Separable Representation of Proton-Nucleus Optical Potentials
L. Hlophe, V. Eremenko, Ch. Elster, F.M. Nunes, G. Arbanas, J.E. Escher, I.J. Thompson
aa r X i v : . [ nu c l - t h ] S e p Separable Representation of Proton-Nucleus Optical Potentials
L. Hlophe ( a ) , ∗ V. Eremenko ( a,e ) , Ch. Elster ( a ) , † and F.M. Nunes ( b ) , G. Arbanas ( c ) , J.E. Escher ( d ) , I.J. Thompson ( d ) (a)Institute of Nuclear and Particle Physics, and Department of Physics and Astronomy, Ohio University, Athens, OH 45701(b) National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy,Michigan State University, East Lansing, MI 48824, USA(c) Reactor and Nuclear Systems Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA(d) Lawrence Livermore National Laboratory L-414, Livermore,CA 94551, USA (e) D.V. Skobeltsyn Institute of Nuclear Physics,M.V. Lomonosov Moscow State University, Moscow, 119991, Russia (The TORUS Collaboration) (Dated: October 4, 2018)Recently, a new approach for solving the three-body problem for (d,p) reactions in the Coulomb-distorted basis in momentum space was proposed. Important input quantities for such calculationsare the scattering matrix elements for proton- and neutron-nucleus scattering. We present a gener-alization of the Ernst-Shakin-Thaler scheme in which a momentum space separable representationof proton-nucleus scattering matrix elements can be calculated in the Coulomb basis. The viabilityof this method is demonstrated by comparing S-matrix elements obtained for p+ Ca and p+
Pbfor a phenomenological optical potential with corresponding coordinate space calculations.
PACS numbers: 24.10.Ht,25.10.+s,25.40.Cm
I. INTRODUCTION
Deuteron induced nuclear reactions are attractive froman experimental as well as theoretical point of view forprobing the structure of exotic nuclei and as an indi-rect tool in astrophysics (see e.g. [1]). From a theoret-ical perspective, (d,p) reactions are attractive since thescattering problem can be viewed as an effective three-body problem [2]. One of the most challenging aspectsof solving the three-body problem for nuclear reactionsis the repulsive Coulomb interaction between the nucleusand the proton. While exact calculations of (d,p) reac-tions based on Faddeev equations in the Alt-Grassberger-Sandhas (AGS) [3] formulation can be carried out [4] forvery light nuclei, this is not the case for heavier nucleiwith higher charges. The reason for this shortcomingis rooted in implementations of the Faddeev-AGS equa-tions that rely on a screening and renormalization proce-dure [5, 6], which leads to increasing technical difficultiesin computing (d,p) reactions with heavier nuclei [7].In Ref. [8], a three-body theory for (d,p) reactions isderived, where no screening of the Coulomb force is in-troduced. Therein, the Faddeev-AGS equations are castin the Coulomb-distorted partial-wave representation, in-stead of the plane-wave basis. The interactions in the dif-ferent two-body subsystems, including the neutron- andproton-nucleus interactions, are assumed to be of sepa-rable form.Separable forms for nucleon-nucleus interactions havebeen considered in the past (e.g. [9, 10]), but are usuallyof a rank-1 Yamaguchi form and are intended to rep-resent the nuclear forces up to a few MeV. This is not ∗ [email protected] † [email protected] sufficient for scattering of heavy nuclei up to tens of MeV.In addition, adjusting the parameters of Yamaguchi-typeneutron-nucleus form factors to obtain proton-nucleusform factors is not very practical when considering alarger variety of nuclei. Therefore, a systematic schemefor deriving separable representations for proton-nucleusoptical potentials is needed.In Ref. [11] we derived a separable representationof phenomenological neutron-nucleus optical potentials,based on a generalization of the Ernst, Shakin and Thaler(EST) scheme for non-hermitian interactions. In Ref. [12]we presented the first test calculations of form factors inthe momentum-space Coulomb basis, using the neutron-nucleus interaction developed in [11]. In this work wegeneralize these studies for proton-nucleus interactions.The derivations in the original EST work laid outin [13] set up the scattering problem in a completeplane-wave basis, whereas in this work we need to usea complete Coulomb basis. Consequently, when workingin momentum space, we require a solution of the mo-mentum space scattering equation in the Coulomb ba-sis exists. We solve the momentum space Lippmann-Schwinger (LS) equation in the Coulomb basis, followingthe method introduced in Ref. [14] and successfully ap-plied in proton-nucleus scattering calculations with mi-croscopic optical potentials in Ref. [15]. We note thata separable expansion for local potentials with Coulombinteractions was first derived by Adhikari [16] and ap-plied to proton-proton scattering. However, it has neverbeen applied to proton scattering from heavier nuclei.In Sec. II we sketch the important steps needed toderive a separable representation of a phenomenologicalglobal optical potential in the momentum-space Coulombbasis for proton-nucleus scattering. Our numerical cal-culations of S-matrix elements for proton scattering from Ca and
Pb at selected laboratory kinetic energies arediscussed in Sec. III, along with the behavior of the formfactors as a function of the external momentum. Finally,we summarize our work in Sec. IV.
