Imprint of nuclear bubble in nucleon-nucleus diffraction
aa r X i v : . [ nu c l - t h ] S e p Imprint of nuclear bubble in nucleon-nucleus diffraction
V. Choudhary , ∗ W. Horiuchi , † M. Kimura , , ‡ and R. Chatterjee § Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247 667, India Department of Physics, Hokkaido University, 060-0810 Sapporo, Japan and Nuclear Reaction Data Centre, Faculty of Science, Hokkaido University, 060-0810 Sapporo, Japan
Background:
The density of most nuclei is constant in the central region and is smoothly decreasing at thesurface. A depletion in the central part of the nuclear density can have nuclear structure effects leading to theformation of “bubble” nuclei. However, probing the density profile of the nuclear interior is, in general, verychallenging.
Purpose:
The aim of this paper is to investigate the nuclear bubble structure, with nucleon-nucleus scattering,and quantify the effect that has on the nuclear surface profile.
Method:
We employed high-energy nucleon-nucleus scattering under the aegis of the Glauber model to analyzevarious reaction observables, which helps in quantifying the nuclear bubble. The effectiveness of this method istested on Si with harmonic-oscillator (HO) densities, before applying it on even-even N = 14 isotones, in the22 ≤ A ≤
34 mass range, with realistic densities obtained from antisymmetrized molecular dynamics (AMD).
Results:
Elastic scattering differential cross sections and reaction probability for the proton- Si reaction arecalculated using the HO density to design tests for signatures of nuclear bubble structure. We then quantify thedegree of bubble structure for N = 14 isotones with the AMD densities by analyzing their elastic scattering at325, 550 and 800 MeV incident energies. The present analyses suggest O as a candidate for a bubble nucleus,among even-even N = 14 isotones, in the 22 ≤ A ≤
34 mass range.
Conclusion:
We have shown that the bubble structure information is imprinted on the nucleon-nucleus elasticscattering differential cross section, especially in the first diffraction peak. Bubble nuclei tend to have a sharpernuclear surface and deformation seems to be a hindrance in their emergence.
I. INTRODUCTION
Advances of radioactive beam facilities have allowed usto study nuclei with extreme neutron to proton ratios.In fact, close to the neutron drip line, one has discov-ered exotic features like haloes [1, 2] - an extended lowdensity tail in the neutron matter distribution. At leastfor light nuclei, this was thought to be a threshold phe-nomenon resulting from the presence of a loosely boundstate near the continuum. In this context, with currentinterest moving towards the medium mass region, an-other exotic structure that of a depression in the centralpart of nuclear density - called a “bubble” - has attractedconsiderable attention.Systematic studies of electron scattering of stable nu-clei have revealed that the central density of stable nucleiis almost constant, ρ ≈ .
16 fm − [3]. In light nuclei,distinct nuclear orbitals play a role in the emergence ofthe bubble structure. If the s -orbitals are empty, the in-terior density of nuclei becomes depleted. For example,Refs. [4, 5] showed that the central depression of the pro-ton density in Si is about 40% as compared to stable S using several mean-field approaches, originated fromthe proton deficiency in the 1 s / orbit. The possibilityof forming bubble nuclei have also been explored theo-retically in the medium [4, 6–8], and superheavy mass ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] regions [9].The experimental indication of the central depressionof protons in the unstable nucleus Si was recently re-ported using γ -ray spectroscopy [10]. Electron scatter-ing on unstable nuclei is the most direct way to probethe central depression of proton density in bubble nuclei.Recently, the SCRIT electron scattering facility has suc-ceeded in extracting information about the nuclear shapeof Xe [11].However, unlike a hadronic probe, which is sensitive toboth neutrons and protons, the electron scattering hasdifficulty to probe the neutron density distribution evenfor stable nuclei [12]. In this context, it is worth mention-ing that proton-nucleus scattering has been successfullyapplied to deduce the nuclear matter density distribu-tions [13]. Proton scattering can also be extended forunstable nuclei with the use of inverse kinematics mea-surement as demonstrated in Ref. [14]. Indeed, this moti-vates us to inquire if information on the bubble structurein nuclei can be investigated with nucleon-nucleus scat-tering.In this paper, we perform a systematic study to testthe nucleon-nucleus scattering as a probe for the nuclearbubble structure. This paper is organized as follows. Sec-tion II briefly presents the formalism that describes thenucleon-nucleus collision at high incident energy withinthe Glauber model, wherein the elastic scattering andtotal reaction cross sections are evaluated. Using thisformulation, in Sec. III, we discuss how signatures of thenuclear bubble structure are reflected in the cross sectionsby using an example of a simple ideal case, Si. We showthe relationship between the internal depression and thesurface diffuseness, and propose a practical way to eval-uate the bubble structure. For this purpose, the gener-alized “bubble” parameter is introduced as a measure ofthe nuclear bubble structure. In this work, we also exam-ine the structure of N = 14 isotones, O, Ne, Mg, Si, S, Ar, and Ca. Section IV presents details ofthe structure calculation by the antisymmetrized molec-ular dynamics (AMD) model. The formalism is brieflyexplained in Sec. IV A, and the resulting structure in-formation focusing on the bubble structure is given inthe following Sec. IV B. Section V demonstrates how thenucleon-nucleus scattering works for extracting the bub-ble parameter of the nuclear density distributions. Wediscuss the feasibility through a systematic analysis of theelastic scattering differential cross sections with variousdensity profiles. The conclusions of our study are pre-sented in Sec. VI. Some details on how nuclear structureparameters are evaluated in the AMD are in appendix A.
