Exploring continuum structures with a pseudo-state basis

PHYSICAL REVIEW C 82, 024605 (2010) Exploring continuum structures with a pseudo-state basis J. A. Lay,*A. M. Moro,†and J. M. Arias‡ Departamento de FAMN, Facultad de F´ ısica, Universidad de Sevilla, Apdo. 1065, E-41080 Sevilla, Spain J. G´ omez-Camacho§ Departamento de FAMN, Facultad de F´ ısica, Universidad de Sevilla, Apdo. 1065, E-41080 Sevilla, Spain and Centro Nacional de Aceleradores, Avda. Thomas A. Edison, E-41092, Sevilla, Spain (Received 27 April 2010; revised manuscript received 7 July 2010; published 10 August 2010) The ability of a recently developed square-integrable discrete basis to represent the properties of the continuum of a two-body system is investigated. The basis is obtained performing a simple analytic local scale transformation to the harmonic oscillator basis. Scattering phase-shifts and the electric transition probabilities B(E1) and B(E2) have been evaluated for several potentials using the proposed basis. Both quantities are found to be in excellent agreement with the exact values calculated from the true scattering states. The basis has been applied to describe the projectile continuum in the 6He scattering by 12Cand208Pb targets at 240 MeV/nucleon and the 11Be scattering by 12Cat67MeV/nucleon. The calculated breakup differential cross sections are found to be in very good agreement with the available experimental data for these reactions. DOI: 10.1103/PhysRevC.82.024605 PACS number(s): 24.10.Eq, 25.10.+s, 25.45.De I. INTRODUCTION Quantum collisions involving weakly bound systems are known to be influenced by the coupling to the unbound states. In nuclear collisions, this was first evidenced in the pioneering study by Johnson and Soper [1], who recognized the relevance of the breakup channels to understand deuteroninduced reactions. In their survey, the deuteron continuum was represented by a single sstate. Later developments by Rawitscher [2] and Austern et al. [3] helped to introduce a more realistic representation of the continuum, leading to the development of the continuum-discretized coupled-channels (CDCC) method. This method reduces the many-body problem to an effective three-body problem and expands the full three-body wave function in a selected set of continuum wave functions of a given pair subsystem Hamiltonian. Projection of the Schr¨ odinger equation onto the selected internal states gives rise to a set of coupled differential equations. The practical implementation of the method requires a discretization procedure, that is, an approximation of the two-body continuum spectrum by a finite and discrete representation. Although not strictly necessary, it is numerically advantageous to use for this representation a set of L2(that is, square-integrable) functions. The standard method of continuum discretization consists in dividing the continuum into a set of energy or momentum intervals. For each interval, or bin, a representative wave function is constructed by superposition of the scattering states within the interval (the average method). An alternative to the discretization method based on bins is the pseudo-state (PS) method, in which the wave functions describing the internal motion of the projectile are obtained *[email protected] †[email protected] ‡[email protected] §[email protected] as the eigenstates of the projectile Hamiltonian in a truncated basis of square-integrable functions. A variety of PS bases have been proposed in the literature for two-body continuum discretization [4–6] and, more recently, also for the three-body continuum [7–10]. In a recent work [6], we proposed a PS method based on a local scale transformation (LST) of the harmonic oscillator (HO) basis. The LST, adopted from a previous work of Karataglidis et al. [11], is such that it transforms the Gaussian asymptotic behavior into an exponential form, thus ensuring the correct asymptotic behavior for the bound wave functions. The accuracy of this transformed harmonic oscillator (THO) basis was tested for several reactions induced by deuterons and halo nuclei, showing an excellent agreement with the standard binning method and an improved convergence rate. Due to their vanishing asymptotic behavior, it is not obvious that genuine continuum properties, such as the scattering phase-shifts, can be well described using square-integrable states. However, in this case one can make use of integral formulas, which require only the wave function within a finite region. Several prescriptions have been proposed in the literature to extract the phase shifts from continuumdiscretized states [12–14]. In this work, we make use of the stabilization method of Hazi and Taylor [12,15,16] to show that the THO basis reproduces very well the exact phase-shifts. As an additional test of the quality of the THO basis, we calculate several transition probabilities and their associated sum rules. Finally, we apply the THO basis to calculate the breakup of the reactions 6He +12C and 6He +208Pb at 240 MeV/nucleon and 11Be +12Cat67MeV/nucleon, making use of the CDCC formalism. The calculations are compared with existing experimental data for these reactions. The work is structured as follows. In Sec. II we review the THO method based on the parametric LST. In Sec. III we introduce an integral