In the framework of the Glauber multiple scattering theory, the complete expansion ͑which contains 2 AϫB Ϫ1 terms͒ of the Glauber amplitude of elastic scattering between two nuclei A, Bу4 is obtained. Using double Gaussian density consistent with the electron-scattering experiments, the differential cross section (nuclearϩCoulomb) is calculated and compared with the experimental data. It is shown that, in general, higher order terms give a substantial improvement in comparison with the previous optical limit results at relatively large scattering angles. The effect of invoking a phase-variation in the nucleon-nucleon ͑NN͒ amplitude is examined. We found that the presence of the phase variation improves our results, especially at the minima of the diffraction pattern.

Within the formalism of Glauber multiple scattering theory, high-energy elastic collisions between light and medium weight nuclei (A,Bу4) have been investigated by invoking an exact center-of-mass correlation function in each order of the complete expansion of the phase-shift function. The calculated angular distributions (nuclearϩCoulomb), using the developed formulas, show better agreement with the scattering data.

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Elastic scattering of positive and/or negative pions (π + and/or π − ) by nuclei has been studied by means of Glauber's multiple scattering theory (GMST). The differential, total and total elastic cross sections using both the full series expansion and the first order correction (called optical limit results, OLA) of the Glauber amplitude, have been calculated. The in-medium πN amplitude (σ e f f . , σ e f f . ) as well as the free πN one (σ f ree , σ f ree ) are invoked into the full series calculations of the Glauber amplitude and our results were compared with the corresponding experimental data. These calculations are carried out for the elastic collisions, π − − 2 He 4 at incident energy 180 MeV, π ± − 3 Li 6 at 240 MeV, π − − 6 C 12 at incident energies 180, 200 and 260 MeV, π + − 8 O 16 at 240 and 270 MeV and π ± − 20 Ca 40 at energy 292.5 MeV. The comparisons reflect that, except around the minima, the full series calculations with the in-medium πN amplitude are better in describing the scattering data than those using the free πN one and both are better than the optical limit results.

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