Chemistry Reference
In-Depth Information
90
with 5
intervals were actually examined). Since the fiber axis is on the Z-axis of
the sample system, or the sample normal is the w
1
axis of the Eulerian space, the ODF
is independent of the w
1
axis. All 18 (only 4 are shown) cross sections of theODF show
almost identical pattern.
Figure 8.14(d) shows three calculated pole figures from theODF for the same planes
of the measured pole figures. The measured pole figures cover the angular region up to
only x ¼80
approximately, but the calculated pole figure covers all the angles up to
x ¼ 90
. The calculated complete pole figures reveal the pole density distributions
following the angles between planar normals in the cubic system [14]. The (111) pole
figure shows the concentrated intensity distribution in the center and in a ring making a
70.5
angle from the center, since the angle between (111) planes is either 0
or 70
32
0
.
The (200) pole figure shows the concentrated intensity distribution in a ring making a
54.7
angle from the center, since the angle between (111) plane and (200) is 54
74
0
.
The (220) pole figure shows the concentrated intensity distribution in two ringsmaking
35.3
and 90
angles, respectively, from the center, since the angle between (220) and
(111) planes is either 35
16
0
or 90
. Since fiber texture has a pole density distribution
symmetrical about the fiber axis or the sample normal, the fiber texture can be
expressed by a fiber texture plot (FTP). Description and examples of a FTP are given
in Chapter 6 on texture correction for phase identification.
8.7.2 ODF of Fiber Texture
The ODF based on the three Eulerian angles can be significantly simplified for fiber
textures. The generally three-dimensional ODF can be expressed by its two-dimen-
sional cross section, since the dependence on the Eulerian angle w
1
vanishes if the
fiber axis is aligned with the Z-axis in the Eulerian space. Consequently, general two-
dimensional pole figures can be expressed by one-dimensional fiber plots. The ODF
for fiber texture can be determined with much less experimental data and calculation
effort. The ODF calculation for fiber texture of cubic materials has been briefly
discussed in Chapter 7, and more details can be found in Refs [14-16].
Figure 8.15 shows the normalized ODF,
cÞ, of a hot extruded rod of a Cu-Be
alloy [15,16]. Since the ODF for fiber texture is reduced to a two-dimensional
distribution, the ODF can be expressed in a contour plot as in Figure 8.15(a) or a
surface plot as in Figure 8.15(b). The Eulerian angles in this example are expressed as
ff
wðW;
cg adopted from the original literature, which have the same definition as
fw
1
; F;
; W;
w
2
g, respectively. The normalized ODF is calculated from the measured
relative intensities of 17 peaks by a spherical harmonic series up to the 16th order. The
fiber plots of various peaks can be calculated from the ODF, and three calculated fiber
plots are shown in Chapter 7 with comparison to measured data.
8.8 OTHER ADVANTAGES OF XRD
2
FOR TEXTURE
XRD
2
systems have many advantages over conventional one-dimensional diffraction
systems when used for texture measurements. The capability of fast and simultaneous
