Theory of single-charge exchange heavy-ion reactions

PHYSICAL REVIEW C 98, 044620 (2018) Theory of single-charge exchange heavy-ion reactions Horst Lenske,1,*Jessica I. Bellone,2,3,†Maria Colonna,2,‡and José-Antonio Lay2,4,§ (NUMEN Collaboration) 1Institut für Theoretische Physik, Justus-Liebig-Universität Giessen, D-35392 Giessen, Germany 2INFN-LNS, I-95123 Catania, Italy 3Dipartimento di Fisica e Astronomia, Universitá degli studi di Catania, Italy 4Departamento de FAMN, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain (Received 16 February 2018; revised manuscript received 27 August 2018; published 24 October 2018) The theory of heavy-ion single-charge exchange reactions is reformulated. In momentum space, the reaction amplitude factorizes into a product of projectile and target transition form factors, folded with the nucleonnucleon isovector interaction. The multipole structure of the transition form factors is studied in detail for Fermitype non-spin-flip and Gamow-Teller-type spin-flip transitions, also serving to establish the connection to nuclear βdecay. The reaction kernel is evaluated for central and rank-2 tensor interactions. Initial- and final-state ion-ion elastic interactions are accounted for by a distortion coefficient. Since the ion-ion interactions are dominated by the imaginary part of the optical potentials, the distortion coefficients can be evaluated in the strong absorption limit. For a Gaussian potential form factor, the distortion coefficient is evaluated in closed form, revealing the relation to the total reaction cross section. It is shown that at small momentum transfer distortion effects reduce to a simple scaling factor, allowing us to define a reduced forward-angle cross section which is given by nuclear matrix elements of βdecay type. Thus we introduce new unit cross sections, as those traditionally used with light projectiles for spectroscopic purposes, for heavy-ion charge-exchange reactions. Results are discussed for τ±excitations of 18Oand40Ca, respectively. Spectral distributions of nuclear-charge-changing transitions are obtained by self-consistent Hartree-Fock-Bogolubov (HFB) and quasiparticle random phase approximation (QRPA) theory and compared to spectroscopic data. The interplay of nuclear structure and reaction dynamics is illustrated for the single-charge exchange (SCE) reaction 18O+40Ca →18F+40KatTlab =270 MeV, by performing full-scale numerical calculations of the SCE cross section. We also show that the latter compare rather well with the results obtained within the strong absorption limit, thus confirming the possibility to factorize the forward-angle cross section into intrinsic nuclear transition dynamics and reaction dynamics. DOI: 10.1103/PhysRevC.98.044620 I. INTRODUCTION Nuclear charge exchange reactions have been the major source of information on the isospin and spin-isospin modes of excitation in nuclei. The discovery of the giant Gamow- Teller resonance (GTR) by the pioneering experiments at IUCF [1] initiated widespread experimental and theoretical research activities, continuing with increasing intensity until today. Over the years, a wealth of data has been accumulated as reviewed, e.g., in [2–6]. Beyond using nucleonic probes, light-ion reactions as, e.g., (3He,3H) have become another workhorse of the field, now reaching accuracies that allow us to investigate subtle details of spectral distributions in both the τ+and the τ−branches. Soon after the first light-ion studies, heavy ions were used in charge exchange studies as in Refs. [7,8]. It was recognized that peripheral heavy-ion collisions, leading to direct reactions, are as useful for spectral *[email protected] †[email protected] ‡[email protected] §[email protected] studies as light-ion scattering. An especially appealing aspect is the broad range of projectile-target combinations which, for example, allow us to project out selectively specific features, e.g., spin-flip and non-spin-flip transitions [9]. Nuclear spin dynamics and the population of continuum states were central aspects of the (7Be,7Li) reactions considered in Refs. [10,11]. While the past experiments have been focused on singlecharge exchange (SCE) reactions, new territory was entered with the pilot experiment of Cappuzzello et al. [12], which studied a nuclear double-charge exchange (DCE) reaction. The reaction 18O+40Ca →18Ne +40Ar gave strong evidence for a direct reaction mechanism even for double-charge exchange processes. Quite recently, the NUMEN project at LNS Catania was initiated, dedicated to investigations of SCE and DCE heavy-ion reactions that elucidate and optimize their potential for spectroscopic studies [13]. On the theoretical side, light-ion reactions have attracted the greatest attention. SCE reactions are typically well described by the distorted-wave Born approximation (DWBA) methods in conjunction with microscopic nuclear structure input. A comprehensive and still widely used formulation of light-ion SCE reactions was worked out by Taddeucci et al. [14], focusing on the spectroscopic applications. DWBA 2469-9985/2018/98(4)/044620(26) 044620-1 ©2018 American Physical Society LENSKE, BELLONE, COLONNA, AND LAY PHYSICAL REVIEW C 98, 044620 (2018) methods and microscopic nuclear structure input are also essential for the quantitative description of heavy-ion SCE reactions. Spectral results from shell model and quasiparticle random phase approximation (QRPA) calculations were indeed used already from the beginning, for example, in Refs. [7–9] and also in the later investigations in Refs. [10,11]. An important difference between light- and heavy-ion collisions lies in the reaction mechanism. In principle, heavy-ion SCE reactions are proceeding by two competing reaction mechanisms. The direct SCE process is a collisional process, mediated by the exchange of charged mesons between projectile and target, as given by the isovector nucleon-nucleon interaction. Hence, the direct SCE reaction mechanism is probing directly the isospin structure of the ions. A competing reaction mechanism is the sequential exchange of protons and neutrons. This is at least a second-order process, mainly probing the mean-field structure of the ions. In Refs. [7,9,15], the two reaction scenarios have been discussed. Direct SCE provides access to nuclear transition matrix elements with spin-isospin vertices as in weak interaction βdecay. This is not the case for the transfer SCE mechanism. By a proper choice of projectile and target, however, the transfer branch can be suppressed. In this paper, we consider only the theory of collisional direct SCE reactions. In heavy-ion scattering above the Coulomb barrier, a large number of reaction channels are available, absorbing a considerable part of the incoming flux. Thus, heavy-ion SCE reactions occur in a strongly absorptive environment, although the SCE cross sections are a minor contribution to the total reaction cross section. As any direct nuclear reaction, peripheral heavy-ion charge exchange processes incorporate reaction and nuclear structure dynamics. Both aspects are intimately interwoven and require their own specific theoretical treatment. In this paper, we present a concise formulation of the theory. For the reaction part, our guiding principle is direct nuclear reaction theory. In Sec. II, we recapitulate the theory of charge exchange reactions, based on the distortedwave Born approximation (DWBA). Initial- and final-state ion-ion interactions are described by optical potentials. Microscopic optical potentials are used, obtained in the impulse approximation, by folding projectile and target Hartree- Fock-Bogolubov (HFB) ground-state densities with free space nucleon-nucleon Tmatrices. In this way, we overcome the lack of elastic scattering data, inhibiting the determination of empirical optical potentials. Until the availability of elastic scattering data, we are left, however, with an open flank of unchecked input. Special emphasis is laid on a suitable formulation of the residual ion-ion charge exchange interaction. We show that momentum representation is of special advantage, allowing us to separate the interaction into operators acting in projectile and target, respectively. We include central rank-0 and rank-2 tensor interactions. It is shown that these two types of interactions can be represented by the same set of operators. Expressions for the reaction amplitudes and differential cross sections are derived. In Sec. III, the intrinsic nuclear operators are evaluated in second quantization, leading to a representation of the residual ion-ion interaction in terms of one-body density operators. Thus, we arrive at a formulation treating interactions and form factors in the general scheme of second quantization, independent of a particular nuclear structure model. The model dependence is introduced only at the very end of the theoretical process when the one-body density matrices have to be evaluated with respect to specific nuclear wave functions. The relation of the form factor and SCE reaction amplitudes to nuclear βdecay is clarified by considering the limit of low momentum transfer. The momentum space techniques are found to provide the proper approach to disentangle reaction and structure aspects. In heavy-ion SCE reactions, also the projectile may have a number of bound excited states, requiring to consider the combined spectral distributions of both ions. Finally, we derive explicit expressions for forward angle cross sections, showing that heavy-ion SCE reactions are indeed providing access to nuclear matrix elements relevant also for βdecay. The results derived in this section are of special importance since they show explicitly the potential of heavy-ion SCE reactions for spectral investigations, including the deduction of nuclear matrix elements for βdecay. Section IV as a central part of the paper is concerned with the investigation of initial- and final-state interactions. They are of crucial importance for an understanding of the reaction yields on a quantitative level. Their effect is, of course, treated properly on the quantum mechanical level by direct reaction theory. However, in order to disentangle SCE reaction dynamics and nuclear dynamics, a deeper insight into elastic ion-ion interactions is necessary. For that purpose, the momentum space formulation of the reaction amplitude is of great advantage. The final result is surprisingly simple: The main effect of ion-ion elastic interactions is finally to act as a (momentum-dependent) weight factor in the folding of the nuclear transition form factors, suitably described by a distortion coefficient. The latter is found to be closely related to the elastic optical model Smatrix. In the strong absorption limit, analytic expressions are derived in Gaussian approximation for the transition potentials in Sec. IV C.In leading order, elastic ion-ion interactions reduce to a simple scaling law, suppressing, however, the SCE cross section by many orders of magnitude with respect to the plane-wave Born limit. SCE reactions in the strong absorption limit and an interesting connection to eikonal theory is discussed in Sec. V. Last but not least, the theoretical results are illustrated in Sec. VI by numerical calculations. QRPA results for nuclear SCE response functions are presented for 18O→18F and 40Ca →40K, respectively. We discuss results for optical potentials and elastic scattering angular distributions, addressing distortion effects due to ion-ion dynamics and examples of charge exchange cross sections. As a case of physical interest, we consider throughout the SCE reaction 18O+40Ca → 18F+40KatTlab =15 A MeV. Mathematical-theoretical details have been shifted into a few appendices. The paper closes in Sec. VII with a summary and an outlook for future work. II. THEORY OF HEAVY-ION SINGLE-CHARGE EXCHANGE REACTIONS Charge-changing reactions by strong interactions are offshell processes mediated by the exchange of virtual particles. 