II. FORMAL CONSIDERATIONS
The scattering between a proton and a nucleus is gov-erned by a potential w = v C + u s , (1)where v C is the repulsive Coulomb potential and u s anarbitrary short range potential. For the proton-nucleussystem u s consists of an optical potential, which describesthe nuclear interactions, and a short-ranged Coulombpotential, traditionally parameterized as the potentialof a charged sphere with radius R of which the pointCoulomb force is subtracted [17]. Since the scatteringproblem governed by the point Coulomb force has an ana-lytic solution, the scattering amplitude for elastic scatter-ing between a proton and a spin-zero nucleus is obtainedas the sum of the Rutherford amplitude f C ( E p , θ ) andthe Coulomb distorted nuclear amplitude given by M CN ( E p , θ ) = f CN ( E p , θ ) + ˆ σ · ˆn g CN ( E p , θ ) , (2)with f CN ( E p , θ ) = − πµ ∞ X l =0 e iσ l ( E p ) P l (cos θ ) × h ( l + 1) h p | τ CNl + ( E p ) | p i + l h p | τ CNl − ( E p ) | p i i , (3) g CN ( E p , θ ) = − πµ ∞ X l =0 e iσ l ( E p ) P l (cos θ ) × h h p | τ CNl + ( E p ) | p i − h p | τ CNl − ( E p ) | p i i . (4)Here E p = p / µ is the center-of-mass (c.m.) scatter-ing energy which defines the on-shell momentum p , and σ l = arg Γ(1 + l + iη ) is the Coulomb phase shift. TheSommerfeld parameter is given by η = αZ Z µ/p with Z and Z being the atomic numbers of the particles,and α the Coulomb coupling constant. The unit vector ˆn is normal to the scattering plane, and ˆ σ/ ′ + ′ and ′ − ′ correspond to atotal angular momentum j = l + 1 / j = l − / j = l + 1 / t -matrix elementis given by h p | τ CNl ( E p ) | p i , which is the solution of aLS type equation, h p | τ CNl ( E p ) | p i = h p | u sl | p i + (5) R p ′ dp ′ h p | u sl | p ′ ih p ′ | g c ( E p + iε ) | p ′ ih p ′ | τ CNl ( E p ) | p i . Here g − c ( E p + iε ) = E p + iε − H − v C is theCoulomb Green’s function, and H the free Hamilto-nian. The Coulomb distorted nuclear t -matrix element h p | τ CNl ( E p ) | p i is related to the proton-nucleus t -matrix h p | t l ( E p ) | p i by the familiar two-potential formula h p | t l ( E p ) | p i = h p | t Cl ( E p ) | p i + e iσ l ( E p ) h p | τ CNl ( E p ) | p i , (6)where h p | t Cl ( E p ) | p i is the point Coulomb t -matrix.When the integral equation in Eq. (6) is solved in thebasis of Coulomb eigenfunctions, g c acquires the form ofa free Green’s function and the difficulty of solving it isshifted to evaluating the potential matrix elements in thisbasis.For deriving a separable representation of the Coulombdistorted proton-nucleus t -matrix element, we generalizethe approach suggested by Ernst, Shakin, and Thaler(EST) [13], to the charged particle case. The basic ideabehind the EST construction of a separable representa-tion of a given potential is that the wave functions calcu-lated with this potential and the corresponding separa-ble potential agree at given fixed scattering energies E i ,the EST support points. The formal derivations of [13]use the plane wave basis, which is standard for scatter-ing involving short-range potentials. However, the ESTscheme does not depend on the basis and can equallywell be carried out in the basis of Coulomb scatteringwave functions. In order to generalize the EST ap-proach to charged-particle scattering, one needs to beable to obtain the scattering wave functions or half-shellt-matrices from a given potential in the Coulomb basis,and then construct the corresponding separable represen-tation thereof. A. The half-shell t -matrices in the Coulomb basis In order to