in memorial Rolf Hosemann
The theory of the paracrystal, which has been developed by R. Hosemann about 50 years ago, offers an elegant method to describe more or less distorted structures at least approximative.
The one-dimensional paracrystal model is mainly based on a statistical distribution of the distance of the next nearest neighbour and the a priori probability, which means the each of the particles of an assembly has in the average the same environment.
The structure of an "Ideal" (one-dimensional) paracrystal can be expressed by an convolution polynom the Fourier transform of which leads to a simple mathematical formula, which allows a transparent evaluation of diffraction pattern. While the one-dimensional paracrystal can be represented by a close formulation the two- and three-dimensional paracrystal leads to significant problems, because different pathways of construction of the position of the next nearest neighbour consecute in inconsiderable statistics.
Hosemann published two solutions of construction. While the "Ideal" paracrystal requires an infinite structure, Hosemann introduced a "Real" paracystal which contains microdomains called microparacrystals. He postulated the $\alpha^{*}$ - law, which connects the degree of distortion to the size of the microparacrystals. It could have been proved for several different materials.
Furthermore a Monte Carlo computer simulation of paracrystalline structures using the present authors potential field model is introduced. Some examples of calculations are demonstrated.