Coulomb corrected eikonal description of the breakup of halo nuclei
aa r X i v : . [ nu c l - t h ] O c t Coulomb corrected eikonal description of the breakup of halonuclei
P. Capel, ∗ D. Baye, † and Y. Suzuki ‡ Physique Quantique, C.P. 165/82 and Physique Nucl´eaire Th´eorique et Physique Math´ematique,C.P. 229, Universit´e Libre de Bruxelles, B 1050 Brussels, Belgium Department of Physics, Niigata University, Niigata 950-2181, Japan (Dated: October 24, 2018)The eikonal description of breakup reactions diverges because of the Coulombinteraction between the projectile and the target. This divergence is due to theadiabatic, or sudden, approximation usually made, which is incompatible with theinfinite range of the Coulomb interaction. A correction for this divergence is analysedby comparison with the Dynamical Eikonal Approximation, which is derived with-out the adiabatic approximation. The correction consists in replacing the first-orderterm of the eikonal Coulomb phase by the first-order of the perturbation theory. Thisallows taking into account both nuclear and Coulomb interactions on the same foot-ing within the computationally efficient eikonal model. Excellent results are foundfor the dissociation of Be on lead at 69 MeV/nucleon. This Coulomb CorrectedEikonal approximation provides a competitive alternative to more elaborate reactionmodels for investigating breakup of three-body projectiles at intermediate and highenergies.
PACS numbers: 24.10.-i, 25.60.Gc, 03.65.Nk, 27.20.+nKeywords: Halo nuclei, Dissociation, eikonal approximation, Coulomb interaction, Be I. INTRODUCTION
Halo nuclei are among the most peculiar quantum structures [1, 2, 3]. These light neutron-rich nuclei exhibit a very large matter radius when compared to their isobars. This extendedmatter distribution is due to the weak binding of one or two valence neutrons. Thanks totheir low separation energy, these neutrons tunnel far inside the classically forbidden region,and have a high probability of presence at a large distance from the other nucleons. In asimple point of view, they can be seen as very clusterized systems: a core that containsmost of the nucleons, and that resembles a usual nucleus, to which one or two neutrons areloosely bound, and form a sort of halo around the core [4]. The Be, C, and C isotopesare examples of one-neutron halo nuclei. Examples of two-neutron halo nuclei are He, Li,and Be. In addition to their two-neutron halo, these nuclei also exhibit the Borromeanproperty: the three-body system is bound although none of the two-body subsystems is [5].Since their discovery in the mid 80s [6], these nuclei have thus been the focus of manyexperimental [1, 2, 3] and theoretical [7, 8, 9] studies. Due to their short lifetime, halo nu- ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] lei cannot be studied with usual spectroscopic techniques, and one must resort to indirectmethods to infer information about their structure. Breakup reactions are among the mostused methods to study halo nuclei [10, 11, 12]. In such reactions, the halo dissociates fromthe core through interaction with a target. In order to extract valuable information fromexperimental data one needs an accurate reaction model coupled to a realistic description ofthe projectile. Various techniques have been developed with this aim: perturbation expan-sion [13, 14], adiabatic approximation [15], eikonal model [16, 17, 18], coupled channel witha discretized continuum (CDCC) [19, 20, 21], numerical resolution of a three-dimensionaltime-dependent Schr¨odinger equation (TDSE) [22, 23, 24, 25, 26, 27], and more recently,dynamical eikonal approximation (DEA) [28, 29, 30].Some of these techniques (perturbation expansion, adiabatic approximation, and eikonalmodel) are based on approximations that lead to easy-to-handle models. Their main ad-vantage is their relative simplicity in use and interpretation. However, the approximationson which they are built usually restrain their validity domain. For example, perturbativeand adiabatic models are restricted to the sole Coulomb interaction between the projectileand the target. The eikonal method on the contrary diverges for that interaction and canbe used only for reactions on light targets. The adiabatic, or sudden, approximation madein the usual eikonal model is responsible for that divergence. It indeed assumes a very briefcollision time, that is incompatible with the infinite range of the Coulomb interaction.The more elaborate models (CDCC, TDSE, and DEA) are not restricted in the choiceof the projectile-target interaction. However, they lead to complex and time-consumingimplementations. First calculations were therefore limited to simple descriptions of theprojectile (i.e. two-body projectiles with local core-halo interactions). Recently, severalattempts have been made to improve the description of the projectile. For example Summers,Nunes, and Thompson have developed an extended version of the CDCC technique, baptizedXCDCC, in which the description of the halo nucleus includes excitation of the core [31].Other groups are developing four-body CDCC codes, i.e. a description of the breakup ofthree-body projectiles, with the aim of modeling the dissociation of Borromean nuclei [32,33]. These techniques albeit promising, require large computational facilities, and are verytime-consuming.Alternatively one could try to extend the range of simpler descriptions of breakup re-actions. Among these descriptions, the eikonal model is of particular interest. It indeedallows taking into account, at all orders and on the same footing, both nuclear and Coulombinteractions between the projectile and the target. Moreover it gives excellent results fornuclear-dominated dissociations [17, 29]. Its only flaw is the erroneous treatment of theCoulomb interaction. A correction to that treatment has been proposed by Margueron,Bonaccorso, and Brink [34] and developed by Abu-Ibrahim and Suzuki [35]. The basic ideaof this Coulomb corrected eikonal model (CCE) is to replace the diverging Coulomb eikonalphase at first-order by the corresponding first-order of the perturbation theory [36]. Thelatter, being obtained without adiabatic approximation, does not diverge. The CCE is muchmore economical than more elaborate techniques (a gain of a factor 100 in computationaltime can be achieved between this CCE and the DEA). It could therefore constitute a com-petitive alternative for simulating the breakup of Borromean nuclei at intermediate and highenergies. However efficient it seems, this correction has never been compared to any otherreaction model.In this work, we aim at evaluating the validity and analyzing the strengths and weaknessesof this correction by comparing it with the DEA. The chosen test cases are the breakup of2 Be on Pb and C so as to see the significance of the correction for both heavy and lighttargets. The considered energy is around 70 MeV/nucleon. This corresponds to RIKENexperiments [11, 12], with which the DEA is in excellent agreement [28, 29].Our paper is organized as follows. In Sec. II, we recall the basics of the eikonal descriptionof reactions, and detail the Coulomb correction proposed in Refs. [34, 35]. The numericalaspects of our calculations are summarized in Sec. III. The results for Be on Pb aredetailed in Sec. IV, while those corresponding to a carbon target are given in Sec. V. Thefinal section contains our conclusions about this model.
