bei48482_FM

The Solid State 353

Example 10.3

The resistivity of copper at 20C is 1.72     10 ^8 

m. Estimate the mean free path 

between collisions of the free electrons in copper at 20C.

Solution

In Example 9.8 we found that the free electron density in copper is n8.48  1028 m^3 , and

earlier in this section we saw that the Fermi velocity there is F1.57     106 m /s. Solving

Eq. (10.16) for gives



3.83   10 ^8 m38.3 nm

The ions in solid copper are 0.26 nm apart, so a free electron travels past nearly 150 of them,

on the average, before being scattered.

The scattering of free electron waves in a metal that leads to its electric resistance

is caused both by structural defects and by ions out of place as they vibrate. Imper-

fections of the former kind do not depend on temperature but on the purity of the

metal and on its history. The resistivities of cold-worked metals (such as “hard drawn”

wires) are lowered by annealing because the number of defects is thereby decreased.

On the other hand, lattice vibrations increase in amplitude with increasing tempera-

ture, and their contribution to resistivity accordingly goes up with temperature. Thus

the resistivity of a metal is the sum it, where idepends on the concentra-

tion of defects and tdepends on temperature.

Figure 10.18 shows how the resistivities of two sodium samples vary with temper-

ature. The top curve corresponds to the sample with the higher concentration of de-

fects, which accounts for its upward displacement. In very pure and almost defect-free

samples, iis small, and at low temperatures, tis also small. When both these

conditions hold in copper, for example, the mean free path may be 10^5 times the value

found in Example 10.3.

(9.11   10 ^31 kg) (1.57   106 m /s)



(8.48   1028 m^3 )(1.60    10 ^19 C)^2 (1.72  10 ^8 

m)

mF



ne^2 

5 × 10 –^3

4 × 10 –^3

3 × 10 –^3

2 × 10 –^3

1 × 10 –^3

Relative resistivity,

ρ/

ρ(290 K)

Temperature, K

0 5 10 15 20

Figure 10.18Resistivities of two sodium samples at low temperatures relative to their resistivities at

290 K. The upper curve corresponds to the sample with the higher concentration of impurities.

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