  Four sets of necessary and sufficient conditions are obtained for the
first-order rigidity of a periodic bond-node framework \C in R^d which is of
crystallographic type. In particular, an extremal rank characterisation is
obtained which incorporates a multi-variable matrix-valued transfer function
\Psi_\C(z) defined on the product space C^d_* = (C\{0})^d. In general the
first-order flex space is shown to be the closed linear span of polynomially
weighted geometric velocity fields whose geometric multi-factors in C^d_* lie
in a finite set. Paradoxically, first-order rigid crystal frameworks may
possess nontrivial nondifferentiable continuous motions. The examples given are
associated with aperiodic displacive phase transitions between periodic states.