II. THEORETICAL FRAMEWORKA. Eikonal description of breakup reactions
To describe the breakup of a halo nucleus, we consider the following three-body model.The projectile P is made up of a fragment f of mass m f and charge Z f e , initially bound toa core c of mass m c and charge Z c e . This two-body projectile is impinging on a target T ofmass m T and charge Z T e . The fragment has spin I , while both core and target are assumedto be of spin zero. These three bodies are seen as structureless particles.The structure of the projectile is described by the internal Hamiltonian H = p µ cf + V cf ( r ) , (1)where r is the relative coordinate of the fragment to the core, p is the corresponding momen-tum, µ cf = m c m f /m P is the reduced mass of the core-fragment pair (with m P = m c + m f ),and V cf is the potential describing the core-fragment interaction. This potential includes acentral part, and a spin-orbit coupling term (see Sec. III).In partial wave lj , the eigenstates of H are defined by H φ ljm ( E, r ) = Eφ ljm ( E, r ) , (2)where E is the energy of the c - f relative motion, and j is the total angular momentumresulting from the coupling of the orbital momentum l with the fragment spin I . Thenegative-energy solutions of Eq. (2) correspond to the bound states of the projectile. Theyare normed to unity. The positive-energy states describe the broken-up projectile. Theirradial part u klj are normalized according to u klj ( r ) −→ r →∞ cos δ lj F l ( kr ) + sin δ lj G l ( kr ) , (3)where k = p µ cf E/ ~ is the wave number, δ lj is the phase shift at energy E , and F l and G l are respectively the regular and irregular Coulomb functions [37].The interactions between the projectile constituents and the target are simulated by op-tical potentials chosen in the literature (see Sec. III). Within this framework the descriptionof the reaction reduces to the resolution of a three-body Schr¨odinger equation that reads, inthe Jacobi set of coordinates illustrated in Fig. 1, (cid:20) P µ + H + V P T ( R , r ) (cid:21) Ψ( R , r ) = E T Ψ( R , r ) , (4)3 br Z TPc f
FIG. 1: Jacobi set of coordinates: r is the projectile internal coordinate, and R = b + Z b Z is thetarget-projectile coordinate. where R is the coordinate of the projectile center of mass relative to the target, P is thecorresponding momentum, µ = m P m T / ( m P + m T ) is the projectile-target reduced mass,and E T is the total energy. The projectile-target interaction V P T ( R , r ) = V cT (cid:18) R − m f m P r (cid:19) + V fT (cid:18) R + m c m P r (cid:19) , (5)is the sum of the optical potentials V cT and V fT (including Coulomb) that simulate thecore-target and fragment-target interactions, respectively. The projectile impinging on thetarget is initially bound in the state φ l j m of energy E . We are therefore interested insolutions of Eq. (4) that behave asymptotically asΨ( R , r ) −→ Z →−∞ e i { KZ + η ln[ K ( R − Z )] } φ l j m ( E , r ) , (6)where Z is the component of R in the incident-beam direction and η = Z T Z P e / (4 πǫ ~ v )is the P - T Sommerfeld parameter (with Z P = Z c + Z f ).In the eikonal description of reactions, the three-body wave function Ψ is factorized asthe product of a plane wave by a new function b Ψ [16, 17, 18],Ψ( R , r ) = e iKZ b Ψ( R , r ) , (7)where K is the wavenumber of the projectile-target relative motion related to the totalenergy E T by E T = ~ K µ + E . (8)With factorization (7), the Schr¨odinger equation (4) reads (cid:20) P µ + vP Z + H − E + V P T ( R , r ) (cid:21) b Ψ( R , r ) = 0 , (9)4here v = ~ K/µ is the initial projectile-target relative velocity. The first step in the eikonalapproximation is to assume the second-order derivative P / µ negligible with respect to thefirst-order derivative vP Z . The function b Ψ is indeed expected to vary weakly in R when thecollision occurs at sufficiently high energy [16, 17, 18]. This leads to the DEA Schr¨odingerequation [28, 29] i ~ v ∂∂Z b Ψ( b , Z, r ) = [( H − E ) + V P T ( R , r )] b Ψ( b , Z, r ) , (10)where the dependence of the wave function on the longitudinal Z and transverse b partsof the projectile-target coordinate R has been made explicit (see Fig. 1). This equationis mathematically equivalent to a time-dependent Schr¨odinger equation with straight-linetrajectories, and can be solved using any algorithm valid for the time-dependent Schr¨odingerequation (see e.g. Refs. [22, 23, 24, 25, 26, 27]). However, contrary to time-dependentmodels, it is obtained without semiclassical approximation: the projectile-target coordinatecomponents b and Z are quantal variables in DEA. This advantage over time-dependenttechniques allows taking into account interferences between solutions obtained at different b s.The DEA reproduces various breakup observables quite accurately for collisions of loosely-bound projectiles on both light and heavy targets [29, 30].The second step in the usual eikonal model is to assume the collision to occur during avery brief time and to consider the internal coordinates of the projectile to be frozen whilethe reaction takes place [17]. This second assumption, known as the adiabatic, or sudden,approximation leads to neglect the term H − E in Eq. (10) which then reads i ~ v ∂∂Z b Ψ( b , Z, r ) = V P T ( R , r ) b Ψ( b , Z, r ) . (11)In these notations, the asymptotic condition (6) becomes b Ψ( b , Z, r ) −→ Z →−∞ e iη ln[ K ( R − Z )] φ l j m ( E , r ) . (12)The solution of Eq. (11) exhibits the well-known eikonal expression [16] b Ψ( b , Z, r ) = exp (cid:20) − i ~ v Z Z −∞ V P T ( b , Z ′ , r ) dZ ′ (cid:21) φ l j m ( E , r ) . (13)This expression is only valid for short-range potentials. The Coulomb interaction requires aspecial treatment that is detailed in the next section. Let us point out that this treatmentallows taking properly account of the projectile-target Rutherford scattering. The Coulombdistortion in Eq. (12) is therefore simulated in the phase of Eq. (13). After the collision, thewhole information about the change in the projectile wave function is thus contained in thephase shift χ that reads χ ( b , s ) = − ~ v Z ∞−∞ V P T ( R , r ) dZ. (14)Due to translation invariance, this eikonal phase depends only on the transverse components b of the projectile-target coordinate R and s of the core-fragment coordinate r .5 . Coulomb correction to the eikonal model The eikonal model gives excellent results for nuclear-dominated reactions [17, 29]. How-ever, it suffers