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A Minimum Property for Cuboidal Lattice Sums

Shaun Cooper, Peter Schwerdtfeger

TL;DR

The paper addresses the minimisation of Epstein's zeta function over a natural one-parameter family of cuboidal lattices in three dimensions, showing a local minimum at the body-centered cubic configuration for all $s>3/2$. It constructs the family $oldsymbol\Lambda(u,v)$ with $A=u^2/v^2$, normalises to unit minimum distance, derives the Gram matrix and the quadratic form $g(A;i,j,k)$, and expresses the Epstein zeta function as $L(A;s)=\sum'\bigl(\frac{A+1}{A(i+j)^2+(j+k)^2+(i+k)^2}\bigr)^s$. A central result is that $\frac{\partial}{\partial A}L(A;s)\big|_{A=1/2}=0$ and $\frac{\partial^2}{\partial A^2}L(A;s)\big|_{A=1/2}>0$ for $s>3/2$, establishing a local minimum at $A=1/2$ corresponding to bcc. The work also clarifies limiting behavior ($A\to\infty$ yields a lattice congruent to $\mathbb{Z}^2$) and situates the result among known lattice-density configurations, highlighting the physical relevance for lattice sums in models such as Lennard–Jones potentials.

Abstract

We analyse a family of lattices considered by Conway and Sloane and show that the corresponding Epstein zeta function attains a local minimum for the body-centred cubic lattice.

A Minimum Property for Cuboidal Lattice Sums

TL;DR

The paper addresses the minimisation of Epstein's zeta function over a natural one-parameter family of cuboidal lattices in three dimensions, showing a local minimum at the body-centered cubic configuration for all . It constructs the family with , normalises to unit minimum distance, derives the Gram matrix and the quadratic form , and expresses the Epstein zeta function as . A central result is that and for , establishing a local minimum at corresponding to bcc. The work also clarifies limiting behavior ( yields a lattice congruent to ) and situates the result among known lattice-density configurations, highlighting the physical relevance for lattice sums in models such as Lennard–Jones potentials.

Abstract

We analyse a family of lattices considered by Conway and Sloane and show that the corresponding Epstein zeta function attains a local minimum for the body-centred cubic lattice.
Paper Structure (4 sections, 1 theorem, 21 equations, 2 figures, 1 table)

This paper contains 4 sections, 1 theorem, 21 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Cuboidal lattices
  3. A minimum property
  4. The limiting cases $\boldsymbol{A\rightarrow 0}$ and $\boldsymbol{A\rightarrow\infty}$

Key Result

Theorem 1

Suppose $s>3/2$. The Epstein zeta function $L(A;s)$ defined by E:Epstein satisfies

Figures (2)

  • Figure 1: Graph of $\Delta$ as a function of $A$ given by \ref{['E:density']}.
  • Figure 2: Graphs of $y=L(A;s)$ for $1/3\leq A \leq 1$ given by \ref{['E:Epstein']} for (from top to bottom) $s=3$, ${s=6}$, ${s=20}$ and $s=\infty$. In the limiting case $s\rightarrow\infty$ we have $L(A;\infty)=\;$kiss$(A)$ where kiss$(A)$ is the kissing number as a function of $A$ as given in Table \ref{['T:1']}.

Theorems & Definitions (1)

  • Theorem 1