formula suitable for the calculation of scattering phase-shifts with PS functions. In Sec. IV we recall some useful formulas to evaluate the dipole and quadrupole 0556-2813/2010/82(2)/024605(11) 024605-1 ©2010 The American Physical Society LAY, MORO, ARIAS, AND G ´ OMEZ-CAMACHO PHYSICAL REVIEW C 82, 024605 (2010) transition probabilities from the scattering states and from the pseudo-states. In Sec. Vthese formulas are applied to study the continuum of the deuteron and 6He nuclei. In Sec. VI the method is applied to the scattering of 6He by 12C and 208Pb at 240 MeV per nucleon and by 11Be +12Cat67MeV/nucleon. Finally, in Sec. VII we summarize the main results of this work. II. THE ANALYTIC LST In this section, we briefly review the features of the PS basis used in this work. This basis was originally developed in Ref. [11] to describe the single-particle orbitals within a meanfield approach. In a later work [6], we adopted this method to discretize the continuum of a two-body system within the context of the continuum-discretized coupled-channels method. The starting point is the HO basis in angular momentum representation. The radial part of the nth HO function for a given partial wave is here denoted as φHO n, (s). These functions are orthogonal and constitute a complete set and, therefore, they can be used to expand the eigenstates (bound and unbound) of an arbitrary potential. For a finite well, the boundstate wave functions decay exponentially at large distances and hence the HO basis does not provide a suitable representation due to its Gaussian asymptotic form. A possible approach to overcome this limitation, while retaining the appealing properties of the HO basis, is to perform a LST that converts the Gaussian behavior into an exponential one [17,18]. This gives rise to the so-called transformed harmonic oscillator (THO) basis. We denote the radial part of these basis states as φTHO n, (r)=ds drφHO n, [s(r)].(1) Note that, by construction, the family of functions φTHO n, (r) is orthogonal and constitutes a complete set with the following normalization: ∞ 0 drφTHO n, (r) 2=1.(2) Moreover, they decay exponentially at large distances, thus ensuring the correct asymptotic behavior for the bound wave functions. In practical calculations a finite set of functions (1) is retained, and the internal Hamiltonian of the projectile is diagonalized in this truncated basis with Nstates, giving rise to a set of eigenvalues and their associated eigenfunctions, denoted, respectively, as {εn}and {ϕ(N) n, (r)}(n=1,...,N). As the basis size is increased, those eigenstates with negative energy will tend to the exact bound states of the system, while eigenstates with positive eigenvalues can be regarded as a finite representation of the unbound states. With the aforementioned criterion, the LST is indeed not unique. In Ref. [19] the LST is defined in such a way that the first HO state is exactly transformed into the exact ground-state wave function. Therefore, by construction, this wave function is exactly recovered for any arbitrary size of the basis. In a more recent work [6] we adopted the parametric form of Karataglidis et al. [11], s(r)=1 √2b1 1 rm+1 γ√rm1 m ,(3) which depends on the parameters m,γ, and the oscillator length b. Following Ref. [6], the oscillator length bis treated as a variational parameter used to minimize the ground-state energy. Asymptotically, the function s(r) behaves as s(r)∼ γ br 2and hence the functions obtained by applying this LST to the HO basis behave at large distances as exp(−γ2r/2b2). Therefore, the ratio γ/b can be related to an effective linear momentum, keff =γ2/2b2, which governs the asymptotic behavior of the THO functions; as the ratio γ/bincreases, the radial extension of the basis decreases and, consequently, the eigenvalues obtained upon diagonalization of the Hamiltonian in the THO basis tend to concentrate at higher energies. Therefore, γ/b determines the density of PS as a function of the excitation energy. This property was used in Ref. [6] to determine a suitable value for the ratio γ/b in scattering calculations. For a more quantitative measurement of the density of states we define the magnitude: ρ(N)(k)= N  n=1ϕ(k)ϕ(N) n, ,(4) with |ϕ(k)denoting the scattering wave function for a momentum k. With this definition the integral of the density with respect to the momentum is just the number of basis states, that is, ∞ 0 dkρ(N)(k)=N, (5) regardless of the choice of the parameters of the LST. In all the calculations presented in this work, the power m is just taken as m=4. This was one of the choices made in Ref. [11] and, in fact, the authors of that work found a very weak dependence of the results on this parameter. III. EXTRACTING THE PHASE SHIFTS FROM THE THO BASIS The properties of the continuum states are completely determined by the phase shifts. In a two-body problem, the phase shifts are readily obtained from the asymptotics of the radial part of the wave function. Ignoring spins for simplicity, the radial part corresponding to a partial wave can be written at large distances as ϕ(k,r)→2 π[cos δ(k)F(kr)+sin δ(k)G(kr)] ,(6) where Fand Gare the regular and irregular Coulomb functions. If the potential is real, the functions ϕas well as the phase shifts δare also real. Equation (6) cannot be applied to PS functions to extract the phase shifts, because these functions vanish asymptotically. However, the phase shifts can be also obtained from integral expressions, which require only the interior part of the wave 024605-2 EXPLORING CONTINUUM STRUCTURES WITH A PSEUDO- ... PHYSICAL REVIEW C 82, 024605 (2010) functions. Here, we make use of the integral formula proposed by Hazi and Taylor [12,16], who applied this formula to extract the phase shifts in a one-dimensional scattering problem using a harmonic oscillator representation. We have generalized this formula to three-dimensional cases. The formula so obtained reads tan δ(k)=−∞ 0ϕ(k,r)[E−H]f(r)F(kr)dr ∞ 0ϕ(k,r)[E−H]f(r)G(kr)dr .