Breakup reaction models for two- and three-cluster projectiles

Breakup reaction models for two- and three-cluster projectiles

D. Baye and P. Capel D. Baye Physique Quantique, C.P. 165/82 and Physique Nucléaire Théorique et Physique Mathématique, C.P. 229, Université Libre de Bruxelles, B 1050 Brussels, Belgium, 22email: dbaye@ulb.ac.beP. Capel National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing MI-48824, USA, 44email: capel@nscl.msu.edu
Abstract

Breakup reactions are one of the main tools for the study of exotic nuclei, and in particular of their continuum. In order to get valuable information from measurements, a precise reaction model coupled to a fair description of the projectile is needed. We assume that the projectile initially possesses a cluster structure, which is revealed by the dissociation process. This structure is described by a few-body Hamiltonian involving effective forces between the clusters. Within this assumption, we review various reaction models. In semiclassical models, the projectile-target relative motion is described by a classical trajectory and the reaction properties are deduced by solving a time-dependent Schrödinger equation. We then describe the principle and variants of the eikonal approximation: the dynamical eikonal approximation, the standard eikonal approximation, and a corrected version avoiding Coulomb divergence. Finally, we present the continuum-discretized coupled-channel method (CDCC), in which the Schrödinger equation is solved with the projectile continuum approximated by square-integrable states. These models are first illustrated by applications to two-cluster projectiles for studies of nuclei far from stability and of reactions useful in astrophysics. Recent extensions to three-cluster projectiles, like two-neutron halo nuclei, are then presented and discussed. We end this review with some views of the future in breakup-reaction theory.

1 Introduction

The advent of radioactive ion beams has opened a new era in nuclear physics by providing the possibility to study nuclei far from stability. In particular the availability of these beams favoured the discovery of halo nuclei Tan85b (). Due to the very short lifetime of exotic nuclei, this study cannot be performed through usual spectroscopic techniques and one must resort to indirect methods. Breakup is one of these methods. In this reaction, the projectile under analysis dissociates into more elementary components through its interaction with a target. Many such experiments have been performed with the hope to probe exotic nuclear structures far from stability Tan96 (); Jon04 ().

In order to get valuable information from breakup measurements, one must have not only a fair description of the projectile, but also an accurate reaction model. At present, a fully microscopic description of the reaction is computationally unfeasible. Simplifying assumptions are necessary. First, we will discuss only elastic breakup, i.e. a dissociation process leaving the target unchanged in its ground state. Other channels are simulated through the use of optical potentials. Second, we assume a cluster structure for the projectile. The projectile ground state is assumed to be a bound state of the clusters appearing during the breakup reaction. The bound and continuum states of the projectile are thus described by a few-body Hamiltonian involving effective forces between the constituent clusters. Theoretical reaction models are therefore based on this cluster description of the projectile and effective cluster-cluster and cluster-target interactions.

Even within these simplifying model assumptions, a direct resolution of the resulting many-body Schrödinger equation is still not possible in most cases. In this article, we thus review various approximations that have been developed up to now.

We begin with the models based on the semiclassical approximation AW75 () in which the projectile-target relative motion is described by a classical trajectory. This approximation is valid at high energies. It leads to the resolution of a time-dependent Schrödinger equation. In a primary version, the time-dependent equation was solved at the first order of the perturbation theory AW75 (). Then, as computers became more powerful, it could be solved numerically KYS94 (); EBB95 (); TW99 (); MB99 (); LSC99 (); CBM03c (). We present both versions indicating their respective advantages and drawbacks.

We then describe the eikonal approximation Glauber () and its variants. The principle is to calculate the deviations from a plane-wave motion which are assumed to be weak at high energy. By comparison with the semiclassical model, it is possible to derive the dynamical eikonal approximation (DEA) that combines the advantages of both models BCG05 (); GBC06 (). The standard eikonal approximation is obtained by making the additional adiabatic or sudden approximation, which neglects the excitation energies of the projectile. With this stronger simplifying assumption, the final state only differs from the initial bound state by a phase factor. This approach is mostly used to model reactions on light targets at intermediate and high energies. Its drawback is that the Coulomb interaction leads to a divergence of breakup cross sections at forward angles. This problem can be solved using a first-order correction of the Coulomb treatment within the eikonal treatment. A satisfactory approximation of the DEA can then be derived MBB03 (); AS04 (): the Coulomb-corrected eikonal approximation (CCE), which remains valid for breakup on heavy targets. It reproduces most of the results of the DEA, although its computational time is significantly lower CBS08 () which is important for the study of the breakup of three-cluster projectiles.

