Page 18 - Modern physical chemistry

1.B Miller Indices 1

1.7 Standard Crystal Lattices

The symmetry that a crystal displays has long been known and its cause discussed.
In 1690, Huygens suggested that the external form was determined by how particles com-
posing the given crystal were arranged. In 1813, Wollaston wrote that the particles could
be replaced by points having suitable properties. Bravais in 1848 adopted this view and
discussed the 14 different ways in which the points could be arranged in space.
Since 1912, the Braggs and others have used diffraction methods to study the various
parallel planes within crystals. They could thus find internal planes not parallel to exter-
nal faces and determine all pertinent interplanar spacings. It was found that three sets
of prominent planes intersect each other to divide a crystal into small parts. Such a part
that can serve as a representative unit of a crystal is called a unit ceU.
Only a few kinds of geometric solids can be unit cells in 3- dimensional space. Solids
having cross sections with 5-fold, 7-fold, 8- fold, or higher-fold axes leave empty regions,
as figure 1.5 illustrates. The possibilities that are observed appear in figure 1.6.
For each, we let the lower front left comer be the origin and the three edges that
intersect at this point be the x, y, and z axes. The distance that the unit cell extends along
the x axis is labeled a, the distance along the y axis b, and the distance along the z axis
c. We designate the angle between the y and z axes a, the angle between the z and x axes
f3 , and the angle between the x and y axes y. The cells in figure 1.6 are then described
by the conditions in table 1.1.
Because the X-ray mirrors are planes of high average atomic density, the highest den-
sities appear where the planes intersect. So one expects to find an atom or molecule at
each comer of a unit cell, where three prominent planes intersect. Bravais noted that
particles may lie at other positions also, as table 1.2 indicates. He considered only point
lattices, lattice structures in which all the structural units (atoms or molecules) are alike
and equivalent to points. More general lattices are described in terms of these, however.

I.B Miller Indices
Each set of parallel reflecting planes is characterized by the distance between suc-
cessive planes and by the orientation of anyone of them. The former is given by the
number d. The latter, on the other hand, is described by the intercepts on the x, y, z axes,
or by their reciprocals.
Consider a typical plane of the given set. Suppose that the axes are drawn as in the
appropriate figure 1.6 and that the plane cuts the axes at points A, B, and G, as in figure 1.7.
If the unit cell edges are taken to be the unit lengths, then the intercepts are OA/a,oBlb,OGIc
units from the origin, while the reciprocals of these measures are aIOA, blOB, clOG.
As long as the crystal is formed only of equivalent unit cells, these reciprocals are
rational. This result is called the law of rational indices. On multiplying the reciprocals
by an appropriate factor, one obtains three small whole numbers. The smallest of these
are the MiUer indices for the set of planes.

FIGURE 1.S Failure of 5-, 7-, and 8-sided
figures to fill space