Low-energy cluster vibrations in N = Z nuclei
F. Mercier, A. Bjelčić, T. Nikšić, J.-P. Ebran, E. Khan, D. Vretenar
LLow-energy cluster vibrations in N = Z nuclei
F. Mercier, A. Bjelˇci´c, T. Nikˇsi´c, J.-P. Ebran,
3, 4
E. Khan, and D. Vretenar IJCLab, Universit´e Paris-Saclay, IN2P3-CNRS, F-91406 Orsay Cedex, France Department of Physics, Faculty of Science, University of Zagreb, Bijeniˇcka c. 32, 10000 Zagreb, Croatia CEA,DAM,DIF, F-91297 Arpajon, France Universit´e Paris-Saclay, CEA, Laboratoire Mati`ere en Conditions Extrˆemes, 91680, Bruy`eres-le-Chˆatel, France
Significant transition strength in light α -conjugate nuclei at low energy, typically below 10 MeV,has been observed in many experiments. In this work the isoscalar low-energy response of N=Znuclei is explored using the Finite Amplitude Method (FAM) based on the microscopic frameworkof nuclear energy density functionals. Depending on the multipolarity of the excitation and theequilibrium deformation of a particular isotope, the low-energy strength functions display prominentpeaks that can be attributed to vibration of cluster structures: α + C+ α and α + O in Ne, C+ C in Mg, 4 α + C in Si, etc. Such cluster excitations are favored in light nuclei withlarge deformation.
I. INTRODUCTION
A number of experiments have observed a significantincrease of the E0 strength at excitation energies be-low the giant monopole resonance in relatively light nu-clei [1–6]. Theoretical studies using, e.g., the clustermodel [7–15], or the Antisymmetrized Molecular Dy-namics (generally combined with Generator CoordinateMethod (GCM)) [16–20], consistently interpret these ob-servations as excitations of cluster structures. Clusterexcitations can also occur with higher multipoles [3, 21–24]. For instance, a low-energy E1 excitation has beenassociated with a reflection-asymmetric vibration of an α cluster against the O core in Ne [15, 18], with astrength that is enhanced in comparison to similar exci-tations contributing in the E0 and E2 response.Valuable information about the structure of a nucleuscan be obtained by analyzing how the system responds toan external perturbation with a given multipolarity (see,for instance, [25, 26]). A useful theoretical frameworkfor such studies is provided by the Random-Phase Ap-proximation (RPA), and the Quasiparticle-RPA (QRPA)which extends the former to superfluid systems. (Q)RPAcalculations on top of reference mean-field states com-puted using Energy Density Functionals (EDFs), havedemonstrated the capacity to describe excitation modesranging from tens of keV to tens of MeV [27, 28].The method has also been extended to charge-exchangemodes [29–31]. There are many ways to derive theQRPA equations, e.g. by linearizing the Hartree-Fock-Bogoliubov (HFB) equations and then solving an eigen-value problem [32]. A major issue in QRPA calculationsare the dimensions of the matrix system which can be-come very large, especially when the HFB reference stateis allowed to spontaneously break the symmetries of thenuclear Hamiltonian.Several methods have been developed to circumventthese numerical difficulties [33–36], here in particular wefocus on the Finite Amplitude Method (FAM) [37]. Itis also based on the linearization of the Hartree-Fock(HF) equations but avoids the solution of a matrix eigen- value problem. The FAM has been extended to super-fluid systems (QFAM) [38] for Skyrme interactions andrelativistic functionals [39, 40]. The Skyrme-based FAMhas been applied to photoabsorption cross sections [41],higher multipole excitation modes [42], giant dipole res-onances in heavy nuclei [44], and β − decay studies [43].The present study is based on the relativistic QFAM[40]. Relativistic EDFs have successfully been used todescribe