Continuum-Discretised Coupled Channels methods

• Prof. Filomena Nunes, National Superconducting Cycltron Laboratory and Department of Physicis and Astronomy Michigan State University

One of the most common methods to study rare isotopes with large proton/neutron asymmetry, is through breakup reactions. These exotic nuclei are so loosely bound they can break up very easily while leaving the target in its ground state. While the first efforts to study breakup assumed the process could be treated perturbatively, it was soon realized that multi-step effects are important. Also, it is not simple to isolate Coulomb effects from nuclear effects. A non perturbative method that treats breakup to all orders, and includes Coulomb and nuclear effects on equal footing is the Continuum discretized coupled channel method (CDCC). Over the last two decades this method has become increasingly popular. Here we present the method as it was originally introduced, address implementation aspects and provide some applications. We also discuss limitations of the method as well as new developments that allow to solve more complicated problems.

A brief history

Breakup reactions are an important tool to study rare isotopes. Unstable nuclei have small separation energies and thus break up very easily. Breakup cross sections have been used with aim of extracting structure properties but also of providing information relevant to astrophysics. The proximity of the ground state to threshold also means that the breakup states of the rare isotope (i.e. its continuum) influences all other reactions that can take place. Understanding the breakup is therefore an essential step in the study of reactions at the limits of stability.

Already in the seventies, with the experiments using deuteron beams, it became clear that often the breakup process could not be well described using Born approximation, and required a more careful treatment of the continuum. Pioneering work by Johnson and Soper formulated the problem in terms of a two channel coupled equation(Soper, 1970)#soper. Later, developments by Rawitscher (Rawitscher, 1974)#rawitscher and Austern et al. (Austern, 1987; Yahiro, 1986; Austern, 1996)#Aus87#yahiro#austern introduced a more realistic representation of the continuum. All these approaches avoided the complications of the full Faddeev equations (Faddeev, 1961; Alt, 1967)#faddeev#ags and described the breakup as a single-particle inelastic excitation of the projectile into the continuum.

This work provides a quick review of the method and does not attempt to provide all the details nor a comprehensive list of references. It does however provide the variety of problems that CDCC can address. In section II we discuss the method as originally developed for deuteron breakup. Next, in section III, we discuss details on implementation and convergence. Later we provide some examples (section IV) and conclude by listing some of the important limiting factors of the method, as well as a few recent extensions to solve more complicated problems (section V).

The CDCC method on paper

Let us consider the reaction \(p+t \rightarrow v+c+t\ ,\) where an initial bound state of the projectile \(p\) breaks up into \(v+c\) under the influence of the target \(t\ .\) The relevant coordinates to describe this process are the Jacobi coordinates \((\vec r,\vec R)\) introduced in Figure 1, where \(\vec r\) is the vector that connects the centers of mass of \(c\) and \(v\ ,\) and \(\vec R\) connects the centers of mass of the target and the (c+v) system. The starting point for the CDCC method is the three-body Hamiltonian: \[\tag{1} H_{\rm 3b}= \hat T_r + \hat T_R + V_{vc} + V_{vt} + V_{ct} , \]

where \(\hat T_{r} + \hat T_{R}\) are the kinetic-energy operators. In general, the potential \(V_{vc}\) describes the bound state spectrum and low energy scattering states of the projectile, and hence is real. However, the potentials \(V_{vt}\) and \(V_{ct}\) describe the scattering of these fragments off the target, and therefore will in general contain an imaginary part.

Figure 1: Jacobi coordinates used in the CDCC method.

Next we define the internal Hamiltonian of the projectile as \[ H_{proj}=\hat T_r + V_{vc} \]

We denote the eigenstates of the projectile \(\phi_{(0,\vec k)}\) satisfying \((H_{proj} - \epsilon)\phi =0\ .\) For simplicity we consider that the projectile has only one bound state \(\phi_0(\vec{r})\) and then, for all other solutions, the internal wave function depends on a continuous variable, the momentum \(\vec k\) (\(k=\sqrt{\frac{2\mu_{vc}\epsilon}{\hbar^2}}\)), representing the scattering states \(\phi_{\vec k}(\vec r)\ .\) Here \(\mu_{vc}\) is the reduced mass of \(v+c\ .\) A generalization for several bound states is trivial.

We use the standard angular momentum decomposition for the eigenstates \(\phi\ :\)

\(\tag{2}
\phi_{(0, k)}(\vec{r})=
\frac{u_{(0,k)}(r)}{r}\, \left[\left[Y_\ell(\hat{\vec{r}})\otimes
\mathcal{X}_s \right]_j \otimes \mathcal{X}_{I_c} \right]_{I_p M} ,
\)

where \(\ell\) is the orbital angular momentum of \(v\) relative to \(c\ ,\) \(s\)(\(I_c\)) is the spin of the fragment \(v\)(c), and the total angular momentum of the projectile is \(I_p\) with projection \(M\ .\) Assuming that the effective interaction between the core \(c\) and fragment \(v\) is central, the radial wave functions \(u_{(0,k)}(r)\) are solutions of the radial equation

\[\tag{3} \left[ -\frac{\hbar^2}{2\mu_{vc}} \left (\frac{d^2}{d r^2} - \frac{\ell(\ell+1)}{r^2} \right) + V_{vc}(r) - \epsilon \right] u_{(0,k)}(r)= 0 , \]

for both the bound state with \(\epsilon_0<0\ ,\) which decays exponentially at large distances, and also the continuum states, with energies \(\epsilon_k>0\ ,\) which oscillate to infinity according to \[ u_k(r) \to e^{i \delta_{\ell}} (\cos \delta_{\ell} F_{\ell}(\eta,kr)+ \sin \delta_{\ell} G_{\ell}(\eta,kr)). \]

Here \(F\) and \(G\) are the regular and irregular Coulomb functions.

