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Insolid-state physics, thenearly free electron model (orNFE model andquasi-free electron model) is aquantum mechanical model of physical properties ofelectrons that can move almost freely through thecrystal lattice of a solid. The model is closely related to the more conceptualempty lattice approximation. The model enables understanding and calculation of theelectronic band structures, especially ofmetals.

This model is an immediate improvement of thefree electron model, in which the metal was considered as anon-interacting electron gas and theions were neglected completely.

The nearly free electron model is a modification of thefree-electron gas model which includes aweakperiodicperturbation meant to model the interaction between theconduction electrons and theions in acrystalline solid. This model, like the free-electron model, does not take into account electron–electron interactions; that is, theindependent electron approximation is still in effect.

As shown byBloch's theorem, introducing a periodic potential into theSchrödinger equation results in awave function of the form

$${\displaystyle {\psi }_{\mathbf{k}}(\mathbf{r})=u_{\mathbf{k}}(\mathbf{r})e^{i\mathbf{k}\cdot \mathbf{r}}}$$

where the function${\displaystyle u_{\mathbf{k}}}$ has the same periodicity as thelattice:

$${\displaystyle u_{\mathbf{k}}(\mathbf{r})=u_{\mathbf{k}}(\mathbf{r}+\mathbf{T})}$$

(where${\displaystyle T}$ is a lattice translation vector.)

Because it is anearly free electron approximation we can assume that

$${\displaystyle u_{\mathbf{k}}(\mathbf{r})\approx \frac{1}{\sqrt{{\mathrm{\Omega }}_r}}}$$where${\displaystyle {\mathrm{\Omega }}_r}$ denotes the volume of states of fixed radius${\displaystyle r}$ (as described inGibbs paradox).^{[clarification needed]}

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

$${\displaystyle ({\lambda }_{\mathbf{k}}-\epsilon )C_{\mathbf{k}}+\sum _{\mathbf{G}}{U}_{\mathbf{G}}{C}_{\mathbf{k}-\mathbf{G}}=0}$$

where${\displaystyle \epsilon }$ is the total energy, and the kinetic energy${\displaystyle {\lambda }_{\mathbf{k}}}$ is characterized by

$${\displaystyle {\lambda }_{\mathbf{k}}{\psi }_{\mathbf{k}}(\mathbf{r})=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2{\psi }_{\mathbf{k}}(\mathbf{r})=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2(u_{\mathbf{k}}(\mathbf{r})e^{i\mathbf{k}\cdot \mathbf{r}})}$$

which, after dividing by${\displaystyle {\psi }_{\mathbf{k}}(\mathbf{r})}$, reduces to

$${\displaystyle {\lambda }_{\mathbf{k}}=\frac{{\hslash }^2{k}^2}{2m}}$$

if we assume that${\displaystyle u_{\mathbf{k}}(\mathbf{r})}$ is almost constant and${\displaystyle {\mathrm{\nabla }}^2{u}_{\mathbf{k}}(\mathbf{r})\ll k^2.}$

The reciprocal parameters${\displaystyle C_{\mathbf{k}}}$ and${\displaystyle U_{\mathbf{G}}}$ are theFourier coefficients of the wave function${\displaystyle \psi (\mathbf{r})}$ and thescreened potential energy${\displaystyle U(\mathbf{r})}$, respectively:

$${\displaystyle U(\mathbf{r})=\sum _{\mathbf{G}}{U}_{\mathbf{G}}{e}^{i\mathbf{G}\cdot \mathbf{r}}}$$$${\displaystyle \psi (\mathbf{r})=\sum _{\mathbf{k}}{C}_{\mathbf{k}}{e}^{i\mathbf{k}\cdot \mathbf{r}}}$$

The vectors${\displaystyle \mathbf{G}}$ are thereciprocal lattice vectors, and the discrete values of${\displaystyle \mathbf{k}}$ are determined by the boundary conditions of the lattice under consideration.

Before doing the perturbation analysis, let us first consider the base case to which the perturbation is applied. Here, the base case is${\displaystyle U(x)=0}$, and therefore all the Fourier coefficients of the potential are also zero. In this case the central equation reduces to the form

$${\displaystyle ({\lambda }_{\mathbf{k}}-\epsilon )C_{\mathbf{k}}=0}$$

This identity means that for each${\displaystyle \mathbf{k}}$, one of the two following cases must hold:

1. ${\displaystyle C_{\mathbf{k}}=0}$,
2. ${\displaystyle {\lambda }_{\mathbf{k}}=\epsilon }$

If${\displaystyle \epsilon }$ is anon-degenerate energy level, then the second case occurs for only one value of${\displaystyle \mathbf{k}}$, while for the remaining${\displaystyle \mathbf{k}}$, the Fourier expansion coefficient${\displaystyle C_{\mathbf{k}}}$ is zero. In this case, the standard free electron gas result is retrieved:

$${\displaystyle {\psi }_{\mathbf{k}}\propto e^{i\mathbf{k}\cdot \mathbf{r}}}$$

If${\displaystyle \epsilon }$ is a degenerate energy level, there will be a set of lattice vectors${\displaystyle {\mathbf{k}}_1,\dots ,{\mathbf{k}}_m}$ with${\displaystyle {\lambda }_1=\cdots ={\lambda }_m=\epsilon }$. Then there will be${\displaystyle m}$ independent plane wave solutions of which any linear combination is also a solution:

$${\displaystyle \psi \propto \sum _{j=1}^m{A}_j{e}^{i{\mathbf{k}}_j\cdot \mathbf{r}}}$$

Now let${\displaystyle U}$ be nonzero and small. Non-degenerate and degenerate perturbation theory, respectively, can be applied in these two cases to solve for the Fourier coefficients${\displaystyle C_{\mathbf{k}}}$ of the wavefunction (correct to first order in${\displaystyle U}$) and the energy eigenvalue${\displaystyle \epsilon }$ (correct to second order in${\displaystyle U}$). An important result of this derivation is that there is no first-order shift in the energy${\displaystyle \epsilon }$ in the case of no degeneracy, while there is in the case of degeneracy (and near-degeneracy), implying that the latter case is more important in this analysis. Particularly, at theBrillouin zone boundary (or, equivalently, at any point on aBragg plane), one finds a twofold energy degeneracy that results in a shift in energy given by:^{[clarification needed]}

$${\displaystyle \epsilon ={\lambda }_{\mathbf{k}}\pm |U_{\mathbf{G}}|}$$.

Thisenergy gap between Brillouin zones is known as theband gap, with a magnitude of${\displaystyle 2|U_{\mathbf{G}}|}$.

Introducing this weak perturbation has significant effects on the solution to theSchrödinger equation, most significantly resulting in aband gap betweenwave vectors in differentBrillouin 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:

1. 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 thePauli 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.
2. Furthermore, the core electronsshield the ion charge magnitude "seen" by the conduction electrons. The result is aneffective nuclear charge experienced by the conduction electrons which is significantly reduced from the actual nuclear charge.

• Empty lattice approximation
• Electronic band structure
• Tight binding model
• Bloch's theorem
• Kronig–Penney model

Wikimedia Commons has media related toDispersion 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.

• Electronic band structures
• Quantum models

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