  We settle a general expression for the Hamiltonian of the electron-phonon
deformation potential (DP) interaction in the case of non-polar core-shell
cylindrical nanowires (NWs). On the basis of long range phenomenological
continuum model for the optical modes and by taking into account the bulk
phonon dispersions, we study the size dependence and strain-induced shift of
the electron-phonon coupling strengths for Ge-Si and Si-Ge NWs. We derive
analytically the DP electron-phonon Hamiltonian and report some numerical
results for the frequency core modes and vibrational amplitudes. Our approach
allows for the unambiguous identification of the strain and confinement
effects. We explore the dependence of mode frequencies and hole-DP scattering
rates on the structural parameters of these core-shell structures, which
constitute a basic tool for the characterization and device applications of
these novel nanosystems.
