High-field emission is essentially quantum-mechanical tunneling through the potential barrier at the surface of the cathode. The current density is given by the Fowler-Nordheim equation J = CE2exp(-D/E), where C = [6.2 x 10-6/(φ + EF)](EF/φ)1/2 A/V2, and D = 6.8 x 109φ3/2 V/m. EF is the Fermi energy and φ is the work function, both in volts. E is the field in V/m. For tungsten, φ = 4.52V and EF = 8.95V. The fields required are very high. A field of 2 x 107 V/cm produces emission of only 1.7 μA/cm2 in tungsten. However, the current increases quite rapidly with electric field. With a field of 3 x 107 V/cm, the current is already 0.2 A/cm2. At atmospheric pressure, the mean free path is about 10-5 cm in air, and if the cathode drop is 10V, the resulting electric field if the cathode drop occurs over one mean free path is 106 V/cm. This is about a factor of 20 less than is required, so some investigators have questioned the importance of high-field emission in arcs. However, it is at least close to the required value, and some local strengthening of the field by the arrangment of adsorbed positive ions may make up the difference.
The Richardson-Dushman equation for thermionic emission, J = AT2exp(-b/T) is very similar in form to the Fowler-Nordheim equation, with the absolute temperature T replacing the field strength E. A = 60.2 A/cm2, and b = 11600φ. At the boiling point of tungsten, 5993K, J = 3.45 x 105 A/cm2, a quite ample result. Even at the melting point, the current density is 541 A/cm2. Mercury has about the same work function as tungsten, 4.5V, but the thermionic emission at its boiling point, 630K, is only about 10-29 A/cm2, which is wholly inadequate. Therefore, high-field emission seems to be the only alternative.