The ratio method: a new tool to study one-neutron halo nuclei
aa r X i v : . [ nu c l - t h ] S e p The ratio method: a new tool to study one-neutron halo nuclei
P. Capel ∗ Physique Nucl´eaire et Physique Quantique (C.P. 229),Universit´e Libre de Bruxelles (ULB), B-1050 Brussels, Belgium
R. C. Johnson † Department of Physics, University of Surrey, Guildford GU2 7XH, United Kingdom andNational Superconducting Cyclotron Laboratory and Department of Physics and Astronomy,Michigan State University, East Lansing, Michigan 48824, USA
F. M. Nunes ‡ National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy,Michigan State University, East Lansing, Michigan 48824, USA (Dated: September 24, 2018)Recently a new observable to study halo nuclei was introduced, based on the ratio between breakupand elastic angular cross sections. This new observable is shown by the analysis of specific reactionsto be independent of the reaction mechanism and to provide nuclear-structure information of theprojectile. Here we explore the details of this ratio method, including the sensitivity to bindingenergy and angular momentum of the projectile. We also study the reliability of the method withbreakup energy. Finally, we provide guidelines and specific examples for experimentalists who wishto apply this method.
PACS numbers: 21.10.Gv, 25.60.Bx, 25.60.GcKeywords: Halo nuclei, angular distribution, elastic scattering, breakup
I. INTRODUCTION
One of the most intriguing phenomenon revealed bythe studies with rare isotope beams is that of halo nuclei[1]. Since the early experiments on reaction cross sections[2], we have built a good understanding of the exotic fea-tures that results from the proximity to threshold and theabsence of repulsive barriers [3]. For very loosely-boundnucleons, which do not suffer the constraint of the cen-trifugal or Coulomb barrier, the wavefunctions developlong tails, extending well into the classically forbiddenregion. A primary signature of the halo phenomenon hasbeen the sudden increase of the matter radius within agiven isotopic chain [4]. In this respect, precision mea-surements of nuclear radii in traps open new possibilities[5, 6]. Narrow momentum distributions are also an indi-cation of the large spatial extension of the wavefunctions[7, 8]. While identifying a halo can be in itself a chal-lenge, one would also like to have a better understandingof the structure of the valence orbital and its wavefunc-tion, such as done in ( d, p ) studies [9]. A new observablebased on a ratio of cross sections, and therefore referredto as the ratio method [10], offers this possibility. In thiswork, we explore the ratio method in detail and provideguidelines for its application.Over the last two decades, many nuclear halos havebeen discovered. Examples include the one neutron halo ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] of Be [11] and the one proton halo B [12], and bothhave been the focus of many studies. In the recent years,experiments have been exploring heavier halos. After themeasurement of C reaction cross section [4], it was ex-pected that O would also exhibit a two-neutron halostructure, but now it has become clear that this halo-to-be is indeed unbound [13]. More recently, total re-action cross section measurements on Ne have beenrather inconclusive [14] as to whether Ne is a halo. Along sequence of theoretical studies [15–19] demonstratethat the structure information can depend strongly onthe model used in the analysis of the reaction and nonintegrated observables may be necessary. Another possi-bility of a halo has been identified in the Mg chain [20]and others will surely surface with new technical devel-opments.As the mass increases, there are a limited number ofisotopes for which l = 0 , Be(1 / − )and F(1 / +1 ). Interest in determining halo propertiesof excited states continues [21] but no optimum probehas been found. In principle, the new observable herediscussed, can be generalized to characterize halo excitedstates too.Halo nuclei are challenging from the nuclear structurepoint of view. Fortunately, many-body methods are nowable to adequately treat the asymptotic behavior of thewavefunction of the loosely-bound nucleons [22, 23]. Halonuclei offer a unique testing ground for the understand-ng of the nuclear force and represent an opportunity tounderstand the density dependence of nuclear matter.Due to its loosely-bound nature, the most likelymethod to explore halo structure is through breakup re-actions. There have been important developments in thetheory for the breakup of halo nuclei (see Refs. [24–29]),and today it is understood that non-perturbative non-classical approaches, which treat nuclear and Coulombon equal footing, are needed to obtain reliable angulardistributions [30, 31]. What is critical to understand isthat the information to be extracted is model dependent,whether the process is inclusive or exclusive, whether it isnuclear or Coulomb dominated. When non-perturbativemethods are used to solve the problem, the model de-pendence arises primarily from uncertainties in the core-target interaction [32]. This interaction is often poorlyknown, specially when the core itself is radioactive, andplays a very important role in the breakup process.In Ref. [33], Capel et al. realized that the elastic andbreakup angular distributions of halo nuclei exhibit verysimilar features. In Ref. [10] the idea of taking the ratioof these angular distributions is introduced, drawing onthe Recoil Excitation and Breakup model (REB) devel-oped earlier [34–36]. The main advantage of this newreaction observable is that it is nearly independent ofthe reaction mechanism and that its sensitivity to thecore-target interaction is strongly reduced. The first ap-plication presented in Ref. [10] is very encouraging. Herewe explore this ratio method in more detail.This paper is organized in the following manner. InSec. II we provide the theoretical framework. A discus-sion of various possible ratios is presented in Sec. III. InSec. IV we demonstrate the validity of the ratio methodand its range of validity. In Sec. V we study the structureinformation contained in this new observable. Finally,conclusions are drawn in Sec. VI. II. THEORETICAL FRAMEWORK
