Physical Chemistry Third Edition

28.1 The Structure of Solids 1161

The simplest X-ray diffraction experiment is carried out with a finely powdered

sample of the crystalline material placed in an X-ray beam. Since there are many

small crystals with many different orientations, a collimated X-ray beam strikes some

crystals at any given angle, and a number of diffracted beams come from the sam-

ple in cones of directions concentric with the incident beam. A photographic film or

other detector is placed to intercept these beams and record their positions, allow-

ing one to calculate the diffraction angles. Analysis of the pattern of the diffraction

angles allows one to determine from the extinction conditions whether one has a

primitive lattice, a face-centered lattice, or a body-centered lattice. From the wave-

length of the radiation and the diffraction angles, one can determine the unit cell

dimensions.

We have discussed the diffraction of X-rays as though each atom in the crystal lattice

were a point from which the X-rays are scattered. In fact, the principal scattering is

from electrons, which are distributed over the entire unit cell. The scattering from the

different parts of the unit cell interferes constructively and destructively in ways that are

determined by the electron density in the unit cell. Analysis of the relative intensities

of the different diffracted beams allows in some cases for the reconstruction of the

electron density as a function of position in the unit cell. This is a complicated process,

which we do not describe. The first such structure determinations were done before the

advent of programmable computers, with many hours of hand calculation. Present-day

calculations are done automatically by computer programs, using intensity data taken

with automated computer-driven diffractometers.

An edge-type dislocation

A step defect

A screw-type dislocation

Figure 28.6 Some Crys-

tal Defects (Schematic).

These types of defects are

the most common simple

types of crystal defects.

Modern studies in surface catalysis often use single crystals with an exposed face

whose Miller indices are known,^2 and it is sometimes found that different planes have

different catalytic activities. There can also be a variety of defects in a real crystal, some

of which are schematically depicted in Figure 28.6, and these defects can be involved

in catalysis.

PROBLEMS

Section 28.1: The Structure of Solids

28.1Gold (Au) forms a face-centered cubic lattice. Its density

is 19.3 g cm−^3 at 300 K. Find the unit cell dimension and

the interatomic distance at this temperature.

28.2Chromium (Cr) forms a body-centered cubic lattice. Its

density is 7.19 g cm−^3. Find the unit cell dimension and

the interatomic distance.

28.3Explain why a single crystal has a greater mechanical

strength than a polycrystalline sample of the same

material.

28.4Tennis racquet frames and golf club shafts are commonly

made of a composite material consisting of a polymer in

which fibers of graphite are embedded. Explain why this

makes a stronger material than the polymer alone or the

graphite alone.

28.5Gallium crystallizes in a primitive orthorhombic lattice.

Its density is 5.92 g cm−^3. Find the unit cell volume.

28.6What is the basis for a crystal of argon, which forms a

face-centered cubic lattice? What is the number of atoms

per unit cell? What is the number of bases per unit cell?

28.7The CsCl crystal has a cubic unit cell with a cesium ion at

each corner and a chloride ion at the center.

a.To which Bravais lattice does it belong?

(^2) D. W. Goodman,Ann. Rev. Phys. Chem., 37 , 425 (1986).