Neutron Scattering

5 .1 Pole Figure Inversion
The détermination of thé ODF f(g) from equation (5) is called pole figure inversion . Different
mathematical procedures have been developed to calculate thé ODF from experimental pole
figures . Depending on thé method, thé calculated ODF is a continuons or a step function . A
widely applied method in texture analyses of high symmetry materials (metals) is thé seriesexpansion
or harmonic method developed by H . J. Bunge [6] (compare chapter 3 .0) . Analogous
to thé classical procedure in single crystal structure determination, thé ODF is expanded
into its corresponding Fourier orthogonal series using surface spherical harmonic functions
with coefficients C (see equation (2) . A similar expansion is performed with thé experimental
pole densities P yielding thé coefficients F(hkl) . A system of linear equations and appropriate
transformations of coefficients are used to détermine thé unknown coefficients C from thé
experimentally known coefficients F . Routine computer programs are available to perform
these calculations . The calculated coefficients C are those texture coefficients C which have
been used in equation (3) to describe thé anisotropy of macroscopic physical properties of a
texturized polycrystal .
After thé ODF has been determined it is possible to calculate pole figures of all planes
(hkl), also of those which have not been or cannot been measured, for instance, because of
extinction. It is also useful to recalculate experimental pole figures from thé ODF in order to
estimate or control thé reliability ofthé texture analysis performed .
A different mathematical approach to thé ODF calculation is thé discretization method
based on thé maximum entropy concept [7] using a finit series expansion into indicator
functions . This method was introduced into thé program MENTEX by H . Schaeben [8] . The
so-called WIMV-method [9] which is rather common in geological texture analysis is based
on certain probability assumptions of f(g) which may then be further improved by itérative
refnements .
8c=o
Fig . 18.19 :
Representation ofa
texture component g`
by a Gaussian
distribution function
f(g) ofhafvidth b °. af
denotes the déviation
of crystalltie
orientations from g°
(atof - 09.
n ° represents a sample
fzxed axis (see [101)