Thermionic constants of metals and semiconductors. IV. Monovalent metals (continued)

In a monovalent metal, in which the valency electrons may be regarded as forming a free-electron assemblage, the assumption of a temperature-independent energy barrier at the surface of the metal is shown to be equivalent to taking the free electrons in the condensed phase, namely, the metal, and the electrons in the gaseous phase in thermal equilibrium with it, as forming a homogeneous single component system. The temperature variation of the work function is then determined by the temperature variation of the thermodynamic potential of the electrons in the condensed phase, when the external pressure is kept constant at the value of the saturation vapour pressure of the electrons, which is equivalent to keeping the pressure of the electron assemblage in the condensed phase also constant, since the energy barrier at the surface is independent of temperature. It is further shown that for a degenerate, or nearly degenerate, electron assemblage the specific heat at constant pressure is the same as that at constant volume, and it is easily calculated. The temperature coefficient of the work function calculated therefrom corresponds to an apparent lowering of about 8 to 10% in the value of the A coefficient in thermionic emission. This agrees with observation. On the other hand, the thermal expansion of the lattice is found to be about 25 to 50 times that to be expected thermodynamically for the electron assemblage in the condensed phase. This result, when viewed against the nearly normal observed value of the A coefficient, shows that the energy barrier at the surface of the metal should decrease with increase of temperature by the same amount by which the thermodynamic potential of the electrons in the condensed phase decreases as a result of the thermal expansion of the lattice. A detailed calculation is made of the effect of both the thermal expansion of the lattice, and the increased thermal oscillations of the atoms in the lattice, associated with the rise in temperature, on the energy of the barrier at the surface. The net effect is found to be a lowering of this energy of the required magnitude