Low-energy M1 excitations in 208 Pb and the spin channel of the Skyrme energy-density functional
V. Tselyaev, N. Lyutorovich, J. Speth, P.-G. Reinhard, D. Smirnov
LLow-energy M1 excitations in
Pband the spin channel of the Skyrme energy-density functional
V. Tselyaev, N. Lyutorovich, J. Speth, P.-G. Reinhard, and D. Smirnov St. Petersburg State University, St. Petersburg, 199034, Russia ∗ Institut f¨ur Kernphysik, Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany Institut f¨ur Theoretische Physik II, Universit¨at Erlangen-N¨urnberg, D-91058 Erlangen, Germany (Dated: March 6, 2019)We investigate the spin dependent part of the Skyrme energy-density functional, in particular itsimpact on the residual particle-hole interaction in self-consistent calculations of excitations. Testcases are the low-energy M1 excitations in
Pb treated within the self-consistent random-phaseapproximation based on the Skyrme energy-density functional. We investigate different parametriza-tions of the functionals to find out which parameters of the functional have strongest correlationswith M1 properties. We explore a simple method of the modification of the spin-related param-eters which delivers a better description of M1 excitations while basically maintaining the gooddescription of ground state properties.
I. INTRODUCTION
The aim of this paper is to explore the description ofnuclear magnetic excitations by an energy-density func-tional (EDF) of Skyrme type [1] taking the low-lyingmagnetic dipole (M1) excitations in
Pb as test case.The random phase approximation (RPA) and its variousextensions is the most often used method for the inves-tigation of nuclear excitation spectra. It takes as inputdata single-particle ( sp ) energies, sp wave functions and aparticle-hole ( ph ) residual interaction. Early calculationsas, e.g., Migdal’s Theory of Finite Fermi Systems (TFFS,see Refs. [2–4]) started with an effective single-particlemodel whose parameters are adjusted to experimental sp -properties and used (in nearly all numerical applica-tions) a density-dependent zero-range ph -interaction. Itrequires only a few parameters, coined Landau-Migdal(LM) parameters, which are adjusted to electric and mag-netic nuclear excitations and which turn out to be uni-versal in the sense that the same values apply throughoutthe chart of nuclei [5]. In self-consistent nuclear models,one obtains the sp -properties as well as the ph -interactionfrom one and the same effective Hamiltonian, or EDF re-spectively. The parameters of the Skyrme EDF are pri-marily adjusted to bulk properties of the nuclear groundstate. An appropriate ph residual interaction is not a pri-ori guaranteed. For example, the first realistic Skyrmeparametrizations [6, 7] had an incompressibility of theorder of 350 MeV and produced therefore the breath- ∗ [email protected] ing mode in Pb at around 17 MeV (which was offby 3 MeV from the experimental value measured someyears later). Including data specific to excitations, onecould later on develop parametrizations which also per-form well for breathing mode and isoscalar quadrupoleresonance [8, 9]. In general, there is sufficient flexibil-ity in the Skyrme EDF to accommodate all modes withnatural parity, isoscalar as well as isovector resonances[10]. The LM parameters for natural-parity excitationsderived from such Skyrme EDFs agree nicely with longtested LM parameters of TFFS [11].For magnetic modes, self-consistent models as, e.g.,Skyrme EDFs have not yet reached that high level of de-scriptive power while TFFS has been adapted very wellalso for these excitation channels. The plan for this pa-per is thus to explore the chances for a better descriptionof magnetic modes with a Skyrme EDF exploiting yetloosely determined aspects of the functional. Here welet us guide from the large body of experience gatheredwithin the TFFS. It tells us that the spin dependent ph -interaction is weak for the isoscalar part and is stronglyrepulsive for the isovector part. This agrees with theexperimental findings: There are no isoscalar collectivemagnetic resonances known over the whole periodic sys-tem but there exist strong Gamow-Teller resonances inheavy nuclei which are created by the spin-isospin depen-dent part of the residual interaction. We also know fromsuch investigations that the M1 states in Pb representan ideal test case. Experimental data on the distribu-tion of the M1 strength in this nucleus at the excitationenergies up to 8.4 MeV are known since the work of [12– a r X i v : . [ nu c l - t h ] M a r Pb consists oftwo marked features: an isoscalar 1 + state with E =5.844 MeV and a broad isovector M1 resonance in theinterval 6.6–8.1 MeV. Strong fragmentation of the M1resonance was one of the reasons of the difficulties withidentification in the early experiments (see, e.g., Ref. [16]for discussion). Moreover, several states which had beenoriginally identified as M1 turned out to be E1 after ex-periments with polarized photons were available.The numerous theoretical papers devoted to the mi-croscopic description of M1 excitations in Pb can bedivided into two main groups. The first group includesthe papers in which the nuclear excitations are treated assuperposition of the one-particle–one-hole (1 ph ) configu-rations, that is within the RPA or the Tamm-Dancoffapproximation (see, in particular, Refs. [3, 4, 17–24]). In the papers of the second group, various ver-sions of beyond-RPA approaches are used in which theRPA configuration space is enlarged by adding the 2 ph ,1 ph ⊗ phonon or two-phonons configurations (see, e.g.,[25–31]). Most of the earlier work as mentioned be-fore was performed within the TFFS. Using experimen-tal single-particle energies as input for the mean-fieldpart and properly tuning the interaction parameters (LMparameters) in the spin-spin channel, they managed toprovide an appropriate description of peaks and M1strengths. Beyond-RPA treatments, properly includingthe coupling of 1 ph states to 2 ph configurations, werenecessary to describe the spectral fragmentation of theM1 resonance around 7.5 MeV [31].Fully self-consistent RPA calculations as done in [20–24] did not yet reach that level of description. In fact,there is no published Skyrme parametrization which candescribe simultaneously position and strength of M1modes in Pb and other nuclei [21, 22]. Already
Pbalone seems to pose insurmountable difficulties. It is hardto get the lower M1 peak and the M1 resonance simulta-neously at their correct energies, not to mention a properprediction of M1 strength. Inappropriate strengths ofspin-orbit coupling were identified as one major sourceof the problem [21, 22]. We had applied a recently op-timized phonon-coupling model on top of self-consistentRPA [32, 33] to M1 modes and, unfortunately, did notfind any improvement concerning spectral separation oflow and upper mode nor sufficiently strong fragmenta-tion. The problem has first to be cleared at RPA level be- fore invoking more advanced approaches. The first taskto be solved is thus to develop a Skyrme parametrizationwhich describes energies and strengths of the leading M1modes correctly. And this is what we will attack in thepresent paper, namely to work out the crucial handles inthe Skyrme energy functionals which have most impactin the M1 spectrum and to try to tune them to delivercorrect M1 spectra without spoiling the high quality withrespect to nuclear ground state observables.The paper is organized as follows: Section II providesthe formal background of RPA, the Skyrme functional,the magnetic operators, and the numerical scheme. Sec-tion III discusses M1 modes in the context of SkyrmeEDFs and works out the leading mechanisms definingthese modes. In Section IV we try a moderate readjust-ment of Skyrme parameters which leads to better de-scription of M1 modes. The last section contains theconclusions.
