Structure of ^{20}Ne states in the resonance ^{16}O+α elastic scattering
D. K. Nauruzbayev, V. Z. Goldberg, A. K. Nurmukhanbetova, M. S. Golovkov, A. Volya, G. V. Rogachev, R. E. Tribble
aa r X i v : . [ nu c l - e x ] A p r Structure of Ne states in the resonance O+ α elastic scattering D. K. Nauruzbayev,
1, 2, ∗ V. Z. Goldberg, A. K. Nurmukhanbetova, † M. S. Golovkov,
A. Volya, G. V. Rogachev, and R. E. Tribble National Laboratory Astana, Nazarbayev University, Astana, 010000, Kazakhstan Saint Petersburg State University, Saint Petersburg, 199034, Russia Cyclotron Institute, Texas A & M University, College station, Texas, 3366, USA Joint Institute for Nuclear Research, Dubna, 141980, Russia Dubna State University, Dubna, 141980, Russia Department of Physics, Florida State University, Tallahassee, Florida, 32306, USA
Background:
The nuclear structure of the cluster bands in Ne presents a challenge for different theoreticalapproaches. It is especially difficult to explain the broad 0 + , 2 + states at 9 MeV excitation energy. Simultaneously,it is important to obtain more reliable experimental data for these levels in order to quantitatively assess thetheoretical framework. Purpose:
To obtain new data on Ne α cluster structure. Method:
Thick target inverse kinematics technique was used to study the O+ α resonance elastic scattering andthe data were analyzed using an R matrix approach. The Ne spectrum, the cluster and nucleon spectroscopicfactors were calculated using cluster-nucleon configuration interaction model (CNCIM).
Results:
We determined the parameters of the broad resonances in Ne: 0 + level at 8.77 ± + level at 8.75 ± ±
120 keV; the width of9.48 MeV level of 65 ±
20 keV and showed that 9.19 MeV, 2 + level (if exists) should have width ≤
10 keV. Thedetailed comparison of the theoretical CNCIM predictions with the experimental data on cluster states was made.
Conclusions:
Our experimental results by the TTIK method generally confirm the adopted data on α clusterlevels in Ne. The CNCIM gives a good description of the Ne positive parity states up to an excitation energyof ∼ I. INTRODUCTION
It is well recognized that the α particle interaction withatomic nuclei is important in astrophysics [1]. Even ifastrophysical reactions involving helium do not proceedthrough the strong α -cluster states (because of their highexcitation energy), these states can provide α width tothe states that are closer to the region of astrophysicalinterest through configuration mixing.For a long time, the surprising alpha cluster structurehas been a stimulus for the development of classical shellmodel approaches (see [2] for new results). Additionally,work by the authors of Ref. [3] recently “strengthenedthe theoretical motivation for experimental searches ofalpha cluster states in alpha-like nuclei” [3]. The authorsof Ref. [3] related the nuclear structure in light systems ofeven and equal numbers of protons and neutrons with thefirst-order transition at zero temperature from a Bose-condensed gas of alpha particles ( He nuclei) to a nuclearliquid. Ne nucleus presents a famous example of the mani-festation of the alpha-cluster structure, and therefore itmakes this nucleus a touchstone for ab initio approaches[4]. The nucleus Ne is a benchmark case for the tra-ditional shell model and its extension into algebraic and ∗ [email protected] † [email protected] clustering domains. The well-established effective inter-action Hamiltonians such as [5] not only shows an out-standing agreement with experimental data for sd -shellnuclei but also generates configuration mixing that showstransition to deformation and clustering.The remarkable feature of Ne nucleus shows up in thefact that almost all the observed states below 10 MeV canbe classified into several overlapping rotational-like