Der Symmetrieeinfluss auf den allgemeinen Lösungsansatz eindimensionaler Fehlordnungs-probleme

Abstract

X-ray intensities of one-dimensionally disordered crystals are usually calculated by the method of difference equations or the equivalent matrix-method. In this paper a new group-theoretical method is developed, which has advantages in cases where the problem is one of pure disorder of positions, or of simultaneous disorder of kinds and positions of layers. A general demonstration is given that a strict solution for the diffracted intensities can be obtained with the aid of operators reducing the secular equation to a set of equations which can be solved independently, provided the sym- metry-group is cyclic. Other groups admit solutions from their representation with cyclic factor- groups, or by separation into sub-groups. In the ease of several kinds of tetragonal layers with possibilities of (0, 0), (½, ½), (½, O) and (0, ½) for their positions, we find three independently soluble equations; for hexagonal layers with (0, 0), ~), and (~, ½) as possible positions there are two equations. In both cases it is not necessary to introduce a prohibition of positions for neighbouring layers. The operators solving the secular equation may be used directly for calculating the X-ray intensities. The constants of the char- acteristic values may be calculated separately for each equation. Each physically significant operator describes the intensity distribution on one kind of reciprocal-lattice row.