Origin of fine structure of the giant dipole resonance in sd-shell nuclei
R. W. Fearick, B. Erler, H. Matsubara, P. von Neumann-Cosel, A. Richter, R. Roth, A. Tamii
OOrigin of fine structure of the giant dipole resonance in sd -shell nuclei R. W. Fearick, B. Erler, H. Matsubara,
3, 4
P. von Neumann-Cosel, ∗ A. Richter, R. Roth, and A. Tamii Department of Physics, University of Cape Town, Rondebosch 7700, South Africa Institut f¨ur Kernphysik, Technische Universit¨at Darmstadt, D-64289 Darmstadt, Germany Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan Tokyo Women’s Medical University, 8-1 Kawada-cho, Shinjuku-ku, Tokyo 162-8666, Japan (Dated: September 3, 2018)A set of high resolution zero-degree inelastic proton scattering data on Mg, Si, S, and Caprovides new insight into the long-standing puzzle of the origin of fragmentation of the Giant DipoleResonance (GDR) in sd -shell nuclei. Understanding is achieved by comparison with Random PhaseApproximation (RPA) calculations for deformed nuclei using for the first time a realistic nucleon-nucleon interaction derived from the Argonne V18 potential with the unitary correlation operatormethod and supplemented by a phenomenological three-nucleon contact interaction. A waveletanalysis allows to extract significant scales both in the data and calculations characterizing the finestructure of the GDR. The fair agreement for scales in the range of a few hundred keV supports thesurmise that the fine structure arises from ground-state deformation driven by α clustering. I. INTRODUCTION
The isovector giant dipole resonance is the best knownof the fundamental collective excitations of the nucleus[1]. A giant resonance can be understood macroscop-ically as a bulk nuclear vibration, and microscopicallyin terms of coherent particle-hole excitations. The grossproperties of the GDR such as the centroid in energy ofthe excitations and the strength in terms of sum rulesare well understood. Less well understood are, however,the details of decay processes of the resonance. Variouscontributions to the width of the giant resonances havebeen identified [1, 2]: direct decay out of the continuumleading to an escape width, coupling to two-particle two-hole states (2 p h ) and then to many-particle many-hole( npnh ) states giving rise to a spreading width, and frag-mentation of the elementary 1 p h states that form theresonance called Landau damping. These processes con-tribute to the total width of the resonance and manifestthemselves experimentally by different structures in theexcitation spectra.A fragmentation of the GDR in p - and sd -shell nucleion the scale of several MeV is long-established and hasalready early been interpreted as configurational splitting[3]. Recently, it has been argued that the strength distri-bution of the GDR in C and O reveals informationon the role of different α -cluster configurations [4]. How-ever, the observation of finer structures of the GDR inlight nuclei on the scale of several hundred keV remainsa puzzle. In general, the physical origin of this kind ofstructure must be related to the existence of complexconfigurations, different time scales in decay processes,or the removal of the angular-momentum substate de-generacy due to deformation. Taking Si as an exampleof an sd -shell nucleus, structure on the finest scales wasobserved in reaction cross sections [5] and identified with ∗ [email protected] Ericson fluctuations [6]. These are essentially a manifes-tation of the spreading width.Fine structure of the GDR has also been observedin heavy nuclei [7, 8] and in other giant resonancessuch as the isoscalar giant quadrupole resonance (GQR)[9, 10], the Gamow-Teller resonance [11], or the magneticquadrupole resonance [12, 13]. Some progress has beenmade in the understanding of the fine structure by com-parison between experiments and theoretical calculationsof the distribution of resonance strength. In the case ofthe GQR it has been demonstrated that the fine structurehas its origin in the coupling of the 1 p h excitations thatconstitute the resonance to low-lying surface vibrations[9, 10], a mechanism discussed in Ref. [14]. However, inrecent studies of lighter nuclei ( Si, Ca) it was shownthat Landau damping plays a role in the formation of finestructure [15, 16]. The importance of Landau dampingwas also demonstrated for the GDR in