Finally, we present the continuum-discretized coupled-channel method (CDCC) Kam86 (); TNT01 (), in which the full projectile-target Schrödinger equation is solved approximately, by representing the continuum of the projectile with square-integrable states. This model leads to the numerical resolution of coupled-channel equations, and is suited for low- as high-energy reactions.

All the aforementioned models have been developed initially for two-body projectiles. However, the physics of three-cluster systems, like two-neutron halo nuclei, is the focus of many experimental studies and must also be investigated with these models. We review here the various efforts that have been made in the past few years to extend breakup models to three-cluster projectiles BCD09 (); EMO09 (); RAG09 ().

In Sec. 2, we specify the general theoretical framework within which the projectile is described. The semiclassical model and approximate resolutions of the time-dependent Schrödinger equation are described in Sec. 3. Sec. 4 presents the eikonal approximation as well as the related DEA and CCE models. Next, in Sec. 5, the CDCC method is developed. In Sec. 6, we review applications of breakup reactions to two-body projectiles. In particular, we emphasize the use of breakup to study nuclei far from stability and as an indirect way to infer cross sections of reactions of astrophysical interest. Sec. 7 details the recent efforts made to extend various reaction models to three-body projectiles. We end this review by presenting some views of the future in breakup-reaction theory.

2 Projectile and reaction models

We consider the reaction of a projectile of mass and charge impinging on a target of mass and charge . The projectile is assumed to exhibit a structure made of clusters with masses and charges ( and ). Its internal properties are described by a Hamiltonian , depending on a set of internal coordinates collectively represented by notation . With the aim of preserving the generality of the presentation of the reaction models, we do not specify here the expression of . Details are given in Secs. 6 and 7, where applications for the breakup of two- and three-body projectiles are presented.

The states of the projectile are thus described by the eigenstates of . For total angular momentum and projection , they are defined by

(2.1)

where is the energy in the projectile centre-of-mass (c.m.) rest frame with respect to the dissociation threshold into clusters. Index symbolically represents the set of all additional quantum numbers that depend on the projectile structure, like spins and relative orbital momenta of the clusters. Its precise definition depends on the number of clusters and on the model selected when defining . We assume these numbers to be discrete, though some may be continuous in some representations when there are more than two clusters. To simplify the notation, the parity of the eigenstates of is understood. In the following, any sum over implicitly includes a sum over parity.

The negative-energy solutions of Eq. (2.1) correspond to the bound states of the projectile. They are normed to unity. The positive-energy states describe the broken-up projectile with full account of the interactions between the clusters. They are orthogonal and normed according to . To describe final states when evaluating breakup cross sections, we also consider the incoming scattering states . They correspond to positive-energy states of describing the clusters moving away from each other in the projectile c.m. frame with specific asymptotic momenta and spin projections. These momenta are not independent, since the sum of the asymptotic kinetic energies of the clusters is the positive energy . However, within that condition, their directions and, if , their norms can vary. By , we symbolically denote these directions and wave numbers, as well as the projections of the spins of the clusters. These incoming scattering states are thus solutions of the Schrödinger equation

(2.2)

They can be expanded into a linear combination of the eigenstates of Eq. (2.1) with the same energy as

(2.3)

where the coefficients depend on the projectile structure. These scattering states are normed following .

The interactions between the projectile constituents and the target are usually simulated by optical potentials chosen in the literature or obtained by a folding procedure. Within this framework the description of the reaction reduces to the resolution of an -body Schrödinger equation

(2.4)

where is the coordinate of the projectile centre of mass relative to the target, is the corresponding momentum, is the projectile-target reduced mass, and is the total energy in the projectile-target c.m. frame. The projectile-target interaction is expressed as the sum of the optical potentials (including Coulomb) that simulate the interactions between the projectile constituents and the target,

(2.5)

where is the relative coordinate of the projectile cluster with respect to the target.