044620-2 THEORY OF SINGLE-CHARGE EXCHANGE HEAVY-ION … PHYSICAL REVIEW C 98, 044620 (2018) FIG. 1. Graphical representation of a single-charge exchange heavy-ion reaction by hadronic interactions corresponding to νβ processes. Both (n, p)-type (left) and (p, n)-type (right) reactions, as seen in the A→Btransition in target system, are displayed, indicating also the exchanged meson. They require two reaction partners, which are acting mutually as the source or sink, respectively, of the charge-changing virtual meson fields, as depicted in Fig. 1. For experimental reasons, the projectile-like ejectile should be preferentially in a particle-stable state (see, however, (d,2He) reactions [16]), thus simplifying the detection. If the ejectile has only a single bound state below the particle emission threshold, the calculations and the interpretation of the spectroscopic data are especially simple. Here, we consider ion-ion SCE reactions according to a za+A ZA→a z±1b+A Z∓1B, (1) which change the charge partition by a balanced redistribution of protons and neutrons. The differential SCE cross section for a reaction connecting the channels αand βis defined as d2σαβ =mαmβ (2π¯h2)2 kβ kα 1 (2Ja+1)(2JA+1) × Ma,MA∈α;Mb,MB∈β|Mαβ (kα,kβ)|2d,(2) where kα(kβ) and mα(mβ) denote the relative 3-momentum and reduced mass in the entrance (exit) channel α={a,A} (β={b, B}). {JaMa,J AMA...}and {JbMb,J BMB...}account for the full set of (intrinsic) quantum numbers specifying the initial and final channel states, respectively. In the distorted-wave approximation, the direct charge exchange reaction amplitude is given by the expression Mαβ (kβ,kα)=χ(−) β,bB|TNN|aA,χ(+) α.(3) The distorted waves, denoted by χ(±) α,β for asymptotically outgoing and incoming spherical waves, respectively, depend on the respective channel momenta kα,β and the optical potentials, thus accounting for initial-state (ISI) and final-state (FSI) interactions. The charge-changing process is described by the nucleonnucleon (NN)Tmatrix TNN. Antisymmetrization between target and projectile nucleons is taken care of by the standard procedure of attaching the spin and isospin exchange operators to the Tmatrix and treating space exchange in local momentum approximation; see, e.g., Refs. [17–19]. The antisymmetrized Tmatrix is then given in nonrelativistic momentum representation by TNN(p)= S=0,1,T =0,1V(C) ST (p2)[σa·σB]S +δS1V(Tn) T(p2)S12(p)[τa·τA]T,(4) including isoscalar and isovector central spin-independent (S=0) and spin-dependent (S=1) interactions with form factors V(C) ST (p2), respectively, and rank-2 tensor interactions with form factors V(Tn) T(p2). The form factors are complexvalued scalar functions. Denoting the nucleon isospinors by |pand |n, respectively, we use the convention p|τ0|p= +1, which implies τ−|p=|n. The standard definition of the rank-2 tensor operator is S12(p)=1 p2(3σa·pσA·p−σa·σAp2),(5) but for applications to nuclear reactions an equivalent, more suitable representation is used, given by the scalar product of two rank-2 tensors, namely the spherical harmonic Y2M(ˆp) and the rank-2 spin operator S2M=[σ1⊗σ2]2M= m1m2 (1m11m2|2M)σ1m1σ2m2(6) such that S12 =24π 5Y2S2=24π 5 M Y∗ 2M(ˆp)S2M,(7) where Y∗ 2M=(−)MY2−M. For the present discussion, we neglect two-body spin-orbit interactions in order not to overload the presentation. An elegant representation of the Tmatrix is obtained in terms of the spin-isospin operators OST(i)=(σi)S(τi)T,(8) which describe the operator structures of both the central and tensor interactions. The operators OST lead to the rather compact representation TNN(p)= S,T V(C) ST (p2)OST(1)OST(2) +δS1V(Tn) T(p2) ×24π 5Y∗ 2(ˆp)·[OST(1) ⊗OST(2)]2,(9) where scalar products are indicated as dot products and the rank-2 tensorial coupling affects, of course, only the spin degrees of freedom. The subset of isovector operators, corresponding to Fermi-type S=0,T =1 and Gamow- Teller-type S=1,T=1 operators, contributes to the chargechanging reaction amplitudes. The matrix element of a single-charge exchange reaction, Eq. (1), can be written in slightly different form as Mαβ (kα,kβ)=χ(−) β|Uαβ |χ(+) α.(10) 044620-3 LENSKE, BELLONE, COLONNA, AND LAY PHYSICAL REVIEW C 98, 044620 (2018) The nuclear structure information on multipolarities, transition strength, and interactions are contained in the (antisymmetrized) transition potential Uαβ (rα,rβ)=JbMbJBMB|T(C) NN +T(Tn) NN ...|JaMaJAMA (11) depending on the channel coordinates rα,β . If recoil effects due to the change of the mass partitions can be neglected and antisymmetrization is taken into account by an equivalent effective local interaction, one can just consider the local transition potential Uαβ (r), where r=rα=rβ. Obviously, by means of Eq. (11), the reaction amplitude, Eq. (10) can be rewritten in terms of a sum of reaction amplitudes defined by the tensorial rank rof the NN interaction, Mαβ (kα,kβ)=M(C) αβ (kα,kβ)+M(Tn) αβ (kα,kβ)+··· = r=C,T n... M(r) αβ (kα,kβ).(12) III. SCE CROSS SECTION AND NUCLEAR MATRIX ELEMENTS A. Momentum representation In order to obtain a deeper insight into the interplay of nuclear structure and β-decay matrix elements on the one side and heavy-ion reaction dynamics on the other side, a more detailed study of the process is necessary. A convenient approach is to consider the reaction amplitude in momentum representation. A considerable advantage of that representation is that the transition potential Uαβ becomes separable into target and projectile transition form factors. The intrinsic nuclear transitions in either target or projectile are induced by one-body operators of the type RST(p,r)=1 4πeip·rOST,(13) where r=rA,a indicates the intrinsic target and projectile nuclear coordinates, respectively. For convenience, we have introduced a normalization to the surface area of the unit sphere. Transitions X→Yare described by the momentum space form factors F(XY ) ST (p)=JYMY|RST(p,rX)|JXMX.(14) The form factors contain the full spectroscopic information on the intrinsic nuclear transitions. In Appendix A, second quantization is used to derive explicit expressions in terms of one-body transition densities. The reaction kernel is given by products of these form factors K(ST ) αβ (p)=(4π)2V(C) ST (p2)F(ab)† ST (p)·F(AB) ST (p) +δS124π 5V(Tn) ST (p2)Y∗ 2(ˆp) ·F(ab)† ST (p)⊗F(AB) ST (p)2,(15) where the rank-2 tensorial coupling relates to the spin degrees of freedom only. In the central interaction part, the scalar product indicates the contraction of the projectile and target form factor with respect to the spin and isospin degrees of freedom. The isospin degrees of freedom are, of course, projected by the nuclear transitions to the proper combination of τ±operators. In terms of the reaction kernels, the transition potential is found as Uαβ (p)= ST K(ST ) αβ (p).(16) It remains to evaluate the integration over the relative motion degrees of freedom, i.e., taking matrix elements with distorted waves. Thus, Mαβ (kα,kβ)=d3p (2π)3χ(−) β|e−ip·rUαβ (p)|χ(+) α(17) and by defining the distortion coefficient Nαβ (kα,kβ,p)=1 (2π)3χ(−) β|e−ip·r|χ(+) α,(18) which will be discussed in detail below in Sec. IV, we obtain the full reaction amplitude Mαβ (kα,kβ)=d3pUαβ (p)Nαβ (kα,kβ,p),(19) showing the dressing of the nuclear transition potential by initial- and final-state ion-ion interactions. Formally, the above relation is fully equivalent to the DWBA amplitude in coordinate space. The momentum representation, however, has the important advantage that the intrinsic nuclear transition dynamics and the reaction dynamics appear in factorized form. The projectile and target transition form factors, Eq. (14), are of a very general structure accounting for the complete set of multipoles as contained in the plane waves. The integration over the nuclear intrinsic coordinates, however, will project on a subset of multipoles according to the multipolarity of the transitions a→band A→B, respectively. A special feature is encountered in the rank-2 tensor amplitudes M(Tn) αβ as discussed, e.g., Ref. [9]. By evaluating the integrals explicitly, one finds that the presence of the spherical harmonics of order 2 induces a corresponding rank-2 tensorial coupling of the nuclear transition multipoles, Eq. (15). Except for transitions involving only s-wave proton and neutron orbitals, Gamow-Teller-like excitations are typically a mixture of a leading multipolarity L1and a subleading one with L2=|L1±2|.βdecay strongly favors the multipolarity with the lower value of L1,2. That selectivity is missing in strong interactions. Since in heavy-ion charge exchange reactions especially processes with large angular momentum transfer are favored, the whole spectrum of multipolarities becomes visible. The recoupling procedure, following standard rules [20] and discussed in Appendix B, leads to the reduced form factors FJ1J2 LS and HJ1J2 LS for central and tensor interactions, respectively. Both are scalar functions of p2;J1,2denote the total angular momentum transfer in projectile and target, respectively; Lis the total orbital momentum transfer seen 044620-4 THEORY OF SINGLE-CHARGE EXCHANGE HEAVY-ION … PHYSICAL REVIEW C 98, 044620 (2018) on the reaction α→β, and S=0,1 indicates as before the spin structure of the transitions. These form factors determine the multipole structure observed in angular distributions. In general, they are given by superpositions of target and projectile intrinsic form factors, as already evident from Eq. (15). Anticipating the result, the reaction kernels become K(ST ) αβ (p)= J1M1,J2M2,LM (JaMaJbMb|J1M1)(JAMAJBMB|J2M2)(J1M1J2M2|LM)iLYLM (ˆ p) ×V(C) ST (p2)FJ1J2 LS (p2)+δS1V(Tn) T(p2)HJ1J2 LS (p2),(20) including central and rank-2 tensor interactions. Correspondingly, for the reaction amplitude, we obtain the expression Mαβ (kα,kβ)= J1M1,J2M2,LM (JaMaJbMb|J1M1)(JAMAJBMB|J2M2)(J1M1J2M2|LM) ×d3pNαβ (kα,kβ,p)iLYLM (ˆ p)M(JaJA,JbJB) LJ1J2(p2),(21) where M(JaJA,JbJB) LJ1J2(p2)= S,T δT1V(C) ST (p2)FJ1J2 LS (p2)+δS1V(Tn) T(p2)HJ1J2 LS (p2).