from two divergence problems when the Coulomb interaction becomes signif-icant. The first is the well-known logarithmic divergence of the eikonal phase describing theCoulomb elastic scattering [16, 17, 18]. The second is caused by the adiabatic approximationused in the eikonal treatment of the Coulomb breakup [17]. To explain this, let us dividethe eikonal phase (14) into its nuclear, and Coulomb contributions χ ( b , s ) = χ N ( b , s ) + χ C ( b , s ) + χ CP T ( b ) . (15)The Coulomb term χ C for a one-neutron halo nucleus reads (the extension to the case of acharged fragment is immediate) [29, 35] χ C ( b , s ) = − η Z ∞−∞ | R − m f m P r | − R ! dZ (16)= η ln − m f m P b b · s b + m f m P s b ! , (17) b b denotes a unit vector along the transverse coordinate b . In Eq. (16), we subtract the term1 /R corresponding to a Coulomb interaction between the projectile center of mass and thetarget. The phase χ C therefore corresponds to the Coulomb tidal force that contributes tothe breakup. Moreover, this subtraction leads to a faster decrease of the potential at largedistances, which enables us to obtain the analytic expression (17). This is compensated bythe addition of the elastic Coulomb phase χ CP T χ CP T ( b ) = − η Z Z max − Z max dZR . (18)This phase describes the Rutherford scattering between the projectile and the target. Theintegral is truncated, for it otherwise diverges (note that the integral in Eq. (17) doesnot diverge and therefore does not require the same treatment). This truncation basicallycorresponds to Glauber’s screened Coulomb potential [16]. Other truncation techniques [16]and other ways to deal with this divergence [18] exist. All lead to the same expression of theelastic Coulomb phase but for an additional constant phase that does not affect the crosssections [16]. The truncation considered in Eq. (18) leads to χ CP T ( b ) ≈ η ln b Z max . (19)This elastic Coulomb phase correctly reproduces Rutherford scattering, indicating that thefirst of the two aforementioned divergences can be easily corrected [16, 17, 18]. The nuclearterm χ N is then by definition the difference between the eikonal phase (14) and the Coulombcontributions (17) and (19).In addition to the divergence in elastic scattering, the Coulomb interaction is responsiblefor a divergence in breakup. The aim of the present paper is to analyse a way to correctthis divergence. It is due to the slow decrease of χ C in b . Indeed, when expanded in powersof χ C , the exponential of the Coulomb eikonal phase reads e iχ C = 1 + iχ C −
12 ( χ C ) + · · · , (20)6here the explicit dependence on the coordinates has been omitted for clarity. When in-tegrated over b in the calculation of the cross sections (see Sec. II C), the 1 /b asymptoticbehavior of the first-order term iχ C will lead to divergence.This divergence problem arises from the incompatibility between the infinite range of theCoulomb interaction and the adiabatic, or sudden, approximation: no short collision timecan be assumed if the Coulomb interaction dominates. Renouncing the use of the adiabaticapproximation solves this divergence: the DEA, which corresponds to the eikonal modelwithout this approximation [see Eq. (10) and Refs. [28, 29]], does not diverge. The excellentresults obtained within the DEA for collisions of loosely-bound projectiles on both light andheavy targets [29, 30] confirm that, when dynamical effects are considered, both nuclear andCoulomb interactions can be properly taken into account on the same footing.To avoid this divergence, a cutoff at large b could be made. In Ref. [38], Abu-Ibrahim andSuzuki proposed to limit the values of b to be considered in the cross-section calculations at b max = ~ v | E | . (21)This cutoff is obtained by requiring the characteristic time of internal excitation ~ / | E | tobe shorter than the collision time b/v . The factor of two is proposed as a qualitative guide.However this treatment is rather artificial and not very satisfactory [35].Alternatively, it has been proposed by Margueron, Bonaccorso, and Brink [34], and de-veloped by Abu-Ibrahim and Suzuki [35], to replace the first-order term iχ C in Eq. (20),which leads to the divergence, by the first-order term of the perturbation theory iχ F O [36] χ F O ( b , r ) = − η Z ∞−∞ e iωZ/v | R − m f m P r | − R ! dZ, (22)where ω = ( E − E ) / ~ , with E the c - f relative energy after dissociation. Since no adia-batic approximation is made in perturbation theory, this term does not diverge. When theadiabatic approximation is applied to Eq. (22), i.e. when ω is set to 0, one recovers exactlythe Coulomb eikonal phase (16). This suggests that without adiabatic approximation thefirst-order term in Eq. (20) would be iχ F O (22). Furthermore, a simple analytic expres-sion is available for each of the Coulomb multipoles in the far-field approximation, i.e. for m f r/m P < R [39]. The idea of the correction is therefore to replace the exponential of theeikonal phase according to e iχ → e iχ N (cid:16) e iχ C − iχ C + iχ F O (cid:17) e iχ CPT . (23)With this Coulomb correction, the breakup of halo-nuclei can be described within theeikonal model taking on (nearly) the same footing both Coulomb and nuclear interactions atall orders. This correction can also be seen as an inexpensive way to introduce higher-ordereffects and nuclear interactions in the first-order perturbation theory.In this work, we analyse the validity of this CCE model by comparing results obtainedwith the correction (23) to results of the DEA. The latter is chosen as reference calculation,since it does not make use of the adiabatic approximation that leads to the divergence inthe eikonal description of breakup. It is also in good agreement with experiments [29, 30].Calculations performed in the usual eikonal model, and at the first-order of the perturbationtheory will also be presented to emphasize the effects of the correction. We focus on the case7f Be breakup. In that case, only the dipole term of the Coulomb interaction is significant[40]. We thus restrict the correction to that multipole. The perturbative correction thenreads [35] χ F O ( b , r ) = − η m f m P ωv (cid:20) K (cid:18) ωbv (cid:19) b b · s + iK (cid:18) ωbv (cid:19) z (cid:21) , (24)where K n are modified Bessel functions [37]. Of course, in other cases, like in B Coulombbreakup, the quadrupole term may no longer be negligible [14, 30], it should then be includedin the correction.