Pb [8].Here we turn attention to the GDR in lighter nucleiwith equal proton and neutron numbers Z and N , re-spectively. In the sd shell these nuclei are deformedand according to a recent theoretical study [17] a keydriver of deformation is the underlying α -cluster struc-ture. Here we focus on the nuclei Mg, Si, S and Ca, which allows to compare nuclei with prolate de-formed ( Mg, S), oblate deformed ( Si), and spheri-cal ground states ( Ca). Does this structural featuremanifest itself in the fragmentation of the GDR? An an-swer to this question has become possible by the conflu-ence of two advances: (i) The recent availability of high-resolution zero degree inelastic proton scattering datafrom a series of light nuclei [18, 19] permits fine struc-ture in the spectra to be resolved while providing a highdegree of selectivity towards 1 − states, and (ii) the avail-ability of microscopic calculations of the GDR strengthusing the random phase approximation (RPA) on top ofa deformed ground state with modern nucleon-nucleoninteractions. These calculations do not yet include cou-pling to the continuum or to more complex configurationsbut probe the effect of deformation on the fine struc- a r X i v : . [ nu c l - e x ] A p r d σ / d Ω d E ( m b / s r M e V ) Mg
16 1801 B ( E )( e f m ) Mg
16 18 Excitation Energy (MeV)01 Mg Si
16 18 Si
16 18 Si S
16 18 S S Ca
16 18 Ca Ca FIG. 1. Top row: High-resolution (20 −
30 keV FWHM) spectra of the ( p, p (cid:48) ) reaction at E = 295 MeV and θ lab = 0 . ◦ for Mg, Si, S, and Ca. The excitation energy range shown encompasses the GDR. Middle row: Theoretical B ( E
1) strengthdistributions calculated in a deformed-basis RPA with the UCOM interaction [17]. Bottom row: Theoretical B ( E
1) strengthdistributions in Mg and Si computed in a HFB+QRPA approach with the D1S Gogny force [42]. ture of the GDR in sd -shell nuclei. As shown below, adetailed comparison yields good agreement with experi-ment, which leads us to the conclusion that deformationplays a key role in the formation of fine structure in these sd -shell nuclei.A central part of this work is a quantitative charac-terization of scales of fragmentation in the GDR region.Various measures have been proposed in the past, viz. av-eraging of spectra at various scales [5], Fourier analysis[20], correlation analysis [21], the entropy index method[22, 23], local scaling dimension [24, 25], and waveletanalysis [26]. We have chosen wavelet analysis as it of-fers a quantification of the energy scales of fine structureswhile resolving the strength of fine structures in both ex-citation energy and energy scale. Thus structures canbe localized within the excitation energy region of theGDR. The wavelet analysis of the experimental spectraand corresponding theoretical strength distributions thenpermits us to make comparisons of the derived energyscales in different nuclei. Given the complexity of nu-clear behavior, such comparisons are necessarily of semi-quantitative nature. We do, however, expect some in-sight into the physical origin of the scales of structures. II. EXPERIMENT
Measurements of inelastic proton scattering at highresolution and at forward angles including 0 ◦ have onlyrecently become feasible [18, 27]. The present data weretaken at RCNP, Osaka, Japan with the Grand Raidenmagnetic spectrometer [28] with 295 MeV proton beams.Dispersion-matching techniques were applied to achievehigh energy-resolution of the order 20 to 30 keV full widthat half maximum (FWHM) at angles near zero degree[18]. Targets consisted of isotopically enriched thin foilswith areal densities of a few mg/cm . The spectrome-ter placed at 0 ◦ covered an angular accpetance of ± . ◦ .Additional data were taken with the spectrometer placedat larger angles covering an angular range up to 15 ◦ .The spectra analyzed in the present work correspond toa mean scattering angle of 0 . ◦ , where the cross sectionsfor excitation of 1 − states by relativistic Coulomb exci-tation dominate [29]. The momentum acceptance of thespectrometer permitted to cover a range of roughly 5 −
12 14 16 18 20 22 24Energy [MeV]0123456789 d σ / d Ω d E [ m b / ( s r M e V ) ] Ca FIG. 2. Multipole decomposition of the Ca( p, p (cid:48) ) cross sec-tions at Θ lab = 0 ◦ − . ◦ (blue histogram) for 200 keV bins.Purple: E1. Green: Quasifree scattering. Red: Isoscalar gi-ant resonances (E0+E2+E3). structure visible is related to the GDR. Figure 2 displaysby way of example a multipole decomposition analysis(MDA) of the angular distribution for the case of Ca.Details of the MDA approach are described in the analy-sis of comparable data on heavier nuclei [30–33]. Here weclosely follow a similar analysis applied recently to Ca[34]. As in Ref. [34] we neglect M1 contributions sincethe M1 strength in Ca is concentrated in a single tran-sition at 10.318 MeV [35]. Besides Coulomb excitationof the GDR and isoscalar E0, E2, and E3 collective ex-citations, we allow for a nuclear background (dominatedby quasifree scattering), whose angular distribution is as-sumed to be constant [36].The resulting decomposition is presented in Fig. 2 forthe 0 ◦ spectrum covering the full angular acceptance.Contributions from E0, E2 and E3 modes are small con-sistent with findings in Ca [34] and in heavier nuclei[30–33]. The MDA confirms the excitation of the GDRpeak by relativistic Coulomb excitation. The magnitudeof the nuclear background of about 2 mb/(sr MeV) isagain consistent with the MDA results in heavier nucleiand with other measurements at similar incident protonenergies [36, 37].
