Effects of density dependence of the effective pairing interaction on the first 2 + excitations and quadrupole moments of odd nuclei
S. V. Tolokonnikov, S. Kamerdzhiev, D. Voytenkov, S. Krewald, E. E. Saperstein
aa r X i v : . [ nu c l - t h ] A ug Effects of density dependence of the effective pairing interaction on the first + excitations and quadrupole moments of odd nuclei S. V. Tolokonnikov
Kurchatov Institute, 123182 Moscow andMoscow Institute of Physics and Technology, 123098 Moscow, Russia
S. Kamerdzhiev and D. Voytenkov
Institute for Physics and Power Engineering, 249033 Obninsk, Russia
S. Krewald
Institut f¨ur Kernphysik, Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany
E. E. Saperstein
Kurchatov Institute, 123182 Moscow
Excitation energies and transition probabilities of the first 2 + excitations in even tin and leadisotopes as well as the quadrupole moments of odd neighbors of these isotopes are calculated withinthe self-consistent Theory of Finite Fermi Systems based on the Energy Density Functional byFayans et al. The effect of the density dependence of the effective pairing interaction is analyzedin detail by comparing results obtained with volume and surface pairing. The effect is found to benoticeable. For example, the 2 + -energies are systematically higher at 200-300 keV for the volumeparing as compared with the surface pairing case. But on the average both models reasonably agreewith the data. Quadrupole moments of odd-neutron nuclei are very sensitive to the single-particleenergy of the state λ under consideration due to the Bogolyubov factor ( u λ − v λ ). A reasonableagreement with experiment for the quadrupole moments has been obtained for the most part of oddnuclei considered. The method used gives a reliable possibility to predict quadrupole moments ofunstable odd nuclei including very neutron rich ones. PACS numbers: 21.10.-k, 21.10.Jx, 21.10.Re, 21.60-n
I. INTRODUCTION
Presently there are two theoretical approaches whichcan quantitatively describe the bulk properties of nuclearisotope chains with a small number of effective couplingconstants: selfconsistent mean field theories and density-functional theory. The successes and open problems ofthe mean field approaches are reviewed in Refs. [1–3].The Kohn-Sham density functional theory was originallyproposed for chemistry and solids [4, 5]. Important theo-retical developments have been made: an extension of theHohenberg-Kohn theorem to pairing degrees of freedomby Oliveira, Gross, and Kohn allowed studies of super-fluids [6] and the generalization of functional theory tostudy excited states made it possible to investigate theelectromagnetic response of correlated electron materials[7]. In nuclear physics, a self-consistent Theory of Fi-nite Fermi Systems (TFFS) was derived by Khodel andSaperstein [8] on the basis of the TFFS by Migdal [9] sup-plemented with the many-body theory self-consistencyrelation [10] for the nucleon mass operator. As it wasshown in [11], the self-consistent TFFS for nuclei with-out pairing can be reformulated as a particular versionof the density functional method with a rather compli-cated ρ -dependence of the energy functional. It containsalso τ -dependent terms but with rather small strengthresulting for the effective mass in a small difference of | m ∗ n,p ( r ) − m | ≃ . m . In a series of articles by Fayans et al. [12, 13], (see also [14] and Refs. therein) the energydensity functional (EDF) method was generalized for su-perfluid nuclei. Just as in the original Kohn–Sham ap-proach, the identity m ∗ = m was imposed. A fractionalform of the density dependence for the central part ofthe normal component of the EDF was introduced. Thecoordinate dependence of it resembled that of [11] butthe functional form was much simpler making the self-consistent QRPA calculations easier. Note that a recentgeneralization of the Skyrme force in [15] contains a newterm with a density dependence resembling that in [14].In addition, the velocity dependent force in [15] is ratherweak leading to the effective mass m ∗ ≃ . m . Thus,the selfconsistent mean field approaches may eventuallyconverge with the density functional methods.The non-relativistic versions of the self-consistentmean field theories introduce three-nucleon forces whichare often expressed as a density dependent two-body in-teraction. In general, one assumes a fractional power ofthe density dependence. Recent advances in EffectiveField Theory open the possibility to connect the densityfunctional with the effective two- and three- nucleon sys-tems which are determined from two-nucleon scatteringand few-nucleon reactions. Reviews about the currentstatus of such attempts are given in Refs.[16, 17].The question arises whether the pairing interactionshould have an analogous dependence on the normal nu-clear density. Several studies derived pairing interactionsfrom free two-nucleon interactions. Baldo et al. solvedthe gap equation in semi-infinite nuclear matter [18], nu-clear slab [19], and finite nuclei [20, 21]. The Paris andArgonne v N N potentials were used, results being al-most identical. To make results more appropriate forpractical nuclear self-consistent calculations dealing withpairing in a model space, the pairing problem was treatedin a two-step way. The gap equation was solved in amodel space with limiting energy E = 30 ÷
40 MeVwith the use of the effective pairing interaction. The lat-ter is found in the subsidiary sub-space in terms of a free
N N potential. For all systems under consideration andthe two
N N potentials the effective pairing interactionfound is much stronger, up to ten times, at the surfacethan inside. The Milan group concentrated on the
Snnucleus, a traditional benchmark for the nuclear pairingproblem, and solved the gap equation starting from theArgonne v potential [22]. In addition to the free N N interaction, they included corrections due to exchangewith low-lying surface vibrations (“phonons”) [23] andhigh-lying excitations, mainly spin-dependent ones, [24].In the last article, a local 3-parameter density-dependenteffective pairing interaction is constructed for the modelspace with E = 60 MeV which reproduce approximatelyexact gap values. Qualitatively, it is similar to that de-scribed above. Without all corrections, it consists of astrong surface attraction and very weak attraction inside.Taking into account of the phonon exchange makes theinner interaction repulsive. At last, inclusion of the spin-dependent excitations makes the inner repulsion ratherstrong. Thus, the ab initio calculations of the effectivepairing interaction predict essential density dependencewith strong surface attraction.As an alternative to consideration of the gap equationwith complete realistic N N interactions, Bulgac and Yuused the fact that this equation depends mainly on thelow- k behavior of N N force which can be approximatedwith a rather simple analytical function. It helped todevelop a renormalization scheme for the gap equationwithout any cutoff in terms of zero-range interactionswith explicit coordinate dependence of the effective pair-ing interaction and to suggest an EDF for superfluid nu-clei [25, 26].The calculations by Fayans et al. employed both vol-ume pairing and surface pairing interactions. The bind-ing energies and the proton and neutron separation en-ergies were found to be insensitive to the type of pairingforce used. But the odd-even staggering of charge radiican be quantitatively reproduced