Page 39 - Electrical Properties of Materials

22 The electron as a particle

Table 1.2 Electrical and thermal conductivities measured at 293 K

–8
–1
7
–1
–2
–1
–1
Metal σ(10 m ) κ(Wm K ) C WF (10 W K )
Silver 6.15 423 2.45
Copper 5.82 387 2.37
Aluminium 3.55 210 2.02
Sodium 2.10 135 2.18
Cadmium 1.30 102 2.64
Iron 1.00 67 2.31
only in the x direction. If there is a heat flow the average energy may change
slightly from point to point. Taking an interval from x – to x + (remember
is the mean free path) the average at the two boundaries will be E –(dE/ dx)
and E +(dE/dx) respectively. Referring now to a result from the kinetic the-
ory of gases that the number of particles flowing in a given direction per unit
1
surface per unit time is N e v th , the net flow across the plane at x is
6

dE
net energy flow = (1/3)N e v th . (1.79)
dx
According to the simple theory of heat, the flow of heat energy is proportional
to the gradient of temperature where the proportionality constant is the heat
conductivity, κ, yielding

dT

net heat energy flow = κ . (1.80)
dx
Equating now eqn (1.79) with (1.80) we obtain

1
κ = N e v th k B . (1.81)
2
We may now relate the heat conductivity to the electrical conductivity as
follows:
1 2
κ 2 N e v th k B 3 k B
= = T, (1.82)
2
σ N e (e τ)/m 2 e
where the relation = v th τ (neglect v D in eqn (1.11)) has been used. The
functional relationship is exactly the same: the ratio of the two conductivities is
indeed proportional to T as was stipulated by the empirical formula. But what
is the value of the constant? Inserting the values of e and k B into eqn (1.82), we
–2
–8
–1
obtain for the Wiedemann–Franz constant a value of 1.22 × 10 WS K ,
about a factor 2 smaller than the experimental value.
Up to now we have talked only about the electronic contribution to the spe-
3
cific heat and quoted it as being N e k B , but classically the lattice will also
2
∗
∗ Valid at room temperature but fails at contribute a term 3Nk B where N is the density of atoms. Thus, we should ex-
low temperatures. pect an alkali metal (in which N e = N) to have a 50% greater specific heat than