  An investigation of possibilities for superconductivity mediated by spin
fluctuations in some elementary metals is motivated by the recent discovery of
superconductivity in the hcp high-pressure phase of iron.
  The electronic structure, the electron-phonon coupling ($\lambda_{ph}$) and
the coupling due to spin-fluctuations ($\lambda_{sf}$) are calculated for
different phases and different volumes for four elementary metals. The results
show that such possibilities are best for systems near, but on the non-magnetic
side of, a magnetic instability. Fcc Ni, which show stable magnetism over a
wide pressure range, is not interesting in this respect. Ferro- and
antiferro-magnetic fluctuations in hcp Fe contribute to a relatively strong
coupling in the pressure range where superconductivity is observed. The absence
of fluctuations at large q-vectors makes fcc Pd only moderately interesting
despite its large exchange enhancement for q=0. Fcc Co at high pressure ($\sim$
0.5 Mbar) behaves as an improved version of Pd, where the fluctuations extend
to larger q. The estimations of T$_C$, which reproduce the experimental
situation in Fe quite well, suggest a measurable T$_C$ for the high-pressure
phase of fcc Co, while the estimate is lower for the ambient-pressure phase of
fcc Pd.