(7) This formula can be derived following the same arguments outlined in Ref. [12] for the one-dimensional case. We note that, if the exact wave functions are used for ϕ(k,r), this expression becomes an alternative to Eq. (6) to calculate the exact phase shifts. The function f(r) appearing in Eq. (7) verifies the following properties: f(r)r→∞ −→1; f(0) =f(0) =0.(8) Following Ref. [12], we adopt the explicit form f(r)=1− exp(−βr2), with β>0. The aim of this function f(r)isto avoid evaluating the function G(kr) at the origin, where it becomes singular. Therefore the parameter βshould be small enough to make f(r)≈0 for distances of the order of the nuclear range. In the cases studied here, we have chosen β= 0.01 fm−2. IV. ELECTRIC TRANSITION PROBABILITIES IN THE PSEUDO-STATE BASIS The accuracy of the PS basis to represent the continuum can be studied by comparing the ground-state to continuum transition probability due to a given operator. Here we consider the important case of the electric dissociation of the initial nucleus ainto the fragments b+c. This involves a matrix element between a bound state (typically the ground state) and the continuum states. The electric transition probability between two bound states, |(is)jiand |(fs)jf(assumed here to be normalized to unity), is given by the reduced matrix element B(Eλ;i→f)=2jf+1 2ji+1|(fs)jf||M(Eλ)||(is)ji|2, (9) where Mis the multipole operator, M(Eλµ)=Z(λ) eff erλYλµ(ˆ r),(10) with the effective charge Z(λ) eff =Zbmc mb+mcλ +Zc−mb mb+mcλ .(11) In the case of a transition to a continuum of states, |k(fs)jf, the preceding definition is replaced by (see, e.g., Ref. [20]) dB(Eλ) dε =2jf+1 2ji+1 µbck (2π)3¯h2 ×|k(fs)jf||M(Eλ)||(is)ji|2,(12) with k=√2µbcε/¯h. Note that the extra factor appearing in Eq. (12) with respect to Eq. (9) is consistent with the convention k(s)j|k(s)j=δ(k−k) and the asymptotic behavior of Eq. (6). In the calculations presented in this work, we ignore for simplicity the internal spins of the clusters and hence s=0, ji=i, and jf=f. In addition, we consider only transitions from the ground state, so r|(is)ji=ϕg.s.(r) (where the index iis omitted for shortness). This reduces Eq. (12)to dB(Eλ) dε =2f+1 2i+1 µbck (2π)3¯h2|ϕf(k)||M(Eλ)||ϕg.s.|2. (13) The reduced matrix element is given by ϕf(k)||M(Eλ)||ϕg.s.=4π kZeffeD(λ) i,fRλ i,f(k),(14) with D(λ) i,fbeing a geometric factor [20] and Rλ i,f(k)the radial integral Rλ i,f(k)=∞ 0 dr ϕf(k, r)rλϕg.s.(r).(15) Using a finite basis, one may calculate only discrete values for the transition probability. According to Eq. (9), the B(Eλ) between the ground state and the nth PS is given by B(N)(Eλ;g.s.→n)=2f+1 2i+1ϕ(N) n,fM(Eλ)ϕg.s. 2. (16) To relate this discrete representation with the continuous distribution (13) one may use the simple approximation dB(Eλ) dε ε=εn ≃1 n B(N)(Eλ;g.s.→n),(17) where n=(εn+1−εn−1)/2 is an estimate for the energy width of the nth PS. This expression provides the B(Eλ) values only for the PS eigenvalues εn. Alternatively, one may derive a continuous approximation to Eq. (13) by introducing the identity in the truncated PS basis, that is, IN = N  n=1ϕ(N) n, ϕ(N) n, .(18) For N→∞ this expression tends to the exact identity operator for the Hilbert space spanned by the eigenfunctions of the considered Hamiltonian. By inserting Eq. (18) into the exact expression (13), we obtain the approximated continuous distribution: dB(Eλ) dε ≃2f+1 2i+1 µbck (2π)3¯h2 × N  n=1ϕ(k)ϕ(N) n, ϕ(N) n, |M(Eλ)||ϕg.s. 2 . (19) 024605-3 LAY, MORO, ARIAS, AND G ´ OMEZ-CAMACHO PHYSICAL REVIEW C 82, 024605 (2010) To test the accuracy of the THO basis to describe the continuum, we calculate also the following magnitudes: (i) The non-energy-weighted sum rule (NEWSR): NEWSR ≡dεdB(Eλ) dε =2f+1 2i+1D(λ) i,f2r2λg.s.,(20) with r2λg.s.≡ϕg.s.|r2λ|ϕg.s.. (ii) The energy-weighted sum rule (EWSR): EWSR ≡dεdB(Eλ) dε (ε−εg.s.) =¯h2 2µbc λ(2λ+1)2f+1 2i+1D(λ) i,f2r2λ−2g.s.. (21) (iii) The energy-inverse-weighted integral (or polarizability): α≡8π 9dε 1 (ε−εg.s.) dB(Eλ) dε .