bandswith the first one based on the ground 0 +1 state. Thereare three other bands based on 0 + levels: on 0 +2 at 6.725MeV, on a very narrow 0 +3 at 7.191 MeV, on a very broad0 +4 at ∼ +2 and 0 +4 bands have α + O core structureas can be seen from their reduced α particles widths, andprobably the 0 +3 band has predominant C+ Be struc-ture which manifests used itself in the selectivity of the Be transfer reactions [7]. As the ground state band andthe 0 +2 band in Ne can be related with similar struc-tures in O and C, the “additional” structure of 0 +4 states is not understood [4]. The cluster approaches [8]related the 0 +4 band with large α -widths, starting withthe so-called “ O+ α ” higher nodal band, which has onemore nodal point in “ O+ α ” relative wave function thanthe lower bands have. However it appeared that there aretoo many bands with a similar structure.The α particle decay threshold in Ne is 4.73 MeV,while the threshold for proton decay is at 12.8 MeV (neu-tron decay threshold is even higher). Therefore resonance α particle scattering should be considered as an evident θ c . m . = ◦ α p E ( MeV )0102030405060 T i m e ( n s ) FIG. 1. E-T spectrum for the zero degrees detector. Alphaparticles dominate; one can see also a weaker proton locusbelow the alpha particles. way to obtain data on the natural parity levels in Neup to 13 MeV excitation energy. Indeed the majority ofadopted data [6] on level properties in the region in ques-tion based on a resonance work and an analysis made in1960 year. More recently the α + O resonance scatter-ing experiment were developed further to backward an-gles and the data were reanalyzed using R matrix codeMulti 6 [9]. The authors [9] obtained quite different re-sults from those used in for many levels (see Table 1);in particular the broad 0 +4 and 2 +4 levels appeared evenmuch more broader. The authors [9] noted difficulties ofthe fit in the region of 6-8 MeV excitation energies, ina region mainly free from narrow resonances. Evidentlystrong states of over 1 MeV width should influence a verybroad excitation region (see for instance [10]).T. Fortune et al. [11, 12] were the first who recognizedthe importance of the fact that single particle structure ofthe broad states is in drastic contradiction with the shellmodel predictions. They [11, 12] also proposed the idea ofmixing different configurations to explain the effect. Thesame idea was used in [13]. However, the authors of Refs.[11, 12] used old data for the 0 +2 state and used someestimates for the properties of the broad states proposedin Ref. [14] to support the idea. Later measurements [6]gave the width of 19 keV for the 6.72 MeV state (whichis about 25% larger than that used in work [11]).The experimental aim of this work is to obtain newinformation on the structure of Ne states, especiallythe broad 0 + , 2 + states. Unlike other experimentalists,we used the Thick Target Inverse Kinematics method(TTIK) (see [15–19] and references therein) to study theexcitation functions for the O( α , α ) O elastic scatter- d σ / d Ω ( m b / s r ) E CM ( MeV ) DataF it [11][9]FIG. 2. The O( α , α ) O elastic scattering excitation func-tion at 180 ◦ cm. The excitation energies in Ne, E x inTable 1, are related with cm energy, E cm , by expression,E x =E cm +4.73 MeV. The bold (red) line is the R matrix fitwith the parameters of the present work. The dot (cyan) lineis a fit with the 0 +4 excitation energy of 8.3 MeV [11], and dot(black) line is a fit with the 0 +4 energy excitation energy of8.62 MeV and the width of 1.472MeV [9]. ing in the Ne excitation region of 5.5-9.6 MeV and in abroad angular interval. On the theoretical side, we alsoused multi configuration shell model calculation to un-derstand the limits of this approach in a description ofthe cluster states.
