Periodic intensity distribution (PID) of mica polytypes: symbolism, structural model orientation and axial settings

Abstract

Following a preliminary revisitation of the nomenclatures in use for mica polytypes, the properties of the periodic intensity distribution (PID) function, which represents the Fourier transform of the stacking sequence, are analysed. On the basis of the relative rotations of neighbouring layers, mica polytypes are classi®ed into three types; for each type, the PID exists in different subspaces of the reciprocal space. A revised procedure to compute the PID, in which further restrictions on the structural model orientation are introduced, is presented. A unifying terminology based upon the most common symbols used to describe mica polytypes (RTW, Z and TS) is derived; these symbols represent the geometrical basis for the computation of the PID. Results are presented for up to four layer polytypes and are compared with the re¯ection conditions derived by means of Zvyagin's functions. Both the PID values and the re¯ection conditions are expressed in suitable axial settings and compared with previous partial reports, revealing some errors in previous analyses. A computer program to compute PID from the stacking symbols is available.

Producing highly complicated materials. Nature does it better

“…In the real structure, because of the difference in the lateral dimensions of the T and O sheets, tetrahedra rotate around [001] * by an amount that depends mainly on the chemical composition but also external conditions like temperature and pressure, so that the symmetry of the T sheet is reduced to ditrigonal. The model that takes into account this rotation is known as the trigonal model (Nespolo et al 1999).…”

Geometric crystal chemical models for structural analysis of micas and their stacking polytypes

“…where two of the octahedral sites are occupied by the same kind of crystallochemical entity and the third by a different one; and (3) hetero-octahedral micas have an octahedral sheet of layer symmetry P(3)12, where each of the three octahedral sites is occupied by a different crystallochemical entity [layer symmetry notation after Dornberger-Schiff (1959), as cited by Nespolo et al (1999)]. On the other hand, Weiss et al (1985Weiss et al ( , 1992 identifi ed three octahedral sheet types based on the values of the average octahedral cation-anion bond lengths <M1-O>, <M2-O>, and <M3-O>: type-I, <M1-O> = <M2-O> = <M3-O>; type-II, any two are equal and the third is different; and type-III, the three average octahedral site bond lengths are different.…”

Perturbative Theory of Mica Polytypism. Role of the M2 Layer in the Formation of Inhomogeneous Polytypes

“…As shown by Ross et al (1966), a portion of the stacking sequence of the inhomogeneous mica polytypes coincides with one or more periods of one of axes (a, b) of the space-fixed reference (Cj setting) and of the structure-related references in the six possible orientations (a~-a 6 ) , and corresponding Z symbols. The direction of the intralayer displacement (ID) vector and of its (001) projection (Z vector) is indicated by Z symbol i (i = 1-6) when the a~ axis of the corresponding half-layer is parallel to the space fixed a axis (modified after Nespolo et al, 1999a).…”

“…The inclusion of stacking faults in the basic structures would add flexibility to the model and give more simple routes to explain the observed stacking sequences, but to illustrate the method in the most direct way, the presence of stacking faults is not considered in these examples. Two polytypes were identified by PID analysis (periodic intensity distribution: Takeda, 1967;Takeda and Ross, 1995;Nespolo et al, 1999a) of the X-ray diffraction patterns (Ross et al, 1966), whereas the third polytype was determined by HRTEM . Both techniques reveal the stacking sequence as described within the homo-octahedral approximation (i.e., assuming all M1 layers).…”