Role of continuum in nuclear direct reactions with one-neutron halo nuclei: A one-dimensional model

PHYSICAL REVIEW C 103, 014604 (2021) Role of continuum in nuclear direct reactions with one-neutron halo nuclei: A one-dimensional model L. Moschini ,1,*A. M. Moro ,2and A. Vitturi 3,4 1Department of Physics, Faculty of Engineering and Physical Sciences, University of Surrey, Guildford, Surrey GU2 7XH, England, United Kingdom 2Departamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain 3Dipartimento di Fisica e Astronomia “G. Galilei”, Università degli Studi di Padova, Via Marzolo 8, 35131 Padova, Italy 4Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy (Received 11 September 2020; accepted 21 December 2020; published 8 January 2021) Background: The problem of the scattering of a one-neutron halo nucleus by another nucleus might involve an extremely complicated solution, particularly when breakup and rearrangement channels are to be considered. Purpose: We construct a simple model to study the evolution of a single-particle wave function during the collision of a one-dimensional potential well by another well. Method: Our one-dimensional model provides the essential three-body nature of this problem, and allows for a much simpler application and assessment of different methods of solution. To simplify further the problem, we assume that the potential well representing the projectile moves according to a predetermined classical trajectory, although the internal motion of the “valence” particle is treated fully quantum mechanically. This corresponds to a semiclassical approach of the scattering problem, applicable in the case of heavy projectile and target. Different approaches are investigated to understand the dynamics involving one-body halo-like systems: the “exact” time-dependent solution of the Schrödinger equation is compared to a numerical continuum-discretized coupled-channels (CC) calculation presenting various model cases including different reaction channels. Results: This framework allows us to discuss the reaction mechanism and the role of the continuum, the inclusion of which in the CC calculation results to be crucial to reproduce the exact solution, even when the initial and final states are well bound. Conclusions: The dynamical situations under study can be linked to analogous problems solved in a threedimensional (3D) CC framework, so the present model provides a simple tool to understand the main challenges experienced in the usual 3D models with the treatment of the continuum. DOI: 10.1103/PhysRevC.103.014604 I. INTRODUCTION One of the most relevant research lines in contemporary nuclear physics is the investigation, both experimentally and theoretically, of nuclei under extreme conditions and, in particular, nuclei far from the stability valley. Along this line, truly enticing and sometimes striking novel nuclear structure phenomena are being observed. In particular, these so-called exotic nuclei present low proton or neutron separation energy, short lifetime, and a radius which noticeably deviates from the A1/3dependence of stable nuclei, a fact that is related to a skin or a halo structure [1,2]. The excess of neutrons or protons also leads to a different arrangement which is generally described as deformation: this is indicated by new single-particle character of states and new magic numbers related to new shell closure [3,4]. Because of the large spatial separation between the center of mass and the center of charge, in presence of an external electric field low-energy electric dipole excitations can result, as well as very clear *[email protected] cluster effects [5]. Examples of exotic systems are nuclei with large neutron excess, with the barely bound outermost ones creating an extended density distribution, referred to as a halo. There can be different kinds of halo systems: a nucleus with one valence neutron is for example 11Be, whereas 8Bis an isotope presenting a one-proton halo. Nowadays, measuring the properties of such nuclei is the goal of the main experimental nuclear facilities around the world [6,7]. Due to their short lifetimes targets of such nuclei are not feasible, so they cannot be studied by usual spectroscopic techniques. Thus, the challenge to measure nuclei on the drip lines has triggered the development of radioactive nuclear beams [8]. Most of our present knowledge of stable and exotic nuclei stems from the analysis of nuclear direct reactions. Direct reactions are characterized by different channels—elastic and inelastic scattering, transfer of nucleons between the colliding nuclei, and breakup (i.e., excitation to positive energy states)—and they will feed a particular channel in a way that depends sensitively on its character [9]. In particular, inelastic scattering excites collective states strongly; one-neutron transfer probes the single-particle character of states; two-nucleon transfer goes preferentially to 2469-9985/2021/103(1)/014604(13) 014604-1 ©2021 American Physical Society L. MOSCHINI, A. M. MORO, AND A. VITTURI PHYSICAL REVIEW C 103, 014604 (2021) states that exhibit strong pairing correlations; pairing could also be tested via multinucleon transfer, as well as