Microscopic analysis of low-energy spin and orbital magnetic dipole excitations in deformed nuclei
V.O. Nesterenko, P.I. Vishnevskiy, J. Kvasil, A. Repko, W. Kleinig
aa r X i v : . [ nu c l - t h ] F e b Microscopic analysis of the low-energy M K = 1) spin and orbital scissors modes V.O. Nesterenko , , , P.I. Vishnevskiy , , , J. Kvasil , A. Repko and W. Kleinig Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow region, 141980, Russia State University ”Dubna”, Dubna, Moscow Region, 141980, Russia Moscow Institute of Physics and Technology, Dolgoprudny, Moscow region, 141701, Russia ∗ Institute of Nuclear Physic Almaty, Almaty Region, Kazakhstan Institute of Particle and Nuclear Physics, Charles University, CZ-18000, Praha 8, Czech Republic and Institute of Physics, Slovak Academy of Sciences, 84511, Bratislava, Slovakia (Dated: March 1, 2021)A low-energy M K = 1) spin-scissors resonance (SSR) located just below the ordinary orbitalscissors resonance (OSR) was recently predicted in deformed nuclei within the Wigner Function Mo-ments (WFM) approach. We analyze this prediction using fully self-consistent Skyrme QuasiparticleRandom Phase Approximation (QRPA) method. The Skyrme forces SkM*, SVbas and SG2 are im-plemented to explore SSR and OSR in , , Dy and
Th. The calculations show that isotopes , , Dy indeed have at 1.5-2.4 MeV (below OSR) K π = 1 + states with a large M K = 1)spin strength. These states are dominated by pp [411 ↑ , ↓ ] and nn [521 ↑ , ↓ ] spin-flip config-urations corresponding to pp [2 d / , d / ] and nn [2 f / , f / ] structures in the spherical limit. Sothe predicted SSR is actually reduced to low-orbital (l=2,3) spin-flip states. Moreover, following ouranalysis and in contradiction with the spin-scissors treatment of WFM, the deformation is not theprinciple origin of the low-energy spin M K = 1) states but only a factor affecting their features.In Th, the M K = 1) spin strength is found very small. The spin and orbital strengths aregenerally mixed and exhibit the interference: weak destructive in SSR range and strong constructivein OSR range. The two groups of 1 + states observed experimentally at 2.4-4 MeV in , , Dyand at 2-4 MeV in
Th are rather explained by fragmentation of the orbital strength than by theoccurrence of spin-flip states. The best agreement with the experimental data is obtained for theforce SG2.
PACS numbers: 13.40.-f, 21.60.Jz, 27.70.+q, 27.80.+w
I. INTRODUCTION
Magnetic dipole ( M
1) excitations in nuclei provide im-portant information on the nuclear spin and orbital mag-netism [1–3]. For a long time, these excitations weremainly represented by M K = 1) spin-flip giant reso-nance located at the energy E ≈ A − / MeV [1–3] andlow-energy M ≈ δA − / MeV [4–9] where δ is the parameter ofnuclear axial quadrupole deformation. Both resonancesare isovector. In deformed nuclei, they are formed by K π = 1 + states ( K is the projection of the total nu-clear moment to the symmetry z-axis) with enhanced M K = 1) transitions to the ground state.The spin-flip and orbital resonances have a differentorigin. The former is produced by particle-hole spin-fliptransitions between spin-orbit partners in the proton andneutron single-particle spectra. This resonance is relatedto the spin nuclear magnetic properties and it exists inboth spherical and deformed nuclei [2, 3]. Instead, OSR istreated as scissors-like out-of-phase oscillations of protonand neutron deformed subsystems, see Fig. 1a. Thisisovector mode can exist only in deformed nuclei [4–6].It is not caused by spin excitations and so represents aremarkable example of nuclear orbital magnetism. ∗ Electronic address: [email protected] p p n n a) OSR n p n p b) SSR-I n pn p c) SSR-II
FIG. 1: The schemes for the members of the scissors triple[26]: OSR (a), SSR-I (b) and SSR-II (c). The neutron (pro-ton) axially deformed fractions are shown by light (dark) bars.The spin direction of nucleons is indicated by arrows. Eachmode in the triple exhibits scissors-like oscillations of twoblades: neutrons vs protons in OSR, spin-up vs spin-down nu-cleons in SSR-I (spins of neutrons and protons in each bladehave the same direction), and SSR-II oscillations where theneutron and proton spins in each blade have opposite direc-tions.
Last years, the spin-flip resonance was widely appliedto test various self-consistent schemes (with Skyrme,Gogny and relativistic functionals) in the spin channel[3, 10–14], to explore effects of tensor forces [10, 11, 15]and spin-orbit interaction [10–12, 16]. The study of OSRwas also very active, see e.g. [17–19]. Note that OSR isthe remarkable example of mixed-symmetry states [20].A decade ago, E.B. Balbutsev, I.V. Molodtsova, and P.
