  The calculation of crystal structure from X-ray diffraction data requires
that the phases of the ``structure factors'' (Fourier coefficients) determined
by scattering be deduced from the absolute values of those structure factors.
Motivated by a question of Herbert Hauptman, we consider the problem of
determining phases by direct algebraic means in the case of crystal structures
with $n$ equal atoms in the unit cell, with $n$ small. We rephrase the problem
as a question about multiplicative invariants for a particular finite group
action. We show that the absolute values form a generating set for the field of
invariants of this action, and consider the problem of making this theorem
constructive and practical; the most promising approach for deriving explicit
formulas uses SAGBI bases.
