Isoscalar monopole and dipole transitions in 24 Mg, 26 Mg and 28 Si
P. Adsley, V. O. Nesterenko, M. Kimura, L. M. Donaldson, R. Neveling, J. W. Brümmer, D. G. Jenkins, N. Y. Kheswa, J. Kvasil, K. C. W. Li, D. J. Marin-Lámbarri, Z. Mabika, P. Papka, L. Pellegri, V. Pesudo, B. Rebeiro, P.-G. Reinhard, F. D. Smit, W. Yahia-Cherif
IIsoscalar monopole and dipole transitions in Mg, Mg and Si P. Adsley,
1, 2, 3, 4, ∗ V. O. Nesterenko,
5, 6, 7
M. Kimura,
8, 9
L. M. Donaldson,
3, 1
R. Neveling, J. W. Br¨ummer, D. G. Jenkins, N. Y. Kheswa, J. Kvasil, K. C. W. Li, D. J. Mar´ın-L´ambarri,
3, 12, 13
Z. Mabika, P.Papka, L. Pellegri,
3, 1
V. Pesudo,
3, 12
B. Rebeiro, P.-G. Reinhard, F. D. Smit, and W. Yahia-Cherif School of Physics, University of the Witwatersrand, Johannesburg 2050, South Africa Department of Physics, Stellenbosch University,Private Bag X1, 7602 Matieland, Stellenbosch, South Africa iThemba Laboratory for Accelerator Based Sciences, Somerset West 7129, South Africa Institut de Physique Nucl´eaire d’Orsay, UMR8608,IN2P3-CNRS, Universit´e Paris Sud 11, 91406 Orsay, France 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 Department of Physics, Hokkaido University, 060-0810 Sapporo, Japan Reaction Nuclear Data Centre, Faculty of Science, Hokkaido University, 060-0810 Sapporo, Japan Department of Physics, University of York, Heslington, York, YO10 5DD, United Kingdom Institute of Particle and Nuclear Physics, Charles University, CZ-18000, Praha 8, Czech Republic Department of Physics, University of the Western Cape, P/B X17, Bellville 7535, South Africa Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico,Apartado Postal 20-364, 01000 Cd. M´exico, M´exico Institut f¨ur Theoretische Physik II, Universit¨at Erlangen, D-91058, Erlangen, Germany Universit´e des Sciences et de la Technologie Houari Boumediene (USTHB),Facult´e de Physique, B.P. 32 El-Alia,16111 Bab Ezzouar, Algiers, Algeria (Dated: October 19, 2020)
Background
Nuclei in the sd -shell demonstrate a remarkable interplay of cluster and mean-field phenomena. The N = Z nuclei, such as Mg and Si, have been the focus of the theoretical study of both these phenomena in the past. A varietyof different cluster structures in these nuclei, characterized by isoscalar dipole and monopole transitions, are predicted.For example, low-energy isoscalar vortical dipole states were predicted in Mg. The cluster and vortical mean-fieldphenomena can be probed by excitation of isoscalar monopole and dipole states in scattering of isoscalar particles suchas deuterons or α particles. Purpose
To investigate, both experimentally and theoretically, the isoscalar dipole ( IS
1) and monopole ( IS
0) strengths inthree different light nuclei with different properties: stiff prolate Mg, soft prolate Mg and soft oblate Si, and toanalyze possible manifestations of clustering and vorticity in these nuclei.
Methods
Inelastically scattered α particles were momentum-analysed in the K600 magnetic spectrometer at iThemba LABS,Cape Town, South Africa. The scattered particles were detected in two multi-wire drift chambers and two plasticscintillators placed at the focal plane of the K600. In the theoretical discussion, the Skyrme Quasiparticle Random-PhaseApproximation (QRPA) and Antisymmetrized Molecular Dynamics + Generator Coordinate Method (AMD+GCM) wereused. Results
A number of isoscalar monopole and dipole transitions were observed in the nuclei studied. Using this information,suggested structural assignments have been made for the various excited states. IS IS K = 1, ii) the dipole (monopole) states should have strongdeformation-induced octupole (quadrupole) admixtures, and iii) that near the α -particle threshold, there should exista collective state (with K = 0 for prolate nuclei and K = 1 for oblate nuclei) with an impressive octupole strength.The results of the AMD+GCM calculations suggest that some observed states may have a mixed (mean-field + cluster)character or correspond to particular cluster configurations. Conclusion
A tentative correspondence between observed states and theoretical states from QRPA and AMD+GCM wasestablished. The QRPA and AMD+GCM analysis shows that low-energy isoscalar dipole states combine cluster andmean-field properties. The QRPA calculations show that the low-energy vorticity is well localized in Mg, fragmentedin Mg, and absent in Si. ∗ Electronic address: [email protected]
I. BACKGROUND
Light nuclei demonstrate a remarkable interplay ofcluster and mean-field degrees of freedom, see e.g. re-views [1–4]. The exploration of this interplay is a de-manding problem which is additionally complicated bythe softness of these nuclei and related shape coexistence a r X i v : . [ nu c l - t h ] O c t [4]. The low-energy isoscalar monopole ( IS
0) and dipole( IS
1) states in light nuclei can serve as fingerprints ofclustering [5, 6]. IS IS IS N = Z nuclei can manifest itself inlow-lying IS J π = 0 + states [6–8]. Recenttheoretical work has suggested that IS i.e. thelow-lying 0 + states caused by asymmetric clusters have1 − partner states, thus forming inversion doublets whichindicate the symmetry of the cluster configuration [6].In N (cid:54) = Z nuclei, the asymmetric clustering may resultin enhanced electric dipole transitions between isoscalarstates.In addition to this clustering behaviour, mean-fieldstructures may also exist. Individual low-lying vorti-cal IS Be [12, 13], C [14], O [15], Ne[10], and Mg [9–11, 16].Such individual low-lying vortical states can be dif-ferentiated from the neighbouring excitations and muchmore easily resolved in experiment. Note that the intrin-sic electric vortical flow of nucleons, though widely dis-cussed in recent decades, is still very poorly understood[17–22]. The experimental observation and identificationof vortical states remains a challenge for the modern ex-perimentalist [11]. In this respect, exploration of indi-vidual low-lying IS e, e (cid:48) ) reaction has been recently suggested to iden-tify the vortical response of nuclei [11]. The complemen-tary ( α, α (cid:48) ) reaction may be used to locate candidates forthe IS e, e (cid:48) ) measurements.The light nuclei Mg, Mg, and Si have essentiallydifferent properties and thus represent a useful set for thecomparative investigation of the interplay between themean-field and cluster degrees of freedom. These nucleidiffer by
N/Z ratio, softness to deformation (stiff Mgand soft Si and Mg), and sign of deformation (pro-late Mg and oblate Si). Therefore, it is interestingto compare the origin and behavior of low-lying IS IS e.g. Refs. [7, 23, 24] for a general view and Refs. [25–31, 33–35] ( Mg), [30, 36–42] ( Mg), [31–33, 35, 40, 42–47]( Si) for particular studies). We now provide compara-tive experimental and theoretical analyses of these nuclei.In this paper, we report IS IS Mg, Mg and Si, determined from α -particle inelas-tic scattering at very forward scattering angles (includ-ing zero degrees). The data were obtained with the K600 magnetic spectrometer at iThemba LABS (Cape Town,South Africa). The data are limited to excitation energy E x <
16 MeV so as to avoid the regions dominated by gi-ant resonances, where identification of individual states isdifficult without observation of charged-particle decays.The theoretical analysis is performed within the QRPAmodel for axially deformed nuclei [48–52] and theAMD+GCM model [12–16] which can take into accountboth axial and triaxial quadrupole deformations and de-scribe the evolution of nuclear shape with excitation en-ergy. Moreover, AMD+GCM includes the ability to de-scribe the interplay between mean-field and cluster de-grees of freedom. Despite some overlap of QRPA andAMD+GCM, the models basically describe different in-formation on nuclear properties; QRPA treats excitedstates with a mean-field approach and is therefore suit-able for investigation of the nuclear vorticity. Mean-while, AMD+GCM highlights cluster properties. Alto-gether, QRPA and AMD+GCM supplement one anotherand comparison of their results is vital for light nuclei.Our analysis mainly focuses on possible manifestationsof clustering and vorticity in IS IS IS IS IS IS IS II. EXPERIMENTAL DETAILS