clustering; the role of the continuum is investigated through breakup reactions. From a theoretical point of view, the conventional theories valid for the description of stable nuclei have difficulties to describe exotic systems, thus indicating the necessity to improve them and include the description of the new phenomena taking place at the limits of stability, taking into account the new features. The theoretical description of halo nuclei is strongly characterized by its weakly bound nature: the valence nucleons are so weakly bound that the addition of any correlation to the simple mean field model inevitably involves the inclusion of the continuum in the system description. For this reason the description of their structure or dynamics is more involved, even considering inert cores. Direct reactions are usually studied applying coupled-channels (CC) calculations; in the case of processes involving exotic systems the continuum discretized coupled-channels (CDCC) description is applied [10–13]. However, it is often difficult to model the continuum, e.g., when transfer or breakup channels are dominant [14–16]. Time-dependent approaches are often used to disentangle the reaction mechanism (see, e.g., Refs. [17–20]) and following this line we construct a simple framework to study the physics of nuclear direct reactions with heavy ions, and in particular to understand how the continuum impacts on the reaction. To better focus on our aim, our model is kept as simple as possible: we consider “semiclassical” motion, and reduce the problem to one dimension. This allows us to concentrate on the study of the role of the continuum in a more systematic and generalized way (varying the mean field properties of the colliding nuclei and the quantum representation of the continuum), and to test CC techniques directly comparing the results to the solution of the time-dependent Schrödinger equation. So, we consider the problem of one particle [21], initially bound in a one-dimensional potential well (the “target” T), subject to an external field, which is also represented by a potential well (the “projectile” P), and which moves in one dimension according to a predetermined classical equation of motion. This situation describes schematically the collision between two nuclei, each one modeled by a potential well, and it permits us to understand the mechanisms which govern a nuclear reaction by following the evolution of the wave function associated to the particle. Initially, since we assume the particle to be a valence nucleon of the target, it coincides with an eigenstate of the target potential well. During the reaction the particle will feel the interaction with the incident nucleus, i.e., the projectile potential well. Depending on the parameters characterizing the reaction, the particle will more likely remain in its initial state (elastic scattering), jump to an excited bound level of the target (inelastic scattering), be transferred to a bound level of the other nucleus, or it could leave the initial nucleus and escape to the continuum (breakup). As a consequence, the particle wave function will change, according to the probability to excite the different reaction channels. This paper is structured as follows: in Sec. II we detail the model for the numerical “exact” solution (Sec. II A)aswell as the coupled-channels description (Sec. II B). In particular, in Sec. II B, we discuss the treatment of the continuum in our CC calculation. In Sec. III we apply our models to different dynamical situations; finally we summarize our results and give our conclusions in Sec. IV. II. THE MODEL Let us start the model description by specifying the Hamiltonian and the initial conditions. The system is described with the Hamiltonian H(x,t)=−¯h2 2μ ∂2 ∂x2+VT(x)+VP(x−xp(t)),(1) where we include two potentials VTand VP, chosen with a Woods-Saxons shape, and associated to target and to projectile, respectively, and μrepresents the reduced mass of the valence particle. In our semiclassical model the target is at rest and only the projectile potential well moves according to a classical trajectory. We use the trajectory proposed in Ref. [22]: xp(t)=x0+ρ2+(vt)2−ρ, (2) that accounts for the projectile motion with fixed asymptotic velocity vat large distances, corresponding to an asymptotic energy EP=1/2mv2, which determines the effective duration of the reaction (the higher the asymptotic energy, the faster is the reaction), and a distance of closest approach x0between the two centers of the potentials. This trajectory allows the projectile to change its acceleration over the distance ρ(in our model of the order of 2 fm), thus simulating the nuclear interaction with the target; at t=±∞the trajectory tends to a uniform motion with zero acceleration. The turning point of the collision corresponds to t=0 ps. Each process is divided in τsteps, so the total duration is τdt. We determine the time interval as dt =dx/4v, small enough to provide a stable solution for a spatial grid step length of dx =0.2 fm at maximum and the asymptotic energies studied in the present paper. Upon variation of these parameters, one can simulate different kinematical conditions due to different bombarding