  We study all of the leading-order contributions to spin relaxation of
\textit{conduction} electrons in silicon due to the electron-phonon
interaction. Using group theory, $k\cdot p$ perturbation method and rigid-ion
model, we derive an extensive set of matrix element expressions for all of the
important spin-flip transitions in the multi-valley conduction band. The
scattering angle has an explicit dependence on the electron wavevectors, phonon
polarization, valley position and spin orientation of the electron. Comparison
of the derived analytical expressions with results of empirical pseudopotential
and adiabatic band charge models shows excellent agreement.
