Three-body model for the analysis of quasifree scattering reactions in inverse kinematics

PHYSICAL REVIEW C 92, 044605 (2015) Three-body model for the analysis of quasifree scattering reactions in inverse kinematics A. M. Moro* Departamento de F´ ısica At´ omica, Molecular y Nuclear, Facultad de F´ ısica, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain (Received 29 July 2015; published 12 October 2015) A new method to calculate cross sections for (p,pn)and(p,2p) reactions measured under inverse kinematics conditions is proposed. The method uses the prior form of the scattering transition amplitude and replaces the exact three-body wave function appearing in this expression with an expansion in terms of p-nor p-pstates, covering the physically relevant excitation energies and partial waves. A procedure of discretization, similar to that used in continuum-discretized coupled-channels calculations, is applied to make this expansion finite and numerically tractable. The proposed formalism is nonrelativistic, but several relativistic kinematical corrections are applied to extend its applicability to energies of current interest. The underlying optical potentials for the entrance and exit channels are generated microscopically by folding an effective density-dependent Gmatrix with the density of the composite nucleus. Numerical calculations for 12C(p,2p), 12C(p,pn), and 23O(p,pn)at ∼400 MeV/nucleon are presented to illustrate the method. The role of final-state interactions and Pauli principle between the outgoing nucleons is also discussed. DOI: 10.1103/PhysRevC.92.044605 PACS number(s): 25.60.Gc,24.10.Eq,25.45.De,25.40.Ep I. INTRODUCTION Quasifree scattering (QFS) experiments of the form (p,pn) and (p,2p) [hereafter (p,pN)] have been used extensively as a tool to extract spectroscopic information of proton-hole and neutron-hole states in nuclei, such as separation energies, spin-parity assignments, and occupation probabilities. In these reactions, an energetic proton beam (E>100 MeV) collides with a stable target nucleus, removing one or more nucleons, and leaving a residual nucleus, either in its ground state or in an excited state. Recently, the technique has been extended to the study of unstable nuclei, using inverse kinematics, i.e., bombarding a hydrogen target with an energetic radioactive beam. This technique is analogous to the knockout experiments with composite targets used extensively in the past years [1–5]. Although both kinds of experiments are meant to provide similar information, there are important differences between them. In knockout reactions, a nucleon is suddenly removed from the fast-moving projectile after colliding with a light target nucleus, like 9Be. Owing to the strongly absorptive nature of the core-target interaction at these energies, the process is highly peripheral and hence mainly dependent on the tail of the wave function of the removed nucleon or, more correctly, on the overlap function between the projectile and residual core wave functions. The norm of this overlap is the spectroscopic factor, a quantity that is directly related to the occupation probability of a given single-particle orbital. Because this norm depends on the full overlap, and not just on its tail, this raises the question of the reliability of the spectroscopic information extracted from a process that is only sensitive to a small piece of the wave function. However, (p,pN) reactions are expected to be more sensitive to deeper portions of the wave function and so they should help to reduce these ambiguities. Consequently, the information obtained *[email protected] from these experiments will be complementary to that obtained from heavy-ion knockout reactions at intermediate energies and from transfer reactions at lower energies. Some leading facilities, such as GSI (Germany), RIKEN (Japan) and NSCL/MSU (USA), have plans to perform inverse kinematics experiments with exotic beams. Consequently, it is of timely importance to revisit theoretical methods to analyze these kinds of processes. Theoretical analyses of the (p,pN) reactions with stable nuclei have been commonly done using the distorted-wave impulse approximation (DWIA) [6]. Roughly speaking, the impulse approximation (IA) means that the binding potential of the removed particle can be neglected in comparison with the projectile-target kinetic energy (see, e.g., Ref. [7], Chap. 11). At the energies usually employed in QFS experiments (several hundreds of MeV per nucleon) this approximation is expected to be justified. The DWIA method is usually formulated in terms of the nucleon-nucleon transition matrix (Tmatrix hereafter); the IA involves the replacement of this operator with a free Tmatrix between the incident proton and the struck nucleon. Practical implementations of the DWIA formalism commonly involve further approximations, such as the substitution of this Tmatrix by its on-shell value, or its representation by a zero-range operator. If reliable structure information is to be extracted from these experiments, these approximations need to be revisited and tested. Consequently, the validity of the DWIA formalism should be investigated comparing with more elaborate reaction theories [8–10]. A three-body reaction framework