Jump to navigation Jump to search This article is about the solid-state model for metals. For elektronentheorie der Metalle PDF model of a free electron gas, see Fermi gas. This article needs additional citations for verification. In solid-state physics, the free electron model is a simple model for the behaviour of charge carriers in a metallic solid.
Författare: H. Bethe.
The free electron model solved many of the inconsistencies related to the Drude model and gave insight into several other properties of metals. The model considers that metals are composed of a quantum electron gas where ions play almost no role. The model can be very predictive when applied to alkali and noble metals. Free electron approximation: The interaction between the ions and the valence electrons is mostly neglected, except in boundary conditions. The ions only keep the charge neutrality in the metal. Unlike in the Drude model, the ions are not necessarily the source of collisions. Independent electron approximation: The interactions between electrons are ignored.
The electrostatic fields in metals are weak because of the screening effect. The collisions do not depend on the electronic configuration. Pauli exclusion principle: Each quantum state of the system can only be occupied by a single electron. The name of the model comes from the first two assumptions, as each electron can be treated as free particle with a respective quadratic relation between energy and momentum. Many physical properties follow directly from the Drude model as some equations do not depend on the statistical distribution of the particles.
Taking the classical velocity distribution of an ideal gas or the velocity distribution of a Fermi gas only changes the results related to the speed of the electrons. Other quantities that remain the same under the free electron model as under Drude’s are the AC susceptibility, the plasma frequency, the magnetoresistance, and the Hall coefficient related to the Hall effect. Many properties of the free electron model follow directly from equations related to the Fermi gas, as the independent electron approximation leads to an ensemble of non-interacting electrons. The Fermi energy defines the Fermi level, i. For metals the Fermi energy is in the order of units of electronvolts.
In three dimensions, the density of states of a gas of fermions is proportional to the square root of the kinetic energy of the particles. This formula takes into account the spin degeneracy but does not consider a possible energy shift due to the bottom of the conduction band. For 2D the density of states is constant and for 1D is inversely proportional to the square root of the electron energy. The perturbative approach is justified as the Fermi temperature is usually of about 105 K for a metal, hence at room temperature or lower the Fermi energy and the chemical potential are practically equivalent. For an ideal gas to have the same energy as the electron gas, the temperatures would need to be of the order of the Fermi temperature. This expression gives the right order of magnitude for the bulk modulus for alkali metals and noble metals, which show that this pressure is as important as other effects inside the metal.
For other metals the crystalline structure has to be taken into account. One open problem in solid-state physics before the arrival of the free electron model was related to the low heat capacity of metals. If this was the case, the heat capacity of a metal could be much higher due to this electronic contribution. Nevertheless, such a large heat capacity was never measured, rising suspicions about the argument. The good estimation of the Lorenz number in the Drude model was a result of the classical mean velocity of electron being about 100 larger than the quantum version, compensating the large value of the classical heat capacity. The free electron model calculation of the Lorenz factor is about twice the value of Drude’s and its closer to the experimental value. The free electron model can be improved in this sense by adding the lattice vibrations contribution.
The linear term comes from the electronic contribution while the cubic term comes from Debye model. At high temperature this expression is no longer correct, the electronic heat capacity can be neglected, and the total heat capacity of the metal tends to a constant. Notice that without the relaxation time approximation, there is no reason for the electrons to deflect their motion, as there are no interactions, thus the mean free path should be infinite. The mean free path is then not a result electron-ion collisions but instead is related to imperfections in the material, either due to defects and impurities in the metal, or due to thermal fluctuations. The free electron model presents several inadequacies that are contradicted by experimental observation. Temperature dependence The free electron model presents several physical quantities that have the wrong temperature dependence, or no dependence at all like the electrical conductivity.