only if the strong den-sity dependence of the pairing force is introduced [14].In this work, we investigate the excitation energies andtransition rates of the low-lying 2 + -states in spherical nu-clei with the aim to analyze the sensitivity of those ob-servables to the details of the pairing interaction. We willcompare two opposite limits, the “volume pairing” withdensity independent effective pairing interaction F ξ andthe case of the function F ξ with the surface dominance.The latter will be named for brevity the “surface pair- ing”. Several sets of calculations of these characteristicsof the first 2 + -excitations were carried out recently withinthe QRPA method with Skyrme force [27, 28] and withinthe Generator Coordinate Method with the Gogny force[29]. No systematic analysis of the density dependence ofthe pairing force was performed in these studies. Deal-ing with low-laying quadrupole excitations, it is naturalto include into analysis also quadrupole moments of oddnuclei which give test of static quadrupole polarization.In this paper, we use the EDF method [14] with thefunctional DF3-a [30]. In the latter the spin-orbit andeffective tensor terms of the original functional DF3 [13,14] were modified. All the QRPA-like TFFS equationsare solved in the self-consistent basis ( ε λ , ϕ λ ) obtainedwithin the EDF method with the functional DF3-a. II. BRIEF OUTLINE OF THE FORMALISM
For completeness, we describe shortly the EDF methodof [14] using mainly the notation of [31]. In this method,the ground state energy of a nucleus is considered as afunctional of normal and anomalous densities, E = Z E [ ρ n ( r ) , ρ p ( r ) , ν n ( r ) , ν p ( r )] d r. (1)The normal part of the EDF E norm contains the cen-tral, spin-orbit and effective tensor nuclear terms andCoulomb interaction term for protons. The main, cen-tral force, term of E norm is finite range with Yukawa-typecoordinate dependence. It is convenient to extract the δ ( r − r ′ )-term from the Yukawa function separating therest of D ( r − r ′ ) = 14 πr | r − r ′ | exp (cid:18) − | r − r ′ | r c (cid:19) − δ ( r − r ′ ) (2)to generate the “surface” part E s which vanishes in in-finite matter with ρ ( r ) = const . The Yukawa radius r c is taken the same for the isoscalar and isovector chan-nels. The “volume” part of the EDF, E v ( ρ ), is takenin [13, 14, 31] as a fractional function of densities ρ + = ρ n + ρ p and ρ − = ρ n − ρ p : E v ( ρ ) = C (cid:20) a v+ ρ f v+ ( x ) + a v − ρ − f v − ( x ) (cid:21) , (3)where f v ± ( x ) = 1 − h v1 ± x h v2 ± x . (4)Here x = ρ + / (2 ρ ) is the dimensionless nuclear den-sity where ρ is the density of nucleons of one kindin equilibrium symmetric nuclear matter. The factor C = ( dn/dε F ) − in Eq. (3) is the usual TFFS nor-malization factor, inverse density of states at the Fermisurface.To write down the surface term in a compact formsimilar to (3), the “tilde” operator was introduced in [31]denoting the following folding procedure: g φ ( r ) = Z D ( r − r ′ ) φ ( r ′ ) d r ′ . (5)Then we obtain E s ( ρ ) = C h a s+ ( ρ + f s+ ) ^ ( f s+ ρ + ) + a s − ( ρ − f s − ) ^ ( f s − ρ − ) i , (6)where f s ± ( x ) = 11 + h s ± x . (7)All the above parameters, a v ± , a s ± , h v1 ± h v2 ± , h s ± , are dimen-sionless.In the momentum space, the operator (2) reads D ( q ) = − ( qr c ) qr c ) . (8)In the small r c limit it reduces to D ( q ) = − ( qr c ) , andEq. (6) could be simplified to a Skyrme-like form pro-portional to ( ∇ ρ ) .The spin-orbit interaction reads F sl = C r ( κ + κ ′ τ τ ) [ ∇ δ ( r − r ) × ( p − p )] · ( σ + σ ) , (9)where the factor r is introduced to make the spin-orbit parameters κ, κ ′ dimensionless. It can be ex-pressed in terms of the above equilibrium density, r =(3 / (8 πρ )) / .In nuclei with partially occupied spin-orbit doublets,the so-called spin-orbit density exists, ρ τsl ( r ) = X λ n τλ h φ τ ∗ λ ( r )( σ l ) φ τλ ( r ) i , (10)where τ = n, p — is the isotopic index and averagingover spin variables is carried out. As it is well known, seee.g. [8], a new term appears in the spin-orbit mean fieldinduced by the tensor forces and the first harmonic ˆ g ofthe spin Landau–Migdal (LM) amplitude. We combinethose contributions into an effective tensor force or firstspin harmonic, F s = C r ( g + g ′ τ τ ) δ ( r − r )( σ σ )( p p ) . (11)In Table 1, we present all parameters of the normalpart of the EDF DF3-a we use. Note that the majorpart of these parameters is identical to the ones used inthe DF3 functional [14]. With one exception, all parame-ters for the central force part remained the same and onlythe spin-orbit and the first spin harmonic are changed ac-cording [30]. Application of the volume part (3) to equi-librium nuclear matter, with the equilibrium relation, i.e.vanishing pressure p ( ρ ) = ρ ∂ ( E /ρ ) /∂ρ , permits to findthe parameters a v+ , h v1+ and h v2+ in terms of the nuclear matter density ρ , the chemical potential µ , and thecompression modulus K = 9 dp/dρ . The asymmetry en-ergy parameter β yields a relation between the constants a v − , h v1 − and h v2 − . They are given in the upper half of Ta-ble 1. The radius r introduced above is shown insteadof ρ . The value used in Ref. [14] was recalibrated inRef. [30] to obtain a more accurate description of nu-clear charge radii [32]. One more remark should be madeconcerning the table. The “natural” TFFS normalizingfactor C = 2 ε / (3 ρ ) = 308 . correspondingto parameters of nuclear matter in the third column ofthe table differs from the one, C = 300 MeV fm , recom-mended in the second edition of the Migdal’s textbook onthe TFFS [33]. To make a comparison with other articleswithin the TFFS, we recalculated all the strength param-eters to the latter. It explains a small difference of somevalues in the second column in the table from the originalthose in [14]. An essential difference between DF3 andDF3-a functionals takes place for the “spin-dependent”sector in the bottom of the table. As we found in [30], thesecond one describes the spin-orbit splitting of doubletsbetter.The anomalous component of the EDF [14]reads E an ( r ) = X τ F ξ,ττ ( r ; [ ρ ]) | ν τ ( r ) | , (12)where the effective pairing interaction reads: F ξ = C f ξ = C (cid:16) f ξ ex + h ξ x / + f ξ ∇ r ( ∇ x ) (cid:17) . (13)The first two terms are similar to those in the TFFS[34, 35] or in the SHFB method [36]. The third in (13) isa new one introduced in [13]. In this paper we use an ap-proximate version of (13) with f ξ ∇ = 0. We will comparetwo models for nuclear pairing: the “volume” pairing( h ξ = 0) and the “surface” pairing where both the pair-ing parameters f ξ ex and h ξ are nonzero. One more remarkshould be made concerning the pairing problem. In theapproach [14] pairing was considered in the coordinaterepresentation explicitly, solving the Gorkov equationswith the method developed in Ref. [37]. However, itwas found that the results are practically equivalent tothose obtained within a more simple BCS-like scheme us-ing the gap ∆ λλ ′ = ∆ λ δ λλ ′ in a sufficiently large modelspace, ε λ < E max . The effective pairing interaction (13)for the BCS approximation is a little stronger than thatin the coordinate representation (at ≃ ÷ E max ). For the systematic calculations in thisarticle we use this simplified method of considering thepairing problem with E max = 36 MeV. We do not applythis method for nuclei close to the dripline for which thediagonal approximation doesn’t work [14].Within the TFFS, the response of a