LAUSR

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 and open crystals with multiatom 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} affine displacements representing local strains and rotations of the ideal reference configuration. In addition, there exist a number of nonaffine 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 nonaffine displacements may be computed within harmonic theory. We show that for compact crystals, such as the square and triangular crystals in d=2 and the simple body-centered-cubic and face-centered-cubic crystals in d=3, a single set of d-fold degenerate modes always dominates the nonaffine subspace and is separated from the rest by a large gap. These modes may be identified with specific precursor configurations that lead to lattice defects. In open crystals, such as the honeycomb and kagome lattices, there is no prominent gap, although soft nonaffine modes continue to be associated with known floppy modes representing localized defects. Higher-order coupling between affine and nonaffine components of the displacements quantifies the tendency of the lattice to be destroyed by large homogeneous strains. We show that this coupling is larger by almost an order of magnitude for open lattices as compared to compact ones. 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.