Abstract
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.
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References
Kirkwood, J. G., J. Chem. Phys. 7, 506 (1939).
Pitzer, S. J., J. Chem. Phys. 8, 711 (1940).
Liang, C. Y., S. Krimm and G. B. B. M. Sutherland, J. Chem. Phys. 25, 543 (1956).
Bowers, W. and H. B. Rosenstock, J. Chem. Phys. 18, 1056 (1950).
van Hove, L., Phys. Rev. 89, 1189 (1953).
Maradudin, A. A., E. W. Montroll, G. H. Weiss and I. P. Ipatova, Solid State Phys. Suppl. 3, 1 (1971)
Wunderlich, B., J. Chem. Phys. 37, 1207 (1962).
Gilat, G. and G. Dolling, Phys. Lett. 8, 304 (1964).
Piseri, L. and G. Zerbi, J. Chem. Phys. 48, 3561 (1968).
Kitagawa, T. and T. Miyazawa, Bull. Chem. Soc. Japan 43, 372 (1970).
Gilat, G. and L. J. Raubenheimer, Phys. Rev. 144, 320 (1966).
Schmid, C. and H. Hölzl, J. Phys. C: Solid State Phys. 6, 2401 (1973).
Hölzl, K. and C. Schmid, J. Phys. C: Solid State Phys. 8, 2235 (1975).
Schmid, C. and K. Hölzl, J. Polym. Sci. A2 10, 1881 (1972).
Zerbi, G., Conference of the European Physical Society on “Radiation Scattering of Bulk Polymers', Strasburg, 1972.
Zbinden, R., Infrared Spectroscopy of High Polymers. (New York, 1964.)
Hölzl, K. and C. Schmid, to be published.
Zerbi, G., Applied Spectroscopy Reviews (New York, 1969).
Faddeev, D. K. and V. N. Faddeeva, Computational Methods of Linear Algebra. (San Francisco-London, 1963.)
Kitagawa, T. and T. Miyazawa, Adv. in Polymer Sci. 9, 335 (1972).
Schmid, C., J. Phys. C: Solid State Phys. 6, L458 (1973).
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© 1975 Dr. Dietrich Steinkopff Verlag GmbH & Co. KG
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Schmid, C. (1975). Smooth frequency distribution derived from complex phases. In: Polymere Aspekte. Progress in Colloid & Polymer Science, vol 58. Steinkopff. https://doi.org/10.1007/BFb0117119
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DOI: https://doi.org/10.1007/BFb0117119
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Print ISBN: 978-3-7985-1796-7
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