II. EXPERIMENT
The experiment was performed at the DC-60 cyclotron(Astana) [17] which can accelerate heavy ions up to the1.9 MeV/A energy. While the TTIK method can’t com-pete with a classical approach in terms of energy resolu-tion, the possibility of observing excitation functions atand close to 180 degrees, where the resonance scatteringdominates over the potential scattering, enables one toobtain more reliable information on the broad states. Inthe TTIK technique the inverse kinematics is used; andthe incoming ions are slowed in a helium target gas. Thelight recoils, α particles, are detected from a scatteringevent. These recoils emerge from the interaction withthe beam ions and hit a Si detector array located at for-ward angles while the beam ions are stopped in the gas,as α particles have smaller energy losses than the scat-tered ions. The TTIK approach provides a continuousexcitation function as a result of the slowing down of thebeam.For the present experiment, the scattering chamberwas filled with helium of 99.99% purity. The 30 MeV O beam entered the scattering chamber through a thinentrance window made of 2.0 µ m Ti foil. Eight mon-itor Si detectors were placed in the chamber to detect O ions elastically scattered from the Ti foil at 21 ◦ an-gle. This array monitors the intensity of the beam withprecision better than 4%. Fifteen 10x10 mm Si detec-tors were placed at a distance of ∼
500 mm from theentrance window in the forward hemisphere at differentlaboratory angles starting from zero degrees. The gaspressure was chosen to stop the beam at distance of 40mm from the zero degrees detector. The detector energycalibration and resolution ( ∼
30 keV) were tested witha
Ra,
Rn,
Po and Po α -source. The experi-mental set up was similar to that used before [19], andmore details can be found in Ref. [17, 19]. The mainerrors in the present experimental approach are relatedto the uncertainties of the beam energy loss in the gas.To test the energy loss, we placed a thin Ti foil (2.0 µ m)at different distances from the entrance window. Thiscan be used during the experiment without cycling vac-uum. We found that the data [20] for energy loss of Oin helium are correct. The details of these tests will bepublished elsewhere. As a result, we estimated that theuncertainties in the absolute cross section are less than6%. This conclusion was tested by comparison with theRutherford cross sections at low energies. The agreementwith the Rutherford scattering is within 5% error bars atall angles (see Fig.3).Together with the amplitude signal, the Si detectorsprovided for a fast signal. This signal together with a“start” signal from RF of the cyclotron was used for theTime-of-Flight measurements. This E-T combination isused for particle identification in the TTIK approach [16–18]. Of course, only α particles should be detected as aresult of the interaction of O with helium at the cho-sen conditions. However, protons can be created in theTi window, and protons can appear due to hydrogen ad-mixtures in the gas. Indeed, we have observed a weakproton banana, likely as a result of reactions in the win-dow. These protons were easily identified by TF andseparated from the α particles, as seen in Fig.1. III. EXPERIMENTAL RESULTS ANDDISCUSSION
The experimental excitation functions were analyzedusing multilevel multichannel R matrix code [21]. Thecalculated curves were convoluted with the experimentalenergy resolution. The experimental energy resolutionwas ∼
30 keV at zero degrees and deteriorated up to ∼
100 keV with angles estranging from zero degrees. Wedid not notice a deterioration of the energy resolutionwith the energy loss of the beam in the chamber. As itseen in Table 1 the excitation energies of the resonancesof the present work agree with the adopted ones [6] within a ) ◦ ≤ θ c . m . ≤ ◦ b ) ◦ ≤ θ c . m . ≤ ◦ d σ / d Ω ( m b / s r ) ( c ) ◦ ≤ θ c . m . ≤ ◦ d ) ◦ ≤ θ c . m . ≤ ◦ E CM ( MeV )( e ) ◦ ≤ θ c . m . ≤ ◦ DataF it [11][6]FIG. 3. R matrix fit (bold red curve) of the excitation func-tions for the α + O elastic scattering. (a) The dashed (cyan)curve presents the data of the 0 + level at the 8.3 MeV excita-tion energy [11] and (e) dot (black) line is a fit with 2 + levelat the excitation energy of 9.0 MeV [6]. R matrix fit. Fig.2 demonstrates the TABLE I. Ne levelsN TUNL data [6] H. Shen et al. [9] This work CNCIME x J π Γ α E x Γ α E x Γ α γ α E x J π SF p SF α (MeV) (keV) (MeV) (keV) (MeV) (keV) (MeV)1 0 0 +1 - - 0 Large 0 0 + +1 - - 1.63 Large 2.242 2 + +1 - - 4.25 Large 4.58 4 + − (28 ± − - - 4.45 0.03 1.45 6.73 0 +2 ± +3 − ± +3 ± +2 +2 ± +3 +3 +2 − ± − ± ≈ +4 >
800 8.62 1470 8.77 ± ± ∼ +4 +1 ± + −
19 8.84 27 8.85 18.015 9.00 2 +4 ≈
800 8.87 1250 8.79 ± ±
120 0.86 8.36** 2 +4 +3 + − + - - (9.29) ≤ + ±
15 9.48 46 9.48 65 ±
20 0.02?20 9.99 4 +4 ±
30 10.02 150 9.97 157 0.38 9.5 4 + − ±
40 10.26 190 10.26 1.922 10.41 3 −
80 10.40 101 10.4123 10.58 2 +
24 10.56 15 10.58 10.2 2+ 0.005 0.0424 10.80 4 +4
350 10.75 400 10.80 10.7 4 + +5
580 10.99 700 10.97 11.9 0 +
26 11.24 1 −