both liquid- and cluster-like nuclear properties[45, 46, 46–48], starting from nucleonic degrees of free-dom. Recently the multi-reference implementation of theGCM based on relativistic EDFs has been employed inthe analysis of spectroscopic properties (energies of ex-cited states, elastic and inelastic form factors) of nucleiwith cluster structures [49, 50]. A QFAM approach basedon relativistic EDFs is hence expected to provide an alter-native consistent and microscopic description of clustervibrations in nuclei.In this work we perform a systematic calculation ofisoscalar multipole ( λ = 0 , , ,
3) strength in α -conjugatenuclei from C to Ni, and analyze the low-energystructure of the strength functions. The calculations arebased on the DD-PC1 parametrization [54] and involvean expansion of the equations of motion in an axially-deformed harmonic oscillator basis. The first nucleus tobe analyzed is Ne whose large equilibrium deformationfavors clusterization, and hence cluster vibration modesare expected to occur at low energy [16]. We will showthat the lowest modes correspond to reflection-symmetric2 α + C and reflection-asymmetric α + O configurationsoscillating around the axially-symmetric deformed equi-librium. The study of Ne is extended to other α -conjugate nuclei, and the evolution of the strength func-tion is analyzed when the quadrupole moment of themean-field reference state is varied from oblate to pro-late deformations.The QFAM formalism is briefly introduced in Sec. II.Section III explores the multipole ( λ = 0 , , ,
3) responseof Ne, as well as the role played by quadrupole defor-mation in the appearance of cluster vibration modes. InSec. IV we extend the study of isoscalar monopole vi-brations to three other α -conjugate nuclei that display a r X i v : . [ nu c l - t h ] J u l pronounced cluster vibrations: Mg, Si and S. Sec-tion V contains a brief summary and conclusions.
II. THEORETICAL FRAMEWORK
Our implementation of the QFAM follows closely theone described in Refs. [51, 52]. The QFAM equationsread: ( E µ + E ν − ω ) X µν ( ω ) + δH µν ( ω ) = − F µν , (1)( E µ + E ν + ω ) Y µν ( ω ) + δH µν ( ω ) = − F µν , (2)where the matrices F and F are calculated from theexternal harmonic perturbation field: F ( t ) = η (cid:0) F ( ω ) e − iωt + F † ( ω ) e + iωt (cid:1) , (3)characterized by the small real parameter η . X µν ( ω ) and Y µν ( ω ) denote the QFAM amplitudes at given excita-tion energy ω , while δH µν ( ω ) and δH µν ( ω ) describe theresponse of the atomic nucleus to the external pertur-bation. The time-dependent density matrix and pairingtensor read: ρ ( t ) = V ∗ V T + η (cid:0) δρ ( ω ) e − iωt + δρ † ( ω ) e + iωt (cid:1) , (4) κ ( t ) = V ∗ U T + η (cid:16) δκ (+) ( ω ) e − iωt + δκ ( − ) ( ω ) e + iωt (cid:17) , (5)where δρ ( ω ) = U X ( ω ) V T + V ∗ Y T ( ω ) U † , (6) δκ (+) ( ω ) = U X ( ω ) U T + V ∗ Y T ( ω ) V † , (7) δκ ( − ) ( ω ) = V ∗ X † ( ω ) V † + U Y ∗ ( ω ) U T . (8)The transition strength at each particular energy is cal-culated from the expression: S ( f, ω ) = − π ImTr (cid:2) f † δρ ( ω ) (cid:3) , (9)where δρ ( ω ) denotes the induced density matrix, and f kl are the matrix elements of the operator F ( ω ) in configu-ration space.To prevent that the QFAM solutions diverge in thevicinity of a QRPA state, a small imaginary part is addedto the energy ω → ω + iγ . This corresponds to foldingthe QRPA strength function with a Lorentzian of widthΓ = 2 γ [53]. The electric isoscalar multipole operator isdefined as f ISJK = A (cid:88) i =1 f JK ( r i ) , (10)with f JK ( r ) = r J Y JK ( θ, φ ). For the monopole modethe operator reads f ( r ) = r , while for the isoscalardipole excitation f K ( r ) = r Y K ( θ, φ ) . Since for an even-even axially