II. FORMAL BACKGROUNDA. Summary of the RPA
Within the RPA one can calculate the spectrum ofthe excitation energies ω n of the even-even nucleus andthe corresponding set of the transition amplitudes Z n where the numerical indices (1 , , , . . . ) stand for the setsof the quantum numbers of some single-particle basis.Generally, this basis can be arbitrary, but it is convenientto suppose that it diagonalizes the single-particle densitymatrix ρ and the single-particle Hamiltonian h whichsatisfy the relations ρ = ρ and [ h, ρ ] = 0. In this casethe following equations are fulfilled h = ε δ , ρ = n δ . (1)In what follows the indices p and h will be used to labelthe single-particle states of the particles ( n p = 0) andholes ( n h = 1) in this basis.The RPA eigenvalue equation has the form (cid:88) Ω RPA12 , Z n = ω n Z n , (2)whereΩ RPA12 , = h δ − δ h + (cid:88) M RPA12 , V , , (3) M RPA12 , = δ ρ − ρ δ , (4) V is the amplitude of the residual interaction and M RPA is the metric matrix. The matrices Ω
RPA and M RPA actin the “ ph + hp ” space. The transition amplitudes Z n are normalized to (cid:88) Z n ∗ M RPA12 , Z n (cid:48) = sgn( ω n ) δ n, n (cid:48) . (5)In the self-consistent RPA, which is supposed in the fol-lowing, the following relations are fulfilled: h = δE [ ρ ] δρ , V , = δ E [ ρ ] δρ δρ , (6)where E [ ρ ] is an energy density functional.The amplitudes Z n allow us to calculate the reducedprobabilities of the transitions caused by the externalfield operator Q αLM according to the formula B n ( αL n ) = (cid:88) M n |(cid:104) Z n | Q αL n M n (cid:105)| , (7)where index α labels different kinds of the operators ofthe multipolarity L (in particular, α = m for the mag-netic transitions). B. The Skyrme energy density functional
As the energy density functional E [ ρ ] in Eqs. (6) wetake the Skyrme EDF of the standard form (see, e.g.,Refs. [34, 35]). It can be represented as the sum of thefollowing terms E Skyrme = E kin + E int + E Coul (8)where E kin = (cid:90) d r (cid:104) (cid:126) m p τ p ( r ) + (cid:126) m n τ n ( r ) (cid:105) , (9) E int = (cid:90) d r E int ( r ) , (10) E Coul = e (cid:90) d r d r (cid:48) ρ p ( r ) ρ p ( r (cid:48) ) | r − r (cid:48) |− e (cid:18) π (cid:19) / (cid:90) d r ρ / p ( r ) . (11)The energy density in Eq. (10) is given by E int = (cid:88) T =0 , (cid:104) C ρT ρ T + C ρ,αT ρ T ρ α + C ∆ ρT ρ T ∆ ρ T + C τT (cid:0) ρ T τ T − j T (cid:1) + C JT (cid:0) J T − s T · T T (cid:1) + C ∇ JT (cid:0) ρ T ∇· J T + s T ·∇× j T (cid:1) + C sT s T + C s,αT s T ρ α + C ∆ sT s T ∆ s T (cid:105) (12) where C ρT , C ρ,αT , C ∆ ρT , C τT , C JT , C ∇ JT , C sT , C s,αT , C ∆ sT ,and α are the constants, ρ T , τ T , J T , s T , T T , and j T are the local densities and currents. These densities andcurrents are divided into two groups (see [1, 36]): time-even ( ρ T , τ T , J T ) and time-odd ( s T , T T , j T ). Theirdefinition through the single-particle density matrix isgiven in Appendix A.In the general case, if the form of the functional E int is constrained only by the conditions of the global sym-metries, the C -constants are the independent parame-ters. Usually, they are determined by fitting the resultsof the Skyrme-Hartree-Fock (SHF) and RPA calculationsto the experimental data on basic nuclear properties withtaking into account the constraints imposed by the nu-clear matter properties. However, if the Skyrme EDF,Eqs. (8)–(12), is derived within the Hartree-Fock ap-proximation from the many-body Hamiltonian contain-ing two-body velocity and density dependent zero-rangeinteraction, the number of the independent C -constantsdecreases. In this case 18 C -constants in Eq. (12) areexpressed through 10 Skyrme-force parameters t , x , t , x , t , x , t , x , W , and x W (see, e.g., [1]). The re-spective formulas are given in Appendix B.Different bias in choosing the data and steady growthof information on exotic nuclei has lead to a great varietyof parametrizations. In order to keep the present surveysufficiently general, we consider a large set of 30 differentparametrizations of the Skyrme EDF: SIII [7], SGII [37],SkM ∗ [8, 9], RATP [38], T5 and T6 [39], SkP [40], SkI3,SkI4, and SkI5 [41], SLy4, SLy5, and SLy6 [42], SKX,SKXm, and SKXce [43], SkO and SkO (cid:48) [44], MSk1 andMSk3 [45], MSk9 [46], SV-bas, SV-K218, SV-kap00, SV-mas07, SV-sym34, and SV-min [10], SV-m56k6 and SV-m64k6 [47], and SAMi [48].Here it should be noted that the time-odd densitiesand currents are equal to zero in the ground states of theeven-even nuclei [36]. So, the constants C sT , C s,αT , and C ∆ sT do not affect the ground-state properties of these nu-clei and the mean field deduced by making use of Eq. (6).Nevertheless, these constants can have an impact on thecharacteristics of the excited states of the even-even nu-clei because in the general case the respective terms ofthe functional E int give the nonzero contribution to theresidual interaction according to Eqs. (6), (8), (10), and(12), even if the time-odd densities and currents are equalto zero. This circumstance allows us to change the con-stants C sT , C s,αT , and C ∆ sT (assuming that they are theindependent parameters) for the purpose of descriptionof nuclear excitations without affecting the ground stateand the self-consistent mean field.It is known that the parameters C sT , C s,αT , and C ∆ sT in most cases have little influence on the characteristicsof the natural parity excitations, but in some cases canlead to the spin instability in the self-consistent RPA andextended RPA calculations. In particular for this rea-son sometimes (including our recent papers [32, 33, 49–51]) they are set to be equal to zero, while the other C -constants are determined by the Skyrme-force param-eters according to Eqs. (B1). However, this choice is notsuitable for the self-consistent description of the magneticexcitations which are the subject of the present paper. Inthis case the terms of the functional E int containing theconstants C sT , C s,αT , and C ∆ sT become relevant. In par-ticular, from Eqs. (6), (8), (10), and (12) it follows thatthe terms containing C sT yield the term V s of the resid-ual interaction V having the form of the Landau-Migdalansatz V s = C N (cid:0) g σ · σ (cid:48) + g (cid:48) σ · σ (cid:48) τ · τ (cid:48) (cid:1) (13)where C N g = 2 C s , C N g (cid:48) = 2 C s , (14) C N is a normalization constant. Just the parameters g and g (cid:48) in Eq. (13) are responsible for the description ofthe unnatural parity excitations in the TFFS (see [2–4]). The method of determining the C -constants of thefunctional E int adopted in the present paper is describedin Sec. IV. C. The M1 operator
The field operator Q in the case of the M1 excitationshas the following (vector) form Q = µ N (cid:114) π (cid:110) ( γ n + γ p ) σ + l + (cid:2) (1 − ξ s ) ( γ n − γ p ) σ − (1 − ξ l ) l (cid:3) τ (cid:111) (15)where l is the single-particle operator of the angular mo-mentum, σ and τ are the spin and isospin Pauli ma-trices, respectively (with positive eigenvalue of τ forthe neutrons), µ N = e (cid:126) / m p c is the nuclear magneton, γ p = 2 .
793 and γ n = − .
