Thermionic Emission - Richardson's Law

Richardson's Law

In any solid metal, there are one or two electrons per atom that are free to move from atom to atom. This is sometimes collectively referred to as a "sea of electrons". Their velocities follow a statistical distribution, rather than being uniform, and occasionally an electron will have enough velocity to exit the metal without being pulled back in. The minimum amount of energy needed for an electron to leave a surface is called the work function. The work function is characteristic of the material and for most metals is on the order of several electronvolts. Thermionic currents can be increased by decreasing the work function. This often-desired goal can be achieved by applying various oxide coatings to the wire.

In 1901 Richardson published the results of his experiments: the current from a heated wire seemed to depend exponentially on the temperature of the wire with a mathematical form similar to the Arrhenius equation. Later, he proposed that the emission law should have the mathematical form

where J is the emission current density, T is the temperature of the metal, W is the work function of the metal, k is the Boltzmann constant, and A_G is a parameter discussed next.

Taking the logarithm of both sides

Thus, the equation showing the relationship between current densities at two temperatures is

In the period 1911 to 1930, as physical understanding of the behaviour of electrons in metals increased, various different theoretical expressions (based on different physical assumptions) were put forwards for A_G, by Richardson, Saul Dushman, Ralph H. Fowler, Arnold Sommerfeld and Lothar Wolfgang Nordheim. Over 60 years later, there is still no consensus amongst interested theoreticians as to what the precise form of the expression for A_G should be, but there is agreement that A_G must be written in the form

where λ_R is a material-specific correction factor that is typically of order 0.5, and A₀ is a universal constant given by

where m and −e are the mass and charge of an electron, and h is Planck's constant.

In fact, by about 1930 there was agreement that, due to the wave-like nature of electrons, some proportion r_av of the outgoing electrons would be reflected as they reached the emitter surface, so the emission current density would be reduced, and λ_R would have the value (1-r_av). Thus, one sometimes sees the thermionic emission equation written in the form

.

However, a modern theoretical treatment by Modinos assumes that the band-structure of the emitting material must also be taken into account. This would introduce a second correction factor λ_B into λ_R, giving . Experimental values for the "generalized" coefficient A_G are generally of the order of magnitude of A₀, but do differ significantly as between different emitting materials, and can differ as between different crystallographic faces of the same material. At least qualitatively, these experimental differences can be explained as due to differences in the value of λ_R.

Considerable confusion exists in the literature of this area because: (1) many sources do not distinguish between A_G and A₀, but just use the symbol A (and sometimes the name "Richardson constant") indiscriminately; (2) equations with and without the correction factor here denoted by λ_R are both given the same name; and (3) a variety of names exist for these equations, including "Richardson equation", "Dushman's equation", "Richardson–Dushman equation" and "Richard–Laue–Dushman equation". In the literature, the elementary equation is sometimes given in circumstances where the generalized equation would be more appropriate, and this in itself can cause confusion. To avoid misunderstandings, the meaning of any "A-like" symbol should always be explicitly defined in terms of the more fundamental quantities involved.

Because of the exponential function, the current increases rapidly with temperature when kT is less than W. (For essentially every material, melting occurs well before kT = W.)