aa r X i v : . [ nu c l - t h ] A ug Charge-exchange dipole excitations in deformed nuclei
Kenichi Yoshida Department of Physics, Kyoto University, Kyoto, 606-8502, Japan (Dated: August 11, 2020)
Background:
The electric giant-dipole resonance (GDR) is the most established collective vibra-tional mode of excitation. A charge-exchange analog, however, has been poorly studied in compar-ison with the spin (magnetic) dipole resonance (SDR).
Purpose:
I investigate the role of deformation on the charge-exchange dipole excitations and ex-plore the generic features as an isovector mode of excitation.
Methods:
The nuclear energy-density functional method is employed for calculating the re-sponse functions based on the Skyrme–Kohn–Sham–Bogoliubov method and the proton-neutonquasiparticle-random-phase approximation.
Results:
The deformation splitting into K = 0 and K = ± K -component. A qualitative argument onthe strength distributions for each component is given based on the non-energy-weighted sum rulestaking nuclear deformation into account. The concentration of the electric dipole strengths in lowenergy and below the giant resonance is found in neutron-rich unstable nuclei. Conclusions:
The deformation splitting occurs generically for the charge-exchange dipole excitionsas in the neutral channel. The analog pygmy dipole resonance can emerge in deformed neutron-richnuclei as well as in spherical systems.
I. INTRODUCTION
Charge-exchange excitations dovetail with the transi-tions from a mother nucleus (
Z, A ) with proton number Z and total nucleon number A to final states in a neigh-boring daughter ( Z ± , A ) in the isospin lowering τ − andraising τ + channels, respectively. They take place eitherin the charged-current nuclear (semileptonic) weak pro-cesses such as the β -decay, charged lepton capture andneutrino-nucleus reactions or in the hadronic reactions of( p, n ) or ( n, p ) type. Therefore, the spin-isospin responsesinduced by the charge-exchange excitations present ac-tive and broad research topics in the fields of fundamen-tal physics [1–9].Response of a nucleus unveils elementary modes ofexcitation emerged by the interactions and correlationsamong constituent nucleons. The nuclear response ischaracterized by the transferred angular momentum ∆ L ,spin ∆ S and isospin ∆ T [10]. The isovector (IV) giantdipole resonance (GDR) represented as ∆ L = 1 , ∆ S =0 , ∆ T = 1 is one of the well studied collective vibra-tional modes of excitation among various types of giantresonance [11]. The GDR is an out-of-phase spatial oscil-lation of protons and neutrons, and thus represented as∆ T z = 0. Recently with the advent of RI-beam technol-ogy, a considerable amount of work has been devoted to aquest for exotic modes of excitation in nuclei far from the β -stability line and the low-energy dipole (LED) mode orthe pygmy dipole resonance (PDR) has attracted a lot ofinterest [12–15]. Furthrermore, the photoresonance canbe seen in a wider perspective when it is considered as asingle component ∆ T z = 0 of the IV dipole modes [16–18]. The additional components ∆ T z = ± β -decay [6, 19] if any andthe single β -decay of heavy neutron-rich nuclei [20–26].From a nuclear structure point of view, the spin-dipoleresonance (SDR) with ∆ L = 1 , ∆ S = 1 , ∆ T = 1 havebeen studied to understand the mechanism for the col-lectivity of a giant resonance and the spin-isospin part ofthe interaction in nuclear medium [1, 10] besides that theGamow–Teller and M1 resonances have been extensivelystudied as the IV magnetic ∆ L = 0 transitions [7]. Themultipole dependence of the SDR elucidates the char-acteristic effects of the tensor force [27, 28], and thestrengths of the dipole resonance are correlated with theneutron-skin thickness [14]. Though the study on thecharge-exchange electric dipole resonance is limited, arecent work investigated a possible appearance of an ana-log of the PDR in neutron-rich nuclei, and the low-lyingexcitation corresponding to − ~ ω in very neutron-richnuclei [29].In this article, I am going to investigate the defor-mation effects on the