  The phase field theory of crystal nucleation described in [L. Granasy, T.
Borzsonyi, T. Pusztai, Phys. Rev. Lett. 88, 206105 (2002)] is applied for
nucleation in hard--sphere liquids. The exact thermodynamics from molecular
dynamics is used. The interface thickness for phase field is evaluated from the
cross--interfacial variation of the height of the singlet density peaks. The
model parameters are fixed in equilibrium so that the free energy and thickness
of the (111), (110), and (100) interfaces from molecular dynamics are
recovered. The density profiles predicted without adjustable parameters are in
a good agreement with the filtered densities from the simulations. Assuming
spherical symmetry, we evaluate the height of the nucleation barrier and the
Tolman length without adjustable parameters. The barrier heights calculated
with the properties of the (111) and (110) interfaces envelope the Monte Carlo
results, while those obtained with the average interface properties fall very
close to the exact values. In contrast, the classical sharp interface model
considerably underestimates the height of the nucleation barrier. We find that
the Tolman length is positive for small clusters and decreases with increasing
size, a trend consistent with computer simulations.