II. NUCLEON-NUCLEUS REACTIONS WITHGLAUBER MODEL
The Glauber theory offers a powerful description ofhigh-energy nuclear reactions [15]. Here we consider thenormal kinematics in which the incident proton is bom-barded on a target nucleus. Thanks to the eikonal andadiabatic approximations, the final state wave functionof the target nucleus after the collision is simplified as | φ f i = e iχ | φ i i , (1)where | φ i i represents the initial wave function of the tar-get nucleus, and e iχ is the phase-shift function, which in-cludes all the information about the nucleon-nucleus col-lision. The elastic scattering amplitude for the nucleon-nucleus reaction is given by F ( q ) = iK π Z d b e i q · b (1 − e iχ N ( b ) ) , (2)where K is the relative wave number of the incident nu-cleon, b is the impact parameter vector perpendicularto the beam direction, and q is the momentum trans-fer vector of the incident nucleon. With this scatteringamplitude, the elastic scattering differential cross sectioncan be evaluated by dσd Ω = | F ( q ) | . (3)The total reaction cross section of the nucleon-nucleuscollision can be calculated by σ R = Z d b P ( b ) (4)with the nucleon-nucleus reaction probability defined as P ( b ) = 1 − | e iχ ( b ) | . (5) Since the evaluation of the phase-shift function is de-manding in general, for the sake of simplicity we employthe optical-limit approximation (OLA). As presented inRefs. [16–19], the OLA works well for many cases of theproton-nucleus scattering so that the multiple scatteringeffects can be ignored. The optical phase-shift functionfor the nucleon-nucleus scattering in the OLA is given by e iχ N ( b ) ≈ exp (cid:20) − Z d r ρ N ( r )Γ NN ( b − s ) (cid:21) , (6)where r = ( s , z ), and s is the two-dimensional vectorperpendicular to the beam direction z . ρ N ( r ) denotes thenucleon density distributions measured from the centerof mass of the system. The crux of any calculation willbe to calculate this density with reliable nuclear structuremodels. This is also the primary point where informationon the bubble structure enters into the Glauber modeland is reflected in the scattering or reaction observables.Γ NN is the profile function, which describes thenucleon-nucleon collisions. The profile function forthe nucleon-nucleon scattering is usually parametrizedas given in Ref. [20]Γ NN ( b ) = 1 − iα NN πβ NN σ tot NN exp (cid:18) − b β NN (cid:19) , (7)where α NN is the ratio of the real part to the imaginarypart of the nucleon-nucleon scattering amplitude in theforward direction, β NN is the slope parameter of the dif-ferential cross section, and σ tot NN is the nucleon-nucleontotal cross section. Standard parameter sets of the pro-file function are listed in Refs. [21, 22]. III. HOW IS NUCLEAR BUBBLE STRUCTUREREFLECTED?
