Theory of (d,p) and (p,d) reactions including breakup: Comparison of methods

PHYSICAL REVIEW C 80, 064606 (2009) Theory of (d,p)and(p,d) reactions including breakup: Comparison of methods A. M. Moro,1F. M. Nunes,2,3and R. C. Johnson2,4 1Departamento de FAMN, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain 2National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824, USA 3Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA 4Department of Physics, University of Surrey, Guildford GU2 7XH, United Kingdom (Received 15 September 2009; published 7 December 2009) There is an increasing interest in studying transfer reactions to probe the nuclear structure of exotic nuclei. For these loosely bound systems, the role of the continuum needs to be well understood. In this study, we concentrate on (p,d)and(d,p) reactions and compare two formulations for the transfer process that take into account breakup states. Applications to 11Be(p,d)10Be at Elab =38.4MeV/nucleon and 10Be(d,p)11Be at Elab =12.5MeV/nucleon are presented, as is a detailed discussion of convergence rates. DOI: 10.1103/PhysRevC.80.064606 PACS number(s): 24.10.Ht, 24.10.Eq, 25.55.Hp I. INTRODUCTION Progress in the production of rare isotope beams has opened new avenues for detailed studies of the structure of exotic nuclei. These studies often involve transfer reactions in inverse kinematics as they provide a flexible tool to determine the angular momentum and spectroscopic strength of a number of states [1]. Reactions on light targets, such as (d,p), are of particular experimental interest because the Coulomb barrier is minimized. Recent measurements have mostly focused on light nuclei [2–6], but interest in the heavier isotopes is increasing [7,8]. Given the rising experimental programs involving transfer reactions, focused theoretical efforts have revisited the topic, addressing issues such as the dependence on the optical potentials, coupling effects, and single-particle parameters [9–14]. As one moves away from stability, the role of the continuum in low-energy nuclear reactions needs to be better understood. Deuterons are themselves loosely bound, and deuteron breakup was early identified as an important ingredient in the description of transfer reactions [15–19]. In Ref. [15], an adiabatic theory for elastic deuteron scattering and transfer reactions was derived that is nonperturbative and includes the effects of coupling to deuteron breakup channels. The modifications introduced for (d,p) and (p,d) calculations were particularly simple and amounted to a prescription for calculating the distorting potential in the deuteron channel in a distorted-wave Born approximation (DWBA)-like matrix element in terms of nucleon optical potentials. This differed significantly from the deuteron optical potential and generated breakup contributions, which went far beyond the Born approximation. Johnson and Soper’s calculation of deuteron elastic scattering was a continuum discretized coupled channel (CDCC) calculation in which the continuum of breakup channels was represented by a single channel. The generalized CDCC method is a powerful tool for including breakup channels in the reaction mechanism [20,21]. It is also nonperturbative and does not make use of the adiabatic approximation; however, it is computationally expensive. There have been a few applications of CDCC to transfer reactions (e.g., Refs. [5,22–26]), and results demonstrate the variety of phenomena that can be addressed within this same framework. While new developments of the CDCC method are taking place [27–29], here we would like to focus on a better understanding of the transfer matrix element when breakup is included. Starting from a given three-body Hamiltonian, there are many ways of writing the amplitude for A(p,d)Bor B(d,p)A, including projectile excitation and breakup. Of these, a preference exists for those methods that can be generalized to the actual many-body case of interest and that can be used to extract nuclear structure information with a minimal uncertainty associated with the treatment of three-body dynamics. However, numerical difficulties should also be considered. In the present work, we compare two specific methods. In Ref. [30], a method is developed for A(p,d)Band B(d,p)A, where A=B+nis a