The projectile being initially bound in the state of negative energy , we look for solutions of Eq. (2.4) with an incoming part behaving asymptotically as

(2.6)

where is the component of in the incident-beam direction. The wavenumber of the projectile-target relative motion is related to the total energy by

(2.7)

The - Sommerfeld parameter is defined as

(2.8)

where is the initial - relative velocity.

A first idea that may come to mind is to solve Eq. (2.4) exactly, e.g., within the Faddeev framework or its extensions. However, the infinite range of the Coulomb interaction between the projectile and the target renders the standard equations ill-defined. Only recently significant progress has been made. For example, in Refs. DFS05c (); DFS05l (), this problem is tackled by using an appropriate screening of the Coulomb force. This technique has been used to successfully describe the elastic scattering and breakup of the deuteron on various targets. However, it has long been limited to light targets (see LABEL:DMC07 for a recent extension to a heavier target). To obtain a model that is valid for all types of target, one must still resort to approximations in the resolution of Eq. (2.4). These approximations are made in the treatment of the projectile-target relative motion, like in the semiclassical (Sec. 3) or eikonal (Sec. 4) approximations, or by using a discretized continuum, like in the CDCC method (Sec. 5).

3 Semiclassical approximation

3.1 Time-dependent Schrödinger equation

The semiclassical approximation relies on the hypothesis that the projectile-target relative motion can be efficiently described by a classical trajectory AW75 (). It is thus valid when the de Broglie wavelength is small with respect to the impact parameter characterizing the trajectory, , i.e. when the energy is large enough. Along that trajectory, the projectile experiences a time-dependent potential that simulates the Coulomb and nuclear fields of the target. The internal structure of the projectile, on the contrary, is described quantum-mechanically by the Hamiltonian . This semiclassical approximation leads to the resolution of the time-dependent equation

(3.1)

The time-dependent potential is obtained from the difference between the projectile-target interaction (2.5) and the potential that defines the classical trajectory

(3.2)

The potential acts as a - scattering potential that bends the trajectory, but does not affect the projectile internal structure. Its interest lies in the fact that decreases faster than . Its effect amounts to changing the phase of the wave function. Usually it is chosen to be the Coulomb potential between the projectile centre of mass and the target, but it may include a nuclear component. At sufficiently high energy, the trajectory is often approximated by a straight line.

For each impact parameter , Eq. (3.1) has to be solved with the initial condition that the projectile is in its ground state,

(3.3)

For each trajectory, the time-dependent wave function must be calculated for the different possible values of .

3.2 Cross sections

From the output of the resolution of Eq. (3.1), the probability of being in a definite state of the projectile can be obtained by projecting the final wave function onto the corresponding eigenstate of . One can for example compute the elastic scattering probability

(3.4)

This probability depends only on the norm of the impact parameter because the time-dependent wave function depends on the orientation of , i.e. on the azimuthal angle , only through a phase that cancels out in the calculation of . From this probability, the cross section for the elastic scattering in direction is obtained as

(3.5)

where is given by the classical relation between the scattering angle and the impact parameter derived from potential . The factor is the elastic scattering cross section obtained from . In most cases is generated from the Coulomb interaction and is thus the - Rutherford cross section.

Likewise, a general breakup probability density can be computed by projecting the final wave function onto the ingoing scattering states of ,

(3.6)

After integration and summation over , the breakup probability per unit energy reads

(3.7)

Similarly to Eq. (3.5), a differential cross section for the breakup of the projectile is given by

(3.8)

The breakup cross section can then be obtained by summing the breakup probability over all impact parameters

(3.9)

Because of the trajectory hypothesis of the semiclassical approximation, the impact parameter is a classical variable. Therefore, no interference between the different trajectories can appear. This is the major disadvantage of that technique since quantal interferences can play a significant role in reactions, in particular in those which are nuclear dominated.

3.3 Resolution at the first order of the perturbation theory

In the early years of the semiclassical approximations, Eq. (3.1) was solved at the first order of the perturbation theory AW75 (). This technique, due to Alder and Winther, was applied to analyze the first Coulomb-breakup experiments of halo nuclei Nak94 ().

