Analyses of \pi^{\pm} nucleus elastic scattering data at T_\pi = 40, 30, 20 MeV using a suggested scaling method

The data for elastically scattered charged pions from few nuclei, namely ^{12}C, ^{16}O, ^{28}Si, ^{32}S, ^{40}Ca, ^{56}Fe, ^{58}Ni, and ^{90}Zr have been analyzed by obtained potentials using a suggested scaling procedure. Originally the \pi ^{\pm}-^{12}C elastic scattering data at 50 MeV was nicely fitted by a parameterized simple local optical potential extracted from available phase shifts using inverse scattering theory. The potential parameters of the  \pi ^{\pm}-^{12}C systems were scaled to  \pi ^{\pm}-^{16}O systems and then successively to other few systems covering the scattering of charged pions from target nuclei, namely \pi ^{\pm}-^{28}Si, \pi ^{\pm}-^{32}S, \pi ^{\pm}-^{40}Ca, \pi ^{\pm}-^{56}Fe, \pi ^{\pm}-^{58}Ni and \pi ^{\pm}-^{90}Zr . The obtained scaled potentials showed a remarkable success in explaining the available measured elastic differential cross sections, and in predicting other ones for the systems under consideration. The reaction cross sections have been calculated for all these systems at the three incident pion's kinetic energies, T\pi = 40, 30, 20 MeV. Unfortunately, experimental reaction cross sections are totally absent or cloudy and unconfident. As such, and at this stage, we consider our calculated values useful and pending for future investigations.  For the systems and energies considered herein, simple scaling relations are well established. This will be beneficial in analyzing similar nuclear scattering data, as low-energy pion-nucleus and kaon-nucleus elastic scattering data; and, hopefully, in explaining pionic atom data.