GEOMETRICAL ASPECTS OF THE DIFFRACTION SPACE OF SERPENTINE ROLLED MICROSTRUCTURES - THEIR STUDY BY MEANS OF ELECTRON-DIFFRACTION AND MICROSCOPY

The geometry of the reciprocal space of cylindrically and conically ro lled microstructures is described. The simpler cylindrical case is fir st discussed, followed by the conical case; in both cases, the observa tions and then the theory are described. The theory is compared with o bservations on chrysotiles, the structural and microstructural feature s of which are briefly recalled. The reciprocal space of an infinite 3 D crystal consists of a lattice of discrete nodes. If a crystalline sh eet is curled up into a cylindrical scroll (or into concentric cylinde rs), the corresponding reciprocal space is obtained by rotating this s et of lattice points about a line parallel to the cylinder axis throug h the origin of reciprocal space. The lattice nodes thereby describe g eometrical loci that, in this simple case, are circles in planes perpe ndicular to the rotation axis. For a general orientation of the rotati on axis, each node produces its own circle. This is the case when the fibre has chiral character. For certain symmetrical orientations of th e axis, 'degeneracy' occurs and two (Or more) nodes may lead to the sa me 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 th e concentric cylinder case. In the diffraction pattern, these produce spots that are streaked in the sense away from the axis. In ideal cyli ndrical scrolls, the structures in successive layers, as viewed along a radial line c, are shifted relative to each other over 2 pi times th e layer thickness; this may lead to superperiods along the normal c to the sheet planes if this shift is commensurate with the lattice vecto rs in the sheet plane, i.e. with its translation symmetry. The superpe riod is clearly related to the sheet thickness, which may be more than one bilayer. If the 2D crystalline sheet is curled up into a cone, th e reciprocal-space loci become curves that are situated on spheres of constant spatial frequency, called spherical spirals instead of the ci rcles in the cylindrical case. Each reciprocal-lattice node describes such a spiral traced out by a node point subject to the coupled rotati ons about the cone axis and about the local normal to the cone surface . The equations of such spirals are derived and their symmetry propert ies 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 sph ere. For certain discrete cone angles, which turn out to be essentiall y determined by the condition of good epitaxic fit between successive sheets of the cone, the spherical spirals become closed curves. The co nditions under which several node points, belonging to the same spatia l frequency, trace out the same spherical spiral are discussed: i.e. t he conditions for degeneracy are formulated. The point symmetries of t he sets of spherical spirals belonging to the same spatial frequency a re found to depend on characteristic values of the semi-apex angle. Al l turns of a conical scroll are in fact formed from a single sheet. Th e structure in any given turn is rotated relative to that in the adjac ent turn over a constant angle, only determined by the semi-apex angle . If this rotation angle is commensurate with 2 pi, superperiods can b e formed, visible as reinforcements in streaks that are parallel to th e generators of the cone formed by the set of normals to the conical s urface. Also, this superperiod depends on the thickness of the sheet a s 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 surfa ce with the Ewald plane leads to V-shaped streaks attached by their ap exes to the spots. They are the homologues of the simple streaks in th e cylindrical case. Under certain conditions of beam incidence, the in tersection is hyperbole branch. Spot positions have been computed for a few characteristic diffraction conditions; they are found to represe nt adequately the observed spot patterns. A Mercator-like projection m ethod is proposed to represent the spherical spirals in a plane and to construct geometrically the intersections with the Ewald plane for di fferent angles of incidence. Throughout the paper, the analogies and t he differences between the diffraction features of cylindrical and con ical scrolls are emphasized and illustrated by observations on chrysot ile.