A self-consistent study of multipole response in neutron-rich nuclei using a modified realistic potential
D. Bianco, F. Knapp, N. Lo Iudice, P. Vesely, F. Andreozzi, G. De Gregorio, A. Porrino
aa r X i v : . [ nu c l - t h ] D ec Self-consistent study of multipole response inneutron rich nuclei using a modified realisticpotential
D Bianco , ‡ , F Knapp , N Lo Iudice , , P Vesel´y , FAndreozzi , , G De Gregorio , , A Porrino , Dipartimento di Fisica, Universit`a di Napoli Federico II, Monte S. Angelo, viaCintia, I-80126 Napoli, Italy Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli, Monte S.Angelo, via Cintia, I-80126 Napoli, Italy Faculty of Mathematics and Physics, Charles University, Prague, CzechRepublicE-mail: [email protected]
Abstract.
The multipole response of neutron rich O and Sn isotopes iscomputed in Tamm-Dancoff and random-phase approximations using thecanonical Hartree-Fock-Bogoliubov quasi-particle basis. The calculations areperformed using an intrinsic Hamiltonian composed of a V lowk potential, deducedfrom the CD-Bonn nucleon-nucleon interaction, corrected with phenomenologicaldensity dependent and spin-orbit terms. The effect of these two pieces on energiesand multipole responses is discussed. The problem of removing the spuriousadmixtures induced by the center of mass motion and by the violation of thenumber of particles is investigated. The differences between the two theoreticalapproaches are discussed quantitatively. Attention is then focused on the dipolestrength distribution, including the low-lying transitions associated to the pygmyresonance. Monopole and quadrupole responses are also briefly investigated. Adetailed comparison with the available experimental spectra contributes to clarifythe extent of validity of the two self-consistent approaches. ‡ present address : Ecole Normale Superieur (ENS) de Cachan, 61 av. du Pr´esident Wilson, Cachan- France elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential
1. Introduction
During the last decades, the quasi-particle random-phase approximation (QRPA)calculations using a self-consistent Hartree-Fock-Bogoliubov (HFB) basis were rootedin the density functional theory. The energy density functionals were usually derivedfrom Skyrme [1, 2, 3, 4, 5, 6, 7, 8] or Gogny forces [9, 10] and from solving relativisticmean field equations [11]. A more exhaustive list of references can be found inRefs.[12, 13]. These approaches were adopted with success to study bulk propertiesof nuclei and nuclear responses of different multipolarities [5, 11].An alternative route based on the use of realistic nucleon-nucleon (NN) potentialswas attempted recently by Roth and coworkers [14, 15]. They applied first [14] afully self-consistent HFB formalism [16] to an intrinsic Hamiltonian using a correlatedUCOM effective interaction [17] derived from the Argonne V18 interaction [18].In a second step [15], the same authors adopted a HFB canonical basisto solve the eigenvalue problem within the framework of QRPA. They usednew effective interactions derived from the same Argonne V18 NN potential bymeans of the similarity renormalization group (SRG) approach [19] and added aphenomenological density dependent term simulating a three-body contact interaction.This phenomenological term was of crucial importance for generating single particlespectra in qualitative agreement with the empirical ones.They showed that the self-consistent QRPA calculation yields a completedecoupling of the center of mass (CM) and number operator spurious states if thesingle particle states are expanded in a sufficiently large harmonic oscillator space.Moreover, their systematic study of the nuclear response for various multipolarities hasproduced some encouraging results. The description of the giant resonances resultedto be of quality comparable to other more phenomenological QRPA studies. Moreover,it was possible to associate the low-lying transitions to the Pygmy modes.Several deviations from experiments still remain. One source of thesediscrepancies is the insufficient spin-orbit splitting resulting from the HFB solutions.As in Ref. [14], we have generated a canonical HFB quasi-particle (qp) basis.We started with using a V lowk potential derived from the CD-Bonn NN interaction[20]. Such a potential was introduced more than a decade ago [21]. It smooths outthe original N N potential and, therefore, is suited for HF and HFB calculations. Itsintimate connection with the renormalization group approaches was pointed out [22].The potential has been used extensively in shell model spectroscopic studies [23].We, then, added to V lowk the same phenomenological