Most natural or artificial solids (rocks, ceramics, metal alloys or polymers) contain of many crystallites of different size, shape and different orientations. They are usually multi-phase substances, i.e. they contain several crystalline phases of different structure. The most important parameter describing the anisotropy of polycrystalline materials is their texture. Via the anisotropy of physical properties due to the lattice structure, a regular texture in which the crystallites of one phase have only a few preferred orientations produces anisotropy of the polycrystalline material as well.

: metals, alloys, intermetallic compounds, ceramic materials, compound materials, polymers, semiconductors, nanocrystals, supraconductors, rocks;__polycrystalline materials__: deformation, recrystallization, phase transformation, synthesis of layers by plasma or laser and ion beams, crystallization at interfaces, rigid particle rotation;*texture-modifying processes*: plasticity, elasticity, hardness, strength, cleavability, thermal expansion and conductivity, electric conductivity, magnetization, corrosion resistance, optical properties.__anisotropic properties__

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The sample coordinate system K_{A} is usually adapted to the process geometry if it is known. Thus,
the rolling direction R is selected parallel to X_{A}
and the normal direction N is selected parallel to Z_{A} for a rolled sample.
The crystal coordinate system K_{B} should be fixed to the basis vectors
a, b and c of the Bravais lattice as follows:

Evidently the crystal symmetry does not lead to a multiplication of the crystallite orientation
since the crystal lattice is projected on itself by the symmetry operations (left figure).
In contrast, the sample symmetry produces the existence of several crystallites
with evidently different orientations which are, however, equivalent in terms of the
sample symmetry.
In the right-hand figure, these are each represented in the same
color for three different orientations.
The sample symmetry is given by the point group C_{6} here.
Imagine a pyramid-shaped sample with a hexagon as basis that lies parallel to
the drawing plane.

A sample symmetry is caused by the texture-modifying process or the environment
of the crystalline phase.
The example shows 13 preferred orientations of a thin film (crystal symmetry D_{6h})
that was produced on an also hexagonal monocrystalline substrate.
Since the c-axis of the substrate crystal is parallel to the sample normal,
there are five more equivalent preferred orientations for each preferred orientation.
In special cases (i.e. for special orientations) the equivalent orientations
can coincide (olive-green crystallite).

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If the rotation is subdivided into three partial rotations to be performed successively (K knot line):

- transforms X
_{A}Þ K, rotation axis : Z (Z_{A}), rotation angle:j_{1}, - transforms Z
_{A}Þ Z_{B}, rotation axis : K, rotation angle: F, - transforms K Þ X
_{B}, rotation axis : Z (Z_{B}), rotation angle:j_{2},

.

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G_{B} | C_{1} | C_{2} |
D_{2} | C_{3} | D_{3} |
C_{4} | D_{4} | C_{6} |
D_{6} | T* | O* |
---|---|---|---|---|---|---|---|---|---|---|---|

F£ | 180° | 180° | 90° | 180° | 90° | 180° | 90° | 180° | 90° | 90° | 90° |

j_{2}£ |
360° | 180° | 180° | 120° | 120° | 90° | 90° | 60° | 60° | 180° | 90° |

Boundaries for j

G_{B} | C_{1} | C_{2} | D_{2} |
---|---|---|---|

j_{1}£ |
360° | 180° | 90° |

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A pole figure is defined by the relationship i.e. it is proportional to the volume fraction of all crystallites whose crystal direction h is parallel to the sample vector y . In the measurement, y describes the direction of the scattering vector of the diffractometer relative to the sample coordinate system. By suitable sample rotation y can comprise the entire angle range (pole sphere). Due to Friedel's law according to which the intensity of the diffracted beam is equal for +y and -y, only 50 % of the angular space must be scanned. Experimental limitations lead to smaller measurement ranges in most practical cases; the results are called incomplete pole figures.

In addition to the crystallite with the represented orientation, all crystallites make a contribution to the "yellow" pole figure value if their orientation can be obtained from the indicated orientation by any rotation around the indicated axis. The totality of these orientations g

This is the basis for the following important statement:

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If these components are suitably chosen, the textures of large sample series can be exactly compared, and texture changes and the processes causing them can be quantitatively described.

Based on the assumption that each real texture can be described, in good approximation, as a superposition of a finite number of texture components of different shape and scattering width (c-component index): a geometric approximation method for the texture calculation from diffraction pole figures was developed, which can also be used for multi-phase samples. The component parameters

- volume fraction I
^{c} - preferred orientation g
^{c} - fiber axis f
^{c} - scattering widths a
^{c}, b^{c}

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