symmetric nucleus the opera-tors f JK and f J − K produce identical strength func-tions, in the code we employ the operator f (+) JK = (cid:0) f JK + ( − K f J − K (cid:1) / √ δ K and assume K ≥ (cid:104) k | V S | k (cid:48) (cid:105) = − Gp ( k ) p ( k (cid:48) ). By assum-ing a simple Gaussian ansatz p ( k ) = e − a k , the twoparameters G and a were adjusted to reproduce the den-sity dependence of the pairing gap at the Fermi surfacein nuclear matter obtained by the Gogny D1S interac-tion [57]. The current implementation of the DIRQFAMsolver employs an expansion of the Dirac spinors in termsof eigenfunctions of an axially symmetric harmonic os-cillator potential. Further details on the QFAM solverDIRQFAM can be found in Ref. [58]. III. ISOSCALAR VIBRATIONS IN Ne We begin our analysis with the isotope Ne. The leftpanel of Fig. 1 displays the prolate deformed ( β ≈ . Ne obtained obtainedwith the DD-PC1 parametrization. The density exhibitscluster structures at the outer ends of the symmetry axiswith density peaks (cid:39) . − , and an oblate deformedcore, reminiscent of a quasimolecular α - C- α structure.The spatial localization and cluster formation in atomicnuclei can also be quantified by using the localizationfunction C τσ ( r ), defined in Ref. [61] for the nuclear case.A value of the localization measure close to 0.5 signalsthat nucleons are delocalized, while a value close to onecorresponds to a localized alpha-like structure at point (cid:126)r in an even-even N = Z nucleus. The localization func-tion for Ne is plotted in the right panel of Fig. 1, andconsistently confirms the alpha-like nature of the local-ized structures appearing in the density.The isoscalar strength function of the monopole op-erator A (cid:88) i =1 r i for Ne is analyzed using the QFAM. Thecalculation has been performed in the harmonic oscillatorbasis with N ( f ) sh = 10 , , ,
16 and 18 major oscillatorshells for the upper component, and N ( g ) max = N ( f ) sh + 1 forthe lower component of the Dirac spinor (see Ref. [62]).In the following discussion the number of shells N sh corre-sponds to the number of major harmonic oscillator shellsused in the expansion of the upper component of theDirac spinor, i.e., N sh ≡ N ( f ) sh . In Fig. 2 we comparethe strength functions of the isoscalar monopole opera-tor for Ne, calculated with N sh =10, 12, 14, 16 and18. The low-energy part of the strength function is fullyconverged even for relatively small values of the N sh .However, for higher energies, the strength function dis- FIG. 1: (Color online) The self-consistent equilibrium den-sity of Ne (left panel), and localization function C τσ (rightpanel) obtained using the RHB model with the DD-PC1 en-ergy density functional. plays a pronounced dependence on the dimension of theharmonic oscillator basis, essentially because these exci-tations involve states in the continuum. Therefore, thehigh-energy part of the strength function is strongly af-fected by the details of single-particle configurations. Wenote, however, that the centroids of the strength distri-bution in the high energy region are much less sensitiveto the basis dimension, as shown in Tab. I. Since thisstudy is focused on the properties of low-lying states, allsubsequent calculations are performed by expanding thelarge component of the Dirac spinors in N ( f ) sh =14 majoroscillator shells. N sh ¯ E low (MeV) ¯ E high (MeV)10 18.4 27.012 18.1 27.014 18.1 27.316 18.0 27.618 18.1 28.0TABLE I: Centroids of the monopole strength function (seeFig. 2) defined as the ratio of moments m /m . The mo-ments of the strength function are m k = (cid:82) E k S ( E ) dE . The¯ E low and ¯ E higs centroids are calculated in the energy inter-vals 10 MeV ≤ E ≤ < E ≤