C. Breakup cross sections
To evaluate breakup cross sections within the CCE we proceed as explained in Ref. [29],replacing the DEA breakup amplitude by S ( m ) kljm ( b ) = e i ( σ l + δ lj − lπ/ χ CPT ) D φ ljm ( E ) (cid:12)(cid:12)(cid:12) e iχ N (cid:16) e iχ C − iχ C + iχ F O (cid:17)(cid:12)(cid:12)(cid:12) φ l j m ( E ) E , (25)where σ l is the Coulomb phase shift [37]. The breakup amplitudes for the usual eikonalmodel are obtained in the same way but without the correction.In the following, we consider two breakup observables. The first is the breakup crosssection as a function of the c - f relative energy E after dissociation [see Eq. (52) of Ref. [29]] dσ bu dE = 4 µ cf ~ k j + 1 X m X ljm Z ∞ bdb | S ( m ) kljm ( b ) | . (26)This energy distribution is the observable usually measured in breakup experiments [11, 12].It corresponds to an incoherent sum of breakup probabilities computed at each bdP bu dE ( E, b ) = 4 µ cf ~ k j + 1 X m X ljm | S ( m ) kljm ( b ) | . (27)The second breakup observable is the parallel-momentum distribution [see Eq. (53) ofRef. [29]] dσ bu dk k = 8 π j + 1 X m Z ∞ bdb Z ∞| k k | dkk X νm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X lj ( lIm − νν | jm ) Y m − νl ( θ k , S ( m ) kljm ( b ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (28)where θ k = arccos( k/k k ) is the colatitude of the c - f relative wavevector k after breakup.Contrary to the energy distribution, the parallel-momentum distribution corresponds to acoherent sum of breakup amplitudes. This observable is therefore sensitive to interferencesbetween different partial waves. Consequently, it constitutes a particularly severe test forreaction models. 8 II. NUMERICAL ASPECTS
For these calculations, we use the same description of Be as in Ref. [41]. The halonucleus is seen as a neutron loosely bound to a Be core in its 0 + ground state. The Be-ninteraction is simulated by a Woods-Saxon potential plus a spin-orbit coupling term (seeSec. IV A of Ref. [41]). The potential is adjusted to reproduce the first three levels of the Be spectrum. The
12 + ground state is seen as a 1 s / − excited stateis described by a 0 p /
52 + resonance at 1.274 MeVabove the one-neutron separation threshold is simulated in the d / Be-Pb potential is scaled from a parametri-sation of Bonin et al. [43] that describes elastic scattering of 699 MeV α particles on lead[potential (1) in Table III of Ref. [27]]. For the Be-C interaction, we use the potentialdeveloped by Al-Khalili, Tostevin, and Brooke, which reproduces the elastic scattering of Be on C at 59.4 MeV/nucleon [44] (potential ATB in Table III of Ref. [41]). In both cases,we neglect the possible energy dependence of the potential. We model the neutron-targetinteraction with the Becchetti and Greenlees parametrisation [45].To evaluate the breakup amplitude (25) within the CCE or the usual eikonal model, weneed to compute the eikonal phase (15). The nuclear part is evaluated numerically, whilethe Coulomb part is obtained from its analytic expression (17). The numerical integral over Z is performed on a uniform mesh from Z min = −
20 fm up to Z max = 20 fm with step∆ Z = 1 fm. The corrected phase (23) is then numerically expanded into multipoles of rank λ . We use a Gauss quadrature on the unit sphere similar to the one considered to solve thetime-dependent Schr¨odinger equation in Ref. [27]. The number of points along the colatitudeis set to N θ = 12, and the number of points along the azimuthal angle is N ϕ = 30. Unlessotherwise stated, we perform all calculations with multipoles up to λ max = 12.The eigenfunctions of the projectile Hamiltonian H (1) are computed numerically withthe Numerov method using 1000 radial points equally spaced from r = 0 up to r = 100 fm.The same grid is used to compute the radial integral in Eq. (25). For Coulomb (nuclear)breakup, the integrals over b appearing in Eqs. (26) and (28) are performed numericallyfrom b = 0 up to b = 300 (100) fm with a step ∆ b = 0 . N θ = 8 (12) points along the colatitude θ , and N ϕ = 15 (23) pointsalong the azimuthal angle ϕ . This corresponds to an angular basis that includes all possiblespherical harmonics up to l = 7 (11). The radial variable r is discretized on a quasiuniformmesh that contains N r = 800 (600) points and extends up to r N r = 800 (600) fm. The timepropagation is performed with a second-order approximation of the evolution operator. It isstarted at t in = −
20 (10) ~ / MeV with the projectile in its initial bound state, and is stoppedat t out = 20 (10) ~ / MeV ( t = 0 corresponds to the time of closest approach). The time stepis set to ∆ t = 0 . ~ / MeV in both Coulomb and nuclear cases.The evolution calculations are performed for different values of b . These values range from0 up to 300 (100) fm with a step ∆ b varying from 0.5 (0.25) fm to 5.0 (2.0) fm, depending9 OEik.CCEDEA b (fm) d P bu / d E ( M e V − ) b (fm) d P bu / d E ( M e V − ) − − − − FIG. 2: Breakup probabilities as a function of transverse coordinate b for Be impinging on
Pbat 69 MeV/nucleon. Three energies E are shown: 0.5 MeV, 1.274 MeV, and 3.0 MeV. The resultsare obtained within DEA (full lines), CCE (dotted lines), usual eikonal approximation (dashedlines), and first-order perturbation theory (dash-dotted lines). The upper part displays the valuesat small b , while the lower part focuses on the asymptotic region. on b . The integrals over b are performed numerically. IV. BREAKUP OF BE ON PB AT 69 MEV/NUCLEON
We first consider the breakup of Be on lead at 69 MeV/nucleon, which corresponds tothe experiment of Fukuda et al. at RIKEN [12]. These data are fairly well reproduced by theDEA [29], that we use as reference calculation. Since we focus on the comparison of models,we do not display Fukuda’s measurements. A comparison with experiment would indeedrequire a convolution of our results, which would hinder the comparison between theories.In Fig. 2, we compare the breakup probability (27) obtained with the DEA (full lines),the CCE (dotted lines), the usual eikonal model (Eik., dashed lines), and the first-orderperturbation theory (FO, dash-dotted line). They are depicted as a function of the transversecoordinate b for three Be-n relative energies: E = 0 .
