In this expression, Nis the total number of atomic sites, Q
v
is the energy required
for the formation of a vacancy, Tis the absolute temperature
1
in kelvins, and k
is the gas or Boltzmann’s constant. The value of kis 1.38 ⫻10
⫺
23
J/atom-K, or
8.62 ⫻10
⫺
5
eV/atom-K, depending on the units of Q
v
.
2
Thus, the number of
vacancies increases exponentially with temperature; that is, as Tin Equation 5.1
increases, so does also the expression exp ⫺(Q
v
/kT). For most metals, the fraction
of vacancies N
v
/N just below the melting temperature is on the order of 10
⫺
4
; 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 tempera-
ture similar to that of Equation 5.1.
Aself-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 5.1. In metals, a self-interstitial introduces
relatively large distortions in the surrounding lattice because the atom is substan-
tially larger than the interstitial position in which it is situated. Consequently,
the formation of this defect is not highly probable, and it exists in very small
concentrations, which are significantly lower than for vacancies.
E
XAMPLE
P
ROBLEM
5.1
Calculate the equilibrium number of vacancies per cubic meter for copper at
1000⬚C. The energy for vacancy formation is 0.9 eV/atom; the atomic weight
and density (at 1000⬚C) for copper are 63.5 g/mol and 8.40 g/cm
3
, respectively.
S
OLUTION
This problem may be solved by using Equation 5.1; it is first necessary, however,
to determine the value of N, the number of atomic sites per cubic meter for
104 ●Chapter 5 / Imperfections in Solids
F
IGURE
5.1 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.
Reprinted by permission of John
Wiley & Sons, Inc.)
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⫽
8.31 J/mol-K, or 1.987 cal/mol-K.