Describing the reaction of an exotic projectile P im-pinging on a target T is a complex many-body problem.While many-body techniques have made important ad-vances to handle a number of reactions, for example cap-ture reactions on light nuclei [37–39] and nucleon elas-tic scattering [40], these techniques are not able to han-dle the general heavy-ion reaction P + T problem. Inthat case, it is imperative to identify the relevant degreesof freedom and address the problem within a few-bodyframework.If the nucleus under study (the projectile) has a one-neutron halo, there is a strong decoupling of the coredegrees of freedom from the valence neutron [1]. Theprojectile can then be described as a two-body system:a neutron n loosely bound to a core c , i.e. P = c + n . Insuch a scenario, one can describe the states of the sys-tem with a mean field V cn that reproduces basic features,such as binding energy and radius, excited states or res- onances, etc. Assuming the target is well bound andfocusing on elastic breakup only, the implicit inclusion oftarget excitation through the imaginary part of opticalpotentials should be sufficient. In this case, one can re-duce the reaction P + T to a three-body problem. Thisis the approach considered here. To retain simplicity inour discussion, we consider the target and the core to bestructureless and of spin zero although all the formalismcan be extended to include target and core spins. A. The three-body model for nuclear reactions
In this model, the projectile P = c + n is described bythe Hamiltonian H = − ~ µ ∆ + V cn ( r ) , (1)where r is the c - n relative coordinate and µ = m n m c / ( m n + m c ) is the c - n reduced mass, with m n and m c the masses of the neutron and of the core, respec-tively. As mentioned above, V cn is a phenomenologicalmean field that captures some essential aspects of thehalo projectile and in principle could be microscopicallyderived (we assume it has central and spin-orbit terms,see Appendix B for details).The eigenstates of H are solutions of H φ ljm ( E, r ) = Eφ ljm ( E, r ) , (2)where E is the c - n relative energy. The total angular mo-mentum j results from the coupling of the orbital angularmomentum l and the spin of the valence neutron; m isits projection. Negative-energy states correspond to c - n bound states. They are normalized to unity. We denoteby { φ l i j i m i } i =0 , ,... these bound states of energy E i < i = 0 corresponding to the projectile ground state, i = 1 to its first excited (bound) state, etc. Positive-energy states describe the c - n continuum. Their radialpart is normalized according to u lj ( E, r ) −→ r →∞ r µπ ~ k [cos δ lj F l ( kr ) + sin δ lj G l ( kr )] , (3)where k = p µE/ ~ is the wave number, δ lj is the nu-clear phase shift at energy E , and F l and G l are regularand irregular Coulomb functions, respectively [41], takenfor zero Sommerfeld parameter. This normalization hasbeen chosen so that h φ ljm ( E ) | φ ljm ( E ′ ) i = δ ( E − E ′ ) . (4)With this two-body description for the projectile, the P - T collision reduces to a three-body problem whoseHamiltonian reads H ( R , r ) = ˆ T R + H ( r ) + U cT ( R c ) + U n T ( R n ) , (5)where R is the coordinate of the projectile center of massrelative to the target. In Eq. (5), additional optical po-tentials have been introduced to describe the scattering2f the core off the target U cT and the neutron off thetarget U nT . These optical potentials are typically phe-nomenological and contain an important imaginary termto account for other reaction channels not explicitly in-cluded in this description. Since they are not uniquelydefined, these potentials may induce significant uncer-tainties in the analysis of the reactions modeled withinthis framework [32].In order to study the reactions of P on T we need tosolve the three-body Schr¨odinger equation H Ψ( R , r ) = E tot Ψ( R , r ) . (6)As customary, we align the initial momentum K withthe Z axis and assume the projectile to be initially in itsground state, so that:Ψ( R , r ) −→ Z →−∞ e iK Z φ l j m ( r ) . (7)The total energy of the system is then given by E tot = ~ K / µ P T + E , where µ P T is the P - T reduced mass. B. The Dynamical Eikonal Approximation
It is important to identify a method for solving thethree-body problem (6), that reliably describes elas-tic and breakup of loosely-bound nuclei. While themomentum-space integral Faddeev method [42, 43] isconsidered exact, its present implementation is limitedto d + T reactions where the target charge is Z T ≤ E ≥
