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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 Van Hove singularity ： ウィキペディア英語版 Van Hove singularity A Van Hove singularity is a singularity (non-smooth point) in the density of states (DOS) of a crystalline solid. The wavevectors at which Van Hove singularities occur are often referred to as critical points of the Brillouin zone. (The critical point found in phase diagrams is a completely separate phenomenon). For three-dimensional crystals, they take the form of kinks (where the density of states is not differentiable). The most common application of the Van Hove singularity concept comes in the analysis of optical absorption spectra. The occurrence of such singularities was first analyzed by the Belgian physicist Léon Van Hove in 1953 for the case of phonon densities of states.〔L. Van Hove, ("The Occurrence of Singularities in the Elastic Frequency Distribution of a Crystal," ) Phys. Rev. 89, 1189–1193 (1953).〕 == Theory == Consider a one-dimensional lattice of ''N'' particles, with each particle separated by distance ''a'', for a total length of L = ''Na''. Instead of assuming that the waves in this one-dimensional box are standing waves, it is more convenient to adopt periodic boundary conditions:〔See equation 2.9 in http://www2.physics.ox.ac.uk/sites/default/files/BandMT_02.pdf From $\backslash phi(x+L)=\backslash phi(x)$ we have $kL=2n\backslash pi$〕 :$k=\backslash frac=n\backslash frac$ where $\backslash lambda$ is wavelength, and ''n'' is an integer. (Positive integers will denote forward waves, negative integers will denote reverse waves.) The smallest wavelength possible is ''2a'' which corresponds to the largest possible wave number $k\_=\backslash pi/a$ and which also corresponds to the maximum possible |n|: $n\_=L/2a$. We may define the density of states ''g(k)dk'' as the number of standing waves with wave vector ''k'' to ''k+dk'':〔 *M. A. Parker(1997-2004)("Introduction to Density of States" ''Marcel-Dekker Publishing'' ) p.7. 〕 :$g(k)dk\; =\; dn\; =\backslash frac\backslash ,dk$ Extending the analysis to wavevectors in three dimensions the density of states in a box will be :$g(\backslash vec)d^3k\; =\; d^3n\; =\backslash frac\backslash ,d^3k$ where $d^3k$ is a volume element in ''k''-space, and which, for electrons, will need to be multiplied by a factor of 2 to account for the two possible spin orientations. By the chain rule, the DOS in energy space can be expressed as :$dE\; =$ \fracdk_x + \fracdk_y + \fracdk_z = \vecE \cdot d\vec where $\backslash vec$ is the gradient in k-space. The set of points in ''k''-space which correspond to a particular energy ''E'' form a surface in ''k''-space, and the gradient of ''E'' will be a vector perpendicular to this surface at every point.〔 *〕 The density of states as a function of this energy ''E'' is: :$g(E)dE\; =\; \backslash iint\_g(\backslash vec)\backslash ,d^3k\; =\; \backslash frac\backslash iint\_dk\_x\backslash ,dk\_y\backslash ,dk\_z$ where the integral is over the surface $\backslash partial\; E$ of constant ''E''. We can choose a new coordinate system $k\text{'}\_x,k\text{'}\_y,k\text{'}\_z\backslash ,$ such that $k\text{'}\_z\backslash ,$ is perpendicular to the surface and therefore parallel to the gradient of ''E''. If the coordinate system is just a rotation of the original coordinate system, then the volume element in k-prime space will be :$dk\text{'}\_x\backslash ,dk\text{'}\_y\backslash ,dk\text{'}\_z\; =\; dk\_x\backslash ,dk\_y\backslash ,dk\_z$ We can then write ''dE'' as: :$dE=|\backslash vecE|\backslash ,dk\text{'}\_z$ and, substituting into the expression for ''g(E)'' we have: :$g(E)=\backslash frac\backslash iint\backslash frac$ where the $dk\text{'}\_x\backslash ,dk\text{'}\_y$ term is an area element on the constant-''E'' surface. The clear implication of the equation for $g(E)$ is that at the $k$-points where the dispersion relation $E(\backslash vec)$ has an extremum, the integrand in the DOS expression diverges. The Van Hove singularities are the features that occur in the DOS function at these $k$-points. A detailed analysis〔 * This book contains an extensive discussion of the types of Van Hove singularities in different dimensions and illustrates the concepts with detailed theoretical-versus-experimental comparisons for Ge and graphite.〕 shows that there are four types of Van Hove singularities in three-dimensional space, depending on whether the band structure goes through a local maximum, a local minimum or a saddle point. In three dimensions, the DOS itself is not divergent although its derivative is. The function g(E) tends to have square-root singularities (see the Figure) since for a spherical free electron Fermi surface :$E\; =\; \backslash hbar^2\; k^2/2m$ so that $|\backslash vecE|\; =\; \backslash hbar^2\; k/m\; =\; \backslash hbar\; \backslash sqrt\}$. In two dimensions the DOS is logarithmically divergent at a saddle point and in one dimension the DOS itself is infinite where $\backslash vecE$ is zero. 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「Van Hove singularity」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.