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Stability of Objective Structures: General Criteria and Applications

Bernd Schmidt, Martin Steinbach

TL;DR

Stability of Objective Structures develops a Fourier-analytic framework to assess stability of objective structures, extending lattice results to nonperiodic, symmetry-driven systems. The approach expresses stability constants as lower bounds on the Hessian $E''(\chi_{\mathcal{G}}x_0)$ via representations of the underlying symmetry group and the interaction potential, yielding explicit computable formulas in the Fourier domain. The main contributions are explicit stability criteria and second-order energy bounds, an implementable stability algorithm, and demonstrations on a chain model and a chiral carbon nanotube, highlighting the method's ability to handle buckling and chirality. This provides a rigorous, practically applicable tool for verifying stability of complex nanoscale assemblies and for guiding design of objective-structure materials.

Abstract

We develop a general stability analysis for objective structures, which constitute a far reaching generalization of crystal lattice systems. We show that these particle systems, although in general neither periodic nor space filling, allow for the identification of stability constants in terms of representations of the underlying symmetry group and interaction potentials. Our main results provide general stability criteria and second order energy bounds for equilibrium configurations. In particular, a general computational algorithm to test objective structures for their stability is derived. By way of example we show that our method can be applied to verify the stability of carbon nanotubes with chirality.

Stability of Objective Structures: General Criteria and Applications

TL;DR

Stability of Objective Structures develops a Fourier-analytic framework to assess stability of objective structures, extending lattice results to nonperiodic, symmetry-driven systems. The approach expresses stability constants as lower bounds on the Hessian via representations of the underlying symmetry group and the interaction potential, yielding explicit computable formulas in the Fourier domain. The main contributions are explicit stability criteria and second-order energy bounds, an implementable stability algorithm, and demonstrations on a chain model and a chiral carbon nanotube, highlighting the method's ability to handle buckling and chirality. This provides a rigorous, practically applicable tool for verifying stability of complex nanoscale assemblies and for guiding design of objective-structure materials.

Abstract

We develop a general stability analysis for objective structures, which constitute a far reaching generalization of crystal lattice systems. We show that these particle systems, although in general neither periodic nor space filling, allow for the identification of stability constants in terms of representations of the underlying symmetry group and interaction potentials. Our main results provide general stability criteria and second order energy bounds for equilibrium configurations. In particular, a general computational algorithm to test objective structures for their stability is derived. By way of example we show that our method can be applied to verify the stability of carbon nanotubes with chirality.
Paper Structure (20 sections, 28 theorems, 181 equations, 2 figures)

This paper contains 20 sections, 28 theorems, 181 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Objective structures
  3. Discrete subgroups of the Euclidean group and their orbits
  4. Deformations and local rigidity seminorms
  5. Energy, criticality and stability
  6. Configurational energy and stability constants
  7. Main results: Stability criteria and second order bounds
  8. A stability algorithm and applications
  9. The general algorithm
  10. Example: a chain of atoms
  11. Example: a carbon nanotube
  12. Proofs
  13. Derivatives of the energy
  14. A sufficient condition for a local minimum
  15. Second order bounds on the energy
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $\mathcal{G}<\mathop{\mathrm{E}}\nolimits(d)$ be discrete, $d\in{\mathbb N}$. There exist $d_1,d_2\in{\mathbb N}_0$ such that $d=d_1+d_2$, a $d_2$-dimensional space group $\mathcal{S}$ and a discrete group $\mathcal{G}'<\operatorname{{\mathrm O}}(d_1)\oplus \mathcal{S}$ such that $\mathcal{G}$ i

Figures (2)

  • Figure 2: As described in Example \ref{['Example:Nanotube']}, the orbit of the point $x_{a_0}$ under the action of the group $\mathcal{G}_{a_0,\alpha_0}$ is a $(5,1)$ nanotube. We have a natural bijection between the group elements and the atoms. $x_{a_0}$ and its nearest neighbor bonds to atoms in $\mathcal{N}\cdot x_{a_0}$ are highlighted.
  • Figure 5: For the nanotube as described in Example \ref{['Example:Nanotube']}, the point $(a^*,\alpha(a^*))$ and the graph of the angle $\alpha(a)$ dependent on the scale factor $a$ are plotted on the left. The point $(a^*,0)$ and the graphs of $\mathop{\mathrm{Relative\,difference}}\nolimits\lparen[\rparen]{\lambda_{\textnormal{a}}(a,\alpha^*),\lambda_{\textnormal{a}}(a,\alpha_a)}$ (blue) and $\mathop{\mathrm{Relative\,difference}}\nolimits\lparen[\rparen]{\lambda_{\textnormal{a},0,0}(a,\alpha^*),\lambda_{\textnormal{a},0,0}(a,\alpha_a)}$ (orange) dependent on the scale factor $a$ are plotted on the right.

Theorems & Definitions (85)

  • Theorem 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5: The Plancherel formula
  • Definition 2.6
  • Lemma 2.7
  • Definition 2.8
  • Remark 2.9
  • Lemma 2.10
  • ...and 75 more