  We generalize, and then use, a recently introduced formalism to study thermal
fluctuations of atomic displacements in several two and three dimensional
crystals. We study both close packed as well as open crystals with multi atom
bases. Atomic displacement fluctuations in a solid, once coarse-grained over
some neighborhood may be decomposed into two mutually orthogonal components. In
any dimension $d$ there are always $d^2$ {\em affine} displacements
representing local strains and rotations of the ideal reference configuration.
In addition, there exists a number of {\em non-affine} localized displacement
modes that cannot be represented as strains or rotations. The number of these
modes depends on $d$ and the size of the coarse graining region. All
thermodynamic averages and correlation functions concerning the affine and
non-affine displacements may be computed within a harmonic theory. We show that
for compact crystals, such as the square and triangular in $d=2$ and the
simple, body-centered and face-centered cubic crystals in $d=3$, a single set
of $d-$fold degenerate modes always dominate the non-affine sub-space and are
separated from the rest by a large gap. These modes may be identified with
specific precursor configurations that lead to lattice defects. Deformation
mechanisms such as lattice slips and stacking faults in close packed crystals
can also be understood within this framework. The qualitative features of these
conclusions are expected to be independent of the details of the atomic
interactions.