charge-exchange dipole resonanceswith both ∆ S = 0 and 1, and explore the generic fea-tures of dipole resonances as an IV mode of excitation.Furthermore, the roles of neutron excess is studied in de-tails for the electric excitations and a possible appearanceof the low-lying states is discussed. The present study isconsidered as an extension of the previous work on spher-ical nuclei [29] to deformed cases. From light to heavynuclei are taken as a target of investigation to extractuniversal features associated with nuclear deformation.To this end, I employ the nuclear energy-density func-tional (EDF) method, which is a theoretical model beingcapable of handling nuclides with arbitrary mass numberin a single framework [30, 31].This paper is organized in the following way: The the-oretical framework for describing the ground state of amother nucleus, the excited states of a daughter nucleusand the transitions between them is given in Sec. II anddetails of the numerical calculation are also given; Sec. IIIis devoted to the numerical results and discussion basedon the model calculation; the electric dipole resonanceis studied in Sec. III A, and the SDR in Sec. III B; then,summary is given in Sec. IV. II. FRAMEWORKA. KSB and pnQRPA for deformed nuclei
Since the details of the formalism can be found inRef. [32], here I show only the gist of the basic equa-tions relevant to the present study. In a framework ofthe nuclear energy-density functional (EDF) method Iemploy, the ground state of a mother (target) nucleus isdescribed by solving the Kohn–Sham–Bogoliubov (KSB)equation [33]: (cid:20) h q ( r σ ) − λ q ˜ h q ( r σ )˜ h q ( r σ ) − h q ( r σ ) + λ q (cid:21) (cid:20) ϕ q ,α ( r σ ) ϕ q ,α ( r σ ) (cid:21) = E α (cid:20) ϕ q ,α ( r σ ) ϕ q ,α ( r σ ) (cid:21) , (1)where the KS potentials h and ˜ h are given by the func-tional derivative of the EDF with respect to the parti-cle density and the pair density, respectively. The su-perscript q denotes ν (neutron, t z = 1 /
2) or π (proton, t z = − / z -component of the angularmomentum.The excited states | i i of a daughter nucleus are de-scribed as one-phonon excitations built on the groundstate | i of the mother nucleus as | i i = ˆΓ † i | i , (2)ˆΓ † i = X αβ n X iαβ ˆ a † α,ν ˆ a † β,π − Y iαβ ˆ a ¯ β,π ˆ a ¯ α,ν o , (3)where ˆ a † ν (ˆ a † π ) and ˆ a ν (ˆ a π ) are the neutron (proton) quasi-particle (qp) creation and annihilation operators that aredefined in terms of the solutions of the KSB equation (1)with the Bogoliubov transformation. The phonon states,the amplitudes X i , Y i and the vibrational frequency ω i , are obtained in the proton-neutron quasiparticle-random-phase approximation (pnQRPA). The residualinteractions entering into the pnQRPA equation are givenby the EDF self-consistently. For the axially symmetricnuclei, the pnQRPA equation is block diagonalized ac-cording to the quantum number K = Ω α + Ω β . B. Numerical procedures
To describe the developed neutron skin and the neu-trons pair correlation coupled with the continuum statesthat emerge uniquely in neutron-rich nuclei, I solvethe KSB equation in the coordinate space using cylin-drical coordinates r = ( ρ, z, φ ) with a mesh size of∆ ρ = ∆ z = 0 . ρ max , z max ) = (14 . , .
4) fm. Since I assume furtherthe reflection symmetry, only the region of z ≥ / N ≃ Z nuclei [37–39]. III. RESULTS AND DISCUSSIONA. Dipole excitations
In Fig. 1, presented are the transition-strength distri-butions in Mg as an example of deformed nuclei forthe isovector dipole operators as functions of the excita-tion energy E with respect to ground state of the mother(target) nucleus: S ± ( E ) = X K S ± K ( E ) (4)= X K X i γ/ π R ± i,K { E − [ ~ ω i ± ( λ ν − λ π )] } + γ / , (5) R ± i,K = |h i | ˆ F ± K | i| = |h | [ˆΓ i , ˆ F ± K ] | i| , (6)where λ ν ( λ π ) is the chemical potential for neutrons (pro-tons) and the mass difference between a neutron and aproton is ignored. The charge-exchange dipole operators E (MeV) S t r eng t h ( f m / M e V ) τ -1 τ τ +1 Mg FIG. 1. Calculated distributions of the IV dipole transi-tion strengths by employing