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Nearly Free Electron Theory Pdf Creator
Nearly Free Electron Theory Pdf Creator

nearly free electron theory pdf creator


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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. This identity means that for each k, one of the two following cases must hold:. 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. {displaystyle nabla ^{2}u{mathbf {k}}({mathbf {r}})ll k^{2}.} . See also[edit]. The reciprocal parameters Ck and UG are the Fourier coefficients of the wave function (r) and the screened potential energy U(r), respectively:. Some free electron bands in a FCC crystal structure according to the Empty Lattice Approximation . Elliott, Stephen (1998). 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. ( λ k − ϵ ) C k = 0 {displaystyle (lambda {mathbf {k}}-epsilon )C{mathbf {k}}=0} . 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. 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:. E k = 1 Ω r ∫ Ω r e − i k ⋅ r [ ℏ 2 k 2 2 m V ( r ) ] e i k ⋅ r d r = ℏ 2 k 2 2 m ⟨ V ⟩ {displaystyle E{k}={frac {1}{Omega {r}}}int {Omega {r}}e^{ -i{mathbf {k}}cdot {mathbf {r}}}left[{frac {hbar ^{2}k^{2}}{2m}} V({mathbf {r}})right]e^{i{mathbf {k}}cdot {mathbf {r}}}d{mathbf {r}}={frac {hbar ^{2}k^{2}}{2m}} langle Vrangle } . The model is closely related to the more conceptual 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. When the energy ϵ {displaystyle epsilon } is equal to this value of , there will be m independent plane wave solutions of which any linear combination is also a solution:. in which T {displaystyle T} is the kinetic and V {displaystyle V} is the potential energy. This energy gap between Brillouin zones is known as the band gap, with a magnitude of 2 U K {displaystyle 2U{K}} . In this case the central equation reduces to the form. H = T V = − ℏ 2 2 m ∇ 2 V ( r ) {displaystyle H=T V=-{frac {hbar ^{2}}{2m}}nabla ^{2} V({mathbf {r}})} . u k ( r ) = u k ( r T ) {displaystyle u{mathbf {k}}({mathbf {r}})=u{mathbf {k}}({mathbf {r}} {mathbf {T}})} . Non-degenerate and degenerate perturbation theory can be applied in these two cases to solve for the Fourier coefficients Ck of the wavefunction (correct to first order in U) and the energy eigenvalue (correct to second order in U). ( λ k − ϵ ) C k ∑ G U G C k − G = 0 {displaystyle (lambda {mathbf {k}}-epsilon )C{mathbf {k}} sum {mathbf {G}}U{mathbf {G}}C{{mathbf {k}}-{mathbf {G}}}=0} . 6c2930289c

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