2p M1 M1 M1spin-orbitsplitting deformationsplittingspin-fliptrans. orb. sciss.trans. (cid:512)(cid:512)
FIG. 2: The scheme of single-particle levels of 2 p subshell inthe spherical (left) and deformed (right) cases. The schemecorresponds to the proton 2 p subshell in Dy, calculatedwith the Skyrme force SG2. In the deformed case, the axialquadrupole deformation β =0.346 is used. The spin-flip andorbital scissors M Schuck have predicted (within the WFM method) thatOSR should be supplemented by a low-energy spin scissormode (SSR) [21]. The next WFM calculations, with in-clusion of the pairing [22–24] and isoscalar-isovector cou-pling in the spin-spin residual interaction [25, 26], haveshown that SSR should have two branches, (see Fig. 1b,c)lying below OSR. Thus altogether the nuclear scissorsmode should have a triple structure: OSR + two SSRbranches. All the scissors states should demonstrate sig-nificant M K = 1) transitions to the ground state. Be-cause of the scissors nature, these states can exist onlyin deformed nuclei.Following the WFM calculations, SSR should exist inmedium and heavy axial deformed nuclei, typically atthe excitation energy E < K π = 1 + states at E < , , Dy,
Th and , U [24–26], thenuclei
Dy and
Th are the most promising candi-dates for SSR. Indeed low-energy 1 + states in these nu-clei form two distinctive groups which in principle couldbe attributed to SSR and OSR.The aim of the present paper is to scrutinize the WFMprediction of SSR from the microscopic viewpoint. It iswell known that both orbital and spin-flip M1 transitionscan be explained using single-particle schemes [2, 32].The example of such scheme for 2 p -subshell is shown inFig. 2. This is the fraction of the proton scheme in Dy, calculated with the Skyrme parametrization SG2[33]. The computed equilibrium axial quadrupole defor- mation is β =0.346. The left part of the figure showsthe splitting of 2 p -subshell into 2 p / and 2 p / levelsdue to the spin-orbit interaction. Already in this spher-ical case, a spin-flip M p / and upshifts the level 2 p / (right part of Fig.2). In this case, two M K = 1) transitions are pos-sible: spin-flip 3 / − [301 ↑ ] → / − [301 ↓ ] and orbital1 / − [310 ↑ ] → / − [301 ↑ ]. The former connects thespin-orbit partners, the latter relates the levels arisingdue to the deformation splitting. So we get two natu-ral candidates for SSR and OSR. Because of the largedeformation splitting, the orbital transition has a largerenergy than the spin-flip one. So SSR should lie lower byenergy than OSR.As seen in Fig. 1 b-c, spins of neutrons and protonsin the scissors blades have different directions. So WFMcorrectly predicts SSR as spin-flip transitions in neutronand proton schemes (though this was not mentioned inthe WFM publications). However, Fig. 2 shows that, incontradiction with WFM picture, the deformation is notthe primary origin of SSR (though it can significantlyaffect its features). Indeed, low-energy spin-flip transi-tions can exist in spherical nuclei as well and we do notneed here the WFM treatment in terms of deformation-induced scissors oscillations.The main aim of the present study is to show thatthe predicted SSR can be reduced to ordinary spin-flipstates which in principle can exist in both deformed andspherical nuclei. Moreover, we will propose another, ascompared to WFM calculations, explanation of the avail-able experimental data. Our analysis is performed foraxially deformed nuclei , , Dy and
Th. Two ofthese nuclei,
Dy and
Th are considered by WFM aspromising candidates for SSR. The calculations are per-formed within fully self-consistent Quasiparticle RandomPhase Approximation (QRPA) [36–41] with the Skyrmeforces SG2 [33], SkM* [42], and SVbas [43]. As shownbelow, the spin and orbital low-energy M K = 1) ex-citations are strongly mixed. So we will analyze bothSSR and OSR. Moreover, to demonstrate the accuracyof our calculations, we will also present the results forthe spin-flip giant resonance.The paper is organized as follows. In Sec. II, thecalculation scheme is outlined. In Sec. III, results of thecalculations are presented and discussed. In Sec. IV, theconclusions are done. In Appendix A, the descriptionof M K = 1) spin-flip giant resonance is illustrated.In Appendix B, the expressions for the orbit and spintransition matrix elements are given. II. CALCULATION SCHEME
The calculations were performed within the self-consistent QRPA model [38–41] based on the Skyrmefunctional [36, 37]. The model is fully self-consistent.Both mean field and residual interaction are derived from
TABLE I: Isoscalar effective mass m ∗ , isoscalar and isovectorspin-orbit parameters b and b ′ , proton and neutron pairingconstants G p and G n , and the type of pairing in the Skyrmeforces SkM*, SVbas, and SG2.force m ∗ b b ′ G p G n pairingMeV fm MeV fm MeV fm MeV fm SkM* 0.79 65.0 65.0 279.1 259.0 volumeSVbas 0.90 62.32 34.11 674.6 606.9 surfaceSG2 0.79 52.5 52.5 279.0 259.0 volumeTABLE II: The calculated parameters β of the equilibriumaxial quadrupole deformation vs the experimental values [46].Nucleus β SkM* SVbas SG2 Exper.
