Physical Principles of Electron Microscopy: An Introduction to TEM ...

106 Chapter 4
4.5 Diffraction Contrast from Polycrystalline Specimens
Many inorganic materials, such as metals and ceramics (metal oxides), are
polycrystalline: they contain small crystals (crystallites, also known as
grains) within which the atoms are arranged regularly in a lattice of rows
and columns. However, the direction of atomic alignment varies randomly
from one crystallite to the next; the crystallites are separated by grain
boundaries where this change in orientation occurs rather abruptly (within a
few atoms).
TEM specimens of a polycrystalline material can be fabricated by
mechanical, chemical or electrochemical means; see Section 4.10. Individual
grains then appear with different intensities in the TEM image, implying that
they scatter electrons (beyond the objective aperture) to a different extent.
Because this intensity variation occurs even for specimens that are uniform
in thickness and composition, there must be another factor, besides those
appearing in Eq. (4.15), that determines the amount and angular distribution
of electron scattering within a crystalline material. This additional factor is
the orientation of the atomic rows and columns relative to the incident
electron beam. Because the atoms in a crystal can also be thought of as
arranged in an orderly fashion on equally-spaced atomic planes, we can
specify the orientation of the beam relative to these planes. To understand
why this orientation matters, we must abandon our particle description of the
incident electrons and consider them as de Broglie (matter) waves.
A useful comparison is with x-rays, which are diffracted by the atoms in
a crystal. In fact, interatomic spacings are usually measured by recording the
diffraction of hard x-rays, whose wavelength is comparable to the atomic
spacing. The simplest way of understanding x-ray diffraction is in terms of
Bragg reflection from atomic planes. Reflection implies that the angles of
incidence and reflection are equal, as with light reflected from a mirror. But
whereas a mirror reflects light with any angle of incidence, Bragg reflection
occurs only when the angle of incidence (here measured between the
incident direction and the planes) is equal to a Bragg angle �B that satisfies
Bragg’s law:
n � = 2 d sin�B
(4.20)
Here, � is the x-ray wavelength and d is the spacing between atomic planes,
measured in a direction perpendicular to the planes; n is an integer that
represents the order of reflection, as in the case of light diffracted from an
optical diffraction grating. But whereas diffraction from a grating takes place
at its surface, x-rays penetrate through many planes of atoms, and diffraction
occurs within a certain volume of the crystal, which acts as a kind of three-