(22) Due to their respective weight factors, the EWSR and the polarizability are useful quantities to test the accuracy of the basis to describe high-energy and low-energy parts of the spectrum, respectively. Note that the closed expression for the EWSR is only valid for angular-momentum-independent Hamiltonians. We note also that there is no closed expression for the polarizability, but to calculate this quantity with the desired accuracy, as well as the EWSR for angular-momentum- dependent Hamiltonians, one can directly evaluate Eq. (22) using the exact continuum states integrated up to a sufficiently high excitation energy. V. APPLICATION TO NUCLEAR STRUCTURE A. Application to the deuteron As an illustration of the expressions derived in the preceding sections, we first consider the case of the p-nsystem with a central potential. Following Ref. [21], the interaction between the proton and the neutron is parametrized in terms of the Poeschl-Teller potential, Vpn(r)=− V0 cosh(ar)2,(23) with V0=102.706 MeV and a=0.9407 fm−1. With these values, the ground-state energy is 2.2245 MeV, in agreement with the experimental value. The oscillator length was chosen to minimize the groundstate energy obtained upon diagonalization of the Hamiltonian in a small THO basis. This yields the value b=1.5 fm. Once the value of bis fixed, the ratio γ/bdetermines the extension of the PS eigenstates; increasing the value of γreduces the radial extension and pushes the eigenvalues to higher energies. This is better seen in terms of the density of states, defined according to Eq. (4). This magnitude is plotted in Fig. 1for the =0 continuum, using a basis of N=30 states, and three different choices of γ, namely, γ=1fm 1/2,2.48fm 1/2, and 5fm 1/2. It is seen that small values of γ(which correspond 0123 k (fm-1) 0 5 10 15 20 25 30 ρ(k) γ=1 fm1/2 γ=2.48 fm1/2 γ=5 fm1/2 HO FIG. 1. (Color online) Density of states for the =0 continuum obtained with a THO basis with N=30 states, defined according to Eq. (4) for different choices of the γparameter. The dot-dashed curve is the density obtained with a HO basis with 30 states and b=2fm. See text for details. to an extended THO basis in configuration space) produce a fine description of the continuum at low energy. This is useful, for example, to study Coulomb breakup. Increasing the value of γwill decrease the density of states at low energies, which is compensated by an increase of the density at higher excitation energies. The most suitable choice for this parameter will depend on the problem at hand, depending on the energy region of interest. We emphasize, however, that the dependence on γis not critical and, in the applications shown here, different values of γconverge to the same results for sufficiently large bases. For comparison, we include also in Fig. 1the density obtained with a HO basis with N=30 states and b=2.0fm, which minimizes the ground-state energy for the HO basis. We next consider the scattering phase-shifts. These are displayed in Fig. 2as a function of the relative p-nenergy. The top, middle, and bottom panels correspond to the s,p, and dwaves. The solid line is the calculation using the asymptotics of the exact scattering states, whereas the circles represent the calculation obtained with Eq. (7), using a THO basis with N=30 states. The LST was generated with the parameters b=1.5 fm and γ=2.48 fm1/2. In the three cases, we find an excellent agreement between the exact and the approximate phase-shifts in the whole energy range. Note that the calculated phase shifts are consistent with the Levinson theorem (see, e.g., Ref. [22]), which establishes that the phase shift at zero energy is given by δ(0) =nπ, where nis the number of bound states for the partial wave . So, because the swave supports a bound state (the deuteron), we have δ0(0) =π, whereas for =1 and =2wehave δ1(0) =δ2(0) =0. We now consider the electric transition probabilities, B(E1) and B(E2). These are shown in Fig. 3. The solid line corresponds to the calculation using the scattering states Eq. (13), the solid circles correspond to the discrete approximation using the THO basis Eq. (17), and the dashed line is the calculation obtained folding the discrete distribution with the continuum states [Eq. (19)]. Both the discrete and folded approximations 024605-4 EXPLORING CONTINUUM STRUCTURES WITH A PSEUDO- ... PHYSICAL REVIEW C 82, 024605 (2010) 0 20 40 60 80 100 0 10 20 30 δ0(ε) 0 20406080100 ε (MeV) 0 5 10 15 20 δ1(ε) 0 20406080100 0 60 120 180 240 δ0(ε) Scattering states THO: N=30 l=0 l=2 l=1 (a) (b) (c) FIG. 2. (Color online) Phase shifts for the deuteron system as a function of the relative p-nenergy. The upper, middle, and bottom panels correspond to =0, 1, and 2 continuum states, respectively. See text for details. 0102030 ε (MeV) 0 1 2 3 4 5 dB(E2)/dε (e2fm4/MeV) 0102030 0 0.05 0.1 0.15 0.2 dB(E1)/dε (e2fm2/MeV) Scattering states THO: N=30 (folded) THO: N=30 HO: b=2 fm, N=30 HO: b=2 fm, N=30 (folded) d=p+n (a) (b) FIG. 3. (Color online) Electric transition probabilities for the d= p+nsystem. The top and bottom panels correspond to λ=1and λ=2 transitions, respectively. See text for details. TABLE I. Convergence of the ground-state energy and the total B(E1) and B(E2) transition probabilities for the deuteron case. NTotal Total εg.s.