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In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, [1] principally by Arnold Sommerfeld, who combined the classical Drude model with quantum mechanical Fermi–Dirac statistics and hence it is also known as the Drude–Sommerfeld model.
Free electron model [ edit ] After taking into account the quantum effects, as in the free electron model , the heat capacity, mean free path and average speed of electrons are modified and the proportionality constant is then corrected to π 2 3 ≈ 3.29 {\displaystyle {\frac {\pi ^{2}}{3}}\approx 3.29} , which agrees with experimental values.
Drude applied the kinetic theory of a dilute gas, despite the high densities, therefore ignoring electron–electron and electron–ion interactions aside from collisions. [ Ashcroft & Mermin 13 ] The Drude model considers the metal to be formed of a collection of positively charged ions from which a number of "free electrons" were detached.
The alkali metals are expected to have the best agreement with the free electron model since these metals only one s-electron outside a closed shell. However even sodium, which is considered to be the closest to a free electron metal, is determined to have a γ {\displaystyle \gamma } more than 25 per cent higher than expected from the theory.
Dispersion relation for the 2D nearly free electron model as a function of the underlying crystalline structure. The nearly free electron model is a modification of the free-electron gas model which includes a weak periodic perturbation meant to model the interaction between the conduction electrons and the ions in a crystalline solid.
The creation of further free electrons is only achieved by impact ionization. Thus Paschen's law is not valid if there are external electron sources. This can, for example, be a light source creating secondary electrons by the photoelectric effect. This has to be considered in experiments. Each ionized atom leads to only one free electron.
The problem with the inaccurate modelling of the kinetic energy in the Thomas–Fermi model, as well as other orbital-free density functionals, is circumvented in Kohn–Sham density functional theory with a fictitious system of non-interacting electrons whose kinetic energy expression is known.
Fowler–Nordheim theory predicted both to be consequences if CFE were due to field-induced tunneling from free-electron-type states in what we would now call a metal conduction band, with the electron states occupied in accordance with Fermi–Dirac statistics. Oppenheimer had mathematical details of his theory seriously incorrect. [15]