(22) Exploiting the completeness and orthogonality relations of Clebsch-Gordan coefficients, the double-differential cross section becomes d2σαβ ddEx= E∗ b,E∗ B mαmβ (2π¯h2)2 kβ kα 1 (2Ja+1)(2JA+1)  LM,J1J2d3pNαβ (kα,kβ,p)YLM (ˆp)M(JaJA,JbJB) LJ1J2(p2) 2 δ(E∗ b+E∗ B−Ex). (23) The Dirac δfunction projects the sum of excitation energies onto the total effective energy loss Ex. The mathematical structure of the SCE amplitudes are strongly simplified for Ja=0+=JA. In that case, J1=Jb and J2=JBand one obtains for the target [see Eq. (A13)] F(AB) ST (p)= L,M (LMSMS|JBMB)f(AB) LSJB(p2)iLYLM (ˆ p). (24) The same kind of relation holds for the projectile. Besides the triangle rule of angular momentum coupling, the allowed values of the orbital angular momentum transfer L are constrained further by parity selection rules. For 0+→ JπJtransitions πJ=(−)Jand πJ=(−)J+1for natural and unnatural parity transitions, respectively, and πJ=(−)Lmust be fulfilled. For natural parity transitions with L=J, nonspin-flip S=0 and spin-flip S=1 transitions are allowed, while for unnatural parity transitions with L=|J±1|only transitions with S=1 will contribute. B. Cross sections at low-momentum transfer In the limit of low momentum transfer qαβ =kα−kβ→ 0, the distortion coefficient, Eq. (18), approaches in leading order a rather simple form: Nαβ (kα,kβ,p)≃¯ Nαβ (kα,kβ)δ(p−qαβ ).(25) Hence, the momentum integration in Eq. (23) can be performed, by which the integrand is projected to p=qαβ →0. The validity of this reduction will be discussed in detail below. The important point is that to a good approximation nuclear dynamics and reaction dynamics are becoming factorized in forward angle cross sections. Thus, once ¯ Nαβ has been determined, nuclear transition probabilities are becoming accessible. They are contained in the form factors F(XY ) ST of Eq. (A13). More specifically, we need to consider the reduced form factors f(XY ) LSJ [see Eqs. (A13) and (A14)]. For small momenta p→0, we find f(XY ) LSJ (p2)∼pL (2L+1)!!∞ 0 drr2+Lρ(XY ) LSJ (r)+O(p2), (26) given in leading order by the matrix element b(XY ) LSJ =∞ 0 drr2+Lρ(XY ) LSJ (r),(27) which is the reduced transition amplitude of the multipole operator B(LST )JM(r)=rL[YL⊗(σ)S]JM(τ)T,(28) carrying the same functional structure as the β-decay transition operators. The corresponding excitation probability is given by B(XY ) LSJ =1 2J+1|JY||BLST J ||JX|2=b(XY ) LSJ  2.(29) Considering specifically transitions when both ions are in 0+ground states, there will be for J>0 in general two contributing multipole form factors, namely those of the S= 0,1 transitions of the same L=Jfor natural parity and those of the L=J±1 transitions with fixed S=1 for unnatural parity. For natural parity transitions, the superposition will 044620-5 LENSKE, BELLONE, COLONNA, AND LAY PHYSICAL REVIEW C 98, 044620 (2018) not modify the low-momentum behavior of the cross sections but has to be taken into account for the extraction of the corresponding transition strengths. At forward angles, the cross section for natural parity transitions in both nuclei will be of the type dσFF d∼q2(Ja+JA) [(2Ja+1)!!(2JA+1)!!]2|¯ Nαβ |2 ×V(C) 01 (0)b(AB) JA0JAb(ab) Ja0Ja+eiφaA V(C) 11 (0)b(AB) JA1JAb(AB) JA1JA 2, (30) where q=|kα−kβ|denotes the momentum transfer at forward direction and φaA accounts for possible relative phase factors of the target and projectile matrix elements. If one of the nuclei undergoes a J=0+monopole excitation, i.e., a0 + g.s. →0+ Extransition, the S=1 components will not contribute and irrespective of the multipolarity of the excitations in the second nucleus; only S=0 transitions will be observed. For unnatural parity states, the multipole mixtures lead to a modification of the momentum dependence because for J= 0−we have two angular momentum transfers, L=J−1 and L=J+1. The forward cross section for unnatural parity transitions in both nuclei behaves as dσGG d∼q2(Ja+JA−2) [(2Ja−1)!!(2JA−1)!!]2V(C) 11 (0) 2|¯ Nαβ |2 × b(AB) JA−11JA+q2 (2JA+1)(2JA+3)b(AB) JA+11JA 2 × b(ab) Ja+11Ja+q2 (2Ja+1)(2Ja+3)b(ab) Ja+11Ja 2. (31) The contributions from the rank-2 tensor interactions are not shown because they will be suppressed at small momentum transfer. The multipole mixtures will change with the effective momentum transfer at forward directions. If there is a 0−transition in one of the two nuclei, the corresponding transition form factor reduces to a single contribution with L=1 and S=1. In addition, there are mixed transitions, combining a natural parity spin-flip excitation in one nucleus with unnatural excitations in the other nucleus. The corresponding cross sections are obtained in a similar way and are easily deduced by an appropriate combination of the above results. IV. INITIAL- AND FINAL-STATE INTERACTIONS A. Distorted waves and distortion coefficient For heavy-ion reactions, the elastic interactions in the initial and the final channels play a key role for a quantitative description of cross sections. In a microscopic description, the optical potentials are obtained in a double-folding approach [17]. In the many cases where elastic scattering data are not available, the folding approach is in fact the only way to obtain information on elastic ion-ion interactions. The doublefolding potential is defined in terms of the NN T matrix and the ground-state densities of the interacting nuclei. Thus, specific contributions, e.g., due to the coupling to breakup and transfer channels or rotational and vibrational excitations, are not included. Experience, however, shows that at kinetic energies above the Coulomb barrier the double-folding potential accounts surprisingly well for the elastic interactions over large ranges of incident energies and projectile-target combinations; see, e.g., Ref. [21]. The reason is that most of the interaction effects are already covered by the multiple scattering series inherent to an elastic amplitude iterated to all orders, as in the case of the solutions of a Schrödingertype wave equation. A commonly used approach is the impulse approximation, amounting to consider the isoscalar and isovector (S=0,T =0,1) parts of the free space NN Tmatrix. Since there are no heavy-ion polarization data available, spin-dependent parts of the optical potential are neglected. Coulomb interactions, of course, must be included as well. They are treated by folding the two-body projectiletarget nucleon Coulomb interaction with the nuclear charge densities. Thus, we use Uopt (r)=V(r)−iW(r)+Uc(r),(32) where the imaginary part must in total correspond to an absorptive potential, guaranteeing a positive absorption cross section. The distorted waves are then defined by wave equations with the generic structure −¯h2 2mγ ∇2+Uγ(r)−E(rel) γχ(±) γ(r,±k)=0 (33) for γ∈{α, β}.Erel α,β =√s−MA,B −Ma,b indicates the kinetic energy available in the projectile-target rest frame, with MA,B and Ma,b denoting target and projectile rest masses. B. Separation approach to the distortion coefficient From Eq. (18), the limiting case of a system without ISI and FSI interactions is immediately found by replacing the distorted waves by plane waves (PW). Then, the distortion coefficient reduces to N(PW) αβ (kα,kβ,p)=δ(kα−kβ−p) (34) and we retrieve the reaction amplitude, Eq. (19), in lowestorder Born approximation as M(B) αβ (kα,kβ)≡Uαβ (qαβ ).(35) In order to establish the connection of the full distorted wave (DW) amplitudes to those of the PW limit, we need to consider the distortion coefficient in more detail. For that purpose, we separate the distorted waves |χ(±) α,β into plane waves |kα,β and a residual distortion amplitude u(±) α,β (kα,β ,r). On very general grounds, such a separation is justified by the representation of an interacting wave in terms of the Møller-wave operator acting on a plane wave [22]. Assuming that u(±) α,β and Uαβ commute, we define the distortion density ηαβ =u(−)† βu(+) α.(36) Then, Eq. (18) is found to be given by the Fourier transform of the distortion density Nαβ (kα,kβ,p)=ηαβ (qαβ −p).(37) 044620-6 THEORY OF SINGLE-CHARGE EXCHANGE HEAVY-ION … PHYSICAL REVIEW C 98, 044620 (2018) Since for a noninteracting system ηαβ (r)→η(PW) αβ (r)=1, it is useful to consider αβ (r)=1−ηαβ (r). This allows to split the distortion coefficient as follows: Nαβ (kα,kβ,p)=N(PW) αβ (kα,kβ,p)−αβ (qαβ −p),(38) where now the ISI and FSI effects are fully contained in the Fourier transform of αβ . Correspondingly, the reaction amplitude divides into Mαβ (kα,kβ)=M(B) αβ (qαβ )−d3pαβ (qαβ −p)M(B) αβ (p) =M(B) αβ (qαβ )−d3qαβ (q)M(B) αβ (qαβ −q). (39) Assuming that αβ is spherical symmetric, we obtain Mαβ (kα,kβ) =M(B) αβ (qαβ )−4π∞ 0 dqq2αβ (q)¯ M(B) αβ (qαβ ,q), (40) where ¯ M(B) αβ (qαβ ,q)=1 4πdqM(B) αβ (qαβ −q) (41) denotes the Born amplitude averaged over the orientations of q. Referring to the definition of the Born amplitude, Eq. (35), the angle integral can be performed analytically, and we obtain ¯ M(B) αβ (qαβ ,q)=d3reiqαβ ·rUαβ (r)j0(qr).(42) The above relations involve in fact different scales, which allow a separation ansatz: The distribution of the momenta qis controlled by the optical model quantity αβ with a typical momentum spread of the order of the potential radius, i.e., qreac ∼1 Ropt ⩽50 MeV/c. The momentum structure of the Born amplitude is determined by the charge-changing nuclear form factors F(ab),(AB). Their overall momentum dependence is closely related to the Fermi momenta of protons and neutrons; thus qnucl ∼kF∼300 MeV/c. Therefore, we introduce the separation ansatz ¯ M(B) αβ (qαβ ,q)≃M(B) αβ (qαβ )hαβ (q),(43) where the separation function hαβ (q) is determined by the variation of the Born amplitude off the physical 3-momentum shell qαβ . Now, we perform the remaining integral and define the absorption index nαβ =4π∞ 0 dqq2αβ (q)hαβ (q).