35 MeV,respectively.
Fig. 3 displays the strength functions for the QFAMresponse to the isoscalar monopole (panel (a)), isoscalardipole (panel (b)), isoscalar quadrupole (panel (c)) andisoscalar octupole (panel (d)) operator. In addition to the K = 0 components, for the multipoles λ = 1 , , K projections sepa-rately, as well as the total strenghts. For the quadrupole K = 1 + strength distribution one notices the appearanceof the spurious state related to the breaking of rotational FIG. 2: (Color online) Evolution of the monopole strengthfunction in Ne with the size of the harmonic oscillator basis. symmetry, and also the ordering of the K = 0 + , K = 1 + ,and K = 2 + peaks in the high energy region above 15MeV is consistent with the prolate deformed ground stateof Ne. Although all strength distributions exhibit pro-nounced fragmentation in the E ≥
10 MeV region, asizeable portion of strength is located at E ≈ ρ ( r , t ) = ρ gs ( r ⊥ , z ) + 2 η Re (cid:2) e − iωt δρ ( ω, r ⊥ , z ) (cid:3) cos ( Kφ ) , (11)where ρ gs ( r ⊥ , z ) denotes the ground-state density and δρ ( ω, r ⊥ , z ) is the transition density at a given excita-tion energy ω . We note that for the K = 0 modes thetime-dependent densities are axially symmetric δρ ( r ) = δρ ( r ⊥ , z ), hence it is sufficient to study their behaviourin the xz plane. Figures 4 and 5 display the snapshotsof the time-dependent density in the xz plane for thelow-energy modes induced by monopole and octupole( K = 0 component) perturbations. Time increases fromthe top to the bottom, with the time step ∆ t = 2 π/ η defined by Eq. (11) equals 0.05 for themonopole and 0.005 for the octupole perturbation, re-spectively. The large value of the intrinsic equilibriumdeformation of Ne leads to cluster formation alreadyin its ground state, and one finds that clusters oscillateagainst the core for both modes shown in Figs. 4 and 5.Furthermore, two different types of vibrations are ob-served: i) the two α clusters oscillate against the Ccore for the J = 0 reflection-symmetric mode, ii) an os-cillation of the α cluster against the O core for the J = 3 reflection-asymmetric mode.The two-dimensional intrinsic transition densities δρ tr ( r ) can be projected onto good angular momentum FIG. 3: (Color online) Ne strength distribution functionsfor the QFAM response to the isoscalar monopole (panel (a)),isoscalar dipole (panel (b)), isoscalar quadrupole (panel (c))and isoscalar octupole (panel (d)) operator. For
J > K = 0 (solid blue), K = 1 (dashed red), K = 2 (dot-dashed green) and K = 3(dotted orange) are plotted separately. The thin dashedcurves denote the total strength. to yield the transition densities in the laboratory frame ofreference. For a particular value of the angular momen-tum J ≥ K , the two dimensional projected transitiondensity can be approximated using its radial part by δρ Jtr ( r ) = δρ Jtr ( r ) Y JK (Ω) , (12)with the radial part defined as δρ Jtr ( r ) = (cid:90) d Ω δρ tr ( r ⊥ , z ) Y ∗ JK (Ω) . (13)Fig. 6 compares the radial parts of the angular-momentum-projected transition densities δρ J =0 tr ( r ), δρ J =2 tr ( r ) and δρ J =4 tr ( r ) that correspond the the low-energy peak of the isocalar monopole response in Ne.The real and imaginary parts of the transition densityare displayed in the left and right panels, respectively.For the real parts we note the characteristic node of thetransition density close to the position of the rms radius.The radial parts of the angular-momentum-projectedtransition densities δρ J =1 tr ( r ), δρ J =3 tr ( r ) and δρ J =5 tr ( r )that correspond the the low-energy peak of the isocalaroctupole response are shown in Fig. 7. In contrastto the volume monopole mode, the isoscalar octupoletransition densities exhibit the predominantly surfacenature of the octupole mode.It is instructive to decompose the excitation modes interms of 2-quasiparticle (2-qp) contributions [63]. This FIG. 4: (Color online) Snapshots of the Ne density oscilla-tions at energy (cid:126) ω = 6 .
75 MeV induced by monopole pertur-bation. Time increases from top to bottom and a full periodis shown.FIG. 5: (Color online) Snapshots of the Ne density oscil-lations at energy (cid:126) ω = 7 .
