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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Nearly free electron model ： ウィキペディア英語版 Nearly free electron model 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. ==Introduction - a heuristic argument== Free electrons are traveling plane waves. Generally the time independent part of their wave function is expressed as :$\backslash psi\_)\; =\; \backslash frac\backslash cdot\backslash bold\}$ These plane wave solutions have an energy of :$E\_k\; =\; \backslash frac$ 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. :$\backslash psi\_)\; =\; \backslash frac\backslash cdot\backslash bold)\; +\; i\; \backslash sin(\backslash bold\backslash cdot\backslash bold)\backslash right\; )$ 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 :$\backslash cos(\backslash bold\backslash cdot\backslash bold)\; =\; \backslash frac\; (e^\}\; +\; e^\})$ 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 :$H\; =\; T\; +\; V=-\backslash frac\backslash nabla^2+V(\backslash bold)$ in which $T$ is the kinetic and $V$ is the potential energy. From this expression the energy expectation value, or the statistical average, of the energy of the electron can be calculated with :$E\; =\; \backslash langle\; H\; \backslash rangle\; =$ \int_\psi_)(+ V )\psi_) d\bold where we integrate in $\backslash bold$ over a single lattice cell. If we assume that the electron is given by a plane wave of wave number $\backslash bold$ (despite the nonconstant potential $V$), the energy of the electron is: :$E\_k\; =\; \backslash frac\backslash int\_\; e^\}$ \left(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. We will look at those regions in $\backslash bold$-space now. Let's assume that we look at the problem from the origin, at position $\backslash bold\; =\; 0$. If $\backslash bold\; =\; 0$ only the cosine part is present and the sine part is moved to $\backslash infty$. If we let the length of the wave vector $\backslash bold$ grow, then the central maximum of the cosine part stays at $\backslash bold\; =\; 0$. The first maximum and minimum of the sine part are at $\backslash bold\; =\; \backslash pm\; \backslash pi\; /\; (2\; \backslash bold)$. They come nearer as $\backslash bold$ grows. Let's assume that $\backslash bold$ is close to the Brillouin zone boundary for the analysis in the next part of this introduction. The atomic positions coincide with the maximum of the $\backslash cos(\backslash bold\backslash cdot\backslash bold)$-component of the wave function. The interaction of the $\backslash cos(\backslash bold\backslash cdot\backslash bold)$-component of the wave function with the potential will be different from the interaction of the $\backslash sin(\backslash bold\backslash cdot\backslash bold)$-component of the wave function with the potential because their phases are shifted. The charge density $\backslash rho\_\}(\backslash bold)|^2$, of the wave function. It is useful to split it into two parts, $\backslash rho\_\}^c(\backslash bold)+\backslash rho\_)$, coming from the $\backslash cos(\backslash bold\backslash cdot\backslash bold)$ and $\backslash sin(\backslash bold\backslash cdot\backslash bold)$ -components. For the former component it is :$\backslash rho\_)\; =\; \backslash frac\; \backslash left(+\; \backslash cos(2\; \backslash bold\backslash cdot\backslash bold)\backslash right)$ and for the $\backslash sin(\backslash bold\backslash cdot\backslash bold)$-component it is :$\backslash rho\_)\; =\; \backslash frac\; \backslash left(-\; \backslash cos(2\; \backslash bold\backslash cdot\backslash bold)\backslash right)$ For values of $\backslash bold$ close to the Brillouin zone boundary, the length of the two waves and the period of the two different charge density distributions almost coincide with the periodic potential of the lattice. As a result the charge densities of the two components have a different energy because the maximum of the charge density of the $\backslash cos(\backslash bold\backslash cdot\backslash bold)$-component coincides with the attractive potential of the atoms while the maximum of the charge density of the $\backslash sin(\backslash bold\backslash cdot\backslash bold)$-component lies in the regions with a higher electrostatic potential between the atoms. 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. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Nearly free electron model」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.