Regularity and Locality of Point Defects in Multilattices

Abstract

We formulate a model for a point defect embedded in a homogeneous multilattice crystal with an empirical interatomic potential interaction. Under a natural, phonon stability assumption we quantify the decay of the long-range elastic fields with increasing distance from the defect.These decay estimates are an essential ingredient in quantifying approximation errors in coarse-grained models and in the construction of optimal numerical methods for approximating crystalline defects.

Geometry equilibration of crystalline defects in quantum and atomistic descriptions

“…We can extend the assumptions to admit multiple species of atoms, e.g. [29]. However, the generalisation will introduce notation that is significantly more complex, so we will not pursue this here.…”

“…Much less is understood about the related geometry optimisation (or, lattice relaxation) problem, [20,12,29] in particular the coupling betweeen electronic and geometry relaxation is essentially open. The present work addresses this coupling and establishes some key results: we give general conditions under which the lattice relaxation problem can be formulated as a variational problem on a Hilbert space, and we establish sharp rates of decay of the discrete equilibrium configurations.…”

Analysis of an atomistic model for anti-plane fracture

“…A general approach to describe a single localised defect embedded in a homogeneous host crystal was rigorously formalised in [HO14, EOS16,BBO17] for point defects and straight dislocations in Bravais lattices, then extended in [OO17] to point defects in multilattices. The overarching idea is to use a continuum model to specify the far-field behaviour where continuum theories such as continuum linear elasticity (CLE) are accurate, while employing the underlying atomistic model to capture the details of the defect core.…”

Some Remarks on Preconditioning Molecular Dynamics

“…In order to estimate σ 2 f further we consider again the d-dimensional lattice model (2.8) and P given by (2.10). For d ≥ 2, since P is a homogeneous discrete elliptic operator, we know from [27] that…”