which does not make use of the IA is the continuum-discretized coupled-channels (CDCC) method [11]. This method has been very successful in the analysis of reactions induced by weakly bound projectiles at low and medium energies. The method has been also applied to one-neutron removal reactions on proton targets at intermediate energies (∼40–70 MeV/nucleon) [12,13]. For a knockout reaction of the form A(p,pN)C, the standard CDCC method aims at expanding the three-body wave function of the system in terms of N−Ceigenstates. To make the 0556-2813/2015/92(4)/044605(13) 044605-1 ©2015 American Physical Society A. M. MORO PHYSICAL REVIEW C 92, 044605 (2015) expansion finite, the N−Ccontinuum must be truncated in energy and angular momentum and discretized. However, the large angular momentum and energy transfer found in these reactions makes that convergence of the observables requires a very large model space. For energies of current interest, of several hundreds of MeV/u, the method becomes unpractical. Benchmark calculations with the Faddeev/AGS method [14]forthe 11Be(p,pn) reaction at ∼35 MeV/u showed that, while the CDCC expansion of the breakup states in terms of n-10Be states converged very slowly with the size of the model space, the alternative expansion in terms of p-nstates required a much smaller model space to reproduce the dominant part of the 10Be inclusive cross sections. This alternative expansion makes use of the prior-form representation of the transition amplitude, in which the three-body wave function is approximated by an expansion in a basis of p+Nstates. A procedure of continuum discretization, similar to that used in standard CDCC calculations, is used for the p-Nstates. The resultant expression is formally similar to the coupled-channel Born approximation (CCBA) expression commonly used in transfer reactions and, therefore, in some previous applications [15], the method has been referred to as transfer to the continuum method. This allows its implementation in standard coupled-channels codes after some suitable modifications. In this paper, I present some exploratory calculations to illustrate the application of this method to the interpretation of (p,2p) and (p,pn) experiments in inverse kinematics. Although the general formulation of the method has been presented before [14,15], it is described in more detail here and some suitable prescriptions for its input ingredients (internal wave functions, optical potentials, etc.), as well as several relativistic kinematic corrections (to make the model applicable to higher energies), are discussed. The role of Pauli principle and final-state interactions of the outgoing nucleons is also discussed. The calculations here presented are for relatively high energies (∼400 MeV/u), but, because the method does not rely on the IA, it might be applicable also to lower energies, for which the DWIA may not be adequate. The paper is structured as follows. In Sec. II the theoretical formulation of the method is presented. This section discusses also the choice of the NN interaction, which is the most responsible for the (p,pN) process, the construction of appropriate optical potentials including in-medium effects, and some relativistic corrections applied to the model. In Sec. III the method is applied to the 12C(p,pN) and 23O(p,pn) reactions at E∼400 MeV/nucleon. In Sec. IV the connection with the DWIA method is discussed. Finally, Sec. Vsummarizes the main results of this work. II. THEORETICAL MODEL A. The transition amplitude Let us consider a reaction of the form A+p→C(α)+p+N, (1) in which an incident composite nucleus A=C+Ncollides with a proton target, losing a nucleon (proton or neutron) and FIG. 1. (Color online) Diagram for a (p,pN) reaction in inverse kinematics, modeled as a binary process. giving rise to a residual core nucleus (C) in some definite state αand two outgoing nucleons (p+nor p+p). The process is schematically depicted in Fig. 1. Using the prior form of the transition amplitude, the exact transition amplitude for this reaction can be written as Tif (α)=(−) α,f |VpN +VpC|φA(ξA)ei KpA  R,(2) where φA(ξA) represents the ground-state wave function of the nucleus A, with ξAdenoting its internal coordinates. The plane wave ei KpA  Rdescribes the relative motion of the p+ Asystem. At this stage, the potential VpC is a many-body operator containing the interaction of the target proton with all the core nucleons. The final function (−) α,f is the exact scattering wave function subject to the boundary conditions consisting of a plane wave (or Coulomb wave) in the channel f(corresponding to some definite state for the relative motion of the three outgoing fragments), with the core in state αand ingoing spherical waves in all other open channels. This wave function is a solution of the many-body Schr¨ odinger equation, [E−−Kr−KR−VpN −V† pC −V† NC](−) α,f (r, R,ξC)=0, (3) where E−=E−i and ξCdenotes the core internal coordinates. Note that the final three-body state has been expressed in terms of the Jacobi coordinates {r, R}(see Fig. 1). Equation (2) gives the exact transition amplitude of the many-body scattering problem. It can be formally reduced to an effective three-body problem, using the approximation, (−) α,f (r, R,ξC)≃3b(−) f(r, R)φα C(ξC),(4) where φα C(ξC) is the core wave function in the state αand 3b(−) f(r, R) is a three-body wave function obtained as a solution of the effective Schr¨ odinger equation, [E−−Kr−KR−VpN −U† pC −U† NC]3b(−) f(r, R)=0, (5) where UpC and UNC are now effective nucleon-nucleus interactions which, in practice, will be replaced with optical model potentials at the appropriate energy per nucleon. 