65 MeV induced by octupole per-turbation ( K = 0 component). Time increases from top tobottom and a full period is shown. can be achieved by using the contour integration pro-cedure introduced in Ref. [64]. The individual QRPAamplitudes corresponding to the excitation mode i are FIG. 6: (Color online) Radial parts of the angular-momentumprojected transition densities that correspond to the low-energy peak of the isocalar monopole response of Ne. Thereal and imaginary parts of the transition density are shownin the left and right panels, respectively. The ground state rms radius is indicated by the vertical dashed line.FIG. 7: (Color online) Same as in the caption to Fig. 6 butfor the isocalar octupole response ( K = 0 component). calculated as X iµν = e − iθ |(cid:104) i | ˆ F | (cid:105)| − πi (cid:73) C i X µν ( ω γ ) dω γ , (14) Y iµν = e − iθ |(cid:104) i | ˆ F | (cid:105)| − πi (cid:73) C i Y µν ( ω γ ) dω γ , (15)where X µν ( ω γ ) and Y µν ( ω γ ) denote the QFAM ampli-tudes for the complex frequency ω γ = ω + iγ , and C i isthe contour in the complex energy plane that encloses thefirst-order pole on the real axis at ω γ = Ω i . We note thatthe common phase e iθ remains arbitrary. The individual2-qp contributions to some particular excitation mode i can be quantified by the following quantity: ξ i qp = (cid:12)(cid:12) X i qp (cid:12)(cid:12) − (cid:12)(cid:12) Y i qp (cid:12)(cid:12) . (16)Fig. 8 displays in a schematic way the most importantneutron 2-qp contributions to the isoscalar monopole ex-citation at (cid:126) ω = 6 . / + state thatoriginates from the spherical 1 d / shell, to the unoccu-pied 1 / + state based on the spherical 2 s / shell. Such a 2-qp excitation can be considered in the context of spon-taneous breaking of rotational symmetry which capturesin an economic way non-trivial correlations as the sourceof collective behavior of the nucleus. This spontaneousbreaking of rotational symmetry leads to the appearanceof new excitation modes commonly referred to as a den-sity wave [65]. Density waves are related to the variationof the modulus of the order parameter of the broken sym-metry. FIG. 8: (Color online) Schematic illustration of the most im-portant neutron 2-qp contributions to the isoscalar monopoleexcitation at (cid:126) ω = 6 . Ne. The area and the num-ber below represent the fraction of the total | X | − | Y | (seeEq.16) for this particular excitation. The Ω π quantum num-bers are listed on the right of the figure. The associated par-tial densities are also plotted for each of the configurations aswell as the total density in the background. The Fermi levelis shown as a red dash-dotted line. Large deformations favor the formation of clusters[59, 60] and the previous discussion also suggests thatthere is a close link between cluster vibrational modesand nuclear deformation. The evolution of the low-energy cluster modes with deformation can be studiedin more detail by performing a deformation-constrainedcalculation. In Fig. 9 we display the isoscalar monopolestrength in Ne for several values of the axial quadrupoleconstraint, from β = 0 .
275 to β = 0 . β = 0 . (cid:126) ω ≈ − β ∼ . d / spherical shellsplits into three levels: 1 / + , 3 / + and 5 / + . In partic- FIG. 9: (Color online) The low-energy isoscalar monopolestrength distribution in Ne isotope. The QFAM responseis calculated for several constrained values of the axialquadrupole deformation β , and the dashed curve correspondsto the equilibrium deformation β = 0 . ular, the occupation probability for the 1 / + level in-creases with deformation thus enabling hole-particle ex-citations to the 1 / + states originating from the spherical2 s / and 1 d / shells. We note that the occupation ofthe 1 / + level based on the 1 d / spherical shell is, ofcourse, also responsible for the formation of clusters inthe ground state of Ne. As shown in Fig. 10, the lowestdeformation for which the low-energy monopole excita-tion is obtained is β ≈ .