52 + resonance energy), and 3.0 MeV. The upper part of Fig. 2 displays the values at small b ,while the lower part, in a semilogarithmic scale, focuses on the asymptotic region.10ver the whole range in b , the CCE results are close to the DEA ones, and this at allenergies. This good agreement suggests the Coulomb correction to be valid for simulating thebreakup of loosely-bound nuclei on heavy targets. In particular, the CCE is superimposedto the DEA results in the asymptotic region. Obviously, the first-order perturbation theoryefficiently corrects the erroneous 1 /b asymptotic behavior of the usual eikonal model.At small b , the agreement between the CCE and DEA seems slightly less good. In par-ticular, at small energy, the corrected eikonal model overestimates the reference calculation.This is due to the far-field approximation used in the first-order perturbation correction. Thisapproximation provides a convenient analytical expression (24) of the phase χ F O . However,it is incorrect at small b : it diverges at b = 0. Nevertheless, in spite of that divergence,the CCE remains close to the DEA. This illustrates that the CCE can also be seen as away to include nuclear interactions within the first-order perturbation theory, and correctits ill-behavior at small b .The breakup cross section (26) computed with the four approximations is displayed inFig. 3(a) as a function of the Be-n relative energy E after dissociation. Contributions ofthe s , p , and d partial waves are shown separately in Fig. 3(b). The small bump at about1.25 MeV is due to the resonance in the d / χ F O .Interestingly, the agreement between CCE and DEA is better for the total cross sectionthan for each partial-wave contribution: The CCE p contribution is larger than the DEAone, while the CCE s and d contributions are smaller than the DEA ones. We interpret thisas a lack of couplings in the continuum in the CCE. In the DEA, these couplings depopulatethe p waves towards the s and d ones without modifying the total cross section [40]. Thedifferences between CCE and DEA partial-wave contributions suggest that this mechanismis hindered in the former.The wrong asymptotic behavior of the Coulomb eikonal phase (17) leads to a divergencein the calculation of the breakup cross sections. To evaluate the energy distribution withinthe usual eikonal model one needs to resort to a cutoff at large b . The cutoff proposed inRef. [38] [see also Eq. (21)] gives here b max = 71 fm. The corresponding cross section isdisplayed in Fig. 3(a) with a dashed line. Its energy dependence is strongly different fromthat of the reference calculation: it is too small at low energy and too large at high energy.The p contribution, which includes the diverging term of the Coulomb eikonal phase (17), isresponsible for that ill-behavior. Contrarily, the s and d contributions are superimposed onthose of the CCE. The use of the Coulomb correction therefore significantly improves theeikonal model when considering collisions with heavy targets.The cross section obtained within the first-order perturbation theory is shown in dot-dashed line. The nuclear interactions between the projectile and the target are described bya mere cutoff at b min = 15 fm. This value has been chosen to fit the DEA energy distributionin the region of the maximum. Here again, the shape of the cross section is very differentfrom that of the reference calculation. However, contrary to the usual eikonal model, itdecreases too quickly with the energy. Moreover, since only the dipole term of the Coulombinteraction is considered, only the p wave is reached from the s ground state, whereas s and d waves are significantly populated through nuclear interactions and higher-order effects.Note that a smaller cutoff b min , in better agreement with the usual choice that correspondsto the sum of the projectile and target radii, does not improve the agreement.11 O bmin=15Eik bmax=71CCEDEA d σ bu / d E ( b / M e V ) (a)Total1.61.41.210.80.60.40.20 E (MeV)(b) ps d FIG. 3: (a) Breakup cross sections for Be impinging on
Pb at 69 MeV/nucleon as a functionof the Be-n relative energy E . The results are obtained within the DEA, the CCE, the usualeikonal approximation with upper cutoff b max = 71 fm, and the first-order perturbation theorywith lower cutoff b min = 15 fm. (b) Contributions of the s , p , and d partial waves. We now consider the parallel-momentum distribution [see Eq. (28)]. This breakup observ-able is more sensitive to interferences and therefore constitutes a more severe test than theenergy distribution. The parallel-momentum distribution computed within the four modelsis displayed in Fig. 4.As in the previous cases, the CCE is in excellent agreement with the DEA in bothmagnitude and shape. We simply note that the former is slightly less asymmetric than thelatter, which is probably a signature of the lack of couplings in the continuum mentionedearlier. On the contrary, both the usual eikonal model and the first-order perturbationtheory lead to rather poor estimates of the momentum distribution. First, they lead to anerroneous magnitude of the cross section. The usual eikonal model gives too large a parallel-momentum distribution. This is related to the too slow decrease obtained for the energydistribution. On the contrary, the first-order perturbation gives too low a cross section;a defect due to the quick decrease in the energy distribution. Lowering the cutoff b min tocure this problem would then lead to too large an energy distribution in the peak region.Second, none of these models exhibits the asymmetry observed in the DEA. This absence ofasymmetry in parallel-momentum distributions of the breakup of loosely-bound projectiles isa well-known problem of the eikonal model [46]. It is fortunate that the Coulomb correction,combining two approximations that lead to perfectly symmetric momentum distributions,restores the asymmetry observed experimentally and in dynamical calculations.12 O b min = 15 fmEik. b max = 71 fmCCEDEA k k (fm − ) d σ bu / dk k ( b f m ) FIG. 4: Breakup cross sections for Be impinging on
Pb at 69 MeV/nucleon as a function of the Be-n relative parallel momentum k k . The figure displays the results obtained within the DEA,the CCE, the usual eikonal approximation with an upper cutoff b max = 71 fm, and the first-orderperturbation theory with a lower cutoff b min = 15 fm. λ max = 12 λ max = 8 λ max = 4 k k (fm − ) d σ bu / dk k ( b f m ) FIG. 5: Convergence of the multipole expansion in λ max of the CCE illustrated on the parallel-momentum distribution computed for Be impinging on
Pb at 69 MeV/nucleon.