913 are the spin gyromagneticratios, ξ s and ξ l are the renormalization constants intro-duced to simulate quenching of the M1 strength that isusually necessary for the description of the experimental data. The nonzero ξ s and ξ l correspond to the effectiveoperator Q . Their standard values are (see [4, 52]) ξ s = 0 . , ξ l = − . . (16)Zero values ξ s = 0 , ξ l = 0 (17)correspond to the bare operator Q (0) .Eq. (15) can be represented as the result of the ac-tion of the effective charge operator e q introduced in theTFFS [2] on the bare operator Q (0) , that is Q = e q Q (0) , (18)where e q = 1 − ( ξ l σ σ (cid:48) + ξ s σ · σ (cid:48) ) τ · τ (cid:48) , (19)and σ is the identity spin matrix. According to theTFFS, the operator e q is universal, i.e. it should act onall the external field operators Q including the opera-tors of the electric type Q e which are proportional to σ .From this it follows that if we impose the condition ofthe invariance e q Q e = Q e , (20)we should set ξ l = 0. The actual values of this con-stant used in the calculations of the magnetic excitationsare very small and thus violate the condition (20) onlyslightly. D. Numerical details
The equations of the RPA for the M1 excitations in
Pb were solved within the fully self-consistent schemeas described in Refs. [49–51].The single-particle basis was discretized by imposingthe box boundary condition with the box radius equal to18 fm. The particle energies ε p were limited by the max-imum value ε max p = 100 MeV. These conditions ensurefulfillment of the RPA energy-weighted sum rule for theisoscalar EL excitations in Pb within 0.1 % for L (cid:54) III. M1 EXCITATIONS IN
Pb IN RPAA. Defining the problem and observables
In order to illustrate the observables for the follow-ing survey, we start with showing in Fig. 1 the distri-bution of M1 strength in
Pb calculated within self-consistent RPA based on the Skyrme EDF with two dif-ferent parametrizations and comparing it with experi-mental data. We employ here the discrete version ofthe RPA because the single-particle continuum plays aminor role in the considered case. The strength func-
FIG. 1. Strength functions of the M1 excitations in
Pbcalculated within RPA using the parametrization SV-bas [10](black dashed line) and SV-bas mx as a modified variantthereof (red solid line) introduced in section IV. Experimentaldata taken from Refs. [13, 15] are shown by the blue dottedline. The low-lying M1 state is at 5.84 MeV, hidden belowthe result from SV-bas mx . The discrete peaks from RPA andthe lower M1 mode have been broadened with a smearingparameter ∆ = 20 keV to represent a smooth distribution. tions were obtained by folding the discrete RPA spec-trum and the discrete experimental mode (lower M1mode) with a Lorentzian of half-width ∆ = 20 keV.The experimental data demonstrate the basic featuresof M1 strength in Pb: there is a very narrow peak atlower energy E = 5 .
84 MeV and a broad resonance at E = 7 .
39 MeV. The height of the lower peak is char-acterized by its integrated B ( M
1) strength. Experi-mental mean energy and strength of the upper M1 reso-nance are computed from moments m k = Σ ν B ν ( M E kν summed/integrated in the interval 6.6–8.1 MeV withthe probabilities B ν ( M
1) and the excitation energies E ν taken from Refs. [13, 15]. We indicate this procedure bythe notation (cid:80) B ( M
1) for that value. Note that we do not include in this interval the state with E = 7.335 MeV(and possible B ( M
1) = 1.8 µ N ) from Ref. [15] becauseof the uncertainty with the identification of its spin. Wealso note that the chosen smearing parameter ∆ = 20keV is sufficiently large to average out the fine structureof the experimental spectrum which is not essential forour analysis, but remains sufficiently small to resolve thespreading widths. The experimental strength distribu-tion is composed from two data sets, below the neutronseparation energy 7.37 MeV from [15] and above from[13]. It is thus not clear whether the dip between thepeaks at 7.26 MeV and 7.47 MeV is a real effect. Inelas-tic proton scattering data [53, 54] seems to indicate thatthe dip does not exist. Anyway, such detailed fragmenta-tion structure cannot be described within RPA. Thus weuse for comparison with RPA the average peak propertiesas explained above. Altogether, we have four observables E , E , B ( M (cid:80) B ( M
1) which we use henceforthto characterize the M1 modes in
Pb.Fig. 1 shows theoretical results from two differentparametrizations. The parametrization SV-bas mx (whichis tuned to data such that theoretical and experimentalcurve for the lower peak at 5.84 MeV coincide) stands atthe end of our investigations and will be discussed later.The results for SV-bas (computed here with the all spin-spin terms included, i.e. η ∆ s = 1) are typical for mostof the available Skyrme parametrizations. They agreequalitatively in that theory also produces two dominantpeaks in the correct energy range. But the position of thepeaks and their strengths differs too much from the data.Reasons for that and possible cures will be discussed inthe following. B. State of the art
It is well known that the properties of the low-energyM1 excitations in
Pb in the RPA are mainly deter-mined by two ph configurations formed by the neutron’s( ν ) and proton’s ( π ) spin-orbit doublets 1 i / − i / and 1 h / − h / . The main characteristics of these con-figurations are the ph energy differences. Since the single-particle spectra produced by the various parametriza-tions of the Skyrme EDF are very different one can tracecorrelations between the values of these energy differ-ences, parameters of the EDF, and the RPA results forthe M1 excitations in Pb.Let us introduce the notations ε νph = ε νp (1 i / ) − ε νh (1 i / ) , (21) ε πph = ε πp (1 h / ) − ε πh (1 h / ) , (22)¯ ε ph = 12 (cid:0) ε νph + ε πph (cid:1) , ∆ ε ph = ε νph − ε πph . (23)The values of ε νph and ε πph along with the energies and thereduced probabilities of the excitation of the (isoscalar)1 +1 state and the mean energies and the summedstrengths of the (isovector) M1 resonance in Pb calcu-lated within the self-consistent RPA for the parametriza-tions of the Skyrme EDF indicated in Sec. II B are pre-sented in Figure 2. The effective M1 operator (15) withthe renormalization constants ξ s and ξ l from Eq. (16) isused. The shifts from mere ε ph to the corresponding RPAenergies E n indicate the strength of residual interactionin the M1 channel. It is generally smaller than for the gi-ant resonances. The figure reveals three main problems:First, some Skyrme-EDF parametrizations used with allspin terms [that means η ∆ s = 1 in Eqs. (B1) and is de-noted by open circles and the label “with s ∆ s ” in Fig. 2]lead to spin instability (imaginary RPA solutions) andthus have no entry in the plot (missing open circles).Second, the reduced probability B ( M