Figure 2: Continuum-continuum quadrupole couplings \(V_{\alpha \alpha'}(R)\) for \(\ell_i=1 \rightarrow \ell_f=1\) in the p-⁷Be system calculated for the breakup reaction of ⁸B on ⁵⁸Ni at 26 MeV (Nunes, 2004)#cont-coupling: R=10 fm, 5 fm, and 1 fm for top, middle and bottom panel respectively. The axis correspond to \(\tilde \epsilon_{p}\) and \(\tilde \epsilon_{p'}\ .\)

The CDCC method assumes that the full three body wave function, which in principle is expressed as a sum of three Faddeev components (Thompson, 2009)#book, can be expanded in a single Jacobi coordinate set \((\vec r, \vec R)\) ( Figure 1), using the complete set of eigenstates of the internal Hamiltonian \(H_{proj}\ :\)

\(\tag{4}
\Psi_{\vec{K}_0}^{(1)}(\vec{r},\vec{R})
= \phi_{0}(\vec{r}) \psi_{0}(\vec{R})+ \int  d \vec{k}\  \phi_{\vec{k}}(\vec{r})
  \psi_{\vec{K}}(\vec{R}).
\)

The momentum \(\vec{k}\) between the internal motion of \(c+v\) is related to the momentum \(\vec{K}\) between the projectile and the target through energy conservation: \[\tag{5} E_{\rm cm} + \epsilon_0 = E= \frac{\hbar^2 k^2}{2 \mu_{vc}} + \frac{\hbar^2 K^2}{2 \mu_{(vc)t}} . \]

This \(E\) is the center of mass energy of the three-body system consisting of the sum of the \(p+t\) relative energy \(E_{\rm cm} \) and the (negative) projectile binding energy \(\epsilon_0\ ;\) where \(\mu_{(vc)t}\) is the \(p+t\) reduced mass.

The three-body wave function now involves an integral over a continuous variable to infinity, as well as sums over angular momenta \( \int_0^\infty d k \sum_{lsjI_cI_p}\ldots\ .\) Due to the integral in k, when replaced into the three-body Schrödinger equation, the expansion (4) is not practical. There are several discretization techniques for dealing with the continuum of the projectile, the most common being the average method which provides a workable square integrable basis. In the average method, the radial functions for the continuum bins \(\tilde u_p(r)\) (with \(p\geq 1\)), are a superposition of the scattering eigenstates within a bin [\(k_{p-1},k_p\)] \[\tag{6} \tilde u_{p}(r) = \sqrt{\frac{2}{\pi N_p }} \int_{k_{p-1}} ^{k_{p}} w_p(k) u_k(r)\, d k~~ \]

where a weight function \(w_p(k)\) has been introduced. The tilde in \(\tilde u_p(r)\) represents the averaging over momentum as defined by Eq.(6). The normalization constant is chosen to be \(N_p = \int_{k_{p-1}} ^{k_{p}} |w_p (k)|^2 \, d k\) to make \(\{ \tilde u_{p}(r) \}\) form an orthonormal set where all the \((k_{p-1},k_p)\) are non-overlapping continuum intervals. The full three-body wave function can now be written in terms of a sum over \(p\ ,\) \[\tag{7} \Psi^{\rm CDCC}(\vec{r},\vec{R}) = \sum_{p=0}^{N} \tilde \phi_{p}(\vec{r}) \psi_p(\vec{R}), \]

with the meaning that for \(p=0\ ,\) the initial ground state of the projectile is included, with quantum numbers \(\{ l_0s_0j_0I_{c0} I_{p0}\}\ ,\) and for \(p \geq 1\) the bin wave functions \(\tilde u_{p}(r) \) are included for specific partial-wave channels and continuum intervals, \(p= \{ lsjI_cI_p;\) \( (k_{p-1},k_p)\}\ .\)

Introducing the basis in Eq.(7) into the three-body Schrödinger equation \[\tag{8} (H_{\rm 3b}-E) \Psi^{\rm CDCC}(\vec{r},\vec{R})=0 \]

and multiplying on the left by the conjugate projectile wave functions \(\tilde \phi^*_{p}(\vec{r})\) and integrating over \(\vec r\ ,\) one can arrive at the coupled channel equations: \[\tag{9} [\hat{T}_R + V_{pp}(R) -E_{p}] \psi_{p}(\vec{R}) + \sum_{p' \ne p} V_{pp'}(R) \psi_{p'}(\vec{R}) = 0, \]

where \(E_{p}= E-\tilde \epsilon_{p}\ ,\) and \(V_{pp'}(R)=\langle \tilde\phi_p(r) | U_{vt} + U_{ct} | \tilde\phi_{p'}(r) \rangle\ .\) Here \(\tilde \epsilon_{p}\) is the energy corresponding to the midpoint momentum of the bin. These equations couple the projectile ground state to its continuum states by \(V_{p0}(\vec R)\ ,\) and also couple projectile states within the continuum, the so-called continuum-continuum couplings.

As in Eq.(2), we use a multipole decomposition for the basis \(\tilde \phi_p\) and a standard multipole decomposition for the relative wavefunction between projectile and target \(\psi_{p}(\vec{R})\) \[ \psi_{p}(\vec{R})=\sum_{L} i^L \chi_{pL}(R) Y_L(\hat{\vec{R}}). \]

Here \(L\) is the orbital angular momentum of the relative motion between the projectile and the target. The final three-body wave function carries total angular momentum and projection \(J_{T},M_{T}\) resulting from the angular momentum coupling \(L \otimes I_p\) (for simplicity we ignore the target spin). It is convenient to introduce \(\alpha = \{p, L\} \equiv \{ lsjI_cI_p; (k_{p-1},k_p), L\}\ .\) Then the CDCC partial wave coupled equations for each \(J_T\) are: \[\tag{10} \left[ -\frac{\hbar^2 }{2 \mu_{(vc)t}} \left ( \frac{d^2 }{ d R^2} - \frac{L(L{+}1)} {R^2} \right ) + V^{J_{\rm tot}}_{\alpha \alpha}(R) + \tilde \epsilon_p - E \right ] \chi_ {\alpha}^{J_{T}} (R) + \sum _ {\alpha'\ne\alpha} i ^ {L' - L} ~ V^{J_{T}} _{\alpha \alpha'}(R) \chi_{\alpha'}^{J_{T}} (R)= 0\ , \]