Fig. 5 illustrates the convergence of the CCE with regard to the number of multipolesconsidered in the breakup computation. The parallel-momentum distributions obtained withmaximum multipolarities λ max = 4, 8, and 12 are displayed. Although all three calculationsare close to one another, λ max = 4 has not yet converged: there remains some 4% differencewith the other two at the maximum. On the contrary, the difference between λ max = 8and 12 is insignificant (about 0.5%). This shows the necessity to include a large number ofpartial waves in dynamical calculations. Note that other breakup observables converge witha lower number of multipoles. In particular, the energy distribution requires only λ max = 4to reach satisfactory convergence.These results confirm the ability of the Coulomb correction to reliably reproduce breakupobservables for collisions of loosely-bound projectiles on heavy targets. It reproduces dy-namical calculations with an accuracy that is unreachable within the usual eikonal modelor the first-order perturbation theory, on which it is based.13 . BREAKUP OF BE ON C AT 67 MEV/NUCLEON
To complete this analysis of the Coulomb correction, we investigate its effect in nuclearinduced breakup. The usual eikonal description of such reactions is known to give excellentresults [17, 29]. The Coulomb interaction between the projectile and the target plays thena minor role and we expect the correction (23) to have much less influence than in theCoulomb breakup case.For this analysis, we consider the breakup of Be on a carbon target at 67 MeV/nucleon,which corresponds to the experiment of Fukuda et al. [12]. The DEA is in excellent agree-ment with Fukuda’s data [29], and therefore constitutes our reference calculation. For thesame reasons as in the previous section, we do not compare directly our calculations withexperiment.Fig. 6 displays the breakup probability (27) obtained at three energies E = 0 .
5, 1.274,and 3.0 MeV within the DEA (full lines), the CCE (dotted lines), and the usual eikonalmodel (dashed lines). Since this reaction is nuclear dominated, we no longer display theresult of the first-order perturbation theory. The upper part of Fig. 6 displays the breakupprobability at small b , while the lower part emphasizes the asymptotic behavior of P bu in asemilogarithmic plot.In this case, all three reaction models lead to similar results. This confirms the validityof the adiabatic approximation in the eikonal description of nuclear-dominated reactions.The difference between the DEA and the other two models is indeed rather small. Onlyat E = 1 .
274 MeV, the energy of the
52 + resonance, does it become significant (up to 10%difference in the vicinity of the peak at b ∼ b = 20 fm, the usual eikonal model and the CCE remain very close to one another,confirming the small role played by the Coulomb interaction in the dissociation. At larger b , where only Coulomb is significant, we observe the 1 /b behavior of the usual eikonalmodel. This ill-behavior is corrected using the CCE, whose breakup probabilities are nearlysuperimposed on the DEA ones in the asymptotic region. However, since this correctionaffects breakup probabilities at two or three orders of magnitude below the maximum, wedo not expect it to significantly influence breakup observables.The breakup cross sections computed within the three models are plotted as functionsof the energy E in Fig. 7. The contributions to the total cross section of the partial waves s , p , and d are shown as well. The large peak at about 1.25 MeV is the signature of thesignificant enhancement of the breakup process by the d / p partialwave, where the Coulomb correction is performed, no significant difference is observed. Thisconfirms that the correction of the eikonal model is not necessary for nuclear-dominated14 ik.CCEDEA b (fm) d P bu / d E ( M e V − ) b (fm) d P bu / d E ( M e V − ) − − − − − − FIG. 6: Breakup probabilities as a function of transverse coordinate b for Be impinging on C at67 MeV/nucleon. Three energies E are shown: 0.5 MeV, 1.274 MeV, and 3.0 MeV. The results areobtained within the DEA (full lines), CCE (dotted lines), and usual eikonal (dashed lines) models.The upper part displays the values at small b , while the lower part emphasizes the behavior in theasymptotic region. reactions due to the small role played by the Coulomb interaction. The cutoff in b proposedin Ref. [38] is therefore sufficient.The parallel-momentum distributions obtained with the three models are displayed inFig. 8. As already mentioned, this observable is a more severe test for reaction models thanthe energy distribution. We observe significant differences between the DEA and the othertwo models. As in the case of Coulomb breakup, the DEA leads to an asymmetric parallel-momentum distribution: The DEA distribution is shifted toward negative k k and presentsa more developed tail on the negative k k side, as observed in Ref. [46].As for the previous observable, the CCE and usual eikonal models lead to very similarparallel-momentum distributions. These distributions are symmetric. As mentioned earlier,this symmetry is due to the lack of dynamical effects in the eikonal description of reactions.Contrary to the Coulomb case, the correction (23) is not able to restore this asymmetry.It indicates that these dynamical effects result from the nuclear interactions between theprojectile and the target.The convergence of the CCE model with the number of multipoles is illustrated in Fig. 9for the parallel-momentum distribution. The CCE distributions computed with λ max = 4–15 ik. b max = 70 fmCCEDEA E (MeV) d σ bu / d E ( b / M e V ) Total ps d
FIG. 7: Breakup cross sections for Be impinging on C at 67 MeV/nucleon as a function of the Be-n relative energy E . Results are obtained within the DEA, the CCE, and the usual eikonalmodel with an upper cutoff b max = 70 fm. Contributions of the s , p , and d partial waves are shownas well. Eik. b max = 70 fmCCEDEA k k (fm − ) d σ bu / dk k ( b f m ) FIG. 8: Breakup cross sections for Be impinging on C at 67 MeV/nucleon as a function of the Be-n relative parallel momentum k k . Results are obtained within the DEA, the CCE, and theusual eikonal approximation with an upper cutoff b max = 70 fm.