Dy 0.339 0.331 0.333 0.334 (2)
Dy 0.351 0.345 0.346 0.341(3)
Dy 0.354 0.348 0.354 0.349(3)
Th 0.256 0.247 0.241 0.248 (6) the initial Skyrme functional. The residual interactiontakes into account all the terms of the Skyrme functionaland Coulomb (direct and exchange) parts. Both particle-hole and particle-particle channels are included [39]. Thespurious admixtures caused by violation of the rotationalinvariance are removed using the technique [41].The representative set of Skyrme forces is used. Weemploy the standard force SkM* [42], recently devel-oped force SVbas [43], and the force SG2 [33] which isoften used in the analysis of magnetic excitations, seee.g. [11, 12, 44]. As seen from Table I, these forces havedifferent isoscalar b and isovector b ′ parameters of thespin-orbit terms in the Skyrme functionals (see defini-tions of the parameters in Refs. [11, 37]). In SkM* andSG2, the usual convection b = b ′ is used while in SVbasthe separate tuning of b and b ′ is done. These threeSkyrme forces generally reproduce the two-hump struc-ture of M1(K=1) spin-flip giant resonance in open-shellnuclei [11, 12]. As shown in Appendix B, SVbas and es-pecially SG2 give a nice description of this resonance. Soperhaps these two Skyrme forces are most robust for thepresent study.The calculations are performed for axially deformednuclei , , Dy and
Th. The nuclear mean fieldand pairing are computed with the code SKYAX [45]using a two-dimensional grid in cylindrical coordinates.The calculation box extends up to three times the nu-clear radii, the grid step is 0.4 fm. The axial quadrupoleequilibrium deformation is obtained by minimization ofthe energy of the system. As seen from Table II, the ob-tained values of the deformation parameters β are in agood agreement with the experimental data [46], espe-cially for SVbas. All the forces reproduce grow of thedeformation from Dy to
Dy.The pairing is described by the zero-range pairing in-
TABLE III: Proton and neutron pairing gaps ∆ p and ∆ n andenergy of 2 +1 state of the ground state rotational band, calcu-lated in Dy and
Th with Skyrme forces SkM*, SVbas,and SG2. The experimental data for the energy E +1 are fromthe database [46].Nucleus SkM* SVbas SG2 exper.∆ p [MeV] 0.55 0.69 0.54 Dy ∆ n [MeV] 0.62 0.95 0.77 E +1 [keV] 67.9 92.7 78.9 80.7∆ p [MeV] 0.53 0.61 0.63 Th ∆ n [MeV] 0.54 0.80 0.72 E +1 [keV] 41.2 57.1 51.9 49.4 teraction [47] V q pair ( r , r ′ ) = G q h − η (cid:16) ρ ( r ) ρ pair (cid:17)i δ ( r − r ′ ) (1)where G q are proton ( q = p ) and neutron ( q = n ) pairingstrength constants. They are fitted to reproduce empiri-cal pairing gaps obtained by the five-point formula alongselected isotopic and isotonic chains. The concrete valuesof G q are shown in Table I. Further, ρ ( r ) = ρ p ( r ) + ρ n ( r )is the sum of proton and neutron densities. We getso-called volume pairing for η =0 and density-dependentsurface pairing for η =1. As indicated in Table I, theformer is used in SkM* and SG2, and the latter is ex-ploited in SVbas. In the latter case, we use SVbas pa-rameter ρ pair =0.2011 fm − . The pairing correlations areincluded at the level of the iterative HF-BCS (Hartree-Fock plus Bardeen-Cooper-Schrieffer) method [39]. Tocope with the divergent character of zero-range pairingforces, energy-dependent cut-off factors are used [39, 47].Table III shows the calculated averaged proton andneutron pairing gaps ∆ p and ∆ n (defined in Eq. (30)of Ref. [47]) in Dy and
Th. Also we exhibit theenergies E +1 = 3 ~ / J (with J being the nuclear momentof inertia) of the first I π = 2 + state in ground staterotational band. These energies are sensitive to bothdeformation and pairing. As seen from Table III, SVbasgives the largest gaps and so should lead to the moststrong pairing effect. The pairing in SVbas is maybetoo strong since this force significantly overestimates theexperimental E +1 -values. The best agreement for theenergies is obtained for SG2.In our calculations, QRPA is implemented in the ma-trix form. The large configuration space is used. Thesingle-particle spectrum extends from the bottom of thepotential well up to 30 MeV. For example, in SG2 cal-culations for Dy, almost 691 proton and 800 neu-tron single-particle levels are used. The two-quasiparticle(2qp) basis in QRPA calculation for K π = 1 + states in-cludes 5270 proton and 9527 neutron configurations.The reduced probability for M K = 1) transitions( M
11 in the short notation) from the ground state | i to SG2 Dy orb a ) B ( M ) [ m N ] B ( M ) [ m N ] B ( M ) [ m N ] b ) B ( M ) [ m N ] Dy Dy c ) d ) spin e ) f ) g ) total h ) i ) j ) E [MeV] exper k ) E [MeV] l ) E [MeV]
FIG. 3: Orbital (a,b,c), spin (d,e,f) and total (g,h,i) low-energy M11 strength in , , Dy, calculated in QRPA with Skyrmeforce SG2. In the bottom panels, the experimental M11 strength for
Dy [27] and , Dy [28] is shown. the excited QRPA state | ν i with I π K = 1 + B ν ( M
11) = 2 | h ν | ˆΓ( M | i | . (2)The coefficient 2 means that contributions of both projec-tions K=1 and -1 are taken into account. The transitionoperator has the formˆΓ( M
11) = µ N r π X qǫp,n [ g qs ˆ s ( µ = 1) + g ql ˆ l ( µ = 1)] (3)where µ N is the nuclear magneton, ˆ s ( µ = 1) and ˆ l ( µ = 1)are µ =1 projections of the standard spin and orbital op-erators, g qs and g ql are spin and orbital gyromagnetic fac-tors. We use the quenched spin g-factors g qs = η ¯ g qs where¯ g ps = 5.58 and ¯ g ns =-3.82 are bare proton and neutrong-factors and η =0.7 is the familiar quenching parameter[2]. The orbital g-factors are g pl = 1 and g nl = 0. In whatfollows, we consider three relevant cases: spin ( g ql = 0),orbital ( g qs = 0), and total (when both spin and orbitaltransitions are taken into account). The expressions forthe orbital and spin M
11 matrix elements are given inthe Appendix B.In deformed nuclei, electric and magnetic states withthe same K π are mixed [2, 8, 9, 32, 48]. In our caseof K π = 1 + states, the modes of multipolarities M11and E21 can be mixed. To estimate this mixing, we alsocalculate the reduced probability of E21 transitions B ν ( E