044605-2 THREE-BODY MODEL FOR THE ANALYSIS OF . . . PHYSICAL REVIEW C 92, 044605 (2015) In this three-body model, the transition amplitude is T3b if (α)=3b(−) fφα C(ξC)VpN +UpCφA(ξA)ei KpA  R.(6) If the potential UpC is taken to be independent of the internal coordinates of C(ξC), as it is usually assumed, one can perform the integral over these internal coordinates to give dξCφα C(ξC)φA(ξA)=Sα,,j ϕα CA(r),(7) where Sα,,j φα CA(r) is an overlap wave function, with φα CA(r) a unit normalized wave function depending on the relative coordinate of the removed particle with respect to the core and Sα,,j the spectroscopic factor. Using Eq. (7)inEq.(6) one finds T3b if (α)=Sα,,j 3b(−) fVpN +UpCϕα CAei KpA  R.(8) As done in transfer calculations, it is convenient to introduce an auxiliary potential in the incoming channel, UpA( R), so the previous equation transforms to T3b if (α)=Sα,,j 3b(−) fVpN +UpC −UpAϕα CAχ(+) pA ,(9) where χ(+) pA is the distorted wave generated by the potential UpA( R). Note that, when 3b fis the exact solution of Eq. (5), the amplitude (9) is strictly independent of the choice of UpA. In practical calculations, in which 3b fmust be approximated somehow, the result will, however, depend on this potential. The usual choice is to use for UpA an optical potential describing the elastic scattering of the p+Asystem. With this choice, one expects that the difference UpC −UpA (the so-called remnant term) will contribute little to the integral and hence the matrix element will be mostly determined by the VpN interaction. To reduce Eq. (5) to a tractable form, 3b(−)is expanded in terms of p+Neigenstates, i.e., 3b(−) f(r, R)= jπdkφjπ(k,r)χj,π( K,  R),(10) where  kis the relative wave number of the p+Npair,  Kis that for the relative motion between the residual nucleus C and the p+Npair, and χj,π( K,  R) is a function describing the relative motion of the p+Nsystem with respect to the residual nucleus, when the former is in a given final state {k,jπ}. (Note that, in actual calculations, this expansion will be done in terms of states of total angular momentum J.To simplify the discussion, the notation used here is somewhat schematic.) The states φjπ(k,r) are the eigenstates of the p+NHamiltonian using the potential VpN (r). In the (p,pn) case, the expansion (10) will contain also a term for the deuteron ground state. This term is omitted for simplicity of the notation. In the (p,2p) case, the p+pwave function must be antisymmetrized to account for the indistinguishability of the outgoing protons. In practice, this restricts the allowed values of jπand introduces a factor of 2 in the calculated cross sections with respect to the unsymmetrized calculation.1 Energy conservation in the final channel (nonrelativistic by now) implies that E=epN +Ec.m.=2k2 2μpN +2K2 2μpN,C ,(11) where Eis the total energy of the system, epN the relative energy of the p+Npair (epN =−2.22 MeV for the deuteron ground state), and Ec.m.the kinetic energy associated with the relative motion of the core with respect to the center of mass (c.m.) of the p+Npair. The functions χj,π( K,  R) could, in principle, be obtained by inserting the expansion (10) into Eq. (5). However, because these functions depend upon the continuous parameter K,this would give rise to an infinite number of equations. In practice, one may use a discretization procedure similar to that used in the CDCC method [11], in which the final p+Nstates are grouped (binned) in energy or momentum intervals as 3b(−) f≈CDCC f= n,j,π φjπ n(kn,r)χn,j,π ( Kn, R),(12) where knare some average values for the discretized p-N energies and φjπ n(kn,r) the bin wave functions. Note that the subscript fin CDCC fretains the information on the final state, corresponding to some definite values of n,j, and π. Details on the construction of these bins can be found elsewhere [11,16]. The angular differential cross section for a given final discretized bin f={n,j,π}and a given core state αis dσn,j,π (α) dc=1 (2sp+1)(2JA+1) ×μiμf (2π2)2 Kn Ki σT3b i,f (α) 2,(13) with μi(μj) the reduced mass of the initial (final) mass partition and T3b i,f (α) the transition amplitude obtained by replacing the CDCC expansion (12) in the transition amplitude (9). The sum in σincludes the spin projections of the outgoing pN pair and of the residual nucleus C. The angle specified by cis the scattering angle of the core in the c.m. frame. The double differential cross section, with respect to the scattering angle of the core and the internal energy of the p+Nsystem, can be obtained at the discretized energies epN =en pN as d2σj,π(α) depN dcepN =en pN ≃1 n dσn,j,π (α) dc ,(14) where nis the width of the bin to which the energy epN belongs. This can be readily transformed to a double differential cross section with respect to the energy of the outgoing core in the overall c.m. frame (Ec), using the usual 1Note that, treating the (p,2p) reaction as a binary process of the form p+A→2He +C, the 2 factor can be regarded as the spectroscopic factor for the overlap 2He|p. 044605-3 A. M. MORO PHYSICAL REVIEW C 92, 044605 (2015) nonrelativistic relation for binary collisions, Ec=m∗ pN m∗ pN +mc Ec.m.,(15) where m∗ pN =mp+mN+epN . Thus, d2σj,π(α) dEcdc=m∗ pN mc+m∗ pN d2σj,π(α) depN dc .