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

TEM Specimens and Images 115
understand the reason for these fringes, imagine the specimen oriented so
that a particular set of lattice planes (indices h k l ) satisfies the Bragg
condition. As electrons penetrate further into the specimen and become
diffracted, the intensity of the (h kl) diffracted beam increases. However,
these diffracted electrons are traveling in exactly the right direction to be
Bragg-reflected back into the original (0 0 0) direction of incidence (central
spot in Fig. 4-12d). So beyond a certain distance into the crystal, the
diffracted-beam intensity decreases and the (000) intensity starts to increase.
At a certain depth (the extinction distance), the (h kl) intensity becomes
almost zero and the (0 0 0) intensity regains its maximum value. If the
specimen is thicker than the extinction depth, the above process is repeated.
In the case of a wedge-shaped (variable-thickness) crystal, the angular
distribution of electrons leaving its exit surface corresponds to the diffraction
condition prevailing at a depth equal to its thickness. In terms of the
diffraction pattern, this means that the intensities of the (0 0 0) and (h kl)
components oscillate as a function of crystal thickness. For a TEM with an
on-axis objective aperture, the bright-field image will show alternate bright
and dark thickness fringes. With an off-axis objective aperture, similar but
displaced fringes appear in the dark-field image; their intensity distribution
is complementary to that of the bright-field image.
A mathematical analysis of this intensity oscillation is based on the fact
that electrons traveling through a crystal are represented by Bloch waves,
whose wave-function amplitude � is modified as a result of the regularlyrepeating
electrostatic potential of the atomic nuclei. An electron-wave
description of the conduction electrons in a crystalline solid makes use of
this same concept. In fact, the degree to which these electrons are diffracted
determines whether solid is an electrical conductor or an insulator, as
discussed in many textbooks of solid-state physics.
4.9 Phase Contrast in the TEM
In addition to diffraction (or scattering) contrast, features seen in some TEM
images depend on the phase of the electron waves at the exit plane of the
specimen. Although this phase cannot be measured directly, it gives rise to
interference between electron waves that have passed through different
regions of the specimen. Such electrons are brought together when a TEM
image is defocused by changing the objective-lens current slightly. Unlike
the case of diffraction-contrast images, a large-diameter objective aperture
(or no aperture) is used to enable several diffracted beams to contribute to
the image.