with scattering boundary conditions \(\chi_{\alpha}(R) \rightarrow i/2 \left[ H_{\alpha}^-(KR) \delta_{\alpha \alpha_i} - H_{\alpha}^+(KR) S_{\alpha \alpha_i}\right]\) for large \(R\ .\) In Eq. (10) we introduced the coupling potentials \(V _{\alpha \alpha'}(R)\) defined by \[\tag{11} V_{\alpha \alpha'}^{J_{\rm tot}}(R) = \langle[\phi_p(\vec{r}) Y_L(\hat{\vec{R}})]_{J_T} | U _ {ct} (\vec{R}_ c) + U _ {vt} (\vec{R}_v )| [\phi_{ p'}(\vec{r}) Y_{L'}(\hat{\vec{R}})]_{J_T}\rangle, \]

where \(U_{ct} (\vec{R}_c) \) and \(U_{vt}(\vec{R}_v)\) are the total (nuclear and Coulomb) interactions between \(c,t\) and \(v,t\) respectively. A detailed study on the properties of these couplings can be found in (Nunes, 2004)#cont-coupling. An example of continuum-continuum coupling potentials is presented in Figure 2. These are quadrupole transitions included in the calculations of the breakup of ⁸B into \(p+\)⁷Be on ⁵⁸Ni at 26 MeV. Here, \(V_{\alpha \alpha'}\) is evaluated for \(\ell_i=1 \rightarrow \ell_f=1\ .\) The contour plots are shown as a function of the relative energy \(\tilde \epsilon_p\) of the initial and final bins. The top, middle and bottom panels refer to \(R=10, 5, 1\) fm, respectively. The plots demonstrate that indeed these couplings have significant off-diagonal contributions, and these go well beyond the range of the nuclear interaction.

Convergence

Figure 3: Convergence studies for the angular distribution of ⁸B breakup on ⁵⁸Ni at 26 MeV: (left) as a function of the angle of the c.m. of ⁸B in the c.m. system, and testing the model space as a function of \(\ell_{max}\) (Nunes, 1999)#nunes99 and (right) as a function of the angle of ⁷Be and testing the model space as a function of \(\epsilon_{max}\) (Tostevin, 2001)#Tos01.

The overall procedure in the CDCC methods consists of first calculating the projectile wave functions (bound and scattering states) and the corresponding continuum bins. Next, calculating the coupling potentials \(V_{\alpha \alpha'}(R)\ ;\) solving the coupled equations (10) to obtain the S matrices; and finally constructing the observables, namely the cross sections. The code FRESCO (Thompson, 1988)#fresco contains an implementation of the CDCC method and this section pertains primarily to the method in FRESCO.

In any CDCC calculation, of any observable, the model space needs to be carefully checked to ensure the results are meaningful (Piyadasa, 1999)#piyadasa. There are many parameters that determine the model space. Below are a few aspects to keep in mind:

• Energy discretization: one needs to pay attention to the maximum energy \(\epsilon_{max}\) included in the model space and the width of the bins \(\Delta k=k_p - k_{p-1}\ .\) Most often, linear spacing in momentum holds better results (momentum bins), although sometimes linear spacing in energy (energy bins) is also used. The agreement between momentum bins and energy bins has been demonstrated (Rubtsova, 2008)#rubtsova.
• Angular momentum: the CDCC model space includes angular momenta up to \(\ell_{max}\) in the relative motion between \(c+v\) and up to \(L_{max}\) in the relative motion between projectile and target. Most methods also use multipole expansion of the couplings \(V_{\alpha \alpha'}\ ,\) where multipoles up to \(\lambda_{max}\) are included. It is usually necessary to increase \(\lambda_{max}\) in pair with \(\ell_{max}\ .\)
• Radial grid: in most implementations, the CDCC coupled channel equations are solved by direct integration in a radial grid out to a large radius \(R_{max}\) for asymptotic matching to Coulomb functions. Here again, checks need to be performed to guarantee that \(R_{max}\) is indeed appropriate and the step size is small enough to capture the details of the coupling potentials in the range of the interactions. For problems with strong Coulomb fields \(R_{max}\) can reach several thousand fm due to the existence of long range Coulomb couplings. To avoid having to integrate the equations out to such large radii, coupled channel Coulomb functions can be used in the intermediate range from \(R_{max} \approx 60\) fm to \(R_{asymptotic}\approx 10^3\) fm. Finally there is also the radial grid associated with the relative motion within the projectile \(c+v\ ,\) which will influence the basis functions \(\phi_i\ .\) Again here one needs to make sure the maximum value \(r_{max}\) is large enough for calculating the bin functions and that the step size \(r_{step}\) is small enough to obtain converged results.

Figure 3(left) (FIG 7 from (Nunes, 1999)#nunes99) shows the convergence of the angular distribution of ⁸B breakup on ⁵⁸Ni at 26 MeV with respect to the relative angular momentum between \(p\) and ⁷Be. Results indicate that at least \(\ell_{max} = 3\) is needed for convergence. Figure 3(right) (FIG 1 from (Tostevin, 2001)#Tos01) shows convergence of the angular distribution for the ⁷Be fragment following the breakup of ⁸B on a ⁵⁸Ni target at 26 MeV with respect to the maximum \(p-\)⁷Be relative energy. For this observable, \(\epsilon_{max}=8\) MeV is sufficient.

Other techniques for discretizing the continuum are: i) the mid point method which consists of taking directly a scattering state \(u_{k_p}(r)\) for a discrete set of scattering energies and ii) many variants of the pseudostate method in which the eigenstates of the internal Hamiltonian \(H_{proj}\) are expanded in terms of some convenient square-integrable basis. Concerning the mid-point method, its main disadvantage is that it is not square integrable, and thus can only be used in a one-step calculation. Concerning the pseudostate methods, some choices for the basis include harmonic oscillator states (as in the shell model), transformed harmonic oscillators (Moro, 2006)#tho, or a large set of Gaussians (Egami, 2004)#kyushu. The main advantage of the pseudostate method is that the basis wavefunctions decay to zero at large distances. The main disadvantage is that the basis has no simple relation to the \(v+c\) scattering solutions \(u_k(r)\ .\) Nevertheless in several examples, the pseudostate approach has proven to be very useful and just as accurate as the average bin method (Moro, 2009)#moro-tho.

