Page 84 - Materials Chemistry, Second Edition

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2.3. The Crystalline State

Another way to represent the structure factor is shown in Eq. 18, where r(r) is the
electron density of the atoms in the unit cell (r ¼ the coordinates of each point in
vector notation). As you may recall, this is in the form of a Fourier transform; that is,
the structure factor and electron density are related to each other by Fourier/inverse
Fourier transforms (Eq. 19). Accordingly, this relation is paramount for the deter-
mination of crystal structures using X-ray diffraction analysis. That is, this equation
enables one to prepare a 3-D electron density map for the entire unit cell, in which
maxima represent the positions of individual atoms. [41]
Ð
ð18Þ F hkl ¼ rðrÞe 2piðhuþkvþlwÞ dV
V
Ð 1 P P P
ð19Þ rðrÞ¼ F hkl e 2piðhuþkvþlwÞ dV = F hkl e 2piðhuþkvþlwÞ
V
h k l
V
It is noteworthy that the units of functions related by a Fourier transform are recipro-
cals of one another. For instance, consider the reciprocal relationship between the
period and frequency of a wave. Whereas the former is the time required for a
complete wave to pass a fixed point (units of sec), the latter is the number of waves
1
passing the point per unit time (units of s ). Similarly, the unit of the structure factor
˚ 1
˚
(A) is the inverse of the electron density (A ). Since the experimental diffraction
2
pattern yields intensity data (equal to |F| ), the spatial dimensions represented by the
diffraction pattern will be inversely related to the original crystal lattice.
Accordingly, the observed diffraction pattern represents a mapping of a secondary
lattice known as a reciprocal lattice; conversely, the “real” crystal lattice may only
be directly mapped if one could obtain high enough resolution for an imaging device
such as an electron microscope. The reciprocal lattice is related to the real crystalline
lattice by the following (as illustrated in Figure 2.43):
(i) For a 3-D lattice defined by vectors a, b, and c, the reciprocal lattice is defined by
vectors a*, b*, and c* such that a* ⊥ b and c, b* ⊥ a and c, and c* ⊥ a and b:

b c a c a b
ð20Þ a ¼ b ¼ c ¼
½a ðb cފ ½a ðb cފ ½a ðb cފ
Recall that the cross-product of two vectors separated by y results in a third
vector that is aligned in a perpendicular direction as the original vectors.
For instance, for b c, the magnitude of the resultant vector would be |b||c|
sin y, and would be aligned along the a axis. This is in contrast to the dot-product
of two vectors, which results in the scalar projection of one vector onto the other,
of magnitude |b||c| cos y for b • c. For instance, for a cubic unit cell, the
denominator terms in Eq. 20,[a •(b c)], would simplify to:

b c ¼ |b||c| sin 90 ¼ |b||c|, aligned along the a axis;

∴ [a • b c] ¼ |a||b||c| cos 0 ¼ |a||b||c|, or the volume of the unit cell
As you might have noticed, the magnitude of a reciprocal lattice vector is in units
of [1/distance], relative to the basis vectors in real space that have units of
distance.