Benchmark calculations are performed aiming to test the use of two different
pseudo-state bases on the the Multiple Scattering expansion of the total
Transition amplitude (MST) scattering framework. Calculated differential cross
sections for p-6He inelastic scattering at 717 MeV/u show a good agreement
between the observables calculated in the two bases. This result gives extra
confidence on the pseudo-state representation of continuum states to describe
inelastic/breakup scattering.Comment: 4 pages, 2 figures. Published in Physical Review 

A. M. Moro

C. J. Joachain

I. J. Thompson

M. Rodríguez-Gallardo

R. Crespo

English

arXiv

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-6 He 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.Fundacao para a Ciencia e a Tecnologia POCTI/1999/FIS/36282Acción  Integrada  HP2003- 0121U.K. by EPSRC  GR/M/8214

Moro Muñoz, Antonio Matías

Rodríguez Gallardo, Manuela

Crespo, Raquel

Thompson, I. J.

idUS. Depósito de Investigación Universidad de Sevilla

PHYSICAL REVIEW C 75, 017603 (2007)Continuum description with pseudostate wave functionsA. M. Moro* and M. Rodrı́guez-Gallardo†Departamento de Fı́sica Atómica, Molecular y Nuclear, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, SpainR. Crespo‡Departamento de Fı́sica, Instituto Superior Técnico, Taguspark, Avenida Prof. Cavaco Silva, P-2780-990 Porto Salvo, Oeiras, PortugalI. 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 multiplescattering expansion of the total transition amplitude scattering framework. Calculated differential cross sectionsfor p-6He inelastic scattering at 717 MeV/nucleon show a good agreement between the observables calculatedin the two bases. This result gives extra confidence on the pseudostate representation of continuum states todescribe inelastic/breakup scattering.DOI: 10.1103/PhysRevC.75.017603 PACS number(s): 24.10.−i, 24.50.+g, 25.40.EpInelastic scattering at intermediate energies can be a usefultool to study multipole excitations of Borromean nuclei (suchas 11Li and 6He). Because of their loosely bound nature, toproperly understand and interpret such reactions, it is crucialto take into account the few-body degrees of freedom. Athigh energies, the multiple scattering expansion of the totaltransition amplitude (MST) is a convenient framework that hasalready been applied to analyze such reactions for elastic [1,2]as well as for inelastic [3,4] scattering. In the latter case, themethod can take into account spin excitations that occur whenscattering from a spin target such as a proton. In these calcu-lations, it is formal and numerically advantageous to representthe continuum states in terms of a basis of square-integrablefunctions, also known as pseudostates (PSs). Unlike thetrue scattering states, the PSs vanish at large distances andhence the method will be only useful if the calculated observ-ables are not sensitive to the asymptotic region. Moreover,calculations performed with different families of states shouldconverge to the same results, provided that enough states areincluded and that the basis is complete within the radial regionrelevant for the process under study.Guided by this motivation, in this Brief Report we presentbenchmark calculations of proton inelastic scattering from6He within the MST framework, making use of two differentPS bases to describe the 6He continuum. We aim to check towhat extent the calculated breakup observables depend on thechoice of the PS functions.For a Borromean system, such as 6He, the wave function fora total angular momentum J (with projection M) and energyε, ϕJMε , can be expressed in terms of the Jacobi coordinates r*Electronic address: moro@us.es†Electronic address: mrodri@cii.fc.ul.pt; present address: Centro deFı́sica Nuclear, Universidade de Lisboa, Avenida Prof. Gama Pinto 2,P-1649-003 Lisboa, Portugal.