  We study the thermodynamics of helium at densities relevant for white dwarf
physics. We find evidence that, as the temperature is increased, there is first
a first order transition between two superconducting phases followed by a
second order transition to the normal state. These transitions occur, for
realistic densities, at temperatures below the crystallization temperature and
the crystalline state is likely to remain as the true ground state of the
system. The calculations are performed with a screening but non-dynamical
electron background and we comment on the impact of this and other
approximations to our result.
