Comparison of Exact Solution With Eikonal Approximation for Elastic Heavy Ion Scattering

SummaryNumerical comparisons are made of an exactsolution with the Eikonal approximation to theLippman-Schwinger equation for an equivalent first-order optical potential for heavy ion scattering. Theexact solution proceeds by using a partial-wave ex-pansion of the Lippmann-Schwinger equation in mo-mentum space. Calculations are made for the to-tal and absorption cross sections for nucleus-nucleusscattering. The results are compared with solutionsin the Eikonal approximation for the equivalent po-tential and with experimental data in the energyrange from 25A to 1000A MeV. The percentage dif-ferences between the exact and Eikonal solutions areshown to be small above 200A MeV, but at lowerenergies, differences become large and corrections tothe first-order optical model are required.IntroductionIn high-energy nucleus-nucleus scattering, manynucleons can interact mutually and the structureof multiple scattering is richer than nucleon-nucleusscattering. A simple approach in nucleus-nucleusscattering is to consider the scattering in terms ofeach constituent of the projectile nucleus interactingwith each constituent of the target nucleus. Otherterms may contribute to the scattering, such as a pro-jectile constituent interacting consecutively with twodifferent constituents of target (i.e., double scatter-ing). Similarly, contributions may come from three,four, and more successive scatterings. The formal-ism using this approach is called multiple scatteringtheory.The nucleus-nucleus scattering processes are con-veniently analyzed by employing an optical potentialtheory. Once the optical potential is determined, theoriginal many-body scattering problem reduces to atwo-body scattering problem. However, the price ofreducing a many-body problem to a two-body situa-tion is that the optical potential will be, in general,a complicated nonlocal, complex operator. Thus, forpractical applications we shall require the approxi-mation method to determine the optical potential.An early generalization of the optical model ideasin nuclear physics to the study of alpha-decay ofnuclei was made by Ostrofsky, Breit, and Johnson(ref. 1). Bethe (ref. 2) introduced the concept of anoptical potential in order to describe low-energy nu-clear reactions within the compound nucleus model.The description of high-energy nuclear collisions bymeans of the optical model formalism was initiatedby Fernbach, Serber, and Taylor (ref. 3), who firsttried to describe elastic nucleon-nucleus scatteringin terms of nucleon-nucleon collisions. They arguedthat at high energies, a nuclear collision should pro-ceed by way of collisions with individual target nucle-ons by using known nucleon-nucleon cross sections.This multiple scattering analysis led to the conclu-sion that particles should move more or less freelythrough nuclear matter at high energies. The factthat the optical potential is complex is worth notic-ing. The imaginary part of the optical potential cor-responds to the absorption of the incident beam bytarget nuclei, and the real part of the optical poten-tial corresponds to the refraction of the beam with-out any disturbance to the target nuclei. Watson(ref. 4) and Kerman, McManus, and Thaler (KMT)(ref. 5) developed the formal theory of the scatter-ing of high-energy nucleons by nuclei in terms of thenucleon-nucleon scattering amplitude.A general multiple scattering theory for the scat-tering of two composite nuclei (neglecting three-bodyinteractions) has been developed by Wilson (refs. 6and 7). Calculations of the reaction and absorptionfor the heavy ion projectile was well developed byWilson and Townsend (refs. 8 and 9) by using anEikonal approximation to a first-order optical model.In the first-order optical model, the excitation of theprojectile or target in intermediate states is neglectedin elastic scattering. The Eikonal approximation isbased on a forward scattering assumption, as well asconsiderations of the strength of the potential (refs. 6and 10). A second-order solution to the Eikonalcoupled-channels (ECC) model was developed byCucinotta et al. (refs. 11 and 12) and was found togive improved accuracy over the first-order solutionsin limited studies for several collision pairs and en-ergies. The Eikonal approximation is computation-ally efficient because it requires only a few numeri-cal integrations for implementation. In this report,neutron-nucleus, alpha-nucleus, and carbon-nucleustotal and absorption cross sections are calculated byusing the Lippmann-Schwinger equation. By com-paring calculations with identical physical inputsthat use the exact solution of the first-order opti-cal model to the Eikonal approximate solutions, weprovide an important validation of databases used incosmic-ray_studies. The Eikonal model is also un-able to account for nuclear medium corrections, al-though such studies may be performed in the futureby using momentum space methods. The treatmentof medium corrections will be required if further im-provements in nuclear databases are needed.