  The spin-flip (SF) Eliashberg function is calculated from first-principles
for ferromagnetic Ni to accurately establish the contribution of Elliott-Yafet
electron-phonon SF scattering to Ni's femtosecond laser-driven demagnetization.
This is used to compute the SF probability and demagnetization rate for
laser-created thermalized as well as non-equilibrium electron distributions.
Increased SF probabilities are found for thermalized electrons, but the induced
demagnetization rate is extremely small. A larger demagnetization rate is
obtained for {non-equilibrium} electron distributions, but its contribution is
too small to account for femtosecond demagnetization.