the SkM* functional as functionsof the excitation energy with respect to the ground-state of Mg. The smearing parameter γ = 2 MeV is used. are defied asˆ F ± K = X σ,σ ′ ,τ,τ ′ Z d r rY K (ˆ r ) h σ | | σ ′ ih τ | τ ± | τ ′ i ˆ ψ † ( r στ ) ˆ ψ ( r σ ′ τ ′ )(7)in terms of the nucleon field operators. Below, I call the∆ T z = ± F ± .A distinct feature seen in the deformed system is theappearance of two-humped peak structure. In spheri-cal nuclei, the giant resonance has a single peak exceptthe shoulder structure due to the pygmy resonance inneutron-rich nuclei as discussed in Ref. [29]. I am thusgoing to discuss the mechanism for the occurrence of thetwo-humped peak shape for the giant resonance. For ref-erence, the strength distribution for the operator ˆ F K isalso shown in Fig. 1, calculated in the like-particle QRPAframework [40, 41]. Here, ˆ F K is defied for τ in Eq. (7),with τ ± , being the spherical components of the nucle-onic isospin: τ ± = ∓ √ ( τ x ± i τ y ) , τ = τ z . It is notedthat the operator ˆ F K is different from the standard IVdipole operator [16, 42]. We assumed here the Mg nu-cleus is unpaired due to the large deformed shell gapof 12. When the Coulomb interaction is discarded, thetransition-strength distributions for τ ± and τ are iden-tical to one another because the ground-state isospin iszero. Therefore, the origin of the two-humped peak struc-ture may be due to the K -splitting that can be seen in thephotoabsorption cross sections of deformed nuclei [16].However, the intuitive picture of the out-of-phase spatialoscillation of protons and neutrons cannot be applied tothe charge-exchange dipole modes, and I am going toinvestigate further the roles of deformation in other sys-tems below. When the Coulomb interaction is turned on,the chemical potential for protons becomes higher thanthat for neutrons; the difference is 4.61 MeV. The spatialdistribution of neutrons are thus shrunk. These structure changes can be seen in the excitation energy and transi-tion strengths. Let me briefly discuss this point beforeinvestigating the heavier systems.The unperturbed mean-excitation energy for theisovector modes with ∆ T z built on the T = 0 state inan N = Z nucleus can be given by E (0) (∆ T z ) ≃ E (0) (∆ T z = 0) − ∆ T z ∆ E Coul , (8)where ∆ E Coul is the shift of the Coulomb energy per unit Z [16]. In the present framework, the Coulomb-energyshift ∆ E Coul is represented approximately as the differ-ence of the chemical potentials of the mother nucleus λ π − λ ν . The energy shifts of the ∆ T z = 0 and ± N = Z nucleus, and they are about5.2 MeV in the present calculation. Thus, the excitationenergy of the ∆ T z = +1 mode is lower than that of the∆ T z = − T z =+1 excitation are larger than those for ∆ T z = −
1. Sincethe strengths are concentrated into the giant resonance,one can apply the argument based on the sum rule to thequalitative understanding of the imbalanced strengths.The model-independent sum rule for the charge-exchangedipole modes in an axially deformed nucleus is given as Z d E [ S − K ( E ) − S + K ( E )]= π [ N h z i N − Z h z i Z ] ( K = 0)2 38 π [ N h ρ i N − Z h ρ i Z ] ( K = ± , (9)where h·i N ( Z ) denotes the expectation value for neutrons(protons) and a factor of two comes from isospin with thedefinition of Eq. (7). In the spherical limit, the sum rulevalue for each K component coincides with π [ N h r i N − Z h r i Z ]. In the present case, N = Z , the difference inthe transition strengths for the ∆ T z = ± T z = ± T z = 0 mode. A simple RPA analysis for a singlenormal mode employing the separable dipole interactiongives the relation for the transition strengths as12 ( S − + S + ) = (cid:20) O (cid:18) N − ZA (cid:19)(cid:21) S , (10)where S is the transition strength of the ∆ T z = 0mode [16]. Though this relation is model dependent, thepresent self-consistent calculation based on the nuclearEDF satisfies it within 2 % accuracy for each K . Thisimplies that the giant resonances calculated here are col-lective; the microscopically computed giant resonance in Mg can be viewed as a single mode.