A detailed description of this experiment has beengiven in two previous papers [8, 40]. A brief summaryof the experimental method is given here.A dispersion-matched beam of 200-MeV α particleswas incident on a target and the reaction products weremomentum-analysed by the K600 magnetic spectrom-eter. The focal-plane detectors consisted of two wirechambers giving horizontal and vertical position infor-mation, and two plastic scintillating paddles which mea-sured energy deposited at the focal plane.The spectrometer was used in two different modes toacquire the data: the zero-degree mode in which scat-tering angles of less than 2 degrees were measured, andthe small-angle mode in which the spectrometer aperturecovered scattering angles from 2 to 6 degrees.For the zero-degree measurement, the background re-sulting from target-induced Coulomb scattering necessi-tated running the spectrometer in a focus mode in whichthe scattered particles were focussed onto a verticallynarrow horizontal band on the focal plane. In order toobtain a spectrum free from instrumental background, astandard technique used with the iThemba K600 [53] and < 2 deg. lab q Mg, ') a , a Mg( + . M e V , + . M e V , + . M e V , + . M e V , - . M e V , - . M e V , . M e V . M e V . M e V < 3 deg. lab q Mg, 2 deg. < ') a , a Mg( + . M e V , + . M e V , - . M e V , . M e V < 6 deg. lab q Mg, 5 deg. < ') a , a Mg( + . M e V , + . M e V , - . M e V , . M e V Excitation Energy [MeV] C oun t s p e r ke V Excitation Energy [MeV] C oun t s p e r ke V FIG. 1: Fitted Mg spectra for: θ < < θ < < θ < < 0 deg. lab q Mg, ') a , a Mg( + . M e V , . M e V + . M e V , < 3 deg. lab q Mg, 2 deg. < ') a , a Mg( + . M e V , . M e V + . M e V , - . M e V , - . M e V , < 6 deg. lab q Mg, 5 deg. < ') a , a Mg( . M e V + . M e V , - . M e V , - . M e V , Excitation Energy [MeV] C oun t s p e r ke V Excitation Energy [MeV] C oun t s p e r ke V FIG. 2: Fitted Mg spectra for: θ < < θ < < θ < the RCNP Grand Raiden [54] magnetic spectrometerswas used, in which background spectra are constructedfrom the regions of the focal plane above and below thefocussed band. These background components are thensubtracted from the signal spectrum. The vertical fo-cussing required for this technique resulted in the loss ofall vertical scattering information and limited the differ-ential cross section for the zero-degree experiment to one point for scattering angles of less than 2 degrees.For the small-angle measurement, the target-inducedCoulomb scattering background was much lower and thespectrometer could be operated in under-focus mode,in which the vertical position on the focal plane corre-sponds to the vertical scattering angle into the spectrom-eter aperture. In this case, the scattering angle could bereconstructed from the angle with which the scattered α particle traversed the focal plane and its vertical po-sition. The angular resolution was around 0.5 degrees(FWHM) and we extracted four points for the differentialcross section between 2 and 6 degrees in the laboratoryframe. The procedure to calibrate the scattering anglesis described in Refs. [8, 53]. III. DATA ANALYSIS
The techniques used for the analysis of the data havebeen described in more detail in Ref. [8]. In summary,the horizontal focal-plane position was corrected for kine-matic and optical aberrations according to the scatteringangle into the spectrometer and the vertical focal-planeposition.The scattering angles into the spectrometer were calcu-lated from the vertical position and the angle with whichthe scattered particle traverses the focal plane; thesequantities were calibrated to known scattering trajecto-ries into the spectrometer using a multi-hole collimatorat the spectrometer aperture.Horizontal focal-plane position spectra were gener-ated for each angular region. The calibration of thefocal-plane position to excitation energy used well-knownstates in Mg, Mg and Si [55, 56]. Corrections weremade according to the thickness of the relevant targetsusing energy losses from SRIM [58].The spectra were fitted using a number of Gaussianswith a first-order polynomial used to represent back-ground and continuum. The resolution was around 75(65) keV (FWHM) for the zero-degree (finite-angle) data.An additional quadratic term was used at E x < p ( α, α ) p elastic-scattering reac-tions from target contaminants. The fitted spectra for Mg and Mg at some angles are shown in Figures 1and 2. The Si spectra along with a description of theassociated fitting procedures can be found in Ref. [8].To quantify contamination in the targets, elastic-scattering data were taken in the small-angle mode. Pop-ulation of low-lying states in nuclei contained in the tar-get was observed. For the natural silicon target, smallquantities of hydrogen, C, O and , Si were ob-served. For the Mg and Mg targets, hydrogen, Cand O were also observed but at much lower levels thanfor the silicon target. From previous experimental stud-ies with the K600 (see e.g.
Ref. [59]), the locations ofthe C and O states are well known and excluded.
IV. EXPERIMENTAL RESULTS
The focus of this paper is on the location and strengthof cluster and vortical states. We report monopole ( J π =0 + ) and dipole ( J π = 1 − ) states in , Mg and Si.In addition, we discuss states which have received firmor tentative monopole or dipole assignments in previous experimental studies but have not been observed in thepresent measurement.The differential cross sections were extracted from thefitted spectra using: dσd
Ω =
YN Iη ∆Ω , (1)where N is the areal density of target ions, I is the inte-grated charge as given by the current integrator (includ-ing the livetime fraction of the data-acquisition system), η is the focal plane efficiency and ∆Ω is the solid angle ofthe spectrometer aperture at that scattering angle. Thetotal efficiency, η , is the product of the efficiencies foreach wire plane per Ref. [8]. The uncertainties in thedifferential cross sections are a combination of the fittingerror and Poissonian statistics.By comparing the experimental differential cross sec-tions to DWBA calculations performed using the codeCHUCK3 [62], (cid:18) dσd Ω (cid:19) exp = β R,λ (cid:18) dσd Ω (cid:19) DWBA , (2)the transition factors β R,λ were extracted for each dipole( λ = 1) and monopole ( λ = 0) state. The contribution ofthe states to the isoscalar dipole and monopole energy-weighted sum rules (EWSRs) were computed. The calcu-lations were performed in accordance with Refs. [23, 56],more details are given in Appendix A.There is a systematic ∼
20% uncertainty due to thechoice of the optical-model potentials. In the presentanalysis, we, e.g. find that the well-known E x = 7 . J π = 1 − state in Mg exhausts 2 . e.g. choices of optical-model potentials whencompared with previous results of 3 . e.g. around the 0 + states in Mg in the region of E x = 13 . −
14 MeV,where a third state lies between the two 0 + states, orat the minima of the differential cross section where thebackground is similar in size to the cross section fromthe state of interest. In these cases, to avoid biasing theextracted transition strengths, a subset of points fromthe angular distributions has been used for comparisonto the DWBA calculation.Below, in Tables I-V and Figs. 3-4, the monopole anddipole spectra for Mg, Mg, and Si are reported.Some states are discussed in separate subsections; thisis done where assignments have been updated or knownstates have not been observed. A. Mg A typical differential cross section for a J π = 0 + statein Mg is shown in Figure 3 and for a J π = 1 − state cm q [ m b / s r ] W d s d FIG. 3: Differential cross section for the J π = 0 + state at10.68 MeV in Mg. The data are represented by points withthe horizontal error bar delineating the angular range cov-ered. The calculated angle-averaged differential cross sectionis shown in red. cm q [ m b / s r ] W d s d FIG. 4: As Figure 3 but for the J π = 1 − state at 11.86 MeVin Mg. in Mg in Figure 4. Similar shapes were used to iden-tify other monopole and dipole states. The J π = 0 + and J π = 1 − levels are summarized in Tables I and II,respectively; states with the corresponding J π listed inthe ENSDF database [63] are included even when notobserved.