The time-dependent wave function is expanded upon the basis of eigenstates of in Eq. (2.1). At the first order of the perturbation theory, the resulting equation is solved by considering that is small. With the initial condition (3.3), the wave function at first order is given by AW75 (); Ba08 ()

(3.10)

Following Eq. (3.6), the general breakup probability density reads

(3.11)

where . The breakup probability per unit energy reads

(3.12)

With Eq. (3.10), exact expressions can be calculated when considering a purely Coulomb - interaction for straight-line trajectories in the far-field approximation EB02 (), i.e. by assuming that the charge densities of the projectile and target do not overlap during the collision. One obtains

(3.13)

where are the electric multipoles operators of rank , and are time integrals (see, e.g., Eq. (13) of LABEL:CB05) that can be evaluated analytically as EB02 ()

(3.14)

where is a modified Bessel function AS70 ().

If only the dominant dipole term E1 of the interaction is considered, the breakup probability (3.12) reads SLY03 ()

(3.15)

The last factor is the dipole strength function per energy unit SLY03 (),

(3.16)

Since modified Bessel functions decrease exponentially, the asymptotic behaviour of for is proportional to .

In the case of a purely Coulomb - interaction, the first order of the perturbation theory exhibits many appealing aspects. First, it can be solved analytically. Second, the dynamics part () and structure part (matrix elements of ) are separated in the expression of the breakup amplitudes (3.13). This first-order approximation has therefore often been used to analyze Coulomb-breakup experiments by assuming pure E1 breakup (see, e.g., LABEL:Nak94). However, as will be seen later, higher-order and nuclear-interaction effects are usually not negligible, and a proper analysis of experimental data requires a more sophisticated approximation.

3.4 Numerical resolution

The time-dependent Schrödinger equation can also be solved numerically. Various groups have developed algorithms for that purpose KYS94 (); EBB95 (); TW99 (); MB99 (); LSC99 (); CBM03c (); KYS96 (); Fal02 (). They make use of an approximation of the evolution operator applied iteratively to the initial bound state wave function following the scheme

(3.17)

Although higher-order algorithms exist (see, e.g., LABEL:BGC03), all practical calculations are performed with second-order approximations of . Various expressions of this approximation exist, depending mainly on the way of representing the time-dependent projectile wave function. However they are in general similar to CBM03c ()

(3.18)

With this expression, the time-dependent potential can be treated separately from the time-independent Hamiltonian , which greatly simplifies the calculation of the time evolution when the wave functions are discretized on a mesh CBM03c ().

The significant advantage of this technique over the first order of perturbation is that it naturally includes higher-order effects. Moreover, the nuclear interaction between the projectile and the target can be easily added in the numerical scheme TS01r (). However, the dynamical and structure evolutions being now more deeply entangled, the analysis of the numerical resolution of the Schrödinger equation is less straightforward than its first-order approximation. The numerical technique is also much more time-consuming than the perturbation one. The first order of the perturbation theory therefore remains a useful tool to qualitatively analyze calculations of Coulomb-dominated reactions performed with more elaborate models. Moreover, as will be seen in Sec. 4.4, it can be used to correct the erroneous treatment of the Coulomb interaction within the eikonal description of breakup reactions.

Fig. 1 illustrates the numerical resolution of the time-dependent Schrödinger equation for the Coulomb breakup of Be on lead at 68 MeV/nucleon CBM03c (). It shows the breakup cross section as a function of the relative energy between the Be core and the halo neutron after dissociation. The full line corresponds to the calculation with both Coulomb and nuclear - interactions. The dashed line is the result for a purely Coulomb potential, in which the nuclear interaction is simulated by an impact parameter cutoff at  fm. A calculation performed with an impact parameter cutoff at  fm simulating a forward-angle cut is plotted as a dotted line. The experimental data from LABEL:Nak94 are multiplied by a factor of 0.85 as suggested in LABEL:FNA04 after a remeasurement.

Figure 1: Breakup cross section of Be on Pb at 68 MeV/nucleon as a function of the relative energy between the Be core and the neutron. Calculations are performed within the semiclassical approximation with or without nuclear interaction CBM03c (). Experimental data Nak94 () are scaled by 0.85 FNA04 (). Reprinted figure with permission from LABEL:CBM03c. Copyright (2003) by the American Physical Society.