energies and impact parameters (corresponding to partial waves in a quantum-mechanical description), while the choice of the parameters of the two potentials accounts for the different masses of the colliding nuclei, the Qvalues of the different final channels, as well as the possibility of simulating weak binding conditions. By solving separately the time-independent Schrödinger equation for each well J=(T,P) HJJ(x)=EJJ(x),(3) with the Hamiltonian HJ=−¯h2 2μ d2 dx2+VJ(x),(4) we obtain two sets of bound levels J(x) associated to negative eigenvalues [23](EJ<0), and a continuum associated to positive energies (EJ>0). By applying the discretization methods detailed in Ref. [24], we can also define a set of square-integrable functions associated to a 014604-2 ROLE OF CONTINUUM IN NUCLEAR DIRECT REACTIONS … PHYSICAL REVIEW C 103, 014604 (2021) finite number of positive energies, the so-called discretized continuum pseudostates. Each wave function is normalized as |(x)|2dx =1. Depending on the kind of reaction under study we will select one of the bound levels as the initial state of our singleparticle wave function. For example, to describe a pick-up reaction we will choose as initial state a single-particle level in the target. In the simulations presented here we will always choose a target wave function as the initial one. The problem can be solved in many different ways. Two such methods are considered and compared here. On one hand, we consider the exact solution of the problem by numerically solving the corresponding time-dependent Schrödinger equation (Sec. II A). On the other hand, we consider an approximate solution using the coupled-channels method, which is usually applied in solving three-dimensional scattering problems (Sec. II B). From this comparison, we expect to get further insight on the accuracy and limitations of the coupledchannels approach to quantum collisions. A. Exact time-dependent solution In the case of the exact solution, we proceed to compute the time evolution of the valence neutron wave function (x,t)by numerically solving the time-dependent Schrödinger equation i¯h∂ ∂t(x,t)=H(x,t)(x,t),(5) with the Hamiltonian (1). The wave function is confined within an interval containing the two wells. It is calculated at fixed points with coordinates xjseparated by dx. According to Ref. [25], one can solve the problem by using a finite-difference approximation of the Hamiltonian, giving rise to the following tridiagonal form: Hjk =− ¯h2 2μdx2(δjk+1+δjk−1−2δjk) +δjk[VT(xj)+VP(xj−xp(t))],(6) and the time evolution of the wave function is governed by the so-called Padé approximation of the evolution operator: (t+dt)=1+idt 2¯hH−11−idt 2¯hH(t),(7) where dt is a finite time step, and His the matrix in Eq. (6) at the intermediate time t+dt/2. Note that the evolution operator is a unitary operator. An alternative approach is to integrate the differential equation using a finite-difference method, such as the Runge-Kutta method. For that, in this paper we make use of the routines D02PVF and D02PCF of the NAG library [26]. Although this solution prevents us from a complete control of the code, it was found to be faster than the Padé method. We have also verified that both methods lead to identical results. At the end of the time evolution, when t=tf, we can compute the final probabilities for each reaction channel by projecting the final wave function (x,tf) onto the corresponding ith eigenstates of each well obtained by solving Eq. (4) for each potential (target bound states J=Tfor elastic and inelastic, projectile bound states J=Pfor transfer channels): Pelastic =T i=g.s.(x)(x,tf) 2,(8) Pinelastic =T i=g.s.(x)(x,tf) 2,(9) Ptransfer =P i(x−xp(tf))(x,tf) 2.(10) We can also evaluate the breakup probability either by direct subtraction, Pbreakup =1−Pelastic −Pinelastic −Ptransfer,(11) or by projecting the final wave function onto a complete set of continuum states ϕ(k,x) depending on the asymptotic wave number k=± 2μE ¯h2: Pbreakup =dEμ 2E¯h2P(k)=dk|ϕ(k,x)|(x,tf)|2. (12) In one dimension, for each positive energy there are two degenerate continuum wave functions with momentum k, one incoming from the left and the other from the right. For each energy we consider the symmetric and antisymmetric combinations of the momentum normalized continuum wave functions [24,27]. B. Approximate solution within the coupled-channels method The same problem can be solved with the so-called coupled-channels method, which is a popular framework used to describe quantum collision problems in atomic, molecular, and nuclear physics. For this calculation we follow the formulation of Ref. [22], and we take into account two finite sets of wave functions, related to the target and the projectile potentials: T j(x) and P j(x), of NTand NPstates, respectively. For collisions among tightly bound systems, the basis expansion is usually restricted to bound states of the projectile and target systems. However, when one of them is weakly bound, it is important to include also continuum states. For that, it is convenient to use a discrete representation of square-integrable functions, as those discussed in Ref. [24] (we will discuss in more detail the inclusion of the