Statistische Probleme der Theorie des Parakristalls

Examination of the basis of the statistical concept of the paracrystalline disorder model reveals a number of contradictions which result in a considerable limitation of the applicability of the model. The assumption of the statistical independence of the dimensions of the particles and the lengths and directions of their distances can only be maintained in the case of very small fluctuations of size and distance. The assumption of the statistical independence of distances between neighbouring particles is only meaningful in connexion with further ordering conditions such as the requirement of having lattice cells with parallelogram shapes in the case of the 'ideal' paracrystal or a chess-board-like arrangement of identical but alternate rotated lattice cells in the case of the 'real' paracrystal. The additional ordering requirement in the case of the 'real' paracrystal is, however, only a necessary condition for the absence of contradictions and the validity of the concept of the 'real' paracrystal is sufficiently verified only by the definition of the corresponding collective statistics. It is shown that such a statistics cannot exist.