1) of excitation ofthe first 1 + state significantly exceeds its experimentalvalue for the most parametrizations, despite the quench-ing produced by the effective M1 operator. Third, themismatch starts already at the level of pure 1 ph energies ε νph which are definitely too large (upper panel) which canbe tracked down to the fact that all parametrizations givetoo large values of ∆ ε ph as compared to the experiment(see Figure 3). As a result, none of the parametrizationslisted in Figure 2 gives a satisfactory description of bothM1-modes simultaneously. These problems were alreadyfound in earlier publications and the spin-orbit couplingwas identified as one mechanism driving the M1 proper-ties [22]. We will now discuss that in more detail andexplore ways for a solution. C. Spin stability
Spin stability is a crucial issue in the constructionof Skyrme parametrizations [42, 55]. The first is tocheck the stability of bulk matter which is done easily interms of the LM parameters of the residual interaction.The LM parameters are related with the C -constants ofthe Skyrme-EDF by the following equations (see, e.g., S III S G II S k M * RA T P S k I S k I S k I S L y4 S L y6 S K XS K X m S K X ce S k O SV - b as SV - K SV - ka p SV - m as07 SV - sy m SV - m i n SV - m SV - m T T S k PS L y5 S k O ’ M S k1 M S k3 M S k9 S A M i exp. η J =0 η J =1 E [ M e V ] with s ∆ sno s ∆ s ε ph B ( M ) [ µ N ] with s ∆ sno s ∆ s567891011 S III S G II S k M * RA T P S k I S k I S k I S L y4 S L y6 S K XS K X m S K X ce S k O SV - b as SV - K SV - ka p SV - m as07 SV - sy m SV - m i n SV - m SV - m T T S k PS L y5 S k O ’ M S k1 M S k3 M S k9 S A M i exp. η J =0 η J =1 E [ M e V ] with s ∆ sno s ∆ s ε ph ∫ B ( M ) [ µ N ] with s ∆ sno s ∆ s FIG. 2. RPA results for energies E n and B n ( M
1) values ofthe two leading M1-modes in
Pb for a variety of publishedSkyrme parametrizations as listed at the end of section II B.For the energies, we show also the leading 1 ph excitations ε πph or ε νph , respectively. Experimental values are indicated byhorizontal dotted lines. The parametrizations are grouped inthose which omit tensor spin-orbit ( η J = 0, C JT = 0) and thosewhich use it ( η J = 1, C JT (cid:54) = 0). RPA results are consideredfor two options concerning the spin gradient terms ∝ s ∆ s : η ∆ s = 1 ( C ∆ sT (cid:54) = 0) and η ∆ s = 0 ( C ∆ sT = 0). Refs. [56, 57]) F = 2 N [ C ρ + ( α + 1)( α + 2) C ρ,α ρ α eq + C τ k ] , (24a) F (cid:48) = 2 N [ C ρ + C ρ,α ρ α eq + C τ k ] , (24b) G = 2 N [ C s + C s,α ρ α eq − C J k ] , (24c) G (cid:48) = 2 N [ C s + C s,α ρ α eq − C J k ] , (24d) F = − N C τ k , F (cid:48) = − N C τ k , (24e) G = 2 N C J k , G (cid:48) = 2 N C J k , (24f)where N = 2 m ∗ k F / ( π (cid:126) ) , k F = (3 π ρ eq / / is theFermi momentum, and ρ eq is the equilibrium density ofthe infinite nuclear matter (INM). Eqs. (24) coincidewith the definitions of Ref. [37] if the C -constants are ex-pressed through the parameters of the Skyrme force bythe standard formulas. However, Eqs. (24) produce G L and G (cid:48) L at variance with Ref. [37] for those parametriza-tions in which the J terms are omitted ( η J = 0 and C JT = 0) as noted in [58]. In particular, the parameters G and G (cid:48) are exactly equal to zero if the J terms areabsent in the Skyrme EDF. To ensure stability, the LMparameters should satisfy the following inequalities (see[2]) F L L + 1 > − , F (cid:48) L L + 1 > − , (25a) G L L + 1 > − , G (cid:48) L L + 1 > − . (25b)Table I shows the LM parameters corresponding to theSkyrme-EDF parametrizations listed in Figure 2. Thevalues of the spin-orbit parameter x W which will be dis-cussed in Sec. III D are also given. The conditions (25)are fulfilled for all parameters from Table I except forthe parameter G of SkO (cid:48) . However, as can be seenfrom Figure 2, the parametrizations T5, SkI4, SkO, SV-mas07, SV-sym34, SV-min, SV-m56k6, and SV-m64k6,for which the INM is stable, lead to the spin instabilityof the ground state of Pb in the case of η ∆ s = 1, inspite of bulk stability as proven by Table I. This insta-bility appears only in certain finite nuclei and is gener-ated by the spin surface terms ∝ C ∆ sT , not contained inEqs. (24) for the LM parameters (see also Ref. [59] wherethis question is discussed in more detail). On the otherhand, Figure 2 shows that the inclusion of the terms pro-portional to C ∆ sT into the Skyrme EDF usually decreasesthe energy of the 1 +1 state (compare open with filled cir-cles). Exceptions from this general trend are SkP, SKX,and SKXce for which E (1 +1 ) slightly increases if η ∆ s = 1.If the downshift by the C ∆ sT terms grows too large, itdrives the finite nucleus to instability. All the Skyrme-EDF parametrizations shown in Figure 2 except for SkO (cid:48) provide a stable ground state for Pb in case of η ∆ s = 0which is in agreement with the INM properties resultingfrom Table I.Note that the instability generated by the EDF SkO (cid:48) disappears in the modified parametrization SkO (cid:48) m , inwhich the C -constants are determined by Eqs. (B1) with η s = η s,α = η ∆ s = 0, C N = 300 MeV · fm , g = 0 . g (cid:48) = 1 .
39. In this case we have G = − . G (cid:48) = 2.24.The parameters F , , F (cid:48) , , G , and G (cid:48) are not changed.Thus, the nuclear matter becomes stable. The parame-ters g and g (cid:48) in SkO (cid:48) m have been adjusted to reproducewithin the RPA the experimental energies of the M1 ex-citations in Pb, E = 5.84 MeV and E = 7.39 MeV.The B ( M
1) values for the 1 +1 state and the isovector M Pb in this parametrization are equal to1.9 µ N and 16.9 µ N , respectively. D. The impact of spin-orbit parameters
Figure 2 indicates that problems appear already at thelevel of the 1 ph energies. This becomes even more ap-parent when looking at the average and difference 1 ph energies (23) as shown in Figure 3. First, ∆ ε ph ex-ceeds for most parametrizations the experimental value(0.29 MeV) by a factor of 3.4 (SAMi) to 7.5 (SkM ∗ ), ex-cept for SkI4, SkO, and SkO (cid:48) for which the spin-orbitparameter is x W < -1012345678 S III S G II S k M * RA T P S k I S k I S k I S L y4 S L y6 S K XS K X m S K X ce S k O SV - b as SV - K SV - ka p SV - m as07 SV - sy m SV - m i n SV - m SV - m T T S k PS L y5 S k O ’ M S k1 M S k3 M S k9 S A M i exp. aver. ε ph exp. ∆ε ph η J =0 η J =1 ph e n e r g i es [ M e V ] ∆ε ph aver. ε ph FIG. 3. Average 1 ph energy and difference as defined in Eq.(23) for the same selection of published Skyrme parametriza-tions as in Figure 2. Experimental values are indicated byhorizontal dotted lines. parametrizations with x W >
0, the value of B ( M
1) cal-culated with η ∆ s = 0 is greater than its experimentalvalue (2.0 µ N ) by a factor of 2.2 (SkP) to 10 (SLy5).This together suggests that the values of x W and ∆ ε ph are key agents determining the RPA results for the M1excitations in Pb.