15 20 25 30 35 4004080120160200240 E (MeV) A= (a) Nd S t r eng t h ( f m / M e V ) K =0 K =1
15 20 25 30 35 40 E (MeV) A= (b) Sm SkM*
15 20 25 30 35 40 E (MeV) A= (c) Sm SLy4
FIG. 2. Calculated charge-exchange ( τ − ) dipole strength distributions (shifted) in the (a) Nd and (b) Sm isotopes with theSkM* functional, and the (c) Sm isotopes with the SLy4 functional. The strengths for K = 0 and 1 excitations are drawn withthe dashed and dotted line, and the strengths for K = ± γ = 2 MeV isused. Let me investigate further the roles of deformationin the charge-exchange dipole resonance. It is observedabove that a two-humped peak structure of the charge-exchange dipole resonance may have the same origin tothat seen in the photoresonance in a deformed nucleus.It is well established that the photoresonance is split intotwo components with K = 0 and | K | = 1 in a de-formed nucleus with axial symmetry, corresponding tothe oscillations in the direction of the symmetry axisand those in the perpendicular directions. The split-ting is proportional to the magnitude of deformation, andthe shape evolution has been measured in the photoab-sorption cross sections in the rare-earth nuclei [10, 16].Furthermore, the nuclear EDF describes well the shapechange of the GDR in accordance with the developmentof nuclear deformation [41, 43–48]. One can thus ex-pect the shape evolution to see similarly in the calculatedcharge-exchange dipole resonance.Figure 2 shows the transition-strength distributions forthe operator ˆ F − in the Nd and Sm isotopes undergoingthe gradual increase in deformation. In this figure, thestrengths for the K = 0 and | K | = 1 excitations areseparately drawn. Note that the transition strengths for K = ± K = 0 and K =1 are identical to each other at N = 82 and 84. The K -splitting starts to appear at N = 86 in consonancewith the appearance of deformation as shown in Fig. 1 ofRef. [44]. With an increase in the neutron number, thesplitting gets gradually larger as deformation develops.This is akin to the photoresonance characteristic of therare-earth nuclei with shape evolution. The strength distributions in the Nd and Sm isotopescalculated with the use of the SkM* functional shownin Figs. 2(a) and 2(b) are indicative of the similar nu-clear structure of each isotone, such as the single-particlelevels, unperturbed matrix elements and magnitude ofdeformation. In the spherical isotones with N = 82and 84, one sees a shoulder structure. As long as thetotal-strength distributions are observed, the shoulderstructure is indistinguishable from the K -splitting in theweakly-deformed nuclides with N = 86 and 88. The ap-pearance of the shoulder is also seen in the calculatedphotoabsorption cross sections with SkM* and it is sup-pressed in the calculation with the SLy4 [49] and SkP [33]functionals [44]. The detailed structure of single-particlelevels affects the shape of the resonance through the Lan-dau damping mechanism [50]. Figure 2(c) shows thestrength distributions in the Sm isotopes calculated withthe use of the SLy4 functional. Indeed a single peakshows up in the spherical nuclides with SLy4. Compar-ing Figs. 2(b) and 2(c), one further finds that the peakenergy calculated with SLy4 is slightly lower than thatobtained with SkM* likewise in the calculated photoab-sorption cross sections in Ref. [44].To see the relation between the magnitude of deforma-tion and the evaluated splitting energy, I show in Fig. 3the K -splitting in the deformed Sm isotopes. Here, themean excitation energy is calculated by the moments as¯ E = P ES ( E ) P S ( E ) (11)in the energy interval of E < E < E . Here, E and E are set to 15 MeV and 40 MeV. The change of the energy Δ E ( M e V ) β SkM*SLy4SGII Mg U - S m Mg FIG. 3. K -splitting energy for the ∆ T z = − , Mg and
U obtained by using the SkM* functional. -5 0 5 10 15 200246810 0 5 10 15 20 25 Sm total K =0 K =1 S t r eng t h ( f m / M e V ) E (MeV) E (MeV) Sm FIG. 4. As Fig. 2 but for the charge-exchange ( τ +1 ) dipoleoperator in Sm and