In this section, we discuss how the nuclear bubble getsreflected in the proton-nucleus scattering at high incidentenergies, where the Glauber model works fairly well. Forthe sake of simplicity, we use the averaged
N N profilefunction given in Ref. [21] and ignore the Coulomb inter-action. Note that the difference between the pp and pn cross sections in the profile functions can be neglected inthe total reaction cross section calculations at the inci-dent energy of E &
300 MeV [17].
A. Density distribution of Si Here, we discuss the density distribution of Si withinthe harmonic-oscillator (HO) model. First, we considertwo types of configurations, (0 d ) and (0 d ) (1 s ) , andcalculate their density distributions with the center-of-mass correction [18], which are denoted as ρ d ( r ) and ρ s ( r ), respectively. Note that ρ d ( r ) shows the mostprominent bubble structure because of the vacancy of r (fm) ρ (r) (f m - ) α = 0.0, G = 0.34 α = 0.33, G = -0.06 α = 0.66, G = -0.47 α = 1.0, G = -0.89 FIG. 1. Matter density distributions of Si with various bub-ble parameters ( G ). The arrow indicates the reference radius,1.8 fm. See text for more details. the 1 s -orbit, while ρ s ( r ) does not. Then, we interpolatethese two densities as ρ ( α ; r ) = (1 − α ) ρ d ( r ) + αρ s ( r ) , (8)where the mixing parameter α (0 ≤ α ≤
1) controlsthe occupation probability of the 1 s -orbit. Consequently, α = 0 yields the most bubbly density, whereas α = 1yields non-bubble density. For a given value of α , thesize parameter of HO is chosen to reproduce the observedpoint-proton root-mean-square (rms) radius, 3.01 fm [23].To quantify a degree of “bubble”, we introduce thebubble parameter ( G ) as, G = ρ ( D ) − ρ (0) ρ ( D ) , (9)where, D denotes the reference radius at which the ρ d ( r )or ρ ( α = 0; r ) takes its maximum value. In the case of Si, D = 1 . ρ (0) and ρ ( D ) represent the densitiesat r = 0 and D , respectively. We remark that this is anextension of the bubble parameter (depletion fraction)given in Ref. [4], where it is defined only by positive val-ues. This extension enables us to quantify the degree ofthe bubble structure for any nuclear density distributionirrespective of whether it exhibits a bubble or not.Figure 1 displays how the matter density distributionof Si and the corresponding G value change dependingon the mixing parameter α . In the present case of Si,the values of G range from 0.34 ( α = 0) to − α = 1),allowing for negative values which signify that the centraldensity is higher than the density at the reference radius.Apparently, the bubble degree is maximized at G = 0 . α = 0, which clearly exhibits a strong depression ofthe central density, thereby suggesting the bubble struc-ture. The value of G decreases with increasing the mixingof the 1 s -orbits. An almost flat behavior of the densitydistribution ( G ≈
0) is obtained with α = 0 . N = 14 isotones from O to Ca. We calculate ρ d ( r ) and ρ s ( r ) as the density distributionsof the (0 d ) A − and (1 s ) (0 d ) A − configurations with22 ≤ A ≤
34. These two densities are interpolated as inEq. (8) and used for the reaction calculation in the fol-lowing sections. The reference radius D and the bubbleparameters are also defined in the same way. B. Bubble structure in proton- Si reactions
How are the different density profiles displayed in Fig. 1reflected in the reaction observables? To address thisquestion, we calculated the reaction probability P ( b )given in Eq. (5), which is the integrand of the total reac-tion cross section [Eq. (4)]. Figure 2 shows the reactionprobability multiplied by 2 πb for proton- Si scatteringas a function of the impact parameter. The density dis-tributions with α = 0 ( d -dominance, G = 0 .
34) and 1(maximum s configuration, G = − .
89) are examinedto see the difference between the two extreme configura-tions. The reaction probabilities for these configurationsare almost identical at small impact parameters up to ≈ b ≈ α = 0) includesonly the d -wave configuration, while the latter ( α = 1)includes the 1 s / configuration which has a longer tailin the asymptotic region. Consequently, we expect thatthe bubble structure gives the larger cross section at thefirst diffraction peak.To confirm this numerically, we calculated the proton- Si elastic scattering differential cross sections. Figure 3plots the elastic scattering differential cross sections ofproton- Si reactions at 325, 550, and 800 MeV with var-ious bubble parameters. As expected, the cross sectionat the first peak position is largest for the ideal bubbleconfiguration with α = 0 and it decreases with increas-ing α and decreasing G . This suggests a practical way toidentify the bubble structure using a hadronic probe. α = 0.0, G = 0.34 α = 1.0, G = -0.89 π b P ( b ) (f m ) b (fm) p + Si (c) 800 MeV(b) 550 MeV(a) 325 MeV FIG. 2. Reaction probabilities multiplied by 2 πb for proton- Si reactions at (a) 325, (b) 550, and (c) 800 MeV. Theconfigurations with α = 0 ( G = 0 .