40 MeV/nucleon [31]. Because this method isless computationally intensive, DEA is used in Ref. [10]and here to demonstrate the ratio method.A well known and useful approach to reactions at highenergies is the eikonal approximation [28, 29, 45]. Mo-tivated by the boundary form (7), the three-body wavefunction Ψ( R , r ) is factorized asΨ DEA ( R , r ) = e iK Z b Ψ( R , r ) . (8)At high energies, one expects a weak dependence on R of b Ψ. Using the factorization (8) in Eq. (6) and neglect-ing second-order derivatives of b Ψ with respect to R , weobtain [29] i ~ K µ P T ∂∂Z b Ψ( Z, b , r )= [( H − E ) + U cT ( R c ) + U nT ( R n )] b Ψ( Z, b , r ) , (9)where Z and b are the longitudinal and transverse com-ponents of R , respectively. In the standard eikonal im-plementation, a subsequent adiabatic approximation is performed to solve Eq. (9). That approximation corre-sponds to neglect the excitation energy of the projec-tile compared to the beam energy. In DEA, no such anadiabatic approximation is made and Eq. (9) is solvednumerically for each b imposing the condition: b Ψ( Z →−∞ , b , r ) = φ l j m ( r ), in agreement with condition (7).The S-matrix is then extracted from the asymptotic be-havior b Ψ( Z → + ∞ , b , r ) as detailed in Ref. [29]. Notethat since this does not imply any semiclassical hypoth-esis, DEA is a fully quantal model [29, 31]. C. The recoil excitation and breakup model
In the nineties, Johnson et al. realized that a simplefactorization of the scattering amplitude can be obtainedwhen a neutron halo projectile interacts with the target[34]. The key ingredients to the so-called Recoil Exci-tation and Breakup (REB) model are i) neglecting thevalence particle’s interaction with the target, and ii) as-suming the excitation energy of the projectile is smallcompared to the beam energy (the adiabatic approxi-mation). When these two conditions are satisfied, theelastic-scattering cross sections becomes [34, 35]: (cid:18) dσd Ω (cid:19) el = | F , ( Q ) | (cid:18) dσd Ω (cid:19) pt (10)where | F , ( Q ) | is a form factor accounting for the ex-tension of the halo [see Eq. (11) below], and (cid:0) dσd Ω (cid:1) pt is across section for a point-like projectile with mass µ P T ,scattered by the core-target interaction U cT . The rela-tion (10) is often mistaken for the first-order perturbationtheory although it does not involve the Born approxima-tion. Note that (cid:0) dσd Ω (cid:1) pt is similar to the experimentalcore-target elastic scattering, but for a different projec-tile mass.The form factor is defined by: | F , ( Q ) | = 12 j + 1 X m (cid:12)(cid:12)(cid:12)(cid:12)Z | φ l j m ( r ) | e i Q · r d r (cid:12)(cid:12)(cid:12)(cid:12) , (11)and represents the Fourier transform of the halo groundstate density. Here Q = m n m c + m n ( K b Z − K ′ ) is propor-tional to the momentum transferred during the scatteringprocess. It modulates the diffraction pattern containedin the point-like cross section, determining which are therelevant scattering angles to be considered in the process Q = 2 m n m c + m n K sin( θ/ . (12)In Ref. [33], it was realized that the elastic and breakupcross sections have similar diffraction patterns, a factonly fully understood with the subsequent work on theratio method [10]. In Ref. [10], we used the fact that thefactorization (10) can be generalized to angular distri-butions for the excitation of the projectile to any of its3tate, either bound or not [35, 36]. For inelastic scatter-ing with excitation to bound state i >
0, we can definethe form factor | F i, ( Q ) | = 12 j + 1 X m X m i (cid:12)(cid:12)(cid:12)(cid:12)Z φ l i j i m i ( r ) φ l j m ( r ) e i Q · r d r (cid:12)(cid:12)(cid:12)(cid:12) , (13)while for breakup to energy E , we use the form factor | F E, ( Q ) | = 12 j + 1 X m X ljm (cid:12)(cid:12)(cid:12)(cid:12)Z φ ljm ( E, r ) φ l j m ( r ) e i Q · r d r (cid:12)(cid:12)(cid:12)(cid:12) , (14)where φ ljm ( E, r ) is the eigenstate of H at positive en-ergy E in the partial wave ljm [see Eq. (2)]. The REBprediction for the inelastic cross section, i.e. the angu-lar distribution for the projectile excited to state i whilescattered in direction Ω reads (cid:18) dσ i d Ω (cid:19) inel = | F i, ( Q ) | (cid:18) dσd Ω (cid:19) pt . (15)Similarly, we get the following angular distribution forbreakup, i.e. the cross section for the projectile beingbroken up at an energy E in the c - n continuum with itscenter of mass scattered in direction Ω (cid:18) dσdEd Ω (cid:19) bu = | F E, ( Q ) | (cid:18) dσd Ω (cid:19) pt . (16)Neglecting the small difference in magnitude betweenthe outgoing momenta for elastic and inelastic processes,the point-like cross section ( dσ/d Ω) pt is identical for allthree processes (10), (15), and (16). This first explainsthe result obtained in Ref. [33], where it was observedthat the angular distributions for elastic scattering andbreakup exhibit very similar patterns. Indeed, most ofthe angular dependence of these cross sections is due tothat point-like cross section. Second, the similarity of theexpressions (10), (15), and (16) is at the core of the ratiomethod. If we now consider the ratio between Eqs. (16)and (10), the point-like cross sections cancel out, leavingan observable which, within the REB model, is just theratio of form factors R el ( E, Q ) = ( dσ/dEd Ω) bu ( dσ/d Ω) el (17) (REB) = | F E, ( Q ) | | F , ( Q ) | . (18)Therefore, according to the REB predictions, this ratioshould be sensitive only to the structure of the projectileand be independent of the reaction mechanism. In par-ticular, considering the ratio (17) automatically removesthe dependence on the core-target interaction, which isthe most ambiguous input in reaction modeling. III. RATIOS OF CROSS SECTIONS
