Interactions and Elastic Moduli

A crystal's linear elastic response to small distortions is described by its
Lamé coefficients
and
[32].
If we consider only nearest-neighbor interactions in an FCC crystal
characterized by isotropic pair interactions, *U*(*r*),
we find

and

The bulk modulus is related to and the shear modulus through

For the screened-Coulomb FCC crystal,

The shear modulus appears to vanish when , the golden mean. However, interactions would be sufficiently long ranged under these conditions that second- and higher-nearest-neighbor interactions ought to be taken into account.

Substituting values for the dense crystal into Eqs. (30) and (31) we obtain and . Even these fabulously small elastic moduli are enormous compared to the values for the dilute crystal, and . Since we have ignored next-nearest-neighbor interactions and entropic contributions to the isotropic compressibility, these values should be considered lower bounds on the elastic moduli for defect-free colloidal crystals.