34) and 1 ( G = − .
89) areemployed.
IV. BUBBLE STRUCTURE OF N = 14 ISOTONES
We have seen that the difference between the bubbleand non-bubble nuclei can be detected in the elastic scat-tering differential cross sections. To demonstrate the fea-sibility of this idea, we take the density distributions ob-tained from a microscopic structure model, the AMD,and try to extract the information on the nuclear bub- θ c.m. (deg) -2 d σ | / d Ω ( m b / s r) α = 0.0, G = 0.34 α = 0.33, G = -0.06 α = 0.66, G = -0.47 α = 1.0, G = -0.89
325 MeV550 MeV800 MeV (x 10 )(x 10 ) p + Si FIG. 3. Elastic scattering differential cross sections of proton- Si reactions at 325, 550, and 800 MeV with various bubbleparameters. ble from the reaction observable. Here the ground-statedensity distributions of N = 14 isotones are examined asthey exhibit the bubble structure in its isotone chain [4]. A. Framework of AMD
The AMD [28, 29] is a fully microscopic approach andoffers a non-empirical description of light to medium nu-clei. Here we briefly explain how we obtain the densitydistributions for the N = 14 isotones within the AMDframework. The Hamiltonian for a nucleon system withthe mass number A is given by H = X i t ( i ) − T cm + X ij v NN ( ij ) , (10)where t ( i ) is the kinetic energy of the single nucleonand the center-of-mass kinetic energy T cm is exactly re-moved. The Gogny D1S parameter set [30] is employedas a nucleon-nucleon effective interaction v NN , which isknown to give a fairly good description for this mass re-gion [31–33].The variational basis function of the AMD is repre-sented by a Slater determinant projected to the positive-parity state as Φ = 1 + P x A { ϕ , ..., ϕ A } , (11)where P x is the parity operator, and ϕ i is a Gaussiannucleon wave packet defined by ϕ i = Y σ = x,y,z (cid:18) ν σ π (cid:19) / exp { − ν σ ( r σ − Z iσ ) }× ( α i χ ↑ + β i χ ↓ )( | p i o r | n i ) . (12)The centroids Z and width ν vectors of the Gaussian andthe spin variables α i and β i are the variational parame-ters. They are determined by the the frictional coolingmethod [34] in such a way to minimize the energy of thesystem under the constraint on the quadrupole deforma-tion parameter β .To describe the ground state of the N = 14 isotones,the wave functions obtained by the frictional coolingmethod are projected to the angular momentum J = 0and superposed employing β as a generator coordinate(generator coordinate method; GCM [35]),Ψ = X i g i P J =0 Φ( β i ) , (13)where P J =0 represents the angular momentum projectorand the amplitudes g i are determined by the diagonaliza-tion of the Hamiltonian. In the present study, the valueof β is chosen from 0.0 to 0.6 with an interval of 0.025.The deformation parameter γ is determined variationally,and hence it takes an optimal value for each Φ( β i ). Fi-nally, the ground-state density distribution is calculatedas ρ ( r ) = h Ψ | P Ai =1 δ ( r i − r cm − r ) | Ψ ih Ψ | Ψ i . (14)Note that the resulting density distribution is free fromthe center-of-mass coordinate r cm . We also evaluatethe quadrupole deformation parameters and occupationprobabilities of the 1 s -orbit according to the proceduredescribed in the appendix A. B. Density distributions of N = 14 isotones Figure 4 plots the matter, neutron and proton densitydistributions of N = 14 isotones obtained with the AMD.The rms matter radii, quadrupole deformation parame-ters, and occupation probabilities of the 1 s -orbit are sum-marized in Table I. The bubble parameters G AMD werealso calculated from the AMD densities using Eq. (9),where the reference radius ( D ), for each isotone, wasderived from an effective density ρ d ( r ). This ρ d ( r ) isessentially a HO density [as in Sec. (III A)] whose sizeparameter is adjusted so