density dependent potentialused in Ref. [15] in order to reduce the too large spacing between qp levels producedby the V lowk potential.We further included a phenomenological spin-orbit term. The latter piece ismeant to enhance the splitting between spin-orbit partners and, thereby, obtain abetter detailed agreement with the empirical single particle spectra.The canonical HFB basis was then adopted to solve the eigenvalue problemin QRPA and within an upgraded qp Tamm-Dancoff approximation (QTDA). Thisupgraded QTDA adopts a basis of states orthogonalized by Gramm-Schmidt to the CMand number operator spurious states within the two quasi-particle space. Althoughmore involved than standard QTDA, it allows to eliminate exactly the spuriousadmixtures induced by either the CM motion or by the violation of the number ofparticles, without being forced to use spaces of huge dimensions.The comparative analysis of QTDA and QRPA calculations gives us the elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential E Pb[27]. Thus, generating the QTDA phonons in a HFB basis represents the first step fora fully self-consistent calculation to be carried out within the EMPM.We will consider different multipole responses within both QTDA and QRPA forsets of neutron rich O and Sn isotopes. We will concentrate mainly on the structureof the giant dipole resonance (GDR) and pay attention also at the low-energy dipoletransitions near the neutron decay threshold. These excitations have been extensivelystudied in the lasts decades and tentatively interpreted as manifestation of the pygmydipole resonance (PDR) promoted by a translational oscillation of the neutron skinagainst a N = Z coreThe first tentative experimental evidence of the PDR was gained in Coulombexcitation experiments on neutron rich oxygen isotopes [28, 29, 30]. The nature of theobserved low-lying peaks was also investigated theoretically in shell model [31] as wellas in a QRPA plus phonon-coupling model [32].The low-lying dipole excitations were investigated extensively also in Sn isotopesin several experiments [33, 34, 35] and several theoretical approaches [36, 37, 38, 39,40]. A review may be found in Refs. [25, 41].It is to be pointed out that the purpose of the present work is not to compete withthe approaches based on energy density functionals, but rather to try to identify thephenomenological terms which are to be added to an effective potential derived froma realistic N N interaction in order to obtain a description of the nuclear response infair agreement with experiments. The identification of these terms may give usefulhints on how realistic NN potentials should be modified in order to be usefully spentfor reliable studies of the nuclear response. The ultimate goal is, in fact, to performnuclear spectroscopy calculations based on the exclusive use of these potentials.
2. Hartree-Fock-Bogoliubov
The HFB transformation is defined [16] as β † r = X s (cid:16) U sr c † s + V sr c s (cid:17) ,β r = X s (cid:16) U ∗ sr c s + V ∗ sr c † s (cid:17) . (1)It transforms the creation and annihilation particle operators c † s and c s , respectively,into the corresponding quasi-particle operators β † r and β r . Its coefficients fulfill theconditions U † U + V † V = I, U U † + V ∗ V T = I, (2) U T V + V T U = 0 , U V † + V ∗ U T = 0 , (3)which ensure that the anticommutation relations are preserved. elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential | i . Let us consider a Hamiltonian composed of a kinetic term T and a two-bodypotential V . Its ground state expectation value E = h | H | i = h | ( T + V ) | i (4)is a functional of the the density matrix and the pairing tensor defined respectively as ρ rs = h | c † s c r | i = ( V ∗ V T ) rs , (5) κ rs = h | c s c r | i = ( V ∗ U T ) rs . (6)By performing a variation of E with respect to ρ and κ , under the constraint trρ = N (7)ensuring that the number N of particles is conserved on average, one obtains the HFBeigenvalue equations (cid:18) h − λ ∆ − ∆ ∗ − h ∗ + λ (cid:19) (cid:18) UV (cid:19) = E qp (cid:18) UV (cid:19) , (8)where h rs = t rs + X tq V rtsq ρ qt , (9)∆ rs = 12 X qt V srtq κ qt . (10)These are a set of non linear equations to be solved self-consistently.For practical purposes it is convenient to adopt the canonical basis. As shown inRef. [16], in such a basis the density matrix ρ is diagonal with eigenvalues v r , whilethe coefficient u r is deduced from v r through the normalization conditions (2) whichbecome u r + v r = 1 . (11)One can define the qp energy E r = p ( ǫ r − λ ) + ∆ r , (12)where ǫ r = h rr and ∆ r = ∆ r ¯ r . The chemical potential λ is fixed by the numberconserving condition (7), which in the canonical basis becomes X r v r = N. (13)The Bogoliubov coefficients assume the BCS-like expressions u r = 12 (cid:18) ǫ r − λE r (cid:19) , v r = 12 (cid:18) − ǫ r − λE r (cid:19) . (14)It is to be pointed out that the energies E r are the diagonal matrix elements of theHFB Hamiltonian in the canonical basis. They do not coincide in general with the qpHFB eigenvalues E qp obtained from solving the HFB equations (8). elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential
3. QTDA and QRPA in the HFB canonical basis
When expressed in terms of the canonical qp operators α † r and ¯ α r = ( − ) j r + m r α rj r − m r ,the starting Hamiltonian becomes H = E + H + V , (15)where E is the HFB ground state energy, H is a one-body qp Hamiltonian, and V a two-body potential describing the interaction among quasi-particles. The one-bodypiece in the angular momentum coupled scheme has the expression H = X rs [ r ] / E rs [ α † r × ¯ α s ] , (16)where [ r ] = 2 j r + 1 and the symbol × denotes coupling of two tensor operators toangular momentum Ω. The other quantity is E rs = ( ǫ rs − λδ rs )( u r u s − v r v s ) + ∆ rs ( u r v s + v r u s ) , (17)where ǫ rs = t rs + Γ rs (18)and Γ rs = 1[ r ] / X t [ t ] / F rstt v t , (19)∆ rs = 12 1[ r ] / X t [ t ] / V rstt u t v t (20)are the Hartree-Fock and pairing potentials, respectively. We have introduced thequantity F νrtsq = X Ω ( − ) r + q − ν − Ω W ( rstq ; Ω ν ) V Ω rstq , (21)where W is a Racah coefficient.It is to be pointed out that the one-body qp Hamiltonian does not have thestandard diagonal structure as it would have been the case, had we used the full HFBbasis.The quasi-particle two-body piece can be written in the synthetic form V = − X rstq Ω [Ω] / V Ω rstq : h ( c † r × c † s ) Ω × (¯ c t × ¯ c q ) Ω i : , (22)where V Ω rstq are unnormalized antisymmetric two-body matrix elements and : . . . :denotes normal order with respect to the HFB qp vacuum. The QRPA creation operator is O † λ = X r ≤ s h Y λrs A † rsλ + Z λrs ¯ A rsλ i , (23)where A † rsλ = ζ rs (cid:16) α † r × α † s (cid:17) λ , ¯ A λrs = − ζ rs (cid:16) ¯ α r × ¯ α s (cid:17) λ , (24) elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential ζ ab = (1 + δ ab ) − / . The QTDA operator is obtained by putting Z = 0.The QRPA eigenvalue equations are derived by standard techniques (cid:18) A λ B λ − B λ ∗ − A λ ∗ (cid:19) (cid:18) Y λ Z λ (cid:19) = ω λ (cid:18) Y λ Z λ (cid:19) , (25)where ω λ = E λ − E . The block-matrices are defined as A λrstq = h | [ A rsλ , [ H, A † tqλ ]] | i , (26) B λrstq = h | [ A rsλ , [ H, A tqλ ]] | i . (27)The block-diagonal matrix A is the QTDA Hamiltonian matrix. It is composed of twopieces A λrstq = ζ rs ζ tq h h ( r × s ) λ | H | ( t × q ) λ i + h ( r × s ) λ | V | ( t × q ) λ i i . (28)The first piece is h ( r × s ) λ | H | ( t × q ) λ i = δ sq E rt + δ rt E sq − ( − ) r + s − λ h δ st E rq + δ rq E st i . (29)It is to be noticed that the above matrix element is non diagonal as it would be thecase if computed in the HFB basis.The second term is the standard QTDA two-body matrix element h ( r × s ) λ | V | ( t × q ) λ i = V λrstq (cid:16) u r u s u t u q + v r v s v t v q (cid:17) + (30) F λrstq (cid:16) u r v s v t u q + v r u s u t v q (cid:17) − ( − ) t + q − λ F λrsqt (cid:16) u r v s u t v q + v r u s v t u q (cid:17) . The off-diagonal block, entering the QRPA matrix only, is given by B λrstq = ζ rs ζ tq h V λrstq (cid:0) u r u s v t v q + u t u q v r v s (cid:1) (31) − F λrstq (cid:0) u r v s u t v q + u s v r u q v t (cid:1) + ( − ) t + q − λ F λrsqt (cid:0) u r v s u q v t + u s v r u t v q (cid:1)i . The expression of the A matrix written above is valid for standard QTDA. As weshall see in Sect. 4.3, A becomes more involved in our upgraded QTDA approach,in which the eigenvalue problem is formulated within a space spanned by states, freeof spurious admixtures, which are linear combinations of two quasi-particle states.Consequently, also the two quasi-particle expansion coefficients of the eigenfunctionsof this modified A matrix have a more complex structure with respect to the standardQTDA wavefunctions. The transition amplitude for a generic multipole operator is < λ k M λ k > = X r ≤ s < r k M λ k s > ζ rs h u r v s + ( − ) λ u s v r ih Y ( λ ) ∗ rs − ( − ) λ Z ( λ ) ∗ rs i . (32)The standard Eλ multipole operator is M ( Eλµ ) = e A X i =1 (1 − τ i ) r λi Y λµ ( ˆ r i ) = M ( Eλµ ) + M ( Eλµ ) , (33) elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential τ = 1 for neutrons and τ = − M τ ( Eλµ ) are the isoscalar( τ = 0) and isovector ( τ = 1) components.In order