical and experimental studies were devoted to its properties in the recent years PDRexp (); ReRPA (); Tarpanov (); Sarchi (); SRPA (); ReRPA1 (); QPM1 (); QPM2 (). The PDR has often been interpreted as a manifestation of the oscillation of the weakly-bound neutron skin against the isospin-symmetric core of protons and neutrons, although this picture is rather sketchy. As a matter of fact, the nature of the PDR still remains an issue open to debate. Different theoretical models still do not agree on its collectivity and coherence. For example, while the relativistic random-phase approximation predicted a prominent peak below 10 MeV in Sn and Zr ReRPA (), which is identified as the collective PDR, the results of calculations including monopole pairing within the quasiparticle RPA (QRPA) show only a group of slightly collective states Tarpanov () in Sn, whereas for Sn Sarchi () no collective states are even seen in the low-energy region. As the PDR is located in a low energy region, the effect of GSC beyond the RPA, which are included in the phRRPA, should be taken into account in the study of the PDR. This study should be complementary to those, which include 2p-2h excitations such as the second RPA SRPA (), the relativistic RPA plus phonon-coupling model ReRPA1 (), the quasiparticle-phonon model QPM1 (), a version of which includes even couplings upto 3p3h configurations in terms of three-phonon components QPM2 (). However, the major drawback, as mentioned above as well as in Ref. Gamba (), is that all extensions of the RPA to the RRPA so far violate the energy-weighted sum rules (EWSRs) for IS and IV dipole excitations.
Obviously, in order to restore the sum rule, one should derive and solve the complete set of RRPA equations that include all the ph, as well as pp and hh configurations. The transition probabilities, which are calculated based on these solutions, should also include all ph, pp, and hh transitions. This is undoubtedly a formidable task, which we are now working on. In the present paper, as a preparatory study, we would like to see if the problem can be resolved in an approximate and simple way, which can nonetheless be applied in practical calculations of electric dipole excitations. To this end, we modify the computer code for self-consistent RPA with Skyrme-type interactions, which has been developed by Colò and collaborators Colo (), to include the effect of GSC beyond the RPA following the method proposed in Ref. CaDaSa (). The dipole EWSRs are restored by perturbatively taking into account also the contribution of pp and hhole configurations, which have been neglected so far in all practical calculations within the RRPA.
Ii The RRPA for particle-hole vibrations (Rrpa)
The renormalized RPA for particle-hole vibrations, which is referred to as the phRRPA, has been discussed in details in Refs. Hara (); CaDaSa (); Dukelsky (); Rowe (); Schuck (); Lenske (); Catara (). Therefore, we summarize here only its main formulation within the ph representation including all angular momentum couplings in spherical basis, which is necessary for its application to the calculations in the present paper.
The RPA operator is constructed as a superposition of ph-pair operators in the form Ring ()
where is the ph-pair creation operator
which couples the particle creation operator on the orbital having the angular momentum and projection with the hole annihilation operator having the angular momentum and projection , where (with ) takes values, namely . The notations and are used for time-reversal operators. The quantum numbers and () denote the total angular momentum (the multipolarity of the phonon excitation) and its projection, respectively.
The RPA excited states are defined as
where the RPA correlated ground state is the vacuum of the phonon operator, that is
and the excited states should be mutually orthonormal, namely
The ph-pair operators and satisfy the exact commutation relation
The expectation value of this commutation relation in the RPA correlated ground state, can be approximated as
where the GSC factor is defined as
with being the particle and hole occupation numbers in the correlated ground state for and , respectively.
ii.1 The quasi-boson approximation (QBA)
Within the QBA, the expectation value (7) is replaced with that estimated in the HF ground state, . This yields 1 because 1 and 0, so that the expectation value of in the HF ground state becomes
This equation shows that, within the QBA, the ph-pair operators and behave like ideal boson operators, which mean that the following approximation holds
Within the QBA, the orthogonality condition (5) becomes
The closure relations
ensure the following inverse expression of ph-pair creation operator in terms of phonon operators and
These mappings are exact within the QBA, where the operators and behave like ideal boson operators according to (10). In this case the commutation relations between the right-hand sides of (15) and (16), and are
which are exactly equal to and , respectively.