Before analyzing the structure content of this new ob-servable, we should point out that identical cancellationsof the point-like cross section can be obtained for theratio of any linear combination of breakup, elastic- andinelastic-scattering angular distributions. Therefore weconsider here, in addition to R el (17), other options. Be-cause in some halo systems, there is a nearby excitedstate, hard to disentangle from the ground state, the elas-tic and inelastic contributions may be easier to measuretogether. We then introduce the quasi-elastic ratio R quasi ( E, Q ) = ( dσ/dEd Ω) bu ( dσ/d Ω) quasi (19) (REB) = | F E, ( Q ) | | F , ( Q ) | + P i> | F i, ( Q ) | , (20)where ( dσ/d Ω) quasi = ( dσ/d Ω) el + P i> ( dσ i /d Ω) inel . Be-cause for low Q , elastic scattering is dominant, addingthe breakup does not make much difference to the ratioobservables but simplifies the form factor dependence.Thus, we also consider R sum ( E, Q ) = ( dσ/dEd Ω) bu ( dσ/d Ω) sum (21) (REB) = | F E, ( Q ) | , (22)where the summed cross section reads (cid:18) dσd Ω (cid:19) sum = (cid:18) dσd Ω (cid:19) el + X i> (cid:18) dσ i d Ω (cid:19) inel + Z (cid:18) dσdEd Ω (cid:19) bu dE. (23)We compare in Fig. 1 the REB prediction for R el (18), R quasi (20) and R sum (22) for the reaction of Be on
Pb at 69 MeV/nucleon. The transferred momentum Q has been converted into the center-of-mass scatteringangle following Eq. (12). As expected there is very littledifference between R el and R quasi . Adding the breakupangular distribution to the denominator modifies onlythe large-angle behavior of the ratio. Other possibilitiesfor the ratio are discussed in Appendix A.After close analysis and a number of exploratory cal-culations, we found it optimal to consider the ratio R sum .This ratio leads to the simplest REB prediction (22) andis probably the easiest to measure experimentally. InRef. [10] we quantified this ratio with DEA calculationsthat do not make the approximations of the REB model,and presented the argument that the REB approxima-tions can be justified in realistic cases. Here we focus thediscussion on the source of the small discrepancies foundbetween DEA calculations and REB predictions and thestructure information that can be extracted from R sum .4 sum R quasi R el θ (deg) − − − FIG. 1: Ratios (18), (20) and (22) suggested by the simi-larity between angular distributions for elastic scattering andbreakup. The calculations are performed within REB for Beimpinging on Pb at 69 MeV/nucleon considering a Be- n continuum energy E = 0 . IV. ANALYSIS OF THE CROSS SECTIONRATIO
For the purpose of illustration, we base our calcula-tions on a concrete reaction measured at RIKEN [11],namely the breakup of Be on C and Pb at 67 and69 MeV/nucleon, respectively. In our two-body descrip-tion of the projectile, Be is seen as an inert Be corein its 0 + ground state, to which a neutron is bound by E = − . s / orbit. Unless mentionedotherwise, we take the same inputs as in Ref. [10] (allinteractions are provided in our Appendix B) and per-form calculations within DEA, which is found in excellentagreement with these experimental data [29].In Fig. 2 we show the corresponding summed crosssections (23) as a ratio to Rutherford (dotted lines), theangular distributions for breakup at a continuum energy E = 0 . R sum (21) in units MeV − (solid lines). Thecontinuum energy E = 0 . R sum should agree with | F E, | Eq. (14),as predicted by the REB model (22) (thick grey line).And, indeed, we find the agreement to be very good. Inboth cases, most of the angular dependence of the crosssections has been removed by taking their ratio, leavinga curve varying smoothly with the scattering angle θ .Moreover, this ratio lies nearly on top of its REB predic-tion. As already pointed out in Ref. [10], this implies thatthe ratio R sum removes most of the dependence on thereaction mechanism and hence contains mostly structureinformation. We note the presence of residual oscillationsat forward angles for the C target and at larger angles forthe Pb target. Note also the slower rise at the most for-ward angles in the latter case (see the insets, which focus θ (deg)( dσ/dEd Ω) bu( dσ/d Ω) sum ( dσ/d Ω) R R sum (a) Be + C − − − − − − θ (deg)( dσ/dEd Ω) bu( dσ/d Ω) sum ( dσ/d Ω) R R sum (b) Be + Pb − − − − − − − − FIG. 2: Illustration of the ratio method for Be impingingon: (a) C at 67 MeV/nucleon, and (b) Pb at 69 MeV/nucleon.Summed cross sections (dotted lines) and breakup angulardistributions (dashed lines) computed within DEA are com-pared to their ratio R sum (thin solid lines), which is foundin excellent agreement with its REB prediction | F E, | (thickgrey line). Calculations with U nT = 0 are shown as dash-dotted lines. The insets focus on the forward-angle behaviorof the ratio. on the forward-angle region). In the next subsections weexplore the source for these small discrepancies. A. The REB cross sections
The validity of the ratio (17) depends crucially on theequality of the two point-like cross sections in Eqs. (10)and (16). So here we test explicitly the validity of theseequations. In Fig. 3 we compare the results of Eqs. (10)and (16) with those obtained in the full dynamical cal-culation (DEA), for our two examples, namely Be on C (a) and Be on
Pb (b). The dotted and dashedlines correspond to the DEA angular distributions forthe elastic and breakup cross sections, respectively. The5 dσ/dEd Ω) bu ( dσ/d Ω) el / ( dσ/d Ω) Ruth
REB θ (deg)(a) Be + C − − θ (deg)(b) Be + Pb − − − − FIG. 3: Comparison of REB and DEA predictions for theelastic and breakup angular distributions for Be: (a) on C at 67 MeV/nucleon and (b) on
Pb at 69 MeV/nucleon.The dotted(dashed) line corresponds to the elastic(breakup)scattering within DEA and the grey lines to results with REB. grey lines are obtained with the factorization in Eqs. (10)and (16). For the light target, the REB follows closelythe DEA result in both elastic and breakup processesbut for a slight shift in the oscillatory pattern. Thesame can be seen for the Pb target with the exception ofsmaller angles. In this regime, the breakup cross sectionis not well described by REB. In the next two subsec-tions we will discuss the two approximations present inthe REB model and their imprint on the discrepanciesseen in Fig. 3.
