Polynomials for crystal frameworks and the rigid unit mode spectrum
Polynomials for crystal frameworks and the rigid unit mode spectrum
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晶体骨架多项式和刚性单位模谱
DOI：
10.1098/rsta.2012.0030
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
Stephen C. Power
中科院分区：
文献类型：
--
作者：
Stephen C. Power
对于中的每个离散的平移周期杆关节框架，我们将定义在d环面上的一个矩阵值函数联系起来。的刚性单位模谱(RUM)定义为相位周期无穷小弯曲的多个相位，并证明了它对应于函数的奇点，也对应于在长波极限内能量为零的简谐激励的波矢集合。对于麦克斯韦计数平衡中的晶体骨架，对应于平方，的行列式产生唯一的多元多项式。对于理想沸石，d-环面上的零点的代数变化与朗姆谱重合。矩阵函数与理想框架刚性和柔性的其他方面有关，特别是得到了一个关于超细胞周期软模数目的显式公式。对于二维沸石骨架和三维沸石骨架，给出了最大软模性质(N阶)的直接证明。特别是，立方对称方钠沸石骨架和其他一些理想化的沸石就是如此。
To each discrete translationally periodic bar-joint framework in , we associate a matrix-valued function defined on the d-torus. 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. To a crystal framework in Maxwell counting equilibrium, which corresponds to being square, the determinant of gives rise to a unique multi-variable polynomial . For ideal zeolites, the algebraic variety of zeros of on the d-torus coincides with the RUM spectrum. The matrix function is related to other aspects of idealized 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 two and three, 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 some other idealized zeolites.