This example shows the validity of the semiclassical approximation to describe breakup observables in the projectile c.m. frame for collisions at intermediate energies. It also confirms that for heavy targets the reaction is strongly dominated by the Coulomb interaction. The inclusion of optical potentials to simulate the nuclear - interactions indeed only slightly increases the breakup cross section at large energy. This shows that Coulomb-breakup calculations are not very sensitive to the uncertainty related to the choice of the optical potentials. Nevertheless, since optical potentials can be very easily included in the numerical resolution of the time-dependent Schrödinger equation, they should be used so as to avoid the imprecise impact-parameter cutoff necessary in purely Coulomb calculations.

4 Eikonal approximations

4.1 Dynamical eikonal approximation

Let us now turn to a purely quantal treatment providing approximate solutions of the Schrödinger equation (2.4). At sufficiently high energy, the projectile is only slightly deflected by the target. The dominant dependence of the -body wave function on the projectile-target coordinate is therefore in the plane wave contributing to the incident relative motion (2.6). The main idea of the eikonal approximation is to factorize that plane wave out of the wave function to define a new function whose variation with is expected to be small Glauber (); SLY03 (); BD04 ()

(4.1)

With factorization (4.1) and energy conservation (2.7), the Schrödinger equation (2.4) becomes

(4.2)

where the relative velocity between projectile and target is assumed to be large.

The first step in the eikonal approximation is to assume the second-order derivative negligible with respect to the first-order derivative ,

(4.3)

This first step leads to the second-order equation (but now first-order in ),

(4.4)

where the dependence of the wave function on the longitudinal and transverse parts of the projectile-target coordinate has been made explicit. This equation is mathematically equivalent to the time-dependent Schrödinger equation (3.1) for straight-line trajectories with replaced by . It can thus be solved using any of the algorithms cited in Sec. 3.4. However, contrary to time-dependent models, it is obtained without the semiclassical approximation. The projectile-target coordinate components and are thus quantal variables. Interferences between solutions obtained at different values are thus taken here into account. This first step is known as the dynamical eikonal approximation (DEA) BCG05 (); GBC06 ().

4.2 Cross sections

The transition matrix element for elastic scattering into direction of the final momentum of the projectile in the c.m. frame reads Aus70 ()

(4.5)

where is the exact solution of the Schrödinger equation (2.4) with the asymptotic condition (2.6). By using Eqs. (4.1), (2.1), and (4.4), one obtains the approximation BCG05 ()

(4.6)

where the transfered momentum is assumed to be purely transverse, i.e. , is neglected. The norm of is linked to the scattering angle by

(4.7)

Let us define the elastic amplitude

(4.8)

The transition matrix element (4.6) reads after integration over ,

(4.9)

where is the azimuthal angle characterizing . The phase factor arises from the rotation of the wave functions when the orientation of varies GBC06 (). The integral over can be performed analytically, which leads to the following expression for the elastic differential cross section GBC06 ()

(4.10)

where is a Bessel function AS70 (). From Eq. (4.10), one can see that contrary to the semiclassical approximation (3.5), the eikonal elastic cross section is obtained as a coherent sum of elastic amplitudes over all values. This illustrates that quantum interferences are taken into account in the eikonal framework.

The transition matrix element for dissociation reads

(4.11)

where is the final projectile-target wave vector. One can then proceed as for the elastic scattering. Using Eqs. (4.1), (2.2), and (4.4), taking into account the energy conservation,

(4.12)

and assuming the transfered momentum to be purely transverse, the transition matrix element is expressed as

(4.13)

with the breakup amplitude

(4.14)

The differential cross section for breakup is given by

(4.15)

where the proportionality factor depends on the phase space. Like the elastic scattering cross section (4.10), it is obtained from a coherent sum of breakup amplitudes (4.14), confirming the quantum-mechanical character of the eikonal approximation. Here also, the integral over can be performed analytically and leads to Bessel functions GBC06 ().

By integrating expression (4.15) over unmeasured quantities, one can obtain the breakup cross sections with respect to the desired variables, like the internal excitation energy of the projectile. Since these operations depend on the projectile internal structure, we delay the presentation of some detailed expressions to Secs. 6 and 7 treating of two-body GBC06 () and three-body BCD09 () breakup.