175 11.19 85 11.2427 11.95 8 + (3.5 ± − + psd space. SF is to the first excited state in O; SFs for the ground state in O are ≤ data at 180 ◦ and illustrates the differences in the fits dueto different parameters of the broad 0 +4 resonance. Thedata on the resonances used in the present R -matrix fitare summarized in the Table 1 together with the adopteddata [6]. Data of the last R matrix analysis [9] are alsogiven in Table 1. The analysis [9] resulted in level pa-rameters which are often different from the adopted.Our analysis (Table 1) resulted in small discrepancieswith the data [6] in the detail description of the narrowstates (widths less than 10 keV). The R -matrix code [21]is tuned for the TTIK measurements and for the analysesof states with a width of over 10 keV to accelerate auto-matic fit calculations. Therefore the small disagreementsfor the narrow states are not significant. We focused onthe broader states and on the part of the excitation func-tion changing slowly with energy and angle.Our analysis indicates that all strong alpha clusterstates at 5 ∼ R -matrix fit. There-fore we included in the fit the Ne ground state, thefirst 2 + and 4 + states and 1 − (5.67 MeV) state in thefit, even though below the investigated region (see Table 1). Among these states, only the 1 − state is above the α particle decay threshold in Ne; the reduced widthof this state is known, and it is large. Shell model cal-culations also give very large spectroscopic factors for allmembers of the ground state band. Indeed, a good agree-ment needs large values of the corresponding amplitudes(over 0.7). Above the investigated excitation energy re-gion, high spin α -cluster resonances are mainly known.Each of these resonances (see Table 1) considered sep-arately influences the fit, especially at 180 ◦ . However,their joint influence is much weaker. This cancellationis due to different parities of the spins. Only the in-fluence of the closest to the investigated region, the 4 + (9.99 MeV) resonance, can be noticed. A somewhat bet-ter fit needs the width of this resonance to be slightlylarger ∼
160 keV (well within the quoted uncertainties,see Table 1). Parameters of all other resonances abovethe investigated region were fixed according to the dataof Ref. [6]. A good general fit ( χ =1.1) was reached inthis way without any backward resonance inclusion.All resonances are at the maximum at 180 ◦ . The broadhump at this angle at cm energy of 4 MeV (Fig. 2) is TABLE II. α + C levels in O O level Γ α exp keV [6] γ α (1); -V MeV γ α (2); -V MeV γ α (3); -V MeV1 − ; 9.58 MeV 420 ±
20 0.70; 138.2 0.72; 150.0 0.84; 158.54 + ; 10.36 MeV 26 ± a clear indication for the presence of low spin states, 0 + and 2 + . Higher spin states are narrower. A single level(2 + ) cannot produce the strong peak at different angles,and levels of different parity, such as 2 + and 1 − , interferedestructively at 180 ◦ . The present analysis resulted intwo practically degenerate states at ∼ + are rather close to the adopted values. However, if thislevel is moved to 9.0 MeV excitation energy (as in [6])then the fit becomes worse, especially in the vicinity ofthe dip die to the presence of another 2 + level at 9.48MeV (Fig. 3(e)). The fit becomes even worse with avery broad 2 + level resulting from the parameters of Ref.[9].T. Fortune at al., [11] observed a broad distributionwith a center at 8.3 MeV excitation energy in Ne andrelated it with the 0 + level. Fig. 2 presents R matrixcalculations with the 0 + level to be moved to 8.3 MeV ex-citation energy. This move destroyed the good fit. Fig.2shows also that a very broad 0 + level of the fit [9] destroysthe agreement.The 18 th (2 + ) level of [6] is in the energy region ofthe present investigation. It was observed in a singlework in a study of Na β + decay. We have not foundany reliable evidence for the presence of this state. If itexists, its width should be less than 10 keV. We observea fluctuation of points which might be associated with anarrow 2 + level at excitation energy of 9.29 MeV.We noted that the 19 th level, 2 + , has the adopted [6]excitation energy in our fit but a different width of 65 ± γ decay of this state inthe presence of a large background. A broader level thanin Ref. [22] was also found in work [9] as shown in Table1. We characterized the alpha-cluster properties of thestates above the alpha particle decay threshold bySF= γ α =Γ α exp /Γ α calc , where Γ α calc is the single alphaparticle width calculated in the α -core potential. To nor-malize SFs we calculated these values for the well-knownalpha-cluster states in O.The Woods-Saxon potential was used to calculate thelimit (Γ α cal ) as the widths of single particle states inthe potential. First we tried to fit the widths of knownalpha-cluster states, 1 − and 4 + , in O so that γ α =Γ α exp /Γ α cal ∼ × / ; the Coulomb potential was taken in to account as a chargesphere potential with R coul = R. We made first calcula-tions (1) with r = 1.31 fm and the diffuseness a = 0.65fm, then we set r = 1.23 fm (2), and we finally per-formed the third calculations (3) with r = 1.23 fm anda = 0.6 fm. The results are summarized in Table 2. The γ α calculations for the Ne states were made with thefinal (3) parameters should be compared with SF givenby a theory.