TABLE I. Landau-Migdal parameters of the Skyrme-EDFs listed in Figure 2.EDF η J x W F F (cid:48) G G (cid:48) F F (cid:48) G G (cid:48) N − m ∗ /m k F (MeV · fm ) (fm − )SIII 0 1 0.31 0.87 0.54 0.95 − − − ∗ − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − (cid:48) − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − To explore this further, we consider simultaneous vari-ation of the spin-orbit parameters x W and W . To thatend, we start from the set SV-bas [10], vary x W , keep-ing all other model parameters frozen, and tune W toreproduce the SHF binding energy of Pb at its ex-perimental value 1636.43 MeV within the accuracy of0.2 MeV. This is done for the option η ∆ s = 1. Fig-ure 4 shows the dependence of the RPA results for thefirst and second 1 + states in Pb on the parameter x W obtained in this way. The respective values of ∆ ε ph , ¯ ε ph ,and W are also shown. All these quantities are given inunits of their values obtained for the original set SV-bas[10] and shown in Figures 2 and 3 [ B ( M
1) = 5.5 µ N , B ( M
1) = 17.4 µ N , E = 5.66 MeV, E = 7.95 MeV,∆ ε ph = 1.60 MeV, ¯ ε ph = 6.02 MeV] and the value W = 124.634 MeV · fm . The B ( M
1) shows the strongestdependence on x W . In fact, one can obtain any value of B ( M < µ N by decreasing the parameter x W . Theexperimental value B ( M
1) = 2 µ N is obtained at x W <
0. The values of E , E , and B ( M
1) depend on x W to much lesser extent. The energy difference ∆ ε ph alsoshows a strong dependence on x W , while the value of¯ ε ph is nearly constant (it is changed within 2.2% in theconsidered interval of x W ). The trend of ∆ ε ph with x W is monotonous. This allows to transform the dependen-cies shown in panels (a) and (b) of Fig. 4 into analogousdependencies on ∆ ε ph . The results are shown in Fig. 5,where again we see the crucial dependence of B ( M
1) on∆ ε ph at the constant ¯ ε ph . This dependence explains whythe parametrization SkO (cid:48) m introduced in Sec. III C gives FIG. 4. Dependence of the characteristics of M1 excita-tions in
Pb on the parameter x W of the Skyrme EDF.Parametrization SV-bas [10] is used. (a) The reduced proba-bilities B ( M
1) of the excitation of the first (solid red line) andsecond (dashed black line) 1 + states calculated in the RPA.(b) Same as in panel (a) but for the energies of these states.(c) The values of the energy differences ∆ ε ph (solid red line)and ¯ ε ph (dashed black line), Eqs. (23), and the spin-orbit pa-rameter W (dotted blue line). All the quantities are given inunits of their values obtained for the original parametrization[10]. See text for more details. nice agreement with the experimental value of B ( M x W = − .
58 and thus a value of ∆ ε ph = 0.48 MeV which is closest to the experimental value0.29 MeV. The other Skyrme-EDF parametrizations havegenerally too large ∆ ε ph which leads to significant over- FIG. 5. Same as in Fig. 4 but for the dependence of thereduced probabilities B ( M
1) on the value of the energy dif-ference ∆ ε ph . estimation of the B ( M ε ph on the properties of M1 excitations in Pb waspointed out in [21, 22].
IV. TOWARD BETTER REPRODUCTION OFM1 MODES
The results presented in Sec. III D show that spin-orbitparameters are most decisive for the M1-modes. And, ofcourse, the parameters of the spin-spin terms play anequally important role. This motivates us to check thechances to find a Skyrme functional in standard formwhich provides a good description of M1-modes togetherwith traditionally good modeling of ground state prop-erties. At present stage, it is too early to launch a fullyfledged least-squares fitting scheme [10, 60, 61] particu-larly because a high precision RPA computation of M1-modes is far too expensive. Thus, for a first exploration,we employ a simple-minded, restricted fitting procedure:We start from a given Skyrme parametrization, keep allmodel parameters at their given value except for the spin-orbit parameters C ∇ JT (alias x W , W ) and the spin-spinparameters C sT , C s,αT , and C ∆ sT . The spin-spin param-eters play no role for ground states of even-even nuclei.Thus we exploit here the freedom of not yet fixed param-eters. However, the spin-orbit parameters enter groundstate properties. Here we have to check that re-tuning0 TABLE II. Parameters η J , x W , W , g , and g (cid:48) of the modifiedSkyrme EDFs. Parameters g and g (cid:48) of the Landau-Migdalinteraction (13) are taken from Ref. [19].EDF η J x W W g g (cid:48) (MeV · fm )SkM ∗ m − − − m − − mx − − m − − m − − m − − m − − m − m − − m − − m − − m − − m − − does not destroy ground-state quality.To keep the number of free spin-spin parameters low,we set η s,α = η ∆ s = 0 and determine C sT by Eqs. (B1)with η s = 0 and the fitting parameters g and g (cid:48) at C N = 300 MeV · fm . After all, we have four free pa-rameters x W , W , g , and g (cid:48) which are determined byadjusting four observables in Pb: the binding energyand the RPA results for the M1 energies E and E and the transition probability B ( M
1) to their exper-imental values. Note that here we use, as before, theeffective M1 operator (15) with the renormalization con-stants ξ s and ξ l from Eq. (16). This fitting procedureis applied to a subset of the parametrizations shown inFigure 2. The modified parametrizations thus obtainedare marked by an index “m”. Resulting re-tuned modelparameters and properties of M1-modes are shown inFigure 6 and the corresponding re-tuned spin-orbit andspin-spin parameters are given in quantitative detail inTable II. As expected from the exploration in sectionIII D, all re-tuned x W parameters are negative, most ofthem in the interval between − . − .
5. Excep-tions are SkP m and SLy5 m which have higher x W due tothe J terms in these parametrizations which contributealso to the single-particle spin-orbit potential. The re-tuned parameters W are all rather large. This seem-ingly happens to compensate the negative x W . The leftlower panel of Figure 6 shows also the isoscalar spin-orbit parameter C ∇ J = − (2 + x W ) W . This com- bination shows much less variations over the differentforces and, in particular, remains practically unmodifiedby re-tuning. It is the isovector spin-orbit term propor-tional to C ∇ J = − x W W which makes the difference.Seeing the dramatic differences in spin-orbit parameters,one wonders what happens to the overall quality of theparametrization. This question will be addressed fartherbelow.The spin-spin coupling parameters g and g (cid:48) showsome correlation with the effective mass m ∗ /m ofa parametrization. The sets SkP m , SV-bas m , SV-K218 m , SV-kap00 m , SV-sym34 m , and SV-min m all hav-ing m ∗ /m ≈ g = 0 . g (cid:48) = 0 .