Some applications

Coulomb dissociation

Figure 4: Capture cross sections, multiplied by the energy factor \(E^{1/2}\ ,\) versus neutron energy. The shaded area corresponds to results obtained from Coulomb dissociation and the black circles are the latest direct measurements (see (Summers, 2008b)#c15summers for details).

One of the most important applications of CDCC has been in the analysis of Coulomb dissociation experiments (e.g. (Nunes, 1999; Tostevin, 2001; Davids, 2001; Mortimer, 2002; Summers, 2004; Ogata, 2006; Summers, 2008b; Belyaeva, 2009) #nunes99#Tos01#davids01#mortimer01#summers-be7#ogata06#c15summers#belyaeva). The Coulomb dissociation method, originally proposed by Bauer et al. (Baur, 1986)#baur, consists of measuring Coulomb dissociation cross sections with the aim of extracting direct radiative capture rates of relevance to astrophysics. It thus is an experimental method that relies on reaction theory, the idea being that the cross section of \(c+v \rightarrow p\) can be related to that of \(p + t \rightarrow c+v+t\) using detailed balance and assuming that the dissociation process is 1-step, Coulomb \(E1\) only. Then, first order perturbation provides a factorization of the strength for the dissociation \(p \rightarrow c+v\ .\) After many studies, it has become clear that in real life, when looking at loosely bound nuclei whose wave functions have extended tails, these assumptions are difficult to satisfy. A number of complications to the reaction mechanism need to be considered: the nuclear component is often significant and interferes with the Coulomb component, high-order multipoles of the Coulomb field can contribute significantly and usually bring in partial waves of the \(c+v\) system that may not be well constrained by data, and multi-step effects can be so large that a perturbative expansion no longer converges. CDCC offers a reliable method to make the required connection between the Coulomb dissociation cross section and the capture reaction of interest. An example of the success of CDCC in extracting a neutron capture rate can be found in (Summers, 2008b)#c15summers. Therein the Coulomb dissociation of¹⁵C on Pb measured at 68 MeV/u is used to constrain the \(n+\)¹⁴C parameters, in particular the asymptotic normalization coefficient of the bound state. Finally, the ¹⁴C(n,\(\gamma\))¹⁵C capture rate obtained with these constrained parameters is in complete agreement with the direct measurement (results shown in Figure 4 - FIG 1 of (Summers, 2008b)#c15summers).

A systematic study of the approximations typically used in the analysis of Coulomb dissociation experiments within the CDCC framework can be found in (Hussein, 2006)#hussein. The procedure of scaling and subtracting a nuclear component is proven to be inaccurate and call for an analysis that includes Coulomb and nuclear effects on equal footing and treats them non-perturbatively.

Breakup on light targets

Figure 5: Study of ⁷Be elastic breakup on ¹²C at 25 MeV/ nucleon: (left) cross section as a function of angular momentum (with lines for total, nuclear only and Coulomb only); top scale relates the angular momentum to the impact parameter via the semi-classical relation \(J=Kb\ ,\) where \(K=4.8\) fm^-1, (right) angular distribution of cross sections.

There have also been many applications of CDCC on lighter targets, where the process is mainly mediated by the nuclear interaction. These processes are usually less peripheral than the Coulomb dissociation and large nuclear coupling effects can be present. Nevertheless, if the target has mass \(A>12\ ,\) it is likely that angular distributions at forward angles are dominated by the tail of the wavefunctions \(\phi_i\ .\) Thus, these measurements are sometimes performed with the aim of extracting an asymptotic normalization coefficient for the bound state \(\phi_0\ .\) In such a reaction, it is never the case that one can neglect Coulomb and in some specific cases the Coulomb and nuclear contributions may indeed be of equal importance. The breakup of ⁷Be off a ¹²C target was studied to explore the peripherality condition and the importance of Coulomb. In Figure 5 we show that for this example there is around 20% contribution from the interior (left) and that Coulomb effects are around 25% (right).

Nuclear knockout reactions are usually measured for extracting spectroscopic information. In knockout, the momentum distribution of the heavy fragment is measured, usually in coincidence with the \(\gamma\)-ray. The resulting cross section thus contains elastic breakup (diffraction) as well as transfer and other inelastic processes (stripping). Although the analysis of these reactions is usually performed with eikonal models, CDCC has often been used to determine the diffraction part and evaluate the precision of the eikonal approximation (e.g. (Tostevin, 2002; Bazin, 2009)#tostevin02#bazin09).

Effects of breakup on elastic scattering

Figure 6: (left) The effects of breakup of the projectile on ⁶He elastic scattering on ⁴He for a c.m. energy of 60.4 MeV (Rusek, 2000)#rusek00. (right) ^10,11Be elastic scattering on a proton target at 40 MeV/nucleon, using an optical model fit to the ¹⁰Be elastic and various ¹¹Be reaction models for the ¹¹Be data (Summers, 2008)#be11summers.

When the projectile is loosely bound, not only is the breakup cross section large, but the continuum can influence other processes. The most obvious one is the feedback to the elastic channel. In principle one might expect a reduction of the elastic cross section due to removal of flux into breakup. Such a phenomenon has been observed for example in the scattering of ⁶He on ⁴He. Figure 5(left) illustrates this clearly (FIG 5 of (Rusek, 2000)#rusek00). However other patterns, dependent on the reaction, may emerge (e.g. (Keeley, 1996)#keeley96) and there are puzzling cases which have not to date been understood within the CDCC framework. One such case is the elastic scattering of ¹¹Be off a proton target (Summers, 2008)#be11summers. The various attempts to describe the data fail severely at large angles, as shown in Figure 5(right) (FIG 1 from (Summers, 2008)#be11summers). The goal in the CDCC method is to obtain a good description of the wave function for all physical regions and thus it should attempt to simultaneously describe all channels, including the elastic.

