Sten Andersson

Rod packings and crystal chemistry

Some crystallographic packings of identical infinite cylinders are described. It is shown how certain crystal structures are related to these packings when the cylinders are replaced by rods of atoms or groups of atoms (coordination polyhedra). Emphasis is placed on the densest cubic rod packing and structures based on this principle, which include garnet, Th3P4, β-Mn, Ag4RbI5 and bixbyite.

The (twin) composition plane as an extended defect and structure-building entity in crystals

Abstract The burden of this paper is the recognition of the composition plane as an extended (planar) “defect” in twinned crystals, together with the notion that new structure types may be generated from simple prototype structures by regular and frequent repetition of a twinning operation, usually on a very fine scale. Less regular application of the operation leads to disorder and variation in stoichiometry, without point defects being necessary. The range of compounds whose structures may be described in these terms is very wide, from metal alloys through metal carbides and borides etc. and inorganic materials, such as metal oxides and sulphides, to minerals, including silicates and the “sulphosalts” of the heavy metals (Pb, Bi etc.). Isostructural compounds frequently span this whole gamut of substances. The main reasons for the twinning appear to be (α) the generation of new types of interstices (not present in the prototype) of appropriate size for some of the constituent atoms, and/or ( b ) a variation in the concentration of appropriate interstices, i.e. in the stoichiometric ratio anion/cation. After a general introduction to the relevant aspects of twinning, attention is successively concentrated on reflection-twinned h.c.p., then c.c.p. and finally mixed c.p. arrays: first with “interstitial” atoms only in the composition planes; and then with other interstices also occupied. In each case lamellar twinning is first considered, and then cyclic twinning. Glide-reflection-twinning of similar arrays is then considered. Attention is also drawn to small topological distortions that, for example, convert the octahedra in a c.c.p. array (twinned or even un-twinned) to arrays of capped trigonal prisms (or to cubes). In this way the apparently complex structures of a number of compounds are derived from those of much simpler twinned c.c.p. arrays. Finally, the not-very-large amount of available, relevant, experimental evidence is reviewed. As far as it goes, this confirms that the notion of the composition plane as an extended defect is well founded. But more experimental data are needed. [In most cases specific literature references are provided for each structure. For a few, rather common, structures they are not. In such cases reference may be made to standard texts, e.g. for alloys, to Schubert 99 or Pearson 133 and, for more ionic structures, to Wyckoff 134 or Povarennykh 135 . Diagrams are to scale: 1 cm = 4.0 A.]

Chapter 3 – Molecular Forces and Self-Assembly

This Chapter outlines the nature, delicacy, and specificity of molecular forces, and the ways these forces work together with the geometry of molecules to organize self-assembled molecular aggregates. It discusses the meaning of the size and shape of a molecule and describes the forces that enable the curvature of aggregates of self assembled molecules to be changed. The acute sensitivity of many biological phenomena to temperature can in some cases be correlated with forces. A brief overview of ideas and the physical notions behind self-assembly of surfactant-water systems is presented. The classical intuition on molecular forces is embodied in the Derjaguin–Landau–Verwey–Overbeek theory of colloid stability. It blends two themes: (1) an intervening liquid can be thought of as a structureless continuum with bulk liquid properties, up to a molecular distance from the surface and (2) any object (surface) must perturb proximal liquid structure (density, dipolar orientation, and hydrogen bonding) so that the transmission of force is propagated via a stress field passing from molecule to molecule in much the same way that the electromagnetic field is carried through the vacuum or a dielectric in Maxwells theory of the electromagnetic field.

The intrinsic curvature of solids

The description of periodic minimal surfaces using differential geometry is summarized. It is shown that interpenetrating (and nonintersecting) structures can be described using surfaces parallel to the minimum surfaces. The Gaussian curvatures of such surfaces vary with their normal distances and are simply related. It is also shown that TschebychefPs net can be used to describe atomic positions with surface coordinates. Examples of compounds whose structures are described using these surfaces are the zeolites faujasite and Linde A, W3Fe3C and Nb 6 Fi 5 . It is proposed that a geometrical description of the chemical bond is possible using the concept of curvature. The selective absorption of gases by zeolites is explained with curvature, as well as a new mechanism for the zeolitic catalytic cracking of oil.