75 used previously inthe non-self-consistent TFFS (see [19]) while the otherparametrizations having lower m ∗ /m also produce lower g and g (cid:48) .The LM parameters G and G (cid:48) in Figure 6 stay allsafely above − E n stay by construction at the ex-perimental values. We show them (second panels fromabove) to illustrate the span toward the pure 1 ph energies ε πph (left) and ε νph (right). Let us concentrate first on theisoscalar mode (left). The up-shift by the residual inter-action is small for the parametrizations with m ∗ /m ≈ g or G . In thesecases, the 1 ph energies represent already a good estimateof E and the theoretical ε πph lie close to the experimentalvalue (faint dotted line). Lower effective masses increase ε πph , away from the wanted E , and need more residualinteraction to compensate. The impact of residual in-teraction is much larger for the isovector modes (right),again in accordance with the much larger spin couplingparameter g (cid:48) . In that case, we also have the problem thatall theoretical ε νph are much higher than the experimentalvalue of 5.84 MeV.The upper right panel of Figure 6 shows the B ( M g (cid:48) and theisovector B ( M
1) value: An increase of g (cid:48) reduces the B ( M
1) value. This is due to the increase of the groundstate correlations ( Y -components of the RPA transitionamplitudes) which decreases the transition probabilitiesin the magnetic case in contrast to the electric casewhere the ground state correlations add coherently. Theparametrizations SkP m and SLy5 m behave slightly dif-1 ε ph E [ M e V ] RPA ε ph S k M * m S L y4 m SV - b as m x SV - b as m SV - K m SV - ka p m SV - m as07 m SV - sy m m SV - m i n m SV - m m SV - m m S k P m S L y5 m η J =0 η J =1 W [ M e V f m ] W retunedW original -C retuned-C original-0.5-0.4-0.3-0.2-0.100.1 s p i n - s p i n p a r a m e t e r s gG ε ph E [ M e V ] RPA ε ph -0.500.511.5 S k M * m S L y4 m SV - b as m x SV - b as m SV - K m SV - ka p m SV - m as07 m SV - sy m m SV - m i n m SV - m m SV - m m S k P m S L y5 m η J =0 η J =1 x W retunedoriginal-0.4-0.200.20.40.60.811.2 s p i n - s p i n p a r a m e t e r s g’G’ ∫ B ( M ) [ µ N ] FIG. 6. Results for the re-tuned parametrizations. Lower two panels: spin-orbit parameters x W , W (filled circles), and − C ∇ J (abbreviated − C in the legend) together with their original values (open circles); spin-spin LM parameters G and G (cid:48) together with the interaction parameters g and g (cid:48) defined in Eqs. (B1). Upper two panels: the M1 energies E and E togetherwith their corresponding 1 ph energies ε πph and ε νph and the B ( M
1) strength for the upper M1 mode integrated over the interval6.6–8.1 MeV. Experimental values are indicated by horizontal faint dashed lines. ferent because as mentioned before in these parametriza-tions the J terms are included. These terms have anoticeable impact on the B ( M
1) values that can be es-timated with the help of the single-particle part of the RPA energy-weighted sum rule (EWSR) m s.p.1 . In thecase of the M1 excitations with the operator (15) it has2the form m s.p.1 = 12 Tr (cid:0) ρ (cid:2)(cid:2) Q , h (cid:3) , · Q (cid:3)(cid:1) (26)(see Ref. [62] for more details). In our self-consistentRPA calculations we obtain that this EWSR is fulfilledwithin 0.2% in the case of the Skyrme-EDF parametriza-tions without the J terms ( η J = 0). In the case of theSkP m and SLy5 m parametrizations ( η J = 1), this EWSRis exceeded by 19 and 25%, respectively.Generally, we see in the upper right panel of Figure6 that the theoretical isovector B ( M
1) strengths, evenfor the best parameter sets, are significantly larger thanthe experimental values. Here one has to bear in mindthat the experimental data in Figure 6 have been in-tegrated only up to 8.1 MeV. We know from previousbeyond-RPA calculations within the Landau-Migdal ap-proach [31] that the theoretical strength is distributed bycoupling to 2 ph states up to much higher energies. Suchspectral fragmentation is also seen in data. A recent( p, p (cid:48) ) experiment [53, 54] reports a summed B ( M
1) =20.5(1.3) µ N when integrated up to 9 MeV, a value whichwould fit nicely into the theoretical results of Figure 6.This situation reminds us at the case of the Gamow-Teller resonance in Pb where only half of the sum-rule strength was concentrated in one single strong res-onance and the rest was missing. Calculations within a2 ph model [63] (where one of the authors was involved)predicted a long tail which included the other half of thetotal strength. Ten years later the predicted strengthhad been detected experimentally. Thus, excess of thestrength can be corrected in extended RPA models in-cluding particle-phonon coupling that give also rise to ashift of the RPA strength to higher energies.So far, we have computed the B ( M
1) strengths withthe effective M1 operator using the renormalization con-stants ξ s and ξ l as defined in Eq. (16). This constructionis designed to account for correlation effects not includedin the actual Hilbert space. Thus the ξ s and ξ l can,in principle, be different for the different models. Thiswas exploited in the variant SV-bas mx where ξ s was usedtentatively as further free parameter and the isovector B ( M
1) strength as additional data point. The fittedrenormalization constants for SV-bas mx are ξ s = 0 . ξ l = 0 is chosen in accordance with the condition(20). The results in Figure 6 shows that this strategyallows to produce better B ( M
1) strength while main-taining the quality of the other observables. Note thatthe changes in ξ s and ξ l are, in fact, small which rathersupports the original choices (16). Anyway, this fit of renormalization constants should be considered as an ex-ploration of still loose ends in modeling. Playing withthese values needs yet to be supported by sound many-body theory.As argued above, spin-orbit parameters have not onlyhuge impact on M1 modes, but also on ground stateproperties. Thus a dramatic change of isovector spin-orbit coupling as implied in the re-tuned parametriza-tions could have unwanted side effects on the quality con-cerning the reproduction of ground state properties. Fig- S k M * m S L y4 m SV - b as m x SV - b as m SV - K m SV - ka p m SV - m as07 m SV - sy m m SV - m i n m SV - m m SV - m m S k P m S L y5 m η J =0 η J =1 r . m . s . e rr o r e n e r g y E [ M e V ] originalmodified05101520253035 r . m . s . e rr o r r a d i u s r [f m * - ] originalmodified FIG. 7. Average quality of the re-tuned parametrizationsquantified in terms of root-mean-square deviation of energyand charge radii taken over the set of spherical nuclei from[10]. ure 7 shows the performance of the refitted parametriza-tions with respect to ground state energy and chargeradius. The change of spin-orbit parameters leaves theoverall quality basically conserved. There is no effect atall for the radii. Energy reacts more sensitively whichis little surprise because pairing in semi-magic nuclei ishighly sensitive to level density which, of course, is influ-enced by spin-orbit splitting. Note that particularly themore recent, well fitted parametrizations show a loss ofenergy quality, fortunately in acceptable bounds. Still,the simple minded re-tuning strategy spoils somewhatthe overall quality of the parametrizations, the better3the quality originally the larger the loss. Moreover, thereare more subtle observables as pairing gaps and isotopicshifts of radii. The latter are known to be sensitive to theisovector spin-orbit term [41], for pairing gaps it is likely.All this calls for more continued investigations, more sys-tematic fits, and correlation analysis [61] to clearly workout the impact of information from M1 modes on nucleardensity functionals.So far, we have discussed the properties of M1 modes interms of two energies and B ( M
1) values. Let us finallylook again at the whole spectral distribution as it wasshown in Fig. 1. The results obtained with the freshly re-tuned parametrization SV-bas mx agree, by construction,nicely with experimental data. Comparison with theoriginal SV-bas shows the gain. Similar plots would beobtained when comparing original and re-tuned versionsof the other parametrizations. But Fig. 1 also pointstoward the yet open problems with the upper M1 mode:First, the strength is overestimated, and second, its spec-tral fragmentation is not described at all. Both problemsare related to each other as discussed above. The hope isthat a beyond-RPA modeling within the phonon-couplingmodel could deliver the missing pieces. V. CONCLUSIONS
In the present paper, we investigate the dependenceof the spin-dependent part of the ph -interaction on theparameters of Skyrme energy density functional (EDF).This part is relevant for computing magnetic excitationmodes within the self-consistent random-phase approxi-mation (RPA). We considered here, in particular, mag-netic dipole (M1) modes in Pb as test case. The M1modes are found depend crucially on the spin-orbit termand on the spin-spin interaction. The latter has no influ-ence on ground state properties and generally only weakrelations to natural-parity modes in even nuclei and isthus open to adjustment. The spin-orbit term is to someextend constrained by ground-state properties. However,we find that ground states leave enough leeway in themto accommodate the properties of M1 modes with onlysmall sacrifices on the overall quality of the ground stateproperties. We have tested that on a variety of 12 pub-lished Skyrme EDFs.In the analysis, we guide by the Landau-Migdal (LM)parameters from the Theory of Finite Fermion Systems(TFFS) which are weak in the isoscalar spin part andstrongly repulsive in the isovector part. The re-tuned Skyrme EDFs deliver LM parameters in accordance withthe TFFS. The relations between the LM parameters andthe parameters of the Skyrme-EDF serve also for a quickfirst check of spin stability of the chosen parameter set.As open questions remain the fragmentation and themagnitude of the isovector M1 resonance. Both are con-nected with more complex configurations beyond RPA,e.g., the coupling to the low-lying phonons (strong modesin each angular momentum channel). This, however, re-quires that all relevant phonons, also in the magneticchannels, are correctly described by RPA. The presentsurvey is a first step toward a proper description of mag-netic excitations in the framework of Skyrme-EDF and sopaves the way to subsequent beyond-RPA calculations.