1. The 10.161-MeV state
A state with J π = 0 + has been reported at10.161 MeV in Mg( p, p (cid:48) ) Mg, Na( He, d ) Mg, Mg( He, He) Mg and C( O, α ) Mg reactions (seeRef. [63] and references therein). This state is not ob-served in the present experiment.
TABLE I: J π = 0 + states in Mg. The excitation energies E x are, where possible, taken from Ref. [63] and otherwisefrom the present experiment; energies are only taken fromRef. [63] if a clear correspondence with a known state of thecorrect J π may be made. The β R, is the dimensionless scal-ing factor for the data compared to the DWBA calculations,see Appendix A for details. The S is the percentage of theEWSR exhausted by the state. E x [MeV] a β R, [10 − ] S Comments9 . . . . . . . b . . . . . . . E x = 13 .
13 MeVin present experiment.13 . c . . . c . . c . . c . . T = 2 [63], not observed15 . c . . a From Ref. [63] unless stated otherwise b Strength extracted from 0 ° data alone. c Present experiment
TABLE II: As in Table I but for J π = 1 − states in Mg. E x [MeV] a β R, [10 − ] S Comments7 . . . . . . . . . . . . b . . a From Ref. [63] unless stated otherwise b Present experiment
2. The 13.044/13.13-MeV state A J π = 0 + state is listed at E x = 13 . J π = 0 + state isobserved at E x = 13 .
13 MeV. The cause of this shift isnot clear; it is possible that these are the same state andthe energy has been incorrectly determined in the pastor that this is an additional state. B. Mg Table III summarizes known J π = 0 + states in Mgeither listed in the ENSDF database [63] or observed dur-ing the present experiment.Table IV summarizes known J π = 1 − states along withelectrical B ( E TABLE III: As in Table I but for J π = 0 + states in Mg. E x [MeV] a β R, [10 − ] S Comments7 . J π = (0 , + [63]7 . J π = (0 , + [63]10 . . J π (cid:54) = 0 + . b . . J = 0, parity unknown12 . c . . . . . . . . c . a From Ref. [63] unless stated otherwise b See Ref. [64] for a discussion of the energy of this level. c Present experiment given and are converted to the reduced matrix elementusing the relation:Γ( λ(cid:96) ) = 8 π ( (cid:96) + 1) (cid:96) [(2 (cid:96) + 1)!!] (cid:18) E γ ¯ hc (cid:19) (cid:96) +1 B ( λ(cid:96) ) (3)for a radiation of multipolarity (cid:96) and type (elec-tric/magnetic) λ . E γ is the energy of the γ -ray tran-sition.
1. The 7.062-MeV state
This state is listed in ENSDF [63] but not observedin a previous Mg( α, α (cid:48) ) Mg reaction of Ref. [56]. Inthe present experiment, a state is observed at E x = 7 . J π = 1 − assignment.
2. The 10.159-MeV state
The J π = 0 + state at E x = 10 .
159 MeV listed inRef. [63] is not observed in the present experiment.The state has been previously observed in Mg( t, p ) Mgwith (cid:96) = 0 [66] and in Mg( p, p (cid:48) ) Mg (see Ref. [63] andreferences therein). We assume that the state has T = 1if it is populated in Mg( t, p ) Mg reactions. Therefore,population of this state in Mg( α, α (cid:48) ) Mg is unlikely tobe isospin-forbidden. The reason why this state is notpopulated remains unclear.
3. The 10.74-MeV State
Ref. [56] lists a tentative J π = 0 + state at E x =10 . E x = 10 . J ≥
4. States in the region of 10.80 to 10.83 MeV
A state with J π = 1 − has been identified at 10.805MeV in Mg( γ, γ (cid:48) ) Mg experiments [38]. In a preced-ing paper focussing on a narrow subset of astrophysi-cally important states in Mg, we demonstrated thatthe strong state observed in the Mg( α, α (cid:48) ) Mg reac-tion has J π = 0 + and is, therefore, evidently a differentstate from the J π = 1 − state [40]. The existence of multi-ple states was confirmed by a high-resolution experimentusing the Munich Q3D [36].In the present case, the extraction of the dipolestrength is hindered by the close proximity of the strong J π = 0 + state. A higher-resolution inclusive measure-ment or a coincidence measurement of Mg( α, α (cid:48) γ ) Mgis necessary for the extraction of the isoscalar dipole tran-sition strength for this state.
5. The . -MeV State Notably, one α -particle cluster state in Mg hasbeen identified through direct reactions. The reso-nance at E α = 0 .
83 MeV observed in Ne( α, γ ) Mg[67, 68] and Ne( α, n ) Mg [69–71] reactions clearly hasa Ne+ α cluster structure. However, the spin and par-ity of this state were not clearly assigned in previous Mg( α, α (cid:48) ) Mg reactions including our prior publica-tion [40, 41]. Based on direct measurements of the res-onance strengths and the inferred α -particle width, thestate almost certainly has J π = 0 + or J π = 1 − [72, 73].We do not observe any strong candidate for this statein our present experimental work and, therefore, cannotprovide a monopole or dipole transition for the state.
6. The . - and . -MeV states Both of these states have been identified as J π = 1 − using the reactions of neutrons with Mg. While γ -raypartial widths are available, the branching of these statesis not, and, therefore, the B ( E
1) for the ground-statetransition cannot be determined.
7. The 12.345-MeV State
A state is listed in Ref. [63] as having J = 0 withunknown parity and Γ = 40(5) keV. This state is notobserved in the present experiment. TABLE IV: The same as in Table I but for J π = 1 − states in Mg. Electric dipole reduced transition probabilities B ( E E x [MeV] a β R, [10 − ] S B ( E , + gs → − )(10 − e fm ) [39] B ( E , + gs → − )(10 − e fm ) [61] Comments7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J π = 0 + state10 . . . . . Mg+ n [65]11 . Mg+ n [65]11 . b . . a From Ref. [63] unless stated otherwise b Present experiment
8. A possible J π = 0 + state at E x ∼ . MeV
Visual inspection of the Mg( α, α (cid:48) ) Mg spectra atvarious angles suggests that there is possibly a weak J π =0 + state at E x ∼ . C. Si Data on the states observed in Si have been previ-ously reported in Ref. [8]. In the present paper, wehave extended the analysis up to 16 MeV to cover thesame range as for the magnesium isotopes. Additional J π = 0 + states are observed at 15.02 and 15.76 MeV.The natural silicon target contains some carbon andoxygen contamination. Carbon and oxygen states whichare strongly populated in α -particle inelastic scatteringat E α = 200 MeV are known from previous studies withthe K600 [59] and are excluded from the reported states.