nucleus to the ex-ternal quadrupole field V exp ( iωt ) can be found in termsof the effective field. In systems with pairing correlations,equation for the effective field can be written in a com-pact form as ˆ V ( ω ) = ˆ V ( ω ) + ˆ F ˆ A ( ω ) ˆ V ( ω ) , (14) TABLE I: Parameters of the normal part of the EDFParameter DF3 [14] DF3-a [30] µ , MeV -16.05 -16.05 r , fm 1.147 1.145 K , MeV 200 200 β, MeV 28.7 28.7 a v+ -6.598 -6.575 h v1+ h v2+ a v − h v1 − h v2 − a s+ -11.4 -11.1 h s+ a s − -4.11 -4.10 h s − r c , fm 0.35 0.35 κ κ ′ g g ′ -0.123 -0.308 where all the terms are matrices. In the standard TFFSnotation [9], we have:ˆ V = Vd d , ˆ V = V , (15)ˆ F = F F ωξ F ωξ F ξω F ξ F ξω F ξω F ξω F ξ , (16)ˆ A ( ω ) = L ( ω ) M ( ω ) M ( ω ) O ( ω ) −N ( ω ) N ( ω ) O ( − ω ) −N ( − ω ) N ( − ω ) , (17)where L , M , and so on stand for integrals over ε of theproducts of different combinations of the Green function G ( ε ) and two Gor’kov functios F (1) ( ε ) and F (2) ( ε ). Theycan be found in [9] and we write down here only the firstof them which is of the main importance for us, L = Z dε πi h G ( ε ) G ( ε + ω ) − F (1) ( ε ) F (2) ( ε + ω ) i . (18)Isotopic indices in Eqs. (15-17) are omitted for brevity.In Eq. (16), F is the usual LM amplitude, F = δ E δρ , (19)whereas the amplitudes F ωξ = F ξω stand for the mixedsecond derivatives, F ωξ = δ E δρδν . (20) In the case of volume pairing, we have F ωξ = 0. Theexplicit form of the above equations and (18) is writtendown for the case of the electric ( t -even) symmetry wedeal with. A static moment of an odd nucleus can befound in terms of the diagonal matrix element h λ | V ( ω =0) | λ i of the effective field over the state λ of the oddnucleon.The effective field operator ˆ V ( ω ) has a pole in the ex-citation energy ω s of the state | s i under consideration,ˆ V ( ω ) = (cid:16) ˆ V ˆ A ( ω s )ˆ g s (cid:17) ˆ g s ω − ω s + ˆ V R ( ω ) . (21)The quantity ˆ g s has the meaning of the correspondingexcitation amplitude. It obeys the homogeneous coun-terpart of Eq. (14) and is normalized as follows [9], ˆ g +0 s d ˆ Adω ˆ g s ! ω = ω s = − , (22)with obvious notation.For excitation probabilities, it is more convenient touse the transition density operator which is conjugatedto ˆ g s , ˆ ρ tr0 s = ˆ A ˆ g s . (23)The explicit definition of the normal and anomalouscomponents of ˆ ρ tr0 s is as follows ρ tr(0)0 s ( r , r ′ ) = Z dε πi δG ( r , r ′ ; ε, ω s ) , (24) ρ tr(1 , s ( r , r ′ ) = Z dε πi δF (1 , ( r , r ′ ; ε, ω s ) . (25)The TFFS equation for transition densities for nucleiwith pairing correlations,ˆ ρ tr0 s = ˆ A ( ω s ) ˆ F ˆ ρ tr0 s , (26)is a complete analogue of the QRPA set of equations.Therefore we will often name it the QRPA equation. Thetransition density is normalized due to Eq. (22), and thetransition matrix element for the excitation of the state | s i with the external field V is given by M s = Z ˆ V ˆ ρ tr0 s ( r ) d r . (27) III. CHARACTERISTICS OF THE +1 EXCITATIONS
The formalism described in the previous Section wasused to describe 2 +1 states in two isotopic chains of semi-magic nuclei, lead and tin. We investigate both a puresurface and a pure volume version of pairing. More cal-culational details can be found in Ref. [14]. We usethe so-called developed pairing approximation. In par-ticular, we don’t make any corrections to particle non-conservation effects induced with the Bogolyubov trans-formation. Therefore in the vicinity of double magic nu-clei, the results should be considered as very approxi-mate. As it was found in [14], it is impossible to describeneutron and proton separation energies S n and S p for allnuclei, from calcium up to lead, with sufficient accuracyusing a fixed set of parameters in Eq. (13), the effectivestrength of the pairing interaction should be diminishedwith increasing nucleon number A.
180 185 190 195 200 205 210 2154681012 exp. DF3-a, surf. DF3-a, vol. HFB-17 S n ( M e V ) A Pb FIG. 1: (Color online) Neutron separation energies S n forlead isotopes. The volume pairing corresponds to ( f ξ ex = − . h ξ = 0), the surface one, to ( f ξ ex = − . h ξ = 0 .
100 105 110 115 120 125 130 1352468101214 exp. DF3-a, surf. DF3-a, vol. HFB-17 S n ( M e V ) A Sn FIG. 2: (Color online) Neutron separation energies S n fortin isotopes. The volume pairing corresponds to ( f ξ ex = − . h ξ = 0), the surface one, to ( f ξ ex = − . h ξ = 0 . In this paper, we limit ourselves to two long isotopic chains, the lead and the tin chains. Therefore we dealwith neutron pairing only. A short comment should bemade on the procedure of solving the pairing problem.No particle number projection procedure is used in ourcalculations, i.e. particle number is conserved only onaverage, corresponding to the chosen chemical potential µ for the kind of nucleons under consideration. The accu-racy of this approximation is examined in a lot of papers.For the self-consistent SHF method with the SLy4 force,it was found in recent article [39] that the average differ-ence between exact and approximate gap values is 0.12MeV, the error being bigger in vicinity of magic nuclei.For finding the parameters of the pairing force (13)we use the strategy of “soft” variation of them to obtainbetter values of S n for both the chains under considera-tion. Values of S n for both kinds of pairing are comparedwith the data in Fig. 1 and Fig. 2. Explicit values ofthe pairing parameters are given in the figure captions.Remind that we use the two-parameter version of (13),with f ξ ∇ = 0. For the volume pairing ( h ξ = 0), one pa-rameter remains which is smaller for lead than for tinapproximately at 6%. For the surface pairing we dealwith a two-parameter form of F ξ . The “external” pair-ing parameter f ξ ex is taken A -independent, in accordancewith its physical meaning as the free NN T -matrix takenat negative energy E = 2 µ [18]. As to the second one, h ξ , it increases from the Sn chain to the Pb one at 2%,the resulting pairing attraction again becoming weaker,but only a little. Thus, the A -independence of the pair-ing parameters in the case of surface pairing is weakerthan for the volume one. This finding suggests to fa-vor surface pairing. As we see, the difference betweenthe predictions for neutron separation energies is smallfor both versions and agreement with the experimentaldata is nearly perfect. For comparison, we display thepredictions of the HFB-17 version of the Skyrme force[36] which has a record accuracy in overall descriptionof nuclear masses. We see that for these two chains ouraccuracy in description of neutron separation energies iseven better. Of course, we achieved the agreement by asmall variation of one of two paring parameters whereascalculations [36] are carried out with an universal set ofparameters. However, the pairing part of the HFB-17functional contains five parameters.Fig. 3 demonstrates that the normal neutron density ρ n ( r ) and the anomalous one, ν n ( r ), both are practi-cally insensitive to the kind of paring used in the calcu-lation. On the contrary, the gap itself is very sensitive.For comparison, we took also a “medium” version, with( f ξ ex = − . h ξ = 0 . S n value approxi-mately with the same accuracy as the previous two.Let us now examine to what extent predictions forcharacteristics of 2 +1 states are different for these twoversions of pairing force which are equivalent in describ-ing the S n values. A comment should be made be-fore presenting results of the QRPA calculations. OurQRPA code doesn’t include the spin-orbit (9) and spin(11) terms of the effective interaction, therefore the self- ( M e V ) b ( f m - ) a r (fm) ( - f m - ) c FIG. 3: (Color online) Neutron density (pannel a), gap (b)and anomalous density (c) in