weakly bound system, using the adiabatic approximation for both the excitation of the exotic nucleus Aand the breakup of the deuteron. This leads to a calculation of the transfer matrix element that is similar to a DWBA calculation, but where the distorted waves are not related to elastic scattering of the composite particle. One important feature of the method introduced in Ref. [30] is that the transfer operator contains only the short-range Vnp interaction. This focuses attention on the value of the three-body wave function at small n-pseparations, where it may be simpler to calculate the full three-body A+n+p wave function. Natural extensions of the model [30]are to include a CDCC wave function for the halolike system, avoiding the adiabatic approximation, and a CDCC wave function for the deuteron, no longer needing the Johnson and Soper simplification [15] or its finite-range extension [19]. Another way of calculating the transfer amplitude is by considering the folding potential of the projectile’s constituents and the target as the main distorting field [31]. This is the natural choice in a CDCC formulation because it corresponds to the diagonal coupling potential appearing in the those coupled equations [20]. In this case, the operator in the transfer matrix element can no longer be reduced to Vnp . If the CDCC wave function is a good representation of the full three-body wave function in the region of space 0556-2813/2009/80(6)/064606(8) 064606-1 ©2009 The American Physical Society A. M. MORO, F. M. NUNES, AND R. C. JOHNSON PHYSICAL REVIEW C 80, 064606 (2009) contributing to the transfer matrix element, then the fully converged results should provide the correct transition amplitude. CDCC calculations involving rearrangement channels can become very cumbersome, and in some cases, a fully converged solution may not be feasible (e.g., Ref. [32]). The key questions addressed in this work are as follows: (i) For a given three-body Hamiltonian, are there significant differences between the predictions of transfer cross sections using different forms of the matrix element when CDCC expansions of the three-body wave functions are used? (ii) What are the limitations of the CDCC expansion as a way of representing the full solution of the three-body Schr¨ odinger equation? (iii) Are there any important computational advantages in using specific forms of the transfer amplitude? (iv) All our calculations are based on a three-body model and hence necessarily involve a simplification of the many-body problem that is the real interest of nuclear structure theory. Does the necessity of linking the threebody model to a many-body theory pick out one of the alternative formulations as being of primary interest? Our approach is to seek the answers to these questions in the context of selected reactions involving a halo nucleus for which there are data, namely, 11Be(p,d)10Be at Elab =38.4MeV/nucleon [2] and 10Be(d,p)11Be at Elab = 12.5MeV/nucleon [33]. The article is organized in the following way: The theoretical background is given in Sec. II, results are presented and discussed in Sec. III, and conclusions are drawn in Sec. IV. II. THREE-BODY MODEL OF ( p,d) AND (d,p) REACTIONS We consider the reaction A(p,d)B. Our starting point is a three-body model Hamiltonian, H=K+Vnp +VnB +UpB,(1) where Kis the total kinetic energy operator, Vnp is the protonneutron interaction, VnB is the neutron-Binteraction, and UpB is the proton-Binteraction. We use the notation Vfor real potentials and Ufor complex potentials. We assume that VnB is real because we are interested in transfers from bound neutron states in a real well, and we want all our calculations to derive from a fixed Hamiltonian. Because of the limitations of the CDCC approximation for the three-body wave function, transfer amplitudes cannot be evaluated by examining the CDCC wave function in the appropriate asymptotic region. There are, however, various ways of expressing the transition amplitude for the reaction A(p,d)Bas a matrix element involving the exact solution of the three-body Schr¨ odinger equation in finite regions of configuration space only. An exact prior form amplitude for this process is given by Tprior =(−) dB |Vnp +UpB −U|(+) pA,(2) ′ ′ FIG. 1. Coordinates used for the entrance and