to remove partially the spurious admixtures induced by the center ofmass (CM) motion we will use the modified isovector E M ( E µ ) = e NA Z X p =1 r p Y µ ( ˆ r p ) − e ZA N X n =1 r n Y µ ( ˆ r n ) (34)obtained by subtracting the contribution of the CM operator. This modification isneeded only for the QRPA. As we shall see, it is irrelevant to the our version of QTDAwhose states are completely free of spurious admixtures.The strength function is S ( Eλ, ω ) = X ν B ν ( Eλ ) δ ( ω − ω ν ) ≈ X ν B ν ( Eλ ) ρ ∆ ( ω − ω ν ) . (35)Here ω is the energy variable, ω ν the energy of the transition of multipolarity Eλ fromthe ground to the ν th excited state of spin J = λ and ρ ∆ ( ω − ω ν ) = ∆2 π ω − ω ν ) + ( ∆2 ) (36)is a Lorentzian of width ∆, which replaces the δ function as a weight of the reducedstrength B ν ( Eλ ) = |h ν, λ k M ( Eλ ) k i| . (37)
4. Calculations
We consider an intrinsic Hamiltonian obtained by subtracting the CM kinetic energy T CM from the shell model kinetic operator. The new kinetic term is therefore T int = 12 m X i p i − T CM = T + T , (38)where T = (1 − A ) 12 m X i p i (39)is a modified one-body kinetic term and T = − mA X i = j ~p i · ~p j (40)is a two-body piece which will be incorporated into the potential V . The fullHamiltonian is H = T + V, (41)where V = V so + V + V ρ . (42)Here V = V lowk + T (43) elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential T and a V lowk potential[21] deduced from the NN CD-Bonn force [20] with a cut-off Λ = 1 . f m − . We havegenerated its matrix elements in the harmonic oscillator basis using the code of Ref.[42]. In addition to V , the potential V includes two phenomenological pieces, a spin-orbit term V so = C so X i ~l i · ~s i (44)plus a density dependent two-body potential V ρ = X i Sn. Theparameter of the density dependent potential was such as to reproduce on average thepeaks of the giant dipole resonance (GDR) over the sets of oxygen and tin isotopes.We obtained C ρ = 2600 MeV f m for oxygen and C ρ = 4200 MeV f m for tin. Thespin-orbit parameter was tuned so as to approach the empirical neutron spin-orbitseparations given in Table II of Ref. [15], ∆ ǫ so = 6 . 18 MeV for O and ∆ ǫ so = 6 . Sn. We obtained C so = − . C so = − . The selective effects of the different pieces of the Hamiltonian are investigated for Oand Sn. In these doubly magic nuclei, HFB turns into HF and, therefore, yieldssingle particle energies directly comparable to the empirical ones. This is not the casefor the canonical basis in open shell nuclei.As illustrated in Figures 1 and 2, the density dependent potential V ρ produces anoverall drastic compression of the HFB spectrum resulting from the use of V only, inline with the calculation of Ref. [15].The spin-orbit piece, by increasing the splitting between spin-orbit partners, has afine tuning effect. It pushes down in energy the spin-orbit intruders, thereby enlargingthe gap between major shells, and compresses further the levels within each majorshell, consistently with the empirical single particle spectra.In tin isotopes, however, the levels within a major shell are not sufficiently packedas the empirical spectra would require. This shortcoming does not allow an accuratedetailed description of the positive parity low-lying QTDA or QRPA spectra. Thus,we will focus on the dipole transitions and on the global features of monopole andquadrupole responses. elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential Figure 1. Neutron single particle spectra in O (a) and Sn (b) with i) V ,ii) V + V ρ . The empirical (exp) single particle levels are taken from [45] Figure 2. Neutron single particle spectra in O (a) and Sn (b) with ii) V + V ρ , iii) V + V ρ + V so . The empirical single particle levels are taken from [45] The effect of the reduced level spacing, induced mainly by V ρ ,, gets especiallymanifest in the QRPA E E V lowk plus the two-body kinetic term T are used. It is shifted downward to the experimental region by the density dependentpotential V ρ . The spin-orbit piece induces only a slight redistribution of the strength.An identical result is obtained in QTDA. As we shall see later, the E In a fully self-consistent QRPA, the 1 − and 0 + spurious states lie at zero excitationenergy and collect the total strength induced by the CM and the number operators,respectively. elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential Figure 3. RPA E O ((a) and (b)) and Sn ((c) and (d)) with V only and V + V ρ ((a) and (c)), with V + V ρ and V + V ρ + V so ((b) and (d)).The centroids deduced from the experiments for O and Sn are, respectively, ∼ . ∼ . ∼ . ∼ . Numerically, their complete decoupling from the physical intrinsic states isachieved only if a sufficiently large configuration space is adopted. This was thecase of Ref. [15], where 15 major oscillator shells were considered in order to generatethe HFB basis.Having adopted a smaller space, which includes up to 12 major oscillator shells,we do not achieve such a complete separation. The energies of the lowest 1 − and 0 + QRPA spurious states are imaginary with ℑ ( ω − ) = 0 . 