  We investigate the intrinsic spin relaxation of conduction electrons in
germanium due to electron-phonon scattering. We derive intravalley and
intervalley spin-flip matrix elements for a general spin orientation and
quantify the resulting anisotropy in spin relaxation. The form of the
intravalley spin-flip matrix element is derived from the eigenstates of a
compact spin-dependent $\mathbf{k}$$\cdot$$\mathbf{p}$ Hamiltonian in the
vicinity of the $L$ point (where thermal electrons are populated in Ge). Spin
lifetimes from analytical integrations of the intravalley and intervalley
matrix elements show excellent agreement with independent results from
elaborate numerical methods.