Sm. interval by a few MeV varies the evaluated K -spilling byabout 0.1 – 0.2 MeV. This ambiguity, however, does notaffect the discussions below.One clearly sees a linear correlation between the mag-nitude of deformation, i.e. deformation parameter β ,and the K -splitting energy, ∆ E = ¯ E K =1 − ¯ E K =0 . Here,the deformation parameter β is defined by β = 4 π AR Z d r r Y (ˆ r ) ̺ ( r ) (12)with the root-mean-square radius R rms = q A R d r r ̺ ( r ) and the isoscalar particle density ̺ ( r ). The linear correlation is also observed in the cal-culations employing the SLy4 and SGII [51] functionals.While the deformation property calculated with variousfunctionals can be different, three lines lie close to eachother. Note that the K -splitting energy calculated fora light nucleus Mg deviates from a trend of the Smisotopes.Having the neutron excess, the strengths for ˆ F +1 K aresuppressed compared with those for ˆ F − K as seen fromEq. (9); microscopically one is due to the smaller numberof proton hole states available to the dipole excitations;the other is due to the smaller number of neutron particlestates available to the excitations, that is also regarded as the Pauli blocking. Shown in Fig. 4 is examples ofthe transition strength distributions for ˆ F +1 K in the Smisotopes. The strength distributions for K = 0 and | K | =1 are identical to each other apart from a factor of twoin the spherical nucleus Sm. In a deformed nucleus
Sm, the K -splitting occurs as for ˆ F − K , however, it isdifficult to see as a two-humped peak structure of thegiant resonance; it is rather recognized as broadeneing.One can also see the hindrance of strengths in Sm thanin
Sm due to the neutron excess.The effects of neutron excess could give us a deeperunderstanding of the excitation modes, and in what fol-lows I am going to discuss the charge-exchange dipoleresonances in heavy nuclei and neutron-rich nuclei. Theheavy nuclei in mid shells exhibiting a rotational spec-trum are an ideal system to investigate the effects ofdeformation and neutron excess. I thus take the
Unucleus as such an example, and show in Fig. 5 thetransition-strength distributions. The vibrational fre-quency for the ∆ T z = ± T z = 0 state. In the present case, the energy dif-ference comes from the symmetry potential associatedwith the neutron excess as well as the Coulomb energy.The deformation splitting of the GDR can be seen inthe photoabsorption cross section [52], and in the calcu-lations [53, 54]. In Fig. 5, the strength distribution forthe operator ˆ F K is also shown for reference. The meanexcitation energy of the K = 0 and K = 1 excitationsare 11.3 MeV and 13.9 MeV, respectively. Thus, the K -splitting is evaluated as 2.6 MeV. Here, the mean energywas evaluated in the region of E = 5 MeV and E = 35MeV in Eq. (11). One clearly sees a deformation splittingfor the charge-exchange dipole resonance for the operatorˆ F − K similarly to the strength distributions for ˆ F K . Themean excitation energy of the K = 0 and K = 1 exci-tations are 26.6 MeV and 29.3 MeV, respectively, calcu-lated in the energy interval of 15 MeV < E <
45 MeV.Thus, the K -splitting is 2.7 MeV. As seen in Fig. 3, theproportionality between the K -splitting and the magni-tude of deformation lies close to the trend of the Sm iso-topes. Though the peak energy of the ∆ T z = − T z = 0 state, one finds the sameamount of K -splitting for the giant resonance. One can-not see, however, the K -splitting in the ∆ T z = +1 giantresonance. Though the transition strengths are gatheredin low energy, they are quite hindered compared to thoseof the ∆ T z = 0 and ∆ T z = − E <
10 MeV for the response to the operator ˆ F − K .The low-energy dipole states correspond to the − ~ ω excitation [29] uniquely appearing in nuclei with neutronexcess. The excitation of this type is associated with theFermi levels of protons and neutrons being apart by onemajor shell. The lowest energy particle-hole or 2qp exci-tations are thus negative parity. In the present case, theNilsson orbitals stemming from the j / shell are partlyoccupied by neutrons while those from the i / shell are S t r eng t h ( f m / M e V ) E (MeV) τ -1total K =0 K =1 238 U 00.81.6 τ +1 τ E (MeV) FIG. 5. As Fig. 1 but for U. -20 -10 0 10 20 30024681012 S t r eng t h ( f m / M e V ) E (MeV) τ -1total K =0 K =1 40 Mg 00.040.08 τ +1 τ E (MeV) FIG. 6. As Fig. 1 but for Mg. almost empty for protons. Since the number of 2qp ex-citations satisfying the selection rule for the transition isnot large as seen in Ref. [29], the collectivity is weak andthe excitation energy is sensitive to the details of shellstructure around the