(44) The full reaction amplitude acquires a considerably simplified structure, Mαβ (kα,kβ)=M(B) αβ (qαβ )(1 −nαβ ),(45) given in leading order by the Born amplitude, scaled by a distortion coefficient which should depend only weakly on the momentum transfer for a meaningful factorization of Mαβ . C. Separation function for Gaussian form factors The separation ansatz discussed above can be checked, on an analytical basis, if one adopts a Gaussian shape, UG,forthe transition potential Uαβ (p). Indeed, nuclear SCE transitions are well modeled by surface form factors for which the Gaussian shape is a quite convenient and realistic choice. For the present purpose, it is sufficient to consider a transition potential with a single Gaussian form factor, UG(r,R)=1 4πU0e−(r−R)2 2σ2,(46) which can be adjusted to microscopically derived shapes by an appropriate choice of the centroid parameter Rand the width parameter σ. Considered as classical quantities, Rand σare determined, in principle, by the radii and surface thicknesses of the colliding ions. UGcontains a rich multipole structure UG(r,R)= LM Y∗ LM (ˆ R)ULM (r,R) (47) with the multipole form factors ULM (r,R)=dˆ RYLM (ˆ R)UG(r,R) (48) =U0e−r2+R2 2σ2iL(rR/σ2)YLM (ˆr),(49) where iL(x)=iLjL(ix) is a modified spherical Bessel function. As discussed in Appendix C, the connection to the microscopic structure of the intrinsic nuclear transitions involved in projectile and target is recovered by imposing on YLM (ˆ R) a quantization condition in terms of the projectile and target state operators, similar to the collective model of nuclear excitations. There, it is also shown that within the Gaussian approximation Rand σare determined by the corresponding projectile and target quantities. The strength parameter U0is related to the volume integral of the NN T matrix. However, for the following those details are of minor relevance because state-independent, universal properties of distortion effects in nonelastic ion-ion reactions are investigated. Thus, for simplicity, we neglect the state dependence, choose U0=1, and leave the determination of Rand σfor later. The Fourier-Bessel transform is derived analytically, UG(p,R)=π 2σ3eip·Re−1 2σ2p2,(50) and the momentum space multipoles are obtained as above by projecting on YLM (ˆ R). This amounts to expand the plane wave into partial waves, resulting in ULM (p,R)=4ππ 2σ3e−1 2σ2p2jL(pR)iLYLM (ˆp).(51) According to Eq. (39), we need to evaluate UG(p)atp= qαβ −q. This leads to UG(qαβ −q,R)=UG(qαβ ,R)Hαβ (q,ρ),(52) describing the (partial) separation of the dependencies on the physical momentum transfer qαβ and the momentum shift q due to the ISI-FSI interactions by means of Hαβ (q,ρ)=e−1 2σ2q2e−iq·ρ(53) 044620-7 LENSKE, BELLONE, COLONNA, AND LAY PHYSICAL REVIEW C 98, 044620 (2018) with the pseudoradius ρ=R+iσ2qαβ ,(54) which is shifted into the complex plane by an amount controlled by the width parameter σ.Weuseρ=ρ2, where ρ2=R2−σ4q2 αβ +2iσ2qαβ ·R.(55) Since ρalso depends on the on-shell momentum transfer qαβ , the separation of variables is not yet fully achieved. The function hαβ of Eq. (43) is given as hαβ (q,ρ)=1 4πdˆqHαβ (q,ρ)=e−1 2σ2q2j0(qρ) (56) and the distortion coefficient (1 −nαβ ) is found according to Eq. (44). Further insight into the modification introduced by the ion-ion ISI and FSI interactions is obtained by using the addition theorem for Bessel functions [23] j0(qρ)= λ (2λ+1)Pλ(cos γ)jλ(qR)iλiλ(qqαβ σ2),(57) where γdenotes the angle between Rand qαβ . Furthermore, using the addition theorem of spherical harmonics, we find hαβ (q,ρ) =4πe−1 2σ2q2 λμ iλYλμ(ˆqαβ )Y∗ λμ(ˆ R)jλ(qR)iλ(qqαβ σ2). (58) For momentum transfers in the range qαβ 1/σ, which amounts to about the order of 100 MeV/c, the sum is well approximated by the monopole term, hαβ (q)=e−1 2σ2q2j0(qR)i0(qqαβ σ2),(59) indicating a remaining dependence on the reaction momentum transfer. This derivation, based on the Gaussian form factor, allows one to understand the range of validity of the separation ansatz, Eq. (43). Indeed, for transferred momenta approaching zero, one recovers the complete factorization discussed above, i.e., hαβ (q)→ e−1 2σ2q2j0(qR).(60) V. DISTORTION COEFFICIENT IN THE STRONG ABSORPTION LIMIT A. Strong absorption limit and black disk approximation In the derivation of Eq. (45), the critical step is clearly the treatment of the distortion effects, which we consider next. For strongly absorbing systems like ion-ion scattering, the distorted waves are almost completely suppressed in the overlap region, thus reflecting the large amount of channel coupling which leads to a redirection of the incoming elastic probability flux into a multitude of nonelastic reaction channels. Such systems are described by optical potentials with a strong imaginary part of a strength comparable in magnitude to the real, diffractive part. Under such conditions, the distortion amplitude introduced before resembles in coordinate space a step function, ηαβ (r)∼eiφ(r)(r−Rabs ). In the following, we neglect the phase factor given by φ. This picture coincides with the black disk assumption (BD), where one considers that a major part of the incoming flux is consumed by a (spherical) absorber of radius Rabs, resulting in the total absorption cross section σ(BD) abs (√s)=πR2 abs(√s),(61) and by equating σ(BD) abs and the quantum mechanical reaction cross sections σ(α,β) abs the absorption radius Rabs is obtained. Considering that σabs ∼1–3 b as a representative range of values for ion-ion reaction cross sections at energies of a few 10 A MeV we find Rabs ∼5–10 fm. These values are implying a variation of the function j0(qRabs ) on a momentum scale qreac ∼1/Rabs ∼20–40 MeV/c, thus complying perfectly well with the previous estimates. In the BD limit, we can evaluate the distortion coefficient in closed form. We find (BD) αβ (q)=1 2π2 Rabs q−∂ ∂qj0(qRabs ) (62) and the scaling function is given by n(BD) αβ (Rabs)=2Rabs π∞ 0 dqj0(qRabs)∂ ∂q[qhαβ (q)],(63) which corresponds to a Fourier-Bessel transform of hαβ (q), mapping the dependence on the variable qto the complementary variable Rabs. As discussed in Appendix D,forh(q) given by Eq. (56), the black disk distortion coefficient can be calculated in closed form, resulting in a superposition of error integrals and Gaussians. In hαβ ,seeEq.(56), the parameter σcontrols the slope of the momentum distribution around the physical momentum transfer qαβ . By the arguments given above, we expect σ∼O(1/kF), thus being related to the binding properties of nuclei. Hence, the width of the Gaussian form factor is determined by the surface diffuseness of nuclear density distributions. The (off-shell) diffraction structure of the transition form factors, which is described by R, is more directly affected by the nuclear geometry, which to a large extent is a mean-field effect, thus related to the radii of the nuclear densities, Ra,A. Taking into account the modifications by the folding with the NN interaction, we estimate therefore R∼ O(Rpot ), where Rpot is the radius of the ion-ion potential. B. Relation to eikonal theory The distortion coefficient, Eq. (18), approaches a wellknown limit for p→0, given by the Smatrix [24]. Evaluating explicitly the case α=βgives Nαβ (kα,kβ,p)p=0→ND=1 (2π)3χ(−) α|χ+ α =1 k2 α δ(kα−k α)1 4πSα,(64) resulting in the optical model Smatrix, given by the partialwave S-matrix elements S: Sα=  (2+1)P(cos θ)S.(65) 044620-8 THEORY OF SINGLE-CHARGE EXCHANGE HEAVY-ION … PHYSICAL REVIEW C 98, 044620 (2018) The δfunction takes care of energy conservation, reflecting the on-shell property of the Smatrix. In the on-shell limit, the quantity αβ ,Eq.(38), is determined by the elastic scattering amplitude. Hence, the distortion coefficient can be understood as an off-shell extension of the elastic scattering amplitude. Averaging the modulus squared of Sαover the final momentum direction, we find |Sα|2=  (2+1)|S|2.(66) Removing the divergent part of that summation—reminiscent of the δfunction deriving from the plane-wave components— we end up with the absorption coefficients A=1−|S|2, which determine also the total reaction cross section. Thus, in the case of the strong absorption characterizing heavy-ion collisions, the distortion coefficient is closely related to the reaction cross section. At incident energies well above the Coulomb barrier, eikonal theory is in fact a suitable method to investigate further distortion effects. The absorptive heavy-ion interactions allow us to use eikonal theory already at the relatively low energies as considered here. In the nuclear interior where also the real potential is sizable, the deep imaginary potential suppresses the scattering waves by many orders of magnitude. Thus, to a large extent the reaction is blind against the region where the potentials may be of the same magnitude or even larger than the kinetic ion-ion energy. The scattering process is determined by the regions where the imaginary potential strength has decreased to values smaller than the real part of the potential plus the kinetic energy. In eikonal theory, the Smatrix is given by an integral over impact parameters b Sα=1 2π∞ 0 dbbJ0(qb)eiχα(b),(67) where J0(x) is the ordinary Bessel function of order n= 0,q=|kα−k α|denotes the elastic momentum transfer. The Smatrix is given now by the profile function χα(b) depending on the impact parameter b, as discussed in Appendix E. A projection on vanishing longitudinal momentum transfer qz=0 is obtained reflecting the fact that distortion affects in the first place the transversal components of scattering wave functions. Thus, finally we obtain1 |Sα|2= 1 2π∞ 0 dbbe−2Im(χα(b)).(68) In Fig. 2, the quantum mechanical absorption factor Al= 1−|S|2, evaluated numerically, and the eikonal analog 1 − |S(b)|2are compared for the reaction 18O+40Ca at TLab = 15 A MeV. The same optical potential is used, assuring that the reaction cross section is reproduced in eikonal approximation (see next section). The agreement is striking. The reason that eikonal theory is doing so well at this low energy 1At low energies, Coulomb barrier effects may be taken into account by the additional factor e−2πηS,whereηSis the Sommerfeld parameter. FIG. 2. Comparison of quantum mechanical (DW) and eikonal (EW) results for the absorption factors A=1−|S|2shown as functions of the impact parameter b≃/kα. is the strong absorption by which contributions of Re[χ(b)] are strongly suppressed in the space regions where Im(Uopt ) is sizable. C. Mass and energy dependence of absorption effects An important conclusion from the foregoing discussion is the paramount role of absorption effects for which the absorption radius Rabs is the key quantity. Moreover, according to Appendix E, in the strong absorption limit the distortion coefficient is fixed once Rabs is known, together with the nuclear shape parameters. For our purpose, it is enough to consider the imaginary part of the optical potential W(r). In the present context, W(r) plays the role of an effective eikonal potential which has to be adjusted such that the quantal results are reproduced as close as possible. A spherical-symmetric potential of Gaussian shape is used: W(r)=−W0e−r2/R2 W.