Pb nucleus. Solid and dot-ted lines correspond to the surface and volume versions cor-respondingly, the dashed one, to the medium version ( f ξ ex = − . h ξ = 0 . consistency is not complete and the excitation energy ofthe ghost 1 − -state does not automatically vanish. In thepresent investigation, we fine tuned the parameter a v+ in(3) in order to decouple the ghost state. This change isdifferent for different nuclei but on average the value of | a v+ | increases at ≃
3% in comparison with that given inTable 1.Let us begin with the lead chain. Excitation ener-gies ω are displayed in Fig. 4 and the probabilities B ( E , up), in Fig. 5. Experimental data for both quan-tities are taken from [40]. For comparison, results ofthe QRPA calculations of [27] with the SkM* and SLy4force are shown. Note that they were carried out withdensity independent pairing. We see that the difference δω = ω vol2 − ω surf2 is, on average, of the order of 0.3MeV, that is the effect under discussion is noticeable forthis quantity. The results for volume pairing are sys-tematically higher, with the exception of the , Pbisotopes for which the two versions practically coincide.Agreement with the data is, on average, quite reason-able for both the versions. Predictions of both the SkM*and SLy4 QRPA calculations for ω values have approx-imately the same accuracy as ours. For excitation probabilities the situation is more com-plex. For isotopes heavier than Pb our “surface” and“volume” curves are very close to each other. For lighterpart of the chain the volume pairing generates largerprobabilities than surface pairing does, producing differ-ences up to ≃ L -vibration: ρ tr , BM L = α L dρdr , (28)where α L = 1 / √ ω L B L , and B L is the collective massparameter of the BM model proportional to the nuclearmass. Then one obtains B ( EL, up) = 2 L + 12 ω L B L ( M L ) , (29)where M BM L = (3 Ze/ π ) R L − , R being the nuclear ra-dius. Thus, in the BM model a lower value of the excita-tion energy ω L inevitably leads to a higher value of theexcitation probability. In our calculations, the situationis opposite. In principle, this is not strange. Indeed, evenin magic nuclei the BM model works only qualitatively[8]. If one solves equations of the self-consistent TFFS orany HF+RPA equations for nuclei without pairing, Eq.(28) remains approximately true, but the mass parameterbecomes ω -dependent and deviates from the BM modelprescription significantly [8]. In nuclei with pairing, thesituation becomes even more different from this simplestmodel as the normal component of the transition density(24) depends now from the anomalous transition ampli-tudes ˆ g (1 , s (see Eq. (26)). They strongly depend onthe kind of pairing. As a result, the correlation betweenthe ω L and B ( EL ) values of the BM-type (29) can bedestroyed.Experimental probabilities are known only for foureven − Pb isotopes. For all of them, the SkM* andSLy4 calculations are in perfect agreement with the data.Agreement of our calculations is poorer. It is especiallytrue for the magic
Pb nucleus where there is no anypairing. It should be noted that in this nucleus the collec-tivity of the 2 +1 -state is not high: the B(E2) value is onlyabout 8 single-particle units (spu). For a comparison,the B(E3) value for the 3 − -state exceeds 30 spu. But forexcitations with low collectivity in nuclei without pairingthe RPA solution depends strongly on the single-particlespectrum, and even a small inaccuracy in the positionsof single-particle levels can change results significantly.In any case, some modification of the normal part of thefunctional DF3-a is necessary to obtain better agreementfor the Pb nucleus.
184 188 192 196 200 204 208 21212345 exp. surf. vol. SkM * SLy4 E ( M e V ) A Pb FIG. 4: (Color online) Excitation energies ω (2 +1 ) for lead isotopes. Predictions for mean field approach with the forcesSkM*(dashed blue line) and SLy4(dashed green line) are taken from [27]. The energy density functional results are given bythe solid lines. In the tin chain, see Fig. 6, the situation with the ex-citation spectrum is partially similar to the one in lead.Again, 2 +1 -levels are higher for volume pairing than in thesurface case, and again the difference δω is ≃
300 keV,the surface predictions being closer to the experimentaldata. As to the SkM* spectrum, for isotopes heavier than
Sn it practically coincides with our “surface” one, bothbeing higher than the experimental spectrum by approx-imately 200 ÷
300 keV. For lighter isotopes, it deviatesfrom our surface spectrum significantly in an irregularway whereas the latter practically coincides with the ex-periment in this A region. As to the SLy4 spectrum, italso looks reasonable for the heavy part of the chain butfor isotopes lighter of Sn it strongly oscillates aroundthe experimental curve. In the dip minimum for
Snthe ω value is less than the experimental one at approx-imately 1 MeV and it is close to an instability.The excitation probabilities are displayed in Fig. 7.Here the results show a very complex pattern. For theheavier part of the chain, beginning at the Sn nucleus,our two theoretical curves and the SkM* practically co-incide, all being close to the experiment. The SkM*curve behaves in a non-regular way with strong devia-tions from the experimental data, up to ≃ ÷ B ( E , up) values rather well. For lightertin isotopes, our two curves began to deviate from eachother, the volume one being higher by ≃ ÷ g (1 , (the index “0s”is for brevity omitted). They are displayed in Fig. 8for the Pb nucleus. We see, first, that, for both theversions, the g (1) amplitude value is much bigger than | g (2) | . Second, the coordinate dependence of the main g (1) amplitude is absolutely different for the two versionsunder comparison. In the surface pairing case, a strongsurface maximum dominates whereas in the volume case g (1) is spread over the volume, with rather strong oscil-lations. In addition, it is seen that the integral effect of g (1)surf should be noticeably bigger than that of g (1)vol . Allthis shows some asymmetry for Bogolyubov quasiparti-cles and quasiholes. Such a situation is typical for nuclei
184 188 192 196 200 204 208 2120.00.10.20.30.40.50.60.70.80.9 exp. surf. vol. SkM * SLy4 B ( E , up ) ( e b ) A Pb FIG. 5: (Color online) B ( E , up) values for lead isotopes. Predictions for the SkM* and SLy4 force are taken from [27].
104 108 112 116 120 124 128 1321234 exp. surf. vol. SkM * SLy4 E ( M e V ) A Sn FIG. 6: (Color online) Excitation energies ω (2 +1 ) for tin isotopes. Predictions for the SkM* and SLy4 force are taken from [27].
104 108 112 116 120 124 128 1320.000.050.100.150.200.250.300.350.40 exp. surf. vol. SkM * SLy4 B ( E , up ) ( e b ) A Sn FIG. 7: (Color online) B ( E , up) values for tin isotopes. Predictions for the SkM* and SLy4 force are taken from [27].Experimental data are taken for − Sn from [40], for − Sn from [42], and for − Sn from [43–45]. which are close to the magic core. g (2) g (1) g (2) g (2) g (1) r (fm) g ( , ) ( + ) ( M e V ) Pb g (1) FIG. 8: (Color online) The neutron anomalous transition am-plitudes g (1 , in Pb nucleus. Solid lines correspond to sur-face, dotted to volume, and dashed, to the medium kind ofpairing, see Fig. 3.