(16) The inclusive cross section will be then obtained summing the contributions from all final jπconfigurations, d2σ(α) dEcdc= j,π d2σj,π(α) dEcdc .(17) Note that Eq. (15), along with Eq. (11), establishes a one-to-one correspondence between the energy of the core, in the c.m. frame, and the internal energy of the p+Npair. Note also that Eq. (9) resembles the transition amplitude for a transfer process, analogous to that appearing in the standard CCBA method for binary collisions [17]. Taking advantage of this formal analogy, one can evaluate this transition amplitude using standard coupled-channels codes. For the calculations presented in this work, the code FRESCO [18] has been used, with some suitable modifications described below. B. Relativistic kinematics Ongoing and planned QFS experiments in inverse kinematics are performed at energies of several hundreds of MeV per nucleon. At these energies, relativistic kinematics is clearly important. Although the treatment presented in the previous sections is nonrelativistic, some relativistic corrections can be readily implemented to take into account, at least approximately, these effects. The total energies of the incident nucleus and the proton target in the center-of-momentum frame are given by (c=1 in this section) εA=s+(mA−mp) 2√s;εp=s−(mA−mp) 2√s,(18) where sis the usual Mandelstam invariant, corresponding to the square of the total energy. Assuming that the proton target is initially at rest, s=(mp+mA)2+2mpTLAB,(19) where TLAB is the kinetic energy of the projectile. Analogously, for the exit channel, the total relativistic energy of the outgoing core can be written as εc=s+(mc−m∗ pN ) 2√s,(20) where one has exploited again the analogy of the process under study with a binary process of the form A+p→C+ (pN). Because m∗ pN =mp+mN+epN and Ec=εc−mc, Eq. (20) relates the relative energy of the outgoing p-Npair to the kinetic energy of the core. Using this relation, the generalization of the double differential cross section, Eq. (16), results in d2σj,π(α) dEcdc=√s m∗ pN d2σj,π(α) depN dc ,(21) which reduces to Eq. (16) in the nonrelativistic limit. The modification of the distorted waves owing to relativistic kinematics has been also studied. Unfortunately, there is no unique prescription to incorporate relativistic kinematics within the Schr¨ odinger scheme. Two common prescriptions used in the context of nucleon-nucleus scattering at intermediate energies have been considered here. The first one (see e.g., Ref. [19]) consists of replacing the reduced mass appearing in the kinetic-energy term of the Schr¨ odinger equation (5) with the so-called reduced energy. For example, for the incident channel, μi≡μpA →εpA =εpεA εp+εA .(22) Inserting this definition into the Schr¨ odinger equation and multiplying the full equation by εpA/μpA, one finds an equivalent Schr¨ odinger equation with the usual reduced mass in the kinetic-energy operator, but with the potential scaled by the factor γ=εpA/μpA.(23) The second prescription considered here (see, e.g., Ref. [20]) assumes that the motion of the heavy nucleus in the c.m. frame can be treated nonrelativistically, whereas the light particle (the proton) is treated relativistically. A relativistic Schr¨ odinger-type equation is then generated by reducing the Dirac equation for a massive energetic fermion of mass mpand relativistic wave number kp,movingina central potential U(r), verifying Umpand ∇Ukp, both well satisfied for intermediate-energy proton scattering. In the reduced two-body problem with relativistic projectile but nonrelativistic target the larger component of the wave function verifies a Schr¨ odinger-like equation with a potential renormalized by the factor γ=2(E−MA) E−MA+mp .(24) Note that, in both prescriptions, the momentum to be used is the relativistic one. Finally, the expression for the differential cross section (13) is replaced with its relativistic counterpart (see, e.g., Ref. [21]), dσn,j,π (α) dc=1 (2sp+1)(2JA+1) 1 (2π)2vi ×Kf 2c2(1/εc+1/εpN ) σT3b i,f (α) 2,(25) where viis the initial projectile-target relative velocity and pN =√s−c. Experimental data usually consist of transverse or longitudinal momentum distributions of the residual core. These are readily obtained from the previous differential cross sections by applying the transformation d2σ dEcdc dEcdc=d3σ dp3d3p, (26) 044605-4 THREE-BODY MODEL FOR THE ANALYSIS OF . . . PHYSICAL REVIEW C 92, 044605 (2015) giving rise to d3σ dp3=c2 p 1 (pc)2+(mc2)2 dσ dEcdc .(27) Decomposing the momentum in its transverse (pt) and longitudinal (pz) components, the differential volume becomes d3p=ptdptdφdpz(with φthe azimuthal angle), and dσ dpt=πpt∞ ∞ d3σ dp3dpz,(28) dσ dpz=2πd3σ dp3ptdpt.