exit channels of the transfer reaction A(p,d)B. where (−) dB is the exact (three-body) wave function with incoming boundary conditions corresponding to the final channel. The initial wave function (+) pA is defined in terms of an auxiliary potential U, which may be complex and is the solution of [E+i −Kr−KR−VnB −U](+) pA(r,R) =+ıφA(r)exp(ıKp·R),(3) where rand Rare the n-Band p-AJacobi coordinates defined in Fig. 1,Kpis the incident proton momentum, and φAis the initial n-Bbound state. It is important to emphasize that whatever the choice of the auxiliary potential U, the three-body Hamiltonian remains the same. In Sec. II A, we first examine the formalism for two selected choices of Uin the prior form matrix element Eq. (2). In Sec. II B, we extend our study to post form matrix elements. A. Dependence of the prior form Tmatrix on U 1. The choice U =UpB If (−) dB and (+) pA are exact solutions of their respective three-body Schr¨ odinger equations, the expression in Eq. (2) is independent of Ufor a wide class of possible choices of U. The form of the Tmatrix for specific choices of U may suggest certain approximations to the three-body wave functions, instead of the full solution. Goldberger and Watson [34] introduce the choice U= UpB(RpB)inEq.(2), which was also the choice of Timofeyuk and Johnson [30]. With this choice, Eq. (2) becomes Tprior(Vnp )=(−) dB |Vnp |˜ (+) pA.(4) A derivation of this matrix element from the three-body Alt, Grassberger, and Sandhas (AGS) equations [35] can be found in Ref. [36]. From Eq. (3), we can see that the initial wave function ˜ (+) pA in Eq. (4) is the solution of the three-body Hamiltonian with Vnp removed: [E+i −Kr−KR−VnB −UpB]˜ (+) pA(r,R) =+ıφA(r)exp(ıKp·R).(5) Note that in the limit mn/MB→0, when recoil effects are negligible, ˜ (+) pA becomes a product of the initial bound-state neutron wave function in Aand a proton scattering wave on a target Abut is distorted by the potential UpB. Recoil effects that excite Awere discussed by Timofeyuk and Johnson [30] 064606-2 THEORY OF (d,p) AND (p,d) REACTIONS ... PHYSICAL REVIEW C 80, 064606 (2009) and are referred to as recoil excitation and breakup (REB) effects. In Ref. [30], the following advantages of using Eq. (4) to calculate Tprior were emphasized: (i) Only nucleon-B potentials are needed as input, unlike for the standard DWBA amplitude, for which an optical potential that fits deuteron elastic scattering and an optical potential for p+Ais required, and (ii) Vnp, the interaction that appears in the matrix element is short ranged, which means that (−) dB is only needed within the range of the n-pinteraction. The obvious disadvantage is that there is no simple separation of coordinates in ˜ (+) pA, and thus the solution of Eq. (5) is not trivial, except for heavy targets. In Ref. [30], further simplifications were made in the calculation of Eq. (4). As mentioned in Sec. I,thewave function (−) dB , which contains both elastic deuteron scattering and breakup, is treated using the adiabatic method of Johnson and Soper [15,16] with finite-range corrections included, as discussed by Wales and Johnson [19]. The function ˜ (+) pA, which contains components describing elastic, inelastic, and breakup of Aby the incident proton, was also calculated within an adiabatic approximation [17,37]. Under these two approximations, the transfer amplitude has a structure similar to a DWBA matrix element, tremendously simplifying the calculations, but the distorted waves are not the usual elastic scattering wave functions, as in DWBA. It is important to realize that despite this formal similarity to DWBA, no Born approximation is involved in the derivation of those expressions. In the present work, we use Eq. (4) without making the preceding approximations. Instead, we replace the three-body wave functions, (−) dB and ˜ (+) pA, by their CDCC approximations, which we discuss in Sec. II C. 2. The choice U =Ufold pA Another possible choice for the auxiliary potential is the folded potential, defined by Ufold pA (R)=φnB |Vnp +UpB|φnB .(6) This is the diagonal coupling in the CDCC equations [20]for (+) pA and thus is a natural choice when the CDCC method is used. As this potential depends only on R, the wave function (+) pA has the factorized form (+) pA(r,R)=χ(+) pA (R)φA(r)(7) and describes elastic scattering of an incident proton on the target A, distorted by a folding potential Ufold. The exact Tmatrix in this case is Tfold prior =(−) dB |Vnp +UpB −Ufold pA |χ(+) pA φnB .