ii.2 The phRRPA
The phRRPA method proposes to use Eq. (7) rather than Eq. (9), without imposing the QBA. Because the excited states should be orthonormal, this leads to the renormalized phonon operators in the form CaDaSa ()
It has been discussed in Ref. Rowe () that, if the number-operator method is used to derive the occupation numbers , the expressions for and in Eq. 22 acquire a factor of 1/2 in front of , which reduces the effect of GSC. However, as has been pointed out in Ref. Ellis (), the precise correspondence between the two methods is not immediately apparent, since it is not easy to pinpoint what effects are included in the number operator method. Moreover, the expressions obtained in Ref. Catara () for the same quantities include the term of even higher orders in and amplitudes [See Eqs. (2.20) and (2.21) in Ref. Catara ()], which makes them different from those given in Eq. (22) as well as in the number- operator method [Eq. (54) in Ref. Rowe ()]. However, as has been already mentioned in Ref. Rowe (), in practice, such difference does not appear to be such a serious effect, since only a few low-lying collective excitations contribute coherently to the GSC. For many non-collective state the backward-going amplitudes are small and and take place with random sign. Therefore, we decide to use the expressions in Eq. (22), keeping in mind that the resulting effect of GSC may be slightly overestimated compared to that predicted by the number-operator method.
Notice that, within the phRRPA, Eqs. (17) and (18) acquire the GSC factor in front of on the right-hand sides because, with respect to the correlated ground state , relation (7) is equivalent to the approximation
rather than the approximation (10), which is valid only for the HF ground state .
The QBA is justified only when the sum on the right-hand side of Eq. (24) becomes negligible compared to 1, so that 1.
The phRRPA equation is obtained in the standard way Ring () by using the renormalized phonon operator (20) and Eq. (7) and a model Hamiltonian consisting of a mean field of single-particles with energies on the spherical orbitals () and two-body residual interactions . The matrix form of the phRRPA equation is given as CaDaSa ()
The matrices and have the form
where the residual interactions are renormalized by the factor . Within the QBA, when 1, one recovers from Eqs. (25) and (26) the conventional RPA equation, and from Eqs. (22) the expressions for occupation numbers (19) within RPA.
The phRRPA equation (25) is solved self-consistently with Eq. (24) and the normalization condition (12). First the RPA matrices ( 1) are diagonalized to determine the eigenvalues (phonon energies), and amplitudes and . The latter are then used to calculate following Eq. (24). In the second step, the residual interaction, which is renormalized by are used in diagonalizing Eq. (25) (with in the submatrices A and B) to obtain , , . The process is repeated until the convergency is reached, that is MeV.
ii.3 Transition probability and energy-weighted sum rules
The nuclear vibrational excitations are generated by the electromagnetic field with one-body excitation operators and reduced matrix elements , whose details are given in Refs. Colo (); Bohr (). The reduced transition probabilities between the ground state and one-phonon state is calculated within the RRPA as
In practice, the distribution of values of (27) at discrete phonon energies is also presented as a continuous strength function by using the -function representation , that is
This kind of presentation not only improves the visual comparison of results obtained within the RPA and RRPA, but also mimics the so-called escape width , which arises from the direct decay to the continuum, consisting of a free nucleon and a hole state. This escape width is around few hundred keV, compared to the total GDR width of around 4 - 5 MeV in medium and heavy nuclei.
The energy-weighted sum of rule (EWSR) of the strength distribution generated by the operator is defined in terms of its the first moment as
This first moment is equal to the half of the expectation value of the double commutator if is the exact two-body Hamiltonian and is the exact ground state Ring (). Therefore the EWSR is also referred to as the double commutator sum rule, which is employed to test the validity of any approach to see if the model-predicted first moment at the left-hand side of Eq. (30) fulfills the EWSR.
For the dipole excitations the IS and IV EWSR, and , are
is the model independent Thomas-Reiche-Kuhn (TRK) sum rule, and is the enhancement factor owing to the velocity dependence and exchange mixtures (Wigner force) in the residual interaction. This enhancement factor is calculated according to Eq. (35) of Ref. Colo (). In Eqs. (31) and (32) the corrected mass is used instead of the bare nucleon mass in all Skyrme-HF calculations to take care of the center-of-mass effect on the total energy.