IV. THEORETICAL DESCRIPTION OF THE α CLUSTER STATES IN Ne CNCIM [2] is among the latest developments of theclassical shell model approaches towards clustering. Thismodel targets a combination of classical configuration in-teraction techniques with algebraic methods that emergein the description of clustering. The ability to constructa fully normalized set of orthogonal cluster channels is atthe core of this approach; the overlaps of the shell modelstates with these channels are associated with spectro-scopic factors and compared in Table 1 with the re-duced widths obtained earlier. The CNCIM allows us tostudy clustering features that emerge in models with well-established traditional shell model Hamiltonians. Theseeffective model Hamiltonians are built from fundamentalnucleon-nucleon interactions followed by phenomenolog-ical adjustments to select observables; thus, they gener-ally describe of a broad scope of experimental data withhigh accuracy. Apart from using these phenomenolog-ical shell model Hamiltonians, our study does not in-volve any adjustable parameters. In order to fully ex-plore the problem, we considered several different modelspaces and corresponding Hamiltonians: the sd modelspace with USDB interaction [5]; unrestricted p-sd shellmodel Hamiltonian [23], the same Hamiltonian has beenused in Refs. [2]; WBP Hamiltonian [24] allowing 0 ~ ω ,1 ~ ω and 2 ~ ω excitations in p-sd-pf valence space; andthe sd-pf Hamiltonian [25]. This sequence of Hamiltoni-ans represents and expansions of the valence space from sd to p-sd , to p-sd-pf . All models are in good agreementfor the sd -states; the low-lying negative parity states aswell as positive 2ph excitations are dominated by the p-sd configurations. Thus, in Table 1 we only include theresults from the p-sd Hamiltonian which turned out tobe most representative although the following discussionand conclusions are largely based on comparisons. Thelowest states associated with significant fp shell compo-nent appear at excitation energies above 15 MeV.Shell model calculations for Ne with open shells predict well the ground state band. The structureof this band is based on the dominating SU(3) configu-ration (about 75%) with quantum numbers (8,0). Themodel predicts (Table 1) large SF for all members of theband based on the ground state in Ne. The 0 + , 2 + , and4 + members of the band are below the α particle decaythreshold and do not have observable α particle widths.However, large α cluster SFs for these states provide fora better R matrix fit. While uncertainties for the R ma-trix amplitudes for these states are large, the fit (Fig.2 and 3) requires these amplitudes to be close to thoseof the negative parity states with the known extreme α cluster structure. The α particle widths of the highest6 + and 8 + members of this band are known. There isa long history of attempts and ideas to describe thesewidths using shell model approaches (see, for instance[5]). The most calculations predicted large clustering forthe band but could not explain the decrease of the re-duced width for the 6 + and 8 + members. As one sees inTable 1, the CNCIM calculations are in fine agreementwith the experimental data for these states. All membersof this band have significant clustering that diminishes athigher energies due to configuration mixing. The second0 + state within sd space appears at around 6.7 MeVof excitation (in psd model in Table 1, this is a third0 + state at 6.9 MeV). This level also has a substantialclustering component and absorbs nearly all 15% of theremaining strength of the SU(3) (8,0) component. Thefollowing 2 +2 (in experiment and in sd model, but thirdin p-sd model as discussed in what follows) at about 7.4MeV of excitation energy being a member of the 0 +2 bandcan be described in a similar way.A defrost of the shell (which is filled in O) resultsin the doubling of the levels, as it is observed and isshown in Table 1 new levels 0 + and 2 + , marked withasterisks, appear. In our calculations, the ordering inenergy is reversed for both doublets. Structurally themembers of each doublet are very different which allowsthem to be so close in energy and inhibits configurationmixing and Wigner repulsion. One of the doublet levels(the lower 0 + and 2 + levels) has a large α cluster SFrelative to the first excited state