21) = 2 | h ν | ˆΓ( E | i | (4)with the transition operatorˆΓ( E
21) = e X qǫp,n e q eff r Y ( θ, φ ) (5) where Y ( θ, φ ) is the spherical harmonic and e q eff are ef-fective charges. Here we use e p eff =1 and e n eff =0. III. RESULTS AND DISCUSSIONA. M strength in , , Dy In Figure 3, we compare the calculated orbital, spinand total M11 strengths (2) in , , Dy with the ex-perimental data from the nuclear resonance fluorescence(NRF) reaction, see Refs. [27] for
Dy and [28] for , Dy. The QRPA results are obtained for the forceSG2. Following the discussion in Sec. II and results forthe spin-flip M
11 giant resonance in the Appendix A,this force seems to be most relevant for our analysis.The plots (a-c) of the figure show that M
11 strengthabove 2.4 MeV is mainly orbital. Just this strength con-stitutes the OSR. Instead, a few states at E < + states at E <
Dy [27] give 1 + states B ( M ) [ m N ] SkM * orb SG2 a ) b ) SVbas Dy c ) d ) B ( M ) [ m N ] spin e ) f ) g ) B ( M ) [ m N ] total h ) i ) j ) B ( E ) [ e f m ] E [MeV] k ) E [MeV] l ) E [MeV]
FIG. 4: Orbital (a,b,c), spin (d,e,f) and total (g,h,i) low-energy M11 strength in
Dy, calculated in QRPA with Skyrme forcesSkM* (left), SVbas (middle) and SG2 (right). In the bottom panels, the quadrupole E21 strength is shown. only for
E > + states at a lower exci-tation energy.In , Dy, the recent NRF data [28] give two groupsof 1 + states located above and below 2.7 MeV. The for-mer group is usually treated as OSR. The latter is treatedby WFM as SSR [23–26]. Note that low-energy groupsof 1 + states were observed in various rare-earth nucleialready a long time ago [27]. The recent Oslo ( γ, n ) ex-periments [30] show that, at the energy range 0-4 MeV in Dy, 40 −
60% of M M
11 strength in
Dy at0 - 4 MeV achieves 6.17 µ N [28] which substantially ex-ceeds the values 3 - 4 µ N typical for OSR in well-deformedrare-earth nuclei. This observation was treated by WFMteam as a clear signature of SSR in Dy. However, fol-lowing our results in Fig. 3, the states at 2.4-2.7 MeVgive mainly orbital M
11 transitions and so should alsobelong to OSR. They are omitted in OSR systematicswith the lower boundary 2.7 MeV [48] but taken into ac-count for the lower boundary 2.5 MeV [17]. So, by ouropinion, the findings of Oslo group cannot be consideredas a strong argument in favor of SSR.In Figure 4, we demonstrate the distribution of M11strength in
Dy, calculated with Skyrme forces SkM*,SVbas, and SG2. It is seen that, despite some deviationsin details, all these three forces give qualitatively similarresults. In all the cases, there is the region 0-2.4 MeVwith the essential spin strength and the region 2.4-4.0MeV with the dominant orbital strength. Fig. 4 alsodemonstrates E
21 strength for the same K π = 1 + states.This strength is large at 2.4-4.0 MeV and negligible at 0-2.4 MeV. The former result is typical for OSR [7–9, 48].It means that OSR states are some mixtures of M E
21 modes, which is usual in well deformed nuclei.Further, in Table IV, we show spin, orbital, and totalQRPA strengths P B ( M
11) summed in the SSR (0 -2.4 MeV), OSR (2.4 - 4 MeV) and SSR+OSR (0 - 4MeV) energy intervals. The total QRPA strengths arecompared with NRF experimental data where 1 + stateswere observed at 2.8 - 3.1 MeV in Dy [27]), 2.3 - 3.1MeV in
Dy [28] and 2.5 - 3.8 MeV in
Dy [28].Table IV demonstrates that at 0 - 2.4 MeV the spinstrength dominates over the orbital one. In some cases(SkM* in , , Dy and SG2 in
Dy), the orbitalstates fall into this energy interval and make the orbitalfraction also essential. In OSR region 2.4 - 4 MeV, theorbital M
11 strength strongly dominates in all the con-sidered cases though the minor spin strength is also no-ticeable.Following Table IV, the QRPA total M
11 strengthssummed at 0-4 MeV essentially overestimate the exper-imental values in , Dy but generally correspond tothe experiment in
Dy (SkM* and SG2). Perhaps, asmentioned above, the experimental data for , Dy[27, 28] miss a significant part of M
11 strength. Fur-ther, the present calculation do not take into account thecoupling with complex configurations which can spreadthe strength and so decrease the P B ( M TABLE IV: The calculated orbital, spin, total strengths P B ( M
11) (in µ N ) summed at SSR (0-2.4 MeV), OSR (2.4-4 MeV)and total (0-4 MeV) energy ranges in , , Dy as compared with the experimental data for
Dy [27] and , Dy [28].For each energy range, the interference factors R are shown.Nucleus Force 0-2.4 MeV 2.4-4 MeV 0-4 MeV P B ( M
11) R P B ( M
11) R P B ( M
11) Rorb spin total orb spin total orb spin total expSkM* 0.52 0.96 1.32 0.89 2.79 0.55 4.85 1.45 3.31 1.51 6.16 1.28
Dy SVbas 0.05 0.49 0.23 0.43 2.15 0.51 3.80 1.43 2.20 1.00 4.03 2.42 1.26SG2 0.71 0.85 1.37 0.88 2.04 0.28 4.35 1.88 2.75 1.12 5.72 1.48SkM* 0.80 1.09 1.80 0.95 2.69 0.51 4.63 1.45 3.49 1.60 6.44 1.27
Dy SVbas 0.06 0.73 0.45 0.57 2.35 0.40 4.04 1.47 2.41 1.14 4.49 3.45 1.26SG2 0.04 0.90 0.63 0.67 3.19 0.40 5.16 1.44 3.23 1.30 5.79 1.28SkM* 0.96 1.09 2.11 1.03 2.18 0.40 3.94 1.53 3.14 1.49 6.05 1.31
Dy SVbas 0.06 0.63 0.32 0.47 2.52 0.50 4.37 1.45 2.57 1.13 4.69 6.17 1.27SG2 0.04 0.87 0.57 0.63 3.61 0.43 5.73 1.42 3.65 1.30 6.30 1.27
III in Sec. II), which upshifts a part of M
11 strengthabove 4 MeV. In general, the most reasonable results areobtained for the force SG2.In both SSR and OSR regions, we see the interfer-ence between spin and orbital contributions to the totalstrength (the sum of spin and orbital contributions doesnot equal to the total strength). It is convenient to esti-mate this effect by the interference factor R = P B ( M t P B ( M o + P B ( M s (6)where P B ( M o , P ( M s and P B ( M t aresummed orbital, spin and total reduced transition prob-abilities. The interference is destructive at R <
1, con-structive at
R > R = 1.Table IV shows that the interference is destructivein SSR range (with exception of SkM* case in Dy),strongly constructive in ORS range and somewhat lessconstructive in the total SSR+OSR range.
