The Microscopic Origin of the Macroscopic Dielectric Permittivity of Crystals: A Mathematical Viewpoint

TL;DR

This paper offers a mathematical analysis of how microscopic atomic structures influence the macroscopic dielectric permittivity in crystals, focusing on the Adler-Wiser formula and electronic structure models.

Contribution

It provides a rigorous mathematical framework for understanding the Adler-Wiser formula and reviews models of electronic structure in finite and infinite crystal systems.

Findings

Mathematical derivation of the Adler-Wiser formula

Analysis of electronic structure models for crystals

Insights into the relation between atomic structure and dielectric properties

Abstract

The purpose of this paper is to provide a mathematical analysis of the Adler-Wiser formula relating the macroscopic relative permittivity tensor to the microscopic structure of the crystal at the atomic level. The technical level of the presentation is kept at its minimum to emphasize the mathematical structure of the results. We also briefly review some models describing the electronic structure of finite systems, focusing on density operator based formulations, as well as the Hartree model for perfect crystals or crystals with a defect.