Continuum description with pseudostate wave functions

PHYSICAL REVIEW C 75, 017603 (2007) Continuum description with pseudostate wave functions A. M. Moro*and M. Rodr´ ıguez-Gallardo† Departamento de F´ ısica At´ omica, Molecular y Nuclear, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain R. Crespo‡ Departamento de F´ ısica, Instituto Superior T´ ecnico, Taguspark, Avenida Prof. Cavaco Silva, P-2780-990 Porto Salvo, Oeiras, Portugal I. J. Thompson§ Departament of Physics, University of Surrey, Guildford GU2 7XH, United Kingdom (Received 13 October 2006; published 17 January 2007) Benchmark calculations are performed aiming to test the use of two different pseudostate bases on the multiple scattering expansion of the total transition amplitude scattering framework. Calculated differential cross sections for p-6He inelastic scattering at 717 MeV/nucleon show a good agreement between the observables calculated in the two bases. This result gives extra confidence on the pseudostate representation of continuum states to describe inelastic/breakup scattering. DOI: 10.1103/PhysRevC.75.017603 PACS number(s): 24.10.−i, 24.50.+g, 25.40.Ep Inelastic scattering at intermediate energies can be a useful tool to study multipole excitations of Borromean nuclei (such as 11Li and 6He). Because of their loosely bound nature, to properly understand and interpret such reactions, it is crucial to take into account the few-body degrees of freedom. At high energies, the multiple scattering expansion of the total transition amplitude (MST) is a convenient framework that has already been applied to analyze such reactions for elastic [1,2] as well as for inelastic [3,4] scattering. In the latter case, the method can take into account spin excitations that occur when scattering from a spin target such as a proton. In these calculations, it is formal and numerically advantageous to represent the continuum states in terms of a basis of square-integrable functions, also known as pseudostates (PSs). Unlike the true scattering states, the PSs vanish at large distances and hence the method will be only useful if the calculated observables are not sensitive to the asymptotic region. Moreover, calculations performed with different families of states should converge to the same results, provided that enough states are included and that the basis is complete within the radial region relevant for the process under study. Guided by this motivation, in this Brief Report we present benchmark calculations of proton inelastic scattering from 6He within the MST framework, making use of two different PS bases to describe the 6He continuum. We aim to check to what extent the calculated breakup observables depend on the choice of the PS functions. For a Borromean system, such as 6He, the wave function for a total angular momentum J(with projection M) and energy , ϕJM , can be expressed in terms of the Jacobi coordinates r *Electronic address: [email protected] †Electronic address: [email protected]; present address: Centro de F´ ısica Nuclear, Universidade de Lisboa, Avenida Prof. Gama Pinto 2, P-1649-003 Lisboa, Portugal. ‡Electronic address: [email protected] §Electronic address: [email protected] (the relative coordinate between the valence nucleons) and  R (the relative coordinate from the center of mass of the neutron pair to the core). It is also convenient to introduce a set of hyperspherical coordinates: the hyper-radius ρand five hyperspherical polar angles 5={α, θx,φ x,θ y,φ y}. The former is defined as ρ=x2+y2with scaled coordinates x=2−1/2rand y= (2/√3)  R. The angle α=arctan(x/y) is the hyperangle and θx,φ x,θ y,φ yare the angles associated with the unit spatial vectors ˆ xand ˆ y. Within the PS method, the eigenstates ϕJM (r,  R)are obtained by diagonalization of the Hamiltonian in a basis of normalizable states. These states are conveniently expanded in a basis of hyperspherical harmonics of the form ψJM nβ (r,  R)=Rnβ (ρ)ϒJM β(5),(1) where ϒJM β(5) is the generalized angle-spin basis [5] ϒJM β(5)=YKxyL(5)⊗χs2⊗χs3SJM,(2) with χsithe neutron spin functions and YKxyL(5)the hyperspherical harmonics, YKxyLML(5)=ψKxy(α)Yx(ˆ x)⊗Yy(ˆ y)LML.(3) The functions ψKxy(α) have an explicit form in terms of Jacobi polynomials of the hyperangle α[5]. The set of quantum numbers β={KxyLS}defines a channel, with xand ythe orbital angular momenta associated with the Jacobi coordinates xand y,K=x+y+2ν(ν=0,1,2,...)the hyperangular momentum,  L= x+ ythe total orbital angular momentum, and Sthe spin of the particles related by the coordinate x.InEq.