(29) Writing px=ptcos(φ) and py=ptsin(φ) the transverse momentum distribution can be also decomposed into its x and ycomponents, resulting in dσ dpx=1 2πpt∞ −∞ dσ dpt dpy,(30) and likewise for dσ/dpy. C. Nucleon-nucleon and nucleon-nucleus interactions In the present approach, the nucleon-nucleon (NN) interaction appears in the transition operator [Eq. (2)], as well as in the equation for the three-body (CDCC) wave function (5). For this interaction, the Reid93 parametrization [22], an updated regularized version of the pioneering Reid soft-core potential [23] developed by the Nijmejen group, has been adopted. This potential contains central, spin-orbit and tensor components and reproduces accurately the proton-proton and proton-neutron phase-shifts up to an energy of 350 MeV (χ2/Ndata =1.03). The nucleon-nucleus potentials are calculated microscopically, by folding an effective NN interaction with the nucleus ground-state density. Systematic studies of nucleon-nucleus scattering at intermediate energies show that this procedure provides a good description of the elastic and inelastic data, provided that the effective NN interaction contains some energy and density dependence. Although q-space folding is usually preferable in folding calculations, the presence of this density dependence makes the use of r-space folding necessary, which, for the central part, reads U(r)=dr[ρgs(r)tD(s,ρgs)+j0(ks)tX(s,ρ)ρ(r,r)], (31) where tDand tXcorrespond to the direct and exchange terms of the effective NN interaction, kis the local wave number, rand rare the projectile (nucleon) and target (nucleus) positions, s=|r−r|, and ρgs is the ground-state density, evaluated at the midpoint position |r−r|/2. The mixed transition density is represented by ρ(r,r)=ρgs(r)C(kFs),(32) where Cis a correlation function describing the exchange nonlocality and kFthe Fermi momentum. These folding calculations have been performed with the code LEA [24]. In the calculations presented in this work, the correlation function has been taken as unity, and the ground-state densities are obtained from Hartree-Fock (HF) calculations. The latter were computed with the code OXBASH [25], using the Skyrme Sk20 interaction. It is worth noting that these Skyrme HF densities reproduce well the nuclear radii extracted from electron scattering [26] and are of common use in the analysis of knockout experiments (see, e.g., Refs. [4,5]). D. Bound-state wave functions According to Eq. (7), the transition amplitude contains the overlap function between the initial (A) and the final (C) nuclei. As is usually done in the analysis of transfer and knockout reactions, this overlap function has been approximated by the single-particle wave function obtained as the solution of a Woods-Saxon potential, with the depth adjusted to reproduce the effective separation energy for the final state αof the core, namely, S∗ α=Sn,p +Eα, where Sn,p is the ground-state to ground-state nucleon separation energy and Eαthe excitation energy of the core. Following Ref. [5], the diffuseness parameter of the potential well was fixed at a0=0.7 fm and the radius parameter r0was adjusted so the r.m.s. of the calculated orbital coincided with that obtained from a HF calculation, i.e., √r2 sp=[A/(A−1)]1/2√r2 HF. Because the separation energy predicted by the HF calculation does not necessarily coincide with the experimental one, the HF separation energy is used for this fit. III. RESULTS To illustrate the method, some calculations for the reactions 12C(p,pN) and 22O(p,pn)at∼400 MeV per nucleon are presented and discussed in this section. These calculations are performed with a modified version of the code FRESCO [18], which incorporates the Reid93 NN interaction, and the relativistic kinematics corrections discussed in Sec. II B. Although Eq. (9), with the discrete expansion (12)for the final three-body wave function, provides a numerically tractable form, the calculations can be significantly simplified, with a small loss of accuracy, making use of the additional approximation of neglecting couplings among p-Ncontinuum states with different jπ. The omission of these couplings will alter to some extent the distribution of flux between the final states. However, for an inclusive situation, in which the contribution from all the included jπmust be added together, this redistribution of flux is not expected to affect significantly the core observables. As an additional simplification, the spin-orbit part of the nucleon-nucleus potentials in Eq. (9) has been neglected. A. Application to 12C(p,pN) The 12C(p,2p)11Breaction at 400 MeV/nucleon, which coincides with the actual energy used in a recent experiment performed at GSI for this reaction [27], is considered first. Only those processes leading to bound states of 11Bare considered, which correspond to the removal of protons from the 1p3/2and 1p1/2orbitals, because removal from the deeper 1s1/2orbital will lead to unbound states of 11B. 044605-5 A. M. MORO PHYSICAL REVIEW C 92, 044605 (2015) 0246 -30 -20 -10 0 10 U(r) HF 2pF 0246 r (fm) -40 -20 0 20 40 U(r) Real Imag Real Imag (a) p+12C potential at Ep=200 MeV (b) p+12C potential at Ep=400 MeV FIG. 2. (Color online) p+12C microscopic optical potential generated with the PH density-dependent NN interaction [28,29], folded with the ground-state density of 12C,atEp=200 MeV (top) and 400 MeV (bottom). Solid and dashed lines correspond, respectively, to HF (Sk20 interaction) and empirical densities obtained from electron scattering (the latter parametrized with a 2pF distribution). The required nucleon-nucleus optical potentials are p+ 12C in the incident channel and p+11B in the exit channel. These potentials were generated with the microscopic folding approach described in Sec. II C, using the Paris-Hamburg (PH) G-matrix effective interaction [28,29]. This interaction is energy and density dependent. The p+12C potential was calculated using the incident energy (Ep=400 MeV/nucleon). For the exit channel, the choice is less clear, because the outgoing protons will emerge with a broad range of energies. For a pure QFS collision, one expects an average value of about Elab/2 for each nucleon, and so the outgoing optical potentials were evaluated at Ep=200 MeV in the present case. The ground-state densities of the 12Cand 11Bnuclei were obtained from a HF calculation, using the Skyrme Sk20 interaction. To test the sensitivity of the calculated potentials with respect to this input, a two-parameter Fermi (2pF) parametrization of the 12Cdensity, extracted from electron scattering [30], was also considered. 0 1020304050 10-2 100 102 104 106 dσ/dΩ (mb/sr) PH+Sk20: Rel I PH+Sk20: Rel II 0 1020304050 θc.m. (deg) 10-4 10-2 100 102 104 106 dσ/dΩ (mb/sr) PH+Sk20: Rel I PH+Sk20: Rel II p+12C @ Ep=398 MeV p+12C @ Ep=200 MeV (a) (b) FIG. 3. (Color online) Differential elastic cross section for p+ 12C at 200 MeV (top) and 398 MeV (bottom), using a microscopic folding potential generated with the PH effective NN interaction and the Skyrme HF density. The dashed and solid lines correspond to the relativistic scaling prescriptions given by Eqs. (23)and(24), respectively. The data are from Ref. [31]. Figure 2shows the real and imaginary parts of the p+12C central potential at 200 (top) and 400 MeV (bottom). The solid and dashed lines are for the HF and 2pF densities, respectively. It is interesting to note how, for the higher energy, the potential becomes highly absorptive and the real part, which is mainly attractive at low energies, develops a strong repulsive core. The 2pF density gives a qualitatively similar behavior. The calculated potentials show some differences at small distances (<2 fm) but are very similar at larger distances. Although (p,pN) reactions are expected to explore shorter distances as compared to knockout reactions, they have still a peripheral nature. Therefore, one expects that both potentials give similar (p,pN) cross sections. The quality of the calculated potentials has been assessed, comparing the calculated differential elastic cross section for p+12C at 200 and 398 MeV with the experimental data from Ref. [31]. For these calculations, the folding potentials computed with the HF densities have been used. The comparison is shown in Fig. 3, where the solid and 044605-6 THREE-BODY MODEL FOR THE ANALYSIS OF . . . PHYSICAL REVIEW C 92, 044605 (2015) dashed lines correspond to the relativistic scaling factors of Eq. (23) (prescription I hereafter) and Eq. (24) (prescription II), respectively. Both prescriptions reproduce fairly well the data at both energies, with prescription II providing a slightly better agreement. Therefore, the latter has been also adopted for the (p,pN) calculations presented below. For the (p,2p) calculations, the transition amplitude (9)is evaluated with the CDCC expansion of the final three-body wave function in terms of p-pbins [Eq. (12)]. For this expansion, partial waves j⩽5 and relative energies (epp ) up to the maximum allowed by energy conservation were used. Note that, in the (p,2p) case, T=1 for the outgoing pair and, hence, the generalized Pauli principle restricts the allowed final configurations to +S=even, where is the orbital angular momentum for the p-ppair and Sits total spin ( S=s1+s2). This excludes some jπconfigurations, such as 1+and 3+, which are purely T=0. The bound-state wave function of the struck nucleon in the projectile is generated with a Woods-Saxon potential, with diffuseness a=0.70 fm and the radius adjusted to reproduce the r.m.s. predicted by a HF calculation, as explained in Sec. II D. This procedure yields the potential radius R0=2.953 fm. Finally, the potential depth is adjusted to reproduce the ground-state to ground-state proton separation energy (Sp=15.96 MeV). Before presenting the (p,2p) cross sections, the correspondence between the relative energy of the outgoing p-ppair and the kinetic energy of the core in the c.m. frame, given by Eqs. (11) and (15), is discussed. This is illustrated in Fig. 4. The top panel shows the contribution of the dominant jπp-pconfigurations as a function of the relative energy epp . For simplicity, the spectroscopic factor has been set equal to unity. It is seen that a wide range of energies, from epp =0toepp ≃300 MeV, are populated, with a maximum around ∼180 MeV. It is also seen that most of the cross section comes from the j<3 configurations, with j=1− giving the dominant contribution. It is therefore very important that the underlying NN interaction used in the evaluation of the scattering amplitude reproduces correctly at least these partial waves. In the bottom panel, the same distributions are plotted as a function of the core c.m. energy, according