18 MeV and ℑ ( ω +1 ) = 1 . + .The strength distributions of the isoscalar (CM) M ( E µ ) (Eq. (33)) and of thenumber n op operators are shown in Figs. 4. The isoscalar E − states. Some of them, especially at low energy, collect upto ∼ − e f m . The strength of the number operator is spread over the whole 0 + spectrum in O. It is concentrated at low energy in Sn.Thus, we do not obtain a complete decoupling of the spurious 0 + and 1 − states asachieved in Ref. [15]. The main reason is to be found in the more restricted harmonicoscillator space we adopted (12 major shells versus 15). It is, however, to be pointed elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential Figure 4. Strengths of the transitions induced by the CM (a) and (b) and bynumber operators (c) and (d) in O and Sn. out that a different potential was used and a different treatment of its short rangerepulsion was made in Ref. [15]. This difference might affect the efficiency in theprocedure of eliminating the spurious admixtures. In any case, the use of effectivecharge in the E − is slightly negative. We get for instance ω CM = − . 18 MeV in Oand ω CM = − . 640 MeV in Sn. In both nuclei, the state collects ∼ − 99% ofthe CM coordinate strength. The residual strength contaminating the physical stateis ∼ . 8% in O and ∼ . 5% in Sn. This spurious strength is distributed mainlyamong low energy states, as illustrated in the same Fig. 4. Some of the low-lyingQTDA states get an isoscalar E E + spectrum is stronger. The energy of the lowest 0 + is ω +1 = − . 226 MeV in O and ω +1 = − . 429 MeV in Sn. The number operatorstrength collected by the lowest 0 +1 is ∼ 95% in O and ∼ . 89% in Sn. Theresidual spurious strength is therefore larger, ∼ . 4% in O and ∼ 11% in Sn. Itspreads over the whole spectrum in O. It is instead concentrated mainly in a lowenergy peak in Sn. elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential | i i , has the expression | Φ i = 1 N (1) n X i =1 C i | i i , (48)where N (1) = n X i =1 | C i | (49)is the normalization constant. The orthogonalized states have the expression | Φ k − i = 1 N ( k − N ( k ) h N ( k − | k − i − X i = k,n C k − C i | i i i , (50)where ( k = 2 , , ..n ) N ( k ) = X i = k,n | C i | . (51)For k = n the sum P i = k,n disappears. So we have simply | Φ n − i = 1 N ( n − N ( n ) h N ( n − | n − i − C n − C n | n i i , (52)where N ( n ) = | C n | . (53)The CM spurious state (cid:16) λ = ( κ , − ) (cid:17) is | λ i = 1 N R µ | i , (54)where R µ is the CM coordinate and N the normalization constant. Expanded in thetwo quasi-particle basis states, it acquires the structure | λ i = 1 N R µ | i = 1 N X r ≤ s C λ rs | ( r × s ) − i , (55)where C λ rs are the unnormalized coefficients C λ rs = r π A h r k rY k s i ( u r v s − u s v r ) (56)and the normalization factor is given by N = X r ≤ s | C λ rs | . (57)Similarly, the number operator spurious state ( λ = ( κ , + )) is obtained by applyingthe number operator in normal order to the HFB vacuum. We get | λ i = 1 N X r C λ rr | ( r × s ) + i , (58) elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential C λ rr ( ν ) are the unnormalized coefficients C λ rr = p r ]( u r v r ) (59)and N is the normalization factor N = X r | C λ rr | . (60)The basis states obtained by the just outlined procedure are adopted to construct theHamiltonian matrix {h Φ r | H | Φ s i} . Its diagonalization yields eigenstates rigorouslyfree of spurious admixtures either induced by the CM excitation or by the violationof the particle number. The price we pay is that the Hamiltonian matrix has a moreinvolved structure and the eigenstates are given in term of the orthogonalized states | Φ r i . Their QTDA standard structure is finally recovered by using Eqs. (50) and(52) to express the states | Φ r i in terms of the two quasi-particle states.The orthogonalization procedure eliminates the negative energy spurious 1 − stateresulting from the diagonalization of the Hamiltonian in the space spanned by theunmodified two quasi-particle basis states.It has the additional important effect of removing the residual spuriousness fromthe remaining physical states, thereby modifying the E O and Sn. Figure 5. Strength distribution of the E O (a) and Sn (b). elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential Figure 6. QTDA versus QRPA E O (a) and Sn (b). 