Geometric rigidity and crystallization

May 10th, 2005 at 14:00 in HG G43 (HWZ)

Given by: Gero Friesecke (Warwick)

Summary: Why are the ground states of so many materials
crystalline? That is, why do atoms prefer to arrange themselves
periodically, or nearly periodically? In joint work with F.Theil, we study
this question in the context of minimizing a Lennard-Jones type pair
potential model over atomic positions.

For appropriate classes of potentials, we show that local minimizers
over a uniform neighbourhood of a finite exact crystal (i.e., a finite
subset of a periodic lattice) are crystalline, in the sense that the
mean square deviation of the atomic positions from those in the
exact crystal is only of the order of the surface area of the crystal.
This means that `up to a surface layer', the optimal configuration
is exactly periodic. Easy examples show that the order of the error
is optimal in system size, due to surface relaxation.

The results rely on a recent sharp rigidity theorem (joint with R.D.James
and S.Mueller) which may be viewed, in a sense which will be explained
in the talk, as `continuum crystallization': for any Sobolev function in
H1 from a domain in R^n to R^n, the L2 distance of the gradient from
the nearest constant rotation is bounded by the L2 distance from the
set of all rotations.