is in fact very complex because most compounds, being molecular, will have
many atoms in corresponding unit-cell positions that can contribute to
diffractions.
There is an additional complication depending on the sample’s form: that
of crystal orientation. A sample of material that exists as one single crystal has
specific unit cell orientations. The refractions of X rays will also have certain
specific orientations (thanks in part to Bragg’s law), and experimental deter-
minations of X-ray diffractions use that specificity to determine the structures
of compounds; computers are extremely useful in back-calculating a molecu-
lar structure from the angles of X-ray diffraction. However, if the sample is
powdered or polycrystalline, each tiny crystal in a sample has its own orienta-
tion with respect to the incoming X rays, and so will impose its own unique
direction to the outgoing, diffracted X rays. X-ray diffraction patterns of pow-
dered samples are usually much more complicated than those of single crystal
X-ray diffractions, although the so-called powder patterns are typically easier
to obtain experimentally. (The exception is for cubic unit cells; because the
unit cells are cubic, it does not matter what orientation each tiny microcrys-
talline fragment has.) As you might expect, scientists who perform X-ray dif-
fraction prefer single-crystal samples to determine the molecular structure of
a compound. This can be difficult to provide, especially if the compound is a
large biomolecule that is hard to crystallize.21.6 Miller Indices
The previous section considered diffractions of X rays as if they were done
by a single layer of atoms in a crystal. Actually, they aren’t, as illustrated by
Figure 21.16: diffractions are caused by the constructive interference of re-
flections of X rays by sequential planes of atoms in similar unit-cell positions
throughout the crystal. The collection of planes actually makes X-ray dif-
fraction a three-dimensional phenomenon (even though Figure 21.16 shows
it in two dimensions).
However inaccurate the depiction, the idea of X-ray diffraction does bring
up the concept that planes of atoms are important in an understanding of
crystal structures. In the first approximation, individual planes of atoms re-
flect X rays, and the constructive interference of many reflections from many
planes yields refraction of X rays at the right angle. How do we define a plane
of atoms in terms of the unit cell? There is another consideration, too: at some
point, the supposed infinite array of unit cells must, in fact, stop and make
the surface of the crystal. This surface is usually considered planar. (In fact,
many examples of large crystals are used as examples because they have well-
defined planar surfaces. A well-cut diamond, for example, has a very specific
shape in terms of the planes that terminate the unit cells.) It becomes clear
that we must be able to define planes of corresponding atoms within arrays
of unit cells.
We use Miller indicesto label the different possible planes that correspond-
ing species can make. (Recall that indicesis the plural ofindex.Although this
system was originally established by William Whewell in 1825, it was popular-
ized in an 1839 crystallography text by William Miller, an English mineralo-
gist.) Miller indices are based on the unit cell dimensions a,b, and cacting as
unit vectors. They are also reciprocal indicesin that they relate to the recipro-
cal of the fraction or multiple of unit vectors along each axis where any given
plane intersects.744 CHAPTER 21 The Solid State: Crystals