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Polynomials for crystal frameworks and the rigid unit mode spectrum

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Polynomials for crystal frameworks and the rigid unit mode spectrum. / Power, Stephen.

In: Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences, Vol. 372, No. 2008, 20120030, 13.02.2014.

Research output: Contribution to journalJournal articlepeer-review

Harvard

Power, S 2014, 'Polynomials for crystal frameworks and the rigid unit mode spectrum', Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences, vol. 372, no. 2008, 20120030. https://doi.org/10.1098/rsta.2012.0030

APA

Power, S. (2014). Polynomials for crystal frameworks and the rigid unit mode spectrum. Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences, 372(2008), [20120030]. https://doi.org/10.1098/rsta.2012.0030

Vancouver

Power S. Polynomials for crystal frameworks and the rigid unit mode spectrum. Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences. 2014 Feb 13;372(2008). 20120030. https://doi.org/10.1098/rsta.2012.0030

Author

Power, Stephen. / Polynomials for crystal frameworks and the rigid unit mode spectrum. In: Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences. 2014 ; Vol. 372, No. 2008.

Bibtex

@article{0a66583e79aa47ff9de2961cb12b8266,
title = "Polynomials for crystal frameworks and the rigid unit mode spectrum",
abstract = "To each discrete translationally periodic bar-joint framework $\C$ in $\bR^d$we associate a matrix-valued function $\Phi_\C(z)$ defined on the $d$-torus. The rigid unit mode spectrum $\Omega(\C)$ of $\C$ 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 $z \to \rank \Phi_\C(z)$and also to the set of wave vectors of harmonic excitations which have vanishing energy in the long wavelength limit. To a crystal framework in Maxwell counting equilibrium the determinant of $\Phi_\C(z)$ is defined and gives rise to a unique multi-variable polynomial $p_\C(z_1,\dots ,z_d)$. For ideal zeolites the algebraic variety of zeros of $p_\C(z)$ on the $d$-torus coincides with the RUM spectrum. The matrix function is related to other aspects of idealised framework rigidity and flexibility and in particular leads to an explicit formula for the number of supercell-periodic floppy modes.In the case of certain zeolite frameworks in dimensions $2$ and $3$ direct proofs are given to show the maximal floppy mode property (order $N$). In particular this is the case for the cubic symmetry sodalite framework and other idealised zeolites.",
keywords = "Crystallographic framework, Rigid unit mode, rigidity operator , crystal polynomial ",
author = "Stephen Power",
year = "2014",
month = feb,
day = "13",
doi = "10.1098/rsta.2012.0030",
language = "English",
volume = "372",
journal = "Philosophical Transactions A: Mathematical, Physical and Engineering Sciences ",
issn = "1364-503X",
publisher = "Royal Society of London",
number = "2008",

}

RIS

TY - JOUR

T1 - Polynomials for crystal frameworks and the rigid unit mode spectrum

AU - Power, Stephen

PY - 2014/2/13

Y1 - 2014/2/13

N2 - To each discrete translationally periodic bar-joint framework $\C$ in $\bR^d$we associate a matrix-valued function $\Phi_\C(z)$ defined on the $d$-torus. The rigid unit mode spectrum $\Omega(\C)$ of $\C$ 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 $z \to \rank \Phi_\C(z)$and also to the set of wave vectors of harmonic excitations which have vanishing energy in the long wavelength limit. To a crystal framework in Maxwell counting equilibrium the determinant of $\Phi_\C(z)$ is defined and gives rise to a unique multi-variable polynomial $p_\C(z_1,\dots ,z_d)$. For ideal zeolites the algebraic variety of zeros of $p_\C(z)$ on the $d$-torus coincides with the RUM spectrum. The matrix function is related to other aspects of idealised framework rigidity and flexibility and in particular leads to an explicit formula for the number of supercell-periodic floppy modes.In the case of certain zeolite frameworks in dimensions $2$ and $3$ direct proofs are given to show the maximal floppy mode property (order $N$). In particular this is the case for the cubic symmetry sodalite framework and other idealised zeolites.

AB - To each discrete translationally periodic bar-joint framework $\C$ in $\bR^d$we associate a matrix-valued function $\Phi_\C(z)$ defined on the $d$-torus. The rigid unit mode spectrum $\Omega(\C)$ of $\C$ 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 $z \to \rank \Phi_\C(z)$and also to the set of wave vectors of harmonic excitations which have vanishing energy in the long wavelength limit. To a crystal framework in Maxwell counting equilibrium the determinant of $\Phi_\C(z)$ is defined and gives rise to a unique multi-variable polynomial $p_\C(z_1,\dots ,z_d)$. For ideal zeolites the algebraic variety of zeros of $p_\C(z)$ on the $d$-torus coincides with the RUM spectrum. The matrix function is related to other aspects of idealised framework rigidity and flexibility and in particular leads to an explicit formula for the number of supercell-periodic floppy modes.In the case of certain zeolite frameworks in dimensions $2$ and $3$ direct proofs are given to show the maximal floppy mode property (order $N$). In particular this is the case for the cubic symmetry sodalite framework and other idealised zeolites.

KW - Crystallographic framework

KW - Rigid unit mode

KW - rigidity operator

KW - crystal polynomial

U2 - 10.1098/rsta.2012.0030

DO - 10.1098/rsta.2012.0030

M3 - Journal article

VL - 372

JO - Philosophical Transactions A: Mathematical, Physical and Engineering Sciences

JF - Philosophical Transactions A: Mathematical, Physical and Engineering Sciences

SN - 1364-503X

IS - 2008

M1 - 20120030

ER -