(69) The radius parameter RWand the potential strength are fixed by comparison to quantum mechanical results for the total reaction cross section for two reference systems: 18O+40Ca and 18O+116Sn at Tlab =270 MeV [13]. Denoting the mass numbers of projectile and target by Ap,T , a proper description of the two systems is obtained with RW=r0A 2 3 p+A 2 3 T, where r0=0.783 fm and W0=wA 2 3 p+A 2 3 T−3/4,(70) with w=5902.743 MeV. Interestingly, the potential depth behaves according to the so-called URαlaw, which was found in the early days of the nuclear optical model by Hodgson [25,26], when studying ambiguities of optical potentials. In our case, we have α=3/2. We also note that the eikonal approximation works rather well for shallow optical potentials, as given by our parametrization. For instance, for the system 18O+40Ca we find RW≃3.375 fm and strength W0≃ 660 MeV, resulting in σ(α) abs ≃2.14 b and Rabs ≃8.26 fm. As 044620-9 LENSKE, BELLONE, COLONNA, AND LAY PHYSICAL REVIEW C 98, 044620 (2018) 0 100 200 300 400 500 600 qαβ [MeV/c] 10-4 10-3 10-2 10-1 100 dσ/dσRuth 5 AMeV 15 AMeV 25 AMeV 35 AMeV 55 AMeV FIG. 9. Elastic scattering angular distributions for 18O+40Ca at several beam energies, normalized to the Rutherford cross section, are shown as a function of the momentum transfer. The momentum range corresponds up to twice the Fermi momentum at central nuclear density, qα,β ≈600 MeV/c. sults of Fig. 8. We emphasize, however, that finally the ion-ion potentials have to be adjusted once elastic scattering data will be available. Results for the elastic scattering cross sections, normalized to the Rutherford cross section, are shown in Fig. 9for the 18O+40Ca system at different beam energies. At extremely small momentum transfer (i.e., extreme forward scattering angles) it is dominated by pure Coulomb scattering, but beyond qαβ ≈200 MeV/c,the short-range nuclear parts are taking over. C. Distortion effects in heavy ion SCE cross sections Following the reaction and nuclear structure formalism outlined above, numerical calculations of single charge exchange cross section have been performed [38]. Form factors are derived by folding the transition densities with the projectile-target residual charge exchange interaction where the momentum representation is used [18]. In order to maintain self-consistency as much as possible, we use the same 2QP isovector interaction as in the nuclear structure calculations. The operator structure includes spin-dependent and spin-independent direct and exchange central interactions, together with second-rank tensor terms. The NN spin-orbit interactions have been neglected. The procedure follows closely the approach used successfully in our previous investigations of SCE reactions [8–11,38]. The closest resemblance to nuclear βdecay processes is found in pure Gamow-Teller (spin-isospin flip with JP=1+) or pure Fermi (isospin flip with JP=0+) excitations, respectively. However, strong interaction processes are less selective on multipolarities than weak interactions. Moreover, because of the peripheral character of inclusive heavy-ion reactions, very often transition of higher angular momentum transfer are favored. Thus, heavy-ion SCE reactions enable to probe the whole spectrum of Gamow-Teller-like spin-isospin-flip and Fermi-like isospin-flip multipole transitions, discussed in the previous section, allowing us to study multipolarities 036912 15 Ex [MeV] 10-6 10-5 10-4 10-3 10-2 10-1 100 dσ/dE [mb/MeV] JP = 0+ JP = 1- JP = 2+ JP proj = 0+ 012345678910 θ [deg] 10-5 10-4 10-3 10-2 10-1 100 101 d2σ/dEdΩ [mb/MeV sr] JP = 0+ JP = 1- JP = 2+ FIG. 10. DWBA cross section as a function of target excitation energy, integrated over the full angular range (top panel), and angular distribution for Ex=0 MeV (bottom panel) for several multipoles, contributing to Fermi-like transitions in the target. Calculations are for the reaction 40Ca(18O,18F)40K reaction at Tlab =270 MeV using double-folding optical potentials and QRPA response functions. suppressed otherwise in weak decay processes. From the theoretical discussion it is clear that distortion effects are playing a significant role in heavy-ion SCE reactions. Results for SCE single- and double-differential cross sections in full DWBA are shown in Figs. 10–12, for the reaction 18O+ 40Ca →18F+40K with Qgs =−2.97 MeV. The complementary SCE reaction 18O+40Ca →18N+40Sc is kinematically suppressed because of the large Qgs =−28.22 MeV. For the Gamow-Teller (Fermi) case, we consider transitions leading to the 1+ground state (0+excited state) of 18F and populating several 40K excited states, identified by the spin Jand the excitation energy Ex=E∗ B.Forthe present purpose, we neglect the small variations in excitation energy of the 18F ground-state multiplet, treating the states as energetically degenerate with vanishing excitation energy. From Figs. 10–12, it is straightforward to note that JP=1+ and JP=0+target transitions contribute significantly to the 044620-16 THEORY OF SINGLE-CHARGE EXCHANGE HEAVY-ION … PHYSICAL REVIEW C 98, 044620 (2018) 01.5 34.5 67.5 910.5 12 13.5 15 10-6 10-5 10-4 10-3 10-2 10-1 100 dσ/dE [mb/MeV] Jπ = 1- Jπ = 2+ Jπ proj = 1+ 01.5 34.5 67.5 910.5 12 13.5 15 Ex [MeV] 10-6 10-5 10-4 10-3 10-2 10-1 100 Jπ = 0- Jπ = 1+ Jπ = 2- FIG. 11. DWBA cross section as a function of target excitation energy, integrated in the full angular range, for several multipoles, contributing to Gamow-Teller-like transitions, with natural parity (top panel) and unnatural parity (bottom panel), in the target. The system is the same as in Fig. 10, using again QRPA spectral distributions. cross section at low excitation energies and dominate at small angles. Having in mind to illustrate global features of heavy-ion SCE reactions, we will focus thereafter on pure Gamow- Teller excitations in both projectile and target. The results concerning distortion effects and the relation of the (physical) DWBA cross section to the plane wave counterpart and the β-decay matrix elements is to a large extent independent of the multipolarity, at least at small momentum transfer. Thus, without loss of generality, it is sufficient to consider a single multipolarity. 012345678910 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 dσ/dΩ [mb/sr] Jπ = 1- Jπ = 2+ Jπ proj = 1+ 012345678910 θc.m. [deg] 10-6 10-5 10-4 10-3 10-2 10-1 100 Jπ = 0- Jπ = 1+ Jπ = 2- FIG. 12. DWBA angular distribution for Ex=0 MeV, for several multipoles contributing to Gamow-Teller-like transitions, with natural parity (top panel) and unnatural parity (bottom panel), in the target. The system is the same as in Fig. 10. 036912 15 Ex [MeV] 10-6 10-4 10-2 100 dσ/dE [mb/MeV] PWBA DWBA PW + iW (r) PW + V (r) PW + UC (r) FIG. 13. Cross sections as a function of the target excitation energy, Ex, for the JP=1+transition, for the SCE reaction 40Ca(18O,18F)40K reaction at Tlab =270 MeV, integrated over the full angular range. The different curves show the effect of Coulomb potential [UC(r)], of real [V(r)] and imaginary [W(r)] components of the optical potential, and of the full potential (DWBA), with respect to PWBA calculations. The system is the same as in the previous figures. In order to understand the influence of the elastic ion-ion interactions on SCE processes, we first disentangle the various contributions to the optical potentials. Figure 13 displays the (18O,18F(g.s.)) total cross section σαβ as a function of the target excitation energy, integrated over the full angular range. Calculations are performed in the plane-wave Born approximation (PWBA), as well as considering separately the effects of Coulomb potential and of real and imaginary parts of the nuclear optical potential, and, finally, combining all these potentials in DWBA. Already at the PWBA level, one can appreciate the main excitation peaks contributing to JP=1+transitions in the target. With respect to the latter results, it is observed that the cross section decreases when the effect of the Coulomb repulsion is taken into account or increases when considering the contribution of the (attractive) real part of the nuclear optical potential. However, the most striking feature is the strong suppression, by about a factor 500–600, observed just taking into account the imaginary part of the optical potential, which essentially brings the cross section down to the value associated with the full DWBA calculation. This indicates that the DWBA result is mainly explained in terms of strong absorption effects, as expected in heavy-ion reactions, and justifies the strong absorption approach, underlying the black disk approximation to model the ion-ion initial- and final-state interactions (see Sec. V). Moreover, the calculations show that absorption effects also affect the diffraction pattern of the nonelastic angular distributions: Under strong absorption conditions, the transition occurs effectively in the surface layers of the reacting nuclei which consequently is reflected in the structure of the angular distribution. The effect is most pronounced in transitions of low multipolarities. The interplay between central and tensor terms of the effective interaction is investigated next. Results are shown 044620-17 LENSKE, BELLONE, COLONNA, AND LAY PHYSICAL REVIEW C 98, 044620 (2018) 036912 15 Ex [MeV] 10-6 10-5 10-4 10-3 10-2 10-1 100 101 dσ/dE [mb/MeV] σ PWBA, L = 0 σ DWBA, L = 0 σ PWBA, L = 0, 2 σ DWBA, L = 0, 2 σ PWBA, L = 0, 2 σ DWBA, L = 0, 2 FIG. 14. Cross section as a function of the target excitation energy, integrated over the full angular range. The plot shows the effects of L=0,2 multipolarities, involved in Jπ=1+transition and of central and tensor components of nuclear interaction. Same system as in the previous figures. in Figs. 14 and 15, together with the contributions associated with the two multipolarities (L=0,2) leading to JP=1+ transitions. One can see that the central interaction contribution to the angle-integrated cross section is fully dominated by L=0 transitions. The differential cross section, Fig. 15, is shown for the state at the lowest excitation energy (Ex= E∗ B=0 MeV), that as discussed before is an intruder state for the 40K ground state. The same conclusion holds, as far as the small angles shown on the figure are concerned. The tensor interaction is seen to slightly reduce the cross section in the PWBA case and, in the full DWBA calculations, for the main excitation peaks. Actually, as shown in Fig. 15, the tensor contributions also shift the cross section to larger angles, especially in the PWBA calculations, owing to the dominant role of L=2 transitions in this case. 012345678910 θCM [deg] 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 d2σ/dEdΩ [mb/MeV sr] σC PWBA, L = 0 σC DWBA, L = 0 σC PWBA, L = 0, 2 σC DWBA, L = 0, 2 σC+T PWBA, L = 0, 2 σC+T DWBA, L = 0,2 FIG. 15. Angular distribution for the target state at Ex=0. The plot shows the effects related to central and tensor components of the nuclear interaction, for the two multipolarities allowed by Jπ=1+ transitions: L=0,2. Same system as in the previous figures. 