Chemistry Reference
In-Depth Information
point. These pole figure pixels should not be confused with pixels that simply have
zero pole density. One method to make this distinction is to store all the filled pixels
with the pole density value plus one, while the unmapped pixels possess the value 0. A
linear interpolation within a defined box is sufficient to fill the unmapped pixels in the
pole figure. The pole figure generated by the pixel mapping can then be processed by
interpolation, normalization, and symmetry operation as described in Chapter 8.
Stress analysis by XRD
2
should be most benefited by the PDD method. In the 2D
method described in Chapter 9, each section of the g-range is integrated into a
diffraction profile first, and then the corresponding 2u value is determined by various
profile fitting methods. Since the large size frame is reduced to a set of (g,2u)) values,
the total number of linear equations in the least squares calculation is relatively small.
This is an advantage only when computing power is very limited. With today's fast
increase in computing power, this advantage gradually diminishes. Furthermore,
there are many disadvantages with the integration and fitting method. For instance,
different integration and fitting methods may give different 2u values from the same
raw data. This is a major source of inconsistency when the same stress data are
evaluated by different fitting algorithms. A “poorly” integrated profile may appear at
some g-integration range due to texture, large screens, shadows, or weak diffraction.
The profile at these g angles may not have enough statistics to determine the peak 2u
position with reasonable accuracy. The 2u values from these data points carry large
error, which has big impact on the least squares regression results. That means a
profile with poor statistics has more weight in the stress calculation. The intensity
weighted least squares method introduced in Chapter 9 may overcome the poor
statistics. The PDD direct method would be able to avoid both the discrepancy in
profile fitting and the poor statistics without ambiguity.
With the PDD approach, each pixel within the selected region is treated like a peak
with angular position (2u, g) and intensity. The pixel intensity weighted least squares
(PIWLS) is used to fit all the pixel values to the fundamental equations for stress. The
summed square of residuals is given in Ref. [9]
S ¼
X
I
i
r
i
¼
X
n
n
2
I
i
ðy
i
^
y
i
Þ
ð13
:
17Þ
i¼1
i¼1
where I
i
is the pixel intensity used as weighting factors, n is the number of total pixels
in the selected region for all 2D frames, and S is the sum of squares error to be
minimized in the least squares regression. The selected region is given by the 2u range
(2u
1
-2u
2
) and g range (g
1
-g
2
) as shown in Figure 9.17. The observed response value
is the measured strain corresponding to each pixel.
sin u
0
sin u
i
y
i
¼ ln
ð13
:
18Þ
and the fitted response value is given by the fundamental equation as
^
y
i
¼ p
11
s
11
þp
12
s
12
þp
22
s
22
þp
13
s
13
þp
23
s
23
þp
33
s
33
þp
ph
s
ph
ð13
:
19Þ
where stress coefficients are calculated from the goniometer angles (v, c, f), the elastic
constants, and the pixel angular position (2u
i
, g
i
). For programming convenience, all
