  We investigate from first-principles theory and experiment the generation of
phonons on picosecond timescales and the relaxation of carriers in multiple
conduction band valleys of photo-excited Ge by inter-valley electron-phonon
scattering. We provide a full description of the phonon and electron relaxation
dynamics without adjustable parameters. Simulations of the time-evolution of
phonon populations, based on first-principles band structure and
electron-phonon and phonon-phonon matrix elements, are compared with data from
time-resolved x-ray diffuse scattering experiments, performed at the LCLS x-ray
free-electron laser facility, which measures the diffuse scattering intensity
following photo-excitation by a 50 fs near-infrared optical pulse. Comparing
calculations and measurements show that the intensity of the non-thermal x-ray
diffuse scattering signal, that is observed to grow substantially near the
L-point of the Brillouin zone over 3-5 ps, is due to phonons generated by
scattering of carriers between the $\Delta$ and L valleys. Non-thermal phonon
populations throughout the Brillouin zone are observed and simulated from first
principles without adjustable parameters for times up to 10 ps. With inclusion
of phonon decay through 3-phonon processes, the simulations also account for
other non-thermal features observed in the x-ray diffuse scattering intensity,
which are due to anharmonic phonon-phonon scattering of the phonons initially
generated by electron-phonon scattering.