01.5 34.5 67.5 910.5 12 13.5 15 Ex [MeV] 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 dσ/dE [mb/MeV] σC+T (L = 0, 2) PWBA σC+T (L = 0, 2) DWBA FIG. 16. Cross section as a function of the target excitation energy, integrated over the full angular range, for the reaction 18O+ 116Sn at Tlab =270 MeV. Transitions leading to the 18F g.s. and to Jπ=1+states of the target are considered. PWBA and DWBA results are compared. In Fig. 16, we show the angle-integrated cross section calculated for the system 18O+116Sn, at the same beam energy Tlab =270 MeV. Jπ=1+transitions are considered in both projectile and target. The same QRPA approach, as described above for the 18O+40Ca system, has been employed to evaluate the corresponding form factors. By comparing PWBA and DWBA results, one can notice that a smaller distortion factor is obtained for this reaction, with respect to the 40Ca target case. This trend is in agreement with the mass behavior predicted by the eikonal approximation; see Fig. 4. Guided by the previous analysis, in the following we will consider, for the sake of simplicity, excitations corresponding to L=0 and we will neglect the tensor part of the effective interaction. D. Cross-section factorization As stressed in Sec. IV C, the case when the transition form factors, Eq. (A13), can be approximated by a Gaussian function is of a particular advantage for the separation of the distortion effects. This implies that the spatial transition densities contained in Eq. (A14) correspond to the multipole components of a Gaussian. Following the formalism outlined in Sec. IV C, we parametrize the microscopic transition densities, as obtained from our QRPA calculations for projectile and target, by a superposition of Gaussians. Combining the results of projectile and target and neglecting the momentum dependence of the interaction form factor VC ST(p2), which is quite flat in the low-momentum transfer range corresponding to θ∈[0◦,10◦], one can finally extract the parameters (Rand σ) entering the expressions in Eqs. (50) and (51) for the full reaction amplitude in Born approximation, M(B) αβ (p), which in the Gaussian representation can be evaluated analytically in closed form. Given Ra(RA) and σa(σA) are the Gaussian parameters referring to the projectile (target) transition density, the combined, effective transition 044620-18 THEORY OF SINGLE-CHARGE EXCHANGE HEAVY-ION … PHYSICAL REVIEW C 98, 044620 (2018) 050 100 150 200 250 300 350 400 q [MeV] -0.2 0 0.2 0.4 0.6 0.8 1 UL = 0 (qαβ , q) (arb. units) qαβ = 20 MeV qαβ = 60 MeV qαβ = 100 MeV qαβ = 20 MeV factorized qαβ = 60 MeV factorized qαβ = 100 MeV factorized FIG. 17. Comparison between the monopole component of the Gaussian reaction kernel, obtained in the separation ansatz, with hαβ (q)givenbyEq.(59) (dashed lines), and in the full BD approximation (solid lines). Different values of qαβ are considered. Some of the lines are difficult to distinguish because they are very close to each other. radius is R2=R2 a+R2 A, and for the corresponding form factor width one finds σ2=σ2 a+σ2 A. Then also the averaging over the orientation of the off-shell momentum q, can be performed, leading to the reduced Born amplitude ¯ M(B) αβ (qαβ ,q), which plays a central role for studies of distortion effects. For the reaction under investigation, we find R≈5fm, σ≈1.2 fm. Figure 17 shows the results obtained for the monopole term UL=0, according to the full expressions Eqs. (41) and (42), or adopting the (partial) separation ansatz, as in Eqs. (52) and (59). One can observe that, whereas the separation ansatz works quite well for small values of qαβ (see, for instance, the results corresponding to qαβ ≈20 MeV/c), important deviations from the exact results are seen for larger qαβ values. Let us first consider the case of small momentum transfer (qαβ =20 MeV/c).UsingEq.(63), the distortion factor fBD =|1−nαβ |2is readily obtained in the black disk approximation.2This is shown in Fig. 18 as a function of the absorption radius Rabs. Here, the results obtained with the full expression of h(q), as given in Appendix D, practically coincide with the approximate expressions, Eqs. (59) and (60). Guided by the total reaction cross section obtained numerically by the quantal partial wave method (σabs ≃2.14 b, see Fig. 8), we adopt Rabs =√(σabs/π )≈8 fm. Correspondingly, the suppression factor is found to be fBD(analytical)|Ex=0 θ=0≃8.14 × 10−4, in good agreement with the numerical DWBA/PWBA result, fBD(numerical)|Ex=0 θ=0≃8.35 ×10−4, as it can be extracted from the ratio between DWBA and PWBA calculations at zero angle. As already anticipated above, owing to the important effects associated with the imaginary part of the 2This relation is reminiscent of the aforementioned relation of the on-shell distortion coefficient to the elastic Smatrix; see Sec. VB. 0369 Rabs [fm] 0 0.2 0.4 0.6 0.8 1 fBD FIG. 18. Distortion factor as a function of Rabs, for the separation function hαβ (q) corresponding to Eq. (59) (see text). optical potential, the black disk assumption represents quite well the distortion effects predicted by the full DWBA calculations. Figure 19 shows the distortion factor fBD, extracted by the ratio of DWBA and PWBA forward cross sections for the systems 18O+40Ca and 18O+116Sn, as a function of the beam energy. The distortion factor drops sharply at low beam energy, despite the smaller absorption cross section seen in Fig. 8. This indicates that at beam energies close to the Coulomb barrier the BD assumption is lacking the additional suppression effects coming from the Coulomb barrier. However, as one can see on Fig. 19, at higher energy the distortion factor is increasing rather slowly and a quite good general agreement with the results obtained within the eikonal approximation, Fig. 4, is observed. Hence, at energies well above the Coulomb barrier, DWBA and PWBA forward angle cross sections are related to a good approximation by scaling factors, depending only moderately on the incident energy. 0 100 200 300 400 500 600 700 800 900 1000 Tlab [MeV] 10-12 10-10 10-8 10-6 10-4 10-2 100 fBD 18O + 40Ca 18O + 116Sn FIG. 19. The distortion factor fBD is displayed as a function of the beam energy for two reaction systems: 18O+40Ca and 18O+116Sn. The results were obtained numerically by the ratio of the quantum mechanical DWBA and PWBA forward angle cross sections. 044620-19 LENSKE, BELLONE, COLONNA, AND LAY PHYSICAL REVIEW C 98, 044620 (2018) The question may arise on the range of validity of the separation ansatz at finite momentum transfer. Explicit calculations can be easily performed in the full BD approximation, Eq. (40), and with the approximations introduced in Eq. (43), considering the separation function given by Eqs. (59) and (60). The results confirm that the black disk approximation indeed accounts appropriately for the absorption effects with, however, as a few caveats. So, as to be expected, we find that the scaling factor generally depends on qαβ , limiting, e.g., the separation ansatz of Eq. (60) to low-momentum transfers up to qαβ ≈50 MeV/c. The description is improved by using the partial separation ansatz of Eq. (59), which extends the range of validity over a larger region of qαβ values, in the present case to momentum transfers of about 100 MeV/c. The results discussed here for Ex=0 can be extended easily to transitions leading to other excited states. We conclude that the full cross-section factorization is generally valid for small momentum transfer, i.e., in the case of low-energy excitations and forward angles. Under these conditions, it is possible to isolate in the SCE reaction amplitude the contribution of the Born amplitude, as done in Eq. (45). This is particularly important because it allows us to access direct information on the nuclear matrix elements, which are linked, in turn, to β-decay strengths, as discussed in Appendix F. VII. SUMMARY AND OUTLOOK Heavy-ion reactions are of wide interest by their own because of the possibility to explore several excitation mechanisms by the same experiment. Charge-changing reactions, in particular, open unprecedented perspectives for detailed nuclear structure investigations of the many-body dynamics underlying also β-decay processes. In this paper, we have presented a revised approach to the theoretical modeling of nuclear SCE reactions. In a strictly microscopic picture, we have reformulated the reaction dynamical aspects in the framework of DWBA theory. Central and rank-2 tensor interactions were considered. In momentum representation, the reaction amplitude was separated formally into projectile and target transition form factors and the distortion coefficient, accounting for ISI and FSI ion-ion interactions. HFB theory is used to describe the projectile and target ground states. The charge changing nuclear excitations were described by correlated 2QP configurations including residual quasiparticle interactions. Effects beyond mean-field dynamics were treated by introducing dynamical self-energies of a universal character. Thus, an extended QRPA approach was obtained. The QRPA problem is solved by direct solution of the Dyson equation, which is closely connected to the polarization propagators formalism. Nuclear response functions were introduced as the appropriate method for investigating charge-changing external fields which in the present context are provided by the isovector NN projectile-target interactions. The nuclear structure approach has been illustrated in calculations for charge-changing excitations of 18O and 40Ca. Response functions for multipole operators, intimately connected to weak interactions at low momentum transfer, have been discussed. As illustrative—and typical—examples, results for both τ±branches have been presented. The 18F spectra could be reproduced satisfactorily well. An even better agreement with spectral data was obtained for the heavier system 40K. Our previous investigations of heavy-ion SCE reactions have shown that fully quantum mechanical DWBA calculations with microscopic nuclear structure input describe observed cross sections quantitatively. Thus, from the theoretical side we have a powerful and successful toolbox at hand. However, for the direct extraction of spectroscopic quantities from measured cross sections, a separation of reaction and nuclear dynamics contributions is of great advantage. In this respect, a central goal of our investigation was to explore in detail the interplay of reaction and nuclear structure aspects in heavy-ion SCE cross section, aiming at identifying the conditions under which the two parts factorize, thus giving access to nuclear matrix elements relevant for β-decay processes. We note that this point has been widely investigated for reactions involving light projectiles (such as protons or 3He) at energies of a few hundred MeV per nucleon and found to be a quite useful tool to extract direct information on the β-decay strength of the target. Here, we could show that heavy-ion reactions of a few tens of MeV per particle are in principle as well suited for such explorations. Performing explicit calculations for the reaction 18O+ 40Ca at Tlab =15 A MeV, we have shown that in heavyion reactions the distortion effects are strongly amplified, where the imaginary part of the nuclear optical potential is playing the major role. Indeed, SCE cross sections obtained with only the imaginary potential (i.e., neglecting real part and Coulomb interaction) in the initial- and final-state elastic ion-ion interactions coincide almost perfectly with results by full DWBA calculations. Thus, the distortion effects are in fact mainly absorption effects which are well reproduced in the strong absorption limit by the black disk approximation. Within such a scheme, we have worked out a factorization of the reaction cross section which is well suited for reactions characterized by small momentum transfer. The investigations have shown that heavy-ion SCE reactions indeed allow us to extract from the forward-angle cross section, i.e., at small momentum transfer, a direct information on the product of the β-decay strengths in projectile and target. Hence, also in the case of heavy-ion reactions, we are led to define a “unit” cross section, which allows us to relate the SCE differential cross section to the β-decay strengths. For a given projectile, calculations performed in the black disk approximation suggest a ≈1/A dependence of the distortion factor on the target mass. The dominant role played by ISI and FSI demands for studies of elastic scattering. Angular distribution data in at least one of the involved channels would be of high importance as a countercheck of the accuracy of the microscopically derived ion-ion potentials which are central parts of the presented approach. Eikonal theory provides an elegant approach to investigate universal aspects of the mass and energy dependences of distortion effects. These kinds of predictions are of interest especially for estimates of yields to be expected in future experiments. The intentions of the present eikonal studies is to encircle global trends and variations of cross sections. 