Chemistry Reference
In-Depth Information
both the sample structure and instrumentation. For a perfect random powder sample,
the number of contributing crystallites for a measured line can be given as
Vf
i
v
i
4p
N
s
¼ p
hkl
ð7
:
19Þ
where p
hkl
is the multiplicity of the diffracting planes, V is the effective sampling
volume, f
i
is the volume fraction of the measuring crystallites (f
i
¼1 for single-phase
materials), v
i
is the volume of individual crystallites, and W is the angular window of
the instrument in solid angle. The multiplicity term, p
hkl
, effectively increases the
number of crystallites contributing to the integrated intensity from a particular set of
(hkl ) planes. The volume of individual crystallites, v
i
, is an average of various
crystallite sizes or assumes that all crystallites have the same volume. Assuming
sphere-shaped particles, the term v
i
can be replaced by the particle size, v
i
¼ pd
i
=
6,
where d
i
is the diameter of the crystallite particles. The combination of the effective
sampling volume and the angular window makes up the instrumental window, which
determines the total volume of polycrystalline material making a contribution to a
Bragg reflection. In two-dimensional X-ray diffraction, the instrumental window is
determined not only by the incident beam size and divergence but also by the detective
area and distance of the area detector (g angular coverage).
7.4.1 Effective Sampling Volume
The effective sampling volume is also referred as the effective irradiated volume in
some literature. However, effective sampling volume seems to be a more appropriate
term for two-dimensional diffraction. In conventional diffraction, the effective
irradiated volume can be uniquely calculated from the given incident angle and
2u angle. In Bragg-Brentano geometry, the effective irradiated volume is a constant,
but in two-dimensional diffraction, several 2u angles at various g angles are measured
simultaneously, and the effective sampling volume is a function of both 2u and g.
It can be hard to comprehend that the same instrument setup can result in variable
irradiated volumes. The effective sampling volume can be obtained by multiplying
the beam cross-sectional area with the transmission coefficients given in Chapter 2 for
absorption corrections. For the Bragg-Brentano geometry, it is given as
A
0
2m
V ¼ A
0
A
BB
¼
ð7
:
20Þ
where A
0
is the cross-sectional area of the incident beam, A
BB
¼ 1
=ð2mÞ is the
transmission coefficient for BB geometry, and m is the linear absorption coefficient.
The cross-sectional area A
0
is a constant only if there is no divergence in the
incident and diffracted beams. It means that the divergence slit and receiving slit
have the same or smaller aperture size than the focal spot. For divergent beams
commonly usedwith Bragg-Brentano geometry, the cross-sectional area A
0
should be
measured near the sample surface or given by the irradiated area, A
i
, at a particular 2u
angle as A
0
¼ A
i
=
sin u. For two-dimensional X-ray diffraction, the incident
beam cross-sectional area can be considered as a constant most of the time because