III. RPA CALCULATIONS
For comparison with the experimental measurements,theoretical B ( E
1) strength distributions were calculatedin the random phase approximation (RPA) starting fromaxially deformed Hartree-Fock (HF) ground states andusing explicit angular-momentum projection techniques.Both the HF and the RPA calculations use the same re-alistic nucleon-nucleon interaction derived from the Ar-gonne V18 potential by a unitary transformation in theframework of the Unitary Correlation Operator Method (UCOM) [38, 39] and are supplemented by a phenomeno-logical three-nucleon contact interaction. This Hamilto-nian was introduced and tested in Ref. [40] for ground-state observables and applied for RPA calculations inclosed-shell nuclei in Ref. [41] (we use the version labeled‘S-UCOM(SRG)’). All calculations were performed in aharmonic-oscillator single-particle basis covering 15 os-cillator shells. Further details on the deformed RPA ap-proach employed in this work can be found in Ref. [17].For Mg and Si, theoretical results are also avail-able from a selfconsistent axially-symmetric deformedHartree-Fock-Bogolyubov (HFB) plus quasiparticle RPA(QRPA) calculation with the D1S Gogny force [42].The resulting theoretical strengths are shown in themiddle and bottom rows of Fig. 1, respectively. Thelocation of the peaks in the calculated spectra of Si, S, and Ca (middle row) is roughly consistent withwhat is seen experimentally (top row), i.e. the predictedspreading of the strength in qualitative agreement withthe data. This is somewhat surprising since e.g. the frag-mentation of the GQR – albeit in heavier nuclei [10] –needs a description in terms of a second-RPA approach.For Mg, a discrepancy is observed at higher excita-tion energies, where a cumulation of strength is pre-dicted around 22 MeV without an experimental coun-terpart. Both, cluster models [43] and density functionalapproaches [44] indicate the presence of triaxial deforma-tions in Mg, which are not accounted for by our calcu-lation and could explain the larger deviation we observefor this nucleus. The calculations of Ref. [42] for Mgand Si (bottom row) roughly reproduce the centroidsbut show too little spreading of the strength.The model calculations do not include the continuumand so the strength distributions consist of discrete tran-sitions. For the subsequent wavelet analysis the theo-retical distributions hve been folded with a Gaussian ofa width corresponding to the experimental resolution sothat the low-scale cutoff in the wavelet power spectra (seebelow) matched the experimental data. These (Q)RPAstrength distributions and the experimental ones are nowanalyzed with the wavelet method.
