Polynomials for Crystal Frameworks and the Rigid Unit Mode Spectrum

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, which corresponds to \Phi_\C(z) being square, the determinant of \Phi_\C(z) 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 some other idealised zeolites.Comment: Final version with new examples and figures, and with clearer streamlined proof

Three-periodic nets and tilings: natural tilings for nets

Rules for determining a unique natural tiling that carries a given three-periodic net as its 1-skeleton are presented and justified. A computer implementation of the rules and their application to tilings for zeolite nets and for the nets of the RCSR database are described

Nets with collisions (unstable nets) and crystal chemistry

Nets in which different vertices have identical barycentric coordinates (i.e. have collisions) are called unstable. Some such nets have automorphisms that do not correspond to crystallographic symmetries and are called non-crystallographic. Examples are given of nets taken from real crystal structures which have embeddings with crystallographic symmetry in which colliding nodes either are, or are not, topological neighbors (linked) and in which some links coincide. An example is also given of a crystallographic net of exceptional girth (16), which has collisions in barycentric coordinates but which also has embeddings without collisions with the same symmetry. In this last case the collisions are termed unforced

Streaming Algorithm for Euler Characteristic Curves of Multidimensional Images

We present an efficient algorithm to compute Euler characteristic curves of gray scale images of arbitrary dimension. In various applications the Euler characteristic curve is used as a descriptor of an image. Our algorithm is the first streaming algorithm for Euler characteristic curves. The usage of streaming removes the necessity to store the entire image in RAM. Experiments show that our implementation handles terabyte scale images on commodity hardware. Due to lock-free parallelism, it scales well with the number of processor cores. Our software---CHUNKYEuler---is available as open source on Bitbucket. Additionally, we put the concept of the Euler characteristic curve in the wider context of computational topology. In particular, we explain the connection with persistence diagrams

Reticular synthesis and the design of new materials

The long-standing challenge of designing and constructing new crystalline solid-state materials from molecular building blocks is just beginning to be addressed with success. A conceptual approach that requires the use of secondary building units to direct the assembly of ordered frameworks epitomizes this process: we call this approach reticular synthesis. This chemistry has yielded materials designed to have predetermined structures, compositions and properties. In particular, highly porous frameworks held together by strong metal-oxygen-carbon bonds and with exceptionally large surface area and capacity for gas storage have been prepared and their pore metrics systematically varied and functionalized.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/62718/1/nature01650.pd

A route to high surface area, porosity and inclusion of large molecules in crystals

One of the outstanding challenges in the field of porous materials is the design and synthesis of chemical structures with exceptionally high surface areas(1). Such materials are of critical importance to many applications involving catalysis, separation and gas storage. The claim for the highest surface area of a disordered structure is for carbon, at 2,030 m(2) g(-1) (ref. 2). Until recently, the largest surface area of an ordered structure was that of zeolite Y, recorded at 904 m(2) g(-1) (ref. 3). But with the introduction of metal-organic framework materials, this has been exceeded, with values up to 3,000 m(2) g(-1) (refs 4-7). Despite this, no method of determining the upper limit in surface area for a material has yet been found. Here we present a general strategy that has allowed us to realize a structure having by far the highest surface area reported to date. We report the design, synthesis and properties of crystalline Zn4O(1,3,5-benzenetribenzoate)(2), a new metal-organic framework with a surface area estimated at 4,500 m(2) g(-1). This framework, which we name MOF-177, combines this exceptional level of surface area with an ordered structure that has extra-large pores capable of binding polycyclic organic guest molecules-attributes not previously combined in one material.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/62609/1/nature02311.pd

Three-dimensional lanthanide-organic frameworks based on di-, tetra-, and hexameric clusters

Three-dimensional lanthanide-organic frameworks formulated as (CH3)2NH2[Ln(pydc)2] · 1/2H2O [Ln3+ ) Eu3+ (1a) or Er3+ (1b); pydc2- corresponds to the diprotonated residue of 2,5-pyridinedicarboxylic acid (H2pydc)], [Er4(OH)4(pydc)4(H2O)3] ·H2O (2), and [PrIII 2PrIV 1.25O(OH)3(pydc)3] (3) have been isolated from typical solvothermal (1a and 1b in N,N-dimethylformamide - DMF) and hydrothermal (2 and 3) syntheses. Materials were characterized in the solid state using single-crystal X-ray diffraction, thermogravimetric analysis, vibrational spectroscopy (FT-IR and FT-Raman), electron microscopy, and CHN elemental analysis. While synthesis in DMF promotes the formation of centrosymmetric dimeric units, which act as building blocks in the construction of anionic ∞ 3{[Ln(pydc)2]-} frameworks having the channels filled by the charge-balancing (CH3)2NH2 + cations generated in situ by the solvolysis of DMF, the use of water as the solvent medium promotes clustering of the lanthanide centers: structures of 2 and 3 contain instead tetrameric [Er4(μ3-OH)4]8+ and hexameric |Pr6(μ3-O)2(μ3-OH)6| clusters which act as the building blocks of the networks, and are bridged by the H2-xpydcx- residues. It is demonstrated that this modular approach is reflected in the topological nature of the materials inducing 4-, 8-, and 14-connected uninodal networks (the nodes being the centers of gravity of the clusters) with topologies identical to those of diamond (family 1), and framework types bct (for 2) and bcu-x (for 3), respectively. The thermogravimetric studies of compound 3 further reveal a significant weight increase between ambient temperature and 450 °C with this being correlated with the uptake of oxygen from the surrounding environment by the praseodymium oxide inorganic core

Tilings And Symbols -- A Report On The Uses Of Symbolic . . .

In this note, a short report is presented concerning a symbolic approach to tiling theory which has been developed in Bielefeld over the last 15 years