(MeV) B(E1)(e2fm2)B(E2)(e2fm4) HO THO HO THO HO THO 10 −2.157 0 −2.215 0 0.813 80 0.852 86 14.699 3 17.972 9 20 −2.220 1 −2.224 5 0.862 00 0.871 29 19.125 7 20.793 8 30 −2.223 7 −2.224 5 0.869 26 0.871 36 20.291 9 20.829 7 40 −2.224 1 −2.224 5 0.870 79 0.871 36 20.642 2 20.829 7 50 −2.224 2 −2.224 5 0.871 19 0.871 36 20.759 1 20.829 7 60 −2.224 3 −2.224 5 0.871 30 0.871 36 20.801 5 20.829 7 70 −2.224 3 −2.224 5 0.871 34 0.871 36 20.817 8 20.829 7 Exact −2.224 5 0.871 36 20.829 7 show an excellent agreement with the exact distribution. We include also the calculation using the HO basis with N=30 states and b=2.0 fm (open circles and dot-dashed line). Both the B(E1) and B(E2) distributions depart significantly from the exact distributions. In addition, the HO basis produces a small density of states at low energy, which might be a drawback for scattering calculations. In Tables Iand II we present the convergence of the groundstate energy and the E1 and E2 sum rules with respect to the basis size. The last row lists the exact values obtained with the closed expressions of Eqs. (20) and (21). It is seen that with a moderately small basis one obtains a very good convergence to the exact values. For comparison, in these tables we include also the calculations using the HO basis. From the quoted numbers, it is clear that the convergence rate is much faster for the THO basis. It is worth noting that, despite the simple Hamiltonian adopted in this work for the p-nsystem, the calculated polarizability is fully consistent with the experimental value, αexp =0.61 ±0.04, quoted in Ref. [23]. B. Application to 6He We now consider a situation in which more complicated continuum structures are present, such as resonances. For this purpose, we take the 6He nucleus, treated as a two-body TABLE II. Convergence of the polarizability and EWSR for the deuteron case. NEnergy-weighted α(fm)3B(E1)(e2fm2MeV) HO THO HO THO 10 0.413 988 0.515 397 8.605 586 8.102 094 20 0.557 520 0.619 350 7.742 321 7.442 755 30 0.599 683 0.620 899 7.526 588 7.437 322 40 0.613 204 0.620 922 7.466 005 7.434 683 50 0.617 923 0.620 931 7.446 418 7.433 209 60 0.620 109 0.620 936 7.439 133 7.432 331 70 0.620 395 0.620 940 7.435 989 7.431 757 Exact 0.620 953 7.429 937 024605-5 LAY, MORO, ARIAS, AND G ´ OMEZ-CAMACHO PHYSICAL REVIEW C 82, 024605 (2010) 0246 ε (MeV) 0 60 120 180 δ2(ε) 120 150 180 δ1(ε) Scattering states THO basis: N=50 01020 01020 120 180 240 300 360 δ0(ε) l=0 6He=α+2n l=2 l=1 (a) (b) (c) FIG. 4. (Color online) Phase shifts for the 6He system as a function of the relative α+2nenergy. The top, middle, and bottom panels correspond to =0, 1, and 2 continuum states, respectively. See text for details. system α+2n. Following Ref. [24], the interaction between the two clusters is described with a Woods-Saxon shape, with R=1.9 fm and a=0.39 fm. For =0 states, the depth of this potential is adjusted to give the effective separation energy of 1.6 MeV between the two clusters. It was shown in Ref. [24] that using this effective binding energy, instead of the two-neutron separation energy (S2n=0.97 MeV), provides a more realistic description of the ground-state wave function. For =2, the intercluster potential is adjusted to yield a resonance at an excitation energy of Ex=1.8 MeV. For =1, we simply took the depth found for =0. A THO basis with N=50 states was used, and the LST was generated with the parameters b=1 fm and γ=1.89 fm1/2. The s-, p-, and d-wave phase shifts are displayed in Fig. 4 as a function of the α+2nrelative energy, ε.For=2, the energy scale has been restricted to the energy interval ε= 0–6 MeV to emphasize the region of the resonance. Again, we find a perfect agreement between the exact (solid line) and the approximate (circles) phase shifts in the whole energy range. Note that, in this case, the s-wave potential supports two bound states, the Pauli forbidden 1Sstate and the 2S ground state. Consistently, the phase shift at zero energy is given by δ0(0) =2π. Analogously, for =1 the phase shift tends to π, due to the presence of a (Pauli forbidden) bound state in this partial wave. Finally, for =2 no bound states are supported by this potential and, therefore, δ2(0) =0. For the d-wave, the phase shift crosses abruptly π/2atε=0.20 MeV, reflecting the presence of a narrow 2+resonance. Interestingly, this behavior is also observed in the THO basis, where there 020406080 r (fm) -0.2 0 0.2 0.4 0.6 0.8 ϕ2(r) Scattering wavefunction at ε=0.20 MeV THO: N=50, n=1 (ε=0.16 MeV) THO: N=50, n=2 (ε=0.20 MeV) THO: N=50, n=3 (ε=0.41 MeV) FIG. 5. (Color online) Radial part of the continuum wave function at the resonance for the 6He system. is a PS that appears exactly at the nominal energy of the resonance. It is then tempting to conclude that this PS will carry most of the character of the resonance, and in fact this is confirmed in Fig. 5, where we show the radial part of the exact scattering wave function, calculated at the energy of the resonance (solid line) along with the radial part of the PS eigenstate that appears at the energy of the resonance (dashed line). The former has been arbitrary normalized so that the two wave functions coincide at the maximum. It is seen that both wave functions are very similar up to very large distances. For comparison, we have included also the PS eigenstates associated with the eigenvalues just below (dotted-dashed line) and above (dotted line) the resonant one. They are very different from the scattering wave function at the resonance. In particular, it can be seen that a significant part of the norm of the resonant wave function is concentrated in the interior, as expected for a resonance, whereas for the nonresonant PS eigenstates, the probability in the interior is very small. This leads to the conclusion that in this case the character of the resonance is very well described by a single PS eigenstate. Indeed, if the basis is increased, the resonant character will be distributed