Chemistry Reference
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and the incident X-ray beam is 90
þu and the apex angle of a vector cone is 90
u.
It is apparent that diffraction vector cones can only exist on the X
L
side of the
diffraction space.
The diffraction vector is given in laboratory coordinates by
2
3
2
3
s
x
s
0x
s
y
s
0y
s
z
s
0z
cos 2u 1
sin 2u sin g
sin 2u cos g
s
s
0
l
1
l
1
l
4
5
¼
4
5
H ¼
¼
ð2
:
6Þ
The direction of each diffraction vector can be represented by its unit vector
given by
2
3
2
3
h
x
h
y
h
z
sin u
cos u sin g
cos u cos g
H
jHj
¼
4
5
¼
4
5
h
L
¼
ð2
:
7Þ
where h
L
is a unit vector expressed in laboratory coordinates and the three
components in the brackets are the projections of the unit vector on the three axes
of laboratory coordinates, respectively. If g takes all values from 0 to 360
at a given
Bragg angle 2u, the trace of the diffraction vector forms a diffraction vector cone.
Since the possible values of u are within 0 to 90
, the trace of the unit vector of
diffraction vectors for all possible u and g values forms a hemisphere with radius 1.
ItcanalsobeseenfromEq.(2.7)thath
x
takes only negative values. If g takes only
the values within the diffractometer plane, that is, g ¼90
on the negative Y
L
side
and g ¼ 270
on the positive Y
L
side, the diffraction vectors stay within the
diffractometer plane. This is the case in the conventional diffractometer and
h
Z
¼0. Since a diffraction vector is always perpendicular to the corresponding
crystal planes, the diffraction vector and its expression in the laboratory coordinates
are mostly used for analyzing orientation-sensitive diffraction data, texture or
stress, for instance.
The diffraction vector has been defined based on the Bragg condition. Therefore,
a diffraction vector should be normal to the lattice planes with a given d-spacing. To
analyze all the X-rays measured by an area detector, we extend the concept to all
scattered X-rays from a sample regardless of the Bragg condition. In general
physics, the diffraction vector, also referred to as scattering vector, is defined as
the difference between thewave vectors of the scattered wave and the incident wave.
Although there seems to be a preference in different fields between the terms
“diffraction vector” and “scattering vector,” we will use diffraction vector or
scattering vector alternatively. We can simply describe a diffraction vector as a
vector that takes the direction bisecting the incident beam and the scattered beam,
and has a dimension of an inverse length given by 2 sin u=l.Here2u is the scattering
angle from the incident beam. When the Bragg condition is satisfied, the diffraction
vector is normal to the diffracting lattice planes and itsmagnitude is reciprocal to the
d-spacing of the lattice planes. In this case, the diffraction vector is equivalent to the
reciprocal lattice vector. Each pixel in an area detector measures scattered X-rays in
a given direction with respect to the incident beam. We can calculate a diffraction