‡Electronic address: Raquel.Crespo@tagus.ist.utl.pt§Electronic address: I.Thompson@surrey.ac.uk(the relative coordinate between the valence nucleons) and R(the relative coordinate from the center of mass of the neutronpair to the core).It is also convenient to introduce a set of hypersphericalcoordinates: the hyper-radius ρ and five hyperspherical polarangles 5 = {α, θx, φx, θy, φy}. The former is defined asρ =√x2 + y2 with scaled coordinates x = 2−1/2r and y =(2/√3) R. The angle α = arctan(x/y) is the hyperangle andθx, φx, θy, φy are the angles associated with the unit spatialvectors x̂ and ŷ.Within the PS method, the eigenstates ϕJMε (r, R) areobtained by diagonalization of the Hamiltonian in a basis ofnormalizable states. These states are conveniently expanded ina basis of hyperspherical harmonics of the formψJMnβ (r, R) = Rnβ(ρ)ϒJMβ (5), (1)where ϒJMβ (5) is the generalized angle-spin basis [5]ϒJMβ (5) ={YKxyL(5) ⊗[χs2 ⊗ χs3]S}JM, (2)with χsi the neutron spin functions and YKxyL(5) thehyperspherical harmonics,YKxyLML(5) = ψKxy (α)[Yx (x̂) ⊗ Yy (ŷ)]LML. (3)The functions ψKxy (α) have an explicit form in terms ofJacobi polynomials of the hyperangle α [5]. The set of quantumnumbers β = {KxyLS} defines a channel, with x andy the orbital angular momenta associated with the Jacobicoordinates x and y,K = x + y + 2ν (ν = 0, 1, 2, . . .) thehyperangular momentum, L = x + y the total orbital angu-lar momentum, and S the spin of the particles related by thecoordinate x. In Eq. (1), Rnβ(ρ) are the hyper-radial functionsand n is an index that labels the basis states within a givenchannel β. These functions are orthogonalized such that∫ ∞0dρρ5Rnβ(ρ)Rn′β(ρ) = δnn′ . (4)0556-2813/2007/75(1)/017603(4) 017603-1 ©2007 The American Physical Societybrought to you by COREView metadata, citation and similar papers at core.ac.ukprovided by idUS. Depósito de Investigación Universidad de SevillaBRIEF REPORTS PHYSICAL REVIEW C 75, 017603 (2007)The aim of the present work is to compare two differentchoices for the functions Rnβ(ρ) in the calculation of breakupobservables within the MST framework. First, we considerthe Gauss-Laguerre (GL) basis [6], whose hyper-radial part,RGLn (ρ), is given byRGLn (ρ) = ρ0−3[n!/(n + 5)!]1/2L5n(z) exp(−z/2) , (5)with z = ρ/ρ0, L5n the generalized Laguerre polynomials, andρ0 a 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-bodysystem. The THO method is based on the idea of transformingthe bound ground-state wave function of the system into theground-state wave function of the harmonic oscillator (HO),defining a local scale transformation (LST). The ground-statewave function can be written as a linear combination of thebasis functions (1),ϕJ0M00 (r, R) =∑βRε0β (ρ)ϒJ0Mβ (5), (6)where we have introduced the abbreviated notation ϕJ0M00 ≡ϕJ0M0ε0 . Then, the equation that defines the LST for each channelβ is ∫ ρ0dρ ′ρ ′5∣∣Rε0β (ρ ′)∣∣2 =∫ s0ds ′s ′5∣∣RHO0K (s ′)∣∣2, (7)where RHO0K (s) is the hyper-radial part of the HO ground statefor the hyperangular momentum K. Then, the THO basis isconstructed for each channel by applying the LST, sβ(ρ), tothe HO basisRTHOnβ (ρ) = Rε0β (ρ)LK+2n(s2β(ρ)), (8)where LK+2n are generalized Laguerre polynomials of degreen. For channels not included in the ground state, informationfrom one of the known (ground-state) channels with the closestquantum labels to the channel of interest is used to constructthe LST, as explained in Ref. [7].Neither the GL nor the THO functions are eigenstates ofthe Hamiltonian, but they provide a complete and orthonormalset in which the Hamiltonian can be diagonalized. Forthis purpose, the basis is truncated by setting a maximumvalue of the index n (n = 0, . . . , N ) as well as a maximumhyperangular momentum Kmax. Upon