Chemistry Reference
In-Depth Information
or
Ið2uÞ¼
g
2
g
1
Jð2u
; gÞ
2u
1
2u 2u
2
ð6
:
10Þ
where J(2u, g) represents the two-dimensional intensity distribution in the 2D frame
and I(2u) is the integration result as a function of intensity versus 2u. g
1
and g
2
are
the lower limit and upper limit of integration, which are constants for g-integration.
Due to the discrete nature of the diffraction frame, Eq. (6.10) is used to sum the counts
within each D2u step. Slice integration, in principle, is the same as g-integration
except that the lower limit and upper limit of integration are given by a constant
vertical pixel range, for instance, 200 pixels in this example. Slice integration can be
expressed as
ð
g
2
ð2u;y
2
Þ
Ið2uÞ¼
Jð2u
; gÞdg
2u
1
2u 2u
2
ð6
:
11Þ
g
1
ð2u
;
y
1
Þ
or
Ið2uÞ¼
X
g
2
ð2u
;
y
2
Þ
Jð2u
; gÞ
2u
1
2u 2u
2
ð6
:
12Þ
g
1
ð2u;y
1
Þ
where the lower limit g
1
is a function of 2u and constant y
1
and the upper limit g
2
is a
function of 2u and constant y
2
.
A 2Ddiffraction frame collected with a 2Ddetector consists of a series of diffraction
rings as shown in Figure 6.8. The 2D detector is the Hi-Star detector positioned at a
sample-to-detectordistanceD¼7.5 cmandswinganglea¼60
.Theframe format is
512512 pixels. The sample is corundum (a-Al
2
O
3
) powder with a diffraction pattern
thatmatches thePDFcard46-1212. The 2u range is from20
to60
and the2u integration
stepsize is0.05
.Figure6.8(a)showsg integrationfrom60
to120
. Figure6.8(b) shows
sliceintegrationwiththesame2urange, from20
to60
andwitha2u integrationstepsize
of 0.05
. The lower limit y
1
is the lowest pixel position in the integration range, which is
100 pixels below the center of the detector and the upper limit y
2
is the highest pixel
position in the integration range, which is 100 pixels above the detector center. A
diffraction profile similar to the conventional diffraction result can be obtained by either
g-integration or slice integration over a select 2u range. Phase ID can then be done with
conventional search/matchmethods.Discrepancies intherelative intensitiesbetweenthe
profiles of two different integration methods are due to the preferred orientation.
It can be observed from Figure 6.8(a) and (b) that overall intensity of both
diffraction patterns are about the same although the g integration ranges are
significantly different. This is because the diffraction profiles are normalized by the
solid angle. To reduce or eliminate the dependence of the integrated intensity on
the integration interval, the integrated value at each 2u step can be normalized by the
number of pixels, the arc length or the solid angle. g-Integration with normalization