as to reproduce the rms mat-ter radius, for each isotone, as obtained from the AMDdensity.We clearly see a prominent bubble structure in O inwhich both the proton and neutron density distributionsexhibit depressed central densities. Consequently, it hasthe largest bubble parameter among the N = 14 isotones. TABLE I. Rms matter radii r m and the quadrupole deforma-tion parameters β, γ , and neutron (proton) occupation prob-abilities of the 1 s -orbit P s ( n )[ P s ( p )] for N = 14 isotonesobtained by the AMD. The bubble parameter G AMD are ex-tracted from the matter density distributions shown in Fig. 4.See text for more details. r m (fm) β γ P s ( n ) P s ( p ) G AMD22
O 2.90 0.20 60 ◦ Ne 2.97 0.37 60 ◦ Mg 3.06 0.40 37 ◦ Si 3.11 0.40 60 ◦ − Si(sph.) 2.98 0.00 – 0.01 0.01 0.34 S 3.11 0.27 43 ◦ − Ar 3.21 0.27 60 ◦ − Ca 3.26 0.12 60 ◦ − This is due to the almost spherical closed-shell configu-ration of this nucleus and the resultant small occupa-tion probabilities of the 1 s -orbit. As the proton numberincreases, the nuclear quadrupole deformation becomesstrong, which mixes the s -, d - and g -orbits and effectivelyincreases the occupation probabilities of the 1 s -orbit. Asa result, the bubble structure in the matter density dis-tributions is weakened in Ne and Mg, and dimin-ished in Si which is most strongly deformed among the N = 14 isotones. Indeed, Table I shows that the bubbleparameter strongly correlates with the quadrupole defor-mation parameter β and neutron occupation probability P s ( n ). The bubble parameter decreases as a function ofthe proton number and becomes negative (non-bubble)from Si. We also see that, if we restrict the AMD cal-culation to spherical shape, Si also shows the bubblestructure as displayed in Fig. 4 (e). This confirms astrong impact of the nuclear deformation on the bubblestructure. With further increase of the proton number,in S, Ar and Ca, the bubble structure in the mat-ter density distributions are not seen since the centraldensities of protons are already filled by the excess pro-tons, while the neutron density distributions still keepthe bubble structure.Thus, the present AMD calculation suggests that Ohas both proton and neutron bubble structure. We note,however, this is in contradiction to the conclusion of themean-field calculations [4]. It was shown that the bubblestructure of O is rather model dependent, and pairingcorrelation tends to diminish the bubble structure as itincreases the neutron occupation of 1 s orbit. Since thepresent AMD calculation does not handle the pairing cor-relation explicitly, the stability of the bubble structure of O shown in Fig. 4 needs to be investigated. To checkthe reliability of the AMD densities, we calculated themirror nucleus of Ca, i.e., Si for which many cal-culations predicted proton bubble structure [4, 36] and matter densityneutron densityproton density r (r) (f m - ) r (fm) (a) O (b) Ne (c) Mg(d) Si (e) Si (sph.) (f) S(i) Si(g) Ar (h) Ca FIG. 4. Density distributions of N = 14 isotones, (a) O, (b) Ne, (c) Mg, (d) Si, (e) Si (spherical), (f) S, (g) Ar,and (h) Ca obtained from the AMD wave function. The arrows indicate the reference radii for each isotone. See text fordetails. The density distribution of Si which is the mirror nucleus of Ca and has prominent proton bubble structure is alsoshown in the panel (i) for comparison. indirect experimental evidence was obtained [10]. Thecalculated density shown in panel (i) of Fig. 4 clearly ex-hibits the proton bubble structure, which is very similarto that of Ca and also that obtained by the mean-fieldcalculations [4]. Therefore, we conclude that the proton(neutron) bubble structure of Si ( Ca ) is robust, whilethe bubble structure of O is somewhat model depen-dent.