Effects of breakup on transfer reactions

Transfer reactions are an important tool to study exotic nuclei. If the breakup probability is large, it is likely that the transfer cross section will be affected. Here we consider one nucleon transfer only but the method can equally be applied to cluster transfer. Typically, if the projectile is loosely bound, the exact one-nucleon transfer amplitude can be written in such a way that the exact three-body wavefunction appears in the initial channel and is then replaced by the CDCC wavefunction (Moro and Nunes, 2009)#moro09. One nucleon transfer reactions induced by deuterons have traditionally been used as a tool for spectroscopy. Since the deuteron is a loosely bound system, the CDCC technique has been broadly applied to describe deuteron breakup effects in (d,p) reactions (e.g. (Masaki, 1994; Hirota, 1998; Huu-tai, 2006; Iijima, 2007) #masaki#hirota#chau#iijima). Here we provide an example for ¹¹⁸Sn(d,p)¹¹⁹Sn in Figure 7(left) (FIG 4 of (Iijima, 2007)#iijima). In this comprehensive study, the energy dependence of the cross section is discussed with particular emphasis on the importance of including the Coulomb couplings.

Another application to understand the effects of breakup on transfer angular distributions is presented in (Moro, 2002)#moro-tr. There the reaction ¹³N(⁷Be,⁸B)¹²C at 84 MeV is studied. When refitting the interactions such that the elastic scattering within CDCC produces the same distribution as the single channel calculation, breakup effects on the cross section is found to be small (Moro, 2002)#moro-tr.

Figure 7: (left) Energy dependence of cross section for ¹¹⁸Sn(d,p)¹¹⁹Sn: comparing zero-range DWBA and CDCC with and without Coulomb couplings (CBU). See (Iijima, 2007)#iijima for more detail.(right) Energy dependence of the total fusion cross section for the fusion of ⁶He+⁵⁹Co (the dashed and solid lines are with and without continuum-continuum coupling respectively. For more detail see (Beck, 2007)#beck07.

Effects of breakup on fusion

Finally there have been a few applications of CDCC to fusion. Fusion with loosely bound systems is an unresolved puzzling topic, however here we only introduce the basic method to calculate fusion using the CDCC method and show one example. Original efforts of studying fusion with loosely bound systems used CDCC calculations to obtain a dynamic polarization potential (e.g. (Keeley, 2001)#keeley01). Subsequently, using the barrier penetration model, fusion cross section could be extracted. An improvement on that method is to solve the CDCC equations using the incoming boundary condition method (Beck, 2007)#beck07. A practical equivalent of the incoming boundary condition method is to introduce a strong short range imaginary part in the projectile-target interactions to ensure the particles get trapped. Fusion processes are sometime incomplete, meaning that a fraction of the nucleons of the projectile/target were lost in the fusion process. Tracking complete and incomplete fusion require a time-dependent description of the process. Within CDCC it is not possible to separate incomplete fusions from complete fusion and thus only the total fusion is meaningful. In Figure 7(right) we show the energy dependence of the total fusion cross section for ⁶He on ⁵⁹Co obtained through the CDCC method. Also, the effect of continuum-continuum couplings is highlighted. For the sake of comparison, the fusion of ⁶Li is also shown. (FIG 8 of (Beck, 2007)#beck07).

Limitations and extensions

As the method expands and develops, we also begin to better understand some limitations. There are certainly conditions associated with specific implementations and not generally with the method itself. For example, recently it has been pointed out the importance of including a realistic NN interaction in the CDCC couplings when using the CDCC method for reactions of loosely bound projectiles on protons (Cravo, 2010)#cravo. Such extensions can and should be done at little cost. Another aspect that has been discussed is relativity (Ogata and Bertulani, 2009)#bertulani. The CDCC method is derived from non-relativistic quantum mechanics and this was considered appropriate for low-energy nuclear physics. However some facilities nowadays have beam energies of the order of GeV. Even for reactions at 100 MeV/u relativistic corrections in the kinematics is important. Relativistic corrections to the dynamics can be effectively introduced in the CDCC method and have been studied in detail (Ogata and Bertulani, 2009)#bertulani.

Other features of the method may require major upgrades or even new theories. Of immediate concern, three limitations have recently become apparent: the first associated with the asymptotic behavior of the CDCC wavefunction for three-charged particles, the second which we would call true three-body dynamics, and the third associated with the non-locality of the nuclear optical potential.

• Asymptotics of three charge particles
  

The CDCC method decomposes the wavefunction into \(c+v\) and \(p+t\ ,\) which means that the asymptotics of the final system is a product of the asymptotics of these two relative motions. Particularly when three-charged particles are involved, the asymptotic behavior written in this form is not correct (Alt, 2005)#alt05. One should be careful when applying CDCC to problems where all bodies \(c+v+t\) are charged (e.g. (Ogata, 2009b)#ogata09).

• Three-body dynamics
  

The original CDCC method attempts to solve the three-body problem focusing on the expansion on a chosen Jacobi coordinate set. However, if there are bound states in other subsystems, the correct formulation for the problem, introduced by Faddeev (Faddeev, 1961)#faddeev, consists in solving the Faddeev coupled channel set, either in coordinate space (Faddeev, 1961)#faddeev or in momentum space using the so-called Alt-Grassberger-Sandhas (AGS) equations (Alt, 1967)#ags. The implicit assumption in the CDCC method is that the coupling between the transfer and breakup channel is small, and therefore breakup can be decoupled from the transfer. In this sense, while transfer can contain breakup contributions, there is not feedback of the transfer channels into the breakup channel. Recently the CDCC method has been benchmarked against the AGS method for a few nuclear reactions (Deltuva, 2009)#deltuva. Although the CDCC results for deuteron breakup on ¹²C and ⁵⁸Ni agree well with the AGS results, the results for reactions of ¹¹Be on protons (elastic, transfer and breakup) demonstrate that the CDCC assumption is not always valid. In this case, there are two subsystems for which we include bound states (the projectile and the final deuteron). It should be noted that in the standard Faddeev formalism one approaches the problem from a single Hamiltonian, whereas when using the CDCC wavefunction in a transfer T-matrix, two Hamiltonians are implicit. For example in (d,p) there deuteron incoming wave is determined from a Hamiltonian where the nucleon-target interaction are evaluated at half the beam energy whereas in the final state the neutron-target interaction has no absorption (to allow for the final bound state) and the proton interaction is determined at the energy of the outgoing proton. This leads to some ambiguity in the choice of the interactions. There have been some efforts to further investigate this problem (Moro and Nunes, 2009)#moro09 but questions remain.

