# Coupled-channels analysis of pion interactions with rotational nuclei near the (3,3) resonance

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## Abstract

A coupled-channels code, JPPIOM, has been developed to model pion interactions with rotational nuclei. The code is an extensive modification of the JUPITOR code published by Tamura (Tamura, 25) in 1967. JPPION may be used to develop an axially-symmetric Legendre expansion of the Laplacian optical potential using phenomenologically deformed expressions for either the Woods-Saxon or modified Gaussian nuclear density. Thus, the optical potential is expanded in terms of the nuclear rotational angular momenta, and a mechanism is provided to couple the pion partial wave arid nuclear rotational angular momenta. For each value of total spin and parity of the pion-nucleus system, the resulting set of coupled Klein-Gordon differential equations is solved for the radial wavefunctions, and matching to the external solutions yields the generalized S-matrix elements. The cross-sections and angular distributions follow by application of the partial wave channel summation theorems. To date, no provision is made in JPPION to deal with target polarization or with separate proton and neutron densities, although these features may Be included readily in the code. An analysis of the reaction ([pi][raised -]+[raised 12]C -> [pi][raised -]+[raised 12]C) has been made in the vicinity of the (3,3) resonance using JPPION to predict cross-sections and angular distributions for scattering from the 0+ (ground) and 2+ (4.43 MeV) excited states of [raised 12]C. Results are compared to the data of Binon et. al. (Binon, 3), to the DWBA predictions of Edwards and Rost Edwards, 71, and to the elastic results of Sternheim and Auerhach (Sternheim, 22). The parameters used were the Kisslinger Fermi-averaged b-parameters, a radius R[lowered 0]=1.50 f for the modified Gaussian density, a radius R[lowered 0]=2.05 f and a diffusivity [alpha]=0.40 f for the Woods-Saxon density, and a hard sphere Coulomb radius R[lowered c]=2.05 f. Using either density, the ground state cross-sections and distributions were found to be nearly independent of the quadrupole deformation parameter [beta] and in good agreement with the previous elastic scattering predictions. However, the angular distributions are typically lower than the experimental data for angles in the central maximum. The total cross-sections also peak at a theoretical energy much lower than the experimental value of 153 MeV. Fitting the 2+ angular distributions gave energy-averaged values for the deformation parameter of [better]=0.53 for the modified Gaussian density and [beta]=0.68 for the Woods-Saxon density. The difference in the values of 6 is explainable in terms of the greater intrinsic diffusivity of the Woods-Saxon density. Finally, the Kisslinger parameter b[lowered 1] was varied using a chi-square fitting routine taken from the code DAMIT [Hungerford, 10], yielding an improved fit to the ground state and a phenomenological energy-dependence for the parameter b[lowered 1]. Fits to the 2+ states also improved in some cases, and the energy-dependence of the cross-sections was significantly improved. The overall coupled-channels results for [raised 12]C are quite similar to the results of DWBA analysis, indicating that the well-deformed nature and well-defined rotational energy spectrum of [raised 12]C allow accurate treatment by first-order approximations. The merit of the coupled-channels calculation should Be more apparent in the consideration of heavier, more complex nuclei.