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In solid-state physics, the **nearly free electron model** (or **NFE model**) or **quasi-free electron 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 calculation of the electronic band structures, especially of 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.

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.

As shown by Bloch's theorem, introducing a periodic potential into the Schrödinger equation results in a wave function of the form

where the function has the same periodicity as the lattice:

(where is a lattice translation vector.)

Because it is a *nearly* free electron approximation we can assume that

where denotes the volume of states of fixed radius (as described in Gibbs paradox).

A solution of this form can be plugged into the Schrödinger equation, resulting in the **central equation**:

where the kinetic energy is given by

which, after dividing by , reduces to

if we assume that is almost constant and

The reciprocal parameters and are the Fourier coefficients of the wave function and the screened potential energy , respectively:

The vectors are the reciprocal lattice vectors, and the discrete values of 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. Here, the base case is with , and therefore all the Fourier coefficients of the potential are also zero. In this case the central equation reduces to the form

This identity means that for each , one of the two following cases must hold:

- ,

If the values of are non-degenerate, then the second case occurs for only one value of , while for the rest, the Fourier expansion coefficient must be zero. 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 with . When the energy is equal to this value of , there will be independent plane wave solutions of which any linear combination is also a solution:

Non-degenerate and degenerate perturbation theory can be applied in these two cases to solve for the Fourier coefficients of the wavefunction (correct to first order in ) and the energy eigenvalue (correct to second order in ). An important result of this derivation is that there is no first-order shift in the energy in the case of no degeneracy, while there is in the case of near-degeneracy, implying that the latter case is more important in this analysis. 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:

This **energy gap** between Brillouin zones is known as the band gap, with a magnitude of .

Introducing this weak perturbation has significant effects on the solution to the Schrödinger equation, most significantly resulting in a band gap between wave vectors in different Brillouin zones.

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:

- The force between the ions and the electrons is greatest at very small distances. However, the conduction electrons are not "allowed" to get this close to the ion cores due to the Pauli exclusion principle: the orbitals closest to the ion core are already occupied by the core electrons. Therefore, the conduction electrons never get close enough to the ion cores to feel their full force.
- Furthermore, the core electrons shield the ion charge magnitude "seen" by the conduction electrons. The result is an
*effective nuclear charge*experienced by the conduction electrons which is significantly reduced from the actual nuclear charge.

In particle physics, the **Dirac equation** is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1⁄2 massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine structure of the hydrogen spectrum in a completely rigorous way.

The **Schrödinger equation** is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

In mathematics, the **unitary group** of degree *n*, denoted U(*n*), is the group of *n* × *n* unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(*n*, **C**). **Hyperorthogonal group** is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.

The **ground state** of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. In quantum field theory, the ground state is usually called the vacuum state or the vacuum.

The **Dirac sea** is a theoretical model of the vacuum as an infinite sea of particles with negative energy. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by the Dirac equation for relativistic electrons. The positron, the antimatter counterpart of the electron, was originally conceived of as a hole in the Dirac sea, before its experimental discovery in 1932.

In condensed matter physics, **Bloch's theorem** states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who discovered the theorem in 1929. Mathematically, they are written

In atomic physics, the **fine structure** describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom by Albert A. Michelson and Edward W. Morley in 1887, laying the basis for the theoretical treatment by Arnold Sommerfeld, introducing the fine-structure constant.

The **Stark effect** is the shifting and splitting of spectral lines of atoms and molecules due to the presence of an external electric field. It is the electric-field analogue of the Zeeman effect, where a spectral line is split into several components due to the presence of the magnetic field. Although initially coined for the static case, it is also used in the wider context to describe the effect of time-dependent electric fields. In particular, the Stark effect is responsible for the pressure broadening of spectral lines by charged particles in plasmas. For most spectral lines, the Stark effect is either linear or quadratic with a high accuracy.

In solid-state physics, the **electronic band structure** of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have.

In quantum physics, **Fermi's golden rule** is a formula that describes the transition rate from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of a weak perturbation. This transition rate is effectively independent of time and is proportional to the strength of the coupling between the initial and final states of the system as well as the density of states. It is also applicable when the final state is discrete, i.e. it is not part of a continuum, if there is some decoherence in the process, like relaxation or collision of the atoms, or like noise in the perturbation, in which case the density of states is replaced by the reciprocal of the decoherence bandwidth.

In quantum mechanics, a **two-state system** is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.

**Møller–Plesset perturbation theory** (**MP**) is one of several quantum chemistry post–Hartree–Fock ab initio methods in the field of computational chemistry. It improves on the Hartree–Fock method by adding electron correlation effects by means of Rayleigh–Schrödinger perturbation theory (RS-PT), usually to second (MP2), third (MP3) or fourth (MP4) order. Its main idea was published as early as 1934 by Christian Møller and Milton S. Plesset.

In electromagnetism, **charge density** is the amount of electric charge per unit length, surface area, or volume. **Volume charge density** is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m^{−3}), at any point in a volume. **Surface charge density** (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m^{−2}), at any point on a surface charge distribution on a two dimensional surface. **Linear charge density** (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m^{−1}), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.

The **Franz–Keldysh effect** is a change in optical absorption by a semiconductor when an electric field is applied. The effect is named after the German physicist Walter Franz and Russian physicist Leonid Keldysh.

The **Poisson–Boltzmann equation** is a useful equation in many settings, whether it be to understand physiological interfaces, polymer science, electron interactions in a semiconductor, or more. It aims to describe the distribution of the electric potential in solution in the direction normal to a charged surface. This distribution is important to determine how the electrostatic interactions will affect the molecules in solution. The Poisson–Boltzmann equation is derived via mean-field assumptions. From the Poisson–Boltzmann equation many other equations have been derived with a number of different assumptions.

In quantum mechanics the **delta potential** is a potential well mathematically described by the Dirac delta function - a generalized function. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it takes an infinite value. This can be used to simulate situations where a particle is free to move in two regions of space with a barrier between the two regions. For example, an electron can move almost freely in a conducting material, but if two conducting surfaces are put close together, the interface between them acts as a barrier for the electron that can be approximated by a delta potential.

In solid-state physics, the **k·p perturbation theory** is an approximated semi-empirical approach for calculating the band structure and optical properties of crystalline solids. It is pronounced "k dot p", and is also called the "**k·p** method". This theory has been applied specifically in the framework of the Luttinger–Kohn model, and of the Kane model.

In physics and engineering, the **envelope** of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude into an **instantaneous amplitude**. The figure illustrates a modulated sine wave varying between an *upper envelope* and a *lower envelope*. The envelope function may be a function of time, space, angle, or indeed of any variable.

The **Holstein–Herring method**, also called the **surface Integral method**, or **Smirnov's method** is an effective means of getting the exchange energy splittings of asymptotically degenerate energy states in molecular systems. Although the exchange energy becomes elusive at large internuclear systems, it is of prominent importance in theories of molecular binding and magnetism. This splitting results from the symmetry under exchange of identical nuclei.

In quantum mechanics, the **variational method** is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. This allows calculating approximate wavefunctions such as molecular orbitals. The basis for this method is the variational principle.

Wikimedia Commons has media related to Dispersion relations of electrons .

- Ashcroft, Neil W.; Mermin, N. David (1976).
*Solid State Physics*. Orlando: Harcourt. ISBN 0-03-083993-9. - Kittel, Charles (1996).
*Introduction to Solid State Physics*(7th ed.). New York: Wiley. ISBN 0-471-11181-3. - Elliott, Stephen (1998).
*The Physics and Chemistry of Solids*. New York: Wiley. ISBN 0-471-98194-X.

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