Chemistry Reference
In-Depth Information
where A is the transmission coefficient,
y
is the extinction coefficient [
22
], a is the
correction for thermal diffuse scattering (TDS) [
23
],
v
is the sample volume, B is
the
background
, and
p
m
Q
(S
m
) is the contribution to the wave scattered along the
direction S
i
from all other vectors S
m
through the so-called multiple scattering. The
integrated reflectivity
Q
(S) per volume of the crystal is
a
2
l
3
V
2
P
sin 2y
F
Bragg
ð
2
Q
ð
S
i
Þ¼
S
i
Þ
:
(7)
e
2
/
mc
2
, l is the wavelength,
V
is the cell volume,
P
is the polarization
factor, and F is the structure factor. After applying several corrections, it is then
possible to obtain the Bragg intensity.
The structure factor is the Fourier transform of the thermally averaged electron
density:
a
¼
ð
V
h
e
2p
i
S
i
r
dr
F
ð
S
i
Þ¼
r
ð
r
Þi
:
(8)
Within an atomistic approximation, the structure factor can be expressed in
terms of the atomic form factors, mean positions and mean-square displacements:
X
F
ð
S
i
Þ¼
f
a
ð
S
Þ
exp
ð
2p
i
S
i
r
a
Þ
T
a
ð
S
i
Þ:
(9)
a
T(
S
)
is the Debye-Waller factor introduced in (2). The atomic form factors are
typically calculated from the spherically averaged electron density of an atom in isolation
[
24
], and therefore they do not contain any information on the polarization induced by
the chemical bonding or by the interaction with electric field generated by other atoms or
molecules in the crystal. This approximation is usually employed for routine crystal
structure solutions and refinements, where the only variables of a least square refinement
are the positions of the atoms and the parameters describing the atomic displacements.
For more accurate studies, intended to determine with precision the electron density
distribution, this procedure is not sufficient and the atomic form factors must be modeled
more accurately, including angular and radial flexibility (Sect.
4.2
).
As outlined above (6), there are many factors that affect the measured intensities,
and therefore in a typical X-ray diffraction experiment there are many sources of
systematic errors. The accuracy of the parameters obtained by X-ray crystal structure
analysis depends on the measuring procedure, the strategy of the data-collection,
the treatment of measured intensities to extract Bragg structure factors, the quality of
the crystal sample, and its handling. It is important to recognize that cooling the
sample cannot solve all inherent defects of a diffraction experiment, although it is true
that it could enormously improve the quality of the data. First and most important,
lower temperature reduces the atomic displacements, and consequently produces
more Bragg scattering. Moreover, radiation damage of the crystals and dynamical
disorder (if present) may be significantly reduced. Therefore, it is typical to read that
low temperature
improves the quality of the data, but the meaning of
low
depends on
the material under study. One could use the Debye temperature as a kind of benchmark.