diagonalization in thetruncated basis, the eigenstates are obtained asϕJMεi (r, R) =∑nβCJεinβ ψJMnβ (r, R), (9)where {εi} are their associated eigenvalues.From this derivation, it becomes apparent that the GL basisis obtained in a more straightforward way than the THObasis. However, the latter has some appealing properties thatcould make it more suitable in some situations. In particular,the THO basis has the advantage of being constructed fromthe ground-state wave function of the system. Thus, whenwe diagonalize the Hamiltonian in a finite THO basis, theground state is recovered for any size of the basis. By contrast,in the GL representation a large basis may be required toobtain a good description of the ground state. Also, note thatthe hyper-radial part of the GL basis is the same for all thechannels β whereas in the THO basis a different hyper-radialpart is calculated for each channel, with the correct behaviorat the origin (ρK).For a meaningful comparison between the two bases, weuse the same three-body Hamiltonian to generate the GL andTHO eigenstates for 6He. In particular, we use the n-n potentialof Gogny, Pires, and de Tourreil [8] with spin-orbit and tensorcomponents and we take the n-4He potential from Ref. [9].Besides the pairwise interactions, an effective three-bodypotential is included, with matrix elements of the form [5]V 3Bβ ′β(ρ) =δβ ′βV3BJ1 + (ρ/5)3 . (10)The J = 0 strength of this effective potential is tuned toreproduce the experimental three-body separation energy andthe J > 0 strength is adjusted to obtain the 2+1 resonance atthe experimental energy.We now consider the scattering process of 6He, originallyin its ground state, |ϕJ0M00 〉, to a final continuum state |ϕJMε 〉,at excitation energy ε and with total angular momentumJ (projection M), by means of its interaction with a proton,with initial (final) linear momentum k1(k ′1) in the nucleon-nucleus c.m. frame and spin S1 = 1/2 with projection σ (σ ′).The double differential cross section for this process can beformally expressed asd2σJJ0ddε= 1(Ŝ1)21(Ĵ0)2[h̄24π2µNA]2×∑σσ ′∑MM0∣∣〈k′1χσ ′S1 ; ϕJMε ∣∣T ∣∣k1χσS1 ; ϕJ0M00 〉∣∣2, (11)where T denotes the transition amplitude operator [10]. Thisoperator can be expressed as a multiple expansion series inthe transition amplitudes t̂I for proton scattering from eachprojectile subsystem I [11]. At high energies and for smallmomentum transfers, this expansion is expected to convergequickly. If only the leading term of the series is retained, thesingle scattering approximation (SSA) is obtained [3,12]:T SSA =4∑I=2t̂1I (12)with I = 2, 3 for the halo neutrons, and I = 4 for the core.The proton-I subsystem transition amplitude satisfies theLippmann-Schwinger equationt̂1I = v1I + v1IG0 t̂1I , (13)with v1I the interaction between the nucleon and I subsystem.Within the impulse approximation, the propagator G0 =(E+ − K)−1 contains the kinetic energy operators of theproton and all the projectile subsystems. Here E is the kineticenergy, E = h̄2k21/2µNA in the overall c.m. frame, and µNA isthe proton-projectile reduced mass.Within the PS method, the scattering states ϕJMε in Eq. (11)are approximated by the pseudostates ϕJMεi . Hence, the doubledifferential cross section (11) becomes a single differential017603-2BRIEF REPORTS PHYSICAL REVIEW C 75, 017603 (2007)cross section for each pseudostate,dσ iJJ0d= 1(Ŝ1)21(Ĵ0)2[h̄24π2µNA]2×∑σσ ′∑MM0∣∣〈k′1χσ ′S1 ; ϕJMεi ∣∣4∑I=2t̂1I∣∣k1χσS1 ; ϕJ0M00 〉∣∣2,(14)where we have replaced the T matrix operator by its singlescattering approximation. By making use of the impulseapproximation [2], the matrix elements for the scatteringfor each constituent can be further simplified, leading to thefollowing factorized form for the scattering from one valencenucleon (I = 2):〈k′1χσ ′S1 ; ϕJMεi ∣∣t̂12∣∣k1χσS1 ; ϕJ0M00 〉=∑bβt[bβSpS ′p](ω12,) × ρ[bβ;ST S ′T εi ](m3M23, m4M234),(15)with M23 = m2 + m3 and M234 = m2 + m3 + m4 and wherewe have introduced the momentum transfer  = k ′1 − k1 andthe energy parameter ω12 [2] and where Sp = {S1σ }(S ′p ={S1σ ′}) are the incoming (outgoing) spin of the nucleonand its projection and ST = {J0M0}(S ′T = {JM}) the initial(final) total spin of the halo valence pair. The amplitudet[bβSpS ′p] is given in terms of the tensor components of thenucleon-nucleon transition amplitude [2,13]. The transitiondensity form factors, ρ[bβ;ST S ′T εi ], depend exclusively on thestructure of the composite system. Its explicit expression as afunction of the hyper-radial parts of the wave functions of theinitial and final states can be found in Ref. [2].The scattering from the core, assumed here as spinless, canequivalently be written as〈k′1χσ ′S1 ; ϕJMεi ∣∣t̂14∣∣k1χσS1 ; ϕJ0M00 〉= t[00SpS ′p](ω14,) × ρ[00;ST S ′T εi ](0,M23M234), (16)where, as before, ω14 is the appropriate energy parameter[2]. The angular differential cross section for 6He inelasticscattering (breakup) is then obtained by summing all excited-state contributions,dσ inelJJ0d=εmaxi∑εidσ iJJ0d. (17)For the evaluation of Eqs. (15) and (16) one needsthe (free) transition amplitudes for proton scattering fromthe valence nucleons and the core. For the former, weused the NN Paris interaction. The transition amplitude forthe α core was generated from a phenomenological opticalpotential, of Woods Saxon form, with parameters obtainedby fitting existing data for the elastic scattering of p+4He atEp = 700 and 800 MeV, as detailed in Ref. [2].We first study the convergence of the breakup observableswith respect to the basis size. For this purpose, we consider theTHO basis, truncated at different values of N . The maximum0 5 10 15 20 25 30θc.m. (deg)10-210-1100101102dσinel/dΩ (mb/sr) N=4N=3N=26He(p,p’)6He*(0+) @ Ep=717 MeV THO basisFIG. 1. (Color online) Angular differential cross section for thebreakup of 6He on protons at 717 MeV per nucleon, leading to J π =0+ continuum states of the 6He nucleus. The three lines represent thecalculation with the THO basis, for several values of N (indicated bythe labels). Each curve includes the contribution from eigeinstates upto 10 MeV in excitation energy, according to Eq. (17).hyperangular momentum was set to Kmax = 20. This yieldsthe three-body force parameters V 3B0 = −2.4 MeV, for J = 0,and V 3BJ = −0.85 MeV, for J > 0.In Fig. 1 we show the angular distribution of the calculatedinelastic differential cross sections. For simplicity, only theJπ = 0+ continuum is included, and the Coulomb interactionbetween the proton and the α core is ignored. The three linesrepresent the SSA calculation for different values of the basissize, according to the choice of the parameter N . The threecases are in almost perfect agreement, indicating that in thisreaction the convergence with the basis size is very fast.Next, we study the dependence of the breakup observableson the choice of the basis, by comparing the calculations inthe GL and THO representations. As before, the maximumhyperangular momentum was set to Kmax = 20, and onlyeigenstates below 10 MeV are considered. The index nwas truncated to N = 20 and N = 4 for the GL and THObases, respectively. With this model space, the number ofpseudostates in the GL (THO) basis is 31 (30) for 0+, 63(86)for 1−, 53(49) for 1+, and 79 (81) for 2+. For the GLbasis, the range parameter was set to ρ0 = 0.25 fm, whichprovides a basis that extends up to about 20 fm in thehyper-radius. With these parameters, the ground state obtainedafter diagonalization of the Hamiltonian appears at −0.9781and −0.9549 MeV, for the GL and THO bases, respectively.In Fig. 2 we compare the inelastic angular distributionscalculated in the GL and THO bases. The