(8) It is important to notice that the use of Ufold to calculate χ(+) pA (R) does not imply an approximation: In fact, Eq. (8) is exact if (−) dB is the exact three-body wave function. In our applications, we take the CDCC approximation to this wave function, as in Sec. II A1. Here, again, no additional potentials are introduced, and thus the problem is fully determined by the initial threebody Hamiltonian. The main advantage of this method over Tprior(Vnp ) is the simplification in (+) pA. The disadvantage resides in the complicated operator UpB −Ufold pA , which enhances the nonlocality of this matrix element. B. Dependence of the post form Tmatrix on U The three-body Schr¨ odinger equation can be used to transform the transition matrix element Eq. (2) into its post form, which also involves an auxiliary potential U. This will be referred to as Tpost and is given by Tpost =(−) dB |VnB +UpB −U|(+) pA ,(9) where (+) pA is the exact solution of the three-body Schr¨ odinger equation corresponding to a plane wave proton incident on A and outgoing waves in all other channels, and where (−) dB satisfies [E−i −Kr−KR−Vnp −U∗](−) dB (r,R) =−ıφd(r)exp(ıKd·R).(10) As in Sec. II A, we test the dependence in the calculation of the transition amplitude in Eq. (9) on the auxiliary potential by comparing results for two choices of U. 1. The choice U =UpB This case is the analog of the Timofeyuk-Johnson Tprior(Vnp ). Here the binding potential in the initial state appears as the transition operator in the matrix element. The Tmatrix is Tpost(VnB )=˜ (−) dB |VnB |(+) pA .(11) The potential VnB does not appear in the equation satisfied by final-state wave function [E−i −Kr−KR−Vnp −U∗ pB]˜ (−) dB (r,R) =−ıφd(r)exp(ıKd·R).(12) The state ˜ (−) dB describes a nonphysical situation in which the neutron and proton in the outgoing deuteron interact with each other, but only the proton interacts with B. Nevertheless, if (+) pA is represented by an accurate solution of the three-body Schr¨ odinger equation, the results obtained with Eq. (11) should be the same as those from Eq. (4). 2. The choice U =Ufold dB Similarly to the prior case, one can choose the folding potential UdB as the auxiliary potential: Ufold dB (R)=φd|VnB +UpB|φd.(13) This is the diagonal coupling in the CDCC equations [20] for (+) dB and therefore corresponds to a standard choice when CDCC is used in the post form of the transition amplitude. Because Ufold dB does not depend on r,(−) dB factorizes in a distorted wave and the deuteron ground state. The post form 064606-3 A. M. MORO, F. M. NUNES, AND R. C. JOHNSON PHYSICAL REVIEW C 80, 064606 (2009) Tmatrix is then given by Tfold post =χ(−) dB φnp |VnB +UpB −Ufold dB |(+) pA ,(14) where χ(−) dB is the deuteron distorted wave associated with the potential Ufold dB (R). This matrix element is exact if the initial three-body wave function (+) pA is exact. In our applications, we use the CDCC approximation to calculate (+) pA . Even though there are formal similarities between Eq. (8) and Eq. (14), the difference in the operator produces a tremendous difference in the model space needed for convergence. First, the binding potential appearing in the prior form (Vnp ) has a much shorter range than that appearing in the post form (VnB ). Second, the operator UpB −Ufold pA in Tprior(fold) is rather different from UpB −Ufold dB in Tpost(fold). The difference between deuteron and proton potentials is much more significant than the difference between proton potentials of neighboring nuclei (even if Ais a halo). For these two reasons, we can expect nonlocality effects in the transfer matrix elements to be more important in Tpost(fold) than in Tprior(fold). C. Using CDCC for the three-body wave function All the Tmatrices in this article are calculated using the CDCC method [20]. More specifically, in the case of the prior form matrix elements introduced earlier, both (−) dB and ˜ (+) pA in Eq. (4) are obtained using the CDCC expansion and by solving the CDCC coupled channel equation, and the same CDCC solution for (−) dB is used in Eq. (8). For the post form matrix