The normal proton and neutron amplitudes g (0) forthe same nucleus are displayed in Fig. 9. As we see,for this quantity the influence of the kind of pairing used is minimal. Thus, evidently, the rather big value of thedifference δω ≃
300 keV for this nucleus is explainedwith different contributions of the anomalous amplitude g (1) which is much stronger in the case of surface pair-ing. For the transition densities, see Fig. 10, the effect israther small but a little bigger than for the normal ampli-tudes g (0) . This additional enhancement of the surfacemaximum of ρ tr (0) ( r ) in the surface pairing case againoriginates from the term with g (1) in Eq. (23). In itsturn, it explains the increase of the B ( E
2) value in thisnucleus for the surface case.Let us go to the tin chain. Figs. 11–13 present for the
Sn nucleus the same quantities which were displayedin Figs. 8–10 for the
Pb nucleus. This nucleus is in themiddle of the chain, and all properties of the “developed”pairing, in particular, particle-hole symmetry should takeplace. Indeed, now (see Fig. 11) the amplitudes g (1) and g (2) possess a similar form and absolute value and,being of the opposite sign. In the result, we have | g ( − ) = g (1) − g (2) | ≫ | g (+) = g (1) + g (2) | as it should be [9].Again, as in the Pb case, the effect of the kind ofpairing on the magnitude of g (1 , is drastic. As to thatfor the normal amplitudes g (0) and transition densities ρ tr (0) , again it is rather moderate but of the another sign.Now in the volume case, the surface peaks in both thesequantities are higher and, correspondingly, the B ( E r (fm) g ( ) ( + ) ( M e V ) Pb pn FIG. 9: (Color online) The proton and neutron normal transi-tion amplitudes g (0) in Pb nucleus. Solid lines correspondto surface pairing, dotted ones, to volume pairing. r (fm) t r ( - f m - ) Pb n p FIG. 10: (Color online) The proton and neutron transitiondensities ρ tr (0) in Pb nucleus. Solid lines correspond tosurface pairing, dotted ones, to volume pairing. destructive interference between normal and anomalouscontributions to solutions of the equations of Section 2.To summarize, we see an effect of the type of pairingon the characteristics of the 2 +1 -states in spherical nuclei.The excitation energies ω are systematically lower in thesurface case to δω ≃
300 keV, and the surface values are,as a rule, closer to the data. For B ( E
2) values, the effectis not so regular and here the volume version predictionson average look better. Thus, the present analysis iscompatible with both volume and surface pairing.The issue could be naturally raised to what extent thesmall differences seen in the observables can be traced tothe kind of pairing employed. In other words, is it possi-ble to fine tune the interaction parameters such that theboth volume and surface pairing produce indistiguishableresults, while keeping the mass differences and separation g (2) g (2) g (1) r (fm) g ( , ) ( + ) ( M e V ) Sn g (1) FIG. 11: (Color online) The neutron anomalous transitionamplitudes g (1 , in Sn nucleus. Solid lines correspond tosurface pairing, dotted ones, to volume pairing. r (fm) g ( ) ( + ) ( M e V ) Sn p n FIG. 12: (Color online) The same as in Fig. 9 but for the
Sn nucleus. energies close to the experimental data? We carried outsuch an analysis for the tin chain. We consider the doublemass differences D ( N ) = 12 (cid:18) S n ( N ) −
12 ( S n ( N −
1) + S n ( N + 1)) (cid:19) , (30) N even, which is very sensitive to the value of pairinggap. Note that the approximate relation D ( N ) ≃ ¯∆takes place where ¯∆ is an average gap value. We cal-culate the average difference between theoretical and ex-perimental values of this quantity, h δD i = s N X N ( D th ( N ) − D exp ( N )) , (31) N even. We include into the analysis isotopes from Sn1 r (fm) t r ( - f m - ) Sn np FIG. 13: (Color online) The same as in Fig. 10 but for the
Sn nucleus.TABLE II: Dependence of the 2 +1 -state characteristics of the Sn nucleus on the strength of pairing force.version h δD i (MeV) ω (MeV) B ( E , up)(e b )surface 0.10 1.216 0.172vol. f ξ = − .
33 0.11 1.470 0.206vol. f ξ = − .
32 0.16 1.375 0.193vol. f ξ = − .
34 0.11 1.570 0.216 till
Sn for which the developed pairing approximationseems to be reasonable. Results are presented in TableII, for the surface pairing and for three versions of thevolume pairing with different values of the strength pa-rameter f ξ . For all of them the characteristics of the2 +1 -state in the example Sn nucleus are given. It isseen that with increase of | f ξ | from the optimal value f ξ = − .
33 deviations from the surface version predic-tions grow. With decrease of | f ξ | they become less, butthis effect is much less than the initial difference evenfor the value f ξ = − .
32 for which description of themass differences is essentially worse than for the optimalvalue. An additional decrease of | f ξ | will absolutely de-stroy the nuclear mass description. In other isotopes ofthe tin chain, influence of variation of the f ξ parameterto values of ω and B ( E
2) is quite similar. Thus, the ef-fect under discussion originates mainly due to the surfacenature of pairing versus the volume one.In conclusion of this Section we compare in Fig. 14 thecharge transition density ρ trch ( r ) in the Sn nucleus withthe experimental transition charge density found with amodel independent analysis of the elastic electron scat-tering in [46]. The theoretical charge density is obtainedfrom ρ tr p ( r ) and ρ tr n ( r ) functions displayed in Fig. 13 withtaking into account relativistic corrections [47]. For bothversions of pairing the agreement with the data is quitereasonable, and it is a little better in the surface case. r (fm) t r c h ( - e f m - ) Sn FIG. 14: (Color online) The charge transition densities ρ tr(0)ch in Sn nucleus. Solid lines correspond to surface pairing,dotted ones, to volume pairing.
IV. QUADRUPOLE MOMENTS OF ODDNUCLEI V d n+ V n V p r (fm) V ( + ) ( f m ) Pb FIG. 15: (Color online) Static effective fields V n V p and d + n in Pb nucleus. Solid lines correspond to surface pairing,dotted ones, to volume pairing.
