In solid-state physics , the nearly free electron model or NFE model is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model is closely related to the more conceptual empty lattice approximation.
The model enables understanding and calculating the electronic band structure of especially metals. This model is an immediate improvement of the free electron model , in which the metal was considered as a non-interacting electron gas and the ions were neglected completely. Free electrons are traveling plane waves. Generally the time independent part of their wave function is expressed as. The expression of the plane wave as a complex exponential function can also be written as the sum of two periodic functions which are mutually shifted a quarter of a period.
In this light the wave function of a free electron can be viewed as the sum of two plane waves. Sine and cosine functions can also be expressed as sums or differences of plane waves moving in opposite directions. Assume that there is only one kind of atom present in the lattice and that the atoms are located at the lattice points.
The potential of the atoms is attractive negative and concentrated near the lattice points. In the remainder of the cell the potential is close to zero. The Hamiltonian is expressed as.
From this expression the energy expectation value , or the statistical average , of the energy of the electron can be calculated with. If the potential is very small we get the empty lattice approximation.
This isn't a very sensational result and it doesn't say anything about what happens when we get close to the Brillouin zone boundary. For the former component it is. As a result the aggregate will be split in high and low energy components when the kinetic energy increases and the wave vector approaches the length of the reciprocal lattice vectors.
Solid State Physics - Lecture 15: Nearly Free Electron Model
The potentials of the atomic cores can be decomposed into Fourier components to meet the requirements of a description in terms of reciprocal space parameters.
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.
This model, like the free-electron model, does not take into account electron—electron interactions; that is, the independent-electron approximation is still in effect. The vectors G are the reciprocal lattice vectors , and the discrete values of k are determined by the boundary conditions of the lattice under consideration.
In any perturbation analysis, one must consider the base case to which the perturbation is applied. In this case the central equation reduces to the form.
In this non-degenerate case, the standard free electron gas result is retrieved:. In the degenerate case, however, there will be a set of lattice vectors k 1 , Non-degenerate and degenerate perturbation theory can be applied in these two cases to solve for the Fourier coefficients C k of the wavefunction correct to first order in U and the energy eigenvalue correct to second order in U.
Particularly, at the Brillouin zone boundary or, equivalently, at any point on a Bragg plane , one finds a twofold energy degeneracy that results in a shift in energy given by:.
Free electron model
In this model, the assumption is made that the interaction between the conduction electrons and the ion cores can be modeled through the use of a "weak" perturbing potential. This may seem like a severe approximation, for the Coulomb attraction between these two particles of opposite charge can be quite significant at short distances. It can be partially justified, however, by noting two important properties of the quantum mechanical system:.
From Wikipedia, the free encyclopedia. Physical model of solid metals as electron gases. Categories : Electronic band structures. Namespaces Article Talk.
Coulson—Fischer theory Generalized valence bond Modern valence bond. Wikimedia Commons has media related to Dispersion relations of electrons.