separatecontributions for Jπ = 0+, 1−, 1+, and 2+ final states arealso shown. As before, the Coulomb interaction is neglected.The thick lines are the incoherent sum of all these Jπcontributions. Solid and dashed lines correspond, respectively,to the calculations with the GL and THO bases. For eachcurve, the contribution of eigenstates up to εmax = 10 MeV areadded incoherently, according to Eq. (17). We consider onlythe forward angles θc.m. < 30◦ since the SSA is not expectedto work well at large momentum transfers [2]. We see that, atthese angles, the dominant contribution to the breakup crosssection comes from the 1− states, whereas for θc.m. > 25◦,017603-3BRIEF REPORTS PHYSICAL REVIEW C 75, 017603 (2007)0 5 10 15 20 25 30θc.m. (deg)10-210-1100101102dσinel/dΩ (mb/sr)Total1-2+0+1+FIG. 2. (Color online) Calculated contributions for breakup dif-ferential cross section leading to final states with J π = 0+, 1−, 1+,and 2+, using the SSA. Solid and dashed lines refer to the GL andTHO basis, respectively.the 2+ excitation becomes dominant. Finally, the populationof the unnatural parity 1+ states is almost negligible at allangles. We notice that this excitation mode requires spin-fliptransitions, which, according to these calculations, are verysmall in this reaction.For all these contributions, the GL and THO bases providevery similar results, suggesting that the calculated observablesdo not depend on the choice of the continuum representation,provided that enough states are included.In summary, in this Brief Report we have calculated protoninelastic scattering from 6He at 717 MeV/nucleon, usingas scattering framework the single-scattering approximationand two different pseudostate representations of the 6Hecontinuum: the GL and the THO. Provided that enough statesare included, both bases predict essentially the same inelasticdifferential cross section. Furthermore, the studied observablesconverge very quickly with the size of the basis. These resultssupport the reliability of the pseudostate method as a usefuland convenient tool to treat scattering problems dealing withcontinuum states. This analysis could be extended to otherPS bases and reactions. Furthermore, it could be applied toother scattering frameworks, for which the PS method has alsobeen implemented, such as the continuum discretized coupledchannels method [14,15].We are grateful to J. Gómez-Camacho and J. M. Arias foruseful discussions. This work was supported by the Fundaçãopara a Ciência e Tecnologia (Portugal) through Grant No.POCTI/1999/FIS/36282, by the Acción Integrada HP2003-0121, and in the U.K. by EPSRC Grant No. GR/M/82141.A.M.M. acknowledges a research grant by the Junta deAndalucı́a.[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. Phys. Commun.161, 87 (2004).[7] M. Rodrı́guez-Gallardo, J. M. Arias, J. Gómez-Camacho, A. M.Moro, I. J. Thompson, and J. A. Tostevin, Phys. Rev. C 72,024007 (2005).[8] D. Gogny, P. Pires, and R. de Tourreil, Phys. Lett. B32, 591(1970).[9] J. Bang and C. Gignoux, Nucl. Phys. A313, 119 (1979).[10] C. J. Joachain, Quantum Collision Theory (North-Holland,Amsterdam, 1987).[11] K. M. Watson, Phys. Rev. 105, 1388 (1957).[12] R. Crespo and R. C. Johnson, Phys. Rev. C 60, 034007(1999).[13] R. Crespo and A. M. Moro, Phys. Rev. C 65, 054001(2002).[14] A. M. Moro, F. Pérez-Bernal, J. M. Arias, and J. Gómez-Camacho, Phys. Rev. C 73, 044612 (2006).[15] T. Matsumoto, E. Hiyama, K. Ogata, Y. Iseri, M. Kamimura,S. Chiba, and M. Yahiro, Phys. Rev. C 70, 061601(R) (2004).017603-4

Continuum description with pseudostate wave functions

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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.</p

The continuum description with pseudo-state wave functions

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