The interfer-ence greatly increases the role of the minor spin fractionin the OSR range.
For example, in
Dy (SG2), the in-terference results in the total strength 4.35 µ N which ismore than twice larger than the orbital strength 2.05 µ N .Our results generally agree with the study of low-energy (0 - 4 MeV) 1 + states in , , Dy, per-formed within the Quasiparticle-Phonon Nuclear Model(QPNM) [49]. This model is not self-consistent but hasthe advantage to take into account the coupling withcomplex configurations. In agreement with our results,QPNM also predicts in Dy isotopes a well separatedgroup of 1 + states located at 2-2.6 MeV and carryinga noticeable fraction of spin M1 strength. The totalstrength of these states is mainly orbital. The only ex-ception is two states at 2.0 -2.1 MeV in Dy wherespin contribution to M + states. They are shown for Dy in TableV. In particular, we inspect two states with the largestspin strength B ( M s and one state with the largestorbital strength B ( M o . In the spin states, we have B ( M s > B ( M o . Their main 2qp components, pro-ton [411 ↑ , ↓ ] and neutron [521 ↑ , ↓ ], are of thespin-flip character and correspond to the particle-hole(1ph) transitions. Note that the same spin-flip configu-rations were found in QPNM calculations [49] for low-energy 1 + states in Dy and
Dy. In the spher-ical limit, these configurations are reduced to spin-flippartners 2 d / , d / and 2 f / , f / , i.e. we deal withlow-moment subshells l =2 and 3. Altogether this meansthat so called SSR states are actually ordinary low-energynon-collective spin-flip excitations which do not need fortheir explanation the scissors-like treatment . In what fol-lows, we will call these states as low-moment spin-flip(LMSF) ones.The orbital and spin-flip M
11 transitions in
Dycan be illustrated using the neutron and proton single-particle level schemes. In Fig. 5, we show the pro-ton scheme for 2 d subshell, calculated with SG2 at theequilibrium deformation β =0.346. This scheme demon-strates the same physical mechanisms as in Fig. 2 butnow for the case including the proton spin-flip transi-tion 3 / + [411 ↑ ] → / + [411 ↓ ] of our interest. We seethat the low-energy spin-flip transition 2 d / → d / cantake place already in the spherical case. In the deformedcase, two spin-flip and three orbital M1 transitions arepossible. However, only two of these transitions are of1ph character and so not suppressed (other transitionscan appear only due to the pairing). They are spin-flip 3 / + [411 ↑ ] → / + [411 ↓ ] and orbital 3 / + [411 ↑ ] → / + [402 ↑ ] (marked by red in the figure). Asseen from Table V, the proton spin-flip 2qp configuration TABLE V: Characteristics of some relevant low-energy K πν = 1 + ν states in Dy, calculated within QRPA with the forcesSkM*, SVbas and SG2. For each state, we show the excitation energy E , orbital, spin and total reduced transition probabilities B ( M
11) and main 2qp components (contribution to the state norm in %, structure in terms of Nilsson asymptotic quantumnumbers, position of the involved single-particle states relative to the Fermi level F, and original quantum subshells in thespherical limit). Force ν E B ( M
11) [ µ N ] main 2qp components[MeV] orb spin total % [ N, n z , Λ] F-position spher. limitSkM* 3 1.95 0.05 0.29 0.11 69 pp [411 ↑ , ↓ ] F − , F + 1 2 d / , d /
30 nn [521 ↑ , ↓ ] F − , F + 2 2 f / , f / ↑ , ↓ ] F − , F + 2 2 f / , f /
28 pp [411 ↑ , ↓ ] F − , F + 1 2 d / , d / ↑ , ↑ ] F − , F + 4 2 f / , f /
25 pp [411 ↑ , ↑ ] F − , F + 4 2 d / , d / SVbas 1 1.88 0.05 0.54 0.27 97 pp [411 ↑ , ↓ ] F, F + 1 2 d / , d / ↑ , ↓ ] F − , F + 2 2 f / , f / ∼ ↑ , ↓ ] F − , F + 2 2 f / , f / ↑ , ↓ ] F, F + 1 2 d / , d / ↑ , ↑ ] F − , F + 4 2 f / , h /
16 pp [413 ↓ , ↓ ] F − , F + 4 1 g / , g / SG2 1 1.83 0.04 0.62 0.35 98 pp [411 ↑ , ↓ ] F, F + 1 2 d / , d / ∼ ↑ , ↓ ] F − , F + 2 2 f / , f / ↑ , ↑ ] F − , F + 4 2 f / , h /
32 pp [413 ↓ , ↓ ] F − , F + 4 1 g / , g / spin-orbitsplitting deformationsplittingspin-fliptransitions orb. scissorstransitions
1/ 2 [411 ]1/ 2 [420 ] FF-6F+1
F+5
F+9
FIG. 5: The calculated (SG2) scheme of spin-flip (left emptyarrows) and orbital scissors (right filled arrows) M
