Absence of diagonal force constants in cubic Coulomb crystals

TL;DR

This paper reveals that in cubic Coulomb crystals, the ion-ion potential can lack diagonal force constants, challenging the traditional spring model and affecting stability analyses across different electronic regimes.

Contribution

It demonstrates the absence of diagonal force constants in cubic Coulomb systems and explores implications across ionic, tight-binding, and nearly-free electron models.

Findings

Zero diagonal force constants in ionic model due to Poisson's law

Localized electrons stabilize crystal structures in tight-binding model

Delocalized electrons in jellium counterbalance ion-ion force constants

Abstract

The quasi-harmonic model proposes that a crystal can be modeled as atoms connected by springs. We demonstrate how this viewpoint can be misleading: a simple application of Gauss' law shows that the ion-ion potential for a cubic Coulomb system can have no diagonal harmonic contribution and so cannot necessarily be modeled by springs. We investigate the repercussions of this observation by examining three illustrative regimes: the bare ionic, density tight-binding, and density nearly-free electron models. For the bare ionic model, we demonstrate the zero elements in the force constants matrix and explain this phenomenon as a natural consequence of Poisson's law. In the density tight-binding model, we confirm that the inclusion of localized electrons stabilizes all major crystal structures at harmonic order and we construct a phase diagram of preferred structures with respect to core and valence electron radii. In the density nearly-free electron model, we verify that the inclusion of delocalized electrons, in the form of a background jellium, is enough to counterbalance the diagonal force constants matrix from the ion-ion potential in all cases and we show that a first-order perturbation to the jellium does not have a destabilizing effect. We discuss our results in connection to Wigner crystals in condensed matter, Yukawa crystals in plasma physics, as well as the elemental solids.