The pyrochlore structure and its relatives

Abstract The structures of pyrochlore, W 3 Fe 3 C, Sb 2 O 3 (cub), KTaWO 6 ·H 2 O, RbNbTeO 6 , and Mg 3 Cr 2 Al 18 are discussed. Ideal atomic parameters are derived and compared with those observed.

Magnesium nitride fluorides

Abstract Three magnesium nitride fluorides, Mg3NF3, LMg2NF and HMg2NF, have been prepared at temperatures between 900–1350°. Mg3NF3 is cubic with a = 4.216 A, space group Pm3m, and its structure is related to the structure of both MgO and MnMg6O8. LMg2NF is tetragonal with a = 4.186A and c = 10.042 A and the space group is I4 1 amd . Its structure is intermediate between the structure types represented by zinc blende and sodium chloride (or MgO). LMg2NF transforms at 20 kb and 1300°C into HMg2NF, which is nearly isostructural with MgO. The anions are ordered in Mg3NF3 and LMg2NF, and the X-ray data of HMg2NF indicates also an ordering of the anions for this compound. LMg2NF and Mg3NF3 decompose at 1000–1150°C in an argon atmosphere to Mg-vapour, N2 and MgF2.

Structures related to the β-tungsten or Cr3Si structure type

Abstract Using traditional crystallographic operations such as translation, rotation and reflection, and intergrowth, it is shown how several structures, among them the tetrahedrally close-packed, can be accurately derived from the β-tungsten structure type. Examples are Zr 4 Al 3 , Fe 2 B, Mo 3 CoSi, Ga 2 Mg 5 , GaMg, Ga, α- and β-V 3 S, W 2 CoB 2 , CeAl, W 5 Si 3 , the σ-phase, the Friauf-Laves phases, the μ-phase, the P and M phases, the (Co 0.57 Si 0.43 ) 3 V 2 phase, the χ-phase, and the ν-phase.

The martensite transition and differential geometry

The martensite transition is explained by a Bonnet transformation operating on the fcc structure to form a bcc packing of atoms. In terms of differential geometry, the F-minimal stuface (with atoms on flat points) is bent (conserving Gaussian curvature) to form the gyroid minimal surface, also with atoms on flat points. The current description of phase transitions as two Step transformations: rotation and translation, is W e given as one homogeneous operation, the elliptic Bonnet transformation.

Electron microscope studies on a quenched FeMo alloy

Abstract An alloy of FeMo (50 atom%) quenched from the melt was studied using a high-resolution electron microscope. Three phases were frequently identified, viz , the σ-phase, the μ-phase, and the P -phase. The latter two were found to be rich in planar defects of intergrowth and twin type. The resolution obtained was better than 4 A, as shown with calculated images. It was possible to derive the detailed atomic structure of the defects observed.

Chemical fourling on the unit cell level as a structure-building operation in the solid state

Abstract Chemical fourling on the unit-cell level is introduced as a structure-building operation in the solid state. A chemical fourling structure is achieved when two twin planes are perpendicular to each other. Each fourling unit has infinite extension in only one direction and can be related to a well-known structure type or packing of atoms. The structures of the tetragonal tungsten bronzes M x WO 3 and Pd(NH 3 ) 4 Cl · H 2 O are presented as examples of chemical fourlings of cubic close packing. The structures of SeO 2 , Mn 2 Hg 5 , and Zr 2 F 7 O are shown to be a sequence of chemical fourlings of primitive cubic packing. Chemical fourling units of hexagonal close packing are found in the structures of tetragonal Ti 3 Sb, α-V 3 S and Sb 6 O 7 (SO 4 ) 2 . The structures of CuAl 2 , SeTl, NbTe 4 , Ti 3 Sb, and MnU 6 all have the same chemical fourling unit. Cr 23 C 6 is described as a bounded chemical fourling of cubic close packing.