ACKNOWLEDGMENTS
V.T. and N.L. acknowledge financial support fromthe Russian Science Foundation (project No. 16-12-10155). Research was carried out using computationalresources provided by Resource Center “Computer Cen-ter of SPbU”.
Appendix A: Local densities and currents
Let us introduce the isoscalar ( T = 0) and isovector( T = 1) single-particle density matrices ρ T ( r , σ ; r (cid:48) , σ (cid:48) ) = ρ n ( r , σ ; r (cid:48) , σ (cid:48) )+ ( − T ρ p ( r , σ ; r (cid:48) , σ (cid:48) ) , (A1)where ρ n ( r , σ ; r (cid:48) , σ (cid:48) ) and ρ p ( r , σ ; r (cid:48) , σ (cid:48) ) are the neutron’sand proton’s density matrices. The expressions for thelocal densities and currents entering Eq. (12) in terms ofthese matrices read ρ T ( r ) = (cid:88) σ ρ T ( r , σ ; r , σ ) , (A2) τ T ( r ) = (cid:88) σ ∇ · ∇ (cid:48) ρ T ( r , σ ; r (cid:48) , σ ) (cid:12)(cid:12) r = r (cid:48) , (A3) J T ( r ) = i (cid:88) σ, σ (cid:48) (cid:2) ( σ ) σ (cid:48) , σ × ∇ (cid:3) ρ T ( r , σ ; r (cid:48) , σ (cid:48) ) (cid:12)(cid:12) r = r (cid:48) (A4)for the time-even quantities and s T ( r ) = (cid:88) σ, σ (cid:48) ( σ ) σ (cid:48) , σ ρ T ( r , σ ; r , σ (cid:48) ) , (A5) T T ( r ) = (cid:88) σ, σ (cid:48) ( σ ) σ (cid:48) , σ ∇ · ∇ (cid:48) ρ T ( r , σ ; r (cid:48) , σ (cid:48) ) (cid:12)(cid:12) r = r (cid:48) , (A6) j T ( r ) = i (cid:88) σ (cid:0) ∇ (cid:48) − ∇ (cid:1) ρ T ( r , σ ; r (cid:48) , σ ) (cid:12)(cid:12) r = r (cid:48) (A7)4for the time-odd quantities.For the local densities τ p ( r ), τ n ( r ), and ρ p ( r ) in Eqs.(9) and (11) we have τ p = ( τ − τ ) / τ n = ( τ + τ ) / ρ p = ( ρ − ρ ) / Appendix B: Parameters of the Skyrme EDF
The following equations establish the relation betweenthe C -constants in Eq. (12) and the parameters of theSkyrme force t , x , t , x , t , x , t , x , W , and x W C ρ = t , C ρ = − t ( + x ) ,C ρ,α = t , C ρ,α = − t ( + x ) ,C ∆ ρ = − t + t + t x , C ∆ ρ = (cid:2) t ( + x ) + t ( + x ) (cid:3) ,C τ = t + t + t x , C τ = − (cid:2) t ( + x ) − t ( + x ) (cid:3) ,C J = (cid:2) t (cid:0) − x (cid:1) − t (cid:0) + x (cid:1)(cid:3) η J , C J = ( t − t ) η J ,C ∇ J = − (2 + x W ) W , C ∇ J = − x W W ,C s = C N g − t (cid:0) − x (cid:1) η s , C s = C N g (cid:48) − t η s ,C s,α = − t ( − x ) η s,α , C s,α = − t η s,α ,C ∆ s = (cid:2) t (cid:0) − x (cid:1) + t (cid:0) + x (cid:1)(cid:3) η ∆ s , C ∆ s = (3 t + t ) η ∆ s . (B1)The formulas for the spin-orbit constants C ∇ JT imply theparametrization introduced in [41, 64] in which the spin-orbit term of the interaction is treated in the Hartreeapproximation. The parameters W and x W are relatedwith the constants b and b (cid:48) of Ref. [41] by the formulas: W = 2 b , x W = b (cid:48) /b . The parameter η J = 1 if the J terms are included in the Skyrme EDF and η J = 0if not. In the standard parametrizations, the parameters x W , η s , η s,α , and η ∆ s are equal to 1, the parameters g and g (cid:48) are equal to 0. [1] M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev.Mod. Phys. , 121 (2003).[2] A. B. Migdal, Theory of Finite Fermi Systems and Ap-plication to Atomic Nuclei (Wiley, New York, 1967).[3] P. Ring and J. Speth, Phys. Lett. B , 477 (1973).[4] I. N. Borzov, S. V. Tolokonnikov, and S. A. Fayans, Sov.J. Nucl. Phys. , 732 (1984).[5] J. Speth and J. Wambach, in Electric and Magnetic Gi-ant Resonances in Nuclei , Vol. International Review ofNuclear Physics, Vol. 7, edited by J. Speth (World Sci-entific, 1991) pp. 2–87.[6] D. Vautherin and D. Brink, Phys. Rev. C , 626 (1972).[7] M. Beiner, H. Flocard, N. Van Giai, and P. Quentin,Nucl. Phys. A , 29 (1975).[8] J. Bartel, P. Quentin, M. Brack, C. Guet, and H.-B.H˚akansson, Nucl. Phys. A , 79 (1982).[9] M. Brack, C. Guet, and H.-B. H˚akansson, Phys. Rep. , 275 (1985).[10] P. Kl¨upfel, P.-G. Reinhard, T. J. B¨urvenich, and J. A.Maruhn, Phys. Rev. C , 034310 (2009).[11] J. Speth, S. Krewald, F. Gr¨ummer, P. G. Reinhard,N. Lyutorovich, and V. Tselyaev, Nucl. Phys. A928 ,17 (2014).[12] K. Wienhard, K. Ackermann, K. Bangert, U. E. P. Berg, C. Bl¨asing, W. Naatz, A. Ruckelshausen, D. R¨uck,R. K. M. Schneider, and R. Stock, Phys. Rev. Lett. ,18 (1982).[13] R. K¨ohler, J. A. Wartena, H. Weigmann, L. Mewissen,F. Poortmans, J. P. Theobald, and S. Raman, Phys.Rev. C , 1646 (1987).[14] R. M. Laszewski, R. Alarcon, D. S. Dale, and S. D.Hoblit, Phys. Rev. Lett. , 1710 (1988).[15] T. Shizuma, T. Hayakawa, H. Ohgaki, T. Toyokawa,T. Komatsubara, N. Kikuzawa, A. Tamii, andH. Nakada, Phys. Rev. C , 061303(R) (2008).[16] G. F. Bertsch, Nuclear Physics A , 157c (1981).