The Foundations of Chemistry

Figure 13-19 (a) X-ray diffraction

by crystals (schematic). (b) A

photograph of the X-ray diffraction

pattern from a crystal of the enzyme

histidine decarboxylase (MW


37,000 amu). The crystal was rotated

so that many different lattice planes

with different spacings were moved

in succession into diffracting

position (see Figure 13-20).

13-14 Amorphous Solids and Crystalline Solids 511

In 1913, the English scientists William (1862–1942) and Lawrence (1890–1971) Bragg
found that diffraction photographs are more easily interpreted by considering the crystal as
a reflection grating rather than a diffraction grating. The analysis of the spots is somewhat
complicated, but an experienced crystallographer can determine the separations between
atoms within identical layers and the distances between layers of atoms. The more electrons
an atom has, the more strongly it scatters X-rays, so it is also possible to determine the
identities of individual atoms using this technique.
Figure 13-20 illustrates the determination of spacings between layers of atoms. The X-
ray beam strikes parallel layers of atoms in the crystal at an angle. Those rays colliding
with atoms in the first layer are reflected at the same angle. Those passing through the
first layer may be reflected from the second layer, third layer, and so forth. A reflected beam
results only if all rays are in phase.
For the waves to be in phase (interact constructively), the difference in path length must
be equal to the wavelength, , times an integer, n. This leads to the condition known as the
Bragg equation.

n 2 dsin or sin 

It tells us that for X-rays of a given wavelength , atoms in planes separated by distances d
give rise to reflections at angles of incidence. The reflection angles increases with
increasing order, n1, 2, 3,....

n



2 d

(a) Crystal

Slit system

X-ray tube

Photographic film

or recording device

(
)

()

(b)

d

Figure 13-20 Reflection of a

monochromatic beam of X-rays by

two lattice planes (layers of atoms)

of a crystal.