Fractional dimensionality and fractional derivative spectra of interband optical transitions

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Abstract
The model of fractional-dimensional space is used to study optical properties associated with electron interband transitions near van Hove critical points in anisotropic systems. Bloch electrons in an anisotropic solid are treated as an isotropic fractional-dimensional free gas, where the dimension is determined by the degree of anisotropy. Density of states and optical spectra are obtained as a function of spatial dimension. Fractional derivative spectra (FDS) are developed for analyzing the dimensionality of solids from the measured interband optical spectra in van Hove singularity regions. Using fractional differentiation, the dimension, as well as critical-point parameters, of a solid is straightforwardly determined from the derivative order that yields a symmetric line shape, Lorentzian or its derivative, in FDS. The fractional dimension determined by FDS is related to the anisotropic electron-lattice interactions and quantitatively describes the degree of anisotropy.