Introduction to Nanotechnology

3.2. STRUCTURE 39
Figure 3.2. Two-dimensional cubic lattice showing projections of pairs of (1 10) and (1 20) planes
(perpendicular to the surface) with the distances d between them indicated.
so higher index planes have larger Bragg dimaction angles 8. Figure 3.2 shows the
spacing d for 110 and 120 planes, where the index 2 = 0 corresponds to planes that
are parallel to the z direction. It is clear from this figure that planes with higher
indices are closer together, in accordance with Eq. (3.3), so they have larger Bragg
angles 8 from Eq. (3.2). The amplitudes of the X-ray lines from different crystal-
lographic planes also depend on the indices hkl, with some planes having zero
amplitude, and these relative amplitudes help in identifying the structure type. For
example, for a body-centered monatomic lattice the only planes that produce
observed diffraction peaks are those for which h + k + I = n, an even integer, and
for a face-centered cubic lattice the only observed diffraction lines either have all
odd integers or all even integers.
To obtain a complete crystal structure, X-ray spectra are recorded for rotations
around three mutually perpendicular planes of the crystal. This provides compre-
hensive information on the various crystallographic planes of the lattice. The next
step in the analysis is to convert these data on the planes to a knowledge of the
positions of the atoms in the unit cell. This can be done by a mathematical procedure
called Fourier transformation. Carrying out this procedure permits us to identify
which one of the 230 crystallographic space groups corresponds to the structure,
together with providing the lengths of the lattice constants a,b,c of the unit cell, and
the values of the angles cl,fl,y between them. In addition, the coordinates of the
positions of each atom in the unit cell can be deduced.
As an example of an X-ray diffraction structure determination, consider the case
of nanocrystalline titanium nitride prepared by chemical vapor deposition with the
grain size distribution shown in Fig. 3.3. The X-ray diffraction scan, with the various
lines labeled according to their crystallographic planes, is shown in Fig. 3.4. The fact
that all the planes have either all odd or all even indices identifies the structure as
face-centered cubic. The data show that TIN has the FCC NaCl structure sketched in
Fig. 2.3c, with the lattice constant a = 0.42417nm.
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