Recently magnetic moments of odd spherical nucleihave been calculated [48, 49] within the same self-consistent approach as the one used here. A reasonabledescription of the data for more than hundred of spher-ical nuclei was obtained. Especially high accuracy wasreached for semi-magic nuclei considered in the “single-quasiparticle approximation” where one quasiparticle inthe fixed state λ = ( n, l, j, m ) with the energy ε λ is addedto the even-even core. According to the TFFS, a quasi-particle differs from a particle of the single-particle modelin two respects. First, it possesses the local charge e q (inour case, we have e pq = 1, e nq = 0), and, second, the coreis polarized due to the interaction between the particle2 V d n+ V n V p r (fm) V ( + ) ( f m ) Sn FIG. 16: (Color online) Static effective fields V n V p and d + n in Sn nucleus. Solid lines correspond to surface pairing,dotted ones, to volume pairing. and the core nucleons via the LM amplitude. In otherwords, the quasiparticle possesses the effective charge e eff caused by the polarizability of the core which is found bysolving the above TFFS equations. In the many-particleShell Model [50], a similar quantity is introduced as aphenomenological parameter which describes polarizabil-ity of the core consisting of outside nucleons.In non-magic nuclei, the term quasiparticle takes adouble meaning. In addition to the initial LM conceptwe consider the Bogolyubov quasiparticles with occu-pation numbers n B λ =( E λ − ε λ ) / E λ and energies E λ = p ( ε λ − µ ) + ∆ λ and solve the set of the QRPA equa-tions (14) instead of one RPA equation.The success of the single-quasiparticle approximationin describing the magnetic moments of semi-magic nucleimakes it of interest to try to use the same approach forquadrupole moments. In this article, we do such anal-ysis limiting ourselves with odd neighbors of the eventin and lead isotopes considered in the previous Section.To our knowledge, there is no systematic calculations ofquadrupole moments of these nuclei.The static quadrupole moment of an odd nucleus inthe single particle state λ can be found in terms of theeffective field (14) with the static external field V = p π/ r Y ( θ ) as follows [9, 51]: Q p,nλ = ( u λ − v λ ) V p,nλ , (32)where u λ , v λ are the Bogolyubov coefficients and V λ = − j − j + 2 Z V ( r ) R nlj ( r ) r dr. (33)The j -dependent factor in (33) appears due to the angu-lar integral [52]. For j > / − V ( ω =0), thatis V n,p ( r ) and d + n ( r )= d (1) n ( r ) + d (2) n ( r ), are displayed inFigs. 15, 16 for Pb and
Sn nuclei, correspondingly.Note that the identity d − ( ω =0)=0 takes place [9]. Onecan see large surface maxima of the quantities V n,p ( r )similar to those in Figs. 9, 12 for the BM-like transitionamplitudes g (0) n,p ( r ). In-volume (“quantum”) correctionsare relatively small, therefore the integral in Eq. (33)is always positive. For protons, it is noticeably largerthan the similar integral with the bare field V , see thediscussion on the effective charges below.Diagonal matrix elements (33) of the proton effectivefield are displayed in Fig. 17 for the tin isotopes and inFig. 18, for the lead ones. As it is seen, for a major part ofthe tin isotopes, the difference between values of protonmatrix elements V pλ surface and volume pairing is quitesmall. Only for − Sn nuclei it reaches 10%. In thelead region, the difference is more pronounced reaching ≃ ÷
40% for 9 / − and 11 / − states.Corresponding quadrupole moments for nuclei withodd proton number Z = 50 ± Z = 82 ± ± / − states. If to suppose that they are single-quasiparticle1 h / states, they should have essentially higher excita-tion energies than it takes place.Experimental data are taken from the compilation [53].From several cases of proton excited isomeric states welimit ourselves with only two, the 1 g ∗ / state in the Sband 2 d ∗ / state in Tl nuclei, for which the hypothe-sis on the single-quasiparticle structure seems to us moreor less safe. Again, we presented results for both thekinds of nuclear pairing (the quantities Q surfth and Q volth for surface and volume pairing, correspondingly). In the6-th column of the table, the single-particle quadrupolemoment is presented which is found from Eqs. (32), (33)with substitution V → V . As it follows from Fig. 17, forodd-proton neighbors of the tin isotopes, difference be-tween values of quadrupole moments for surface and vol-ume pairing is quite small, in limits of 10%. In the leadregion, see Fig. 18, the difference is more pronounced,but here the number of the data is very small, only 4.In addition, only in the , Bi and
Tl case neutronpairing exists. For these nuclei, the effect under discus-sion reaches ≃ ÷ ≃ ÷ Bi iso-tope where pairing is absent experimental data are con-tradictory. We think that the main reason of existing3
TABLE III: Quadrupole moments Q (e b) of odd-proton nu-clei.nucl. λ Q exp Q surfth Q volth Q e surfeff e voleff105 In 1 g / +0.83(5) +0.83 +0.90 +0.18 4.6 5.0 In 1 g / +0.81(5) +0.98 +1.07 +0.18 5.4 5.9 In 1 g / +0.84(3) +1.11 +1.14 +0.18 6.2 6.3 In 1 g / +0.80(2) +1.16 +1.10 +0.19 6.1 5.8 In 1 g / +0.80(4) +1.12 +1.02 +0.19 5.9 5.4 In 1 g / +0.81(5), 0.58(9) +1.03 +0.97 +0.19 5.4 5.1 In 1 g / +0.829(10) +0.96 +0.95 +0.19 5.1 5.0 In 1 g / +0.854(7) +0.91 +0.92 +0.19 4.8 4.8 In 1 g / +0.814(11) +0.83 +0.84 +0.19 4.4 4.4 In 1 g / +0.757(9) +0.74 +0.74 +0.19 3.9 3.9 In 1 g / +0.71(4) +0.66 +0.74 +0.19 3.8 3.9 In 1 g / +0.59(3) +0.55 +0.49 +0.19 2.9 2.6 Sb 2 d / -0.36(6) -0.88 -0.81 -0.14 6.3 5.8 Sb 2 d / -0(2) -0.82 -0.77 -0.14 5.9 5.5 Sb 2 d / -0.37(6) -0.77 -0.76 -0.14 5.5 5.4 Sb 2 d / -0.36(4), -0.45(3) -0.72 -0.73 -0.14 5.1 5.21 g ∗ / -0.48(5) -0.81 -0.81 -0.17 4.8 4.8 Sb 1 g / -0.49(5) -0.74 -0.74 -0.17 4.4 4.4 Tl 3 d ∗ / Bi 1 h / -0.68(6) -1.32 -0.91 -0.25 5.3 3.6 Bi 1 h / -0.59(4) -0.94 -0.72 -0.25 3.8 2.9 Bi 1 h / -0.37(3), -0.55(1) -0.34 -0.34 -0.25 1.4 1.4-0.77(1), -0.40(5) disagreements is neglecting the phonon coupling effects.Let us go to odd-neutron nuclei, the odd tin and leadisotopes. The results are presented in Table IV and Figs.19,20. In selecting nuclei for the table, we used the sameconcept as for protons. In this case, we included into theanalysis twelve excited states, in addition to the groundones. With the only exception of the Pb nucleus, allthe nuclei under consideration exhibit pairing effects andthe factor ( u λ − v λ ) in Eq. (32) becomes non-trivial. Itchanges permanently depending on the state λ and thenucleus under consideration. Note that in the case ofmagnetic moments the factor of ( u λ + v λ ) = 1 appearsin the relation analogous to (32) [51]. In our case, thisfactor determines the sign of the quadrupole moment.In all cases when the sign of the experimental moment isknown the theoretical sign is correct. This permits to useour predictions to determine the sign when it is unknown.The factor under discussion depends essentially on valuesof the single-particle basis energies ε λ reckoned from thechemical potential µ as we have ( u λ − v λ ) = ( ε λ − µ ) /E λ .Keeping in mind such sensitivity, we found this quantityfor a given odd nucleus ( Z, N + 1), N even, with takinginto account the blocking effect in the pairing problem[51] putting the odd neutron to the state λ under con-sideration. For the V λ value in Eq. (32) we used thehalf-sum of these values in two neighboring even nuclei.We consider agreement with the data reasonable if wehave | Q th − Q exp | < . ÷ . h / inSn isotopes and 1 i / in Pb isotopes originate just fromtheir too distant positions from the Fermi level. Thus,the Q values depend strongly on the single-particle levelstructure. Again, as for protons, the difference betweenpredictions of the two models under consideration is, asa rule, rather small, and only for 1 i / -states in the leadchain it reaches ≃ ÷ e p,n eff = V p,nλ / ( V p ) λ . It is a direct characteristic of the core polar-izability by the quadrupole external field. In these tables,there are only two nuclei, Bi and
Pb, with a double-magic core, and in this case the polarizability is rathermoderate, e p eff = 1 . , e n eff = 0 .
