In solid-state physics, the nearly free electron model (or NFE model and 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.
Mathematical formulation
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
ψ k ( r ) = u k ( r ) e i k ⋅ r {\displaystyle \psi _{\mathbf {k} }(\mathbf {r} )=u_{\mathbf {k} }(\mathbf {r} )e^{i\mathbf {k} \cdot \mathbf {r} }}
where the function u k {\displaystyle u_{\mathbf {k} }} has the same periodicity as the lattice:
u k ( r ) = u k ( r + T ) {\displaystyle u_{\mathbf {k} }(\mathbf {r} )=u_{\mathbf {k} }(\mathbf {r} +\mathbf {T} )}
(where T {\displaystyle T} is a lattice translation vector.) Because it is a nearly free electron approximation we can assume that
u k ( r ) ≈ 1 Ω r {\displaystyle u_{\mathbf {k} }(\mathbf {r} )\approx {\frac {1}{\sqrt {\Omega _{r}}}}}
where Ω r {\displaystyle \Omega _{r}} denotes the volume of states of fixed radius r {\displaystyle r} (as described in Gibbs paradox). A solution of this form can be plugged into the Schrödinger equation, resulting in the central equation:
( λ k − ε ) C k + ∑ G U G C k − G = 0 {\displaystyle (\lambda _{\mathbf {k} }-\varepsilon )C_{\mathbf {k} }+\sum _{\mathbf {G} }U_{\mathbf {G} }C_{\mathbf {k} -\mathbf {G} }=0}
where ε {\displaystyle \varepsilon } is the total energy, and the kinetic energy λ k {\displaystyle \lambda _{\mathbf {k} }} is characterized by
λ k ψ k ( r ) = − ℏ 2 2 m ∇ 2 ψ k ( r ) = − ℏ 2 2 m ∇ 2 ( u k ( r ) e i k ⋅ r ) {\displaystyle \lambda _{\mathbf {k} }\psi _{\mathbf {k} }(\mathbf {r} )=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi _{\mathbf {k} }(\mathbf {r} )=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}(u_{\mathbf {k} }(\mathbf {r} )e^{i\mathbf {k} \cdot \mathbf {r} })}
which, after dividing by ψ k ( r ) {\displaystyle \psi _{\mathbf {k} }(\mathbf {r} )} , reduces to
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