(1), Rnβ (ρ) are the hyper-radial functions and nis an index that labels the basis states within a given channel β. These functions are orthogonalized such that ∞ 0 dρρ5Rnβ (ρ)Rnβ(ρ)=δnn.(4) 0556-2813/2007/75(1)/017603(4) 017603-1 ©2007 The American Physical Society BRIEF REPORTS PHYSICAL REVIEW C 75, 017603 (2007) The aim of the present work is to compare two different choices for the functions Rnβ (ρ) in the calculation of breakup observables within the MST framework. First, we consider the Gauss-Laguerre (GL) basis [6], whose hyper-radial part, RGL n(ρ), is given by RGL n(ρ)=ρ0−3[n!/(n+5)!]1/2L5 n(z)exp(−z/2) ,(5) with z=ρ/ρ0,L 5 nthe generalized Laguerre polynomials, and ρ0a parameter that sets the radial scale of the basis. The second choice is the transformed harmonic oscillator (THO) basis, recently introduced in Ref. [7] for a three-body system. The THO method is based on the idea of transforming the bound ground-state wave function of the system into the ground-state wave function of the harmonic oscillator (HO), defining a local scale transformation (LST). The ground-state wave function can be written as a linear combination of the basis functions (1), ϕJ0M0 0(r,  R)= β R0 β(ρ)ϒJ0M β(5),(6) where we have introduced the abbreviated notation ϕJ0M0 0≡ ϕJ0M0 0. Then, the equation that defines the LST for each channel βis ρ 0 dρρ5R0 β(ρ)2=s 0 dss5RHO 0K(s)2,(7) where RHO 0K(s) is the hyper-radial part of the HO ground state for the hyperangular momentum K. Then, the THO basis is constructed for each channel by applying the LST, sβ(ρ), to the HO basis RTHO nβ (ρ)=R0 β(ρ)LK+2 ns2 β(ρ),(8) where LK+2 nare generalized Laguerre polynomials of degree n. For channels not included in the ground state, information from one of the known (ground-state) channels with the closest quantum labels to the channel of interest is used to construct the LST, as explained in Ref. [7]. Neither the GL nor the THO functions are eigenstates of the Hamiltonian, but they provide a complete and orthonormal set in which the Hamiltonian can be diagonalized. For this purpose, the basis is truncated by setting a maximum value of the index n(n=0,...,N) as well as a maximum hyperangular momentum Kmax. Upon diagonalization in the truncated basis, the eigenstates are obtained as ϕJM i(r,  R)= nβ CJ i nβ ψJM nβ (r,  R),(9) where {i}are their associated eigenvalues. From this derivation, it becomes apparent that the GL basis is obtained in a more straightforward way than the THO basis. However, the latter has some appealing properties that could make it more suitable in some situations. In particular, the THO basis has the advantage of being constructed from the ground-state wave function of the system. Thus, when we diagonalize the Hamiltonian in a finite THO basis, the ground state is recovered for any size of the basis. By contrast, in the GL representation a large basis may be required to obtain a good description of the ground state. Also, note that the hyper-radial part of the GL basis is the same for all the channels βwhereas in the THO basis a different hyper-radial part is calculated for each channel, with the correct behavior at the origin (ρK). For a meaningful comparison between the two bases, we use the same three-body Hamiltonian to generate the GL and THO eigenstates for 6He. In particular, we use the n-npotential of Gogny, Pires, and de Tourreil [8] with spin-orbit and tensor components and we take the n-4He potential from Ref. [9]. Besides the pairwise interactions, an effective three-body potential is included, with matrix elements of the form [5] V3B ββ(ρ)=δββV3B J 1+(ρ/5)3.