9. In nuclei with an unfilledneutron shell it becomes much stronger, e eff ≃ ÷ V , virtual transitions inside the un-filled shell begin to contribute and small energy denomi-nators appear in the propagator L n (18) playing the mainrole in the problem under consideration. It enhances theneutron response to the field V and, via the strong LMneutron-proton interaction amplitude F np , the proton re-sponse as well. Results for the chain , , Bi showhow the polarizability grows with increase of the numberof neutron holes. Keeping in mind this physics, one canrepresent the effective charges as e p eff = 1+ e p pol , e n eff = e n pol where e p,n pol is the pure polarizability charge. To separate TABLE IV: Quadrupole moments Q (e b) of odd-neutron nu-clei.nucleus λ Q exp Q surfth Q volth e surfeff e voleff109 Sn 2 d / +0.31(10) +0.25 +0.27 3.5 3.7 Sn 1 g / +0.18(9) +0.05 +0.10 4.0 3.9 Sn 1 h ∗ / Sn 1 g ∗ / h ∗ / Sn 1 h ∗ / -0.42(5) -0.59 -0.58 3.9 3.7 Sn 2 d ∗ / +0.094(11), -0.03 -0.02 3.0 2.9-0.065(5),-0.061(3)1 h ∗ / Sn 2 d / -0.02(2) +0.06 +0.08 2.9 2.91 h ∗ / -0.14(3) -0.29 -0.29 3.3 3.3 Sn 1 h / +0.03(4) -0.12 -0.10 3.0 2.9 Sn 1 h / +0.1(2) +0.04 +0.06 2.7 2.7 Pb 1 i ∗ / +0.085(5) +0.0004 +0.10 5.3 5.9 Pb 1 i ∗ / +0.195(10) +0.33 +0.39 6.5 5.5 Pb 1 i ∗ / +0.306(15) +0.69 +0.66 6.6 5.2 Pb 3 p / -0.08(17) +0.19 +0.14 5.2 3.81 i ∗ / +0.38(2) +0.98 +0.78 6.4 4.6 Pb 3 p / +0.08(9) +0.27 +0.19 4.5 3.1 Pb 2 f / -0.01(4) +0.14 +0.09 4.2 2.8 Pb 2 f / +0.10(5) +0.28 +0.22 3.2 2.3 Pb 2 f / +0.23(4) +0.34 +0.28 2.6 2.01 i ∗ / Pb 2 g / -0.3(2) -0.26 -0.26 0.9 0.9
100 104 108 112 116 120 124 128 132 136-1.2-1.0-0.8-0.6-0.4-0.2 Sn V ( b ) A + surf. 9/2 + vol. 7/2 + surf. 7/2 + vol. 5/2 + surf. 5/2 + vol. FIG. 17: (Color online) Diagonal matrix elements V pλ of the effective proton quadrupole field in the tin isotopes. Solid linescorrespond to surface pairing, dotted ones, to volume pairing.
192 196 200 204 208-2.4-2.0-1.6-1.2-0.8-0.4 + surf. 3/2 + vol. 11/2 - surf. 11/2 - vol. 9/2 - surf. 9/2 - vol. Pb V ( b ) A FIG. 18: (Color online) The same as in Fig. 17, but for the lead isotopes.
103 107 111 115 119 123 127 131-0.8-0.6-0.4-0.20.00.20.4 + surf. 7/2 + vol. 3/2 + surf. 3/2 + vol. 11/2 - surf. 11/2 - vol. 5/2 + surf. 5/2 + vol. Sn Q ( e b ) A FIG. 19: (Color online) Quadrupole moments of odd tin isotopes. Solid lines correspond to surface pairing, dotted ones, tovolume pairing. Experimental data are shown with N for 3 / + , H for 11 / − , △ for 5 / + , and ▽ for 7 / + states.
191 193 195 197 199 201 203 205 207-0.8-0.6-0.4-0.20.00.20.40.60.81.01.2 3/2 - surf. 3/2 - vol. 13/2 + surf. 13/2 + vol. 5/2 - surf. 5/2 - vol. Pb Q ( e b ) A FIG. 20: (Color online) Quadrupole moments of odd lead isotopes. Solid lines correspond to surface pairing, dotted ones, tovolume pairing. Experimental data are shown with N for 13 / + , H for 3 / − , and △ for 5 / − states. V. DISCUSSION AND CONCLUSIONS
The effect of the density dependence of the pairinginteraction to low-lying quadrupole excitations in spher-ical nuclei is analyzed for two isotopic chains of semi-magic nuclei. Static quadrupole moments of neighbor-ing odd nuclei are also examined. The complete set ofthe QRPA-like TFFS equations for response functions issolved in a self-consistent way within the EDF approachto superfluid nuclei with previously fixed parameters ofthe functional. The DF3-a functional [30] is used whichis a small modification of the functional DF3 [13, 14].Specifically, spin-orbit and effective tensor terms of theinitial EDF DF3 were changed. Two models for effectivepairing force are considered, the surface and the volumeones, which give rise to approximately the same accu-racy in reproducing mass differences. A noticeable effectin excitation energies ω is found: predictions for thevolume model are systematically higher than the surfaceones by δω ≃ ÷
300 keV. As to the excitation prob-abilities B ( E , up), the effect is not so regular, however,as a rule, the volume values are also higher. Thus, thecorrelation between these two quantities typical for theBM model, where a higher frequency always results ina lower probability, is destroyed. On the average, bothmodels reasonably agree with the data. In addition, theyboth reproduce rather well the model-independent chargedensity ρ trch (2 +1 ) for the Sn nucleus.Comparison with recent QRPA calculations [27] withthe Skyrme force SkM* and SLy4 shows that for the leadchain they agree with the data a little better than ourresults but for the tin chain the situation is opposite andour predictions occur to be essentially better. The sur-face model is systematically better in describing the en-ergies ω whereas the excitation probabilities are, as arule, reproduced better with the volume model.Whereas the charge radii study [14] and ab initio the-ory of paring [18, 24] favor the surface pairing, the ω (2 +1 )and B ( E , up) data do not allow to prefer any of the twokinds of pairing.A reasonable agreement with experiment for thequadrupole moments of odd neighbors of the even tinand lead isotopes has been obtained for the most partof nuclei considered. For odd-proton this confirms thatthe single-quasiparticle approximation works sufficientlywell. For odd-neutron isotopes under consideration, va-lidity of this approach was checked previously with theanalysis of magnetic moments [49]. In the case we con-sider, the problem is more complicated than for oddproton isotopes as the Bogolyubov factor ( u λ − v λ ) =( ε λ − µ ) /E λ comes to the quadrupole moment value, in addition to the matrix element of the effective field V λ .This factor makes the quadrupole moment value very sen-sitive to accuracy of calculating the single-particle energy ε λ of the state under consideration, especially near theFermi surface as the quantity Q λ vanishes at ε λ = µ . Forsuch a situation, the influence of the coupling of single-particle degrees of freedom with phonons, see [49, 55],should be especially important. This rather complicatedproblem will be considered separately.As to the effect of the density dependence of pairing,for quadrupole moments it is, on the average, less thanfor quadrupole transitions. It depends on a nucleus ex-amined and on the odd-nucleon state as well. In the tinregion, it is, as a rule, of the order of ≃ ≃ ÷
50% for , Bi and
Pb.
