Callister - An introduction - 8th edition

18.8 Electrical Resistivity of Metals • 729
Electrical resistivity (10 –8 Ω.m)
6
5
4
3
2
1
Temperature (°F)
–400 –300 –200 –100 0 +100
Cu + 3.32 at% Ni
Cu + 2.16 at% Ni
Deformed
d
Cu + 1.12 at% Ni
i
"Pure" copper
Figure 18.8 The
electrical resistivity versus
temperature for copper
and three copper–nickel
alloys, one of which has
been deformed. Thermal,
impurity, and deformation
contributions to the
resistivity are indicated
at 100C. [Adapted from
J. O. Linde, Ann. Physik,
5, 219 (1932); and C. A.
Wert and R. M. Thomson,
Physics of Solids, 2nd
edition, McGraw-Hill
Book Company, New
York, 1970.]
t
0
–250 –200 –150 –100 –50 0
Temperature (°C)
+50
Matthiessen’s rule—
for a metal, total
electrical resistivity
equals the sum of
thermal, impurity,
and deformation
contributions
Matthiessen’s rule
Dependence of
thermal resistivity
contribution on
temperature
Impurity resistivity
contribution (for
solid solution)—
dependence on
impurity
concentration (atom
fraction)
mechanisms act independently of one another. This may be represented in mathematical
form as follows:
total t i d (18.9)
in which t , i , and d represent the individual thermal, impurity, and deformation
resistivity contributions, respectively. Equation 18.9 is sometimes known as
Matthiessen’s rule. The influence of each r variable on the total resistivity is demonstrated
in Figure 18.8, a plot of resistivity versus temperature for copper and several
copper–nickel alloys in annealed and deformed states. The additive nature of
the individual resistivity contributions is demonstrated at 100C.
Influence of Temperature
For the pure metal and all the copper–nickel alloys shown in Figure 18.8, the resistivity
rises linearly with temperature above about 200C. Thus,
(18.10)
where and a are constants for each particular metal. This dependence of the thermal
resistivity component on temperature is due to the increase with temperature
in thermal vibrations and other lattice irregularities (e.g., vacancies), which serve as
electron-scattering centers.
Influence of Impurities
For additions of a single impurity that forms a solid solution, the impurity resistivity
is related to the impurity concentration c i in terms of the atom fraction
(at%/100) as follows:
r i
r 0
r t r 0 aT
i Ac i (1 c i ) (18.11)