12 are displayed. The convergence is much slower than for Coulomb-dominated breakup(see Fig. 5). The relative difference between λ max = 10 and λ max = 12 is indeed about3% at the maximum. This is due to the rapid variation of the nuclear potential with theprojectile-target coordinates. It confirms the need of a larger number of partial waves inthe dynamical calculation of nuclear-dominated dissociation. Note that the convergence isfaster for the energy distribution. For that observable, an acceptable convergence is reachedat λ max = 6. 16 = 12 λ = 10 λ = 8 λ = 6 λ = 4 k k (fm − ) d σ bu / dk k ( b f m ) FIG. 9: Convergence in λ max of the CCE illustrated on the parallel-momentum distribution for thebreakup of Be on C at 67 MeV/nucleon.
VI. CONCLUSION
The eikonal description of reactions is a useful tool to simulate breakup and strippingreactions on light targets at intermediate and high energies [16, 17, 29]. This model is inter-esting because of its relative simplicity in implementation and interpretation with respectto other elaborate models, like CDCC or DEA. Unfortunately, it suffers from a divergenceproblem associated with the treatment of the Coulomb interaction between the projectileand the target. This divergence is due to the incompatibility of the adiabatic, or sudden, ap-proximation which is made in the usual eikonal model, and the infinite range of the Coulombinteraction. One way to cure this problem is not to make this adiabatic approximation. Thisleads to the DEA [28, 29]. However, like other elaborate models, the DEA is computationallyexpensive. Another way to solve this problem is to substitute the diverging Coulomb phaseat the first-order of the eikonal model by the corresponding first-order of the perturbationtheory [34, 35].In this work, we study the validity of this Coulomb correction by comparing it to theDEA, which does not present the divergence problem of the usual eikonal model. Thechosen test cases are the dissociation of Be on Pb and C at about 70 MeV/nucleon. Thesecorrespond to RIKEN experiments [11, 12] that are well reproduced by the DEA [29].In the case of the Coulomb breakup, the CCE gives results in excellent agreement with theDEA. The combination of the eikonal model with the first-order perturbation theory indeedsolves the divergence problem due to the Coulomb interaction. Moreover, it correctly takesinto account the nuclear interaction between the projectile and target. The breakup observ-ables (energy and parallel-momentum distributions) obtained within the DEA are accuratelyreproduced using the CCE. This agreement is obtained while both CCE ingredients—usualeikonal and first-order perturbation—fail to describe the reaction. First they both requirea rather arbitrary upper or lower cutoff in b in order not to diverge. Second they do not re-produce the shape of the breakup cross sections. In particular the CCE gives an asymmetricparallel-momentum distribution, in agreement with the dynamical calculation. Contrarily,both the usual eikonal and the perturbative models lead to perfectly symmetric distributions.This suggests that CCE restores dynamical effects that are missing in its ingredients.The Coulomb correction has much less effect on the nuclear-dominated breakup. This17as expected because of the much smaller influence of the Coulomb interaction in reactionsinvolving light targets. This result indicates that in this case the correction is not essential.It also implies that the CCE suffers the same lack of dynamical effects as the usual eikonalmodel in nuclear dominated reactions.The CCE successfully combines the positive aspects of both the eikonal model and thefirst-order perturbation theory. It allows describing accurately the nuclear interaction whilecorrectly reproducing Coulomb-induced effects. Moreover the CCE restores some of thedynamical effects, which are totally absent in other simple models. It therefore providesa reliable description of the breakup of loosely-bound projectiles at intermediate and highenergies. Its simplicity in use and interpretation suggests it as a competitive alternative tomore elaborate models to describe the breakup of Borromean nuclei. Acknowledgments
This work has been done in the framework of the agreement between the Japan Societyfor the Promotion of Science (JSPS) and the Fund for Scientific Research of Belgium (F.R. S.-FNRS). Y. S. acknowledges the support of the Grant for the Promotion of NiigataUniversity Research Projects (2005–2007). P. C. acknowledges travel support of the Fondsde la Recherche Scientifique Collective (FRSC) and the support of the F. R. S.-FNRS. Thistext presents research results of the Belgian program P6/23 on interuniversity attractionpoles initiated by the Belgian-state Federal Services for Scientific, Technical and CulturalAffairs (FSTC). [1] P. G. Hansen, A. S. Jensen, and B. Jonson, Annu. Rev. Nucl. Part. Sci , 591 (1995).[2] I. Tanihata, J. Phys. G , 157 (1996).[3] B. Jonson, Phys. Rep. , 1 (2004).[4] P. G. Hansen and B. Jonson, Europhys. Lett. , 409 (1987).[5] M. V. Zhukov, B. V. Danilin, D. V. Fedorov, J. M. Bang, I. J. Thompson, and J. S. Vaagen,Phys. Rep. , 151 (1993).[6] I. Tanihata, H. Hamagaki, O. Hashimoto, S. Nagamiya, Y. Shida, N. Yoshikawa, O. Yamakawa,K. Sugimoto, T. Kobayashi, D. E. Greiner, et al., Phys. Lett. B , 380 (1985).[7] I. J. Thompson and Y. Suzuki, Nucl. Phys. A 693 , 424 (2001).[8] J. Al-Khalili and F. M. Nunes, J. Phys. G , R89 (2003).[9] G. Baur, K. Hencken, and D. Trautmann, Prog. Part. Nucl. Phys. , 487 (2003).[10] T. Kobayashi, S. Shimoura, I. Tanihata, K. Katori, K. Matsuta, T. Minamisono, K. Sugimoto,W. M¨uller, D. L. Olson, T. L. M. Symon, et al., Phys. Lett. B , 51 (1989).[11] T. Nakamura, S. Shimoura, T. Kobayashi, T. Teranishi, K. Abe, N. Aoi, Y. Doki, M. Fujimaki,N. Inabe, N. Iwasa, et al., Phys. Lett. B , 296 (1994).[12] 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).[13] S. Typel and G. Baur, Phys. Rev. C , 2104 (1994).[14] H. Esbensen and G. F. Bertsch, Nucl. Phys. A 600 , 37 (1996).[15] J. A. Tostevin, S. Rugmai, and R. C. Johnson, Phys. Rev. C , 3225 (1998).