• Non-local interactions
  

It is understood that the effective interactions between the complex bodies \(c+v+t\) are in principle non-local. Traditionally, local interactions have been used to reproduce the known observables of the subsystems and therefore non-local correction introduced à la Perey and Buck (Perey, 1962)#perey have always carried a certain ambiguity. However, some efforts have been initiated on determining the form of the non-locality from microscopic calculations [Thompson (2010)]. . In addition, recent AGS reaction calculations including non-locality show that the effect on many reaction observables can be very large (Deltuva, 2009)#deltuva-nl. These exploratory works may eventually prove that a local representation of the interactions is not sufficient. Of course, the CDCC method was originally developed to include local interactions only. If time demonstrates that non-local interactions are essential to provide the required level of accuracy, new developments will be needed. The R-matrix method mentioned next is one avenue to achieve this goal.

Regardless of the limitations, the new extensions of the CDCC method hold promise to the field. In the last few years, there have been a number of efforts to expand the CDCC method from its original formulation (Yahiro, 1986)#yahiro based on a three-body problem with inert cores, to more complex systems. Below we list a few different directions taken by the community.

• Core excitation in CDCC
  

In many breakup reactions of \(p=c+v\) the core \(c\) is left in an excited state. This may be because the wavefunction of the projectile already contained core excited components, but also because dynamic core excitation occurred during the influence of the target. An extension of the CDCC method to address this possibility called XCDCC has been implemented (Summers, 2006a)#xcdcc and the few available applications prove it is a useful framework.

• Four-body CDCC
  

In the standard CDCC method, the projectile is described in terms of a two-body model. However, in many halo nuclei, there are two valence nucleons orbiting the core, and thus a three-body model is the natural choice. The four-body CDCC method has been developed independently by two groups (Matsumoto, 2006; Rodríguez-gallardo, 2008)#cdcc4b-1#cdcc4b-2 and been applied mostly to reactions with ⁶He. This method is computationally expensive and some developments will be needed in the future to make this method useful for reactions with other projectiles (e.g. ¹¹Li, ¹⁴Be, ²²C, etc).

• R-matrix CDCC
  

In the most commonly used implementation of the CDCC method (Thompson, 1988)#fresco, the CDCC coupled channels equations are solved by direct integration. There are of course other methods of solving the equations. Recently the R-matrix method has been considered (Druet, 2010)#rmatrix. This method has the great advantage that it can easily handle non-local interactions.

• Transfer to the continuum
  

Transfer reactions can often populate unbound states in the final nucleus. The reactions commonly referred to as transfer to the continuum are particularly interesting when we wish to explore structures beyond the limits of stability and have large applicability in nuclear astrophysics, in connection to resonant capture reactions. If the reaction transfers to a narrow resonance, the wavefunction is often treated as a bound state. Methods to solve the problem more generally are lacking. The CDCC technique can be used for the non-resonant continuum or for broad non-overlapping resonances. The idea is to use the CDCC expansion in the exact T-matrix but now in the final state (Moro, 2006)#moro06. An application of the CDCC method to the reaction ⁹Li(d,p)¹⁰Li can be found in (Jeppesen, 2006)#li10rex. The purpose of this study was to investigate the structure of the unbound nucleus ¹⁰Li.

Acknowledgements

We thank Pierre Capel for useful discussions and detailed comments to the manuscript. This work was partially supported by the National Science Foundation grant PHY-0555893, the Department of Energy under contract DE-FG52-08NA28552 and the topical collaboration TORUS DE-SC0004087.