calculate the half-shell t -matrix of Eq. (6),we evaluate the integral equation in the Coulomb ba-sis as suggested in [14], and note that in this case theCoulomb Green’s function behaves like a free Green’sfunction. Taking | Φ cl,p i to represent the partial waveCoulomb eigenstate, the LS equation becomes h Φ cl,p | τ CNl ( E p ) | Φ cl,p i = h Φ cl,p | u sl | Φ cl,p i + ∞ Z h Φ cl,p | u sl | Φ cl,p ′ i p ′ dp ′ E p − E p ′ + iε h Φ cl,p ′ | τ CNl ( E p ) | Φ cl,p i≡ h p | τ CNl ( E p ) | p i , (7)which defines the Coulomb distorted nuclear t -matrix ofEq. (6). To determine the short-range potential matrixelement, we follow Ref. [14] and insert a complete set ofposition space eigenfunctions h Φ cl,p ′ | u sl | Φ cl,p i = 2 π ∞ Z h Φ cl,p ′ | r ′ i r ′ dr h r ′ | u sl | r i r dr h r | Φ cl,p i = 2 πp ′ p ∞ Z rr ′ drdr ′ F l ( η ′ , p ′ r ′ ) h r ′ | u sl | r i F l ( η, pr ) . (8)The partial wave Coulomb functions are given in coordi-nate space as h r | Φ cl,p i ≡ e iσ l ( p ) F l ( η, pr ) pr , (9)where F l ( η, pr ) are the standard Coulomb functions [18]and η ( η ′ ) is the Sommerfeld parameter determined withmomentum p ( p ′ ).For our application we consider phenomenological op-tical potentials of Woods-Saxon form which are local incoordinate space. Thus the momentum space potentialmatrix elements simplify to h Φ cl,p ′ | u sl | Φ cl,p i = 2 πp ′ p ∞ Z dr F l ( η ′ , p ′ r ) u sl ( r ) F l ( η, pr ) . (10)We compute these matrix elements for the short-rangepiece of the CH89 phenomenological global optical po-tential [17], which consists of the nuclear and shortrange Coulomb potential. The nuclear potential isparameterized using Woods-Saxon functions. For theshort range Coulomb interaction, the potential of a uni-formly charged sphere is assumed, from which the pointCoulomb force is subtracted. The integral can be carriedout with standard methods, since u sl ( r ) is short rangedand the coordinate space Coulomb wavefunctions are welldefined. The accuracy of this integral can be tested byreplacing the Coulomb functions with spherical Besselfunctions and comparing the resulting matrix elementsto the partial-wave decomposition of the semi-analyticFourier transform used in [11]. For the cases under study,and a maximum radius of 14 fm, 300 grid points are suf-ficient to obtain matrix elements with a precision of sixsignificant digits. B. EST representation of the proton-nucleus t -matrix in the Coulomb basis Extending the EST separable representation to theCoulomb basis involves replacing the neutron-nucleushalf-shell t -matrix in Eqs. (14)-(16) of Ref. [11] by theCoulomb distorted nuclear half-shell t -matrix. This leadsto the separable Coulomb distorted nuclear t -matrix τ CNl ( E p ) = X i,j u sl | f cl,k Ei i τ cij ( E p ) h f c ∗ l,k Ej | u sl , (11)with τ cij ( E p ) being constrained by X i h f c ∗ l,k En | u sl − u sl g c ( E p ) u sl | f cl,k Ei i τ cij ( E ) = δ nj (12) X j τ CNij ( E p ) h f c ∗ l,k Ej | u sl − u sl g c ( E p ) u sl | f cl,k Ek i = δ ik . Here | f cl,k Ei i and | f c ∗ l,k Ei i are the regular radial scatter-ing wave functions corresponding to the short range po-tentials u sl and ( u sl ) ∗ at energy E i . The constraints of l separable p-space r-space0 -0.0512 0.3765 -0.0518 0.3768 -0.0523 0.37672 0.3805 0.0420 0.3809 0.0421 0.3808 0.04276 -0.0445 0.0170 -0.0457 0.0118 -0.0462 0.011110 0.9818 0.0248 0.9814 0.0253 0.9814 0.0253TABLE I. The partial wave S-matrix elements obtained fromthe CH89 [17] phenomenological optical potential for j = l +1 / l calculated for p+ Caelastic scattering at E