(10) The J=0 strength of this effective potential is tuned to reproduce the experimental three-body separation energy and the J>0 strength is adjusted to obtain the 2+ 1resonance at the experimental energy. We now consider the scattering process of 6He, originally in its ground state, |ϕJ0M0 0, to a final continuum state |ϕJM , at excitation energy and with total angular momentum J(projection M), by means of its interaction with a proton, with initial (final) linear momentum  k1( k 1) in the nucleonnucleus c.m. frame and spin S1=1/2 with projection σ(σ). The double differential cross section for this process can be formally expressed as d2σJJ 0 dd =1 ( S1)2 1 ( J0)2¯h2 4π2µNA 2 × σσ MM0 k 1χσ S1;ϕJM T k1χσ S1;ϕJ0M0 02,(11) where Tdenotes the transition amplitude operator [10]. This operator can be expressed as a multiple expansion series in the transition amplitudes ˆ tIfor proton scattering from each projectile subsystem I[11]. At high energies and for small momentum transfers, this expansion is expected to converge quickly. If only the leading term of the series is retained, the single scattering approximation (SSA) is obtained [3,12]: TSSA = 4  I=2 ˆ t1I(12) with I=2,3 for the halo neutrons, and I=4 for the core. The proton-Isubsystem transition amplitude satisfies the Lippmann-Schwinger equation ˆ t1I=v1I+v1IG0ˆ t1I,(13) with v1Ithe interaction between the nucleon and Isubsystem. Within the impulse approximation, the propagator G0= E+−K−1contains the kinetic energy operators of the proton and all the projectile subsystems. Here Eis the kinetic energy, E=¯h2k2 1/2µNA in the overall c.m. frame, and µNA is the proton-projectile reduced mass. Within the PS method, the scattering states ϕJM in Eq. (11) are approximated by the pseudostates ϕJM i. Hence, the double differential cross section (11) becomes a single differential 017603-2 BRIEF REPORTS PHYSICAL REVIEW C 75, 017603 (2007) cross section for each pseudostate, dσi JJ 0 d =1 ( S1)2 1 ( J0)2¯h2 4π2µNA 2 × σσ MM0 k 1χσ S1;ϕJM i 4  I=2 ˆ t1I k1χσ S1;ϕJ0M0 02, (14) where we have replaced the Tmatrix operator by its single scattering approximation. By making use of the impulse approximation [2], the matrix elements for the scattering for each constituent can be further simplified, leading to the following factorized form for the scattering from one valence nucleon (I=2):  k 1χσ S1;ϕJM iˆ t12 k1χσ S1;ϕJ0M0 0 = bβ t[bβSpS p](ω12,)×ρ[bβ ;STS Ti]m3 M23  , m4 M234  , (15) with M23 =m2+m3and M234 =m2+m3+m4and where we have introduced the momentum transfer  = k 1− k1and the energy parameter ω12 [2] and where Sp={S1σ}(S p= {S1σ}) are the incoming (outgoing) spin of the nucleon and its projection and ST={J0M0}(S T={JM}) the initial (final) total spin of the halo valence pair. The amplitude t[bβSpS p]is given in terms of the tensor components of the nucleon-nucleon transition amplitude [2,13]. The transition density form factors, ρ[bβ;STS Ti], depend exclusively on the structure of the composite system. Its explicit expression as a function of the hyper-radial parts of the wave functions of the initial and final states can be found in Ref. [2]. The scattering from the core, assumed here as spinless, can equivalently be written as  k 1χσ S1;ϕJM iˆ t14 k1χσ S1;ϕJ0M0 0 =t[00SpS p](ω14,)×ρ[00;STS Ti]0,M23 M234  ,(16) where, as before, ω14 is the appropriate energy parameter [2]. The angular differential cross section for 6He inelastic scattering (breakup) is then obtained by summing all excitedstate contributions, dσinel JJ 0 d = max i  i dσi JJ 0 d .(17) For the evaluation of Eqs. (15) and (16) one needs the (free) transition amplitudes for proton scattering from the valence nucleons and the core. For the former, we used the NN Paris interaction. The transition amplitude for the αcore was generated from a phenomenological optical potential, of Woods Saxon form, with parameters obtained by fitting existing data for the elastic scattering of p+4He at Ep=700 and 800 MeV, as detailed in Ref. [2]. We first study the convergence of the breakup observables with respect to the basis size. For this purpose, we consider the THO basis, truncated at different values of N. The maximum 0 5 10 15 20 25 30 θc.m. (deg) 10-2 10-1 100 101 102 dσinel/dΩ (mb/sr) N=4 N=3 N=2 6He(p,p’)6He*(0+) @ Ep=717 MeV THO basis FIG. 1. (Color online) Angular differential cross section for the breakup of 6He on protons at 717 MeV per nucleon, leading to Jπ= 0+continuum states of the 6He nucleus. The three lines represent the calculation with the THO basis, for several values of N(indicated by the labels). Each curve includes the contribution from eigeinstates up to 10 MeV in excitation energy, according to Eq. (17). hyperangular momentum was set to Kmax =20. This yields the three-body force parameters V3B 0=−2.4 MeV, for J=0, and V3B J=−0.85 MeV, for J>0. In Fig. 1we show the angular distribution of the calculated inelastic differential cross sections. For simplicity, only the Jπ=0+continuum is included, and the Coulomb interaction between the proton and