among several PS. These results clearly show that a distinctive feature of the continuum such as the resonant structures are very well accounted for by the PS basis, despite its wrong asymptotic behavior. In Fig. 6we compare the E1 and E2 transition probabilities obtained from the scattering states by means of Eq. (13) (solid line), with the approximate distributions calculated with the THO basis. The circles correspond to the discrete expression of Eq. (17), whereas the dashed line is the smooth distribution obtained with Eq. (19). Both the discrete and smooth distributions are in excellent agreement with the exact distribution. In Table III we display the convergence of the ground-state energy and the E1 and E2 sum rules with respect to the basis size. As in the deuteron case, both observables converge very fast to their exact value, given by the sum rule of Eq. (20). Finally, in Table IV we present the convergence of the integrated-energy-weighted B(E1) and the polarizability. For 024605-6 EXPLORING CONTINUUM STRUCTURES WITH A PSEUDO- ... PHYSICAL REVIEW C 82, 024605 (2010) 0123456 ε (MeV) 0 5 10 15 20 dB(E2)/dε (e2fm4/MeV) 0246810 0 0.1 0.2 0.3 0.4 dB(E1)/dε (e2fm2/MeV) Scattering states THO: N=50 (folded) THO: N=50 6He=α + 2n(a) (b) FIG. 6. (Color online) Electric transition probabilities for the 6He =α+2nsystem. The top and bottom panels correspond to λ=1andλ=2 transitions, respectively. See text for details. the former, we cannot use the closed expression of Eq. (21), because the 6He Hamiltonian depends on the angular momentum. The exact value listed in this Table corresponds to the explicit calculation using the scattering states up to a high excitation energy. The same holds for the polarizability, because for this observable there is no closed expression. VI. APPLICATIONS TO NUCLEAR REACTIONS A. Application to the reactions 6He +12C and 6He +208Pb at 240 MeV/nucleon The THO basis considered in this work is intended to provide a suitable discrete representation of the continuum spectrum of a loosely bound system, which can be useful for TABLE III. Convergence of the ground-state energy and the total B(E1) and B(E2) transition probabilities for the 6He =α+2n system. Nε g.s.Total B(E1) Total B(E2) (MeV) (e2fm2)(e 2fm4) 10 −1.591 3 1.353 8 8.552 4 20 −1.599 9 1.385 4 9.747 1 30 −1.599 9 1.385 5 9.747 1 40 −1.599 9 1.385 5 9.747 1 50 −1.600 0 1.385 5 9.747 1 80 −1.600 0 1.385 5 9.747 1 Exact −1.600 0 1.385 5 9.747 1 TABLE IV. Convergence of the polarizability and energyweighted B(E1) for the 6He =α+2nsystem. NαTotal-energy-weighted B(E1) (fm3)(e 2fm2MeV) 20 1.865 2 6.562 3 30 1.874 6 6.546 4 40 1.875 0 6.543 8 50 1.875 2 6.542 5 60 1.875 3 6.541 8 70 1.875 3 6.5412 120 1.875 5 6.540 1 Exact 1.875 6 6.539 3 scattering calculations within the CDCC method [3]. As a test case, we apply the THO basis to the reactions 6He +12C and 6He +208Pb at 240 MeV/nucleon. These reactions were measured by Aumann et al. [25] at the GSI facility with the aim of extracting information on the 6He nucleus. The breakup of 6He on 208Pb was already analyzed using the CDCC method with the THO basis in Ref. [6], showing an excellent agreement with the binning discretization method for the modulus of the breakup Smatrix. In this work, we extend the analysis of Ref. [6] to compare with the data of Ref. [25]. In particular, we consider the exclusive breakup differential cross section as a function of the excitation energy of the projectile dσ/dEx. In Ref. [25], this observable was obtained by reconstructing the kinematics of the 6He c.m. from the measured momenta of the outgoing fragments (4He +n+n) and integrating up to a laboratory scattering angle of 80 mrad. To obtain this observable in our calculations, we first construct the double-differential cross section dσ/ddExfrom the breakup Smatrices. In principle, the breakup Smatrix is a continuous function of the asymptotic momentum k. However, within a PS representation of the continuum, only discrete values of the Smatrix are obtained, corresponding to the eigenvalues εn. A continuous breakup Smatrix can be obtained from the solution of the coupled equations following the procedure used in Refs. [4,6,26], in which the discrete Smatrices are folded with the exact scattering states, similarly to what was done with the B(Eλ) distribution in Eq. (19), that is, Sf:i(k)≈ N  n=1ϕf(k)ϕ(N) n,fˆ S(N) n:i(kn),(24) where ˆ S(N) n:i(kn) are the discrete Smatrix elements resulting from the solution of the coupled-channels equations using a PS basis with Nstates. The subscripts i,n, and f denote the channels {ϕg.s.;Li, i,J},{ϕ(N) n,f;Lf, f,J}, and {ϕf(k); Lf, f,J}, respectively, where Li(Lf) is the initial (final) orbital angular momentum for the projectile-target relative motion and Jis the total angular momentum of the system. We first consider the Pb target. For the 2n-αpotential and the parameters of the THO basis we keep the same values of the preceding section. These parameters were found to provide an appropriate distribution of eigenstates for the 6He + 208Pb reaction at 240 MeV/nucleon [6]. The α+208Pb and 024605-7 LAY, MORO, ARIAS, AND G ´ OMEZ-CAMACHO PHYSICAL REVIEW C 82, 024605 (2010) 2n+208Pb interactions are needed to generate the diagonal and nondiagonal coupling potentials of the CDCC equations. For the α+208Pb