  A recent experiment by Shimizu et al. has provided evidence of a
superconducting phase in hcp Fe under pressure. To study the
pressure-dependence of this superconducting phase we have calculated the phonon
frequencies and the electron-phonon coupling in hcp Fe as a function of the
lattice parameter, using the linear response (LR) scheme and the full potential
linear muffin-tin orbital (FP-LMTO) method. Calculated phonon spectra and the
Eliashberg functions $\alpha^2 F$ indicate that conventional s-wave
electron-phonon coupling can definitely account for the appearance of the
superconducting phase in hcp Fe. However, the observed change in the transition
temperature with increasing pressure is far too rapid compared with the
calculated results. For comparison with the linear response results, we have
computed the electron-phonon coupling also by using the rigid muffin-tin (RMT)
approximation. From both the LR and the RMT results it appears that
electron-phonon interaction alone cannot explain the small range of volume over
which superconductivity is observed. It is shown that
ferromagnetic/antiferromagnetic spin fluctuations as well as scattering from
magnetic impurities (spin-ordered clusters) can account for the observed values
of the transition temperatures but cannot substantially improve the agreeemnt
between the calculated and observed presure/volume range of the superconducting
phase. A simplified treatment of p-wave pairing leads to extremely small ($\leq
10^{-2}$ K) transition temperatures. Thus our calculations seem to rule out
both $s$- and $p$- wave superconductivity in hcp Fe.