4.3 Standard eikonal approximation

In most references, the concept of eikonal approximation involves a further simplification to the DEA HBE96 (); SLY03 (). This adiabatic, or sudden, approximation consists in neglecting the excitation energy of the projectile compared to the incident kinetic energy. It comes down to assume the low-lying spectrum of the projectile to be degenerate with its ground state, i.e. to consider the internal coordinates of the projectile as frozen during the reaction SLY03 (). This approximation therefore holds only for high-energy collisions that occur during a very brief time. This second assumption leads to neglect the term in the DEA equation (4.4) which then reads

(4.16)

The solution of Eq. (4.16) that follows the asymptotic condition (2.6) exhibits the well-known eikonal form Glauber (); BD04 ()

(4.17)

After the collision, the whole information about the change in the projectile wave function is thus contained in the phase shift

(4.18)

Due to translation invariance, this eikonal phase depends only on the transverse components of the projectile-target coordinate and of the projectile internal coordinates . Cross sections within this standard eikonal approximation are obtained as explained in Sec. 4.2, replacing by .

Being obtained from the adiabatic approximation, expressions (4.17) and (4.18) are valid only for short-range potentials. For the Coulomb interaction, the assumption that the reaction takes place in a short time no longer holds, due to its infinite range. The adiabatic approximation thus fails for Coulomb-dominated reactions SLY03 (). Besides imprecise uses of a cutoff at large impact parameters AS00 (), there are two ways to avoid this problem. The first is not to make the adiabatic approximation, i.e. to resort to the more complicated DEA (see Sec. 4.1). The second is to correct the eikonal phase for the Coulomb interaction as suggested in LABEL:MBB03 (see Sec. 4.4). Nevertheless, as shown in LABEL:GBC06, the Coulomb divergence does not affect eikonal calculations performed on light targets at high enough energies. Most of the nuclear-dominated reactions can thus be analyzed within an eikonal model including the adiabatic approximation (see, e.g., LABEL:HT03).

Fig. 2 illustrates the difference between the DEA (full line), the usual eikonal approximation (dashed line) and the semiclassical approximation (dotted line) when Coulomb dominates. It shows the breakup cross section of Be on Pb at 69 MeV/nucleon for a Be-n relative energy of 0.3 MeV as a function of the - scattering angle. As explained above, the usual eikonal approximation diverges for the Coulomb-dominated breakup, i.e. at forward angles. The DEA, which does not include the adiabatic approximation, exhibits a regular behaviour at these angles. Interestingly, the semiclassical approximation follows the general behaviour of the DEA, except for the oscillations due to quantum interferences between different values. The DEA has therefore the advantage of being valid for describing any breakup observable on both light and heavy targets.

Figure 2: Breakup cross section of Be on Pb at 69 MeV/nucleon as a function of the - scattering angle in the - c.m. frame for a Be-n energy  MeV. Calculations are performed within the DEA, usual eikonal, and semiclassical approximations phdGG ().

The nuclei studied through breakup reactions being exotic, it may be difficult, if not impossible, to find optical potentials that describe the scattering of the clusters by the target. One way to circumvent that problem is to resort to what is usually known as the Glauber model GM70 (); HBE96 (); SLY03 (); BD04 (). This model has been mostly used to calculate total and reaction cross sections. At the optical-limit approximation (OLA) of the Glauber model, correlations in the cluster and target wave functions are neglected. The nuclear component of the eikonal phase shift for cluster is then expressed as a function of the densities of the target and of the cluster, and of a profile function that corresponds to an effective nucleon-nucleon interaction. The nuclear component of the eikonal phase shift is approximated by SLY03 ()

(4.19)

where and are the transverse components of the internal coordinates of the target and of cluster , respectively, and is the transverse component of the c.m. coordinate of cluster . The OLA is therefore equivalent to the double-folding of an effective nucleon-nucleon interaction. The density of the target can usually be obtained from experimental data. The cluster density being unknown, it has to be estimated from some structure model, like a mean-field calculation. The profile function is usually parametrized as SLY03 (); AHK08 ()

(4.20)

where is the total cross section for the N-N collision, is the ratio of the real part to the imaginary part of the N-N scattering amplitude, and is the slope parameter of the N-N elastic differential cross section. These parameters depend on the nucleon type (p or n) and on the incident energy. Their values can be found in the literature (see, e.g., LABEL:AHK08). The validity of the Glauber approximation is discussed in LABEL:YMO08.