IV. WAVELET ANALYSIS
The wavelet analysis of the measured spectra is illus-trated by the example of the Si( p, p (cid:48) ) data [Fig. 3(a)].It proceeds via the calculation of a wavelet coefficient C from the measured cross sections σ ( E ) (expressed inCounts/channel) C i ( δE ) ≡ C ( δE, E i ) = 1 √ δE (cid:90) σ ( E )Ψ ∗ (cid:18) E i − EδE (cid:19) dE, (1)where E i is the excitation energy of channel i , δE thewavelet scale, and Ψ the wavelet function. Here, thecomplex Morlet waveletΨ( x ) = π − / e ik x e − x / , (2)
16 18 20 22 24Excitation Energy (MeV)(b)01 C o un t s / C h a nn e l Si(p,p’)E = 295 MeVθ = 0.4 ◦ (a)0.11Wavelet Power (arb. units)0.11 W a v e l e t S c a l e ( M e V ) (c) FIG. 3. Example of the wavelet analysis. (a) Experimentalspectrum of the GDR in Si from the ( p, p (cid:48) ) reaction. (b)Square of the wavelet coefficient C [Eq. (1)] as a function of E x and wavelet scale. The highlighted area (16 −
24 MeV) isselected for projection onto the scale axis. (c) Wavelet powerspectrum. The peaks quantiatively characterize locations andwidths of the fine structure. with k = 5 is employed, which provides optimum bal-ance between resolution of excitation energy and energyscale for the present application (see, e.g., also Ref. [16]).The wavelet decomposition is performed over the wholespectrum with reflective boundary conditions and a re-gion of interest corresponding to the bulk of the GDRstrength. The squares of the wavelet coefficients, repre-senting a measure of the strength of structures resolvedby the wavelets, are displayed in Fig. 3(b).Because of possible contributions to the spectra fromspin- M −
24 MeV). The projected power spectrum P w ( δE ) = 1 N i (cid:88) i = i | C i ( δE ) C ∗ i ( δE ) | , (3)where i and i indicate the boundaries of the region ofinterest, is shown in the lower left hand panel (c). Peaksof strength in this power spectrum are associated withcharacteristic scales of the structures in the region of theGDR. The power spectrum is normalized to the spectralvariance in order to facilitate comparison between dif-ferent nuclei and with theoretical results. The analysisof the fluctuations, if represented as a power, character-izes the variance of the series under consideration. TheFourier transform preserves the variance of the signal andthe CWT does as well (at least approximately) since itis a convolution. Thus, a normalization to the variancefacilitates a comparison of powers deduced from the var-ious spectra despite the absence of an absolute scale.For the case of Si there are several sets of highresolution data available in the literature shown in the W a v e l e t P o w e r ( a r b . un i t s ) C o un t s / C h a nn e l ( a r b . un i t s ) (a) Si(p,p’) 1101000510 (b) Si(e,e’) 11010005 (c) Al(p, γ ) 0.1 1Wavelet Scale (MeV) 11010016 18 20 22Excitation Energy (MeV)05 (d) Al(p, α ) FIG. 4. Left: Experimental data from different high-resolution experiments populating the GDR in Si. Right:Power spectra from the wavelet analysis summed over excita-tion energy regions 16 −
24 MeV (solid line) and 17 −
23 MeV(dashed line). l.h.s. of Fig. 4: (a) present work, (b) the Si( e, e (cid:48) ) re-action [46, 47], (c) the Al( p, γ ) reaction [5], and (d)the Al( p, α ) reaction [48, 49]. It is expected that re-actions (a)-(c) predominantly excite the GDR. Reaction(d) favors isospin T = 0 states in Si and is thereforenot selective towards 1 − levels but may provide a pos-sible window into more general origins of fine structure.These data were analyzed in the same way and the re-sulting wavelet power plots are displayed on the r.h.s.of Fig. 4. Not only can corresponding structures be lo-cated in the experimental spectra, but also in the waveletpower plots there is a good agreement demonstrating theutility and reliability of the wavelet method. This is par-ticularly evident comparing the ( p, p (cid:48) ), ( p, γ ), and ( e, e (cid:48) )reactions. The similarities of the results underline thatthe structures extracted with the wavelet analysis are in-deed intrinsic features of the nuclei involved.We have also investigated the sensitivity to the en-ergy window chosen for the wavelet analysis. The greendashed lines in Fig. 4 show the power spectra resultingfrom a summation of the wavelet coefficients over the en-ergy region 17 −
23 MeV rather than 16 −
24 MeV. Asone can see, the differences are very small. W a v e l e t P o w e r ( a r b . un i t s ) Mg I II III Si I II III S I II III Ca I II III
FIG. 5. Wavelet power spectra of the GDR in Mg, Si, S, and Ca from the experimental and theoretical data of Fig. 1.Blue solid lines: experiment. Short-dashed red lines: HF+RPA calculatons with S-UCOM(SRG) interaction [17]. Long-dashedgreen lines: HFB+QRPA calculations with D1S Gogny interaction [42].