5. QTDA and QRPA E1 response We apply both QTDA and QRPA to the study of the E σ int = Z EE σ ( ω ) dω = 16 π e hc Z EE ω S ( E , ω ) dω. (61)where S ( E , ω ) is the E S ( E , ω ) is evaluated by means of the modified E S ( T RK ) = X n ω n B n ( E 1) = ¯ h m π N ZA e . (62)We have in fact σ int = 16 π hc S ( T RK ) = (2 π ) ¯ h m e ¯ hc N ZA = 60 N ZA ( M eV mb ) . (63)For some specific nuclei we have also computed the isoscalar dipole transitionstrength using the operator M IS ( λ = 1 , µ ) = A X i =1 (cid:16) r i − < r > (cid:17) Y µ ( ˆ r i ) . (64)The second term in the bracket removes, partially, the CM contribution to the QRPAtransitions strength. This term is unnecessary in the improved QTDA used here. The cross section is computed by using a Lorentzian of width ∆ = 3 . E O and the neutron-rich − O isotopes. The theoretical spectra arecompared with the available data [46, 47, 48]. elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential Figure 7. Theoretical versus experimental E − O isotopes.The experimental data are taken from Ref. [46] for O and from [47, 48] forthe other isotopes. The theoretical cross sections were computed in QTDA andQRPA. One can hardly notice any difference between the two cross sections. Even theirintegrated values are close. In fact, the QTDA integrated cross section overestimatesthe TRK sum rule by a factor ∼ . 54 in O and ∼ . 48 in O close to thecorresponding QRPA factors ∼ . 43 and ∼ . 42, respectively. The violation of theTRK sum rule in QRPA, consistent with the results obtained in Ref. [15], is to beascribed to momentum dependent components of the two-body potential.The plots show important discrepancies between theory and experiments. In fact,the computed cross sections compare only qualitatively with the experimental data.In the doubly magic O, the theoretical peak lies ∼ ∼ 10 MeV above. Also the experimental cross sectionsexhibit two peaks, one at ∼ − 18 Mev and the other at ∼ − 26 MeV. The twopeaks, however, are of comparable height.Consistently with experiments, the theoretical cross section below ∼ 15 MeV isnegligible in the N = Z O, exhausting only 1 . 5% of the classical TRK sum rule.It exhausts, instead, a considerable fraction of the classical sum rule in the N > Z isotopes. This fraction is ∼ 10% in O, ∼ 15% in O, ∼ 13% in O. These valuesare only slightly larger than the quantities deduced from the data [28]. These low- elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential Figure 8. Distribution of isoscalar (a) and isovector (b) dipole reduced transitionstrengths in O. lying excitations were tentatively associated to the Pygmy resonance. Their collectivecharacter, however, has been questioned [25].Just to gain a closer insight into the structure and nature of the dipole spectrawe plot in Figure 8 the isovector and isoscalar dipole strength distributions of O.We show only the QTDA spectra since the ones obtained in QRPA are very similar.As pointed out already, the QTDA states are completely free of spurious admixtures.Thus, the CM corrective terms in the isoscalar (Eq. (64)) and isovector (Eq. (34))operators are irrelevant.The low-lying isovector E ω − = 7 . 46 MeV, almost at the neutron decay threshold (7 . 608 MeV), andanother well above the neutron decay threshold at ω − = 12 . − states are B ( E 1) = 0 . e f m and B ( E 1) = 0 . e f m respectively.Experimentally, two E − E elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential P B ( E ∼ . e f m .Thus, almost the total low-energy E E − is strongly excited also by the isoscalar operator(64). This is, actually, the strongest isoscalar dipole peak. The rest of the strength isdistributed almost uniformly in the whole energy interval.It may be of interest to have a quick look at the wave functions. The lowest 1 − state of energy ω − = 7 . 46 MeV is built almost entirely of neutron excitations. In factthe neutron component represents the 94% of the state. The dominant neutron quasi-particle components are | (1 p / ) n (0 d / ) n i and | (1 p / ) n (1 s / ) n i which account for43% and 34%, respectively, of the total wavefunction.These configurations describe excitations of valence neutrons and, therefore,emphasize the role of the neutron skin, suggesting strongly the pygmy nature of thelow-lying transitions. The other strongly excited state at ω − = 12 . | (0 f / ) n (0 d / ) n i witha weight of 66%.We have also evaluated the proton and neutron weights of the 1 − state of energy ω − = 21 . 