in O and much smallerSF relative to the ground state in O (pay attentionthat the theory gives the wrong order for the levels inquestion, see Table 1). Indeed the predicted differencein the structure is supported by population selectivity indifferent nuclear reactions. The 6.72 MeV 0 + and 7.42MeV 2 + are populated much stronger than neighboring7.20 and 7.83 MeV levels in the O( Li, d) reaction [26].The opposite is the case in the C( Be, n) or C( C, α )[27] reactions.The theory gives large single particle spectroscopic fac-tors for the sd states and smaller for the 7.20 and 7.83MeV states. Indeed, one expects that states in Ne witha hole in the p1/2 shell will be weakly excited in thesingle nucleon transfer, F( He, d) reaction in accor-dance with the experimental data [28]. The experiment[11, 12, 28] supports also detailed single particle SF cal- culations giving SF for the ground state smaller than forthe 0 + + than for 2 + member of the band based on the6.73 MeV state.In the sd valence space (USDB) the third 0 + stateappears only at 11.9 MeV, and it has a relatively smallalpha spectroscopic factor; opening of the p -shall in ad-dition to 0 + at 6.27 MeV, leads to 0 + state at 9.7 MeV.However, the predicted state has a low proton spectro-scopic factor which is not as well supported by obser-vations. A similar serious discrepancy is observed witha broad 4 th + state at around 9 MeV, both sd and p-sd shell models produce candidates but with very lowalpha SF. As evident from our studies the only strongcoupling to alpha channels could come from fp shell andhigher shells. The lowest two bands saturate the alphastrength within sd configurations, holes in the p -shell donot lead to a significant contribution due to low levelof core excitation in the ground state of O. While ourmodels predict high excitation energies of states with sig-nificant fp components, we can speculate that strong con-figuration mixing, collective effects such as deformation,and coupling to continuum via super radiance mechanism[29, 30] can enhance admixture needed to reproduce thebroad resonances observed. There is a similar problemwith α cluster negative parity states 1 − and 3 − , the p-sd Hamiltonian produces an acceptable spectrum but thealpha spectroscopic factors are low (see also [31]).
V. CONCLUSIONS
In this work we study α -clustering in Ne. This nu-cleus is a benchmark example of many theoretical tech-niques targeting clustering in light nuclei. Our R -matrixanalysis of TTIK experimental data confirms previouslyknown results and establishes new constraints for the po-sitions and widths of the resonances. We compared ourfindings with those obtained theoretically using cluster-nucleon configuration interaction approach developed inseveral previous works [2] and references therein. Thereis good overall agreement between theoretically predictedand observed spectra. Our theoretical approach describesvery well the ground state band and the band built onthe first 0 + state. Allowing cross shell excitations fromthe p -shell, it was possible to reproduce the band builton the second 0 + state. For these states, all spectro-scopic factors for alpha transitions to the ground state of O and to the first excited state in O as well as pro-ton spectroscopic factors to the ground state of F arewell reproduced. The situation is not as good when itcomes to resonances 1 − − and 4 th + and 4 th + , all ofthese states are broad and have exceptionally large alphaspectroscopic factors.In order to describe strong clustering features thesestates must include configurations from fp shell and fromhigher oscillator shells, however Hamiltonians that we ex-plored predict these contributions to be negligible below15 MeV of excitation. Thus, inability of theoretical mod-els to describe broad states exclusively while working wellelsewhere suggests an additional coupling mechanism un-accounted for in the traditional shell model Hamiltoni-ans. The super radiance suggested in Refs. [29, 30]could provide this mechanism. Alternatively, the prob-lem could be associated with relatively unknown crossshell interactions. Therefore, work shows that the exper-imental study of alpha clustering represents an outstand-ing tool for exploring cross shell excitations especiallythose of multi-particle multi-hole nature. ACKNOWLEDGMENTS