Pseudo-symmetry in multiple twinned crystals having M3M point group symmetry
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I N the last y e a r s , several authors have observed multiple twinning in diffraction patterns of thin f i l m s of fcc m e t a l s .1-6 T h e s e m e t a l s twin a c r o s s the {111} planes. We thought that it would be useful a m a t h e matical analysis, in o r d e r to determine the possible pseudo-symmetries of twins. The idea of pseudo-symmetry a r i s e s from the fact t h a t , if the coincidence of points coming from different orientations of the twin is neglected, the comp o s i t e lattice will m i m i c a point-group symmetry. However, the composite lattice will not conform to space groups appropriate to t h e s e point groups. The importance of pseudo-symmetry in diffraction pattern analysis has been shown.%B We have considered the primary and secondary twinning of a crystal having m3m point group symmetry, and w e have determined the pseudo-symmetry of composite structures formed by several twin o r i e n tations. MATHEMATICAL PROCEDURE
W e consider twinning a c r o s s the {III] planes. W e have considered the c o m p o s i t e structure f o r m e d by the original orientation of the crystal and one, two, three, or the four twin orientations. T h e p r o c e d u r e to d e t e r m i n e the p s e u d o - s y m m e t r y of the c o m p o s i t e lattice is similar to the one followed by Mokievskii et al.;9 the b a s i c relations a p p e a r in the classic g r o u p t h e o r y bibliography, i.e. S c h a n k m a n ,I° and K u r o s h . n T h e point g r o u p m 3 m of the crystal is i s o m o r p h i c to the direct product: [1]
In a crystal with m3m point group symmetry, because of its center of symmetry, the twin reflections are equivalent t o 180 deg rotations. Let these r o t a tions be Mi, w h e r e i = 1,2, 3, 4 corresponds t o the twin axes [111], [111], [111], and [111] respectively. Let {Mi} be the group of o r d e r 2 f o r m e d by Mi and the identity; and: p i : {Mi} × {r~3rn} : {i} × Mi × {432}
gm P;• = Pj'i
[3]
for any j w i t h
P i -- { e / } Relation [3] shows t h a t , as i commutes with any considered symmetry operation, it is only necessary t o take into account the elements of {m3m} which a r e also elements of {432}. If w e now consider the p r e s e n c e of more than one twin orientation, the pseudo-symmetry elements of the composite lattice are a group of the union U of the sets p i corresponding t o the orientations present. Let Ak be any element of {432}. It can be shown that if Mi and Mj are conjugate u n d e r A k , i.e.: AkMiAk 1 = Mj
Primary Twinning
{I} × {432}
If w e consider the lattice f o r m e d by the original and one twin orientation, the pseudo-symmetry operations are defined as those in the corresponding set P* that form the group G of highest o r d e r such t h a t , for every one of its elements gin, w e have:
[4]
then all the elements of p i and PJ fulfill [3]. Therefore, t h e r e are two different kinds of elements of the group U: a) Elements Ak belonging t o the group 432; pseudosymmetry elements of this kind must fulfill [4] for all the orie
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