Hearing the Symmetries of Crystal Lattices from the Integrated Acoustic Spectrum

Abstract

Let $C$ be a crystal and $\varphi$ be a periodic realization of it in ${\mathbb{R}}^n$, also let $L$ be the lattice group of $\varphi (C)$ 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.