Abstract
An introduction and survey is given of some recent work on the infinitesimal dynamics of crystal frameworks, that is, of translationally periodic discrete bond-node structures in ℝd, for d = 2,3,... We discuss the rigidity matrix, a fundamental object from finite bar-joint framework theory, rigidity operators, matrix-function representations and low energy phonons. These phonons in material crystals, such as quartz and zeolites, are known as rigid unit modes, or RUMs, and are associated with the relative motions of rigid units, such as SiO4 tetrahedra in the tetrahedral polyhedral bondnode model for quartz. We also introduce semi-infinite crystal frameworks, bi-crystal frameworks and associated multi-variable Toeplitz operators.
Keywords
- Crystal framework
- rigidity operator
- matrix function
- rigid unit mode
Mathematics Subject Classification (2010). Primary 52C75; Secondary 46T20.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
L. Asimow and B. Roth, The rigidity of graphs, Trans. Amer. Math. Soc., 245 (1978), 279–289.
L. Asimow and B. Roth, Rigidity of graphs II, J. Math. Anal. Appl. 68 (1979) 171– 190.
Ch. Baerlocher and L.B. McCusker, Database of Zeolite Structures: http://www.izastructure. org/databases/
C.S. Borcea and I. Streinu, Periodic frameworks and flexibility, Proc. R. Soc. A 2010 466, 2633–2649.
R.Connelly, P.W. Fowler, S.D. Guest, B. Schulze, W.J. Whiteley, “When is a pinjointed framework isostatic?” International J. of Solids and Structures, 46 (2009), 762–773.
H.S.M. Coxeter, Regular polytopes, Dover, New York, 1973. [7] M.T. Dove, Introduction to lattice dynamics, Cambridge topics in Mineral Physics and Chemistry, C.U.P., 1993.
M.T. Dove, A.K.A. Pryde, V. Heine and K.D. Hammonds. Exotic distributions of rigid unit modes in the reciprocal spaces of framework aluminosilicates, J. Phys., Condens. Matter 19 (2007) doi:10.1088/0953-8984/19/27/275209.
M.D. Foster and M.M.J. Treacy, A database of hypothetical zeolite structures: http://www.hypotheticalzeolites.net
A.P. Giddy, M.T. Dove, G.S. Pawley, V. Heine, The determination of rigid unit modes as potential soft modes for displacive phase transitions in framework crystal structures. Acta Crystallogr., A49 (1993), 697–703.
J. Graver, B. Servatius and H. Servatius, Combinatorial rigidity, Graduate Texts in Mathematics, vol 2, Amer. Math. Soc., 1993.
K.D. Hammonds, H. Deng, V. Heine, and M.T. Dove, How floppy modes give rise to adsorption sites in zeolites, PRL 78 (1997), 3701–3704.
K.D. Hammonds, V. Heine, and M.T. Dove, Rigid-Unit Modes and the quantitative determination of the flexibility possessed by zeolite frameworks, J. Phys. Chem. 102 (1998), 1759–1767.
P.W. Fowler and S.D. Guest, A symmetry extension of Maxwell’s rule for rigidity of frames, International Journal of Solids and Structures 37 (2000), 1793–1804.
S.D. Guest and J.W. Hutchinson, On the determinacy of repetitive structures, Journal of the Mechanics and Physics of Solids, 51 (2003), 383–391.
R.G. Hutchinson and N.A. Fleck, The structural performance of the periodic truss, Journal of the Mechanics and Physics of Solids, 54 (2006) 756–782.
V. Kapko, C. Dawson, M.M.J. Treacy and M.F. Thorpe, Flexibility of ideal zeolite frameworks, Physical Chemistry Chemical Physics, DOI: 10.1039/c003977b.
A.B. Kempe, On a general method of describing plane curves of the ntℎ degree by linkwork, Proc. London Math. Soc. 7 (1876), 213–216.
G. Laman, On graphs and the rigidity of plane skeletal structures, J. Engineering Mathematics, 4 (1970), 331–340.
J. Malestein and L. Theran, Generic combinatorial rigidity of periodic frameworks, Advances in Mathematics, 233(1), 291–331, 2013.
J.C. Owen and S.C. Power, Infinite bar-joint frameworks, Proceedings of the Symposium in Applied Computing, (SAC 2009) March 8–12, 2009, Honolulu, Hawaii, USA.
J.C. Owen and S.C. Power, Frameworks, symmetry and rigidity, Inter. J. Computational Geometry and Applications, 20, (2010), 723–750.
J.C. Owen and S.C. Power, Continuous curves from infinite Kempe linkages, Bull. London Math. Soc., 41 (2009) 1105–1111.
J.C. Owen and S.C. Power, Infinite bar-joint frameworks, crystals and operator theory, New York J. Math., 17 (2011) 445–490.
S.C. Power, Polynomials for crystal frameworks and the rigid unit mode spectrum, Phil. Trans. of Roy. Soc. A, 2013, to appear.
S.C. Power, Crystal frameworks, symmetry and affinely periodic flexes, arXiv: 1103.1914, 2011.
E. Ross, B. Shulze and W. Whiteley, Finite motions from periodic frameworks with added symmetry, Intern. J. of Solids and Structures, 42 (2011), 1711–1728.
B. Schulze, Symmetric versions of Laman’s Theorem, Discrete & Computational Geometry, 44 (2010), 946–972.
B. Schulze, Block-diagonalised rigidity matrices of symmetric frameworks and applications, Contributions to Algebra and Geometry 51 (2010), 427–466.
A.P. Sutton and R.W. Balluffi, Interfaces in crystalline materials, Monographs on the physics and chemistry of materials 51, Oxford University Press, 1995.
I.P. Swainson and M.T. Dove, Low-frequency floppy modes in β-cristobalite, Phys. Rev. Letters, 71 (1993), 193–196.
W. Whiteley, The union of matroids and the rigidity of frameworks, Siam J. Discrete Math. Vol. 1 (1988), 237–255.
F. Wegner, Rigid-unit modes in tetrahedral crystals, Journal of Physics: Condensed Matter, Volume 19, Issue 40 (2007).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Basel
About this paper
Cite this paper
Power, S.C. (2014). Crystal Frameworks, Matrix-valued Functions and Rigidity Operators. In: Cepedello Boiso, M., Hedenmalm, H., Kaashoek, M., Montes Rodríguez, A., Treil, S. (eds) Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Operator Theory: Advances and Applications, vol 236. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0648-0_26
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0648-0_26
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0647-3
Online ISBN: 978-3-0348-0648-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)