lab = 38 MeV. Eqs. (13) ensure that, at the EST support points, the ex-act and separable Coulomb distorted nuclear half-shell t -matrices are identical. We want to point out that the gen-eralization of the EST scheme to complex potentials [11]is not affected by changing the basis from plane wavesto Coulomb scattering states. The separable Coulombdistorted nuclear t -matrix elements are given by h p ′ | τ CNl ( E p ) | p i ≡ X i,j h cl,i ( p ′ ) τ cij ( E p ) h cl,j ( p ) == X i,j h Φ cl,p ′ | u sl | f cl,k Ei i τ cij ( E p ) h f c ∗ l,k Ej | u sl | Φ cl,p i , (13)where the form factor h cl,i ( p ) ≡ h Φ cl,p | u sl | f cl,k Ei i (14)= h f c ∗ l,k Ei | u sl | Φ cl,p i = h p | τ CNl ( E i ) | k E i i is the short-ranged half-shell t -matrix satisfying Eq. (7).For our analysis, and the comparison with coordinate-space calculations, we consider the partial-wave S -matrixelements, which are obtained from the on-shell t -matrixelements by the relation S l ( E p ) = 1 − πiµp h p | τ CNl ( E p ) | p i . Evaluating the separable Coulomb distorted proton-nucleus t -matrix involves integrals over the proton-nucleus form factor h cl,i ( p ). If the short range Coulombpotential is omitted, the functional behavior of theproton-nucleus potential is similar to the one of theneutron-nucleus one, and thus the numerical integrationcan be carried out as discussed in [11]. However, if itis included, the proton-nucleus form factor falls off moreslowly as function of momentum. This implies that largermaximum momenta and an increased number of gridpoints are necessary to obtain a separable representationof the Coulomb distorted proton-nucleus t -matrix withthe same accuracy as the separable representation of theneutron-nucleus t -matrix. III. RESULTS AND DISCUSSION
For studying the quality of the separable representa-tion of t -matrices for proton-nucleus optical potentials weconsider p+ Ca and p+ Pb S -matrix elements in therange of 0-50 MeV laboratory kinetic energy. We use theCH89 global optical potential [17] and its rank-5 separa-ble representation in all calculations. The same support R e S l ( E ) (iii) (iv) -0.3-0.2-0.100.1 I m S l ( E ) -0.400.40.81.2 R e S l ( E ) l -0.4-0.200.20.4 I m S l ( E ) (i) (ii) (a) E=
10 MeV E=
45 MeV (b)(c)(d) Pb FIG. 1. (Color online) The partial wave S -matrix for p+ Pbelastic scattering obtained from the CH89 [17] global opti-cal potential as function of angular momentum j = l + 1 / S -matrix at E p = 10 MeV and panels (c) and (d) providethe same information at E p = 45 MeV: ( i ) S -matrix elementscalculated from the separable representation (crosses); ( ii ) co-ordinate space calculation (open circles); ( iii ) the calculationin which the short-range Coulomb potential is omitted (opendiamonds) and ( iv ) S -matrix elements for n+ Pb elasticscattering (filled circles). points used for the neutron-nucleus separable represen-tation (summarized in Table I of [11]) provide a descrip-tion of equal quality for the proton-nucleus S -matrix el-ements. This is demonstrated for p+ Ca scattering at38 MeV laboratory kinetic energy in Table I, which givesthe S -matrix elements calculated with the separable rep-resentation of the Coulomb distorted proton-nucleus t -matrix, together with the corresponding direct calcula-tions performed either in momentum or coordinate space.Similar results for the p+ Pb S -matrix elements areshown in Fig. 1. The top two panels (a) and (b) showthe real and imaginary parts of the S -matrix elementsat 10 MeV laboratory kinetic energy while the bottomtwo panels (c) and (d) show