Crystal flex bases and the RUM spectrum

@article{Badri2018CrystalFB,
  title={Crystal flex bases and the RUM spectrum},
  author={Ghada Badri and Derek Kitson and S. C. Power},
  journal={Proceedings of the Edinburgh Mathematical Society},
  year={2018},
  volume={64},
  pages={735 - 761}
}
A theory of infinite spanning sets and bases is developed for the first-order flex space of an infinite bar-joint framework, together with space group symmetric versions for a crystallographic bar-joint framework ${{\mathcal {C}}}$. The existence of a crystal flex basis for ${{\mathcal {C}}}$ is shown to be closely related to the spectral analysis of the rigid unit mode (RUM) spectrum of ${{\mathcal {C}}}$ and an associated geometric flex spectrum. Additionally, infinite spanning sets and bases… 
View on Cambridge Press
[PDF] Semantic Reader
1 Citations

One Citation

Linear zero mode spectra for quasicrystals

  • S. Power
  • Mathematics
    Journal of Mathematical Analysis and Applications
  • 2022

References

SHOWING 1-10 OF 22 REFERENCES

The first-order flexibility of a crystal framework

Four sets of necessary and sufficient conditions are obtained for the first-order rigidity of a crystallographic bond-node framework $\C$ in $\bR^d$. In particular, an extremal rank characterisation…

The first-order flexibility of a crystallographic framework

Polynomials for crystal frameworks and the rigid unit mode spectrum

  • S. Power
  • Mathematics
    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2014
The rigid unit mode (RUM) spectrum of is defined in terms of the multi-phases of phase-periodic infinitesimal flexes and is shown to correspond to the singular points of the function and also to the set of wavevectors of harmonic excitations which have vanishing energy in the long wavelength limit.

The Almost Periodic Rigidity of Crystallographic Bar-Joint Frameworks

The almost periodic infinitesimal flexes of C are characterised in terms of a matrix-valued function, ΦC(z), on the d-torus, Td, determined by a full rank translation symmetry group and an associated motif of joints and bars.

Infinite bar-joint frameworks, crystals and operator theory

A theory of fl exibility and rigidity is developed for general infinite bar-joint frameworks (G; p). Determinations of nondeformability through vanishing flexibility are obtained as well as…

Rigid-unit modes in tetrahedral crystals

  • F. Wegner
  • Mathematics
    Journal of physics. Condensed matter : an Institute of Physics journal
  • 2007
The ‘rigid-unit mode’ (RUM) model requires unit blocks, in our case tetrahedra of SiO4 groups, to be rigid within first order of the displacements of the oxygen ions. The wavevectors of the lattice…

The Rigidity of Infinite Graphs

A rigidity theory is developed for countably infinite simple graphs and analogous results are obtained for the classical non-Euclidean $$\ell ^q$$ℓq norms.

The Determination of Rigid-Unit Modes as Potential Soft Modes for Displacive Phase Transitions in Framework Crystal Structures

This paper describes a computational method for the determination of all possible phonon modes in framework crystal structures that leave the fundamental structural units (tetrahedra and octahedra)…

Rigid unit modes in framework silicates

Abstract We describe a model for framework silicates in which the SiO4 (and AlO4) tetrahedra are treated as perfectly rigid and freely jointed. From this model we are able to identify low-energy…

Bulk-Edge Correspondence for Two-Dimensional Topological Insulators

Topological insulators can be characterized alternatively in terms of bulk or edge properties. We prove the equivalence between the two descriptions for two-dimensional solids in the single-particle…