11 transi-tions in the proton 2 d subshell in Dy. As indicated in thetop inscriptions, the left part of the figure demonstrates thespin-orbit splitting into 2 d / and 2 d / levels in the sphericalcase, while the right part exhibits the additional deformation( β =0.346) splitting. In the deformed case, M
11 transitionsform two groups, spin-flip and orbital scissors, as indicatedin the bottom prescriptions. The Fermi level is 3 / + [411 ↑ ].The 1ph transitions, spin-flip 3 / + [411 ↑ ] → / + [411 ↓ ] andorbital 3 / + [411 ↑ ] → / + [402 ↑ ], are marked by red color. [411 ↑ , ↓ ] indeed dominates in the states at 1.95-MeV(SkM*), 1.88-MeV (SVbas), and 1.83-MeV (SG2) states.The orbital configuration 3 / + [411 ↑ ] → / + [402 ↑ ] isfragmented between many states, it is seen e.g. in 3.09-MeV state (SkM*). Since deformations in , , Dy spin-orbitsplitting deformationsplittingspin-fliptransitions orb. scissorstransitions
1/ 2 [521 ] F+4F-8F+2F+9F+10
1/ 2 [530 ]
F+20F-1
FIG. 6: The same as in Fig. 5 but for the neutron 2 f subshellin Dy. The 1ph spin-flip 3 / − [521 ↑ ] → / − [521 ↓ ] andorbital 3 / − [521 ↑ ] → / − [512 ↑ ] transitions are marked byred color. are similar (see Table II), the same results should takeplace for Dy and
Dy as well.The similar analysis can be done for the neutron single-particle scheme in
Dy. The relevant part of thisscheme, namely the 2f subshell, is shown in Fig. 6.We see that again, between many possible spin-flip andorbital M1 transitions, there are only two 1 ph transi-tions: spin-flip 3 / − [521 ↑ ] → / − [521 ↓ ] and orbital3 / − [521 ↑ ] → / − [512 ↑ ]. The corresponding configu-rations are indeed seen in Table V.It is easy to recognize from Fig. 6, that Dy and
Dy whose Fermi levels correspond to F-1 and F+1states of the given neutron scheme, also allow 1ph spin-flip transitions 3 / − [521 ↑ ] → / − [521 ↓ ]. This explainsour result that all three isotopes , , Dy demon-strate rather similar distributions of low-lying spin-flipexcitations. B. M strength in Th In addition to strongly deformed Dy isotopes, the SSRwas also predicted by WFM in less deformed nucleus
Th [23, 24, 26]. In this nucleus, the experiment [31]also gives two separate groups of low-energy 1 + states(see e.g. plot (d) in Fig. 7). The lower group locatedat E <
Th using theforces SVbas and SG2. Note that these forces, especiallySG2, provide a good description of the spin-flip M1 giantresonance in
Th, see Appendix A.In Fig. 7, the computed orbital, spin, and total B ( M
11) strengths in
Th are compared with NRF ex-perimental data [31]. We see that, in this nucleus, thespin strength is much smaller that the orbital one even at
E <
E < P B ( M
11) somewhat overestimate theexperimental data. As mentioned in Sec. III-A, the over-estimation can be caused by i) missing of a significantpart of M
11 strength in the experiment and ii) neglectof the coupling with complex configurations. Also, likein Dy isotopes, we see in
Th the constructive inter-ference of the spin and orbital contributions to the totalstrength.Further, in Table VII, we show the calculated featuresof some representive states with the large spin and orbitalstrength. In SVbas case, the first state is not spin-flipdespite it has the largest spin strength at
E < orb d) orb Th SVbas a) spin b) spin totalc)total exper B ( M ) [ m N ] B ( M ) [ m N ] B ( M ) [ m N ] B ( M ) [ m N ] E [MeV]exper e) SG2 g) f) h) E [MeV]
FIG. 7: The computed (SVbas, SG2) orbital, spin and to-tal low-energy M11 strengths in
Th as compared with theexperimental data [31].TABLE VI: The computed orbital, spin and total B ( M R are the interference factors.Force P B ( M µ N ] Rorbital spin total experSVbas 3.16 0.59 4.70 1.14SG2 4.53 0.81 6.69 4.26 1.25 Despite its large spin strength 0.12 µ N , this state is nev-ertheless mainly orbital (like for SVbas). The third SG2state is spin-flip one. This state is almost fully exhaustedby the neutron configuration [631 ↑ , ↓ ]. In these twostates there is the strong constructive interference of or-bital and spin contributions. The 2.86-MeV state is acollective pure orbital state.Altogether we see that Th does not demonstratea distinctive SSR. Indeed, both level groups, below andabove 2.5 MeV, are strongly dominated by the orbitalstrength. So, these two level groups are explained not byseparation of SSR and OSR modes (as was suggested byWFM) but rather by a fine structure of the OSR alone.