elements, CDCC expansions are used for ˜ (−) dB and (+) pA in Eq. (11) and (+) pA in Eq. (14). In the CDCC method, the exact wave function (−) dB appearing in both Eqs. (4) and (8) is replaced by an expansion in terms of a truncated and discretized set of eigenstates of the n-pHamiltonian Kr+Vnp (r). The coefficients in the expansion are determined from a coupled-channel version of the equation [E−Kr−KR−Vnp −VnB −U∗ pB](−) dB (r,R)=0,(15) with the boundary condition that there be a plane wave in the deuteron channel and incoming scattered waves in all other channels. ˜ (−) dB is obtained in a similar way, but with Vnp removed from Eq. (15). Similarly, in Eqs. (11) and (14), the exact initial-state wave function (+) pA is replaced by an expansion in eigenstates of the n-Bsystem, with Hamiltonian Kr+VnB (r). The solutions approximately satisfy [E−Kr−KR−Vnp −VnB −UpB]˜ (+) pA (r,R)=0,(16) with the boundary condition that there be a plane wave in the incoming proton channel and outgoing scattered waves in all other channels. Again, ˜ (+) pA is calculated using the preceding equation with VnB removed. The CDCC functions for the initial p+Astate and final d+Astate involve expansions in different complete sets and depend on different Jacobi coordinates; therefore we expect TABLE I. Optical potentials used in our applications. Ebeam V0r0a0WvWdriai (MeV) (MeV) (fm) (fm) (MeV) (MeV) (fm) (fm) p-10Be 38.451.21.114 0.57 19.5 0 1.114 0.50 p-10Be 12.568.461 1.188 0.469 0 15.629 1.043 0.328 n-10Be 51.639 1.39 0.52 n-p72.15 1.484 the sizes of the model spaces to be different in the various methods: Tprior(Vnp ) and Tpost(VnB ),T prior(fold) and Tpost(fold). Regarding the methods described by Eq. (4) and Eq. (11), one might naively think that there is double counting of the three-body continuum because there are CDCC discretizations for both initial and final states. This is not the case. The two states involve different three-body Hamiltonians in a matrix element, which emphasizes a particular subspace of the six-dimensional coordinate space. The full three-body Schr¨ odinger equation guarantees that the overlaps of these functions, when evaluated as in Eq. (4) and Eq. (11), produce the correct transition amplitude. III. RESULTS We first consider the reaction 11Be(p,d)10Be at Elab = 38.4MeV/nucleon. Table Icontains the parameters for all interactions used in this application, where V0,r 0, and a0are the depth, radius, and diffuseness of the real part, respectively; Wv,W d,r i, and aiare the depth of the volume and surface imaginary potentials and their radius and diffuseness, respectively. For p+10Be, we use an optical potential of Woods-Saxon shape, which reproduces elastic scattering at Elab/A =39.1MeV[38]. (In our formulation, the Hamiltonian is energy-independent; thus this is a reasonable choice for the reactions under study, although not unique.) The n+10Be interaction is obtained fixing the diffuseness and the radius and adjusting the depth to reproduce a 2sstate with the correct neutron separation energy (Sn=0.504 MeV). For the deuteron, we use a central Gaussian interaction, reproducing the binding energy and the radius of the ground state of the n-psystem. All calculations are performed using the code FRESCO [39]. For the purpose of our study, interactions need to be sensible and qualitatively relate to the physical systems. However, as the objective of this work is not to extract precise structure information, but rather to compare different reaction methods, spin dependence or tensor terms in the interactions would be an unnecessary complication. Also, the n+10Be interaction (VnB ) appearing in the matrix operator of Eq. (14) needs to be real and independent of angular momentum because of the limitations of the FRESCO implementation. In Fig. 2, we show the results for the prior form: Tprior(fold) (Fig. 2, top) and Tprior(Vnp ) (Fig. 2, bottom). In Tprior(fold), deuteron breakup is included through CDCC in the final state. Convergence of the angular distribution up to 60◦is obtained when n-ppartial waves up to lf=4 are included. In Tprior(Vnp ), in addition to the CDCC wave function for the 064606-4 THEORY OF (d,p) AND (p,d) REACTIONS ... PHYSICAL REVIEW C 80, 064606 (2009) 0 1020304050 60 θc.m. (deg) 0.01 0.1 1 10 dσ/dΩ (mb/sr) li=0,1, lf=0-2 