4.4 Coulomb-corrected eikonal approximation

The eikonal approximation gives excellent results for nuclear-dominated reactions SLY03 (); GBC06 (). However, as mentioned above, it suffers from a divergence problem when the Coulomb interaction becomes significant. To explain this, let us divide the eikonal phase (4.18) into its Coulomb and nuclear contributions

(4.21)

In this expression, is the global elastic Coulomb eikonal phase between the projectile and the target. However, Coulomb forces not only act globally on the projectile, they also induce ‘tidal’ effects due to their different actions on the various clusters. The tidal Coulomb phase is due to the difference between the cluster-target and projectile-target bare Coulomb interactions. The remaining phase contains effects of the nuclear forces as well as of differences between Coulomb forces taking the finite size of the clusters into account and the bare Coulomb forces.

At the eikonal approximation, the integral (4.18) defining diverges and must be calculated with a cutoff Glauber (); SLY03 (). Up to an additional cutoff-dependent term that plays no role in the cross sections, it can be written as BD04 ()

(4.22)

where appears the projectile-target Sommerfeld parameter defined in Eq. (2.8). The phase (4.22) depends only on .

The tidal Coulomb phase is computed with Eq. (4.18) for the difference between the bare Coulomb interactions for the clusters in the projectile and the global - Coulomb interaction,

(4.23)

It can be expressed analytically. Because of the long range of the E1 component of the Coulomb force, this phase behaves as at large distances GBC06 (); CBS08 (). In the calculation of the breakup cross sections (4.15), the integration over diverges for small values, i.e. at forward angles, because of the corresponding asymptotic behaviour of the breakup amplitude, as illustrated in Fig. 2. This divergence occurs only in the first-order term of the expansion of the eikonal Coulomb amplitude .

As seen in Sec. 3.3, the first order approximation (3.15) decreases exponentially at large and hence does not display such a divergence. A plausible correction is therefore to replace the exponential of the eikonal phase according to MBB03 (); AS04 ()

(4.24)

where is the result of first-order perturbation theory (3.10),

(4.25)

Note that because of the phase , the integrand in Eq. (4.25) does not exhibit a translational invariance. The first-order phase depends on all internal coordinates of the projectile. When the adiabatic approximation is applied to Eq. (4.25), i.e. when is set to 0, one recovers exactly the Coulomb eikonal phase (4.23). This suggests that without adiabatic approximation the first-order term of would be (4.25) instead of (4.23), intuitively validating the correction (4.24). Furthermore, since a simple analytic expression is available for each of the Coulomb multipoles (see Sec. 3.3), this correction is easy to implement.

With this Coulomb correction, the breakup of loosely-bound projectiles can be described within the eikonal approximation taking on (nearly) the same footing both Coulomb and nuclear interactions at all orders. This approximation has been tested and validated for a two-body projectile in Ref. CBS08 (). Note that in all practical cases AS04 (); CBS08 (); BCD09 (), only the dipole term of the first-order expansion (3.13) is retained to evaluate

Fig. 3 illustrates the accuracy of the CCE for the breakup of Be on lead at 69 MeV/nucleon CBS08 (). The figure presents the parallel-momentum distribution between the Be core and the halo neutron after dissociation. This observable has been computed within the DEA (full line), which serves as a reference calculation, the CCE (dotted line), the eikonal approximation including the adiabatic approximation (dashed line), and the first-order of the perturbation theory (dash-dotted line). The usual eikonal approximation requires a cutoff at large impact parameter to avoid divergence. The value  fm is chosen from the value prescribed in LABEL:AS00. At the first order or the perturbation theory, the nuclear interaction is simulated by an impact parameter cutoff at  fm.

Figure 3: Breakup of Be on Pb at 69 MeV/nucleon. The parallel-momentum distribution between the Be core and the halo neutron is computed within the DEA, the CCE, the eikonal approximation including the adiabatic approximation, and the first-order of the perturbation theory CBS08 (). Reprinted figure with permission from LABEL:CBS08. Copyright (2008) by the American Physical Society.

We first see that the magnitude of the CCE cross section is close to the DEA one, whereas, the other two approximations give too large (eikonal) or too small (first order) cross sections. Moreover the CCE reproduces nearly perfectly the shape of the DEA distribution. In particular the asymmetry, due to dynamical effects, is well reproduced. This result suggests that in addition to solving the Coulomb divergence problem introduced by the adiabatic approximation, the CCE also restores some dynamical and higher-order effects missing in its ingredients, the usual eikonal approximation and the first order of the perturbation theory.

