Chemistry Reference
In-Depth Information
The geometric relations between the incident beam, diffracted beam, and sample
are equivalent in both the u 2u and u u configurations. The incident angle to the
sample varies with the c-tilt angle, so X-ray penetration varies dramatically as well as
the irradiated area on the sample surface due to the defocusing effect. These effects are
more severe at the negative c-tilt angles where v
u. Measurement at low incident
angle is also more sensitive to sample alignment error. Therefore, the iso-inclination
mode should be used for samples with a mild stress gradient in depth and relatively
homogeneous stress distribution on the surface. Measurement of the shear stress
by c-split requires both positive and negative c-tilt angles. By rotating the sample by
180
in the f axis, we have
<
t
fþp
¼½s
13
cosðfþpÞþs
23
sinðfþpÞ ¼ ðs
13
cos fþs
23
sin fÞ¼t
f
ð9
:
61Þ
2
S
fhklg
2
S
fhklg
2
S
fhklg
1
t
fþp
sin 2c
þ
¼
1
t
f
sin 2c
þ
¼
1
t
f
sin 2c
ð9:62Þ
2
2
2
This means that measurement at positive c-tilt angles with 180
sample rotation
will create the same deviation from the straight line in the e
fhklg
fc
sin
2
c plot as at
negative c-tilt angles. Therefore, measurement at negative c-tilts can be replaced
by measurement at positive c-tilts with a 180
rotation of the sample in the f axis.
c-tilt in the side-inclination mode is achieved by a rotation axis within the
diffractometer plane that is perpendicular to themain axes of a goniometer. Therefore,
side-inclination mode requires an additional rotation axis beyond the main axes.
This axis is also called the c axis in Eulerian geometry. In some systems, this axis is
named as the x axis. Hence, the side-inclination mode is also referred to as the
x-method. In a diffractometer with a u 2u configuration, the measurement is done at
v ¼ u, and at u
1
¼ u
2
¼ u for the uu configuration. The c-tilt is directly given by
the goniometer c-angle or x-angle. The incident angle to the sample is given by
a ¼ arcsinðsin u cos cÞ
ð9
:
63Þ
where a is the incident angle. It varies with the c-tilt angle, but not as dramatically as
in the iso-inclinationmode. For example, for {211} planes of Fe with Cr radiation, the
incident angle is u 78
at c ¼ 0, and at the typical maximumc-tilt angle of 45
, the
incident angle is 43.8
. The incident angle is also the same for positive and negative
c-tilt angles. Therefore, X-ray penetration is relatively more consistent with side-
inclination. Side-inclination is also less sensitive to alignment error than the iso-
inclinationmethod. Due to themechanical limitations of the goniometer, it may not be
able to achieve both the positive and negative c-tilt. In this case, measurement at
negative c-tilts can be replaced by measurement at positive c-tilts by rotating the
sample by 180
around the f axis, or vice versa.
9.2.6 Sin
2
c
Method with Area Detector
Diffraction data from area detectors can be used for stress measurement with
conventional methods, such as the sin
2
c method or other algorithms developed for