IV. CONCLUSIONS
The prediction of the Wigner Function Moment(WFM) method on existence of low-energy spin-scissorsresonance (SSR) in deformed nuclei [21, 23–26] was an-alyzed in the framework of the self-consistent Quasipar-ticle Random Phase Approximation (QRPA) approachusing Skyrme forces SkM*, SVbas, and SG2. The calcu-lations were performed for deformed nuclei , , Dy TABLE VII: The same as in Table V but for states in
Th. For each state, the maximal 2qp component is shown.Force ν E B ( M K = 1)) [ µ N ] main 2qp components[MeV] orb spin total % [ N, n z , Λ] F-position spher. limit1 1.79 0.21 0.08 0.54 91 pp [660 ↑ , ↑ ] F, F + 1 1 i / , i / SVbas 14 2.88 1.20 0.02 0.93 51 nn [633 ↓ , ↓ ] F, F + 3 2 g / , g / ↑ , ↑ ] F, F + 1 1 i / , i / SG2 3 2.07 0.02 0.24 0.39 98 nn [631 ↑ , ↓ ] F, F + 3 1 i / , d /
13 2.86 1.41 ∼ ↓ , ↓ ] F − , F + 2 2 g / , g / and Th, proposed by WFM as the promising candi-dates for SSR.The calculations have shown that in strongly deformednuclei like , , Dy indeed there can exist a group of K π = 1 + spin states located at 1.5-2.4 MeV, i.e. belowthe conventional orbital scissor resonance (OSR). Follow-ing our analysis, these states are ordinary spin-flip exci-tations corresponding to M1(K =1) transitions betweenspin-orbit partners in the subshells with a low orbitalmomentum, e.g. 2 d and 1 f . Such low-moment spin-flip (LMSF) states can form a separate low-energy groupif the large deformation shifts OSR to a higher energy.In our calculations, this is the case for well deformed , , Dy but not for less deformed
Th.The LMSF states are non-collective and mainly ex-hausted by one 2qp spin-flip configuration. This can beexplained by basically isovector character of the spin-spin residual interaction which upshifts the collectivityto higher energies. The non-collective character of LMSFstates contradicts with the collective nature of the pre-dicted SSR.Since OSR energy E ≈ δA − / MeV falls with themass number A, this resonance in heavy (actinide) nucleiis located already at the energy relevant for LMSF. Thenthe strong OSR can conceal the weaker LMSF. So heavynuclei are generally not suitable to exhibit LMSF.At the excitation energy
E < + statesdemonstrate the significant interference of spin-flip andorbital contributions to M K = 1) strength. In par-ticular, the interference considerably increases the total M K = 1) strength in the OSR energy range, whichshould be taken into account while comparing the com-puted strengths with the estimations derived merely forthe orbital mode. A part of the orbital strength is down-shifted to the SSR region ( E ≤ + states in , Dy and
Th.These two groups are treated by WFM as SSR andOSR. Our calculations show that the lowest 1 + states in , , Dy are indeed of the spin-flip character. How-ever they are located at E ≤ + states are produced by the fragmenta-tion of the orbital strength. This is even more the case in Th where the low-energy spin strength is almost neg- ligible. So, by our opinion, the available experimentaldata do not confirm the existence of SSR.The WFM scissor-like treatment of SSR requires thenuclear deformation. So, following WFM, SSR can ex-ist only in deformed nuclei. Instead, our calculationsshow that low-energy spin states arise from the spin-orbitsplitting and so can exist even in spherical nuclei. Thus the deformation is not the origin of the low-energy spinstrength but rather the essential factor affecting its prop-erties.The spin-orbit splitting and spin-spin residual interac-tion are of a primary importance in the exploration ofspin excitations. To check the accuracy of our QRPAmethod in description of these factors, we performed cal-culations for the spin-flip M1(K =1) giant resonance in
Dy and
Th and obtained for the forces SVbas andSG2 a good agreement with the experiment. The sametest should be done by WFM as well.The discrepancy between WFM and QRPA predictionsfor SSR in
Th could be clarified by ( p, p ′ ) measure-ments which are sensitive to spin-flip excitations and notso much to orbital ones. If SSR indeed exists in Th,it should be well observed in ( p, p ′ ) reaction.Since LMSF states are often reduced to almost pure2qp excitations, these states can be useful for investiga-tion of low-momentum spin-orbit splitting and its inter-play with nuclear deformation. Besides, such states canbe useful for testing tensor forces. Acknowledgement
We thank Profs. P.-G. Reinhard, P. von Neumann-Cosel and A.V. Sushkov for useful discussions. The workwas partly supported by Votruba - Blokhintsev (CzechRepublic - BLTP JINR) grant (VON and JK) and a grantof the Czech Science Agency, Project No. 19-14048S(JK). VON and WK appreciate the Heisenberg-Landaugrant (Germany - BLTP JINR). A.R. acknowlegdes thesupport by the Slovak Research and Development Agencyunder contract No. APVV-15-0225, Slovak grant agencyVEGA (contract No. 2/0067/21), and the Research andDevelopment Operational Programme funded by the Eu-ropean Regional Development Fund, project No. ITMScode 26210120023.0 ( exp: Gd ) S ( M ) [ m N / M e V ] S ( M ) [ m N / M e V ] S ( M ) [ m N / M e V ] Dy a) SkM*
SkM* d) Th b) SVbas
SVbas e) c) SG2
E [MeV] f) SG2
E [MeV]
FIG. 8: M
11 spin-flip giant resonance in
Dy and
Th,calculated with Skyrme forces SkM*, SVbas and SG2. Theresults are compared with the scaled experimental data for
Th [44, 51] (right plots) and neighbouring nucleus
Gd[50] (left plots). See details in the text.
Appendix A: M K = 1) spin-flip giant resonance The energy and structure of M K = 1) ( M
11 in theshort notation) spin-flip giant resonance in open-shell nu-clei are basically determined by the interplay between thespin-orbital splitting in proton and neutron schemes fromone side and the spin-spin residual interaction from an-other side [3, 11, 12]. In this connection, we present hereour QRPA results for spin-flip giant resonance in
Dyand
Th, obtained with the Skyrme parametrizationsSkM*, SVbas, and SG2. We were not able to find the ex-perimental data for this resonance in
Dy. So, for thisnucleus, we compare our QRPA results with the ( p, p ′ )data for the neighbouring nucleus Gd [50] which hasthe similar quadrupole deformations ( β =0.348) [46]. For Th, we use ( p, p ′ ) data [44, 51].In Fig. 8, the results of our calculations are comparedwith the experimental data. QRPA strength functionsare obtained by averaging transition rates B ν ( M
11) forseparate QRPA states by the Lorentz weight with the av-eraging parameter ∆=1 MeV, see Refs. [11, 12] for moredetail. Only spin part of M11 transition operator (3) isused. The experimental data are properly scaled for theconvenience of the comparison with QRPA strength func-tions. Fig. 8 shows that SVbas, and especially SG2, welldescribe localization and fine structure of the resonancein both nuclei. In SkM*, the distribution of the strengthis too wide and upshifted to higher energies. This dif-ference can be explained by smaller values of spin-flipparameters b and b ′ in SVbas and SG2 sets (see TableI in Sec. II).In Table VIII, the spin reduced transition probabilitiessummed at the energy interval E =0-12 MeV are com- TABLE VIII: The strength B ( M s summed at E=0 - 12MeV in our SkM*, SVbas and SG2 calculations as comparedwith QRPA (SG2) results of Sarriguren et al [44].Nucleus P B ( M s , [ µ N ]SkM* SVbas SG2 Sarriguren [44] Dy 14.5 13.2 12.9 11.4
Dy 14.7 13.4 13.1 12.2
Dy 14.7 13.6 13.3 12.2
Th 17.3 15.6 14.9 14.9 pared with early QRPA results of P. Sarriguren et al [44]obtained with the force SG2. It is seen that the agree-ment is fine for SG2, acceptable for SVbas and worse forSkM*.Altogether, Fig. 8 and Table VIII show that forcesSVbas and SG2 are most relevant for exploration of spin-flip excitations.
