Following
J. J. Thomson's identification of the electron in 1897, the British physicist
Owen Willans Richardson began work on the topic that he later called "thermionic emission". He received a
Nobel Prize in Physics in 1928 "for his work on the thermionic phenomenon and especially for the discovery of the law named after him". From
band theory, there are one or two electrons per
atom in a solid 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 (eV). 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 modified
Arrhenius equation, T^{1/2} \mathrm{e}^{-b/T}. Later, he proposed that the emission law should have the mathematical form : J = A_{\mathrm{G}} T^2 \mathrm{e}^{-W \over k T} 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
AG is a parameter discussed next. In the period 1911 to 1930, as physical understanding of the behaviour of electrons in metals increased, various theoretical expressions (based on different physical assumptions) were put forward for
AG, by Richardson,
Saul Dushman,
Ralph H. Fowler,
Arnold Sommerfeld and
Lothar Wolfgang Nordheim. Over 60 years later, there is still no consensus among interested theoreticians as to the exact expression of
AG, but there is agreement that
AG must be written in the form: : A_{\mathrm{G}} = \; \lambda_{\mathrm{R}} A_0 where
λR is a material-specific correction factor that is typically of order 0.5, and
A0 is a universal constant given by : A_0 = {4 \pi m k^2 q_\text{e} \over h^3} = 1.20173 \times 10^6\,\mathrm{A{\cdot}m^{-2}{\cdot}K^{-2}} where m and -q_\text{e} are the mass and
charge of an electron, respectively, and h is the
Planck constant. In fact, by about 1930 there was agreement that, due to the wave-like nature of electrons, some proportion
rav of the outgoing electrons would be reflected as they reached the emitter surface, so the emission current density would be reduced, and
λR would have the value . Thus, one sometimes sees the thermionic emission equation written in the form: : J = (1-r_{\mathrm{av}})\lambda_\text{B} A_0 T^2 \mathrm{e}^{-W \over k T}. 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
λB into
λR, giving A_{\mathrm{G}} = \lambda_{\mathrm{B}} (1-r_{\mathrm{av}}) A_0 . Experimental values for the "generalized" coefficient
AG 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
λR. Considerable confusion exists in the literature of this area because: (1) many sources do not distinguish between
AG 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
λR 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 "Richardson–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 .) The thermionic emission law has been recently revised for 2D materials in various models. == Schottky emission ==