1. The 11.142- and 11.148-MeV states
As explained in the previous K600 paper on Si( α, α (cid:48) ) Si, the literature lists two unresolved J π =0 + and J π = 2 + states at 11.141 and 11.148 MeV, re-spectively [8]. Further investigation of the existing dataon Si [75] has showed that there is, in fact, only one
TABLE V: As in Table I but for J π = 0 + states in Si. E x [MeV] a β R, [10 − ] S Comments9 . . . . . . . . . . b . . . b . . . b . . E x = 15 . J π = 0 + state [56]. a From Ref. [63] unless stated otherwise b Present experiment
TABLE VI: As in Table I but for J π = 1 − states in Si. E x [MeV] a β R, [10 − ] S Comments8 . . . . . J π = 1 − assignment [56].10 . . . . . J π = 1 − assignment [56].11 . b . . . b . . a From Ref. [63] unless stated otherwise b Present experiment state at this energy which has J π = 0 + and, therefore, itis not necessary to include contributions from two states.
2. The 11.65-MeV state
Ref. [56] reports a tentative J π = 1 − state at E x =11 . V. COMPARISON WITH QRPACALCULATIONSA. Calculation Scheme
We use a fully self-consistent QRPA approach [49, 50]with the Skyrme force SLy6 [76]. This force was foundto be optimal in the previous calculations of dipole ex-citations in medium-heavy nuclei [9, 77]. The nuclearmean field is computed using a two-dimensional mesh incylindrical coordinates. The mesh spacing is 0.7 fm. Thecalculation box extends up to 3 nuclear radii. The equi-librium deformation of nuclei is obtained by minimizationof the nuclear energy. The volume pairing is treated withthe Bardeen-Cooper-Schrieffer (BCS) method [50]. Thepairing was found to be weak (with a pairing gap about1 MeV) in all the cases with the exception of the neu-tron system in Mg. The QRPA is implemented in thematrix form [49]. The particle-hole (1ph) configurationspace extends up to 80 MeV, which allows the calcula-tions to exhaust the isoscalar E E β =0 . , .
355 and − .
354 for , Mg and Si, respec-tively, meaning that , Mg are taken to be prolate nu-clei while Si is treated as oblate. In the SLy6 calcu-lations, Mg has comparable oblate and prolate energyminima. Following the experimental data of Stone [78]as well as AMD+GCM [79] and Skyrme [80] calculations,the ground-state deformation of Mg is prolate and so,we use the equilibrium deformation β = 0.355 from theprolate minimum for Mg.Note that the absolute values obtained for equilib-rium deformations are smaller than the experimentalones ( β exp2 = 0 . , . , − .
412 for , Mg, Si) [81].This is a common situation for deformation-soft nuclei.Indeed, β exp2 are obtained from the B ( E
2) values for thetransitions in the ground-state rotational bands. How-ever, in soft nuclei, B ( E
2) values include large dynamicalcorrelations and so, lead to overestimation of the size ofthe quadrupole deformation, | β | . Therefore, the presentobservation that | β | < | β exp2 | is reasonable.The isoscalar reduced transition probabilities B ( ISλµ ) ν = |(cid:104) ν | M ( ISλµ ) | (cid:105)| , (4)for the transitions from the ground state | (cid:105) with I π K =0 + gs to the excited ν -th QRPA state with I π K = λ π µ are calculated using the monopole ( IS
0) and dipole ( IS K ) transition operators:ˆ M ( IS
0) = A (cid:88) i =1 ( r Y ) i , (5)ˆ M ( IS K ) = A (cid:88) i =1 ( r Y K ) i , K = 0 , Y = 1 / √ π . To investigate the deformation-induced monopole/quadrupole and dipole/octupole mix-ing, we also compute quadrupole B ( IS
20) and octupole B ( IS K ) transition probabilities for isoscalar transitions0 + gs → + ν and 0 + gs → − K ν using transition op-erators ˆ M ( IS
20) = A (cid:88) i =1 ( r Y ) i , (7)ˆ M ( IS K ) = A (cid:88) i =1 ( r Y K ) i , K = 0 , . (8)We now consider the vortical and compression isoscalarstrengths, B ( IS Kv ) ν and B ( IS Kc ) ν , using current-dependent operators from Refs. [9, 11]. We need thesestrengths to estimate the relative vortical and irrora-tional compression contributions to the dipole states.The current-dependent compression operator includes di-vergence of the nuclear current and so, can be reducedto Eq. (6) using the continuity equation. For the sake ofsimplicity, we will further omit the dependence on ν inrate notations. B. IS strength distributions Mg In Figure 5, the ( α, α (cid:48) ) experimental data (transi-tion factors β R, ) for Mg (upper) are compared with B ( IS K ) values (middle) for QRPA states with K = 0(red) and K = 1 (black). We see that experiment andQRPA give the lowest dipole states at a similar en-ergy, 7.56 and 7.92 MeV, respectively. In QRPA, thestates 7.92-MeV (K=1) and 9.56-MeV (K=0) have large B ( IS
1) responses and so, should be well populated in the( α, α (cid:48) ) reaction. However, it is still difficult to establishone-to-one correspondence between these QRPA statesand observed excitations, see the discussion in Ref. [9].In general, QRPA gives many more dipole states between E x = 7 −
16 MeV than the observed spectrum. The cal-culated summed B ( IS
1) strength is given in Table VII.The bottom panel of Figure 5 shows QRPA strengths B ( IS K ) for isoscalar octupole transitions 0 + gs → − K ν . The dipole 1 − K ν and octupole 3 − K ν statesbelong to the same rotational band built on the band-head state | ν (cid:105) . Thus, B ( IS K ) ν represents the level ofdeformation-induced octupole correlations in the band-head | ν (cid:105) . We see that the lowest states 7.92-MeV B ( I S K ) [ f m ] K=0 K=1 b =0.536 QRPA
IS1K S a =9.3 MeV24 Mg b R , [ - ] experiment 6 7 8 9 10 11 12 13 14 15 160100020003000 IS3K B ( I S K ) [ f m ] E [MeV]
K=0 K=1
FIG. 5: Experimental transition factors β R, (upper), QRPAisoscalar dipole compression strength B ( IS K ) for K = 0 and1 (middle), and isoscalar octupole strength B(IS3K) (bottom)in Mg. The α -particle threshold energy S α and QRPA equi-librium deformation β are displayed. (K=1) and 9.56-MeV (K=0) exhibit fundamental oc-tupole strengths: B ( IS
31) = 715 fm (21 W.u.) and B ( IS
30) = 2450 fm (72 W.u.), respectively. Such large B ( IS K ) values originate from two sources: i) collectiv-ity of the states and ii) that the dominant proton andneutron 1 ph components of the states ( pp [211 ↑ − ↑ ], nn [211 ↑ − ↑ ] for 7.92-MeV ( K = 1) state and pp [211 ↓ − ↓ ], nn [211 ↓ − ↓ ] for 9.56-MeV( K = 0) state) fulfill the selection rules for E K transi-tions [82]:∆ K = 0 : ∆ N = ± , ± , ∆ n z = ± , ± , ∆Λ = 0 , ∆ K = 1 : ∆ N = ± , ± , ∆ n z = 0 , ± , ∆Λ = 1 . Here, the single-particle states are specified by Nilssonasymptotic quantum numbers
N n z Λ [83], whilst the ar-rows indicate spin direction. The large B ( IS K ) valuessignify that 7.92-MeV ( K = 1) and 9.56-MeV ( K = 0)states are of a mixed octupole-dipole character. Theirleading 1 ph components correspond to ∆ N = 1 tran-sitions between the valence and upper quantum shells,so these states can belong to the Low-Energy OctupoleResonance (LEOR) [23, 84].As may be seen in Figure 5, both IS K and IS K distributions can be roughly separated into two groups,the first located below (7-10 MeV) and the second lo-cated above (11-14 MeV) the α -particle threshold S α =9 . S α , there is a E x = 9 . K = 0) state with a huge B ( IS
30) strength, whichperhaps signals the octupole-deformation softness of thenucleus at this energy. It is reasonable to treat the statesbelow S α as being of mean-field origin, while the statesclose to and above S α (including the 9.56-MeV ( K = 0)near-threshold state) as including cluster degrees of free-dom. This is confirmed by recent AMD+GCM calcula- TABLE VII: QRPA isoscalar B ( IS
1) compression strength(in fm ) summed at the energy interval 0-16 MeV.Nucleus QRPA B ( IS , K = 0) B ( IS , K = 1) B ( IS , total) Mg 80 82 162 Mg 90 141 230 Si 21 168 189 B ( I S vc ) [f m ] S a =9.3 MeV B ( I S vc ) [f m ] Mg QRPA
K=0
E [MeV] vor comK=1 b =0.536 FIG. 6: QRPA results for isoscalar vortical (black bars) andcompression (red filled squares) dipole strengths in K = 0(middle) and K = 1 (bottom) dipole states in Mg. tions for Mg [16], where similar results were obtained:the lowest mean-field 9.2-MeV ( K = 1) state is of mean-field character and the 11.1-MeV ( K = 0) state has clus-ter properties.In Figure 6, the vortical B ( IS v ) and compression B ( IS c ) strengths for K = 0 and K = 1 dipole branchesin Mg are compared. The states with B ( IS v ) >B ( IS c ) should be considered as vortical in nature, see e.g. the E x = 7 . K = 1) state. Instead, thestates with B ( IS v ) < B ( IS c ) have compressional ir-rotational character. Compressional states can be di-rectly excited in the ( α, α (cid:48) ) reaction [23]. The vorticalstates usually have a minor irrotational admixture and,most probably, are weakly excited in the ( α, α (cid:48) ) reactionthrough this admixture. Figure 6 shows that, in accor-dance with previous QRPA predictions [9, 11], the lowest K = 1 state at 7.92 MeV is mainly vortical. Moreover,for E x <
14 MeV, the K = 1 branch exhibits much more TABLE VIII: QRPA isoscalar vortical B ( IS v ) and compres-sion B ( IS c ) strengths (for K = 0 and K = 1) summed overthe energy interval E x = 0 −
16 MeV. K = 0 K = 1Nucleus B ( IS v ) B ( IS c ) B ( IS v ) B ( IS c ) Mg 0.010 0.0038 0.019 0.0033 Mg 0.012 0.0011 0.028 0.0074 Si 0.015 0.0011 0.029 0.0071 K = 0 branch. The summed B ( IS v )and B ( IS c ) are reported in Table VIII.The vortical character of the lowest dipole state maybe a peculiarity of Mg, potentially unique to that nu-cleus. At least, this is not the case in Mg and Si, asdiscussed below. As mentioned above, the vortical 7.92-MeV ( K = 1) state in Mg is mainly formed by theproton pp [211 ↑ − ↑ ] and neutron nn [211 ↑ − ↑ ]1 ph configurations. Just these configurations produce thevortical flow [10]. The large prolate deformation in Mgdownshifts the energy of these configurations, thus mak-ing the vortical dipole state the lowest in energy [9, 10]. Itis remarkable that the previous AMD+GCM calculations[16] give a very similar result for Mg: that lowest dipolestate at E x = 9 . K = 1) characterand a higher compressional ( K = 0) state at E x = 11 . Mg In Figure 7, we compare the calculated B ( IS K ) and B ( IS K ) responses with the ( α, α (cid:48) ) data. In both exper-iment and theory, we see numerous dipole states above E x ∼ Mg, whichcan be explained by the stronger neutron pairing in Mg(in contrast, the proton pairing in Mg and both protonand neutron pairings in Mg are weak).Again, we see rather large B ( IS K ) values, whichmeans that many of the K = 0 and K = 1 excitationsare of a mixed dipole-octupole character. As in Mg,the states can be separated into two groups, below andabove the threshold S α = 10 . E x = 9 . K = 0) statewith an impressive B ( IS
30) value.As can be seen in Figure 7, the theory suggests anotherpattern for the lowest dipole states in Mg. Unlike Mg,where the lowest dipole K = 1 state is well separated andexhibits a vortical character, the QRPA dipole spectrumin Mg starts with two almost degenerate K = 1 and K = 0 states at E x ∼ . Mg are notvortical.To understand these results, we should inspect thestructure of the lowest 6.60-MeV ( K = 1) and 6.64-MeV( K = 0) QRPA states in Mg. They are dominatedby 1 ph neutron configurations nn [211 ↓ +330] ↑ and nn [211 ↓ − ↑ , respectively. The same content ex-plains the quasi-degeneracy of these states. These 1 ph configurations have low B ( E v ) values and are not vor-tical. The configurations correspond to F + 1 → F + 5transitions, where F marks the Fermi level. Both single-particle levels involved in the transition lie above theFermi level and the transition is active only because of thedeveloped neutron pairing in Mg (but it is suppressedin Mg, where the calculated pairing is negligible).Note that 1 ph excitations pp [211 ↑ − ↑ and b R , [ - ] experimentQRPA Mg b = 0.355 S a =10.6 MeV IS3K
K=0 K=1 B ( I S K ) [ f m ] B ( I S K ) [ f m ] E [MeV]
K=0 K=1
IS1K
FIG. 7: The same as in Figure 5 but for Mg. B ( I S vc ) [f m ] QRPA B ( I S vc ) [f m ] Mg b = 0.355 K=0 S a =10.6 MeV E [MeV] vor com
K=1
FIG. 8: The same as in Figure 6 but for Mg. nn [211 ↑ − ↑ , which produce the vorticity in thelowest K = 1 vortical dipole state in Mg, also ex-ist in Mg, but they are located at a higher energy of E x = 8 . − . ph configurations. Si In Figure 9, we present the experimental data andQRPA results for IS K and IS K strengths in oblate Si. We see that the theory significantly overestimatesthe energy of the lowest K − state: it appears at 8.8 MeVin experiment and at 10.5 MeV in QRPA. So, unlike theexperiment, the theory does not suggest any K − statesbelow the threshold S α = 9 .
98 MeV. Perhaps this dis-crepancy is caused by a suboptimal oblate deformation β = − .
354 used in our calculations. Further, Fig-ure 9 and Table VII show that the dipole and octupolestrengths for K = 1 are much larger than for K = 0. So,in this nucleus, K = 1 states should be more stronglypopulated in ( α, α (cid:48) ) than K = 0 states.1 K=0 K=1 B ( I S K ) [ f m ] Si QRPA b = -0.354 S a =10.0 MeVIS1K b R , [ - ] experiment K=0 K=1
IS3K B ( I S K ) [ f m ] E [MeV]
FIG. 9: The same as in Figure 5 but for Si. B ( I S vc ) [ f m ] vor com B ( I S vc ) [ f m ] Si QRPA K = 0 b = -0.354 S a =10.0 MeVE [MeV] K =1 FIG. 10: The same as in Figure 6 but for Si.
The bottom panel of Figure 9 shows that, with the ex-ception of the E x = 11 . K = 1) state, the nucleus Si does not demonstrate any fundamental octupolestrength. So, for most its K − states, the dipole-octupolecoupling is suppressed. The near-threshold state at 11.2MeV with significant octupole strength has K = 1 butnot K = 0 as in , Mg. Perhaps all these peculiaritiesare caused by the oblate deformation of Si.In our calculations, the pairing in Si is weak. As aresult, the vortical configuration [211] ↑ − [330] ↑ corre-sponding in this nucleus to the transition between parti-cle states is suppressed. So, as seen from Figure 10, thelowest dipole states in Si are not vortical and vorticityappears only above 12 MeV. As in , Mg, the vorticityis mainly concentrated in the K = 1 branch. C. Summary for IS QRPA results
QRPA calculations do not allow one to establish a di-rect correspondence between the calculated and observed 1 − states. Perhaps this is because the present QRPAscheme does not take into account such important factorsas triaxiality, shape coexistence, clustering, and complexconfigurations are omitted.Nevertheless, the QRPA calculations lead to some in-teresting and robust results.1) The strong deformation-induced mixture of thedipole and octupole modes is predicted for most of K π =0 − and 1 − states in , Mg and in a few particular statesin Si. Some mixed states demonstrate impressive oc-tupole transition probabilities B ( IS K ). Perhaps thesestates belong to the so-called LEOR [23, 84].2) In all three nuclei, the collective state with a largeoctupole strength is predicted near the α -particle thresh-olds S α = 9 . − . K = 0 in , Mg and K = 1 in Si. Most probably, the differ-ence is caused by different signs of the axial deformationin these nuclei.3) Above the α -particle thresholds, fragmented vortic-ity is found in K = 1 states in all three nuclei. Below S α ,the picture is different: the vorticity is concentrated inthe lowest dipole state at ∼ Mg, fragmentedbetween several states at ∼ . − . Mg, andfully absent in Si. As was discussed, the vorticity is de-livered by particular 1 ph configurations, which can havea different energy location depending on the nuclear de-formation and other factors, e.g. residual interaction.Moreover, these configurations are active only if theyare of particle-hole character or supported by the pairing(like in Mg). A particular interplay of these factors in , Mg and Si leads to the difference in their vorticitydistribution. D. IS strength distributions In Figure 11, the ( α, α (cid:48) ) data for K π = 0 + states in , Mg and Si (plots (a)-(c)) are compared with QRPAisoscalar monopole strength B ( IS
0) in the energy inter-val 0-16 MeV (plots (d)-(f)).As mentioned above, because of the limitations of theexperimental set up, the present ( α, α (cid:48) ) data cover E x =9 −
16 MeV. Low-energy 0 + states listed in Tables I, IIIand V of Section IV are omitted in Figure 11.Figure 11 shows that, in Mg, the experimental andQRPA strength distributions look rather similar. In Mg, the situation is quite different since QRPA pre-dicts IS E x = 1 − Si,QRPA suggests the onset of 0 + states around E x = 4 − + statesat 9-16 MeV, which is generally in accordance with the( α, α (cid:48) ) data. The QRPA IS E x = 0 −
16 MeV interval are 26.1 fm , 13.76 fm , and12.6 fm in Mg, Mg, and Si, respectively.Note that, in the QRPA calculations, the actual num-ber of 0 + states at E x <
16 MeV is much larger thanmight be seen in Figure 11. In fact, QRPA gives 48( Mg), 53 ( Mg), and 50 ( Si) states. However, most2 b ) Mg QRPA, SLy6b =0.536 exp a ) expexp Mg b R , [ - ] c ) Si d ) IS0IS0 B ( I S ) [ f m ] e )IS0 QRPA, SLy6b =0.355 f ) QRPA, SLy6b = -0.354 g ) B ( I S ) [ f m ] B ( I S ) [ f m ] IS0 h ) IS0IS0 i ) j )IS20 E [MeV] k )IS20 E [MeV] l ) IS20 E [MeV] FIG. 11: Experimental β R, factors (a-c), QRPA B ( IS
0) values for K π = 0 + excitations at 0-16 MeV (d-f) and 0-30 MeV (g-i),QRPA B ( IS
20) values at 0-30 MeV (j-l). of these states are not seen in plots (d)-(f) because oftheir very small B ( IS
0) values.Plots (g)-(i) in Figure 11 show QRPA B ( IS
0) strengthin the larger energy interval 0-30 MeV including theIsoScalar Giant Monopole Resonance (ISGMR). In de-formed nuclei, there is the coupling of monopole andquadrupole modes, see e.g. early studies [23, 85, 86]and recent systematics [87]. In particular, the ISGMRis coupled with the λµ = 20 branch of the IsoScalarGiant Quadrupole Resonance, ISGQR(20). Due to thiscoupling, a part of the IS IS , Mg and Si have largequadrupole deformations and so, we should expect a sig-nificant ISGMR splitting. Indeed, the plots (g)-(i) showthat the ISGMR in these nuclei is split into the narrowdistribution between 15 and 19 MeV and the main wideISGMR distribution between 20 and 30 MeV. The pic-ture is consistent in prolate , Mg and oblate Si.This treatment is justified by the plots (j)-(l), wherethe strength B ( IS
20) of quadrupole isoscalar transitions 0 + gs → + ν from the ground state to the rotationalquadrupole state built on the band-head | ν (cid:105) is exhibited.We see that the ISGQR(20) branch is located at 15-19MeV, i.e. at the same energy as the narrow IS IS K π = 0 + states.Plots (g)-(i) and (j)-(l) highlight some importantpoints. First, the plots (g)-(i) show that the J π = 0 + states in ( α, α (cid:48) ) data lie just below the ISGMR hump,i.e. basically beyond the ISGMR. Only in Mg, thesestates perhaps cover the edge of the ISGMR hump. Sec-ond, from comparison of the plots (g)-(i) and (j)-(l), wecan learn that K π = 0 + states at 0-16 MeV exhibit bothstrong IS IS
20 transitions. They should, there-fore, not be treated as solely monopole states but ratheras strong mixtures of monopole and quadrupole excita-tions.
VI. COMPARISON WITH AMD+GCMCALCULATIONS
In this section, we discuss the comparison between thepresent experimental results and the AMD+GCM calcu-lations for Mg and Si presented in Refs. [5, 6, 89].These calculations do not take into account all the de-grees of freedom of the collective excitations. There-fore, they are not appropriate for the discussion of the3global features of the observed strength distributions.However, AMD+GCM describes the clustering aspectswhich involves many-particle-many-hole excitations, andhence, can offer a different insight into the low-lyingstrengths than that from QPRA. From the mean-fieldside, AMD+GCM takes into account the interplay be-tween axial and triaxial nuclear shapes, which is impor-tant for light nuclei.In Ref. [5], using the AMD+GCM framework, the re-lationship between the monopole strengths in Mg andclustering has been discussed. The α + Ne, Be+ O, C+ C and 5 α cluster configurations were investigatedin addition to the 1 ph single-particle excitations. It wasconcluded that several low-lying monopole transitions atenergies below the giant monopole resonance can be at-tributed to the clustering as summarized in Table IX.As already discussed in previous works on AMD+GCMand QPRA calculations [89–91], the Gogny D1S interac-tion overestimates the energy of the non-yrast states of Mg. Therefore, when we compare the AMD+GCMresults listed in Table IX with the experiment, it is bet-ter to shift down the calculated excitation energies tomatch with the well-known states. For this purpose, Ta-ble IX also lists the calculated excitation energies shifteddown by 2.9 MeV so as to reproduce the observed energy( E exp =6.4 MeV) of the 0 +2 state. Note that this shiftalso changes the calculated excitation energy of the 0 +3 state (11.7 MeV → +5 and 0 +8 states), as theobserved level density is rather high, the experimentalcounterparts in the ENSDF database [92] are ambigu-ous.Table IX should be compared with the present experi-mental data from Table I and Figure 11. We see that the0 +2 state is out of the acceptance of the present experi-ment, but the 0 +3 state is clearly observed and has the en-hanced monopole strengths as predicted by AMD+GCM.In Ref. [5], it was concluded that 0 +3 is a mixture of thecollective and Ne+ α cluster excitations. Consequently,it is interesting to note that the 0 +3 also appears as aprominent peak in the QRPA result (Figure 11). In ad-dition to the 0 +3 state, Table I reports a state at 11.7 MeVand a group of states at 13.0-13.9 MeV with the enhancedmonopole strengths. These states are of particular inter-est because their energies are close to the correspond-ing cluster decay thresholds (9.3 MeV for Ne+ α , 13.9MeV for C+ C, 14.047 MeV for O+2 α and 14.138MeV for O+ Be) as listed in the Ikeda diagram [93].Furthermore, these states are also visible in the excita-tion function reported in another Mg( α, α (cid:48) ) Mg exper-iment and seem not be reproduced by RPA calculations[25, 90]. Therefore, they can be attributed to the clusterresonances. In the AMD+GCM calculations, the candi-dates of the Ne+ α and C+ C cluster configurationswere predicted at 13.2 and 15.3 MeV (10.3 and 12.4 MeVwith the 2.9-MeV shift), respectively. Of course, to firmlyestablish the assignments of these states, more detailed analysis is indispensable. For example, the differentialcross sections of these states should be compared withtheoretical predictions in the future. The present ex-periment probes only a small range of angles and so, isinsufficient for thorough comparison with theory.For Si, AMD+GCM calculations suggest pairs of 0 + and 1 − states pertinent to asymmetric cluster configu-rations, such as Mg+ α , Ne+ Be and O+ C [6].The predicted results are summarized in Table X. Sim-ilar to the Mg case, the Gogny D1S interaction sys-tematically overestimates the energies of the non-yraststates, see Figure 6 in Ref. [6]. Therefore, while compar-ing the AMD+GCM and experimental results, we againuse the downshift of the calculated excitation energies,now by 3.3 MeV, to match the energy of the observed0 +3 state. Note that this well-known prolate-deformedstate should have a large contribution from the O+ Ccluster configuration [94–97]. The value of the energydownshift looks reasonable as it is similar to that intro-duced for Mg. With this shift, the energies of otherwell-known states show reasonable agreement betweenthe AMD+GCM and experimental results. For exam-ple, the 2 + member of the SuperDeformed (SD) band,which has been experimentally identified at 9.8 MeV inRef. [74], agrees well with the shifted AMD+GCM stateat 9.7 MeV. Furthermore, a couple of the Mg+ α clus-ter resonances have been identified around 13 MeV inresonant scattering experiments [98, 99], which are closeto the shifted AMD 0 +6 state at 14.9 MeV.We now examine the cluster configurations listed inTable X and compare to the present experimental data.Since the monopole ( IS
0) and dipole ( IS
1) transitionshave a strong selectivity for the cluster states, the clus-ter configurations can be classified into two groups, whichare strongly populated/hindered in the ( α, α (cid:48) ) reaction.For example, from a simple theoretical consideration, wecan predict that the 0 +3 state that is the band head of theprolate band (the lowest O+ C cluster band) shouldbe hindered. See Ref. [101] for details of the hindrancemechanism. It is interesting that the hindrance of the0 +3 state can also be seen in the QRPA results shownin Figure 11. Unfortunately, this state (which is im-portant for validation of the relationship between themonopole transitions and clustering) is out of the accep-tance of the present experiment. For the same reason, theAMD+GCM predicts that the SD band head expectedat 9.3 MeV should also be hindered. However, in thepresent experiment, the observed 9.7-MeV 0 + state isvery close to the 9.8-MeV 2 + state and, following TableV, has the enhanced monopole strength in contradictionto the AMD+GCM prediction. This new result requiresa more detailed analysis of the SD state in Si.At the same time, AMD+GCM predicts an enhance-ment of the Ne+ Be and Mg+ α cluster configu-rations. The pair of the 0 +5 and 1 − states with the Ne+ Be configuration is predicted at 10 −
11 MeV,and some fractions of IS IS TABLE IX: The cluster configurations with significant B ( IS
0) strengths and their excitation energies E x in Mg calculatedby AMD+GCM and compared with the observed data [92]. The energies E shift are obtained by a downshift of 2.9 MeV so asto adjust E exp for the 0 +2 state.Cluster J π E x [MeV] B ( IS
0) [fm ] E shift [MeV] E exp [MeV] B ( IS exp [fm ]0 +2 ± Ne+ α ) 0 +3 Ne+ α +5 C+ C 0 +8 E x and transition strengths ( B ( IS
0) for the 0 + statesand B ( IS
1) for the 1 − states) in Si, calculated withinAMD+GCM. The experimental counterparts are taken fromRef. [100] and the present experiment (denoted by bold). Theenergies E shift are obtained by a downshift of 3.3 MeV so asto adjust the energy E exp =6.69 MeV for the 0 +3 state.cluster J π E x B ( ISλ ) E shift E exp B ( IS exp +2 Ne+ Be 0 +5 − Mg+ α − +6 − C+ C 0 +3 − M+ α (SD) 0 +4 +5 − be the first indication of the Ne+ Be clustering in Si,which must be confirmed by a more detailed study, e.g. the transfer of Be to Ne. Other states which are pre-dicted to be strongly populated in the ( α, α (cid:48) ) reaction are Mg+ α cluster states. AMD+GCM study a 1 − state at9.6 MeV and 0 + and 1 − states at approximately 15 and17 −
20 MeV. The 1 − states at 17-20 MeV are beyondthe present experiment. However, several 0 + states canbe seen at 9.5 MeV and 15 MeV. It is worthwhile to notethat the α transfer and α + Mg resonant scattering ex-periments [98, 99] also report a group of the α + Mg res-onances with J = 0 + in the same energy region. There-fore, the data of the previous and present experimentsas well as the AMD+GCM results look consistent. Amore detailed comparison between AMD+GCM and ex-perimental results may be conducted in the future. VII. CONCLUSIONS
The isoscalar dipole ( IS
1) and monopole ( IS
0) exci-tations of Mg, Mg and Si at the energy interval E x = 9 −
16 MeV have been measured using the ( α, α (cid:48) ) inelastic-scattering reaction at forward angles (includingzero degrees). The experiment was performed at theK600 magnetic spectrometer at iThemba LABS (CapeTown, South Africa). New monopole and dipole stateswere reported at E x = 13 −
