Thermionic Emission

a) Assume that the evaporation of electrons from a hot wire (Richardson’s effect) is thermodynamically equivalent to the sublimation of a solid. Find the pressure of the electron gas, provided that the electrons outside the metal constitute an ideal classical monatomic gas and that the chemical potential μ_s of the electrons in the metal (the solid phase) is a constant.

b) Derive the same result by using the Clausius–Clapeyron equation

where L is the latent heat of electron evaporation. Neglect the volume occupied by the electrons in the metal

SOLUTION

a) We can consider the electron gas outside the metal to be in equilibrium with the electrons inside the metal. Then the number of electrons hitting the surface from the outside should be equal to the number of electrons leaving the metal. Using the formula for chemical potential of a monatomic ideal gas, we can write

(1)

where g = 2s + 1 = 2 for an electron gas. Rewriting (1), we have

(2)

The state of equilibrium requires that this chemical potential be equal to the potential inside the metal, which we can take as μ_s = -ϕ₀; i.e., the energy ϕ₀ is required to take an electron from the Fermi level inside the metal into vacuum. So, the pressure of the electron gas is given by

(3)

On the other hand, the number of particles of the ideal gas striking the surface per unit area per unit time is

(4)

The current

(5)

where e is the electron charge. Therefore, we can express P from (5):

(6)

Equating (6) with (3), we find the current

(7)

Alternatively, we can calculate the current by considering the electrons leaving the metal as if they have a kinetic energy high enough to overcome the potential barrier.

b) For one particle,

(8)

where E_g and E_s are the energies, and v_g and v_s the volumes per particle, of the gas and solid, respectively. Since v_g >> v_s,  we can rewrite (8) in the form

(9)

Substituting (9) into the Clausius-Clapeyron equation

(10)

we obtain

(11)

We may rewrite (11) as

(12)

Integrating, we recover (2):

where A is some constant, or

(13)