the real and imaginary partsof the S − matrix elements at 45 MeV. At 10 MeV thepartial-wave series converges much faster, thus we donot show matrix elements beyond l = 12. First, wenote that the momentum space S -matrix elements cal-culated with the separable representation (crosses) agreeperfectly with the corresponding coordinate-space calcu-lation (open circles).To illustrate the effects of the short-range Coulombpotential on the S -matrix elements, we show a calcula-tion in which this term is omitted (open diamonds). Asindicated in Fig. 1, only the low l partial waves are af-fected. To demonstrate the overall size of all Coulombeffects for Pb, we also plot the corresponding n+ Pb S -matrix elements at the same energies (filled circles). -0.0200.020.040.060.08 (i) (ii) (iii) (iv) -0.0200.020.040.06 R e (f o r m f ac t o r) [f m ] p [fm -1 ] -0.02-0.0100.010.02 (a)(b)(c) l= 3 l= 6 l= 0 Ca FIG. 2. (Color online) The real parts of the partial waveproton-nucleus form factor for Ca as function of the mo-mentum p for selected angular momenta l : (a) l = 0, (b) l = 3, and (c) l = 6. The form factors are calculated at E c.m. = 36 MeV and based on the CH89 global optical po-tential: full calculations ( i ) are compared to those omittingthe short range Coulomb ( ii ), the neutron-nucleus form factor( iii ) and the Coulomb distorted neutron-nucleus form factor( iv ). -0.0500.05-0.1-0.0500.05 R e (f o r m f ac t o r) [f m ] p [fm -1 ] -0.0200.020.040.06 (i) (ii) (iii) (iv) l= 4 l= 8 l= 0 (a)(b)(c) Pb FIG. 3. (Color online). Same as Fig. 2 but for
Pb. Theform factors for l = 0 (a) and l = 4 are calculated at E c.m. = 21 MeV but for l = 8 (c) these are calculated at E c.m. = 36 MeV. The differences between the crosses and the filled circlesdemonstrate the importance of the correct inclusion ofthe Coulomb interaction.Next we examine the form factors of the separable rep-resentation in detail. In Fig. 2 we compare p+ Ca formfactors for selected angular momenta calculated with theproton-nucleus potential and the short-range Coulombpotential ( i ) to those calculated with the proton-nucleuspotential alone ( ii ), as well as the n+ Ca ( iii ). In addi-tion we show the Coulomb distorted n+ Ca form factor( iv ) obtained with the techniques introduced in [12] to il-lustrate that a Coulomb distorted neutron-nucleus formfactor differs from the corresponding Coulomb distortedproton-nucleus form factor. First, we observe that, withthe exception of l = 0, the form factors already vanishat 3.5 fm − . For l = 0, comparing the solid and dashedlines, we see that the short-range Coulomb potential sig-nificantly modifies the nuclear form factor. The effects ofthe short-range Coulomb potential quickly decrease as l increases.In Fig. 3, we show a similar calculation for the Pbform factors. With the larger charge, the overall observa-tions are maintained but magnified. For l = 0, the shortrange Coulomb force creates a very slow fall-off of theproton form factor, and only for l = 8 is the short-rangeCoulomb potential sufficiently weak to produce a negligi-ble effect on the proton-nucleus form factor. Again we seethat for the angular momenta shown, the Coulomb dis-torted neutron-nucleus form factor does not resemble theCoulomb distorted proton-nucleus form factor, empha-sizing the need for a proper introduction of the Coulombforce in the EST scheme. IV. SUMMARY AND CONCLUSIONS
We have generalized the EST scheme [11, 13] so thatit can be applied to the scattering of charged particleswith a repulsive Coulomb force. To demonstrate thefeasibility and accuracy of our method, we applied thisCoulomb EST scheme to elastic scattering of p+ Caand p+