B. Role of U nT The REB model neglects the contribution of U nT andthis explains why the form factor | F E, | is perfectlysmooth, whereas the DEA ratio exhibits residual oscil-lations. The neutron interaction with the target gives the projectile a minor kick that causes a slight shift inthe diffractive pattern, as already noted by Johnson etal. [34] and confirmed in Fig. 3. A careful analysis ofthe angular distributions shows that this shift dependsslightly on the excitation energy of the projectile. Theoscillatory pattern in the dynamical calculations there-fore differs between elastic, inelastic, and breakup crosssections, leading to the residual oscillations in their ratio.To confirm this analysis, we repeat the DEA calculationssetting U nT = 0 (dash-dotted lines in Fig. 2). The an-gular distributions obtained in this manner are exactlyin phase and their ratios exhibit no residual oscillations.These ratios are in perfect agreement with the REB pre-dictions but for the very forward-angle region on the Pbtarget (see inset of Fig. 2(b)). In that region, setting U nT = 0 does not improve the agreement between DEAand REB. The reason for that difference has to be lookedfor in the second ingredient of the REB model, i. e. theadiabatic approximation (see Sec. IV C).Even though U nT has an effect on the dynamics, whenapplying the ratio method to data it is likely that theresidual oscillations will not be noticeable experimentallywith current angular resolutions. In Fig. 4 we show theratio obtained with DEA (solid line), that predicted bythe REB model (thick grey line) and that obtained af-ter folding the DEA angular distributions with a typicalexperimental resolution (dash-dotted line). To this end,we convolute the theoretical cross sections with a Gaus-sian of standard deviation 0 . ◦ , which corresponds tothe angular resolution of the RIKEN experiment [11]. InFig. 4(a) we show the log plot, and to emphasize the dif-ference we include Fig. 4(b) with the corresponding linearplot. As expected the convolution reduces the residualoscillations to the point where they would no longer bedetectable. C. Role of the adiabatic approximation
In addition to neglecting U nT , the REB model neglectsthe excitation energy of the projectile (adiabatic approx-imation). This second approximation is responsible forthe different slope of the ratio R sum and its REB pre-diction at the most forward angles on the Pb target (seeinset of Fig. 2(b)).The Fig. 3(b) shows very clearly that at very forwardangles, the elastic scattering is well described by REB butthe breakup cross section is not, introducing an unphys-ical divergence. One could arrive at these same conclu-sions directly by analysing the Q -dependence of Eqs. (10)and (16). It is for this reason that the REB ratio (22) ishigher than the correct R sum at forward angles, as shownin the inset of Fig. 2(b). The adiabatic—or sudden—approximation assumes a very brief interaction time withthe target. When the reaction is entirely dominated bythe Coulomb interaction, which is the case for breakup onPb at forward angles, it cannot be treated satisfactorilywithin the adiabatic approximation due to the infinite6 olded R sum | F E, | θ (deg)(a) − − − θ (deg)(b) FIG. 4: Smoothing of the angular distributions by the fold-ing with experimental angular resolution ( Be on Pb at69 MeV/nucleon): (a) log plot; (b) linear plot. range of the Coulomb potential. Note that the overesti-mation of the ratio by the REB is not observed for thecarbon target. In that case the reaction is dominated byshort-ranged nuclear interactions, which allow us to relyon the adiabatic approximation [29].This analysis shows that the effects of the adiabatic ap-proximation upon the ratio are small and limited to thevery forward angles for Coulomb-dominated reactions.Since cross sections in this region can hardly be mea-sured, it is very unlikely that these effects will ever benoticeable. Nevertheless this analysis will help us un-derstand differences between DEA calculations and REBpredictions observed in later subsections.
D. Independence of the ratio on the reactionprocess
In Ref. [10] we showed that the ratio obtained whenconsidering the C target is identical to that obtained
Coul.C.+N. | F E, | θ (deg) R s u m ( M e V − ) − − − − FIG. 5: Ratio computed for Be on Pb at 69 MeV/nucleonusing different interactions: Coulomb plus nuclear (solid line)and purely Coulomb (dashed line) P - T potentials. with a Pb target. In other words, the new observableis independent of the reaction mechanism. To appre-ciate this fact we emphasize the difference between thebreakup and summed distributions obtained on C andon Pb (compare dotted and dashed lines in Fig. 2(a) and(b)). Even though the cross sections are orders of mag-nitude apart, and their diffraction pattern is completelydifferent, still the resulting ratio is very close to the formfactor Eq. (14) as predicted by REB.In Fig. 5 we focus on the reaction on Pb and explorethe interplay between Coulomb and nuclear interactions.In addition to DEA calculations including Coulomb andnuclear interactions (C.+N., solid line), we show resultsobtained from DEA calculations where only the Coulombterm of the U cT optical potential is considered (Coul.,dashed line). As already observed in Ref. [33], the an-gular distributions vary strongly with the P - T potential,indicating the sensitivity of the reaction mechanism tothat potential choice. Nevertheless, both ratios fall ontop of the form factor, confirming the independence ofthe ratio to the reaction process. The residual oscilla-tions are significantly reduced when only the Coulombinteraction is present. This is due to a much smootherbehavior of the angular distributions when no nuclearinteraction is included [33].To complete this analysis of the sensitivity of the ratioto the reaction mechanism, we now turn to its variationwith the beam energy. In Fig. 6, R sum is plotted for Be impinging on Pb at 40, 69, and 100 MeV/nucleon.As detailed in Appendix B, the optical potentials U cT and U nT are adapted to the beam energy, while the pro-jectile description is kept unchanged. To compare allthree ratios to one another, they are plotted as a functionof Q Eq. (12). The most significant difference betweenall three calculations are observed at large Q , wherethe ratios exhibit residual oscillations. As explained inSec. IV B, they are due to U nT , which varies with thebeam energy. Another, though smaller, difference is ob-7 (fm − ) R s u m ( M e V − ) − − − −
100 MeV/nucleon69 MeV/nucleon40 MeV/nucleon 0.30.250.20.150.10.05010.110 − − − − FIG. 6: Ratio computed on Pb at different beam energies. served at very small Q , corresponding to very forwardangle (see inset of Fig. 6). In that region, the DEA un-derestimates its REB prediction because of the adiabaticapproximation (see Sec. IV C). Since the REB approxi-mation is more reliable at high beam energy, the agree-ment between DEA and REB improves at forward angleswhen larger energies are considered. E. Applicability to other one-neutron halo systems
To check the applicability of the ratio method to otherhalo nuclei, we study the case of C. This one-neutronhalo nucleus has been studied experimentally by variousgroups. We choose here the conditions of the RIKEN ex-periment, i. e. performed at 67 MeV/nucleon on a leadtarget [46]. In Fig. 7, the DEA summed (dotted line) andbreakup (dashed line) cross sections are plotted as a func-tion of the scattering angle θ of the C- n center of masstogether with the corresponding ratio R sum (solid line)and its REB prediction (thick gray line). Here we as-sumed the final breakup state to be a non-resonant stateat E = 0 . Be: both angular distri-butions exhibit similar features that are mostly removedwhen taking their ratio. This confirms the validity of theratio method for other one-neutron halo projectiles.