References

• Alt, E O; Grassberger, P and Sandhas, W (1967). Reduction of the three-particle collision problem to multi-channel two-particle Lippmann-Schwinger equations. Nuclear Physics B 2(2): 167 - 180. doi:10.1016/0550-3213(67)90016-8.
• Alt, E O; Irgaziev, B F and Mukhamedzhanov, A M (2005). Three-body Coulomb interaction effects in the final state of the ²⁰⁸Pb(⁸B,⁷Bep)²⁰⁸Pb Coulomb breakup reaction. Phys. Rev. C 71(2): 024605. doi:10.1103/physrevc.71.024605.
• Austern, N et al. (1987). Continuum-discretized coupled-channels calculations for three-body models of deuteron-nucleus reactions. Physics Reports 154(3): 125 - 204. doi:10.1016/0370-1573(87)90094-9.
• Austern, N; Kawai, M and Yahiro, M (1996). Three-body reaction theory in a model space. Phys. Rev. C 53(1): 314--321. doi:10.1103/physrevc.53.314.
• Austern, N; Yahiro, M and Kawai, M (1989). Continuum discretized coupled-channels method as a truncation of a connected-kernel formulation of three-body problems. Phys. Rev. Lett. 63(24): 2649--2652. doi:10.1103/physrevlett.63.2649.
• Baur, G; Bertulani, C A and Rebel, H (1986). Coulomb dissociation as a source of information on radiative capture processes of astrophysical interest. Nuclear Physics A 458(1): 188 - 204. doi:10.1016/0375-9474(86)90290-3.
• Bazin, D et al. (2009). Mechanisms in Knockout Reactions. Phys. Rev. Lett. 102(23): 232501. doi:10.1103/physrevlett.102.232501.
• Beck, C; Keeley, N and Diaz-torres, A (2007). Coupled-channel effects in elastic scattering and near-barrier fusion induced by weakly bound nuclei and exotic halo nuclei. Phys. Rev. C 75(5): 054605. doi:10.1103/physrevc.75.054605.
• Belyaeva, T L et al. (2009). Astrophysical \(S_{17}\)(0) factor extraction from breakup of ⁸B on ⁵⁸Ni at energies near the Coulomb barrier. Phys. Rev. C 80(6): 064617.
• Cravo, E; Crespo, R; Moro, A M and Deltuva, A (2010). Probing nucleon-nucleon interactions in breakup of the one-neutron halo nucleus ¹¹Be on a proton target. Phys. Rev. C 81(3): 031601. doi:10.1103/physrevc.81.031601.
• Davids, B et al. (2001). Electromagnetic dissociation of ⁸B and the rate of the ⁷Be \((p,\gamma)\)⁸B reaction in the Sun. Phys. Rev. C 63(6): 065806.
• Deltuva, A (2009). Three-body direct nuclear reactions: Nonlocal optical potential. Phys. Rev. C 79(2): 021602. doi:10.1103/physrevc.79.021602.
• Deltuva, A et al. (2007). Three-body description of direct nuclear reactions: Comparison with the continuum discretized coupled channels method. Phys. Rev. C 76(6): 064602. doi:10.1103/physrevc.76.064602.
• Druet, T; Baye, D; Descouvemont, P and Sparenberg, J (2010). CDCC calculations with the Lagrange-mesh technique. Nuclear Physics A 845(1-4): 88 - 105. doi:10.1016/j.nuclphysa.2010.05.060.
• Egami, T et al. (2004). Gaussian expansion approach to nuclear and Coulomb breakup. Phys. Rev. C 70(4): 047604. doi:10.1103/physrevc.70.047604.
• Faddeev, L D (1960 [1961]). Scattering theory for a three particle system. Zh. Eksp. i Teor. Fiz.[Translation:Sov. Phys. JETP] 39 [12]: 1459 [1014]. Faddeev equations could be found in the book of E.W. Schmid and H. Ziegelmann, The Quantum Mechanical Three Body Problem (Pergamon, Oxford, 1974).
• Ferey(1962). A non-local potential model for the scattering of neutrons by nuclei. Nuclear Physics 32(): 353 - 380. doi:10.1016/0029-5582(62)90345-0.
• Hirota, K; Aoki, Y; Okumura, N and Tagishi, Y (1998). Deuteron elastic scattering and (d, p) reactions on ²⁰⁸Pb at 22 MeV and j-dependence of T20 in (d,p) reaction. Nuclear Physics A 628(4): 547 - 579. doi:10.1016/s0375-9474(97)00641-6.
• Hussein, M; Lichtenthäler, R; Nunes, F and Thompson, I (2006). Scaling and interference in the dissociation of halo nuclei. Physics Letters B 640(3): 91 - 95. doi:10.1016/j.physletb.2006.07.046.
• Huu-tai, P C (2006). Systematic study of elastic and reaction cross sections of deuteron induced reactions within the CDCC approach. Nuclear Physics A 773(1-2): 56 - 77. doi:10.1016/j.nuclphysa.2006.04.006.
• Iijima, M; Aoki, Y; Ozawa, A and Okumura, N (2007). CDCC analysis of 118Sn(d, p) reaction below Coulomb barrier. Nuclear Physics A 793(1-4): 79 - 95. doi:10.1016/j.nuclphysa.2007.06.013.
• Jeppesen, H et al. (2006). Study of ¹⁰Li via the ⁹Li(²H,p) reaction at REX-ISOLDE. Physics Letters B 642(5-6): 449 - 454.
• Johnson(1970). Contribution of Deuteron Breakup Channels to Deuteron Stripping and Elastic Scattering. Phys. Rev. C 1(3): 976--990. doi:10.1103/physrevc.1.976.
• Keeley(1996). The role of breakup in near-barrier ⁶Li+²⁰⁸Pb scattering. Physics Letters B 375(1-4): 9 - 15. doi:10.1016/0370-2693(96)00263-8.
• Keeley, N; Kemper, K and Rusek, K (2001). Fusion calculations for the ^6,7Li+¹⁶O systems. Phys. Rev. C 65(1): 014601. doi:10.1103/physrevc.65.014601.
• Masaki, M et al. (1994). CDCC analysis of 22 MeV deuteron elastic scattering and (d,p) reactions on 52Cr and 206Pb. Nuclear Physics A 573(1): 1 - 27. doi:10.1016/0375-9474(94)90012-4.
• Matsumoto, T et al. (2006). Coulomb breakup effects on the elastic cross section of ⁶He+²⁰⁹Bi scattering near Coulomb barrier energies. Phys. Rev. C 73(5): 051602. doi:10.1103/physrevc.73.051602.
• Moro, A M; Arias, J M; Gómez-camacho, J and Pérez-bernal, F (2009). Analytical transformed harmonic oscillator basis for continuum discretized coupled channels calculations. Phys. Rev. C 80(5): 054605. doi:10.1103/physrevc.80.054605.
• Moro, A M; Crespo, R; Nunes, F and Thompson, I J (2002). ⁸B breakup in elastic and transfer reactions. Phys. Rev. C 66(2): 024612. doi:10.1103/physrevc.66.024612.