continuum in coupled-channels method in Sec. II B 1). Moreover, they are defined in a one-dimensional spatial grid, the origin of which corresponds to the center of the target potential, which also corresponds to the laboratory frame. A different choice, like the center-of-mass frame of reference in which the two potentials are moving, would have implied a careful treatment of target and projectile wave functions due to the noncovariance of the Schrödinger equation [19,27]. In addition, these two bases are nonorthogonal so we will solve this problem introducing the dual bases ωJ j(x,t), as explained in Refs. [22,27–30]. They are, respectively, associated to each well and conjugated to the channel wave functions of each potential, through the definition I mωJ n=δI,Jδn,m,(13) where I,J=T,Pand n,m=1,2,...,N(T,P). 014604-3 L. MOSCHINI, A. M. MORO, AND A. VITTURI PHYSICAL REVIEW C 103, 014604 (2021) In the CC approach, the wave function describing the entire system is expressed as a combination of target and projectile states: (x,t)= NT  j=1 cT j(t)T j(x)+ NP  j=1 cP j(t)P j(x−xp(t)),(14) and the solution of the problem is reduced to the determination of the time evolution of the coefficients cJ j(t) from the finite set of coupled differential equations: i¯hdcT j(t) dt =cT k(t)ωT jVPT k+cP k(t)ωT jVTP k, i¯hdcP j(t) dt =cT k(t)ωP jVPT k+cP k(t)ωP jVTP k, (15) where the bases functions and potentials related to Tand P depend on (x) and (x−xp(t)), respectively. These equations are solved with the initial conditions cP j(t=−∞)=0 and cT j(t=−∞)=δi,j, where iindicates one of the bound states in the target potential well. To derive Eqs. (15), we first insert Eq. (14) of the total wave function (x,t) into the time-dependent Schrödinger Eq. (5), thus obtaining  j i¯hdcT j(t) dt T j(x)+ j i¯hdcP j(t) dt P j(x−xp(t)) = j cT j[H(x,t)−HT(x)]T j(x) + j cP j[H(x,t)−HP(x−xp(t))]P j(x−xp(t)) (16) where HJis the Hamiltonian corresponding to the potential well J=T,Pof Eq. (4), and His the Hamiltonian of the full system presented in Eq. (1). In problems involving different mass partitions, one may use the so-called prior and post representations of the Hamiltonian, depending on whether one considers the projectile-target combination of the initial or final states; by definition they have to give the same results. Equations (14) and (15) are constructed in prior representation. In post representation we can expand the system wave function on the dual basis: (x,t)= n ˜cT n(t)ωT n(x)+ n ˜cP n(t)ωP n(x−xp(t)),(17) and, following a derivation similar to the one given for the prior representation, we obtain the corresponding set of coupled equations: i¯hd˜cT n(t) dt =˜cT m(t)T nVPωT m+˜cP m(t)T nVPωP m, i¯hd˜cP n(t) dt =˜cT m(t)P nVTωT m+˜cP m(t)P nVTωP m, (18) where the bases functions and potentials related to Tand P depend on (x) and (x−xp(t)), respectively. From the total wave function (x,t), we can also extract amplitudes for excitation and transfer in post and prior representations through the expressions ˜cI n(t)=ωI n, cI n(t)=I n.(19) Due to post-prior symmetry, the amplitudes in the two representations are related by ˜cT n(t)=cT n(t)+ mT nP mcP m(t), ˜cP n(t)=cP n(t)+ mP nT mcT m(t).(20) The probabilities to populate the different final channels are defined as PJ j=cJ j(tf) 2(21) in the prior representation, or as ˜ PJ j=˜cJ j(tf) 2(22) in the post representation. The index jdenotes a label to the quantum number of the final state in one of the two wells. Because of the nonorthogonality of the basis states, the sum of these “probabilities” is not conserved during the collision. If we instead define the probabilities by PJ j=RecJ j(tf)∗˜cJ j(tf),(23) conservation of total probability is always fulfilled within the coupled-channel formalism. This follows from the fact that the matrix governing the time evolution of the amplitudes in the post representation is minus the Hermitian conjugate of the matrix that determines the time evolution of the amplitudes in the prior representation. We shall therefore call Eq. (23)the unitary representation of probabilities. However, there is no guarantee that these quantities are always non-negative during the collision. After the collision, when all overlaps between the basis states in the two wells vanish, the amplitudes for a given transition are the same in the post and prior representation, as evident from Eq. (20). This so-called post-prior symmetry implies that the total probability is conserved once the collision is over, also in a truncated coupled-channel treatment. Inclusion of the continuum in the coupled-channels method In Ref. [22] only bound states were included in the bases and, hence, breakup channels were omitted. This is possibly justified for tightly bound systems, but not for weakly bound ones, for which the coupling to these channels can be very important. In a coupled-channels scheme one cannot include the full continuum spectrum, since these states form a continuum of energies. Moreover, the fact that these states are not square integrable poses numerical problems since the coupling potentials become of infinite range. To overcome these difficulties, 014604-4 ROLE OF CONTINUUM IN NUCLEAR DIRECT REACTIONS … PHYSICAL REVIEW C 103, 014604 (2021) it is customary to resort to an approximate, discrete description of the continuum. To describe the breakup process in CC a set of discretized continuum states defined on a finite spatial range is usually included for one or more subsystems of the reaction constituents. Even if the description of the process is supposed to occur at the end of the reaction, when the distance between the constituents tends to infinity, this constitutes a nonphysical situation. In fact, a proper description of the continuum, even if discretized, should take into account all the fragments involved in the reaction. This is what is done in the Faddeev formalism [31]. In time-dependent approaches, like the one we apply, to include a proper description of the continuum is a challenge. Here, we use the pseudostates introduced in Ref. [24]; in particular, we discretize the continuum in an infinite squarewell basis (BOX method in Ref. [24]). We apply the same approach as in CDCC and provide discretized continuum pseudostates for each potential well. To include them in our coupled-channels calculation, we should ensure that at the end of the time evolution there is no overlap between target and projectile bases. This is due to the fact that the two bases are not mutually orthogonal, so while they overlap the problem loses unitarity, as we mentioned in Sec. II B. After the reaction, when the projectile is far enough for these overlaps to vanish, the problem has restored its unitarity and we can evaluate the final probabilities. Thus, if we include the continuum we need to restrict it to a small range [−xJ b,+xJ b] centered in the corresponding potential J=T,P, in order that xT b+xP b> xp(tf). A limitation of this method is that the choice of xJ bis done a posteriori to include the breakup component into the continuum interval. In situations in which the breakup channel is strongly dominant, it could become impossible to apply this method because the continuum component of the wave function might not be localized close to one of the potential wells. By comparing our CC results to the exact solution of the Schrödinger equation in this simple one-dimensional framework, we are able to understand the limitation of this description of the continuum. III. NUMERICAL RESULTS: COMPARISON OF EXACT AND APPROXIMATE METHODS In previous works we have applied the present approach on a variety of model cases to study different aspects of the direct reaction mechanism, always considering bound and weakly bound systems. In particular, we have tested the model by varying the distance of closest approach between the two potential wells and the Qvalue of the reaction in Refs. [27,32,33], and we have studied the role of the continuum in different configurations in Refs. [27,34,35]. To explore different kinematical situations, we have performed further calculations. In each case, we present the exact timedependent solution (Sec. II A) and compare this result with the approximate solutions obtained with the coupled-channels calculation presented, as described in Sec. II B. The chosen model cases have been selected to illustrate several physical situations in which different reaction channels are favored. For each case we have highlighted an analogous problem solved in a three-dimensional CDCC environment, discussing the -30 -15 015 30 x (fm) -30 -25 -20 -15 -10 -5 0 Energy (MeV) initial state other bound state (a) Target -30 -15 015 30 x (fm) -30 -25 -20 -15 -10 -5 0 (b) Projectile -7.99 MeV -21.6 MeV -1.79 MeV -8.19 MeV FIG. 1. Target (a) and projectile (b) Woods-Saxon potentials for the case A. Their bound states are schematically depicted in correspondence with their binding energy. The initial state, in this case the target ground state, is the dashed red curve. main challenges experienced in the 3D situation within our simplified framework. A. Case A We start with the simplest case of a well-bound target in which we observe a dominance of elastic and inelastic channels. The target and projectile potentials are depicted in Fig. 1 with their respective bound-state wave functions. The lowest state in the target is chosen as initial state for this reaction; its wave function is shown as a dashed red curve in Fig. 1.The projectile follows the trajectory (2) with an asymptotic velocity of 0.1×c, which corresponds to an asymptotic energy of 5.0 MeV. The reduced mass for this system is 1.001 amu; the time evolution is divided in τ=2510 steps. In Fig. 2we present the exact wave-function squared value at different moments of the time evolution. One can see that a small component of the system wave function is transferred and remains bound to the projectile potential well as it moves away after the collision. In Table Iwe present the results obtained by solving the problem using different approaches and different bases. In particular, they correspond to the following, as shown in Table I. TABLE I. Final probabilities for the model case A obtained with the exact and the coupled-channels methods. Exact CC(1) CC(2) CC(3) CC(4) Elastic 86.1% 89.8% 89.4% 74.1% 84% Inelastic 9.8% 10.1% 10.3% 18.4% 11.6% Transfer g.s. 2.5% 6.8% 2.3% Transfer 1 0.8% 0.56% 1.9% Breakup 