Fermi levels.At the end of investigation of the dipole resonances innuclei with neutron excess, I discuss the strength distri-butions in neutron-rich exotic nuclei. The Mg nucleushas attracted interest in a possible quadrupole deforma-tion due to the broken spherical magic number of N = 28near the drip line [55, 56]. Theoretically, the deforma-tion properties of the Mg isotopes close to the drip linehave been explored by the Skyrme [57–61], Gogny [62]and relativistic [63] EDF approaches, and the GDR aswell as the LED/PDR are predicted by the Skyrme EDFcalculations [64, 65]. Thus, I investigate here the charge-exchange dipole responses in Mg to see the effects ofdeformation and an extreme neutron excess.Figure 6 displays the strength distributions in Mg.The K -splitting associated with deformation emerges forthe giant resonances. The mean energy of the K = 0 and K = 1 excitations of the giant resonance is 13.2 MeV and16.7 MeV, respectively. Here, the energy interval is setas 0 MeV < E <
40 MeV though this may include theeffect of LED. The calculated K -splitting energy for the∆ T z = − T z = 0 giant resonance. One of the common features in nuclei with neutron excess is thatthe strengths for the ∆ T z = − T z = +1 excitation are suppressed.Furthermore, one sees occurrence of the − ~ ω excita-tion. In the present case, the Nilsson orbitals stemmingfrom the p / and f / shells are mostly occupied by neu-trons while the sd shell is almost empty for protons. Thisis an ideal situation where the negative-parity excitationsappear in low energy.One sees, however, some distinct features in neutron-rich unstable nuclei. The ordering of the excitation ener-gies of the ∆ T z = ± E ∆ T z =0 ≃ E ∆ T z = − < E ∆ T z =+1 ,while one found E ∆ T z =+1 < E ∆ T z =0 < E ∆ T z = − in sta-ble nuclei. This anomalous behavior is due to the highlyimbalanced Fermi levels of protons and neutrons. As onesees in the figure, the strengths appear around −
20 MeVfor the ∆ T z = − ∼
20 MeV for the ∆ T z = +1 excitation. Furthermore,the concentration of transition strengths and a shoulderstructure emerge below the giant resonance. The concen-tration of transition strengths below the giant resonancefor the ∆ T z = − f / shell and the proton continuum states in the g / shell.Since the LED states below 10 MeV for the ∆ T z = 0 exci-tation are generated by the continuum states of neutronsin the g / shell [64] instead of protons, the shoulderstructure below E ∼
10 MeV for the ∆ T z = − B. Spin dipole excitations
Let me investigate briefly the deformation effects onthe spin-dipole (SD) strengths. I show in Fig. 7 thestrength distributions in
Sm and
Sm. Here, thecharge-exchange rank- λ SD operators are defied asˆ F ± λK = X σ,σ ′ ,τ,τ ′ Z d r r [ Y ⊗ σ ] λK h τ | τ ± | τ ′ i ˆ ψ † ( r στ ) ˆ ψ ( r σ ′ τ ′ ) , (13)where [ Y ⊗ σ ] λK = P µν h µ ν | λK i Y µ h σ | σ ν | σ ′ i withthe spherical components of the Pauli spin matrix ~σ =( σ − , σ , σ +1 ). Since there is no K -dependence in spher-ical nuclei, the strengths for each K -component aresummed up in drawing the strength distribution for Sm. The strengths of ± K are summed up as above.The strengths in the τ + channel [Figs. 7(a), 7(c), 7(e)]are hindered compared with those in the τ − channel[Figs. 7(b), 7(d), 7(f)] as in the case for the electric (non-spin-flip) dipole resonance due to the neutron excess.The deformation effects can be observed as broadening E (MeV) S t r eng t h ( f m / M e V ) (cid:1) -1 λ =0(b) 0 10 20 30 40 5001020304050 (cid:0) -1 λ =1(d) 0 10 20 30 40 5001020304050 (cid:2) -1total K =0 K =1 K =2 λ =2(f) Sm Sm total (cid:3) +1 (a) 061218 (cid:4) +1 (c) 061218 (cid:5) +1 (e) FIG. 7. Calculated spin-dipole strength distributions in
Sm and
Sm obtained by using the SkM* functional. of the resonance. Let me discuss below the strengths inthe τ − channel.Irrespective of deformation, the transition strengthsare scattered into three components of λ = 0 , λ = 0 , Sm (
Sm). Here, the energy intervalis set as 0 MeV < E <
