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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 Reciprocal lattice ： ウィキペディア英語版 Reciprocal lattice In physics, the reciprocal lattice represents the Fourier Transform of another lattice (usually a Bravais lattice). In normal usage, this first lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial function in real-space and is also known as the ''direct lattice''. While the direct lattice exists in real-space and is what one would commonly understand as a physical lattice, the reciprocal lattice exists in reciprocal space (also known as ''momentum space'' or less commonly ''K-space'', due to the relationship between the Pontryagin duals momentum and position.) The reciprocal lattice of a reciprocal lattice, then, is the original direct lattice again, since the two lattices are Fourier Transforms of each other. ==Mathematical description== Consider a set of points R (R is a vector depicting a point in a Bravais lattice) constituting a Bravais lattice, and a standing plane wave defined by: :$e^\}=\backslash cos\; )\}\; +i\backslash sin\; )\}$ If this plane wave has the same periodicity as the Bravais lattice, then it satisfies the equation: :$\backslash begin$ e^} &= e^} \\ \therefore e^} e^} &= e^} \\ \Rightarrow e^} &= 1 \end Mathematically, we can describe the reciprocal lattice as the set of all vectors K that satisfy the above identity for all lattice point position vectors R. This reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice, which reveals the Pontryagin duality of their respective vector spaces. For an infinite two-dimensional lattice, defined by its primitive vectors $(\backslash mathbf\})$, its reciprocal lattice can be determined by generating its two reciprocal primitive vectors, through the following formulae, :$\backslash begin$ \mathbf &= 2\pi \frac - \hat \otimes \hat) \mathbf} \otimes \hat - \hat \otimes \hat) \mathbf} \\ \mathbf &= 2\pi \frac - \hat \otimes \hat) \mathbf} \otimes \hat - \hat \otimes \hat) \mathbf} \end where "$\backslash otimes$" has been used to form the tensor product between the Euclidean unit vectors, $\backslash hat$ and $\backslash hat$. The tensor products displayed here form simple 90 degree rotations. For an infinite three-dimensional lattice, defined by its primitive vectors $(\backslash mathbf\},\; \backslash mathbf$ \mathbf &= 2\pi \frac} \times \mathbf)} \\ \mathbf &= 2\pi \frac} \times \mathbf)} \\ \mathbf &= 2\pi \frac} \times \mathbf)} \end Note that the denominator is the scalar triple product. Using column vector representation of (reciprocal) primitive vectors, the formulae above can be rewritten using matrix inversion: :$\backslash left()^T\; =\; 2\backslash pi\backslash left()^.$ This method appeals to the definition, and allows generalization to arbitrary dimensions. The cross product formula dominates introductory materials on crystallography. The above definition is called the "physics" definition, as the factor of $2\; \backslash pi$ comes naturally from the study of periodic structures. An equivalent definition, the "crystallographer's" definition, comes from defining the reciprocal lattice to be $e^\}=1$ which changes the definitions of the reciprocal lattice vectors to be :$$ \mathbf} \times \mathbf} \times \mathbf and so on for the other vectors. The crystallographer's definition has the advantage that the definition of $\backslash mathbf\}$ in the direction of $\backslash mathbf\}$, dropping the factor of $2\; \backslash pi$. This can simplify certain mathematical manipulations, and expresses reciprocal lattice dimensions in units of spatial frequency. It is a matter of taste which definition of the lattice is used, as long as the two are not mixed. Each point (hkl) in the reciprocal lattice corresponds to a set of lattice planes (hkl) in the real space lattice. The direction of the reciprocal lattice vector corresponds to the normal to the real space planes. The magnitude of the reciprocal lattice vector is given in reciprocal length and is equal to the reciprocal of the interplanar spacing of the real space planes. The reciprocal lattice plays a fundamental role in most analytic studies of periodic structures, particularly in the theory of diffraction. For Bragg reflections in neutron and X-ray diffraction, the momentum difference between incoming and diffracted X-rays of a crystal is a reciprocal lattice vector. The diffraction pattern of a crystal can be used to determine the reciprocal vectors of the lattice. Using this process, one can infer the atomic arrangement of a crystal. The Brillouin zone is a Wigner-Seitz cell of the reciprocal lattice. 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「Reciprocal lattice」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.