2, which coincides with theintersection of the 1 / + [200] level and the Fermi level.A further increase of deformation between β = 0 . β = 0 . / + [010] and 1 / + [101] to the QFAM tran-sition strength. The contribution of these levels to thetotal strength increases from 25% to more than 40%. Theoscillations with constrained deformation are illustratedin Fig.11, where we display the snapshots of the totaldensity oscillations at energy (cid:126) ω and constrained defor-mation β caused by a monopole perturbation. At largerdeformations the cluster structure is, of course, more pro-nounced. The oscillation frequency increases because theenergy splitting of the single-particle levels increases withdeformation.The very low-energy excitation at (cid:126) ω ≈ / + [200] and 3 / + [101] levels. They are competing be-tween β = 0 and β = 0 .
5, at which deformation the1/2+[200] becomes fully occupied. Between these defor-mations, and because these levels are very close to theFermi energy, pairing excitations can occur, dependingon the pairing gap as well as the quasiparticle energies.
FIG. 10: (Color online) Evolution of the leading neutron 2qpcontributions to the low-energy monopole mode with con-strained deformation (upper panel). The lower panel showsthe evolution of the single-particle energies (left) and occupa-tion number (right) in the canonical basis with deformation.The vertical black lines denote the transitions that correspondto the principal 2-qp contribution shown in the upper panel.The thick black curve denotes the Fermi level.FIG. 11: (Color online) Snapshots of Ne total densitymonopole oscillations at energy (cid:126) ω and constrained initial de-formation β . The time flows from the top to the bottom anda full period is shown. IV. ISOSCALAR MONOPOLE RESPONSE OF N = Z NUCLEI
In this section we extend the analysis of low-lyingisoscalar monopole QFAM response to Mg, Si and S. Figure 12 displays the corresponding isoscalarmonopole strength functions for several values of the ax-ial quadrupole constraint β . One notices the appearanceof the low-energy and large prolate deformation peak ofthe strength distribution for all isotopes shown in Fig. 12,similar to the results obtained for Ne in the previoussection. We have also performed corresponding calcu-lations for other light and medium-heavy N = Z nuclei,from C to Ni. The appearance of low-energy strengthis much less pronounced for isotopes in the vicinity ofdoubly closed-shells.
FIG. 12: (Color online) Low-energy isoscalar monopolestrength distribution in N = Z nuclei: Mg, Si and S.The QFAM response is calculated for several values of con-strained axial quadrupole deformation β , and the dashedcurves correspond to the equilibrium deformation for eachnucleus. The structure of the strength distributions can be an-alyzed by considering the principal 2-qp contributions,displayed in Fig. 13. We have selected several low-energypeaks in Mg, Si and S, and the results again in-dicate that these low energy excitations are primarilydetermined by a single 2-qp excitation. In Mg we ob-tain two peaks, one at ∼ ∼
10 MeV, that have already been observed in experi-ment [66]. Similar to the case of Ne, the lower state in Mg (first column of Fig. 13) is mainly determined bythe transition between the 1 / + states originating fromthe 1 d / spherical shell (hole-like) and 2 s / sphericalshell (particle-like). The addition of two neutron andtwo protons leads to the appearance of the second modeat excitation energy (cid:126) ω = 10 .
03 MeV (second column ofFig. 13). This excitation, corresponding to the oscilla-tions of two large clusters ( C + C), is determined bythe transition between the 3 / + states originating fromthe 1 d / spherical shell (hole-like) and 1 d / spherical shell (particle-like). While for Ne the 3 / + [101] statewas not occupied, two more particles in Mg start fill-ing the 3 / + [101] state with the occupation probabilityapproaching 1 for β ≈ .
7. Hence, the mechanism thatdrives the low-energy excitations in Mg isotope is gen-erally the same as for Ne. The splitting of the spher-ical 1 d / and 1 d / levels with deformation allows nowfor two transitions, one between Ω π = 1 / + states, andanother between Ω π = 3 / + states. Similar arguments FIG. 13: (Color online) Upper panel: leading neutron 2-qpcontributions to the low-energy monopole modes in Mg, Siand S isotopes (for detailed description see the caption toFig. 10). Lower panel: snapshots of the corresponding densityoscillations (see the caption to Fig. 11). apply to other low-energy excitations shown in Fig. 13.