[17] J. D. Vergados, Phys. Lett. B , 12 (1971).[18] J. Speth, V. Klemt, J. Wambach, and G. E. Brown,Nucl. Phys. A , 382 (1980).[19] E. Migli, S. Dro˙zd˙z, J. Speth, and J. Wambach, Z. Phys.A , 111 (1991).[20] L.-G. Cao, G. Col`o, H. Sagawa, P. F. Bortignon, andL. Sciacchitano, Phys. Rev. C , 064304 (2009).[21] P. Vesely, J. Kvasil, V. O. Nesterenko, W. Kleinig, P.-G. Reinhard, and V. Y. Ponomarev, Phys. Rev. C ,031302(R) (2009).[22] V. O. Nesterenko, J. Kvasil, P. Vesely, W. Kleinig, P.-G.Reinhard, and V. Y. Ponomarev, J. Phys. G: Nucl. Part. Phys. , 064034 (2010).[23] L.-G. Cao, H. Sagawa, and G. Col`o, Phys. Rev. C ,034324 (2011).[24] P. Wen, L.-G. Cao, J. Margueron, and H. Sagawa, Phys.Rev. C , 044311 (2014).[25] J. S. Dehesa, J. Speth, and A. Faessler, Phys. Rev. Lett. , 208 (1977).[26] S. P. Kamerdzhiev and V. N. Tkachev, Phys. Lett. B , 225 (1984).[27] D. Cha, B. Schwesinger, J. Wambach, and J. Speth,Nucl. Phys. A , 321 (1984).[28] D. T. Khoa, V. Y. Ponomarev, and A. I. Vdovin,Preprint JINR E4-86-198 (1986).[29] S. P. Kamerdzhiev and V. N. Tkachev, Z. Phys. A ,19 (1989).[30] V. I. Tselyaev, Sov. J. Nucl. Phys. , 780 (1989).[31] S. P. Kamerdzhiev, J. Speth, G. Tertychny, andJ. Wambach, Z. Phys. A , 253 (1993).[32] V. Tselyaev, N. Lyutorovich, J. Speth, and P.-G. Rein-hard, Phys. Rev. C , 024312 (2017).[33] V. Tselyaev, N. Lyutorovich, J. Speth, and P.-G. Rein-hard, Phys. Rev. C , 044308 (2018).[34] J. Dobaczewski and J. Dudek, Phys. Rev. C , 1827(1995).[35] J. Dobaczewski and J. Dudek, Acta Phys. Pol. B , 45(1996).[36] Y. M. Engel, D. M. Brink, K. Goeke, S. J. Krieger, andD. Vautherin, Nucl. Phys. A , 215 (1975).[37] N. Van Giai and H. Sagawa, Phys. Lett. B , 379(1981).[38] M. Rayet, M. Arnould, F. Tondeur, and G. Paulus, As-tron. Astrophys. , 183 (1982).[39] F. Tondeur, M. Brack, M. Farine, and J. M. Pearson,Nucl. Phys. A , 297 (1984).[40] J. Dobaczewski, H. Flocard, and J. Treiner, Nucl. Phys.A , 103 (1984).[41] P.-G. Reinhard and H. Flocard, Nucl. Phys. A , 467(1995).[42] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, andR. Schaeffer, Nucl. Phys. A , 231 (1998).[43] B. A. Brown, Phys. Rev. C , 220 (1998).[44] P.-G. Reinhard, D. J. Dean, W. Nazarewicz,J. Dobaczewski, J. A. Maruhn, and M. R. Strayer,Phys. Rev. C , 014316 (1999).[45] F. Tondeur, S. Goriely, J. M. Pearson, and M. Onsi,Phys. Rev. C , 024308 (2000).[46] S. Goriely, M. Pearson, and F. Tondeur, Nucl. Phys. A , 349c (2001). [47] N. Lyutorovich, V. I. Tselyaev, J. Speth, S. Krewald,F. Gr¨ummer, and P.-G. Reinhard, Phys. Rev. Lett. ,092502 (2012).[48] X. Roca-Maza, G. Col`o, and H. Sagawa, Phys. Rev. C , 031306(R) (2012).[49] N. Lyutorovich, V. Tselyaev, J. Speth, S. Krewald,F. Gr¨ummer, and P.-G. Reinhard, Phys. Lett. B ,292 (2015).[50] N. Lyutorovich, V. Tselyaev, J. Speth, S. Krewald, andP.-G. Reinhard, Phys. At. Nucl. , 868 (2016).[51] V. Tselyaev, N. Lyutorovich, J. Speth, S. Krewald, andP.-G. Reinhard, Phys. Rev. C , 034306 (2016).[52] S. Kamerdzhiev, J. Speth, and G. Tertychny, Phys. Rep. , 1 (2004).[53] I. Poltoratska, P. von Neumann-Cosel, A. Tamii,T. Adachi, C. A. Bertulani, J. Carter, M. Dozono,H. Fujita, K. Fujita, Y. Fujita, K. Hatanaka, M. Itoh,T. Kawabata, Y. Kalmykov, A. M. Krumbholz, E. Litvi-nova, H. Matsubara, K. Nakanishi, R. Neveling, H. Oka-mura, H. J. Ong, B. ¨Ozel-Tashenov, V. Y. Pono-marev, A. Richter, B. Rubio, H. Sakaguchi, Y. Sakemi,Y. Sasamoto, Y. Shimbara, Y. Shimizu, F. D. Smit,T. Suzuki, Y. Tameshige, J. Wambach, M. Yosoi, andJ. Zenihiro, Phys. Rev. C , 041304(R) (2012).[54] J. Birkhan, H. Matsubara, P. von Neumann-Cosel,N. Pietralla, V. Y. Ponomarev, A. Richter, A. Tamii,and J. Wambach, Phys. Rev. C , 041302(R) (2016).[55] S. Stringari, R. Leonardi, and D. M. Brink, Nucl. Phys.A , 87 (1976).[56] M. Bender, J. Dobaczewski, J. Engel, andW. Nazarewicz, Phys. Rev. C , 054322 (2002).[57] N. Chamel, S. Goriely, and J. M. Pearson, Phys. Rev. C , 065804 (2009).[58] T. Lesinski, M. Bender, K. Bennaceur, T. Duguet, andJ. Meyer, Phys. Rev. C , 014312 (2007).[59] A. Pastore, D. Tarpanov, D. Davesne, and J. Navarro,Phys. Rev. C , 024305 (2015).[60] M. Kortelainen, T. Lesinski, J. Mor´e, W. Nazarewicz,J. Sarich, N. Schunck, M. V. Stoitsov, and S. Wild,Phys. Rev. C , 024313 (2010).[61] J. Dobaczewski, W. Nazarewicz, and P.-G. Reinhard, J.Phys. G , 074001 (2014).[62] V. I. Tselyaev, N. A. Lyutorovich, and N. A. Belov, Bull.Russ. Acad. Sci. Phys. , 899 (2011).[63] S. Dro˙zd˙z, S. Nishizaki, J. Speth, and J. Wambach, Phys.Rep. , 1 (1990).[64] M. M. Sharma, G. Lalazissis, J. K¨onig, and P. Ring,Phys. Rev. Lett.74