54 MeV, well in the region of the GDR. The neutron component of this stateis 61%, still dominant but not overwhelming. Its largest components are the neutronconfiguration | (0 d / ) n (0 p / ) n i with a weight 53% as well as the proton states | (0 d / ) p (0 p / ) p i and | (1 s / ) p (0 p / ) p i with weights 19% and 13%, respectively.These configurations describe excitations of protons and neutrons from the core and,therefore, qualify the peak as a member of the GDR. QTDA and QRPA cross sections for some typical Sn isotopes are plotted in Figure 9.They were computed using a Lorentzian width ∆ = 2 . ∼ . 54 in both Sn and Snisotopes, close to the QRPA corresponding values ∼ . 46 and ∼ . 47, respectively.In , Sn the computed cross section is ∼ ∼ . ∼ 5% in Sn, ∼ 4% in Sn, ∼ 6% in Sn, ∼ 4% in Sn. These fractions are consistent with the experimental ones, which are7(3)% and 4(3)% in Sn and Sn, respectively [33].In order to elucidate the nature of these low-lying excitations we plot for Snthe strength distribution of the QTDA isovector and isoscalar dipole transitions.As shown in Figure 10, at low energy we have two 1 − states strongly excited by theisovector E ω − = 7 . 49 MeV which carries a strength B ( E 1) = 1 . elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential Figure 9. Theoretical versus experimental E , Sn and from [47, 48] for , Sn e f m and the other at ω − = 9 . 01 MeV collecting a strength B ( E 1) = 1 . e f m .The summed strength up to ω ∼ P n B n ( E 1) = 3 . e f m , close to theexperimental value P n B ( exp ) n ( E 1) = 3 . e f m [33]. Similarly, for Sn, the summedstrength up to ω ∼ P n B n ( E 1) = 2 . e f m in agreement with the measured P n B ( exp ) n ( E 1) = 1 . e f m .Let us now look at the wave functions in Sn. The 1 − excitation at ω − =7 . 49 MeV has the properties of a pygmy mode. Indeed, the neutron component | (2 p / ) n (2 s / ) n i accounts for 85% of this state and the total neutron weight is95%.The state at ω − = 9 . 01 MeV has a more complex and collective structure.Neutrons and protons contribute with a comparable weight, 49% and 51% respectively.The dominant proton components are | (1 d / ) p (1 p / ) p i and | (1 d / ) p (1 p / ) p i withrespective weights 22% and 23%. The dominant neutron states are | (2 p / ) n (1 d / ) n i and | (2 p / ) n (1 d / ) n i with weights 28% and 14%, respectively.The 1 − state of energy ω − = 13 . 13 MeV belonging to the group of GDR peakshas a collective nature with a neutron dominance (68%). Its largest proton componentis | (0 h / ) p (0 g / ) p i with a weight 23%. The dominant neutron configurations are | (1 f / ) n (0 g / ) n i and | (0 i / ) n (0 h / ) n i with weights 27% and 15%, respectively. elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential Figure 10. Distribution of the isoscalar (a) and isovector (b) dipole reducedtransition strengths in Sn. Figure 10 shows that the isoscalar dipole spectrum not only overlaps with theisovector one but covers a larger region. Indeed, the isoscalar probe excites stronglythe states in the ∼ hω region describing a compressional mode as well as the ones inthe low-energy region associated to the pygmy resonance.The coexistence of an isovector and and isoscalar spectrum at low energy isconsistent with a recent experiment [35]. This experiment has detected a very densespectrum in the ∼ . − . γ, γ ′ ) reaction, is of isovector nature, the other, observed in elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential Figure 11. Theoretical versus experimental monopole strength functions in Sn (a) and Sn (b). The theoretical strength functions were computed inQTDA and QRPA. The experimental data are taken from [50, 51] Figure 12. QRPA versus QTDA E Sn (a)and Sn (b) ( α, α ′ γ ), is isoscalar. All the isovector E B ( E ∼ − − − e f m and the integrated strength is P n B n ( E ∼ . e f m .Our calculation is unable to reproduce this highly fragmented strength. In orderto obtain such a rich spectrum it is necessary to go beyond QRPA by resorting forinstance to QPM [37] or the relativistic time blocking approximation (RTBA) [49]. 