Alan L. Mackay

Crystallography and the penrose pattern

The Penrose pattern is a tiling of two-dimensional and of three-dimensional space by identical tiles of two kinds (acute and obtuse rhombi with α = 72° and 144° in two dimensions and acute and obtuse rhombohedra with α = 63.43° and 116.57° in three dimensions). The two-dimensional pattern is a section through that in three dimensions. When joining (or recursion) rules are prescribed, the pattern is unique and non-periodic. It has local five-fold axes and thus represents a structure outside the formalism of classical crystallography and might be designated a quasi-lattice.

Systematic enumeration of crystalline networks

The systematic enumeration of all possible networks of atoms ininorganic structures is of considerable interest. Of particular importance are the 4-connected networks (those in which each atom is connected to exactly four neighbours), which are relevant to a wide range of systems — crystalline elements, hydrates, covalently bonded crystals, silicates and many synthetic compounds. Systematic enumeration is especially desirable in the study of zeolites and related materials, of which there are now 121 recognized structural types, with several new types being identified every year. But as the number of possible 4-connected three-dimensional networks is infinite, and as there exists no systematic procedure for their derivation, the prediction of new structural types has hitherto relied on empirical methods (see, for example, refs 2–4). Here we report a partial solution to this problem, basedon recent advances in mathematical tiling theory. We establish that there are exactly 9, 117 and 926 topological types of, respectively, 4-connected uninodal, binodal and trinodal networks, derived from simple tilings based on tetrahedra. (Here nodality refers to the number of topologically distinct vertices from which the network is composed.) We also show that there are at least 145 more distinct uninodal networks based on a more complex tiling unit. Of the total number of networks that we have derived, only two contain neither three- nor four-membered rings, and most of the binodal and trinodal networks are new.

Magnetite in CI Carbonaceous Meteorites: Origin by Aqueous Activity on a Planetesimal Surface

The composition and morphology of magnetite in CI carbonaceous meteorites appear incompatible with a nebular origin. Mineralization on the meteorite parent body is a more plausible mode of formation. The iodine-xenon age of this material therefore dates an episode of secondary mineralization on a planetesimal rather than the epoch of condensation in the primitive solar nebula.

The geometry of hypothetical curved graphite structures

Abstract Assuming that rings of 5, 7 and 8 carbon atoms are allowable (as well as hexagons) in graphitic sheets, a great variety of finite and infinite structures can be postulated. Besides the shells of the fullerenes, which have positive Gaussian curvature and the cylinders which have zero Gaussian curvature, infinite periodic networks with negative Gaussian curvature are possible. The latter promise to have lower energies than the convex fullerenes.

From Chemical Gardens to Chemobrionics

Chemical gardens in laboratory chemistries ranging from silicates to polyoxometalates, in applications ranging from corrosion products to the hydration of Portland cement, and in natural settings ranging from hydrothermal vents in the ocean depths to brinicles beneath sea ice. In many chemical-garden experiments, the structure forms as a solid seed of a soluble ionic compound dissolves in a solution containing another reactive ion. In general any alkali silicate solution can be used due to their high solubility at high pH. The cation should not precipitate with the counterion of the metal salt used as seed. A main property of seed chemical-garden experiments is that initially, when the fluid is not moving under buoyancy or osmosis, the delivery of the inner reactant is diffusion controlled. Another experimental technique that isolates one aspect of chemical-garden formation is to produce precipitation membranes between different aqueous solutions by introducing the two solutions on either side of an inert carrier matrix. Chemical gardens may be grown upon injection of solutions into a so-called Hele-Shaw cell, a quasi-two-dimensional reactor consisting in two parallel plates separated by a small gap.

Nodal surface approximations to the P, G, D and I-WP triply periodic minimal surfaces

Abstract The cubic P , G , D and I-WP triply periodic minimal surfaces (TPMS) may be closely approximated using periodic nodal surfaces (PNS) with few Fourier terms, thus enabling easy generation of TPMS for use in various chemical and physical applications. The accuracy of such approximations is quantitatively discussed and represented visually using a colour coding.

Crystalline protein in the chorion of insect egg shells

A polycrystalline material has been found to be a common constituent of the insect chorion. In a survey of nearly fifty species, crystalline material was found in the chorion of representatives of six orders (Orthoptera, Odonata, Neuroptera, Hemiptera, Hymenoptera, and Coleoptera). Chemical analysis indicated that the crystalline material is mostly protein. In different insects, either the endochorion, the exochorion, or the entire chorion were crystalline. The interpretation of the electron micrographs is discussed in relation to the shape and packing of the repeating units. It is proposed that the crystallites are constructed from units joined together into strips and that the simplest building block is a pair of strips. Layers in the plane of the shell with a period of about 5 nm are another feature common to most of the material. There is insufficient information available so far to permit reconstruction in any detail.

Triply periodic minimal surfaces decorated with curved graphite

Abstract Hypothetical negatively curved structures derived from graphite are described, in which all carbon atoms rest on triply periodic minimal surfaces (TPMS). The D minimal surface was calculated using the Weierstrass representation. By applying the Bonnet transformation to the D surface, the gyroid and P surfaces were constructed. Curvatures, densities, lattice parameters and energies have been calculated for all structures. The absolute value of the maximum Gaussian curvature is smaller than that for C 60 fullerene. A new periodic graphite net with the same topology as the I-WP minimal surface, using 5-, 6- and 8-membered rings is found possible. The stability of 11 negatively curved graphitic structures has been determined using Tersoffs three-body potential. All the structures described are more stable than C 60 ,mainly because the 120° bond angles in ordinary graphite are almost preserved in the 7- and 8-membered carbon rings. The way is now open to explore the decoration of minimal surfaces with further arrangements of atoms of different elements.

Exact computation of the triply periodic D (`diamond') minimal surface

Parametric expressions for the coordinates of Schwarzs D triply periodic minimal surface, found in many diverse physical, chemical and biological systems, allow us to describe fully the properties of the surface and to demonstrate its straightforward exact computation. Knowledge of the precise coordinates of the surface enables real structures to be quantified in terms of the parameters of the surface. q 1999 Elsevier Science B.V. All rights reserved.

The closest packing of equal spheres on a spherical surface

A table is given of putative solutions to the Fejes problem: to find the maximum value of the smallest angular distance between any two of N points movable on the surface of a sphere. Values of N run without omission up to 27 with six sporadic cases thereafter. Some applications of this system as a model are discussed.