Hearing the Symmetries of Crystal Lattices from the Integrated Acoustic Spectrum

TL;DR

This paper introduces the concept of the acoustic spectrum of crystal lattices and provides an algebraic method to determine lattice symmetries from spectral data, especially for crystals with uniform atomic force constants.

Contribution

It formalizes the acoustic spectrum of crystal lattices and offers an algebraic approach to identify lattice symmetries from spectral information, extending understanding of crystal vibrational properties.

Findings

Defined the acoustic spectrum for crystal lattices.

Provided an algebraic method to recover lattice symmetries.

Solved the spectral problem for crystals with uniform atomic force constants.

Abstract

Let $C$ be a crystal and $\phi$ be a periodic realization of it in $\mathbb{R}^n$, also let $L$ be the lattice group of $\phi(C)$ which preserves the covering space nature of crystal lattices. In this article, firstly, we define the concept of acoustic spectrum of the crystal lattice C and then provide the algebraic formalism of the question of finding the frequencies of the torus $\mathbb{R}^n/L$, when the set of acoustic spectrum is known. An answer for crystals with uniform atomic force constants is given.