ALBERT

Notes: The geometry of the reciprocal space of cylindrically and conically rolled microstructures is described. The simpler cylindrical case is first discussed, followed by the conical case; in both cases, the observations and then the theory are described. The theory is compared with observations on chrysotiles, the structural and microstructural features of which are briefly recalled. The reciprocal space of an infinite 3D crystal consists of a lattice of discrete nodes. If a crystalline sheet is curled up into a cylindrical scroll (or into concentric cylinders), the corresponding reciprocal space is obtained by rotating this set of lattice points about a line parallel to the cylinder axis through the origin of reciprocal space. The lattice nodes thereby describe geometrical loci that, in this simple case, are circles in planes perpendicular to the rotation axis. For a general orientation of the rotation axis, each node produces its own circle. This is the case when the fibre has chiral character. For certain symmetrical orientations of the axis, `degeneracy' occurs and two (or more) nodes may lead to the same circular locus. This is the case for achiral fibres. The curvature often causes disorder in the stacking of successive cylindrical sheets – this leads to `coronae' instead of sharp circles – especially in the concentric cylinder case. In the diffraction pattern, these produce spots that are streaked in the sense away from the axis. In ideal cylindrical scrolls, the structures in successive layers, as viewed along a radial line c, are shifted relative to each other over 2π times the layer thickness; this may lead to superperiods along the normal c to the sheet planes if this shift is commensurate with the lattice vectors in the sheet plane, i.e. with its translation symmetry. The superperiod is clearly related to the sheet thickness, which may be more than one bilayer. If the crystalline sheet is curled up into a cone, the reciprocal-space loci become curves that are situated on spheres of constant spatial frequency, called spherical spirals instead of the circles in the cylindrical case. Each reciprocal-lattice node describes such a spiral traced out by a node point subject to the coupled rotations about the cone axis and about the local normal to the cone surface. The equations of such spirals are derived and their symmetry properties are studied analytically. The spiral's shape is a function of the semi-apex angle of the cone. For an arbitrary cone angle, these curves are not closed; they completely fill a band on the surface of the sphere. For certain discrete cone angles, which turn out to be essentially determined by the condition of good epitaxic fit between successive sheets of the cone, the spherical spirals become closed curves. The conditions under which several node points, belonging to the same spatial frequency, trace out the same spherical spiral are discussed: i.e. the conditions for degeneracy are formulated. The point symmetries of the sets of spherical spirals belonging to the same spatial frequency are found to depend on characteristic values of the semi-apex angle. All turns of a conical scroll are in fact formed from a single sheet. The structure in any given turn is rotated relative to that in the adjacent turn over a constant angle, only determined by the semi-apex angle. If this rotation angle is commensurate with 2π, superperiods can be formed, visible as reinforcements in streaks that are parallel to the generators of the cone formed by the set of normals to the conical surface. Also, this superperiod depends on the thickness of the sheet as well as on its rotation symmetry. Diffuse scattering is found to be concentrated on a V-shaped hyperboloid-like surface, the point of the V being situated on a spherical spiral. The intersection of this surface with the Ewald plane leads to V-shaped streaks attached by their apexes to the spots. They are the homologues of the simple streaks in the cylindrical case. Under certain conditions of beam incidence, the intersection is a hyperbole branch. Spot positions have been computed for a few characteristic diffraction conditions; they are found to represent adequately the observed spot patterns. A Mercator-like projection method is proposed to represent the spherical spirals in a plane and to construct geometrically the intersections with the Ewald plane for different angles of incidence. Throughout the paper, the analogies and the differences between the diffraction features of cylindrical and conical scrolls are emphasized and illustrated by observations on chrysotile.

Notes: Abstract Based on coexisting rolled chrysotile and polygonal serpentine fibers with 15 or 30 sectors each, a crystallographic model for polygonization of chrysotile is proposed. It is based on an assumed chrysotile-to-lizardite transition. Polygonization of chrysotile requires more likely 15 partial dislocations per turn, as required by polytype translational operators for serpentines. The observed number of sectors corresponds to the two most elastically stable arrays of dislocations. Homogeneous shear of the layer stacking arising from intersector kinking results in a cyclic distribution of twins and/or different polytypes. This makes the fiber axis a fivefold symmetry axis and consequently polygonal serpentine and chrysotile to be both forms of serpentine with local fivefold symmetry. This model is alternative to the recent crystallograpic model by Chisholm (1991, 1992).