V. STRUCTURE INFORMATION CONTAINEDIN THE CROSS SECTION RATIO
Now that we have a good understanding of the smalldiscrepancies of the true ratio and the prediction fromthe REB model, we can explore the structure informa-tion contained in this observable. Having shown the ratioto be independent of the reaction process, we expect it tobe more sensitive to the projectile structure than usualreaction observables. Below we discuss the dependenceon the binding energy of the halo neutron E , its orbital θ (deg) C + Pb @ 67AMeV( dσ/dEd Ω) bu( dσ/d Ω) sum ( dσ/d Ω) R R sum − − − FIG. 7: Analysis of the ratio for C on Pb at67 MeV/nucleon: summed cross section (dotted), breakupcross section (dashed) and ratio (solid) versus the REB pre-diction (thick grey). angular momentum l , the details of the c - n radial wave-function, and the final scattering state. For this analysis,we stick to the collision of Be on Pb at 69/nucleon.
A. Binding energy
The ratio R sum is very sensitive to the one-neutronseparation energy E . Because the breakup cross sec-tion is larger for loosely bound systems, the magnitudeof the ratio increases with decreasing binding [10]. InFig. 8, we show the ratio obtained for a Be-like systembound by 50 keV, 0.5 MeV, and 5.0 MeV, respectively.They result from DEA calculations where the depth ofthe Be- n interaction in the s wave is adjusted to repro-duce the appropriate one-neutron separation energy (seeAppendix B). Our results show that changing the bind-ing energy by one order of magnitude produces a changein the ratio by two orders of magnitude. Moreover, theshape of the ratio differs significantly from one bindingenergy to the other. Looking into the details of the an-gular distributions one sees also that the agreement withthe REB prediction deteriorates with increasing bindingenergy. This is to be expected since for large bindingenergy, the excitation energy needed for breakup is largeand the adiabatic approximation is no longer justified.Nevertheless, it is clear that the cross section ratio pro-vides a very accurate indirect measurement of the bindingenergy of the system.8 = − E = − . E = −
50 keV θ (deg) R s u m ( M e V − ) − − − − − FIG. 8: Sensitivity of ratio R sum to the binding energy of theprojectile: E = −
50 keV, − . − B. Orbital angular momentum
Next we investigate the dependence on the orbital an-gular momentum of the initial bound state l . The Be- n interaction is adjusted to reproduce a Be ground stateat E = − . s / configura-tion, a 0 p / and a 0 d / configuration (see Appendix B).DEA calculations are repeated with these new interac-tions, and the resulting ratios are plotted in Fig. 9. Againwe find that the cross section ratio is very sensitive to thisproperty of the projectile initial state. The magnitude ofthe ratio decreases with increasing angular momentum.It is important to note that even though the magnitudefor a 5 MeV bound 1 s / state (Fig. 8) is similar to thatof a 0 . d / state (Fig. 9), the shape of the distri-bution is very different, particularly the slope at largerangles. This feature would make it possible to determineunequivocally both the binding energy and the angularmomentum of the nucleus under inspection. C. Radial wave function
We now turn to the sensitivity of the ratio to details ofthe projectile radial wavefunction. We consider variousgeometries for V cn in the s wave and readjust the depthof the interaction to reproduce the physical neutron sepa-ration energy E = − . R sum folded with experimental resolution and their REBpredictions are plotted in Fig. 10(b). At forward angles,i.e. in the range 1 ◦ to 3 ◦ , the DEA ratios are in excellentagreement with their REB predictions. In that region,the ratio is proportional to the square of the asymptoticnormalization coefficient (ANC). At larger angles, the d / p / s / θ (deg) R s u m ( M e V − ) − − − − − FIG. 9: Sensitivity of ratio R sum to the orbital angular mo-mentum of the halo neutron. Calculations with a valenceneutron bound to a Be core by E = − . s / ,0 p / or 0 d / orbital are compared to one another. discrepancy between DEA and REB increases. Never-theless, the general behavior, and especially the orderingof the curves, is the same in both models. In particular,the REB predicts a crossing of the curves, that actuallytakes place at θ ≃ ◦ in dynamical calculations. At thatangle, the ratio obtained with the initial 0 s / state over-takes the others although it corresponds to the smallestANC. This can only be explained if the ratio is sensitiveto the internal part of the wave function at larger angles.The ratio is thus able to probe different parts of the ra-dial wave function, depending on the scattering angle θ .It is one of the few reaction observables to display sucha property. Elastic-scattering and breakup cross sectionsare indeed purely peripheral, in the sense that they probeonly the tail of the projectile wave function and not itsinterior [47]. Although these details do not affect theratio as significantly as the binding energy and the angu-lar momentum, we expect them to be observable in datawith enough statistics. D. Choice of continuum energy E So far we have fixed the Be- n relative energy in thefinal state to be E = 0 . R sum can bedefined for any relative energy E between the core andthe halo neutron after breakup. We now study the effectof this energy by considering E = 0 .
5, 1.0, and 1.5 MeV.In addition, we also consider the Be- n energy E res =1 .
274 MeV, which corresponds to a 5 / + resonance inthe Be continuum. In our model, that resonance issimulated by a d / E = 0 . for E = 1 . for E res ,and 10 for E = 1 . s / R = 4 fm R = 1 fmOriginal r (fm) u ( f m − / ) (a) θ (deg) R s u m ( M e V − ) (b) FIG. 10: Sensitivity of ratio R sum to the radial wave func-tion of the projectile: (a) radial wave functions of the initial s / states; (b) corresponding DEA ratios R sum folded withexperimental resolution (thin black lines) and their REB pre-dictions (thick gray lines). analysis, the agreement between DEA calculations (solidlines) and REB predictions (thick gray lines) worsenswith increasing continuum energy E . On both targets,the residual oscillations increase at larger E . This effectis caused by the shift due to U nT , which increases with E (see Sec. IV B). The difference in the oscillatory patternbetween the elastic scattering and the breakup angulardistributions therefore increases at larger E , leading tomore significant residual oscillations in the ratio. Thisdisagreement between DEA calculations and the REBpredictions fully disappears when U nT is set to 0 (seeFig. 2).For the heavy target, the rise of the DEA ratio at for-ward angles is slower compared to its REB prediction(see Fig. 11(b)). As explained in Sec. IV C, this slowerrise is due to the adiabatic approximation made in theREB. Accordingly, the agreement between the dynam-ical calculation and the form factor | F E, | at forwardangles gets worse with increasing excitation energy. Thisdiscrepancy is not observed on the light target for whichthe adiabatic approximation is well suited. θ (deg) R s u m ( M e V − ) (a) E = 0 . × E = 0 . × E = 1 . × E res × E = 1 . − − − θ (deg) R s u m ( M e V − ) (b) E = 0 . × E = 0 . × E = 1 . × E res × E = 1 . − − − FIG. 11: Change of the ratio with the energy E in the c - f continuum. (a) Be on C at 67 MeV/nucleon; (b) Be onPb at 69 MeV/nucleon. For convenience, each ratio has beenmultiplied by a factor.