15 MeV.The extracted IS IS IS T = 0 negative-parity clus-ter bands, which are the doublets of the positive-paritybands based on the monopole states [6]. Some tracesof these doublets were found in the comparison of the-oretical calculations and experimental data. IC statesare actually irrotational dipole oscillations of two clusterswhich constitute the nucleus relative to one other. Thesestates originate from the reflection-asymmetric form ofthe nucleus exhibiting the clustering. The negative-parity bands produced by IC states usually have K = 0.Instead, the VMF states in light nuclei are mainly ofmean-field origin [9, 10, 16] and can exist without cluster-ing. They do not need the reflection-asymmetric nuclearshape and the associated the monopole doublets. Follow-ing previous studies [9, 10, 16] and present QRPA calcu-lations, these states produce negative-parity rotationalbands, mainly with K = 1.Both IC and IMF/VMF states exhibit enhanced IS K -mixing. Nevertheless, the relation to the K = 0 or K = 1 bands is perhaps a reasonable indicatorfor an initial discrimination of IC and VMF states.Being strongly deformed, , Mg and Si shouldexhibit a strong coupling between dipole and oc-tupole modes and between monopole and quadrupolemodes. This coupling was confirmed by QRPA calcu-lations, where strong IS K (0 + gs → − K ν ) and IS + gs → + ν ) transitions were found. So, the theoret-5ically explored states are actually dipole/octupole andmonopole/quadrupole mixtures. Further, QRPA pre-dicts that there should exist a specific collective state( K = 0 in prolate and K = 1 in oblate nuclei) with an im-pressive octupole strength near the α -particle threshold.This near-threshold state manifests the onset of stateswith cluster features.Due to triaxiality and significant shape coexistence in , Mg and Si, QRPA results obtained at the fixedaxial deformation should be considered as approximate.In addition, QRPA calculations do not include all thedynamical correlations, e.g. the coupling with complexconfigurations. Nevertheless, the main QRPA prediction- of vortical dipole states with enhanced IS α -particle thresholds S α in , Mg and Si. Following our analysis, the vorticityis concentrated in the lowest dipole state in Mg at ∼ ∼ Mg, and is fully absent in Si. The differenceis explained by different energies of 1 ph configurationsresponsible for the vorticity. Our explorations confirmthe suggestion made in Ref. [9] that Mg is perhaps theunique nucleus with a well-separated low-energy vorticalstate.The present ( α, α (cid:48) ) data do not yet allow confidentassignment of the vortical or cluster character of the ex-citations. However, these data improve our knowledgeof the isoscalar monopole and dipole states at the ex-citation energies where the clustering and vorticity arepredicted. This is a necessary and important step in theright direction. The use of the ( α, α (cid:48) ) reaction at in-termediate energies complements other suggested mech-anisms for populating cluster and vortical states such asthe ( γ, γ (cid:48) ) [88, 102], ( e, e (cid:48) ) [11] and ( d, Li) reactions [88]or ( / Li, d/t ), although detailed information on the inte-rior of the nuclei and the vortical mode is likely only avail-able from the ( e, e (cid:48) ) reaction. Branching ratios and tran-sition strengths of γ -ray transitions from the observeddipole states would provide information on the K assign-ment of the levels and should also be a focus of additionalfuture experimental work. Acknowledgments
The authors thank the Accelerator Group at iThembaLABS for the high-quality dispersion-matched beam pro-vided for this experiment. PA acknowledges supportfrom the Claude Leon Foundation in the form of a post-doctoral fellowship, M. N. Harakeh for providing the bel-gen and fermden codes and helpful advice regardingthe DWBA calculations, and Josef Cseh for useful discus-sions concerning Si. RN acknowledges support from theNRF through Grant No. 85509. VON and JK thank Dr. A. Repko for the QRPA code. The work was partly sup-ported by Votruba - Blokhintsev (Czech Republic - BLTPJINR) grant (VON and JK) and a grant of the CzechScience Agency, Project No. 19-14048S (JK). VON andPGR appreciate the Heisenberg-Landau grant (GermanyDLTP JINR).
Appendix A: Details of DWBA calculations
In past studies, e.g. [25–27], the real part of the po-tential has been calculated using a folding model, andthe imaginary part of the potential has been determinedby fitting to elastic-scattering data. Due to time limita-tions, especially in moving the detectors from the high-dispersion focal plane to the medium-dispersion focalplane of the K600, it was not possible to take elastic-scattering data for this purpose. Instead, the Nolte,Machner and Bojowald optical-model potential was used.For this potential, the reduced radii are r R = 1 .
245 fmand r I = 1 .
570 fm for the real and imaginary part of thepotential, respectively. Other parameters, such as thediffuseness and the depths of the potentials are energy-dependent quantities, which are calculated separately foreach entrance and exit channel.For Mg, we employ the quadrupole deformation β =0.355 from Ref. [28]. Using this deformation and the re-duced radius of the real potential, we get (with the codes belgen and fermden which have been made avail-able at github.com/padsley/KVICodes ) for the ground-state band, the value B ( E ↑ = 0 . b , whichis in good agreement with the experimental value of B ( E ↑ = 0 . b . Using the measured B ( E Mg and Si, we obtain the quadrupole defor-mations of β = 0.295 and β = -0.255, respectively. Thesigns of these deformations (prolate in Mg and oblatein Si) were chosen following the discussion in Sec. V-A. Note that the above parameters of the quadrupoledeformation are much smaller than the absolute valuesfor those from the NNDC database [81] ( β exp2 = 0 . . − .
412 for , Mg and Si) This is becausethe NNDC quadrupole deformation parameters are de-termined assuming a uniform charge distribution, whilewe use the Fermi distribution for the mass.Since the radii of the real and imaginary parts of thepotential are different, we assumed that the deformationlengths for the real and imaginary parts of each of thepotentials are the same: β R R R = β I R I (A1)where R R = r R A / , R I = r I A / , and A is the massnumber of the target [103]. Additionally, following Refs.[23, 56], we assume that the deformation lengths of thepotential and the mass distribution are identical, i.e. β R R p = β m R m (A2)where the mass radius is R m = r m A / and r m isdetermined from the reduced radius for the potential6of Nolte, Machner and Bojowald [104]. Using the de-scription by Satchler [57], the potential radius is R p = r m ( A / + A / P ) where A P is the mass of the projectile.The relation (A2) means that the potential and mass dis-tributions evolve self-consistently.For monopole transitions, we used the formf code[105, 106] to calculate the Satchler type-I form factor[57]. For dipole transitions, we employed the form fac-tors from Ref. [103].For each excitation state, the β R,λ parameters weredetermined by comparing the corresponding experimen-tal and DWBA differential cross sections, see Eq. (2).Then, using Eqs. (A1) and (A2), the values β I and β m were obtained.The percentage of the monopole ( λ = 0) EWSR ex-hausted by a given state is given by [56]: S = β m, β M, , (A3)where β m, is the monopole transition strength deter-mined from Eq. (A2) and β M, = 4 π ¯ h mAE x (cid:104) r (cid:105) (A4)is the total transition strength for the state located atthe excitation energy, E x and exhausting 100% of themonopole EWSR [56]. Here m is the nucleon mass and (cid:104) r (cid:105) is calculated from the Fermi mass distribution usingthe fermden code.For dipole ( λ = 1) transitions, the fraction of theEWSR exhausted by a state is given by [56]: S = β m, β M, , (A5)where β m, is the dipole transition strength, again fromEq. (A2) and β M, = 6 π ¯ h mAE x R m (cid:104) r (cid:105) − (cid:104) r (cid:105) − (cid:15) (cid:104) r (cid:105) (A6)is the total transition strength for the state lying at ex-citation energy, E x and exhausting 100% of the dipoleEWSR [103]. Here (cid:104) r (cid:105) and (cid:104) r (cid:105) are calculated from thereal part of the optical-model potential using fermden ,and R m is the half-density radius of the Fermi mass dis-tribution. 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