Smooth frequency distribution derived from complex phases
The frequency distribution g (ω) (density of states at frequency ω) of an isolated polymer chain (with arbitrary number γ of degrees of freedom per unit cell) is usually calculated approximately as a histogram from the phase frequency curve ω β(φ) (β = 1,..., γ) with a large set of real phases φ). In this paper a method is presented for calculating g (ω) exactly as a rational function of a small set of complex phases φi (ω) (i = 1,…, n 0γ, n 0 - range of interaction). The φi (ω) are the roots of a frequency dependent secular equation the coefficients of which are obtained by the theory of invariants (differentiation of the algorithm of Leverrier-Souriau). The method is of basic importance also for the computation of the Green function in space energy representation for realistic polymers. Analytical results are obtained for a polyatomic linear chain with arbitrary γ and n 0 = 1.
KeywordsDispersion Relation Polynomial Matrix Complex Phasis Solid State Phys Dynamical Matrix
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