Of particular interest is the case where the final stateis a resonance. The calculations displayed in Fig. 11 donot show unusual effects at E res . The ratio rises slightlyfaster at large scattering angles and its residual oscilla-tions are slightly larger than off-resonance. These moresignificant departures from the REB prediction might beseen experimentally and could hence be used to spot res-onant structures in the continuum of exotic nuclei. How-ever such a feature is more clearly observed in energydistributions measured after breakup on a light target[11, 32].This analysis shows that although the agreement be-tween the DEA ratio and its REB prediction remains fair,the ratio method would be better applied at low energy E in the projectile continuum and off resonance. In thecase studied here, the discrepancy remains small up to a Be- n energy E ∼ . I. CONCLUSION AND PROSPECTS
A new observable for halo nuclei has been studied indetail. It consists of the ratio of the breakup angular dis-tribution and the summed angular distribution includingelastic, inelastic and breakup. We show that realistic cal-culations of this observable closely follow predictions bythe REB model. The latter neglects the neutron-targetinteraction and the excitation energy of the projectile. Inthis work we have explored the small discrepancies thatexist between dynamical calculations and their REB pre-dictions: the DEA ratio exhibits residual oscillations andrises more slowly than the REB at very forward anglesfor Coulomb-dominated processes. The residual oscilla-tions are caused by the additional kick the neutron feelsdue to its interaction with the target U nT . The differ-ence observed at forward angles on heavy targets is dueto the adiabatic approximation made in the REB, whichis incompatible with the long range of the Coulomb in-teraction. Nevertheless, these discrepancies remain smallindicating that most of the dependence on the reactionmechanism is removed by taking this ratio of cross sec-tions. Therefore the cross section ratio is an optimal toolto study the structure of exotic nuclei.We next analyze the structure information containedin the cross section ratio. Our results show that the ratiois extremely sensitive to the binding energy of the haloand the angular momentum of the halo orbital. Whilebinding energy and angular momentum can be unequiv-ocally extracted, an experimental error of a few percentswould be necessary for learning about the details of theradial behavior of the halo. Very accurate data couldconstrain simultaneously the asymptotic normalizationcoefficient as well as the internal behavior of the radialwave function. One advantage of the ratio method isthat it provides an additional control variable, the en-ergy of the continuum bin, which can be tuned to bemost appropriate to the case under study. Although themethod works best when the excitation in the continuumis small, we show that even at 1.5 MeV useful results canbe obtained.Thanks to its independence of the reaction mechanism,the cross section ratio enables us to probe in-depth thestructure of one-neutron halo nuclei and brings out in-formation inaccessible to any other reaction observable.According to our analysis, the ratio method works bothon light and heavy targets. It gives better results at highbeam energy and for loosely-bound projectiles, for whichthe adiabatic approximation is more reliable. Appendix A: Other ratios
We have also considered ratios obtained by integratingover the relative energy of the breakup fragments. Thiscould be of interest due to low statistics of the energy dis-tribution of the breakup fragments or other experimen-tal constraints. The resulting expressions are denoted by R R el and R R sum and are defined by R R el ( Q ) = R dσ bu /dEd Ω dEdσ el /d Ω (A1) (REB) = 1 − ( | F , ( Q ) | + P i> | F i, ( Q ) | ) | F , ( Q ) | , (A2)and R R sum ( Q ) = R dσ bu /dEd Ω dEdσ sum /d Ω (A3) (REB) = 1 − ( | F , ( Q ) | + X i> | F i, ( Q ) | ) . (A4)In this case however, all configurations of the finalbreakup fragments are contributing. As detailed inSecs. IV C and V D, the higher the relative energy be-tween the breakup fragments E , the less valid is the adi-abatic approximation assumed in the REB model. There-fore, we found that neither R R el (A1) nor R R sum (A3) areuseful. Appendix B: Two-body interactions
Most of the calculations presented in this text havebeen performed using the DEA [28, 29] with the descrip-tion of Be developed in Ref. [32] and the optical poten-tials of Ref. [29]. This appendix details the form-factorsof these potentials and lists their parameters used in thisstudy.The projectile model is based on the two-body descrip-tion presented in Sec. II A. The V cn potential contains acentral term plus a spin-orbit coupling term V cn ( r ) = V f ( r, R , a ) + V LS l · s r ddr f ( r, R , a ) , (B1)with the Woods-Saxon form factor f ( r, R , a ) = (cid:20) (cid:18) r − R a (cid:19)(cid:21) − (B2)of radius R and diffuseness a . In Eq. (B1), l is therelative orbital angular momentum between the core andthe valence neutron, and s is the spin of the neutron. Forcompleteness, we list in Table I the parameters of the c - n potentials used in this