The Paracrystal as a Model for Liquid Crystals

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Abstract
It is emphasized, that the loss of long range order does not necessarily destroy three-dimensional lattices. Nematic or smectic mesophases as well as special types of liquid crystals, many crystalline or amorphous high polymers, crystals near the melting point, molten substances and mixed crystals are examples in nature. The theory of paracrystals describes such lattices quantitatively in a most convenient mathematical way. A very short survey of the main formulas is given, which are needed for a practical application of the theory to X-ray, electron and neutron diffraction problems. The derivations are given in standard texts.1,2 This publication is confined to the discussion of the integral widths of the reflections. Two reasons for line broadening are discussed: paracrystalline distortions and particle size effects. It will be shown how one can separate these two effects and also the effect of thermal oscillations. These results are applied in a separate section to the analysis of paracrystalline mosaic blocks in single polyethylene crystals and paracrystalline ultrafibrils in stretched polyethylene. The mosaic blocks have sizes of about 300 Å and g-values=2%, the ultrafibrils diameters of 100 Å and g= 3%. 2° C below and some degrees above the melting point lead also consists of such micro-paracrystallites. Their mean size 23° C above the melting point is 30 Å, g= 11.5%, 223° C above the melting point 10 Å, g=12.5%. 2° C below the melting point g drops down to 4%. At the melting point a network of paracrystalline grain-boundaries arises suddenly, whilst the thermal and paracrystalline distortions within the paracrystals increase more or less continuosly with rising temperature. Inside the paracrystals subgroups of atoms exist with a smaller amount of paracrystalline distortion. All liquid metals investigated by us, at least up to 100° C above the melting point, can be explained quantitatively by aggregates of clusters of atoms in paracrystalline distorted hexagonal close packing. Mixed crystals of α-Fe with 15 atom % Al have g-values of 1%, whilst rigid atoms should have g-values about twice as large. This discrepancy can be explained by the deformability of the electron clouds. A Laves phase (Fe2Zr) has g-values up to 0.7%, reduced Fe, Al-spinels (3 atom % AI) have g-values near 1%. The paracrystalline distortions show characteristic anisotropies, which give some information about the arrangement of the foreign atoms or subgroups in the lattice.