VI. ACKNOWLEDGMENT
We thank J. Engel and J. Terasaki for kind supply-ing us with tables of the results of the QRPA calcula-tions [27] with the SkM* and SLy4 force. Four of us, S.T., S. Ka., E. S., and D. V., are grateful to Institut f¨urKernphysik, Forschungszentrum J¨ulich for hospitality.The work was partly supported by the DFG and RFBRGrants Nos.436RUS113/994/0-1 and 09-02-91352NNIO-a, by the Grants NSh-7235.2010.2 and 2.1.1/4540 of theRussian Ministry for Science and Education, and by theRFBR grants 09-02-01284-a, 11-02-00467-a.7 [1] T. Niksic, D. Vretenar, and P. Ring, Prog. Part. Nucl.Phys. , 519 (2011).[2] J. Erler et al . Phys. Part. Nucl. , 851 (2010).[3] M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Mod. , 121 (2003).[4] W. Kohn and L. J. Sham, Phys. Rev. , A 1133 (1965).[5] R. O. Jones, O. Gunnarson, Rev. Mod. Phys. , 689(1989).[6] L. N. Oliveira, E. K. U. Gross, and W. Kohn, Phys. Rev.Lett. , 2430 (1988).[7] D. N. Basov et al ., Rev. Mod. Phys. , 473 (2011).[8] V. A. Khodel and E. E. Saperstein, Phys. Rep. , 183(1982).[9] A. B. Migdal, Theory of finite Fermi systems and appli-cations to atomic nuclei (Wiley, New York, 1967).[10] S. A. Fayans and V. A. Khodel, JETP Lett. , 633(1973).[11] V. A. Khodel, E. E. Saperstein, and M. V. Zverev, Nucl.Phys. A465 , 397 (1987).[12] A. V. Smirnov, S. V. Tolokonnikov, and S. A. Fayans,Sov. J. Nucl. Phys. , 995 (1988).[13] S. A. Fayans, JETP Letters , 169 (1998).[14] S. A. Fayans, S. V. Tolokonnikov, E. L. Trykov, and D.Zawischa, Nucl. Phys. A676 , 49 (2000).[15] J. Erler, P. Kl¨upfel, and P.-G. Reinhard, Phys. Rev. C , 044307 (2010).[16] E. Epelbaum, H. -W. Hammer, U. -G. Meissner, Rev.Mod. Phys. , 1773-1825 (2009). [arXiv:0811.1338 [nucl-th]].[17] J. E. Drut, R. J. Furnstahl, L. Platter, Prog. Part. Nucl.Phys. , 120-168 (2010). [arXiv:0906.1463 [nucl-th]].[18] M. Baldo, U. Lombardo, E. E. Saperstein, and M. V.Zverev, Phys. Lett. B477 , 410 (2000).[19] M. Baldo, U. Lombardo, E. E. Saperstein, M. V. Zverev,Phys. Rep. , 064016 (2010).[21] S. S. Pankratov, M. V. Zverev, M. Baldo, U. Lombardo,and E. E. Saperstein, [arXiv:1103.4137 [nucl-th], to bepublished in Phys. Rev. C.[22] F. Barranco, R. A. Broglia, H. Esbensen, and E. Vigezzi,Phys. Lett. B390, 13 (1997).[23] F. Barranco, R. A. Broglia, G. Colo, et al ., Eur. Phys. J.A , 57 (2004).[24] A. Pastore, F. Barranco, R. A. Broglia, and E. Vigezzi,Phys. Rev. C , 024315 (2008).[25] Aurel Bulgac and Yongle Yu, Phys. Rev. Lett. , 042504(2002).[26] Yongle Yu and Aurel Bulgac, Phys. Rev. Lett. , 222501(2003).[27] J. Terasaki, J. Engel, and G. F. Bertsch, Phys. Rev. C , 044311 (2008).[28] A. P. Severyukhin, V. V. Voronov, and Nguyen Van Giai, Phys. Rev. C , 024322 (2008).[29] G. F. Bertsch, M. Girod, S. Hilaire, J.-P. Delaroche, H.Goutte, and S. P´eru, Phys. Rev. Lett. , 032502 (2007).[30] S. V. Tolokonnikov and E. E. Saperstein, Phys. Atom.Nucl. , 1684 (2010).[31] D. J. Horen, G. R. Satchler, S. A. Fayans, and E. L.Trykov, Nucl. Phys. A600 , 193 (1996).[32] E. E. Saperstein and S. V. Tolokonnikov, Phys. Atom.Nucl. , to be published (2011).[33] A. B. Migdal, Theory of finite Fermi systems and ap-plications to atomic nuclei , Second Edition (in Russian,Moscow, “Nauka”, 1982).[34] E. E. Saperstein and M. A. Troitsky, Sov. J. Nucl. Phys. , 284 (1965).[35] M. V. Zverev and E. E. Saperstein, Sov. J. Nucl. Phys. , 683 (1985).[36] S. Goriely, N. Chamel, and J. M. Pearson, Phys. Rev.Lett. , 152503 (2009).[37] S. T. Belyaev, A. V. Smirnov, S. V. Tolokonnikov, S. A.Fayans, Sov. J. Nucl. Phys. , 014319 (2011).[40] S. Raman, C. W. Nestor Jr., and P. Tikkanen, Atom.Data Nucl. Data Tables , 1 (2001).[41] A. Bohr and B. R. Mottelson, Nuclear Structure (Ben-jamin, New York, Amsterdam, 1974.), Vol. 2.[42] D. C. Radford, et al. , Nucl. Phys.
A752 , 264c (2005).[43] J. Cederk¨all, et al. , Phys. Rev. Lett. , 172501 (2007).[44] C. Vaman, et al. , Phys. Rev. Lett. , 162501 (2007).[45] A. Ekstr¨om, et al. , Phys. Rev. Lett. , 012502 (2008).[46] J. E. Wise, et al. , Phys. Rev. C , 2701 (1992).[47] J. L. Friar, J. Heisenberg, and J. W. Negele, in Proceed-ings of the June Workshop in Intermediate Energy Elec-tromagnetic Interactions , Ed. by A.M. Bernstein (Mas-sachusetts Institute of Technology, 1977), p. 325.[48] I. N. Borzov, E. E. Saperstein, and S. V. Tolokonnikov,Phys. Atom. Nucl. , 469 (2008).[49] I. N. Borzov, E. E. Saperstein, S. V. Tolokonnikov, G.Neyens, and N. Severijns, EPJ A , 159 (2010).[50] M. Honma, T. Otsuka, B. A. Brown, T. Mizusaki, Phys.Rev. C , (2004) 034335.[51] V. G. Soloviev, Theory of Complex Niclei , (Oxford: Perg-amon Press, 1976).[52] A. Bohr and B. R. Mottelson,
Nuclear Structure (Ben-jamin, New York, Amsterdam, 1969.), Vol. 1.[53] N. J. Stone, Atom. Data Nucl. Data Tables, , 75(2005).[54] S. P. Kamerdzhiev, Sov. Nucl. Phys. , 324 (1969).[55] S. P. Kamerdzhiev, A. V. Avdeenkov and D. A.Voitenkov, Phys. Atom. Nucl.73