V. SUMMARY AND CONCLUSION
A systematic analysis of low-lying multipole responsein deformed N = Z nuclei has been performed using thequasiparticle finite amplitude method based on relativis-tic nuclear energy density functionals. It has been shownthat the low-energy modes correspond to cluster vibra-tions for all considered isoscalar multipole operators. Inparticular, in Ne the monopole and quadrupole opera-tors induce oscillations of two α -clusters around the Ccore, while the dipole and octupole operators induce vi-brations of an α -cluster with respect to the O core.To analyze the effect of deformation on the low-lyingstrength distribution, in a first step we have performeda deformation-constrained QFAM calculation for themonopole response in Ne. The appearance of clusteroscillations is closely related to the structure of single-nucleon levels in the canonical basis and, in particu-lar, to the splitting of the 1 d / spherical shell. Themonopole response is governed predominantly by thetransition from the 1 / + state originating from the spher-ical 1 d / shell to the 1 / + state that correspond to thespherical 2 s / shell. We have also extended the analy-sis of the low-lying isoscalar monopole QFAM responsefor light and medium-heavy N = Z nuclei, from C to Ni. It has been found that the low-energy peaks of themonopole strength distribution are more pronounced indeformed isotopes far from closed shells. The results areillustrated by three isotopes with clearly visible clustervibration low-energy modes: Mg, Si and S. Similarto the Ne case, the low-energy excitations in these iso-topes are dominated by single 2-qp excitations. A studyof higher-multipole QFAM response in light and medium-heavy N = Z nuclei is in preparation. Acknowledgments
This work has been supported in part by the Quan-tiXLie Centre of Excellence, a project co-financed by the Croatian Government and European Unionthrough the European Regional Development Fund - theCompetitiveness and Cohesion Operational Programme(KK.01.1.1.01). [1] D. H. Youngblood, Y.-W. Lui, and H. L. Clark, Phys.Rev. C55, 2811 (1997).[2] D. H. Youngblood, Y.-W. Lui, and H. L. Clark, Phys.Rev. C57, 2748 (1998).[3] Y.-W. Lui, H. L. Clark, and D. H. Youngblood, Phys.Rev. C64, 064308 (2001).[4] D. H. Youngblood, Y.-W. Lui, and H. L. Clark, Phys.Rev. C65, 034302 (2002).[5] S. Yildiz, M. Freer, N. Soic, S. Ahmed, N. I. Ashwood,N. M. Clarke, N. Curtis, B. R. Fulton, C. J. Metelko,B. Novatski, N. A. Orr, R. Pitkin, S. Sakuta, and V. A.Ziman, Phys. Rev. C73, 034601 (2006).[6] Y. K. Gupta et al.,Phys. Lett. B748, 343 (2015).[7] T. Tomoda and A. Arima, Nucl. Phys. A303, 217 (1978).[8] E. Uegaki, Y. Abe, S. Okabe, and H. Tanaka, Prog.Theor. Phys.62, 1621 (1979).[9] M. Kamimura, Nucl. Phys. A351, 456 (1981)[10] P. Descouvemont and D. Baye, Phys. Rev. C36, 54(1987).