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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 Axiality and rhombicity ： ウィキペディア英語版 Axiality and rhombicity In physics and mathematics, axiality and rhombicity are two characteristics of a symmetric second-rank tensor in three-dimensional Euclidean space, describing its directional asymmetry. Let ''A'' denote a second-rank tensor in R3, which can be represented by a 3-by-3 matrix. We assume that ''A'' is symmetric. This implies that ''A'' has three real eigenvalues, which we denote by $A\_$, $A\_$ and $A\_$. We assume that they are ordered such that :$A\_\; \backslash le\; A\_\; \backslash le\; A\_.$ The axiality of ''A'' is defined by :$\backslash Delta\; A\; =2\; A\_-(A\_+A\_).\; \backslash ,$ The rhombicity is the difference between the smallest and the second-smallest eigenvalue: :$\backslash delta\; A\; =\; A\_-A\_.\; \backslash ,$ Other definitions of axiality and rhombicity differ from the ones given above by constant factors which depend on the context. For example, when using them as parameters in the irreducible spherical tensor expansion, it is most convenient to divide the above definition of axiality by $$. ==Applications== The description of physical interactions in terms of axiality and rhombicity is frequently encountered in spin dynamics and, in particular, in spin relaxation theory, where many traceless bilinear interaction Hamiltonians, having the (eigenframe) form :$\backslash hat\; H\; =\; \backslash hat\backslash vec\backslash mathbf\backslash cdot\backslash mathbf\backslash cdot\backslash hat\backslash vec\backslash mathbf\; =\; A\_\; \backslash hat\; a\_\; \backslash hat\; b\_\; +\; A\_\; \backslash hat\; a\_\; \backslash hat\; b\_\; +\; A\_\; \backslash hat\; a\_\; \backslash hat\; b\_$ (hats denote spin projection operators) may be conveniently rotated using rank 2 irreducible spherical tensor operators: :$\backslash hat\backslash vec\backslash mathbf\backslash cdot\backslash mathbf\backslash cdot\backslash hat\backslash vec\backslash mathbf\; =\; \backslash frac\; \backslash hat\; T\_\; +\; \backslash frac\; \backslash hat\; T\_\; +\; \backslash frac$ :$\backslash hat\; \backslash hat\; R\_\; (\backslash hat\; T\_)\; =\; \backslash sum\_^2\; \backslash hat\; T\_\; \backslash mathfrak\_^(\backslash alpha,\backslash beta,\backslash gamma)$ where $\backslash mathfrak\_^(\backslash alpha,\backslash beta,\backslash gamma)$ are Wigner functions, $(\backslash alpha,\backslash beta,\backslash gamma)$ are Euler angles, and the expressions for the rank 2 irreducible spherical tensor operators are: :$\backslash hat\; T\_\; =\; +\; \backslash frac\; \backslash hat\; a\_\; \backslash hat\; b\_$ :$\backslash hat\; T\_\; =\; -\; \backslash frac(\; \backslash hat\; a\_\; \backslash hat\; b\_\; +\; \backslash hat\; a\_\; \backslash hat\; b\_\; )$ :$\backslash hat\; T\_\; =\; +\; \backslash sqrt\}(\; \backslash hat\; a\_\; \backslash hat\; b\_\; -\; \backslash frac(\; \backslash hat\; a\_\; \backslash hat\; b\_\; +\; \backslash hat\; a\_\; \backslash hat\; b\_)\; )$ :$\backslash hat\; T\_\; =\; +\; \backslash frac(\; \backslash hat\; a\_\; \backslash hat\; b\_\; +\; \backslash hat\; a\_\; \backslash hat\; b\_\; )$ :$\backslash hat\; T\_\; =\; +\; \backslash frac\; \backslash hat\; a\_\; \backslash hat\; b\_$ Defining Hamiltonian rotations in this way (axiality, rhombicity, three angles) significantly simplifies calculations, since the properties of Wigner functions are well understood. 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「Axiality and rhombicity」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.