the αcore is ignored. The three lines represent the SSA calculation for different values of the basis size, according to the choice of the parameter N.Thethree cases are in almost perfect agreement, indicating that in this reaction the convergence with the basis size is very fast. Next, we study the dependence of the breakup observables on the choice of the basis, by comparing the calculations in the GL and THO representations. As before, the maximum hyperangular momentum was set to Kmax =20, and only eigenstates below 10 MeV are considered. The index n was truncated to N=20 and N=4 for the GL and THO bases, respectively. With this model space, the number of pseudostates in the GL (THO) basis is 31 (30) for 0+,63(86) for 1−,53(49) for 1+, and 79 (81) for 2+.FortheGL basis, the range parameter was set to ρ0=0.25 fm, which provides a basis that extends up to about 20 fm in the hyper-radius. With these parameters, the ground state obtained after diagonalization of the Hamiltonian appears at −0.9781 and −0.9549 MeV, for the GL and THO bases, respectively. In Fig. 2we compare the inelastic angular distributions calculated in the GL and THO bases. The separate contributions for Jπ=0+,1−,1+, and 2+final states are also shown. As before, the Coulomb interaction is neglected. The thick lines are the incoherent sum of all these Jπ contributions. Solid and dashed lines correspond, respectively, to the calculations with the GL and THO bases. For each curve, the contribution of eigenstates up to max =10 MeV are added incoherently, according to Eq. (17). We consider only the forward angles θc.m.<30◦since the SSA is not expected to work well at large momentum transfers [2]. We see that, at these angles, the dominant contribution to the breakup cross section comes from the 1−states, whereas for θc.m.>25◦, 017603-3 BRIEF REPORTS PHYSICAL REVIEW C 75, 017603 (2007) 0 5 10 15 20 25 30 θc.m. (deg) 10-2 10-1 100 101 102 dσinel/dΩ (mb/sr) Total 12+ 0+ 1+ FIG. 2. (Color online) Calculated contributions for breakup differential cross section leading to final states with Jπ=0+,1−,1+, and 2+, using the SSA. Solid and dashed lines refer to the GL and THO basis, respectively. the 2+excitation becomes dominant. Finally, the population of the unnatural parity 1+states is almost negligible at all angles. We notice that this excitation mode requires spin-flip transitions, which, according to these calculations, are very small in this reaction. For all these contributions, the GL and THO bases provide very similar results, suggesting that the calculated observables do not depend on the choice of the continuum representation, provided that enough states are included. In summary, in this Brief Report we have calculated proton inelastic scattering from 6He at 717 MeV/nucleon, using as scattering framework the single-scattering approximation and two different pseudostate representations of the 6He continuum: the GL and the THO. Provided that enough states are included, both bases predict essentially the same inelastic differential cross section. Furthermore, the studied observables converge very quickly with the size of the basis. These results support the reliability of the pseudostate method as a useful and convenient tool to treat scattering problems dealing with continuum states. This analysis could be extended to other PS bases and reactions. Furthermore, it could be applied to other scattering frameworks, for which the PS method has also been implemented, such as the continuum discretized coupled channels method [14,15]. We are grateful to J. G´ omez-Camacho and J. M. Arias for useful discussions. This work was supported by the Fundac¸˜ ao para a Ciˆ encia e Tecnologia (Portugal) through Grant No. POCTI/1999/FIS/36282, by the Acci´ on Integrada HP20030121, and in the U.K. by EPSRC Grant No. GR/M/82141. A.M.M. acknowledges a research grant by the Junta de Andaluc´ ıa. [1] R. Crespo and I. J. Thompson, Phys. Rev. C 63, 044003 (2001). [2] R. Crespo, A. M. Moro, and I. J. Thompson, Nucl. Phys. A771, 26 (2006). [3] R. Crespo, I. J. Thompson, and A. A. Korsheninnikov, Phys. Rev. C 66, 021002(R) (2002). [4] R. Crespo, I. J. Thompson, and A. M. Moro, Phys. Rev. C 74, 044616 (2006). [5] B. V. Danilin, I. J. Thompson, M. V. Zhukov, and J. S. Vaagen, Nucl. Phys. A632, 383 (1998). [6] I. Thompson, F. Nunes, and B. Danilin, Comput. 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