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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 Journal/Magazine › Journal article › peer-review
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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 -