0.7% 0.3% 0.13% 014604-5 L. MOSCHINI, A. M. MORO, AND A. VITTURI PHYSICAL REVIEW C 103, 014604 (2021) FIG. 2. Model case A, with the exact squared total wave function at different times and a zoom for x>20 fm in the inset. Target and projectile potential wells position is indicated by the labels xTand xP, respectively. The elapsed time for each frame is also indicated: this process is composed by 2510 time steps; the turning point t=0 ps occurs at step 1255. (1) Exact: Calculation done by solving numerically the time-dependent Schrödinger Eq. (5). The different results represent the probability for the valence neutron to remain in the ground state (elastic) or to directly excite the corresponding bound states of the two potential wells (inelastic and transfer) or to excite continuum states (breakup). (2) CC(1): Coupled-channels calculation in which only the two target bound states are included. (3) CC(2): Coupled-channels calculation using target bound levels plus the first ten continuum pseudostates obtained in a range equal to the maximum radius of the grid. (4) CC(3): Coupled-channels calculation including only target and projectile bound states. (5) CC(4): Coupled-channels calculation including target and projectile bases composed by bound and five continuum pseudostates calculated in a [−20; 20]-fm range centered in the respective potential. It corresponds to an energy cutoff in the continuum of 4 MeV. By comparing the results of Table I, we note that the coupled-channels calculation without transfer and continuum channels, i.e., CC(1), reproduces rather well the elastic and inelastic probabilities. This is a consequence of the dominance of the elastic channel and, to a lesser extent, the inelastic -15 015 30 45 60 75 x (fm) 0 0.1 0.2 0.3 |ψ|2 (fm-1) exact CC(4) FIG. 3. Final exact wave function (solid red line) and final CC(4) wave function (dotted blue line). The projectile potential is placed at xP(tf)=65 fm. channel in this case. Including the breakup (CC(2)) channel alone does not improve the result, because it is not possible to describe the transfer in this configuration. Interestingly, a calculation with elastic, inelastic, and transfer channels (CC(3))is not enough to reproduce the expected values, even if those are the dominant channels in this case. By including all possible channels simultaneously (CC(4)) we obtain a fine agreement with the numerical solution. In Fig. 3, we compare the final exact wave function (solid red line), already reported in the upper panel of Fig. 2(d),to the wave function obtained from calculation CC(4) using the final coupled-channels coefficients in Eq. (14) (dotted blue line). We observe a quite good agreement between the two results, the slight differences being due to the overestimation of the excitation of the first excited states of both target and projectile wells by the CC calculation. To further understand the role of continuum states in this process, we construct the exact momentum-normalized bases ϕ(k,x)forthetwo potential wells. We then calculate the probability as a function of the momentum P(k) following Eq. (12). The results are displayed in Fig. 4: the exact (solid red) and CC(4) (dashed blue) final wave functions from Fig. 3have been projected onto the target and projectile exact continuum in the upper and lower panels, respectively. We can notice how the “projectile continuum” plays a more relevant role than the “target continuum”, and also how well the CC(4) calculation works in this case. For the target continuum we observe that the coupled-channels calculation does not reproduce accurately the structure observed at lower momenta. Anyway, as already observed, the total strength is negligible with respect to the role of the other well. It should be noticed that the incident energy of 5 MeV corresponds to a threshold of k=0.49 fm−1; the excitation slightly exceeds this limit probably due to the nonconservation of energy in semiclassical descriptions [36]. This case is ideally described within the CC method, given the dominance of the elastic channel. Many authors have found an excellent agreement with elastic and transfer experimental cross sections, provided that breakup channels 014604-6 ROLE OF CONTINUUM IN NUCLEAR DIRECT REACTIONS … PHYSICAL REVIEW C 103, 014604 (2021) 0 0.4 0.8 1.2 1.6 0 0.1 0.2 0.3 P(k) 0 0.4 0.8 1.2 1.6 k (fm-1) 0 1 2 3 4 P(k) exact CC(4) (a) Target (b) Projectile FIG. 4. Breakup probabilities as a function of the momentum in the continuum. The upper and lower panels correspond to the projection on target and projectile states, respectively. Solid red lines indicate the exact final wave function, while dashed blue lines indicate the CC(4) final one. are properly included in the CDCC calculation (see, e.g., Refs. [13,37–39]). Also, some authors have suggested that continuum-continuum couplings have a strong influence in the complete and incomplete fusion reaction for weakly bound nuclei [40–42]. The present model case confirms the crucial role of the continuum, and contributes to demonstrate that even if the system under study is well bound and the breakup component is negligible, the inclusion of continuum states can be essential to get a proper description