Iii The RRPA including the effect of particle-particle and hole-hole correlations
As has been mentioned in the Introduction, all extensions of the RPA in the spirit of phRRPA so far violate the sum rules. The reason comes from the presence of the GSC factors in the transition strengths (27), which are reduced because 1. This major drawback of the phRRPA was noticed for the first time in Ref. Hirsch (), where it has been pointed out that the renormalized quasiparticle RPA (RQRPA) violates the Ikeda sum rule for the charge exchange excitations such as the Gamow-Teller resonance or for the Fermi transitions between the ground state of the initial even-even nuclei and the ground as well as excited states of the odd-odd nuclei. For the dipole excitations, the GSC factor causes the decrease of the EWSRs. To resolve this problem, one needs to take into account the contribution owing to the expectation value of the commutation relation
which is neglected within the phRRPA. In Eq. (33), ( or ) are the correlation factors coming from the pp and hh channels. As the result, the total transition probabilities are the sum of the those associated with the ph, pp, and hh excitations. These transition probabilities are denoted as , , and , respectively.
with and being the vibration amplitudes that correspond to the pp and hh excitations, whereas the GSC factors and are expressed explicitly in terms of the GSC and the amplitudes by using Eqs. (22) and (33) ERPA () as
In principle, the phonon amplitudes and should be obtained by diagonalizing the full RRPA matrix, including all the ph, pp, and hh configurations, similar that obtained in Ref. Sommer () for the FT-RPA. However, doing so will significantly enlarge the size of the RRPA matrix, leading to a sharp increase of computational time. This project is now underway. In the present study, for the feasibility in practical calculations as a preparatory step to test the effect of GSC in the restoration of the GDR sum rule, we employ the extended RRPA (ERRPA), proposed in Ref. ERPA (), which considers a model Hamiltonian with a separable residual interaction in the form of Eq. (7) in Ref. ERPA (). The full RRPA matrix equations with the separable interaction can be easily transformed to obtain the phonon amplitudes in an explicit form, including the and of the pp and hh excitations (Eqs. (18) – (20) in Ref. ERPA ()). All the eigenvalues of the full RRPA equations are obtained by solving the secular equation (21) therein, instead of diagonalizing the full RRPA matrix.
with 1. Because the contribution of and excitations is expected to be small, we treat it in a perturbative way. To this end, instead of solving Eq. (21) of Ref. ERPA () to obtain all the eigenvalues , , and as well as the amplitudes , , and , we approximate the energies and the amplitudes and with the corresponding values obtained within the phRRPA described in Sec. II. Regarding the energies of the new phonon states, which appears between the poles of pp and hh excitations, their energies are considered to be close to the corresponding poles , namely
The energy shift is adjusted to restore the EWSR, which is violated within the phRRPA. In this way, although plays the role of a parameter of the model, it has a physical justification as the difference between the new RRPA solutions due to the pp (hh) configurations and the corresponding pp (hh) poles. The amplitudes and are calculated by using the expression obtained with the separable interaction given by Eqs. (19) and (20) of Ref. ERPA () with , whereas and are approximated with and , respectively, according to Eq. (40). This version of phRRPA, which takes into account the contribution of pp and hh excitations, is referred to as the RRPA hereafter.
The advantage of this approximation is its simplicity, based on the solutions of the phRRPA, avoiding the diagonalization of a large-size matrix. Its shortcoming is the lost of self-consistency, which may not be serious so long as the GSC is not large to justify the validity of the perturbative approximation employed here.
Iv Analysis of the numerical results
The computer code for the self-consistent HF-RPA with Skyrme-type interactions, developed by Colò and collaborators Colo (), is modified to include the effect of GSC beyond RPA. This code is restricted to the calculations of natural-parity states having in spherical nuclei with filled subshells, that is without partial occupancies. The single-particle energy spectra for neutrons and protons are discretized and superfluid pairing is not included. Regarding the effective nucleon-nucleon interaction, the density-dependent Skyrme interactions are employed in the calculations. The code first carries out the HF calculations for a specified nucleus using a given Skyrme-type interaction. The HF single particle energies, wave functions and densities, which are obtained in a radial mesh extending to (fm) and cut-off energy of unoccupied states, are then used in solving the RPA equation. As for the phRRPA and RRPA equations, they are solved by iteration as has been specified respectively in Secs. II and III above. For the details of the HF-RPA code see Ref. Colo ().
In the present paper we restrict ourselves to studying the IS and IV dipole states in Ca and Zr isotopes. For the Skyrme interactions, we employ the SLy5 force, whose parameters have been specified and used in the HF-RPA code in a spherical box with radius 15 fm and the cut-off energy 60 MeV of unoccupied states. This value has been chosen because, for the IV E1 excitations in the nuclei under consideration, the energy of the lowest RPA state decreases very slowly with increasing higher than 60 MeV. A value 0.4 MeV for the smoothing parameter is used in calculating the strength function (29).