V. DISCUSSION
Table I summarizes the significant scales in Si inthe wavelet power spectra for all experimental data andtheoretical calculations. These can be grouped in threeclasses: In all the experimental spectra there is a scale(Class I) at approximately 80 keV. This is similar to theaverage level width due to Ericson fluctuations in Si[50, 51] and we tentatively follow this identification. Itconforms with the absence of a corresponding scale inthe theoretical results. We note that the wavelet analy-sis of the strength distribution from Ref. [42] also showsa scale at about 80 keV. However, the two-dimensionalcorrelation analog to Fig. 3(b) traces its origin back toa series of transitions localized in a small energy intervalof E x (cid:39) −
21 MeV. Thus, the scale does not representa genuine feature of the GDR.At larger energies scales similar in energy to thosein Si( p, p (cid:48) ) are seen in the Si( e, e (cid:48) ) data and in the Al( p, γ ) data. The Al( p, α ) data shows similar num-bers of scales but with slightly shifted energy. These aredenoted Class II and Class III, where Class III scales arelarge scales associated with the spread of the distributionof strength while Class II are intermediate scales in the
TABLE I. Characteristic scales of the fine structure of theGDR in Si from different experiments and the theeoreticalresults of Refs. [17] and [42]. Scales are divided into threeclasses (see text).Spectrum Ref. Scales (MeV)Class I Class II Class III Si( p, p (cid:48) ) present 0.08 0.23 0.36 0.59 1.0 2.9 Si( e, e (cid:48) ) [46, 47] 0.08 0.23 0.36 1.0 3.3 Al( p, γ ) [5] 0.08 0.14 0.24 0.38 0.59 1.1 3.2 Al( p, α ) [48, 49] 0.09 0.12 0.40 1.0 2.6RPA [17] 0.23 0.44 2.1QRPA [42] (0.08) a a Not a genuine scale of the GDR. see text. region 100 keV to 1 MeV . Similar results are obtainedfor the other nuclei chosen for study, Mg, S and Caas illustrated in Fig. 5 although in the case of Ca theClass II scales are rather weak and the power spectrumis dominated by Class III.The power spectra extracted from the data and fromthe theoretical methods are compared in Fig. 5, Thepower spectra from theory reproduce features of the ex-perimental data like the common observation of a scalearound 100 keV and a typical number of 3 − α clustering (for experimental evidence see We note that the choice 100 keV and 1 MeV as borders to distin-guish between the different classes is somewhat arbitrary. Thevalues were chosen to facilitate easy comparison to previous stud-ies of the GDR in
Pb [8] and of the GQR in many nuclei[9, 10]. e.g. [52]) in these nuclei [17]. This suggests that a primesource of the fine structure in the GDR in light nucleiis deformation rather than the coupling to surface vi-brations invoked for the GQR in heavier nuclei [9, 10].This observation is supported by the case of the closedshell nucleus Ca where both experiment and theory ex-hibit only weak fine structures and correspondingly littlewavelet power of Class I and II.
VI. CONCLUSIONS
We have available, for the first time, a set of high res-olution data for the GDR region of N = Z nuclei in the sd shell, together with RPA calculations performed ontop of a deformed ground state with a realistic nucleon-nucleon interaction. This enables us to investigate thelong-standing question of the origin of fine structure ofthe GDR in nuclei in this mass region. A wavelet analysis permits to extract scales characterizing this fine struc-ture and the results for different reactions exciting theGDR in Si shows that good consistency is achieved.Comparisons between experimental data and the RPAcalculations suggest that fine structure at the level of afew hundred keV (class II scales) results mainly from thedeformation of the nuclei driven by α clustering. This isin sharp contrast to the case of the GQR where couplingto 2 p h states is the main source of characteristic scalesin the region 100 keV - 1 MeV [9, 10]. ACKNOWLEDGMENTS
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