interaction we have adopted the first optical potential used in Ref. [27]. For the 2n+208Pb interaction, we used the following single-folding model: U(R)=ρnn(rnn)UnR+rnn 2+UnR−rnn 2drnn, (25) where Unis the neutron-208 Pb optical potential taken from the parametrization of Madland [28], evaluated at the apropriate energy per nucleon, and ρnn(r) is the neutron-neutron density distribution. The latter was calculated by integrating the square of the three-body wave function of the 6He nucleus along the 2n-αcoordinate, that is, ρ(rnn)=r2 2n−α|3b(rnn,r2n−α)|2dr2n−αdnn,(26) where 3b(rnn,r2n−α) is the three-body wave function and nn denotes the angular variables (θnn,φ nn). In the present calculations, the function 3bwas taken from Ref. [24]. Very good convergence of the CDCC calculations was achieved with a basis of N=30 states. In addition, we found that continuum states above 50 MeV have a negligible effect on the scattering observables, and hence these eigenstates were removed from the coupled-channels calculation. This reduces the number of PS included in the CDCC equations to ns=np=14 and nd=15 for s,p, and dwaves. The coupled equations were integrated up to 100 fm and for a total angular momentum up to Jmax =2000. In Fig. 7we show the calculated energy differential breakup cross section, along with the GSI data. The dotted, dashed, and dotted-dashed lines are the separate contributions of the s,p, and dcontinuum states. For this angular range (θlab < 80 mrad) the cross section is largely dominated by the coupling 02468 Ex (MeV) 0 50 100 150 200 dσ/dEx (mb/MeV) GSI data s-wave p-wave d-wave Total Total: convoluted 6He+208Pb @ 240 MeV/nucleon FIG. 7. (Color online) Angle-integrated breakup differential cross section, as a function of the excitation energy, for the reaction 6He +208Pb at 240 MeV/nucleon. The dotted, dashed, and dotteddashed lines are the contribution of the s,p,anddstates. The thin solid line is the sum of these contributions and the thick solid line is the full calculation convoluted with the experimental resolution. The circles are the data from Ref. [25]. to the j=1−states due, mainly, to the strong dipole Coulomb interaction. The thin solid line in Fig. 7is the sum of the s,p, and dcontributions. For a meaningful comparison with the data, this curve has to be convoluted with the experimental energy resolution, which we took from the same work [25]. The result of this folding is shown by the thick solid line. At low excitation energies, this calculation reproduces very well the shape and magnitude of the data. For excitation energies Ex>4 MeV, the calculation underpredicts the data. This discrepancy was also found in the semiclassical calculations reported in Ref. [25]. Note that the narrow peak due to the 2+resonance in 6He disappears in the folded calculation. The dominance of the dipole Coulomb couplings at these very small angles was used in Ref. [25] to extract the dB(E1)/dε distribution, by comparing the measured differential cross section, dσ/dEx, with semiclassical calculations. Although our calculations confirm the dominance of the E1 couplings, we have found that nuclear potentials have a small but not negligible effect on this observable. In addition, starting from the same structure model for the 6He nucleus, our CDCC calculations show some departure from the semiclassical calculations, suggesting that the connection between the energy differential cross section and the underlying E1 probability is more complicated than suggested by the semiclassical approach. These results are potentially very interesting because they may affect the extracted dB(E1)/dε distribution from the cross section data. This analysis is beyond the scope of the present work and we leave it for a separate publication. We consider now the 12 C target. The 2n-αinteraction and the parameters of the LST were the same as those used in the Pb case. For the 6He continuum states, =0–3 waves were included. In this case, we used a THO basis with N=40 states. The α+12C potential was adopted from Ref. [29]. For the 2n+12Cinteraction we used the same single-folding model as for the 208Pb case taking also the neutron-target potential, n+ 12C in this case, from the parametrization of Madland [28]. The coupled equations were integrated up to 100 fm and for a total angular momentum up to Jmax =200. In Fig. 8we compare the CDCC calculations with the experimental data from Ref. [25]. The meanings of the curves are the same as thoseinFig.7. In this case, the low-energy cross section is dominated by the population of the 2+continuum, with s and pwaves contributing to the background. The contribution of the fwave was found to be very small and hence it has not been included in this figure. As in the Pb case, the d-wave cross section shows a narrow peak corresponding to the well-known resonance at Ex=1.8 MeV. The width of the peak in our calculation is significantly smaller than the experimental width of this resonance (exp ≈100 keV). This is a consequence of our simple two-body model adopted for the 6He nucleus. Nevertheless, when the calculation is convoluted with the experimental resolution (thick solid line) it becomes very close to the data. Despite the simplicity of the structure model adopted in these calculations, these results show that the THO discretization method constitutes a useful method to describe accurately detailed structures of the continuum that may show up in scattering observables. 