Chemistry Reference
In-Depth Information
where
D
k
¼ p
11
s
11
þp
21
s
12
þp
22
s
22
þp
13
s
13
þp
23
s
23
þp
33
s
33
It must be noted that when calculating D
0
, u
0
is a constant for all of the diffraction
ring, while u
1
;
are functions of g. The sin u
0
term in the equation should
always be the same. The iteration starts with k ¼ 0 until a stop condition is satisfied
u
2
;
u
3
; ...
du at all g angles
where du is the required accuracy. Typically, one or two iterations can achieve an
accuracy better than 0.01
. The accurate simulation of the diffraction ring is typically
used for displaying the measured data points against the simulated ring from the
measured stress tensor. It is a useful way of observing the scatter of the measured data
points to evaluate the quality of the measurement [28]. If the approximate d-spacing
d
0
has been used, the effect of the pseudohydrostatic term s
ph
should be included in
the equation as
u
k þ 1
u
k
<
D
k
¼ p
11
s
11
þp
21
s
12
þp
22
s
22
þp
13
s
13
þp
23
s
23
þp
ph
s
ph
ð9
:
90Þ
1
where the coefficient p
ph
¼ 12n=
2
S
2
þ3S
1
, and the term p
ph
s
ph
is a constant.
By definition, s
33
¼ 0, so the term p
33
s
33
is omitted.
The simulated diffraction ring can be displayed as a radar chart or a 2u-g plot.
Figure 9.12 shows simulated diffraction rings for iron's (211) peak with Cr-K
a
radiation and E¼210000MPa, n ¼0.28, d
0
¼1.1702A
, and l¼2.2897A
. The sam-
ple orientation is set at v¼90
and c¼0 so that the incident beam is perpendicular to
the sample surface. The stress tensor is s
11
¼1000 MPa and s
22
¼ 1000 MPa. The
top of Figure 9.12 is a radar chart with a stress scale in the radial direction and g in the
azimuthal direction. The perfect circular ring corresponds to 2u
0
¼ 156
, which can
also be considered diffraction rings from a stress-free sample. There are two distorted
rings. The one with the solid line is the diffraction ring, based on the isotropic
assumption, that is calculated from the macroscopic elasticity constants. The one with
the broken line stands for the diffraction ring, based on the anisotropic assumption, that
is calculatedwith theXEC. In thismodel, the XEC are generated from themacroscopic
elasticity constant, lattice plane index {hkl}, and the anisotropic factor A
RX
¼ 1
E ¼
49.
The 2u scale is enlarged to 155.80
-156.25
from the center to the outer circle so that
the 2u shift from the stress-free ring can be easily observed.
At a sample rotation angle f¼0
, the 2u values increase in the horizontal direction
due to the compressive stress component s
11
¼1000 MPa and decrease in the
vertical direction due to the tensile stress component s
22
¼ 1000 MPa. v and
c-rotation will change the shape of the distorted ring, and f-rotation will rotate the
ring above its center but not change the shape, since the rotation axis is perpendicular
to the plane of biaxial stress. The diffraction ring from the above stress condition
at f¼45
is identical to the diffraction ring from shear stress s
12
¼ 1000 MPa (other
components are zero) at f¼0
. It is reasonable that, mechanically, the two stress
conditions are equivalent with a f¼45
offset. It should be noted that the actual
diffraction ring shift is reversed on a 2Ddetector since the diffraction cone apex angles
:
