Chemistry Reference
In-Depth Information
interpolated to fill the gap. The pole figures have been measured up to a x angle of
approximately 85
(a¼5
), so the pole figures are not complete, as is shown by the
broken line circles. The pole densities in the regions of x>85
are very difficult to
measure due to the low incident and reflection angles. Due to crystal symmetry, the
three incomplete pole figures are sufficient to calculate the ODF. The ODF calculation
takes advantage of the redundancy in pole density information and crystal symmetry
from all the measured pole figures, so the ODF contains improved statistics and
integrity of texture information. Therefore, the recalculated three pole figures for
(111), (200), and (220), as shown in row (b) are complete, symmetric, and smooth.
This process has been frequently used to generate complete pole figures from
incompletely measured pole figures or pole density data. Once the ODF is obtained,
pole figures of any other orientation can be calculated. Row (c) of Figure 8.13 shows
the calculated pole figures for crystallographic planes of (311), (331), and (420). The
pole figures of the crystallographic planes with zero structure factors may never be
measured by X-ray diffraction, but can be calculated from the ODF based on available
measured pole figures of other planes.
8.7 FIBER TEXTURE
If the texture has a rotational symmetry with respect to a sample orientation, the
texture is referred to as a fiber texture. The sample orientation containing the
symmetry axis is referred to as the fiber axis. If the fiber axis is aligned to the ND
(S
3
) direction as shown in Figure 8.3(a), the pole density distribution function
becomes independent of the azimuthal angle b. The fiber texture is mostly observed
in two types of materials, metal wires or rods formed by drawing or extrusion and thin
films formed by physical or chemical deposition. The fiber axis is the wire axis for a
wire and normal to the sample surface for thin films. Fiber texture can also be
artificially formed by spinning a sample about its normal. This is a very useful feature
since the relative intensity correction for a texture can be significantly simplified with
sample spinning. The details have been covered in Chapter 7 on phase identification.
8.7.1 Pole Figures of Fiber Texture
Figure 8.14 shows the pole figures and ODF of fiber texture measured on a magnetron
sputter-deposited Cu film onto a Si (111) wafer with the experimental conditions and
data collection strategy illustrated in Figures 8.6 and 8.7. Figure 8.14(a) shows the 3D
surface plots of the three measured pole figures for the (111), (200), and (220) planes.
The pole density distribution for (111) is concentrated in the center of the pole figure.
The pole density distributions of (200) and (220) are concentrated in the regions that
match the corresponding angles of the (200) plane and the (220) plane with respect to
the (111) plane. This means the (111) planes are mostly aligned approximately
parallel to the film, or the [1 1 1] directions are aligned to the fiber axis or the sample
normal direction. This fiber texture is called a (111) fiber texture. Figure 8.14(b)
shows the contour plots of the three measured pole figures. Figure 8.14(c) shows four
cross sections of the ODF at w
1
¼0
,30
,60
, and 90
(19 cross sections from 0
to