of the reaction. In this situation in which the breakup component is not dominant, to include two sets of continuum states, each one related to one potential well, allows us to obtain a result in agreement with the expected exact one. So, this treatment of the continuum could constitute an acceptable approximation in these conditions. B. Case B In this second case the initial target state is extremely weakly bound and consequently the breakup channel is the most relevant. The target and projectile potentials are depicted in Fig. 5with their respective bound-state wave functions. The initial state is the dashed red curve corresponding to the target bound state. The energy of this state is −0.276 MeV and its weakly bound nature is clearly evident from the extended tail of its wave function. The projectile follows the trajectory (2) with an asymptotic velocity of 0.1×ccorresponding to an incident energy of 5.0 MeV, and with a reduced mass of 1.001 amu; the time evolution is divided in τ=4790 steps. The evolution of the exact wave function during the collision is presented in Fig. 6. In this case, both the transfer and the continuum components of the system wave function are clearly evident. The results for case B, obtained applying different methods, are presented in Table II. In particular, they correspond to the following, as shown in Table II. -40 -20 0 20 40 x (fm) -5 -4 -3 -2 -1 0 1 Energy (MeV) initial state (a) Target -40 -20 0 20 40 x (fm) -5 -4 -3 -2 -1 0 1 other bound state (b) Projectile -0.276 MeV -2.87 MeV FIG. 5. Target (a) and projectile (b) Woods-Saxon potentials for the case B. Their bound states are schematically depicted in correspondence with their binding energy. The initial state, in this case the target ground state, is the dashed red curve. FIG. 6. Model case B, with the exact squared total wave function at different times. Target and projectile potential wells position is indicated by the labels xTand xP, respectively. The elapsed time for each frame is also indicated: this process is composed by 4790 time steps; the turning point t=0 ps occurs at step 2395. 014604-7 L. MOSCHINI, A. M. MORO, AND A. VITTURI PHYSICAL REVIEW C 103, 014604 (2021) TABLE II. Final probabilities for the model case B obtained with the exact and the coupled-channels methods. Exact CC(1) CC(2) CC(3) Elastic 20.94% 22.9% 21.9% 25.8% Transfer 6.73% <10−3% 0.3% Breakup 72.32% 77.1% 78.2% 73% (1) CC(1): Coupled-channels calculation in which the target bound state and ten target continuum states within the [−40; 40]-fm range are included. (2) CC(2): Coupled-channels calculation using target and projectile bound levels plus 50 continuum target pseudostates obtained in a radius equal to 40 fm. (3) CC(3): Coupled-channels calculation including target and projectile bases composed by bound and five continuum pseudostates calculated in a [−22; 22]-fm range centered in the respective potential. It corresponds to an energy cutoff in the continuum of about 4 MeV. Including only target basis states, calculation CC(1) cannot describe alone the three channels of this process, even if it reproduces quite well the proportion between elastic and breakup channels. By adding the projectile bound state (CC(2)) the elastic channel is better reproduced, but the transfer is not well accounted for. Including also the projectile continuum improves the result (CC(3)), thus showing that in a perturbation picture the transfer is not a one-step process, but it is reached through successive steps involving the continuum. Multistep processes in direct reactions are traditionally inferred by CDCC calculations, and, as we have seen in the previous section, continuum-continuum couplings play a crucial role [41,43,44]. In Fig. 7we show the final wave functions for the exact (solid red), CC(1) (dashed magenta), and CC(3) (dotted blue) calculations. Neither of the two CC wave functions reproduces the shape of the final exact wave function, al- -60 -40 -20 0 20 40 60 80 x (fm) 0 0.04 0.08 0.12 |ψ|2 (fm-1) exact CC(1) CC(3) FIG. 7. Final exact wave function (solid red line), final CC(1) wave function (dashed magenta line), and final CC(3) wave function (dotted blue line). The projectile potential is placed at xP(tf)= 50 fm. 0 0.3 0.6 0.9 1.2 0 1 2 3 4 P(k) 0 0.3 0.6 0.9 1.2 k (fm-1) 0 1 2 3 4 5 P(k) exact CC(3) (a) Target (b) Projectile FIG. 8. Breakup probabilities as a function of the momentum in the continuum. The upper and lower panels correspond to the projection on target and projectile states, respectively. Solid red lines indicate the exact final wave function, while dashed blue lines indicate the CC(3) final one. though their breakup probabilities are in agreement. This is partly explained by the failure in describing the transfer component, and, more importantly, by the spatial restriction of the pseudostates. This is also seen in Fig. 8, where we show the projection of the exact (solid red) and CC(3) (dashed blue) wave functions onto the exact continuum ϕ(k,x) of target and projectile wells (upper and lower panels, respectively). The coupled-channels calculation does not reproduce properly the exact result: only their magnitudes are in agreement and, in the case of the target