6. Monopole and quadrupole responses As mentioned already, the quasi-particle energies within a major shell are too far fromthe empirical ones to allow a detailed study of the positive parity spectra. Thus, wewill give only a brief account of the global properties of the monopole and quadrupoleresponses.To this purpose we have computed the strength functions (35) for Sn and Snusing a Lorentzian of width ∆ = 2 . E elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential Figure 13. QRPA versus QTDA isoscalar strength function S ( E 2) in Sn(a) and Sn (b) Figure 14. QRPA versus QTDA isovector strength function S ( E 2) in Sn(a) and Sn (b) by using the operator given by the standard formula (33) with the bare charges e p = 1and e n = 0. For the monopole transitions we used the operator M ( λ = 0) = A X i =1 r i Y ( ˆ r i ) . (65)As shown in Figures 11, the QRPA monopole strength function describes fairly wellthe experimental trend [50, 51] and is in fair agreement with the response evaluatedwithin a density functional approach [52]. The QTDA strength, instead, follows theevolution of the data but has a lower peak.Unlike the dipole case, appreciable differences between QTDA and QRPAresponses appear in the low energy sector. These difference are also clearly visiblein the E elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential E 7. Concluding remarks The present calculation has confirmed that the HFB equations, using only a realisticpotential V lowk deduced from the bare NN interaction, generate completely unrealisticquasi-particle spectra. These become compatible with experiments only if one addsto V lowk a density dependent two-body potential V ρ simulating a three-body contactforce, first adopted in Ref. [15].Although the improvement promoted by V ρ is impressive, appreciablediscrepancies between theory and experiments remain. The spin-orbit term added hereimproves the spectra by enhancing the spin-orbit splitting but is not able to reducesufficiently the distance between the quasi-particle configurations within major shellsas it would be necessary in order to fill the gap with the empirical data. This pointdeserves further investigation.The mentioned limitation, however, is expected to affect the low-energy spectraand to have a marginal impact of the multipole responses. We adopted both QTDAand QRPA to study mainly the dipole excitations. We discussed briefly also themonopole and quadrupole transitions.Appreciable differences between the QTDA and QRPA are noticeable only in thelow-energy sectors of the monopole and quadrupole spectra, while the dipole responsesare practically identical in the two approaches.These results indicate that the QRPA ground state correlations affect only theisoscalar modes. In particular, they seem to improve the description of the monopolestrength distribution as our comparative analysis has suggested.The dipole cross sections, whether computed in QTDA or QRPA, are in fairagreement with experiments in tin isotopes apart from a ∼ Pb [27]. The method has been already formulatedin the language of quasi-particles [53] and is therefore suited to the study of complexspectra in open shell nuclei. In fact, a HFB self-consistent EMPM calculation of thedipole response for the nuclei investigated here is under way.In summary, it is possible to improve drastically the QTDA or QRPA descriptionof the multipole nuclear strength distributions starting from a V lowk potential deducedfrom a realistic N N interaction, if one adds to V lowk a density dependent plus aspin-orbit corrective terms. This is just an ad hoc prescription and, therefore, notsatisfactory on theoretical ground. elf-consistent study of multipole response in neutron rich nuclei using a modified realistic potential N N interactions with accompanying three-body terms. The latterterms have been shown to play an important role in nuclear structure [55]. Thephenomenological density dependent term, so crucial for reducing the gap betweentheoretical and experimental spectra, may be one of the elements to be taken intoaccount in the fine tuning parametrization of the chiral three-body force. The spin-orbit corrective term might be one of the inspiring elements for a revision of theparameters of the N N tensor forces which is known to affect strongly the spin-orbitsplitting. This aspect was emphasized recently in a mean field model using a mesonexchange tensor potential [56] and a HF+BCS approach using a Skyrme force [57]. Acknowledgments One of the authors (F. Knapp) thanks for partial financial support the IstitutoNazionale di Fisica Nucleare (INFN) as well as the Czech Science Foundation (GrantNo. 13-07117S) and Charles University in Prague (project UNCE 204020/2012) References [1] Dobaczewski J, Flocard H and Treiner J 1984 Nucl. Phys. A Phys.Rev. C Phys. Rev. C Phys. Rev. C Phys. Rev.C Phys. Rev. Phys. Rev. C Phys. Rev. C Nucl.Phys. A Phys. Rev. C Phys. Rev. C Rev. Mod. Phys. Phys. Rep. Phys. Rev. C Phys. Rev. C The Nuclear Many-Body Problem (Berlin: Springer)[17] Feldmeier H, Neff T, Roth R and Schnack J 1998 Nucl. Phys. A Phys. Rev. C Prog. Part. Nucl. Phys. Phys. 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