Pb. We found that the same EST supportpoints employed to obtain the neutron form factors canbe used for the separable representation of the proton-nucleus potential. We showed that the momentum-space S -matrix elements calculated with the separable repre-sentation of the Coulomb distorted proton-nucleus po- tential agree very well with the corresponding coordinate-space calculation. Since changing from a plane wave to aCoulomb basis preserves the time reversal invariance ofthe separable potential, the separable Coulomb distortedproton-nucleus off-shell t -matrix also obeys reciprocity.We also studied the effects of the short-range Coulombpotential on the proton-nucleus form factor. We foundthat, with the exception of the lowest partial waves ( l =0,1 for Ca and l = 0, 1, 2 for Pb), the form factorsalready vanish at 3.5 fm − . For the lowest partial wavesthe short range Coulomb force creates a very slow fall-off for the proton-nucleus form factor at high momenta.The effects of the short-range Coulomb potential quicklydecrease as l increases and almost vanish for l =6 ( Ca)and l =8 ( Pb).In addition, we demonstrated that the proton-nucleusform factor is very different from the Coulomb distortedneutron-nucleus form factor computed according to [12].Thus, when applying those form factors in a A(d,p)BFaddeev calculation, it will be mandatory to evaluateneutron and proton-nucleus form factors separately.
ACKNOWLEDGMENTS
This material is based on work in part supportedby the U. S. Department of Energy, Office of Scienceof Nuclear Physics under program No. de-sc0004084and de-sc0004087 (TORUS Collaboration), under con-tracts DE-FG52-08NA28552 with Michigan State Uni-versity, DE-FG02-93ER40756 with Ohio University; byLawrence Livermore National Laboratory under Con-tract DE-AC52-07NA27344 and the U.T. Battelle LLCContract DE-AC0500OR22725. F.M. Nunes acknowl-edges support from the National Science Foundation un-der grant PHY-0800026. This research used resourcesof the National Energy Research Scientific ComputingCenter, which is supported by the Office of Science ofthe U.S. Department of Energy under Contract No. DE-AC02-05CH11231. [1] R. Kozub, G. Arbanas, A. Adekola, D. Bardayan,J. Blackmon, et al. , Phys.Rev.Lett. , 172501 (2012).[2] I. J. Thompson and F. M. Nunes,
Nuclear Reactions forAstrophysics (Cambridge University Press, 2009).[3] E. Alt, P. Grassberger, and W. Sandhas, Nucl. Phys. B , 167 (1967).[4] A. Deltuva and A. Fonseca,Phys.Rev. C79 , 014606 (2009).[5] A. Deltuva, A. Fonseca, and P. Sauer,Phys.Rev.
C71 , 054005 (2005).[6] A. Deltuva, A. Fonseca, and P. Sauer,Phys.Rev.
C72 , 054004 (2005).[7] F. Nunes and N. Upadhyay,J. Phys. G: Conf. Ser. , 012029 (2012).[8] A. Mukhamedzhanov, V. Eremenko, and A. Sattarov,Phys.Rev.
C86 , 034001 (2012).[9] G. Cattapan, G. Pisent, and V. Vanzani,Z.Phys.
A274 , 139 (1975).[10] G. Cattapan, G. Pisent, and V. Vanzani, Nucl. Phys.
A241 , 204 (1975). [11] L. Hlophe et al. (The TORUS Collaboration),Phys.Rev.
C88 , 064608 (2013).[12] N. Upadhyay et al. (TORUS Collaboration), Phys. Rev.
C90 , 014615 (2014).[13] D. J. Ernst, C. M. Shakin, and R. M. Thaler,Phys.Rev. C8 , 46 (1973).[14] Ch. Elster, L. C. Liu, and R. M. Thaler,J.Phys. G19 , 2123 (1993).[15] C. R. Chinn, Ch. Elster, and R. M. Thaler,Phys.Rev.
C44 , 1569 (1991).[16] S. K. Adhikari, Phys.Rev.
C14 , 782 (1976).[17] R. Varner, W. Thompson, T. McAbee, E. Ludwig, andT. Clegg, Phys.Rept. , 57 (1991).[18] M. Abramovitz and I. Stegun,