50 MeV. The energies calculatedhere including the even- N Sm follow the system-atic trend with λ = 0 being highest and λ = 2 lowest [66].To investigate the strength distributions, it may behelpful to see the model-independent sum rules for theSD operators [18]. The sum rules generalized to the de-formed systems read Z d E [ S − λK ( E ) − S + λK ( E )]= π [ N h r i N − Z h r i Z ] ( λ = 0 , K = 0)2 38 π [ N h ρ i N − Z h ρ i Z ] ( λ = 1 , K = 0)2 316 π [ N h ρ + 2 z i N − Z h ρ + 2 z i Z ] ( λ = 1 , K = ± π [ N h ρ + 4 z i N − Z h ρ + 4 z i Z ] ( λ = 2 , K = 0)2 316 π [ N h ρ + 2 z i N − Z h ρ + 2 z i Z ] ( λ = 2 , K = ± π [ N h ρ i N − Z h ρ i Z ] ( λ = 2 , K = ± , (14)and coincide with 12 π [ N h r i N − Z h r i Z ] in the sphericallimit for each K -component as given in Ref. [18]. Inderiving the formulae, the time-reversal symmetry of theground state was assumed.The strength distribution for the rank-0 SD operatoris shown in Fig. 7(b). Since the rank-0 operator is scalar,one has no K -dependence even in deformed nuclei. In- deed, the line shapes for Sm and
Sm are similar toeach other. One sees K -splitting in the strength distribu-tions for the rank-1 and 2 operators as shown in Figs. 7(d)and 7(f). The simple geometrical argument for the K -splitting in the electric dipole resonance cannot be ap-plied. In the case of the non-spin-flip dipole resonance,the K -dependence comes from the spherical harmonics Y K representing the nuclear shape in real space. Whenthe spin degree of freedom is involved in the case of theSD resonance, the K quantum number does not directlycharacterize the nuclear deformation represented by µ inthe definition of the operator Eq. (13). However, a qual-itative argument on the K -dependence of the strengthscan be given according to the sum rules. To first orderin deformation, one can express h z i = h r i (1 + δ )and h ρ i = h r i (1 − δ ) with δ representing the defor-mation parameter [16]. The K = 0 strength is reducedby π ( N h r i N − Z h r i Z ) δ while the K = 1 strength isenhanced by π ( N h r i N − Z h r i Z ) δ for λ = 1. For λ = 2, the K = 0 and K = 1 strengths are enhancedby π ( N h r i N − Z h r i Z ) δ and π ( N h r i N − Z h r i Z ) δ ,respectively and the K = 2 strength is reduced by π ( N h r i N − Z h r i Z ) δ . The summed strengths for each λ are unchanged within this approximation.One sees a two-peak structure of the SDR in Sm anda broad resonance structure in
Sm for the λ = 1 excita-tion. As expected from the sum rules, the K = 0 strengthis reduced while the K = 1 strength is enhanced due todeformation. Broadening of the resonance can be seenalso for the λ = 2 excitation. One observes that the splitstates are overlapping. It is clearly seen that the K = 2strength decreases. The precedent nuclear EDF calcula-tions [67–70], though restricted to spherical nuclei, pre-dict that the fragmentation increases with λ . Followingthe early findings, one sees that the strengths for λ = 2are fragmented in Sm. Therefore, the deformation-induced broadening is unlikely to observe experimentallyif the spreading width is & λ and reducing the continuum background to extract thedetails of resonance structure [71–75]. IV. SUMMARY
The deformation effects on the charge-exchange elec-tric (non-spin-flip) and magnetic (spin-flip) dipole ex-citations were investigated by means of the fully self-consistent pnQRPA with the Skyrme EDF. I found thatthe deformation splitting into K = 0 and K = ± K -splitting shows up also for the charge-exchange magnetic dipole resonance. However, a simplegeometrical assertion valid for the electric cases cannotbe applied for explaining the vibrational frequencies ofeach K -component due to the coupling of spin and angu-lar momentum in the magnetic excitations. The model-independent non-energy-weighted sum rules were derivedfor the axially-deformed nuclei, and a qualitative argu- ment on the structure of strength distributions for each K -component was given. The IVGDR and IV low-energyoctupole resonance can couple in deformed nuclei, andthe shoulder structure in the octupole resonance is pre-dicted to appear due to these coupling [41]. It is thusan interesting future study to see if the coupling effectsshow up generally in the charge-exchange octupole res-onances in deformed nuclei. In nuclei with an appre-ciable neutron excess, I found the concentration of thedipole strengths in low energy and a shoulder structurebelow the giant resonance. These modes of excitation areunique in neutron-rich unstable nuclei and can emerge indeformed nuclei as well as in spherical systems [29]. ACKNOWLEDGMENTS
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