V. DISCUSSIONSA. Extraction of bubble parameters for N = 14 isotones In the previous section, we saw that N = 14 iso-tones show remarkable variations in their nuclear densityprofiles with O exhibiting the most prominent bubblestructure, although the strong model dependence was re-ported [4]. We now examine the possibility of extract-ing the degree of the bubble structure from the reactionobservables by performing a numerical test as follows. First, we calculate the elastic scattering differential crosssections using the density distributions obtained by theAMD, which we regard as the experimental data (mock-up data). Then, by assuming spherical HO type densitydistributions defined in Eq. (8), we fit the α (the mixingparameter) and size parameter of the HO to reproducethe position and magnitude of the first diffraction peak ofthe mock-up data. This procedure uniquely determinesthe spherical HO type density distribution from whichwe extract the bubble parameter G . Thus, the obtainedbubble parameter G is compared with that of the originalone, G AMD listed in Table I, to test the feasibility of themethod.Figure 5 plots the bubble parameters of N = 14 iso-tones obtained from the mock-up data at the incident en-ergies of 325 MeV, 550 MeV and 800 MeV, in comparisonwith G AMD . It is noted that all the mock-up data (totalreaction cross sections calculated with AMD densities)are reproduced within 1% differences. The differences ofthe extracted bubble parameters are also less than 1%for all the incident energies. These show the robustnessof this analysis. Although the bubble parameters ex-tracted from the elastic scattering cross sections alwaysundershoot the “exact” bubble parameters G AMD (over-estimate the bubble structure), we do notice similarity intheir behavior as a function of the proton number. Thedisagreement is apparently due to the inappropriate as-sumption of the model density - we assumed spherical HOdensity distributions for all N = 14 isotones. However,most of the nuclei are deformed inducing some deviationsin the bubble parameter extraction. In fact, as we see inthe Fig. 5, the bubble parameter of Si is perfectly re-produced when we constrain the AMD calculation to thespherical configuration. We also see a reasonable descrip-tion of almost spherical nuclei, O ( Z = 8) and Ca( Z = 20). Although it is beyond the scope of this paper,an analysis with more elaborated model density distri-butions including such as nuclear deformation is worthconsidering to obtain more precise determination of thebubble parameters. Z -0.6-0.4-0.200.20.4 B ubb l e p a r a m e t e r ( G ) AMD325 MeV550 MeV800 MeV Si (sph.)
FIG. 5. Comparison of the bubble parameters obtained di-rectly from the AMD densities and the elastic scattering anal-ysis at the incident energies of 325, 550 and 800 MeV.
B. How effective is proton scattering in probingthe nuclear bubble?
One may think that proton scattering does not probethe nuclear bubble structure but only probes the nuclearsurface regions, the nuclear diffuseness. To address thisself-criticism, we performed the same analysis as in theprevious section but with the two-parameter Fermi (2pF)model density, ρ / [1 + exp ( r − R ) /a ], whose parameters( R, a ) are fixed so as to reproduce the first peak posi-tion and its magnitude in the elastic scattering differ-ential cross sections. ρ , the central density, gets fixedfrom the normalization of the density distribution. Ob-viously, the 2pF distribution has no bubble. Note thatwith this analysis the 2pF model density nicely repro-duced the density profile at around the nuclear surfaceof the realistic density distributions obtained from themicroscopic mean-field model [19]. -t [(MeV/c) ] -6 -4 -2 d σ / d | - t | [ m b / ( M e V / c ) ] AMDHO2pF p + O
325 MeV550 MeV800 MeV (x 10 )(x 10 ) FIG. 6. Elastic scattering differential cross sections of p + Owith the AMD, HO, and 2pF densities.
Figure 6 displays the proton- O elastic scattering dif-ferential cross sections with the AMD, HO and 2pFmodel densities as a function of the four momentumtransfer | − t | at the incident energies of 325, 550, and800 MeV. The cross sections are essentially the same upto | − t | ≈ c , which is understandable as both theHO and 2pF model densities are adjusted to reproducethe position and magnitude of the first diffraction peak.However, beyond this limit while the HO and AMD re-sults agree with each other, those with the 2pF modeldensity deviate significantly.We already know in Fig. 2 the fact that the incidentproton cannot probe differences in the internal densitiesbelow ≈ S scattering with bubble andnon-bubble density profiles [37].