[11] Suzuki and S. Hara, Phys. Rev. C39, 658 (1989)[12] M. Chernykh, H. Feldmeier, T. Neff. P. von Neumann-Cosel, and A. Richter, Phys. Rev. Lett. 98, 032501 (2007)[13] M. Ito, Phys. Rev. C83, 044319 (2011).[14] D. S. Delion, R. J. Liotta, P. Schuck, A. Astier, and M.-G. Porquet, Phys. Rev. C85, 064306 (2012).[15] Y. Kanada-En’yo and Y. Shikata, Phys. Rev. C 100,014301 (2019).[16] M. Kimura,Phys.Rev.C69,044319 (2004).[17] Y. Chiba and M. Kimura,Phys. Rev. C91, 061302 (2015).[18] Y. Chiba, M. Kimura, Y. Taniguchi, Phys. Rev. C93,034319 (2016).[19] Y. Kanada-En’yo, Phys. Rev. C 93, 054307 (2016)[20] Y. Kanada-En’yo and K. Ogata, Phys. Rev. C 101,014317 (2020)[21] D. H. Youngblood, Y.-W. Lui, and H. L. Clark, Phys.Rev. C60, 014304 (1999).[22] D. H. Youngblood, Y.-W. Lui, X. F. Chen, and H. L.Clark, Phys. Rev. C80, 064318 (2009).[23] X. Chen, Y.-W. Lui, H. L. Clark, Y. Tokimoto, and D.H. Youngblood, Phys. Rev. C80, 014312 (2009).[24] M. Itoh, H. Akimune et al., Phys. Rev. C84, 054308(2011).[25] T. Nakatsukasa and N. Schunk, Energy Density Func-tional Methods for Atomic Nuclei (6-1 to 6-55), IOP Pub- lishingv (2019).[26] S. P´eru and M. Martini, Eur. Phys. J. A (2014) 50: 88.[27] E. Khan, N. Paar and D. Vretenar, Phys. Rev. C 84,051301 (2011).[28] S. P´eru, H. Goutte, J.F. Berger, Nuc. Phys. A788, 44(2007).[29] S. Fracasso and G. Colo, Phys. Rev. C 72, 064310 (2005).[30] M. Martini, S. P´eru, and S. Goriely, Phys. Rev. C 89,044306 (2014).[31] H. Liang, P. Zhao and J. Meng, Phys. Rev. C 85, 064302(2012).[32] P. Ring and P. Schuck, The Nuclear Many-Body Problem(Springer-Verlag, Berlin, 1980).[33] K. Yoshida and N. V. Giai, Phys. Rev. C78, 64316 (2008).[34] D. Pena Arteaga, E. Khan, and P. Ring, Phys. Rev. C79,034311 (2009).[35] J. Terasaki and J. Engel, Phys. Rev. C82, 034326 (2010).[36] C. Losa, A. Pastore, T. Dossing, E. Vigezzi, and R. A.Broglia, Phys. Rev. C81, 064307 (2010).[37] T. Nakatsukasa, T. Inakura, K. Yabana, Phys. Rev. C76,024318 (2007).[38] P. Avogadro, T.Nakatsukasa, Phys. Rev. C84, 014314,(2011).[39] H. Liang, T. Nakatsukasa, Z. Niu, and J. Meng, Phys.Rev. C87, 054310 (2013).[40] T. Nikˇsi´c et al, Phys. Rev. C88, 044327 (2013).[41] T. Inakura, T. Nakatsukasa, and K. Yabana,Phys. Rev.C80, 044301 (2009).[42] M. Kortelainen, N. Hinohara, and W. Nazarewicz, Phys.Rev. C 92, 051302(R) (2015).[43] M. T. Mustonen and J. Engel, Phys. Rev. C 93, 014304(2016).[44] T. Oishi, M. Kortelainen, and N. Hinohara, Phys. Rev.C 93, 034329 (2016).[45] J.P. Ebran, E. Khan, T. Nikˇsi´c and D.Vretenar, Nature487, 341 (2012).[46] J.P. Ebran, E. Khan, T. Nikˇsi´c and D. Vretenar, Phys.Rev. C 90, 054329 (2014).[47] J.-P. Ebran, E. Khan, T. Nikˇsi´c and D. Vretenar, Phys.Rev. C , 054329 (2014).[48] J.-P. Ebran, E. Khan, R.-D. Lasseri and D. VretenarPhys. Rev. C , 061301(R) (2018).[49] P. Marevi´c, J.-P. Ebran, E. Khan, T. Nikˇsi´c, and D.Vretenar, Phys. Rev. C 97, 024334 (2018), 061301(R) (2018).[49] P. Marevi´c, J.-P. Ebran, E. Khan, T. Nikˇsi´c, and D.Vretenar, Phys. Rev. C 97, 024334 (2018)