Appendix B: Matrix elements of magnetictransitions in axially deformed nuclei
In cylindrical coordinates, the single-particle wavefunction with quantum numbers K π has the spinor formΨ i ( r ) = R (+) i ( ρ, z ) e im (+) i φ R ( − ) i ( ρ, z ) e im ( − ) i φ ! , (B1)for the normal state andΨ i ( r ) = ˆ T Ψ i ( r ) = − R ( − ) i ( ρ, z ) e − im ( − ) i φ R (+) i ( ρ, z ) e − im (+) i φ ! (B2)for the time-reversal state. Here the momentum projec-tion is decomposed as K i = m ( σ ) i + σ with σ = ± M λµ transition operators are [52]ˆ S lλµ = µ N p λ (2 λ + 1) r l g qs { ˆ s Y l } λµ , (B3)ˆ L lλµ = µ N p λ (2 λ + 1) r l g ql λ + 1 { ˆ l Y l } λµ (B4)where l = λ − µ N is the nuclear magneton, ˆ s and ˆ l arestandard spin and orbital operators, g qs and g ql are spinand orbital gyromagnetic factors. Further { ˆ s Y l } λµ = X m X α = − , , C λµlm, α Y lm ˆ s α , (B5) { ˆ l Y l } λµ = X m X α = − , , C λµlm, α Y lm ˆ l α (B6)where Y lm are the spherical harmonics and C λµlm, α areClebsch-Gordan coefficients.The matrix elements for the orbital and spin M λµ transitions from the BCS vacuum | BCS i to the two-quasiparticle (2qp) state α + i α +¯ j | BCS i with the selection1rule | K i − K j | = µ ( K i , K j > , µ ≥
0) have the form h i ¯ j | ˆ L lλµ | BCS i = 2 πµ N p λ (2 λ + 1) 2g l λ + 1 u ( − )ij (B7) · Z dzdρρ { g lµ C λµlµ, [ R (+) i m (+) j R (+) j + R ( − ) i m ( − ) j R ( − ) j ]+ 1 √ g lµ +1 C λµlµ +1 , − [ R (+) i ( ρ ddz − z ddρ − m (+) j zρ ) R (+) j + R ( − ) i ( ρ ddz − z ddρ − m ( − ) j zρ ) R ( − ) j ]+ 1 √ g lµ − C λµlµ − , [ R (+) i ( ρ ddz − z ddρ + m (+) j zρ ) R (+) j + R ( − ) i ( ρ ddz − z ddρ + m ( − ) j zρ ) R ( − ) j ] } , h i ¯ j | ˆ S lλµ | BCS i = 2 πµ N p λ (2 λ + 1)g s u ( − )ij (B8) · Z dzdρρ { g lµ C λµlµ, [ R (+) i R (+) j − R ( − ) i R ( − ) j ]+ 1 √ g lµ +1 C λµlµ +1 , − R ( − ) i R (+) j − √ g lµ − C λµlµ − , R (+) i R ( − ) j } . Here u ( − ) ij = u i v j − u j v i is the combination of Bogoliubovfactors. The ( ρ, z )- dependence in the functions R ( ± ) i , g lµ and g lµ ± is omitted for the sake of simplicity.For the selection rule K i + K j = µ , the matrix elementsfor the transitions to the 2qp state α + i α + j | BCS i read h ij | ˆ L lλµ | BCS i = 2 πµ N p λ (2 λ + 1) 2g l λ + 1 u ( − )ij (B9) · Z dzdρρ { g lµ C λµlµ, [ R ( − ) i m (+) j R (+) j − R (+) i m ( − ) j R ( − ) j ]+ 1 √ g lµ +1 C λµlµ +1 , − [ R (+) i ( ρ ddz − z ddρ + m ( − ) j zρ ) R ( − ) j − R ( − ) i ( ρ ddz − z ddρ + m (+) j zρ ) R (+) j ]+ 1 √ g lµ − C λµlµ − , [ R (+) i ( ρ ddz − z ddρ − m ( − ) j zρ ) R ( − ) j − R ( − ) i ( ρ ddz − z ddρ − m (+) j zρ ) R (+) j ] } , h ij | ˆ S lλµ | BCS i = 2 πµ N p λ (2 λ + 1)g s u ( − )ij (B10) · Z dzdρρ { g lµ C λµlµ, [ R (+) i R ( − ) j + R ( − ) i R (+) j ]+ 1 √ g lµ +1 C λµlµ +1 , − R ( − ) i R ( − ) j + 1 √ g lµ − C λµlµ − , R (+) i R (+) j } . In (B7)-(B10), the functions g lm ( m = µ, µ ±
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