to the relation (20). Although this expression is relativistic, it is clear from this figure that the core energies, in the overall c.m. frame, can be treated nonrelativistically, as assumed in the relativistic prescription II discussed in the previous section. Once the double differential cross sections have been obtained, the transverse and longitudinal momentum distributions are computed by means of Eqs. (30) and (29), respectively. These quantities are shown in Figs. 5(a) and 5(b), respectively. Note that the longitudinal (pz) momentum distribution has been calculated in the projectile rest frame. As in the case of knockout reactions between composite systems, the shape of these distributions is mostly determined by the wave function of the struck nucleon. It is observed that the longitudinal momentum distribution exhibits some asymmetry and is somewhat shifted to negative values of pz. This is mainly a consequence of the negative Qvalue of the reaction, which reduces the available kinetic energy in the final channel. This phenomenon has been analyzed in detail in a recent work in terms of the DWIA formalism [32]. The calculations were 0 100 200 300 epp (MeV) 0.00 0.01 0.02 dσ/depp (mb/MeV) 0+ 012+ 230 102030405060 Ec (MeV) 0 0.1 0.2 dσ/dEc (mb/MeV) 12C(p,2p) @ Ep=400 MeV (a) (b) FIG. 4. (Color online) (Top) Relative energy distribution of the outgoing protons, following the process 12C(p,2p), for several jπ configurations of the p-ppair. (Bottom) Corresponding energy distribution of the core, in the c.m. frame, calculated from Eq. (21). The calculations are done with microscopic distorting potentials (PH +Sk20), the relativistic prescription II, and unit spectroscopic factor for the 12C→11B+pdecomposition. See text for details. repeated using the 2pF density of the 12Cnucleus, but the results are almost identical, so they are not shown here. Similar calculations for the 12C(p,pn)11Creaction have been performed. The ingredients are almost the same as in the (p,2p) case, except for the fact that one needs also the n+11C potential for the exit channel, which was also obtained from the microscopic folding model, with the PH effective nucleon-nucleon interaction and the HF (Sk20) density of the 11Cnucleus. The bound-state wave function was obtained with a Woods-Saxon well with diffuseness a=0.7 fm, radius R0=2.850 fm (extracted from the HF calculation), and the depth adjusted to reproduce the experimental neutron separation energy (Sn=18.72 MeV). Figure 6shows the calculated (p,pn) cross section, as a function of the relative energy p-nenergy, epn (top), and the core energy in the c.m. frame, Ec(bottom). As in the (p,2p) case, the cross section is dominated by the 1−partial wave of 044605-7 A. M. MORO PHYSICAL REVIEW C 92, 044605 (2015) -200 0 200 px (MeV/c) 0 0.02 0.04 dσ/dpx (mb/(MeV/c)) -200 0 200 px (MeV/c) 0 0.02 0.04 -200 0 200 pz (MeV/c) 0 0.02 dσ/dpz (mb/(MeV/c)) -200 0 200 pz (MeV/c) 0 0.02 0.04 12C(p,2p)11B12C(p,pn)11C (a) (b) (c) (d) FIG. 5. Transverse (top) and longitudinal (bottom) momentum of the residual core nucleus for the 12C(p,2p)(left)and 12C(p,pn) (right) reactions at 400 MeV/nucleon. In all cases, the spectroscopic factor has been set to unity. 050 epn 0 50 100 150 200 250 300 epn (MeV) 0 0.01 0.02 dσ/depn (mb/MeV) 0 102030405060 Ec (MeV) 0.04 0.08 0.1 dσ/dEc (mb/MeV) 0+ 01+ 12+ 212C(p,pn) @ Ep=400 MeV (a) (b) FIG. 6. (Color online) (Top) Relative energy distribution of the outgoing nucleons, following the process 12C(p,pn), for some jπ configurations of the outgoing p-npair. (Bottom) Corresponding energy distribution of the core in the c.m. frame. See text for details. the p+nsystem, although the contributions of the 1+,2 +, and 2−waves are also sizable. The epn distribution corresponding to the 1+configuration exhibits a low-energy tail, with a sharp increase near epn =0. This peculiar behavior is a consequence of the final-state interaction (FSI) between the outgoing pand nnucleons. The contribution of the (p,d) channel, leading to the deuteron ground state, is also included, although it was found to be negligibly small at these relatively high energies. This is better seen in the inset, where the low-energy region has been enlarged. The FSI effect owing to the 1+wave is also apparent in the core energy distribution (bottom panel), giving rise to a high-energy tail. The results for the transverse (px) and longitudinal (pz) momentum distributions are shown in Figs. 5(c) and 5(d), respectively. As in the (p,2p) case, the struck neutron is also removed from the p3/2orbital, and so the shapes of the calculated momentum distributions turn out to be very similar in both cases. However, the magnitude of the (p,pn) case is found to be ∼20% larger. Using unit spectroscopic factors, one finds σ(p,pn)=6.6 mb and σ(p,2p)=5.4mb and, hence, σ(p,pn)/σ(p,pp)≃1.2, which is close to the free nucleon-nucleon cross sections at this energy, σpn/σpp =1.27. This is consistent with a QFS interpretation of this reaction. The departure from these free nucleon-nucleon cross sections can be attributed to the difference in separation energies, the Coulomb interaction, and the slight differences in the matter densities (and hence in the nucleon-nucleus potentials). To finish this section, the