VI. CONCLUSIONS
Nuclei with a depression in the central part of theirdensity - the so-called bubble structure - has attractedattention in recent times. Considerable efforts are un-derway to look for suitable probes for these exotic sys-tems. In this work, we have discussed the feasibility ofusing a proton probe to extract the degree of the bubblestructure. We have calculated the structure of even-even N = 14 isotones in the 22 ≤ A ≤
34 mass range us-ing a microscopic structure model, the antisymmetrizedmolecular dynamics (AMD). The Glauber model is thenemployed to evaluate reaction observables of high-energynucleon-nucleus scattering.Due to the strong absorption in the internal region ofthe target nucleus, the bubble structure or the centraldepression of the target density, cannot be directly mea-sured using the proton probe. However, effects of thisstructure are reflected from the middle to the surface re-gions of the nuclear density. They also tend to have asharper nuclear surface. Furthermore, nuclear deforma-tion acts as an hindrance to the emergence of the bubblestructure.We find that the AMD calculation predicts prominentbubble structure of O, which exhibits a small deforma-tion, after analyzing a host of N = 14 isotones. The de-gree of the bubble structure is extracted by a systematicanalysis of the calculated cross sections obtained withthe AMD by using simple harmonic-oscillator type modeldensities. To improve the accuracy of the extraction, it isnecessary to employ a more realistic model density thatcan describe the nuclear deformation.We have shown that the bubble structure informationis imprinted on the nucleon-nucleus elastic scattering dif-ferential cross sections and is possibly extracted by ana-lyzing the cross sections up to the first diffraction peak.Nevertheless a more accurate analysis involving the sec-ond diffraction peak would be a welcome addition. ACKNOWLEDGMENTS
This work was supported by JSPS KAKENHI GrantsNos. 18K03635, 18H04569, 19H05140, and 19K03859,the Collaborative Research Program 2020, InformationInitiative Center, Hokkaido University and the Schemefor Promotion of Academic and Research Collaboration(SPARC/2018-2019/P309/SL), MHRD, India. V.C. alsoacknowledges MHRD, India for a doctoral fellowship and a grant from SPARC to visit the Hokkaido University.
Appendix A: Estimation of the deformationparameters and occupation probabilities from theAMD wave functions
Here, we explain how we estimated the quadrupoledeformation parameters and single-particle occupationprobabilities of the AMD wave functions for N = 14 iso-tones listed in Table I. The AMD wave function given inEq. (13) is, in general, a superposition of the Slater deter-minants with different deformation and different single-particle configurations. Therefore, to estimate thesequantities, we pick up the Slater determinant Φ( β ), whichhas the maximum overlap with the AMD wave function | h P J =0 Φ( β ) | Ψ i | , and regard it as an approximation ofthe AMD wave function Ψ .The deformation parameters β and γ of Ψ may be ap-proximated by those of Φ( β ). The occupation probabili-ties of the 1 s -orbit are also estimated in a similar manner.We calculate the single-particle energies and orbits of de-scribed by Φ( β ) by using the AMD+HF method [38].Because of the nuclear deformation, the single-particleorbits, φ ( r ) , ..., φ A ( r ), are no longer the eigenstates ofthe angular momentum. Therefore, we consider the mul-tipole decomposition of them, φ i ( r ) = X jlj z φ i ; jlj z ( r ) (cid:2) Y l (ˆ r ) × χ / (cid:3) jj z . (A1)The squared amplitudes for the j = 1 / l = 0 compo-nents should give us an estimate of the occupation proba-bility. Assuming the complete filling of the 0 s -orbit, theneutron ( n ) and proton ( p ) occupation probabilities ofthe 1 s -orbit are obtained approximately as P s ( n/p ) = N/Z X i =1 X j z = − (cid:12)(cid:12)(cid:12)D φ i ; j z (cid:12)(cid:12)(cid:12) φ i ; j z E(cid:12)(cid:12)(cid:12) − . (A2) [1] I. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida, N.Yoshikawa, K. Sugimoto, O. Yamakawa, T. Kobayashi,and N. Takahashi, Phys. Rev. Lett. , 2676 (1985).[2] I. Tanihata, H. Savajols, and R. Kanungo, Prog. Part.Nucl. Phys. , 215 (2013).[3] R. Hofstadter, Rev. Mod. Phys. , 214 (1956).[4] M. Grasso, L. Gaudefroy, E. Khan, T. Nikˇsi´c, J.Piekarewicz, O. Sorlin, N. Van Giai, and D. Vretenar,Phys. Rev. C , 034318 (2009).[5] J. J. Li, W. H. Long, J. L. Song, and Q. Zhao, Phys. Rev.C , 054312 (2016).[6] X. Campi and D. W. L. Sprung, Phys. Lett. B 46 , 291(1973).[7] K. T. R. Davis, S. J. Krieger, and C. Y. Wong, Nucl.Phys.
A 216 , 250 (1973).[8] E. Khan, M. Grasso, J. Margueron, and N. Van Giai, Nucl. Phys.
