Twins and allotwins of basic mica polytypes: theoretical derivation and identification in the reciprocal space

Abstract

The geometry of the diffraction pattern from twins and allotwins of the four basic mica polytypes (1M, 2M 1 , 3T, 2M 2 ) is analysed in terms of the`minimal rhombus', a geometrical asymmetric unit in reciprocal space de®ned by nine translationally independent reciprocal-lattice rows. The minimal rhombus contains the necessary information to decompose the reciprocal lattice of twins or allotwins into the reciprocal lattices of the individuals. The nine translationally independent reciprocal-lattice rows are divided into three types (S, D and X): rows of different type are not overlapped by the n  60 rotations about c à , which correspond to the relative rotations between pairs of twinned or allotwinned individuals. A symbolic representation of the absolute orientation of the individuals, similar to that used for layers in polytypes, is introduced. The polytypes 1M and 2M 1 undergo twinning by reticular pseudo-merohedry with ®ve pairs of twin laws: they produce twelve independent twins, of which nine can be distinguished by the minimal rhombus analysis. The 2M 2 polytype has two pairs of twin laws by pseudo-merohedry, which give a single diffraction pattern geometrically indistinguishable from that of the single crystal, and three pairs of twin laws by reticular pseudo-merohedry, which give a single diffraction pattern different from that of the single crystal. The 3T polytype has three twin laws: one corresponds to complete merohedry and the other two to selective merohedry. Selective merohedry produces only partial restoration of the weighted reciprocal lattice built on the family rows and the presence of twinning can be recognized from the geometry of the diffraction pattern.

Identification of polytypic groups in hydrous phyllosilicates using electron backscattering patterns

“…These three categories are also equivalent to the three kinds (S, D, and X) of reciprocal-lattice rows defined by Nespolo et al (2000). }; and (3) others (k 3n).…”

The chromatic symmetry of twins and allotwins

“…An informative description requires the point groupoid (Brandt or This case corresponds to path 2.1 of the flowchart in Fig. The only examples we are aware of are the two 1M-2M 1 allotwins in micas we have reported (Nespolo et al, 2000a), in which the two individuals were rotated by 120 and by 60 , respectively, about the normal to the (001) plane. The only examples we are aware of are the two 1M-2M 1 allotwins in micas we have reported (Nespolo et al, 2000a), in which the two individuals were rotated by 120 and by 60 , respectively, about the normal to the (001) plane.…”

“…The two allotwins reported in the literature are represented by the symbols Z T = 3 5 and Z T = 3 4 , respectively, where the upper and lower digits give the orientation of the first and second individuals with respect to a fixed reference, in which the six possible orientations are numbered from 1 to 5 and the first one is taken with orientation 3 (Nespolo et al, 2000b). The two polytypes 1M and 2M 1 crystallize in space-group types C2/m and C2/c, respectively, so that they have the same type of point group, 2/m.…”

Identification of two allotwins of mica polytypes in reciprocal space through the minimal rhombus unit

“…(i) S rows, corresponding to h = 0(mod 3) and k = 0(mod 3); (ii) D rows, corresponding to h T 0(mod 3) and k = 0(mod 3); (iii) X rows, corresponding to k T 0(mod 3), where S, D and X represent`Single',`Double' and`Sextuple', respectively. These three types of rows are translationally independent and are related by n  60 rotations about c* (Nespolo et al, 1997;Nespolo et al, 2000). All the central reciprocal lattice planes containing c* can be divided into two types: SD planes (containing only S and D rows) and SX planes (containing only S and X rows).…”

“…Application of these criteria showed (Nespolo & Takeda, 1999) that the 24-layer polytype reported by Hendricks & Jefferson (1939) was not a twin of an eightlayer polytype, as previously stated by Smith & Yoder (1956). Nespolo et al (2000) developed a theory to geometrically analyse the diffraction pattern of composite crystals and decompose the twin-/allotwin-weighted reciprocal lattice (w.r.l.) into the w.r.l.…”