li=0,1, lf =0-4 li=0,1, lf=0-6 Tprior (Vnp) 0 1020304050 60 θc.m. (deg) 0.01 0.1 1 10 dσ/dΩ (mb/sr) lf =0-2 lf=0-4 lf=0-6 Tprior (fold) FIG. 2. (Color online) Transfer cross sections for 11Be(p,d)10Be at Ep=38.4 MeV calculated in the prior form: convergence with respect to the number of partial waves for n-10Be (li)andp-n(lf). Data are taken from Ref. [2]. deuteron, we also need the CDCC wave function for the entrance channel [see Eq. (4)]. Convergence in li(n-10Be partial waves) is rapid and requires only sand pwaves. We also have to consider the sum over the projectile-target angular momentum L. Convergence is obtained for Lmax =25. As to the radial grid, integration out to 40 fm with steps of 0.1fm was found sufficient. Figure 2shows that the converged angular distributions obtained with Tprior(fold) and Tprior(Vnp ) are in very good agreement. We also verified that there were no significant differences in either the size or speed of the calculations. We now turn our attention to the post representation. In Fig. 3, we present the convergence study for Tpost(fold) (Fig. 3, top) and Tpost(VnB ) (Fig. 3, bottom). Within Tpost(fold), the 11Be breakup is included through CDCC in the initial state; thus liis the relevant quantum number in the expansion. We find that the angular distribution out to 60◦reaches convergence for li⩽6. For Tpost(VnB ), both entrance and exit channels are expressed in terms of CDCC, so convergence with respect to liand lfshould be considered [see Eq. (11)]. Full convergence in this case was not feasible. It is clear from Fig. 3that lf=6 is still not enough for the outgoing channel, and even though it appears that the calculation has converged with respect to li, the model space for the incoming channel might need to be increased if more partial waves are included in the outgoing channel. We 0 1020304050 60 θc.m. (deg) 0.01 0.1 1 10 dσ/dΩ (mb/sr) li=0-4 lf=0-4 li=0-6 lf=0-4 li=0-8 lf=0-4 li=0-4 lf=0-6 Tpost (VnB) 0 1020304050 60 θc.m. (deg) 0.01 0.1 1 10 dσ/dΩ (mb/sr) li=0-2 li=0-4 li=0-6 Tpost (fold) FIG. 3. (Color online) Transfer cross sections for 11Be(p,d)10Be at Ep=38.4 MeV calculated in the post form: convergence with respect to the number of partial waves for n-10Be (li)andp-n(lf). Data are taken from Ref. [2]. find that the angular distributions for the transfer calculated using Tpost(fold) are in reasonable agreement with those using Tprior(fold), particularly at forward angles, which is the region more relevant for the extraction of spectroscopic information. This is to be expected when starting from the same three-body Hamiltonian. The small differences remaining test the level of accuracy of the CDCC representation of the full three-body wave function. The cross sections calculated with Tpost(VnB )do not agree with Tpost(fold). Because the calculations presented for Tpost(VnB ) are not converged, we do not believe that there is physical significance in this difference. Computationally, the converged Tpost(fold) calculation is somewhat more demanding than the converged Tprior(fold) calculation, regarding both speed and memory requirements, because of the need for a larger model space. The Tpost(VnB ) calculation, in addition to the lack of convergence, is by far the most demanding computationally. This result shows how important the choice of the appropriate representation of the transition amplitude can be in practical calculations, despite the formal equivalence between the different representations. To better understand the characteristics of the matrix elements in these different approaches, we present in Table II the range of nonlocality of the transfer kernels (RNL) required for the four types of calculations as well as a summary of the partial waves used. As Tprior(Vnp ) involves only the 064606-5 A. M. MORO, F. M. NUNES, AND R. C. JOHNSON PHYSICAL REVIEW C 80, 064606 (2009) TABLE II. Convergence parameters for the different methods, where RNL refers to the range of nonlocality in the transfer couplings (i.e., R-R). lilfRNL (fm) Prior(fold) – 4 14 Prior(Vnp )1 4 5 Post(fold) 6 – 21 Post(VnB )6>620 Vnp interaction in the operator, we expect this range to be small as compared with Tprior(fold), which contains the term UpB −Ufold pA . This is verified in practice (see Table II). As mentioned in Sec. II B,fortheTpost(VnB ) and Tpost(fold) formalisms, instead of Vnp , the binding potential for n-11Be is used in the operator. Consequently, the range required is much larger than for the prior methods. As mentioned earlier, we have chosen a central spinindependent effective interaction for the n-10Be system. This interaction does not reproduce the excited 1/2−bound state in 11Be, bound by 0.18 MeV. Using the prior representation, we have checked that this bound state has a negligible effect in the reaction mechanism for this reaction. In Figs. 2and 3, we show data along with the theoretical predictions. This serves as an indication that our starting three-body Hamiltonian, Eq. (1), is not too far from reality. We emphasize, however, that we cannot expect our predicted transfer angular distributions to be appropriate for extracting spectroscopic factors or other nuclear structure information. As discussed earlier, in our model, there is no imaginary part in the n-10Be interaction, when in reality, there should be an imaginary component of similar magnitude to the proton optical potential. Also, our calculations implicitly assume a unit spectroscopic factor (norm of the n-10Be overlap function [1]), but it is not obvious what will be the effect on our results if the spectroscopic factor were to differ from this value because then the effective three-body Hamiltonian would need to include core internal degrees of freedom, and several overlap functions might contribute to the process. We have also studied the 10Be(d,p)11Be reaction at Ed= 25 MeV. All interactions apart from the proton optical potential are kept the same. The proton optical potential taken at half the deuteron energy (shown in Table I) is the same as that used in Ref. [30]. For the (d,p) case, we have only performed calculations in the post representation. Note that the post form matrix elements for the (d,p) reaction are similar to the prior form matrix elements for the (p,d) reaction. Here we only show the converged results. For the Tpost(fold) calculation, convergence was achieved with li=0,2,4 partial waves for the p-ncontinuum, whereas for the Tpost(Vpn) calculation, we needed, in addition, lf⩽ 2forthen-10Be continuum. The odd partial waves for the deuteron continuum were found to have a negligible effect on the (d,p) cross section. As expected, the range of nonlocality required for the Tpost(Vpn) calculation (RNL = 5 fm) was significantly smaller than that for the Tpost(fold) calculation (RNL =15 fm). This difference may represent a 0 1020304050 60 70 θc.m. (deg) 0.1 1 10 100 dσ/dΩ (mb/sr) Tpost (Vpn): li<4, lf<2 Tpost (fold): li<4 Tpost (Vpn): li<4, lf<2, UnB=UpB FIG. 4. (Color online) Transfer cross sections for 10Be(d,p)11Be at Ed=25 MeV: comparison of Tpost(Vnp )andTpost(fold). Data are takenfromRef.[33]. significant numerical advantage of the Tprior(Vpn) representation in practical calculations. In Fig. 4, we compare the calculations using the two post representations, along with the experimental data from Ref. [33]. The two calculations are in good agreement, but they depart considerably from the data. At this stage, it is worth recalling that we have restricted ourselves to a very simple Hamiltonian, with only one complex interaction (UpB) and a very simplified description of the 11Be spectrum because of our fixed n-10Be interaction. We have therefore performed another calculation using a complex n-10Be interaction for the incident (deuteron) channel (UnB = UpB) and a real parity-dependent interaction for n-10Be in the final state, which is identical to the one in Table I, except for the =1 depth, which is adjusted to reproduce the 1p1/2 bound state in 11Be. Such a combination of potentials would have made the comparisons in Sec. II impossible. The result of this calculation, using the Tpost(Vpn) formula, is displayed in Fig. 4with a dash-dotted line. This new calculation brings the shape of the angular distribution in much better agreement with the data, illustrating the dependence of the results on the underlying three-body Hamiltonian. There have been attempts in the past [40,41] to decide on the relative merits of post- and prior-form DWBA matrix elements by comparison with experiment. We emphasize that none of the calculations reported here are DWBA calculations