A 800 , 37 (2008).[9] M. Bender, K. Rutz, P.-G. Reinhard, J. A. Maruhn, andW. Greiner, Phys. Rev. C , 034304 (1999).[10] A. Mutschler, A. Lemasson, O. Sorlin, D. Bazin, C.Borcea, R. Borcea, Z. Dombr´adi, J.-P. Ebran, A. Gade,H. Iwasaki et al. , Nature Physics , 152 (2017).[11] K. Tsukada, A. Enokizono, T. Ohnishi, K. Adachi, T.Fujita, M. Hara, M. Hori, T. Hori, S. Ichikawa, K. Kurita et al ., Phys. Rev. Lett. , 262501 (2017).[12] S. Abrahamyan et al. (PREX collaboration), Phys. Rev.Lett. , 112502 (2012).[13] H. Sakaguchi and J. Zenihiro, Prog. Part. Nucl. Phys. , 1 (2017), and references therein[14] Y. Matsuda, H. Sakaguchi, H. Takeda, S. Terashima, J.Zenihiro, T. Kobayashi, T. Murakami, Y. Iwao, T. Ichi-hara, T. Suda et al. , Phys. Rev. C , 034614 (2013). [15] R. J. Glauber, Lectures in Theoretical Physics , editedby W. E. Brittin and L. G. Dunham (Interscience, NewYork, 1959), Vol. 1, p.315.[16] K. Varga, S. C. Pieper, Y. Suzuki, and R. B. Wiringa,Phys. Rev. C , 034611 (2002).[17] T. Nagahisa and W. Horiuchi, Phys. Rev. C , 054614(2018).[18] B. Abu-Ibrahim, S. Iwasaki, W. Horiuchi, A. Kohama,and Y. Suzuki, J. Phys. Soc. Jpn., Vol. , 044201 (2009).[19] S. Hatakeyama, W. Horiuchi, and A. Kohama, Phys.Rev. C , 054607 (2018).[20] L. Ray, Phys. Rev. C , 1857 (1979).[21] W. Horiuchi, Y. Suzuki, B. Abu-Ibrahim, and A. Ko-hama, Phys. Rev. C , 044607 (2007).[22] B. Abu-Ibrahim, W. Horiuchi, A. Kohama, and Y.Suzuki, Phys. Rev. C , 034607 (2008); ibid ,029903(E) (2009); , 019901(E) (2010).[23] I. Angeli, K. P. Marinova, At. Data Nucl. Tables , 69(2013).[24] R. D. Amado, J. P. Dedonder, and F. Lenz, Phys. Rev.C , 647 (1980).[25] A. Kohama, K. Iida, and K. Oyamatsu, Phys. Rev. C ,064316 (2004).[26] A. Kohama, K. Iida, and K. Oyamatsu, Phys. Rev. C , 024602 (2005).[27] A. Kohama, K. Iida, and K. Oyamatsu, J. Phys. Soc.Jpn. , 094201 (2016).[28] Y. Kanada-Enyo, M. Kimura, and H. Horiuchi, ComptesRendus Physique , 497 (2003).[29] M. Kimura, Phys. Rev. C , 044319 (2004).[30] J. Berger, M. Girod, and D. Gogny, Comp. Phys. Comm. , 365 (1991).[31] T. Sumi, K. Minomo, S. Tagami, M. Kimura, T. Mat-sumoto, K. Ogata, Y. R. Shimizu, and M. Yahiro, Phys.Rev. C , 064613 (2012).[32] S. Watanabe, K. Minomo, M. Shimada, S. Tagami,M. Kimura, M. Takechi, M. Fukuda, D. Nishimura, T.Suzuki, T. Matsumoto, Y. R. Shimizu, and M. Yahiro,Phys. Rev. C , 044610 (2014).[33] S. P`eru and M. Martini, Eur. Phys. J. A , 88 (2014).[34] Y. Kanada-Enyo, M. Kimura, and A. Ono, Prog. Theor.Exp. Phys. , 1A202 (2012).[35] D. L. Hill and J. A. Wheeler, Phys. Rev. , 1102 (1953).[36] T. Duguet, V. Som`a, S. Lecluse, C. Barbieri, and P.Navr´atil Phys. Rev. C , 034319 (2017).[37] T. Furumoto, K. Tsubakihara, S. Ebata, and W. Hori-uchi, Phys. Rev. C , 034605 (2019).[38] A. Dote, H. Horiuchi and Y. Kanada-En’yo, Phys. Rev.C56