role of the different partial waves of the outgoing p-Npair is discussed. This is shown in Fig. 7, where the histogram corresponds to the contribution of each jπto the (p,pN) cross section, and the inset shows the total transverse (px) momentum distribution (summed in jπ). It is seen that most of the contribution comes from the j=1,2 waves. It is to be noted that the pure T=0 configurations 1+ and 3+are absent in the (p,2p) case. It is also seen that the 0+01+ 12+ 23+3-4+ 4jπ 0 1 2 3 4 σjπ (mb) 12C(p,pn)11C 12C(p,2p)11B -400 -200 0 200 400 px (MeV/c) 0 0.01 0.02 0.03 0.04 dσ/dpx (mb/(MeV/c)) (p,pn) SF=1 (p,2p) SF=1 12C(p,2p)11B vs. 12C(p,pn)11C FIG. 7. (Color online) Contribution of each angular momentum/parity (jπ)tothe 12C(p,2p)11Band 12C(p,pn)11Creactions at E=400 MeV/nucleon for a p3/2configuration and unit spectroscopic factor. The inset shows the corresponding transverse momentum distributions, summed over all the jπcontributions. 044605-8 THREE-BODY MODEL FOR THE ANALYSIS OF . . . PHYSICAL REVIEW C 92, 044605 (2015) contribution of j⩾3 is small and hence convergence of these observables requires only a small number partial waves. This is in contrast to the usual DWIA formulation, in which the final states are written in terms of the distorted waves for the outgoing nucleus, and hence a large number of partial waves is expected to be required in both waves to achieve convergence of inclusive observables. B. Application to 23O(p,pn) The 23O(p,pn) reaction at 450 MeV/u, populating bound states of the 22Onucleus, is considered now. In a simple meanfield picture, the single-particle configuration of the outermost neutrons of 23Ois expected to be ν(1p1/2)2(1d5/2)6(2s1/2)1. Hartree-Fock calculations with the Skyrme Sk20 interaction yield the single-particle energies ε(p1/2)=−12.6MeV, ε(1d5/2)=−6.1 MeV, and ε(2s1/2)=−4.3 MeV. Because the neutron separation energy of 22Ois Sn=6.85 MeV, one would expect that only the removal of neutrons from the 2s1/2and 1d5/2orbitals of 23Owould lead to bound states of 22O. Removal from deeper orbits will lead to a residual 22O system with an excitation energy above its neutron separation threshold, which will therefore decay into 21O. However, shell-model calculations, as those presented below, indicate that part of the 1p1/2strength in the 23Onucleus corresponds to bound states of 22O. Consequently, this orbital has been also considered in the calculations of this section. As in the previous examples, the required nucleon-nucleus potentials (p+23O for the incident channel and p/n +22Ofor the exit channel) were obtained by folding the PH NN effective interaction with HF (Sk20) ground-state densities. Relativistic corrections were included according to the prescriptions discussed in Sec. II B, with the scaling factor of Eq. (24) for the potentials. The effective separation energy for each single-particle configuration will depend on the considered state of 22O. In the present calculations, these states have been obtained from a shell-model calculation, performed with the code OXBASH [25] using the WBT effective interaction of Warburton and Brown [33]. The results are summarized in Table I. Guided by these results, for the 2s1/2configuration the effective one-neutron separation energy corresponding to the transition to the 22O ground-state has been considered, whereas for the d5/2and 1p1/2configurations the excitation TABLE I. Shell-model spectroscopic factors (SF) for 23O→ 22O+p, corresponding to 22Obound states. The excitation energies and angular momentum-spin assignment are also those predicted by the shell-model calculation. See text for details. Eα(MeV) Iπ αnj SF 0 (g.s.) 0+ 12s1/20.80 3.4 2+ 11d5/22.08 4.8 3+ 11d5/23.08 4.6 0+ 22s1/20.12 5.8 1− 21p1/20.76 6.1 0− 21p1/20.32 6.5 2+ 21d5/20.24 100 200 300 epn (MeV) 0.0 0.1 0.2 dσ/depn (mb/MeV) s1/2 SF=1 p1/2 SF=1 d5/2 SF=1 0 10203040 Ec (MeV) 1 2 dσ/dEc (mb/MeV) (a) 23O(p,pn)22O (b) FIG. 8. (Color online) (Top) Differential cross section as a function of the outgoing proton-neutron relative energy for the reaction 23O(p,pn) at 445 MeV/nucleon. Solid, dashed, and dotted lines correspond to the removal from 2s1/2,1d5/2,and1p3/2configurations in 23O, assuming in all cases unit spectroscopic factor. (Bottom) Differential cross section as a function of the kinetic energy of the residual nucleus 22Oin the c.m. frame. energies of the 2+ 1and 1− 1states were used, respectively. This gives the values Sn=2.739 MeV, S∗ n=5.939 MeV, and S∗ n=8.539 MeV for the 2s1/2,d5/2, and 1p1/2configurations, respectively. Figure 8shows the calculated (p,pn) differential cross sections as a function of the p-nrelative energy (top panel) and the core energy in the c.m. frame (bottom). The solid, dashed, and dotted lines correspond to the removal from 2s1/2,1d5/2, and 1p3/2orbitals in 23O. The marked dependence is clearly seen, with the =0 configuration giving a much narrower energy distribution. In this case, the p-Npair will be detected with a relatively narrow distribution of energies centered at ∼210 MeV, which corresponds to half of the incident beam energy, in accordance with a quasifree NN scattering. For other values of , the distribution is also approximately centered at this value, but with a larger dispersion. 044605-9