Chemistry Reference
In-Depth Information
s
s
s
s
s
s
1〉
2〉
i
-1〉
i
〉
n
〉
t
1
t
2
X
s
1
X
s
2
s
i
-1
t
t
i-
1
X
t
i
X
s
i
s
n
t
n
X
FIGURE 9.29 The measured average stress value s
hii
for each measurement depth t
i
and the
corresponding calculated stress value s
i
at the depth t
i
.
For a first-order approximation, we can assume that the overall diffraction cone
distortion contributed by each thickness element is proportional to the fraction of the
diffracted intensity from that thickness element. Therefore, the stress value at the
penetration depth t corresponding to 50 percent of the total diffracted intensity is also
the diffraction weighted average stress for the full penetration, assuming a linear
stress depth gradient. In this case, G
t
¼0.5 and
0
693 sin v sin
ð
2u
v
Þ
m½sin vþ sinð2uvÞ
:
t ¼
ð9
:
105Þ
As shown in Figure 9.29, with a series of incident angles we can have stress
measured for t
1
;
t
n
with t
1
corresponding to the lowest incident angle.
The distribution above t extends to a much greater depth, so we consider only the
stress distribution within each depth t
i
.Att
1
, we have
s
1
¼ s
h1i
t
2
; ...;
t
i
; ...;
ð9
:
106Þ
We assume that the measured average stress in t
2
is a superposition of the
contribution from the layers t
1
and t
2
t
1
, that is
t
1
t
2
t
1
t
2
s
h2i
¼ s
h1i
1 exp
þs
2
exp
ð9
:
107Þ
Then, we have a general equation to calculate the stress value at depth t
i
þs
hi 1i
exp
t
i
1
t
i
s
i
¼ s
hii
s
hi 1i
ð9
:
108Þ
The above equation is a rather simplified approximation. More sophisticated
algorithms can be developed by following the strategies used in layer removal
methods [53-57].
