TEXTURE - STATISTICAL CRYSTALLOGRAPHY OF POLYCRYSTALS

H.J. Bunge

Department of Physical Metallurgy, Technical University of Clausthal, Germany

Keywords: Texture, Orientation stereology, Aggregate function g(x), Grain boundaries, Grain interaction, Physical properties, Statistical symmetries, Powder diffraction, Materials science.

Structure of polycrystalline aggregates

Crystallography studies the crystalline state of matter - its ideal structure, lattice defects and physical properties. In very many cases the crystalline state is polycrystalline. Additional to the parameters describing one crystal individuum, aggregate parameters are then needed which characterize, for instance, size, shape, orientation and mutual arrangement of the crystallites. The structure of an aggregate is completely described by the aggregate function g(x) which specifies crystal orientation g in each small (single-crystalline) volume element at the location x in the sample. From this complete description of the aggregate several "Derivative-functions" may be deduced such as, for instance, the volume fraction f(g) of crystals in the orientation g i.e. the orientation distribution function ODF, the grain boundary network, i.e. the discontinuities of g(x), or the misorientation distribution function of crystal orientations across the grain boundaries, MODF.

Two types of aggregates may be particularly distinguished:

1. The crystallites form a loose powder ----> Powder Diffraction
2. The crystallites are rigidly connected ---- > Materials Science

In the first case the aggregate function g(x) or its "derivate" f(g) is needed in order to evaluate powder diffraction diagrams (with non-random orientation distribution).

In the second case the aggregate function plays an important role because it influences virtually all physical properties of the material i.e. all those which are anisotropic in the single crystal.

Experimental methods to measure aggregate parameters

The aggregate function g(x) can be measured by location resolved single crystal diffraction methods. Because of small crystallite sizes this requires virtually always electron diffraction with location resolving powers in the order of 1 mm (in the reflection technique) or 1 nm (in the transmission technique). Recently, these techniques have been fully automatized and are now available as Orientation Imaging Microscopy (OIM) or Automated Crystal Orientation Mapping (ACOM).

The orientation distribution function f(g) can be deduced mathematically from polycrystal diffraction diagrams taken in many different sample orientations (pole figure measurement followed by pole figure inversion). This latter technique has been essentially developed and improved, recently. Particular new methods are: location resolved polycrystal diffraction, the use of position sensitive and area detectors, energy dispersive detectors, highest angular resolving power obtained by parallel beam techniques and others.

Besides the orientation distribution of the integral intensity (pole density distribution functions or "pole figures") also peak shift and peak broadening can be measured as functions of sample direction (generalized pole figures) which contain information about the orientation dependence of crystallite size, strains, and lattice defects.

Physical properties of polycrystalline aggregates

Physical properties of polycrystalline materials depend on the properties of the individual crystallites, i.e. on the ideal crystal structure and on lattice defects, as well as on the aggregate function g(x). The influence of this latter one can be separated into the volume average of the crystal anisotropies which can be expressed by the function f(g) and the mutual interaction of the crystallites which needs the complete function g(x). It has been shown by mathematical model calculations that the interaction effects may reach the same order of magnitude as the simple volume average (without interaction). Hence, the complete "orientation stereology" described by g(x) or at least its statistical derivative functions "Statistical crystallography" are needed in order to understand the properties of polycrystalline materials.

Symmetries of polycrystalline bodies

The aggregate function g(x) posseses statistical symmetries in the "location space" x as well as in the "orientation space" g, the full description of which requires some generalizations compared with the concept of symmetry in single crystals. These are for instance "direction" and "orientation" symmetries, "fuzzy" symmetries with symmetry degrees lower than on, or "symmetroids" which do not form a group. The symmetries of polycrystalline bodies are closely related with the "generation" process of the material i.e. the "production" process if it is man-made or geological processes such as rock deformation. Symmetry analysis can be used as a sensitive indicator for these processes.