044620-20 THEORY OF SINGLE-CHARGE EXCHANGE HEAVY-ION … PHYSICAL REVIEW C 98, 044620 (2018) We emphasize again that for a quantitative analysis of a given reaction a fully quantum mechanical calculation, as indicated, will be the preferred method. Here, we have used physically meaningful but schematic descriptions for the mass and energy dependences of the input quantities, as there are optical potentials and transition potentials. Both quantities were approximated by Gaussian form factors. Within the Gauss-eikonal approximation, the mass and energy dependences of the distortion factor were investigated in the strong absorption limit. An attractive feature of GEA approach is that it allows us to connect the phenomenological concept of black disk (or black sphere) scattering to the microscopic background. The results for the distortion factor are clearly indicating certain constraints on heavy-ion SCE reactions: The magnitude of the cross sections will depend critically on the projectile-target combination. At fixed energy, systems with low total mass are favored, meaning that, e.g., an increase of the target mass will result in a decreasing cross section. The same is true for a variation of the projectile mass. However, this mass effect can be compensated to a large extent by varying the energy because the coefficient fBD increases rapidly with incident energy. Broad space was given to a formulation covering reaction and nuclear structure aspects on the same footing. By practical considerations, the main weight was laid on reactions at energies well above the Coulomb barrier. Such reactions are of high interest for currently active experiments, e.g., within the NUMEN project at LNS Catania [12]. The theoretical results, however, apply to other choices of projectile-target combinations and energy as well. These developments of the theory of heavy-ion SCE reactions open new interesting perspectives for experiments involving unstable nuclei [39] and for studies of double charge exchange (DCE) reactions. The extension to heavy-ion DCE reactions will be the topic of a forthcoming paper. In fact, with appropriate extensions, the methodology developed in this work is a suitably entry point for investigations of second-order processes as the heavy-ion DCE reactions, allowing us to establish their possible relation to double-β-decay processes. ACKNOWLEDGMENTS Part of this work was done during the visits of H.L. to LNS Catania. The warm hospitality of the NUMEN group and the financial support by the NUMEN project are gratefully acknowledged. We wish to thank F. Cappuzzello and M. Cavallaro for fruitful discussions. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 654002 and from the Spanish Ministerio de Ciencia, Innovacion y Universidades and FEDER funds under Project No. FIS2017- 88410-P. APPENDIX A: SECOND QUANTIZATION AND ONE-BODY TRANSITION DENSITIES The spacial multipole components of the transition operator, Eq. (13), lead to the irreducible tensor operators (for S=0,1) RLSJM(r,σ)= ML,MS (LMLSMS|JM)iLYLML(ˆr)σMSS =[iLYL(ˆr)⊗(σ)S]JM,(A1) where YLMLdenotes spherical harmonics functions. In the second quantization, the one-body transition matrix elements for n→ptransitions are given by USTLML jpmpjnmn(p2)= JM (LMLSMS|JM)jpmp|jL(pr) ×RLSJM(r,σ)τ+|jnmn,(A2) where jL(x) are Bessel functions. For p→ntransitions, one can write a similar expression, exchanging nand pindices and replacing τ+by τ−. After we apply the Wigner-Eckardt theorem [20], the matrix elements separate into a Clebsch-Gordan coefficient and a reduced matrix element. This allows us to perform the summation over the proton and neutron magnetic quantum numbers, leading to the one-body transition density operators A† JM(jpjn)= mpmn (jpmpjnmn|JM)a† jpmp˜ ajnmn,(A3) where ˜ ajm =(−)j+maj−mdenotes the conjugated operator. The proton-neutron and the neutron-proton particle-hole operators are related by Hermitian conjugation, A† JM(jnjp)=(−)J+MAJ−M(jpjn),(A4) reflecting charge-conjugation symmetry. The reduced isovector matrix elements is ¯ ULSJ jpjn(p2)=√2 ˆ Jpspjp||jL(pr)RLSJ ||nsnjn,(A5) where ˆ J=√2J+1 and sp=sn=1 2. The factor √2 results from the isospin structure of the isovector nucleon-meson vertices. These steps lead to the representation of the transition operators in terms of irreducible tensor components of conserved total angular momentum J  mpmn USTLML jpmpjnmn(p2)a† jpmpajnmn = JM (LMLSMS|JM)¯ ULSJ jpjn(p2)A† JM(jpjn).(A6) Thus, using second quantization, the transition operator becomes RST(p,a†a)= LMLJM Y∗ LML(ˆp)(LMLSMS|JM) jp,jn¯ ULSJ jpjn(p2)A† JM(jpjn)+¯ ULSJ jnjp(p2)A† JM(jnjp).(A7) 044620-21 LENSKE, BELLONE, COLONNA, AND LAY PHYSICAL REVIEW C 98, 044620 (2018) The target transition form factors are now given as F(AB) SMS(p)= λAμAIANA Y∗ λAμA(ˆp)(λAμASMS|IANA)⎧ ⎨ ⎩ jp,jn ¯ UλASIA jpjn(p2)JBMB|A† IANA(jpjn)|JAMA + jp,jn ¯ UλASIA jnjp(p2)JBMB|A† IANA(jnjp)|JAMA⎫ ⎬ ⎭ (A8) and accordingly in the projectile system. For a given reaction, only one of the terms in Eq. (A8) is effectively contributing to the transition: If, e.g., a p−1n-type transition is occurring in the projectile, only the parts containing operators of a† nap structures give nonvanishing contributions while the complementary a† panoperator branch is only active in the target, and vice versa. The spectroscopy of the charge exchange process is now contained in one-body transition density matrix elements defined as DJM jcjd(JfMf,J iMi)=JfMf|A† JM(jcjd)|JiMi.(A9) The Wigner-Eckardt theorem leads to DJM jcjd(JfMf,J iMi) =(−)Jf−MfJfMfJi−Mi|J−M¯ DJ jcjd(Jf,J i)(A10) with the reduced one-body transition density ¯ DJ jcjd(Jf,J i)=1 ˆ JJf||A† J(jcjd)||Ji.(A11) If the parent state has Ji=0, the result simplifies to DJM jcjd(JfMf,00) =¯ DJ jcjdδJJ fδMMf.(A12) The same simplification is obtained for the case Jf=0. Obviously, the one-body transition densities are the elements of central importance for the spectroscopy of the charge exchange process. They are providing access to the manybody structure of the underlying nuclear wave functions. The evaluation of the one-body transition densities requires knowledge of the structure of the initial and final nuclear states, establishing the connection to nuclear structure theory. By standard angular momentum coupling techniques, Eq. (A8) becomes F(AB) ST (p)= L,ML,J2,M2 (JAMAJBMB|J2M2)(LMLSMS|J2M2) ×f(AB) LSJ2(p2)iLYLML(ˆ p) (A13) and corresponding expressions are found for projectile transitions. In Eq. (A13), the reduced form factors f(XY ) LSJ have been introduced. The latter are related to the corresponding reduced radial transition densities ρ(XY ) LSJ by a Fourier-Bessel transform: f(XY ) LSJ (p2)=∞ 0 drr2ρ(XY ) LSJ (r)jL(pr).(A14) The total angular momentum transfer in the projectile and target system are given by J1,2, defining the set of multipole components which are contributing to a given reaction leading from initial states Ja,A to final states Jb,B. These relations are expressed by the first Clebsch-Gordan coefficient in the above equations. In accordance with the investigations of the previous section, these multipoles carry substructures given by the coupling of orbital (L1,2) and spin (S1,2) angular momentum transfers, as expressed by the second Clebsch- Gordan coefficients in Eq. (A13). APPENDIX B: ANGULAR MOMENTUM DECOMPOSITION OF THE REACTION KERNEL The decomposition of the full reaction kernel and correspondingly the reaction amplitude relies on their decomposition into irreducible tensorial components. For that purpose, the product of the projectile and target form factors, Eq. (14), must be recoupled to total angular momentum. We use the addition theorem of spherical harmonics [20] iL1YL1m1(ˆp)iL2YL2m2(ˆp) = LML (−)L1+L2−L 2 ˆ L1ˆ L2 √4πˆ L(L10L20|L0) ×(L1m2L2m2|LM)iLYLM (ˆp).(B1) Then, for a central interaction, the product of nuclear form factors is obtained as F(αβ ) ST (p)= J1,M1,J2,M2,L,M (JaMaJbMb|J1M1) ×(JAMAJBMB|J2M2)(J1M1J2M2|LM) ×iLYLM (ˆ p)FJ1J2 LS (p2),(B2) with the reduced multipole form factors FJ1J2 LS (p2)= L1L2 ALS (L1L2,J 1J2)f(ab) L1SJ1(p2)f(AB) L2SJ2(p2). (B3) 044620-22 THEORY OF SINGLE-CHARGE EXCHANGE HEAVY-ION … PHYSICAL REVIEW C 98, 044620 (2018) We have introduced the recoupling coefficients ALS (L1L2,J 1J2)=(−)L1+L2−L 2 ˆ L1ˆ L2 √4πˆ L(L10L20|L0)(−)L2+J1−LW(L1J1L2J2;LS)ˆ J1ˆ J2,(B4) where W(abcd;ef ) is a Racah coefficient [20]. The rank-2 tensor component requires additional recoupling of the spin and orbital angular momentum operators in order to comply with the quadrupole character of the vertex. The resulting form factor can be cast into a form similar to Eq. (B2): H(αβ ) ST (p)= J1,M1,J2,M2,L,M (JaMaJbMb|J1M1)(JAMAJBMB|J2M2)(J1M1J2M2|LM)iLYLM (ˆp)HJ1J2 L1(p2).(B5) The reduced form factors, however, are of a somewhat more involved structure: HJ1J2 L1(p2)= L1,L2,L BLL(L1L2,J 1J2)f(ab) L11J1(p2)f(AB) L21J2(p2).