continuum, the main structure is reproduced with the maximum and minimum values placed at similar momenta. Anyway, the similar order of magnitude of the target and projectile distributions suggests the importance of including the continuum of all the possible subsystems of the outgoing particles to properly describe all the reaction channels. Moreover they emphasize the fact that, when both the projectile and target systems are weakly bound, the choice for the continuum representation is not obvious and might strongly affect the final result. We notice also that, since both bases are not mutually orthogonal, including both representations leads to overcompleteness, similar to what is done in the Faddeev formalism [31]. To further improve the coupled-channels solution we should add more continuum pseudostates and enlarge the range over which they are defined. Doing so, the two bases would overlap and we would obtain a nonunitarity solution (as we detailed in the previous section). To avoid that, we should let the projectile evolve further, but in this way the continuum component of the system wave function would get away from the two wells and none of the two continuum representations would describe properly the process. As one can notice, already in the CC(3) calculation a small component of the system wave function has not been included in the range covered by the two bases [around −30 fm in Fig. 6(d)]. Anyway, even if the exact result is not fully reproduced, the 014604-8 ROLE OF CONTINUUM IN NUCLEAR DIRECT REACTIONS … PHYSICAL REVIEW C 103, 014604 (2021) -30 -15 015 30 x (fm) -40 -30 -20 -10 0 Energy (MeV) other bound states initial state (a) Target -30 -15 015 30 x (fm) -40 -30 -20 -10 0 (b) Projectile -1.54 MeV -4.99 MeV -1.96 MeV -20.7 MeV FIG. 9. Target (a) and projectile (b) Woods-Saxon potentials for the case C. Their bound states are schematically depicted in correspondence with their binding energy. The initial state, in this case the target excited state, is the dashed red curve. coupled-channels calculation allows us to infer important information about the reaction mechanism for this case, and in particular about the transfer process. This resembles the case involving weakly bound systems, in which breakup becomes an important reaction channel and the CDCC description becomes challenging. Also in the three-dimensional case, the discrete basis representing the projectile and/or target continuum is spatially constrained by the extension of the basis, and hence the calculated breakup observables are expected to be reliable only in those kinematical situations which are sensitive to that region [45]. C. Case C We treat now a case in which all the possible channels are relevant: elastic, inelastic, transfer, and breakup. The target and projectile potentials are depicted in Fig. 9with their respective bound-state wave functions. The initial state is represented by the red dashed curve corresponding to the first excited state of the target. We assume that the target ground state is Pauli forbidden and the corresponding channel should not be included in the calculation. However, in this case we take into account this channel, calling it “decay to g.s.” The projectile follows the trajectory (2) with an asymptotic velocity of 0.05 ×ccorresponding to an incident energy of 1.0 MeV, and with a reduced mass of 0.975 amu; the time evolution is divided in τ=1450 steps. The evolution of the exact wave function during the collision is presented in Fig. 10. The initial wave function exhibits a node, because the initial state corresponds to the first excited state of the well. At the end of the time evolution [Fig. 10(d)] the component related to the target is asymmetric, and so we expect a probability to deexcite the target to its ground state. One can also clearly identify a continuum component [in the region around −50 fm in Fig. 10(d)], and the presence of transfer to the projectile excited state because the FIG. 10. Model case C, with the exact squared total wave function at different times. Target and projectile potential wells position is indicated by the labels xTand xP, respectively. The elapsed time for each frame is also indicated: this process is composed by 1450 time steps; the turning point t=0 ps occurs at step 725. wave-function component which remains bound to that well presents a node. The results for this case are reported in Table III, and correspond to the following calculations. (1) CC(1): Coupled-channels calculation in which the target and projectile bound states are included. (2) CC(2): Coupled-channels calculation in which the target bound states and ten target continuum states within the [−100; 100]-fm range are included. (3) CC(3): Coupled-channels calculation using the target and projectile bound levels plus the first ten continuum pseu- TABLE III. Final probabilities for the model case C obtained with the exact and the coupled-channels methods. Note that CC(3) and CC(4) total probability exceeds 100%, as expected. Exact CC(1) CC(2) CC(3) CC(4) Decay to g.s. 6.7% 0.85% 23.5% 16% 3% Elastic 29% 99.06% 44.8% 10% 84% Transfer g.s. 1.2% 0.03% 0.6% 0.04% Transfer 1 15.6% 0.06% 36% 0.2% Breakup 47.5% 31.7% 44% 20% 014604-9