5 Continuum-discretized coupled-channel method

The CDCC method is a fully quantal approximation which does not imply some restriction on energies. Its main interest lies in low energies where the previous methods are not valid. The principle of the CDCC method is to determine, as accurately as possible, the scattering and dissociation cross sections of a nucleus with a simplified treatment of the final projectile continuum states. To this end, these states describing the relative motions of the unbound fragments are approximately described by square-integrable wave functions at discrete energies. The relative motion between the projectile and target and various cross sections can then be obtained by solving a system of coupled-channel equations. The number of these equations and hence the difficulty of the numerical treatment increase with increasing energy.

The CDCC method was suggested by Rawitscher Ra74 () and first applied to deuteron + nucleus elastic scattering and breakup reactions. It was then extensively developed and used by several groups Kam86 (); AIK87 (); NT99 (); MAG01 (); RKM08 (); MAG09 (); DBD10 (). Its interest has been revived by the availability of radioactive beams of weakly bound nuclei dissociating into two TNT01 (); NT99 (); MAG01 (); RKM08 (); MAG09 (); DBD10 () or three MHO04 (); RAG08 (); EMO09 (); RAG09 () fragments.

We assume that the breakup process leads to clusters and that the cluster-target interactions do not depend on the target spin. The projectile wave functions describing -body bound states at negative energies and describing -body scattering states at positive energies are defined with Eq. (2.1). Since the total angular momentum of the projectile-target system is a good quantum number, the first step consists in determining partial waves of the -body Hamiltonian (2.4). The general partial wave function for a total angular momentum can be expanded over the projectile eigenstates as

(5.1)

In this expansion, index runs over the bound states of the projectile. The total angular momentum results from the coupling of the orbital momentum of the projectile-target relative motion with the total angular momentum of the projectile state. The relative-motion partial waves and are unknown and must be determined. The parity is given by the product of and the parity of . The first term of Eq. (5.1) represents the elastic and inelastic channels while the second term is associated with the breakup contribution. However, the presence of the continuum renders this expression intractable.

The basic idea of the CDCC method is to replace wave function (5.1) by

(5.2)

where the functions represent either bound states () or square-integrable approximations of continuum wave functions () at discrete energies

(5.3)

Approximation (5.2) resembles usual coupled-channel expansions and can be treated in a similar way.

In practice, two methods are available to perform the continuum discretization. In the “pseudostate” approach, the Schrödinger equation (2.1) is solved approximately by diagonalizing the projectile Hamiltonian either within a finite basis of square-integrable functions or in a finite region of space. In both cases, square-integrable pseudostates are obtained. This approach is simple but there is little control on the obtained energies . Therefore, it is customary to keep only the pseudostates with energies below some limit .

The alternative is to separate the integral over in Eq. (5.1) into a limited number of small intervals, or “bins”, which may depend on and to use in each of them some average of the exact scattering states in this range of energies Kam86 (); Ra74 (); AIK87 (); NT99 (). This “bin” method provides the square-integrable basis functions

(5.4)

where the weight functions may also depend on . Such states are orthogonal because of the orthogonality of the scattering states and they are normed if is the norm of over . Using Eq. (5.4), their energy (5.3) is given by

(5.5)

Here also, a maximum energy is chosen. In practice, these basis states are usually constructed by averaging the scattering states normalized over the wave number , often within equal momentum intervals TNT01 ().

The total wave function (5.2) can be rewritten as

(5.6)

where represents the channel and

(5.7)

By inserting expansion (5.6) in the Schrödinger equation (2.4) and using Eq. (5.3), the relative wave functions are given by a set of coupled equations

(5.8)

where . The sum over is truncated at some value . The sum over the pseudo-states or bins is limited by the selected maximum energy . The CDCC problem is therefore equivalent to a system of coupled equations where the potentials are given by

(5.9)

This matrix element involves a multidimensional integral over and over the internal coordinates . In general, the potentials are expanded into multipoles corresponding to the total angular momentum operator of the system. This may allow an analytical treatment of angular integrals.

System (5.8) must be solved with the boundary condition