16] R. J. Glauber, in
Lecture in Theoretical Physics , edited by W. E. Brittin and L. G. Dunham(Interscience, New York, 1959), vol. 1, p. 315.[17] Y. Suzuki, R. G. Lovas, K. Yabana, and K. Varga,
Structure and Reactions of Light ExoticNuclei (Taylor and Francis, London, 2003).[18] C. A. Bertulani and P. Danielewicz,
Introduction to Nuclear Reactions (Institute of PhysicsPublishing, Bristol, 2004).[19] M. Kamimura, M. Yahiro, Y. Iseri, H. Kameyama, Y. Sakuragi, and M. Kawai, Prog. Theor.Phys. Suppl. , 1 (1986).[20] N. Austern, Y. Iseri, M. Kamimura, M. Kawai, G. Rawitscher, and M. Yahiro, Phys. Rep. , 125 (1987).[21] J. A. Tostevin, F. M. Nunes, and I. J. Thompson, Phys. Rev. C , 024617 (2001).[22] T. Kido, K. Yabana, and Y. Suzuki, Phys. Rev. C , R1276 (1994).[23] T. Kido, K. Yabana, and Y. Suzuki, Phys. Rev. C , 2296 (1996).[24] H. Esbensen, G. F. Bertsch, and C. A. Bertulani, Nucl. Phys. A 581 , 107 (1995).[25] S. Typel and H. H. Wolter, Z. Naturforsch. Teil A , 63 (1999).[26] V. S. Melezhik and D. Baye, Phys. Rev. C , 3232 (1999).[27] P. Capel, D. Baye, and V. S. Melezhik, Phys. Rev. C , 014612 (2003).[28] D. Baye, P. Capel, and G. Goldstein, Phys. Rev. Lett. , 082502 (2005).[29] G. Goldstein, D. Baye, and P. Capel, Phys. Rev. C , 024602 (2006).[30] G. Goldstein, P. Capel, and D. Baye, Phys. Rev. C , 024608 (2007).[31] N. C. Summers, F. M. Nunes, and I. J. Thompson, Phys. Rev. C , 014606 (pages 12) (2006).[32] T. Matsumoto, E. Hiyama, K. Ogata, Y. Iseri, M. Kamimura, S. Chiba, and M. Yahiro, Phys.Rev. C , 061601(R) (pages 5) (2004).[33] M. Rodriguez-Gallardo, J. M. Arias, J. Gomez-Camacho, R. C. Johnson, A. M. Moro, I. J.Thompson, and J. Tostevin, arXiv:0710.0769 (2007).[34] J. Margueron, A. Bonaccorso, and D. M. Brink, Nucl. Phys. A 720 , 337 (2003).[35] B. Abu-Ibrahim and Y. Suzuki, Prog. Theor. Phys. , 1013 (2004); ibid. , , 901 (2005).[36] K. Alder and A. Winther, Electromagnetic Excitation (North-Holland, Amsterdam, 1975).[37] M. Abramowitz and I. A. Stegun,
Handbook of Mathematical Functions (Dover, New-York,1970).[38] B. Abu-Ibrahim and Y. Suzuki, Phys. Rev. C , 034608 (2000).[39] H. Esbensen and C. A. Bertulani, Phys. Rev. C , 024605 (2002).[40] P. Capel and D. Baye, Phys. Rev. C , 044609 (2005).[41] P. Capel, G. Goldstein, and D. Baye, Phys. Rev. C , 064605 (2004).[42] S. Typel and R. Shyam, Phys. Rev. C , 024605 (2001).[43] B. Bonin, N. Alamanos, B. Berthier, G. Bruge, H. Faraggi, J. C. Lugol, W. Mittig, L. Papineau,A. I. Yavin, J. Arvieux, et al., Nucl. Phys. A 445 , 381 (1985).[44] J. S. Al-Khalili, J. A. Tostevin, and J. M. Brooke, Phys. Rev. C , R1018 (1997).[45] F. D. Becchetti, Jr. and G. W. Greenlees, Phys. Rev. , 1190 (1969).[46] J. A. Tostevin, D. Bazin, B. A. Brown, T. Glasmacher, P. G. Hansen, V. Maddalena, A. Navin,and B. M. Sherrill, Phys. Rev. C , 024607 (2002)., 024607 (2002).