(B6) In this case, the recoupling coefficient is given by BLL(L1L2,J 1J2)=24π 5(−)L1+L2−L 2 ˆ L1ˆ L2√5 4πˆ LL10L20|L0L020|L0ˆ J1ˆ J2ˆ L√5⎧ ⎪ ⎨ ⎪ ⎩ L11J1 L21J2 L2L ⎫ ⎪ ⎬ ⎪ ⎭ ,(B7) where the object in the last line is a 9-jsymbol [20]. APPENDIX C: GAUSSIAN FORM FACTORS AND MICROSCOPIC NUCLEAR STRUCTURE The price paid for the advantage of the Gaussian approximation that the dependencies on the ion masses and sizes are directly accessible by closed-form expressions is that the connection to microscopic nuclear structure seems to be lost. However, by reinterpretation of the parametrical dependence on the yet-to-be-specified radius R, that connection can be restored under certain constraints. In leading order, the transition potential Eq. (14) is given by replacing the NN T matrix by a contact interaction where the strength if given by the momentum space amplitude at p=0, i.e., the volume integral. Denoting the intrinsic projectile and target coordinates by r1,2, respectively, and the ion-ion relative coordinate by r, the zero-range assumption implies r1+r−r2=0. For a contact interaction, the folding integral defining the transition potential reduces to the folding of the nuclear transition form factors. For that purpose, we assume that the intrinsic nuclear transitions a→band A→Bare described by Gaussian form factors FN(rN)=CNe−1 2σ2 N|rN−RN|2 ,(C1) where N∈{ab, AB}(N=1,2) and the normalization constant is chosen as CN=(√2πσN)−3such that FNhas a volume integral equal to unity. In coordinate space, we have to evaluate a folding integral of the type F12(r)=C1C2d3r1e−1 2σ2 1 (r1−R1)2 e−1 2σ2 2 (r1+r−R2)2 .(C2) With the substitutions x=r1−R1and ρ=r+R1−R2,the integral becomes F12(r)=C1C2d3xe−1 2σ2 1 x2 e−1 2σ2 2 (x+ρ)2 .(C3) The angle integrations lead to modified Bessel function of order n=0. The remaining integration can be performed in closed form with the final result F12(r,R)=1 (√2πσ)3e−1 2σ2(r−R)2.(C4) The width is given by σ2=σ2 1+σ2 2,(C5) and the centroid radius is found as R=R1−R2,(C6) which plays the role of a scale-defining quantity. Considered as classical mathematical objects, the vectors R1,2are free parameters reflecting the nuclear scales. Thus, we use R1,2∼ A 1 3 1,2. Since the relative orientation of the two centroid vectors is arbitrary, we use the averaging, resulting in R2=R2 1+ R2 2≃A 2 3 1+A 2 3 2. Within the above zero-range approximation, the transition potential, Eq. (46), is given by the superposition of a spinscalar (S=0) and a spin-vector (S=1) component UG(r)= S=0,1,T =1 ISTB(ab,AB) ST F(ab,AB) 12,ST (r,RST ),(C7) where IST ≡V(C) ST (p=0) denotes the volume integral of the interaction. The crucial point is how to incorporate the underlying microscopic nuclear structure physics. The simplest, but rather schematic approach is to use projectile and target spectral distributions averaged over multipolarities. Such a solution is indicated above: The nuclear charge-changing spectral transition strengths for projectile and target are contained in Bab,AB ST , obtained, e.g., by the response function formalism, Sec. VI A. In a refined approach, the multipole structure of the form factors and spectral distributions should be combined explicitly. On a formal level, this is achieved by identifying YLM (ˆ R) 044620-23 LENSKE, BELLONE, COLONNA, AND LAY PHYSICAL REVIEW C 98, 044620 (2018) as a dynamical quantity with an operator structure inducing intrinsic nuclear transitions. Formally, this is achieved by imposing the quantization conditions YLM (ˆ R)→YLM (†) = λκ bλ(ab)bκ(AB)[† λ(ab)† κ(AB)]LM ,(C8) i.e., a representation by the state operators † λ,κ of projectile and target, respectively. λand κinclude spin and orbital angular momenta. The coupling to good total angular momentum transfer Lis indicated. The expansion coefficients are given by nuclear multipole transition amplitudes. Thus, we have obtained a relation similar to the collective model approach to nuclear spectroscopy of Bohr and Mottelsen [40], widely used in the past for nuclear reactions. Thus, the essence of the Gaussian form factor is seen to separate the state-dependent transition form factors into a state-independent spatial form factor ULM (r), Eq. (48), and state-dependent amplitudes bμ, giving rise to the multipole spectral distributions B(aA,bB) L=bB|YLM |aA(C9) and the multipole transition potentials UaA,bB ST,LM (r)=B(aA,bB) LISTULM (r),(C10) where the spectral amplitudes and the reduced form factors will depend in general also on the spin transfer S. APPENDIX D: GAUSSIAN APPROACH TO THE BLACK DISK DISTORTION FACTOR As discussed, the separation function h(q) is well approximated by the modified Gaussian in Eq. (56), h(q)=e−1 2q2σ2j0(qρ).(D1) The parameter σcontrols the slope of the momentum distribution around the momentum transfer p=qαβ . The (off-shell) diffraction structure is determined by ρ.Thus,wehaveto evaluate the integral nBD =2Rabs π∞ 0 dqj0(qRabs)∂ ∂q[qh(q)],(D2) which is given explicitly by the three-parameter form nBD =2Rabs π∞ 0 sin (qRabs) qRabs e−1 2σ2q2 ×−σ2q ρsin (qρ)+cos (qρ)dq. (D3) The absorption radius, Rabs, is fixed by the total reaction cross section. The integral can be performed in closed form, with the result nBD =1 2erf 1 √2σ(R−ρ)+erf 1 √2σ(R+ρ) −2 π σ 2ρe−1 2σ2(R−ρ)2−e−1 2σ2(R+ρ)2(D4) expressed in terms of the scaled (dimensionless) quantities R=Rabs/R,σ=σ/R, q αβ =qαβ R, and ρ=√1−σ4q2 αβ +2iσ2q αβ cos γ.nBD is a complex-valued function, because it depends on the complex pseudoradius ρ. Moreover, nBD contains the full set of multipoles in qαβ . Typical results for nBD are displayed in Fig. 18,forqαβ ≈0. APPENDIX E: DISTORTION COEFFICIENT IN EIKONAL APPROXIMATION For wavelengths λ∼1/k short against the scale Rpot of the interaction zone, i.e., ξ=kRpot 1 semiclassical descriptions become an appropriate description for nuclear reactions. In the case considered here, we have kα≃10 fm−1and Rpot ≃ 5 fm (see Table II), leading to ξ≃40. Thus, despite the rather low energy of Tlab =270 MeV, the kinematical conditions allow us to apply eikonal theory [41]. The distorted waves are given as χ(+) α(kα,r)=eiS(+) α(ρ,z)e+ikα·r,(E1) χ(−)∗ β(kβ,r)=e−iS(−)∗ β(ρ,z)e−ikβ·r,(E2) with the asymptotically in- and outgoing eikonals S(+) α(ρ,z)=z −∞ dζ[Qα(ρ,ζ)−kα],(E3) S(−)∗ β(ρ,z )=z +∞ dζ(Qβ−kβ),(E4) where zand zare directed along the channel momenta kαand kβ, respectively. We have defined the local channel momenta Qγ(ρ,ζ)=k2 γ−2mγ ¯h2Uγ(ρ,ζ)(E5) and ρ,zare oriented such zcoincides with the direction of kα and ρ,z are taken accordingly with respect to kβ. Hence, we identify u(±) α,β =e±iS(±) α,β , leading to ηαβ =u(−)∗ βu(+) α=ei(S(+) α−S(−)∗ β)=eiφαβ −καβ ,(E6) where the (real) phase shift φαβ and the attenuation exponent καβ are given by the sum of the real and imaginary parts, respectively, of the eikonals. For small momentum and energy transfer, we may neglect the differences in the channel momenta and potentials. Furthermore, at small momentum transfer, zand zmay be assumed to be parallel. Under such conditions, the distortion amplitude is given by the (diagonal) distortion phase shift and attenuation exponent φα(ρ)≃+∞ −∞ dζRe[Qα(ρ,ζ)]−kα,(E7) κα(ρ)≃+∞ −∞ dζIm[Qα(ρ,ζ)].(E8) As discussed in Sec. VB, in the strong absorption limit we need to consider φαonly in space regions where καhas decreased to O(1) or less. In those space regions, typically also Re(Uopt ) is already of small magnitude. Hence, we can neglect terms of order Uα/k2 αand obtain the standard eikonal 044620-24 THEORY OF SINGLE-CHARGE EXCHANGE HEAVY-ION … PHYSICAL REVIEW C 98, 044620 (2018) expressions φα(ρ)≃−mα ¯h2kα+∞ −∞ dζRe[Uα(ρ,ζ)],(E9) κα(ρ)≃mα ¯h2kα+∞ −∞ dζIm[Uα(ρ,ζ)],(E10) usually combined in the profile function χα=φα−iκα.For the present purpose, it is sufficient to consider the attenuation exponent. Assuming Gaussian form factors and spherical symmetry, Uα(r)=−U0e−r2/R2 U−iW0e−r2/R2 W(E11) with potential strengths U0>0 and W0>0, the leading-order absorption exponent is obtained in closed form: κα(ρ)=√πmαW0 ¯h2kα RWe−ρ2/R2 W,(E12) depending only on the modulus of ρ. Corresponding expressions are obtained for φα. Within the Gaussian approximation, the reaction cross section can be evaluated in closed form. As shown in Ref. [42], the key point is to consider the continuity equation of the distorted waves from which one derives the relation σ(α) abs =−2mα ¯h2kαd3r|χ(+) α(kα,r)|2Wα(r) =d2ρ(1 −e−2κα(ρ)).(E13) For a Gaussian W(r), the integration can be performed analytically. As anticipated before, the result may be expressed indeed in a form resembling in structure the black disk expression σ(α) abs (√sα)=πR2 abs(√sα),(E14) but where the effective absorption radius is related to the potential radius RWby the shape function f(x) R2 abs(√sα)=f(ξ(Wα,k α))R2 W.(E15) The shape function is given analytically by f(x)=γ+log(x)+Ei(1,x),(E16) where γ=0.5772... denotes Euler’s constant and Ei(1,x)is an exponential integral. f(x) is increasing steadily with x, vanishing at x=0 and diverging logarithmically for x1. The argument ξ(W,k)=√πkRW W0 Tcm ,(E17) depends on the reduced mass, the energy, and the absorption potential. Tcm =(¯hk)2/2mis the kinetic energy in the rest frame, and kRW∼gcorresponds to a grazing angular momentum with respect to the potential W. Hence, the absorption and the potential radius are relate in a nontrivial manner, changing with mass and energy. Results have been shown in Fig. 3. APPENDIX F: UNIT CROSS SECTION AND β-DECAY STRENGTHS In this Appendix, we indicate the connection of the present formulation to the approach developed by Taddeucci et al. [14] for light-ion SCE reactions. As a side aspect, a schematic approach is derived from the full theory. In the Born approximation, the reaction cross section, Eq. (2), is simply given by the product of a kinematical factor and the modulus squared of the reaction amplitude Uα,β in Eq. (16). As shown in Sec. VI D, the distortion effects obtained in DWBA can be accounted for at small momentum transfer by means of the scaling function: fBD(Rabs,R,σ)=|1−nαβ |2. Let us keep considering only L=0 transitions, for both projectile and target, and only the central part of the nuclear interaction. Then, the SCE cross section, for small momentum transfer, can be recast in the form [see also Eqs. (30) and (31)] dσαβ =Kf(Tlab,ω)(2S+1)|V(C) ST (0)|2b(ab) 0SS  2b(AB) 0SS  2 ×exp−1 3q2 αβ (r2a+r2A)fBD(Rabs,R,σ), (F1) where the expansion of the Bessel function in Eq. (A14)at small argument has been considered: j0(x)≈1−1/6x2≈ exp(−1/6x2). Thus, in the above equation, r2aand r2A denote the mean square radius of proton and neutron transition densities, respectively. The kinematical factor Kf(Tlab,ω)is given by Kf(Tlab,ω)=mαmβ (2π¯h2)2 kβ kα .(F2) It essentially depends on the energy loss ω=Ex− (MA+Ma−MB−Mb)=Etot −Qgs, where Ex=E∗ b+ E∗ Bis the total excitation energy. The cross section can be rewritten as dσαβ =F(qαβ ,ω)σUb(ab) 0SS  2b(AB) 0SS  2,(F3) where we define a “unit” cross section, in analogy with what is usually done for SCE reactions involving light projectiles [14], as σU=Kf(Tlab,0)|V(C) ST (0)|2fBD(Rabs,R,σ).(F4) The function F, mainly determining the shape of the cross section, is given by F(qαβ ,ω)=Kf(Tlab,ω) Kf(Tlab,0) exp−1 3q2 αβ (r2a+r2A). (F5) We note that the two equations above retrace the formalism developed in Ref. [14]. From Eq. (F5), it follows that F(qαβ ,ω)→1for(qαβ ,ω)→(0,0), so that the proportionality coefficient between the SCE cross section and the product of the βdecay strengths relative to projectile and 044620-25