analysis. The first two rows con-tain the original potential of Ref. [32] for Be (for evenand odd orbital angular momenta). This potential repro-duces the experimental binding energy E = − . s / orbital to describe the 1 / + ground stateof Be. For the 1 / − excited state, we take the 0 p / bound state of the potential, which requires a differentdepth V for even and odd partial waves.The continuum wave functions appearing in Eq. (14)are obtained with the V cn potential. Note that the 5 / + resonance at 1.274 MeV in the Be- n continuum, is re-produced in the d / partial wave with the original poten-tial of Table I [32]. The other lines of Table I contain the11 V LS R a (MeV) (MeV fm ) (fm) (fm)Original (even l ) 62.52 21.00 2.585 0.6Original (odd l ) 39.74 21.00 2.585 0.6 E = −
50 keV 57.91 21.00 2.585 0.6 E = − R = 1 fm 210.19 21.00 1 0.6 R = 4 fm 30.915 21.00 4 0.60 s / p / d / C- n n potentials (B1). parameters used for the calculations presented in Figs. 8,9, and 10. The last line correspond to the potential usedto describe C in the calculations presented in Fig. 7.The potentials are labeled as the corresponding curves inthe figures. In most of the Be calculations the param-eters listed in Table I are used only in the ground-statepartial wave, the others, including the continuum wavefunctions, being described by the “Original” potential.Only in the 0 p / and 0 d / cases do we use the samepotential in all partial waves. In the C case, all partialwaves are described using the same potential.The nuclear part of the optical potentials used to sim-ulate the interaction between the projectile constituentsand the targets contains real and imaginary volume termsand an imaginary surface term U xT = − V f ( r, R r , a r ) − i (cid:20) W f ( r, R i , a i ) + W d a i ddr f ( r, R i , a i ) (cid:21) . (B3) The Coulomb part of U cT is simulated by the potentialdue to a uniformly charged sphere of radius R C . Theparameters of the optical potentials used in this studyare listed in Table II. We follow Ref. [29] for the choicesof most of these optical potentials. For U nT we use theBecchetti and Greenlees parameterization [48], for thePb target. For the carbon target, we follow Ref. [32] anduse the p -C potential of Comfort and Karp [49]. The Be-Pb potentials are obtained from α -Pb potentialsby merely rescaling the radius for an A = 10 projec-tile [50, 51]. For the Be-C potential, we follow Ref. [32]and use the potential developed by Al-Khalili, Tostevinand Brooke that fits the elastic scattering of Be on Cat 49.3 MeV/nucleon [52]. The C-Pb interaction is de-rived from the potential of Ref. [53] that fits the C-Pbelastic scattering at 390 MeV.
Acknowledgments
We thank I. J. Thompson, B. Tsang, N. Timofeyuk,and the MoNA collaboration for interesting discussionson the subject. This work was supported by the NationalScience Foundation grant PHY-0800026 and the Depart-ment of Energy under contract DE-FG52-08NA28552and de-sc0004087. R. C. J. is supported by the UnitedKingdom Science and Technology Facilities Council un-der Grant No. ST/F012012. This text presents researchresults of the Belgian Research Initiative on eXotic nuclei(BriX), program nr P7/12 on interuniversity attractionpoles of the Belgian Federal Science Policy Office. [1] A. S. Jensen, K. Riisager, D. V. Fedorov, and E. Garrido,Rev. Mod. Phys. , 215 (2004).[2] I. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida,N. Yoshikawa, K. Sugimoto, O. Yamakawa,T. Kobayashi, and N. Takahashi, Phys. Rev. Lett. , 2676 (1985).[3] A. Jensen and K. Riisager, Phys. Lett. B480 , 39 (2000).[4] K. Tanaka, T. Yamaguchi, T. Suzuki, T. Ohtsubo,M. Fukuda, D. Nishimura, M. Takechi, K. Ogata,A. Ozawa, T. Izumikawa, et al., Phys. Rev. Lett. ,062701 (2010).[5] L.-B. Wang, P. Mueller, K. Bailey, G. W. F. Drake, J. P.Greene, D. Henderson, R. J. Holt, R. V. F. Janssens,C. L. Jiang, Z.-T. Lu, et al., Phys. Rev. Lett. , 142501(2004).[6] R. S´anchez, W. N¨ortersh¨auser, G. Ewald, D. Albers,J. Behr, P. Bricault, B. A. Bushaw, A. Dax, J. Dilling,M. Dombsky, et al., Phys. Rev. Lett. , 033002 (2006).[7] M. Zahar, M. Belbot, J. J. Kolata, K. Lamkin,R. Thompson, N. A. Orr, J. H. Kelley, R. A. Kryger,D. J. Morrissey, B. M. Sherrill, et al., Phys. Rev. C ,R1484 (1993). [8] D. Bazin, W. Benenson, B. A. Brown, J. Brown,B. Davids, M. Fauerbach, P. G. Hansen, P. Mantica, D. J.Morrissey, C. F. Powell, et al., Phys. Rev. C , 2156(1998).[9] K. T. Schmitt, K. L. Jones, A. Bey, S. H. Ahn, D. W.Bardayan, J. C. Blackmon, S. M. Brown, K. Y. Chae,K. A. Chipps, J. A. Cizewski, et al., Phys. Rev. Lett. , 192701 (2012).[10] P. Capel, R. Johnson, and F. Nunes, Phys.Lett. B705 ,112 (2011).[11] N. Fukuda, T. Nakamura, N. Aoi, N. Imai, M. Ishihara,T. Kobayashi, H. Iwasaki,T. Kubo, A. Mengoni, M. Notani, et al., Phys. Rev. C , 054606 (2004).[12] M. Smedberg, T. Baumann, T. Aumann, L. Axels-son, U. Bergmann, M. Borge, D. Cortina-Gil, L. Fraile,H. Geissel, L. Grigorenko, et al., Phys. Lett. B452 , 1(1999).[13] E. Lunderberg, P. A. DeYoung, Z. Kohley, H. At-tanayake, T. Baumann, D. Bazin, G. Christian, D. Di-varatne, S. M. Grimes, A. Haagsma, et al., Phys. Rev.Lett. , 142503 (2012). T Energy
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