Page 135 - Materials Science and Engineering An Introduction

4.2 Vacancies and Self-Interstitials • 107

Figure 4.1 Two-dimensional representations of a vacancy
and a self-interstitial.
(Adapted from W. G. Moffatt, G. W. Pearsall, and J. Wulff, The
Structure and Properties of Materials, Vol. I, Structure, p. 77.
Copyright © 1964 by John Wiley & Sons, New York, NY.
Reprinted by permission of John Wiley & Sons, Inc.) Self-interstitial
Vacancy

Scanning probe
micrograph that
shows a vacancy on
a (111)-type surface
plane for silicon. defects. The necessity of the existence of vacancies is explained using principles of
Approximately thermodynamics; in essence, the presence of vacancies increases the entropy (i.e., the
7,000,000 .
(Micrograph courtesy randomness) of the crystal.
of D. Huang, Stanford The equilibrium number of vacancies N y for a given quantity of material (usually
University.) per meter cubed) depends on and increases with temperature according to
Temperature
dependence of the N y = N expa - Q y b (4.1)
equilibrium number kT
of vacancies
In this expression, N is the total number of atomic sites (most commonly per cubic me-
ter), Q y is the energy required for the formation of a vacancy (J/mol or eV/atom), T is
1
Boltzmann’s constant the absolute temperature in kelvins, and k is the gas or Boltzmann’s constant. The value
#
#
2
5
of k is 1.38 10 23 J/atom K, or 8.62 10 eV/atom K, depending on the units of Q y .
Thus, the number of vacancies increases exponentially with temperature—that is, as T
in Equation 4.1 increases, so also does the term exp( Q y /kT). For most metals, the frac-
4
tion of vacancies N y /N just below the melting temperature is on the order of 10 —that
is, one lattice site out of 10,000 will be empty. As ensuing discussions indicate, a number
of other material parameters have an exponential dependence on temperature similar
to that in Equation 4.1.
self-interstitial A self-interstitial is an atom from the crystal that is crowded into an interstitial
site—a small void space that under ordinary circumstances is not occupied. This kind of
defect is also represented in Figure 4.1. In metals, a self-interstitial introduces relatively
Tutorial Video: large distortions in the surrounding lattice because the atom is substantially larger than
Computation of the interstitial position in which it is situated. Consequently, the formation of this defect
the Equilibrium is not highly probable, and it exists in very small concentrations that are significantly
Number of lower than for vacancies.
Vacancies

1 Absolute temperature in kelvins (K) is equal to C 273.
2 #
Boltzmann’s constant per mole of atoms becomes the gas constant R; in such a case, R 8.31J/mol K.