and that the issues discussed here cannot be decided by comparison with experiment. We address solely the question of accurate and practical evaluation of the predictions of three-body models and which methods are most simply generalized to the manybody case, with the aim of extracting credible nuclear structure information. IV. CONCLUSIONS We have found that for the 11Be(p,d)10Be reaction, there is very little to choose between the two Tmatrices Tprior(Vnp ) and Tprior(fold) from the point of view of convergence and 064606-6 THEORY OF (d,p) AND (p,d) REACTIONS ... PHYSICAL REVIEW C 80, 064606 (2009) convenience. The same can be said about the corresponding post form results for the 10Be(d,p)11Be reaction. The agreement between converged results implies that within the subspaces of the Jacobi-coordinate six-dimensional space required for the two methods, the CDCC wave functions represent the exact three-body wave functions with similar accuracy. Our study of 11Be(p,d)10Be using the post form clearly shows the difficulties one might encounter when the transition operator contains interactions with longer range. Our study indicates that for transfer observables, it is best to use the prior form of the matrix elements for (p,d) reactions and the post form for (d,p) reactions. From the point of view of linking three-body models with the underlying many-body theory, the two matrix elements Tprior(Vnp ) and Tprior(fold) have a very different significance [17]. In the many-body generalization of Tprior(Vnp ), the fact that Vnp does not depend on the internal coordinates of the core B(in our case, 10Be) means that nuclear structure information appears entirely through the overlap function φAB (rn)ofA and B. The question of what nuclear structure information can be deduced from a particular transfer experiment then reduces to the question of what features of φAB (spectroscopic factor, asymptotic normalization constant, etc.) an experiment at a particular incident energy or angular range is sensitive to. The same consideration applies when it is necessary to include excitations of Aand/or Bin the matrix element because then, additional overlap functions appear. In the method in which the folding potential is used, Tprior(fold), the transfer matrix element is not as attractive when generalized to the many-body case. Because of the complicated transfer operator in Eq. (8), the matrix element does not involve a single overlap function but, in principle, can involve a large number of overlap functions as well as matrix elements of the many-body interaction VpB between different states of B. This study is particularly relevant for light systems. As the mass of the target increases, both formalisms Tprior(Vnp ) and Tprior(fold) simplify, and the practical differences found in our applications will not arise. For heavy targets, the initial state in Tprior(Vnp ) factorizes as in the CDCC approach, and the complicated operator in Tprior(fold) reduces to Vnp .Inthis limit, these two approaches become equivalent. This same 11Be(p,d) reaction has been used for a comparison between CDCC and the solution of the full threebody integral equations (known as the AGS equations) [38]. The calculations here denoted as Tprior(fold) correspond to CDCC-TR* in Ref. [38], apart from minor differences in the interactions. The comparison with Faddeev shows an ≈15% difference between the two calculations, demonstrating the level of accuracy of Tprior(fold). Writing the matrix element in terms of the short-range Vnp ,asinTprior(Vnp ), held promise in solving this disagreement because then the CDCC representation of dB is at its best. However, our present study suggests that there are still inaccuracies, arising probably from the CDCC expansion of the initial state pA because there is no truncation of the matrix element with regard to the n-10Be distance. ACKNOWLEDGMENTS This work was partially supported by National Science Foundation Grant No. PHY-0555893, D.O.E. Grant No. DEFG52-08NA28552, and the Elsevier Foundation. A.M.M. and R.C.J. wish to acknowledge the generous hospitality of NSCL. The work of R.C.J. is partially supported by the United Kingdom Science and Technology Facilities Council (STFC) under Grant No. ST/F012012. 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