  Phase field crystal (PFC) theory, extensively used for modelling the
structure of solids, can be derived from dynamical density functional theory
(DDFT) via a sequence of approximations. Standard derivations neglect a term of
form $\nabla\cdot[n\nabla L n]$, where $n$ is the scaled density profile and
$L$ is a linear operator. We show that this term makes a significant
contribution to the stability of the crystal, and dropping this term from the
theory forces another approximation, that of replacing the logarithmic term
from the ideal gas contribution to the free energy with its truncated Taylor
expansion, to yield a polynomial in $n$. However, the consequences of doing
this are the presence of an additional spinodal in the phase diagram, so the
liquid is predicted first to freeze and then to melt again as the density is
increased; and other periodic structures are erroneously predicted to be
thermodynamic equilibria. A second approximation is to replace $L$ by a
gradient expansion. This leads to the possibility of solutions failing to exist
above a certain value of the average density. We illustrate these conclusions
with a simple model two-dimensional fluid. The consequences of the PFC
approximations are that the phase diagram is both qualitatively incorrect, in
that it has a stripe phase, and quantitatively incorrect (by orders of
magnitude) regarding the properties of the crystal. Thus, although PFC models
are successful as phenomenological models of crystallisation, we find it
impossible to derive the PFC model as an accurate approximation to DDFT,
without introducing spurious artefacts. However, making a simple one-mode
approximation for the logarithm of the density distribution is surprisingly
accurate, which gives a tantalising hint that accurate PFC-type theories may
instead be derived for the field $\log(\rho(x))$, rather than for the density
profile itself.