• Moro, A M; Nunes, F M and Johnson, R C (2009). Theory of \((d,p)\) and \((p,d)\) reactions including breakup: Comparison of methods. Phys. Rev. C 80(6): 064606. doi:10.1103/physrevc.80.064606.
• Moro, A M; Pérez-bernal, F; Arias, J M and Gómez-camacho, J (2006). Coulomb breakup in a transformed harmonic oscillator basis. Phys. Rev. C 73(4): 044612. doi:10.1103/physrevc.73.044612.
• Moro(2006). Transfer to the continuum and breakup reactions. Nuclear Physics A 767(): 138 - 154. doi:10.1016/j.nuclphysa.2005.12.016.
• Mortimer, J; Thompson, I J and Tostevin, J A (2002). Higher-order and \(E2\) effects in medium energy ⁸B breakup. Phys. Rev. C 65(6): 064619. doi:10.1103/physrevc.65.064619.
• Nunes(1999). Multistep effects in sub-Coulomb breakup. Phys. Rev. C 59(5): 2652--2659. doi:10.1103/physrevc.59.2652.
• Nunes, F M; Mukhamedzhanov, A M; Rosa, C C and Irgaziev, B (2004). Insight into continuum couplings. Nuclear Physics A 736(3-4): 255 - 268. doi:10.1016/j.nuclphysa.2004.03.035.
• Ogata(2009a). Dissociation of Relativistic Projectiles with the Continuum-Discretized Coupled-Channels Method. Progress of Theoretical Physics 121(6): 1399-1406. doi:10.1143/ptp.121.1399.
• Ogata, K et al. (2006). Determination of \(S_{17}\) from ⁸B breakup by means of the method of continuum-discretized coupled channels. Phys. Rev. C 73(2): 024605.
• Ogata, K; Kan, M and Kamimura, M (2009b). Quantum Three-Body Calculation of the Nonresonant Triple-\(\alpha\) Reaction Rate at Low Temperatures. Progress of Theoretical Physics 122(4): 1055-1064. doi:10.1143/ptp.122.1055.
• Perey(1962). A non-local potential model for the scattering of neutrons by nuclei. Nuclear Physics 32(): 353 - 380. doi:10.1016/0029-5582(62)90345-0.
• Piyadasa, R A; Kawai, M; Kamimura, M and Yahiro, M (1999). Convergence of the solution of the continuum discretized coupled channels method. Phys. Rev. C 60(4): 044611. doi:10.1103/physrevc.60.044611.
• Rawitscher, G H (1974). Effect of deuteron breakup on elastic deuteron - nucleus scattering. Phys. Rev. C 9(6): 2210--2229. doi:10.1103/physrevc.9.2210.
• Rodríguez-gallardo, M et al. (2008). Four-body continuum-discretized coupled-channels calculations using a transformed harmonic oscillator basis. Phys. Rev. C 77(6): 064609. doi:10.1103/physrevc.77.064609.
• Rodríguez-gallardo, M et al. (2009). Four-body continuum-discretized coupled-channels calculations. Phys. Rev. C 80(5): 051601. doi:10.1103/physrevc.80.051601.
• Rubtsova, O A; Kukulin, V I and Moro, A M (2008). Continuum discretization methods in a composite particle scattering off a nucleus: Benchmark calculations. Phys. Rev. C 78(3): 034603. doi:10.1103/physrevc.78.034603.
• Rusek(2000). Breakup couplings in ⁶He+⁴He elastic scattering. Phys. Rev. C 61(3): 034608. doi:10.1103/physrevc.61.034608.
• Summers(2004). ⁷Be breakup on heavy and light targets. Phys. Rev. C 70(1): 011602. doi:10.1103/physrevc.70.011602.
• Summers(2008). Erratum: Core excitation in the elastic scattering and breakup of ¹¹Be on protons [Phys. Rev. C 76, 014611 (2007)]. Phys. Rev. C 77(4): 049901. doi:10.1103/physrevc.77.049901.
• Summers(2008a). Erratum: Extracting \((n,\gamma)\) direct capture cross sections from Coulomb dissociation: Application to ¹⁴C\((n, \gamma)\)¹⁵C. Phys. Rev. C 78(6): 069908. doi:10.1103/physrevc.78.069908.
• Summers(2008b). Extracting \((n,\gamma)\) direct capture cross sections from Coulomb dissociation: Application to ¹⁴C\((n,\gamma)\)¹⁵C. Phys. Rev. C 78(1): 011601.
• Summers, N C; Nunes, F M and Thompson, I J (2006a). Core transitions in the breakup of exotic nuclei. Phys. Rev. C 73(3): 031603. doi:10.1103/physrevc.73.031603.
• Summers, N C; Nunes, F M and Thompson, I J (2006b). Extended continuum discretized coupled channels method: Core excitation in the breakup of exotic nuclei. Phys. Rev. C 74(1): 014606. doi:10.1103/physrevc.74.014606.
• Thompson, I J (1988). Coupled reaction channels calculations in nuclear physics. Computer Physics reports 7(4): 167 - 212. doi:10.1016/0167-7977(88)90005-6.
• Tostevin, J A et al. (2002). Single-neutron removal reactions from ¹⁵C and ¹¹Be: Deviations from the eikonal approximation. Phys. Rev. C 66(2): 024607. doi:10.1103/physrevc.66.024607.
• Tostevin, J A; Nunes, F M and Thompson, I J (2001). Calculations of three-body observables in ⁸B breakup. Phys. Rev. C 63(2): 024617. doi:10.1103/physrevc.63.024617.
• Yahiro, M et al. (1986). Chapter III. \ Effects of Deuteron Virtual Breakup on Deuteron Elastic and Inelastic Scattering. Progress of Theoretical Physics Supplement 89(): 32-83. doi:10.1143/ptps.89.32.
• Yahiro, M; Nakano, M; Iseri, Y and Kamimura, M (1982). Coupled-Discretized-Continuum-Channels Method for Deuteron Breakup Reactions Based on Three-Body Model. Progress of Theoretical Physics 67(5): 1467-1482. doi:10.1143/ptp.67.1467.

Recommended reading

• Thompson, I J and Nunes, F M (2009). Nuclear reactions for astrophysics: principles, calculation and applications of low-energy reactions. Cambridge Univ. Press, Cambridge. 480 Pages. ISBN 9780521856355 Website

External links

• Prof. Filomena Nunes webpage.
• Coulomb wave function at Wikipedia and MathWorld.
• Presentation at UNEDF collaboration meeting by I.J. Thompson, East Lansing, June 2010.

See also

Sponsored by: Dr. Nicolas Alamanos, CEA, IRFU; F-91191 Gif-sur-Yvette, France, France

Reviewed by: Anonymous

Accepted on: 2011-09-07 17:45:35 GMT