Pion elastic and inelastic scattering from 24Mg and 26Mg.

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Reported are measurements of angular distributions of resonance-energy positive and negative pions exciting approximately 40 states in $^{24}\mathrm{Mg}$ and $^{26}\mathrm{Mg}$. These include the (ground state, ${0}^{+}$), (1.36 MeV, ${2}^{+}$), (4.14, ${2}^{+}$), (5.93, ${4}^{+}$), (6.44, ${0}^{+}$), (7.34), (7.55, ${3}^{\mathrm{\ensuremath{-}}}$), (8.33, ${3}^{\mathrm{\ensuremath{-}}}$), (9.32, ${4}^{+}$), (9.97, ${5}^{\mathrm{\ensuremath{-}}}$), (11.08, ${3}^{\mathrm{\ensuremath{-}}}$), (12.06), (13.26), (13.96, ${3}^{\mathrm{\ensuremath{-}}}$), (15.1, T=1, ${6}^{\mathrm{\ensuremath{-}}}$), and (15.4) states in $^{24}\mathrm{Mg}$ and the (ground state, ${0}^{+}$), (1.81, ${2}^{+}$), (2.92, ${2}^{+}$), (3.59, ${0}^{+}$), (4.31, ${2}^{+}$+${4}^{+}$), (4.90, ${4}^{+}$), (5.31, ${2}^{+}$), (5.44, ${4}^{+}$), (5.69, ${4}^{+}$), (6.86, ${3}^{\mathrm{\ensuremath{-}}}$), (7.33, ${3}^{\mathrm{\ensuremath{-}}}$), (7.79, ${3}^{\mathrm{\ensuremath{-}}}$), (8.17, ${3}^{\mathrm{\ensuremath{-}}}$), (9.2, possible ${6}^{\mathrm{\ensuremath{-}}}$), (10.30, ${4}^{+}$), and (18.1, T=2, ${6}^{\mathrm{\ensuremath{-}}}$) states in $^{26}\mathrm{Mg}$. The distorted-wave impulse approximation with a Kisslinger form for the optical potential using a \ensuremath{\pi}-nucleon t matrix at a shifted energy of -25 MeV was found to explain the elastic scattering data from $^{24,26}\mathrm{Mg}$ in the energy range 116--292 MeV that is spanned by these data. Inelastic distorted-wave impulse approximation calculations employing collective-model deformation parameters were simultaneously fitted to the ${\mathrm{\ensuremath{\pi}}}^{+}$ and ${\mathrm{\ensuremath{\pi}}}^{\mathrm{\ensuremath{-}}}$ data for each state.The deformation parameters and matrix elements in most cases compare favorably with results from other studies. Published s-d shell-model calculations using one value for the effective charges were found to reproduce the trend of both the strengths and ratios of neutron-to-proton matrix elements for the ${2}^{+}$ and ${4}^{+}$ states. The data at the first maximum in the inelastic angular distributions for $^{24}\mathrm{Mg}$ and that from other studies for $^{12}\mathrm{C}$, $^{28}\mathrm{Si}$, and $^{40}\mathrm{Ca}$ show that the cross section for ${\mathrm{\ensuremath{\pi}}}^{+}$ scattering is equal to that for ${\mathrm{\ensuremath{\pi}}}^{\mathrm{\ensuremath{-}}}$ scattering, which forces the proton deformation parameters to be greater than the neutron deformation parameters and gives a ratio of neutron-to-proton elements to be less than unity. This difference from unity is interpreted as a measure of the failure of the model and a systematic error of 11% is assumed to dominate the errors in the results for $^{26}\mathrm{Mg}$. Coupled-channels calculations employing monopole form factors are compared to data for low-lying ${0}^{+}$ states in $^{24}\mathrm{Mg}$ and $^{26}\mathrm{Mg}$.