024605-8 EXPLORING CONTINUUM STRUCTURES WITH A PSEUDO- ... PHYSICAL REVIEW C 82, 024605 (2010) 1234567 Ex (MeV) 0 2 4 6 8 10 12 dσ/dEx (mb/MeV) GSI data s-wave p-wave d-wave Total Total: convoluted 6He+12C @ 240 MeV/nucleon FIG. 8. (Color online) Energy breakup differential cross section, as a function of the excitation energy, for the reaction 6He +208Pb at 240 MeV/nucleon. The dashed line is the CDCC calculation with the THO discretization method. The solid line is the result of convoluting the CDCC calculation with the experimental resolution. The circles are the data from Ref. [25]. As a final remark on these results, we note that, at these energies, relativistic effects might affect the calculated observables. Some recent exploratory studies [30,31]have shown that these effects produce an increase of about 10%– 15% on the calculated breakup cross section for the 8B+208Pb and 11Be +208Pb reactions at 250 MeV/nucleon. These effects affect only the very small angles and are mostly due to the modification of the Coulomb potential. Therefore, we might expect a similar effect in our 6He +208Pb case. For the 6He +12C case, we do not expect these effects to be important, because dynamical relativistic corrections to the nuclear interaction were found by the same authors to be negligible at these energies. These corrections refer to dynamical effects exclusively. Relativistic kinematics effects were included in the referred works, as well as in our calculations, by using the appropriate relativistic momentum. In any case, the aim of our work is to show the ability of the THO basis to describe the continuum of a two-body system, and so the emphasis of our study is more on the description of the structure, rather than on the reaction mechanism. B. Application to the 11Be +12C reaction at 67 MeV/nucleon As a final example, we consider the scattering of the halo nucleus 11Be on a carbon target. This reaction has been recently measured by Fukuda and collaborators at RIKEN [32], to extract information on the 11Be continuum by measuring neutron-10Be coincidences following the projectile breakup. The angle-integrated differential cross section as a function of the relative energy between the outgoing neutron and 10Be displays a structure dominated by a prominent resonance at Ex=1.78 MeV. This resonance was interpreted as a d5/2 neutron coupled to the 10Be ground state. A second bump was also observed at Ex=3.41 MeV, which was given a tentative assignment of 3/2+, with a small contribution of the d3/2wave coupled to 10Be(0+) and a larger contribution of the 10Be(2+)⊗ν2s1/2configuration. As in previous analyses of this reaction [33,34], we use a two-body model of the projectile, 11Be =10Be(g.s.)+n, and hence those states based on the core excited states are absent from our model space. In the present CDCC calculations, the neutron-10Be interaction was taken from Ref. [35]. This potential reproduces the separation energy of the ground state (1/2+) and first excited state (1/2−) and the position of the 5/2+ 1resonance, assuming the configurations 2s1/2,1p1/2, and d5/2, respectively. The 11Be continuum was described with a THO basis with N=25 states. The LST was generated with the parameters b=2.4fm and γ=3.6fm 1/2. Continuum states with configurations s1/2, p1/2,p3/2,d3/2, and d5/2were considered. After diagonalization of the projectile Hamiltonian in this THO basis, only those eigenstates with excitation energies below E<20 MeV were included in the coupled-channels calculations, because the breakup cross section was found to be very small above this energy. This leaves about 10–11 eigenstates for each partial wave. Following Ref. [34], the n+12Cand 10Be +12C potentials were taken from Refs. [36] and [37], respectively. The coupled-channels equations were integrated up to a matching radius of R=90 fm and for total angular momenta up to J=350. In Fig. 9we compare the experimental [32] and calculated energy differential cross sections as a function of the n-10Be relative energy. Both the data and the calculations correspond to the angular range 0◦⩽θc.m.⩽12◦. The calculated contribution of each partial wave is shown. The symbols correspond to the contribution of specific pseudo-states, whereas the continuous lines are obtained convoluting the discrete Smatrices with the exact continuum states. The low-lying continuum is dominated by the p3/2and d5/2waves, with the latter being responsible for the resonant peak at Ex=1.78 MeV. The dotted line is the sum of the different partial waves, and the thick solid line is the result of folding this full calculation with the instrumental resolution quoted in Ref. [32]. At energies 012345 ε (MeV) 0 20 40 60 dσ/dε (mb/MeV) RIKEN data s1/2 p1/2 p3/2 d3/2 d5/2 Total Total: convoluted 11Be+12C @ 67 MeV/nucleon FIG. 9. (Color online) Energy breakup differential cross section, as a function of the excitation energy, for the reaction 11Be +12Cat 67 MeV/nucleon. The separate contribution of each partial wave is shown. The dotted line is the full contribution, and the thick solid line is the folding of the latter with the instrumental resolution. The experimental data are from Ref. [32]. 024605-9