Chemistry Reference
In-Depth Information
where N is the total number of crystallites in the sample and dN is the total number of
crystallites that possess the orientation g within an element of orientation dg.
8.6.2 ODF Calculation
The ODF cannot be measured directly, but can be obtained through evaluation of a set
of pole figures. Different methodologies exist to obtain the ODF from the pole figures
or measured pole density data [10-12]. One is the harmonic method, which expresses
the ODF as a series of generalized spherical harmonic functions
f ðgÞ¼
¥
l¼0
þ l
þ l
C
mn
l
T
mn
l
ðgÞ
ð8
:
28Þ
m¼l
n¼1
where C
mn
l
represents the ODF coefficients, also referred as C-coefficients, T
m
l
ðgÞ are
the generalized spherical harmonic functions, l is the order of the series, and m and n
denote finite numbers of independent C-coefficients at each order. The three indices l,
m, and n correspond to the dependence of the ODF on the three Eulerian angles with n
related to w
1
and m related to w
2
. The index l has to be a finite number to fit the
experimentally determined pole density data with a finite number of independent
C-coefficients. The series has to be truncated to fit the finite number of measured data
points.
f ðgÞ¼
X
þ l
þ l
L
C
m
l
T
mn
ðgÞ
ð8
:
29Þ
l
l¼0
m¼l
n¼1
where L is the maximum order (also referred as degree or rank in various literature) of
the series. L should be determined by the number of available pole figures and the
resolution of the ODF. Some of the T
m
l
ðgÞ terms vanish due to crystal symmetry;
therefore, a fewer number of pole figures are required for a crystal of high symmetry
of the same order. Table 8.1 lists the order of series that can be solved from the number
of pole figures for each symmetry type [13].
To have an overview of the ODF values, it is typically visualized as contour
plots in a series of cross sections perpendicular to one of the three axes through the
TABLE 8.1 The Order of the Series of Harmonics for the Number of Pole Figures
Number of
Pole Figures (n)
Cubic
Hexagonal
Tetragonal
Trigonal
Orthorhombic
2
22
10
6
4
2
3
34
16
10
8
4
4
34
22
14
10
6
5
34
22
18
14
8
6
34
22
22
16
10
7
34
22
22
20
12
8
34
22
22
22
14
>8
46
22
22
22
2(n 1) if n<13;
22 if n 13
