  We investigate the spin-dependent electron-phonon scatterings of the $L$ and
$\Gamma$ valleys and the band structure near the conduction band minima in
germanium. We first construct a $16\times16$ ${\bm k}\cdot{\bm p}$ Hamiltonian
in the vicinity of the $L$ point in germanium, which ensures the correctness of
the band structure of the lowest three conduction bands and highest two valence
bands. This Hamiltonian facilitates the analysis of the spin-related properties
of the conduction electrons. We then demonstrate the phonon-induced electron
scatterings of the $L$ and $\Gamma$ valleys, i.e., the intra-$\Gamma$/ $L$
valley, inter--$\Gamma$-$L$ valley and inter--$L$-$L$ valley scatterings in
germanium. The selection rules and complete scattering matrices for these
scatterings are calculated, where the scattering matrices for the
intra-$\Gamma$ valley scattering, inter--$\Gamma$-$L$ valley scattering and the
optical-phonon and the separated transverse-acoustic- and
longitudinal-acoustic-phonon contributions to the intra-$\Gamma$ valley
scattering have not been reported in the literature. The coefficients in these
scattering matrices are obtained via the pseudo-potential calculation, which
also verifies our selection rules and wave-vector dependence. We further
discuss the Elliott-Yafet mechanisms in these electron-phonon scatterings with
the ${\bm k}$$\cdot$${\bm p}$ eigenstates at the $L$ and $\Gamma$ valleys. Our
investigation of these electron-phonon scatterings are essential for the study
of the optical orientation of spin and hot-electron relaxation in germanium.
