Abstract
In this paper, a diatomic chain composed of classical harmonic oscillators and an impurity with different mass and Hooke constant is studied by means of the recurrence relations method. The Laplace transform of the momentum autocorrelation function of the impurity has three pairs of resonant poles and three separate branch cuts; however, one pair of poles was found nonphysical, so only two pairs of poles contribute to the momentum autocorrelation function. The three branch cuts give the acoustic and optical branches that are given by a convolution of a sum of sines and an expansion of even-order Bessel functions. The expansion coefficients are integrals of a Jacobian elliptic function along the real axis in a complex plane for the acoustic branch and those along a contour parallel to the imaginary axis for the optical branch, respectively. The ergodicity of momentum of the impurity is also discussed.
Graphical abstract
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.F. Fox, Phys. Rev. A 27, 3216 (1983)
R.I. Cukier, Physica 61, 321 (1972)
P. Mazur, E.W. Montroll, R.B. Porrs, J. Wash. Acad. Sci. 46, 2 (1956)
A.A. Maradudin, E.W. Montroll, G.H. Weiss, inTheory of Lattice Dynamics in the Harmonic Approximation (Academic Press, New York, 1963), p. 179
P. Dean, J. Inst. Math. Appl. 3, 98 (1967)
G. Lucovsky, M.H. Brodsky, E. Burstein, Phys. Rev. B 2, 3295 (1970)
P. Ullersma, Physica 32, 74 (1966)
J. Hori, T. Asahi, Prog. Theor. Phys. (Kyoto) 17, 523 (1957)
R.L. Bjork, Phys. Rev. 105, 456 (1957)
M.H. Lee, Phys. Rev. Lett. 49, 1072 (1982)
M.H. Lee, J. Math Phys. 24, 2512 (1983)
M.H. Lee, J. Hong, J. Florencio Jr., Phys. Scr. T 19, 498 (1987)
U. Balucani, M.H. Lee, V. Tognetti, Phys. Rep. 373, 409 (2003)
M.H. Lee, J. Hong, Phys. Rev. Lett. 48, 634 (1982)
M.H. Lee, J. Hong, Phys. Rev. B 32, 7734 (1985)
E.M. Silva, Phys. Rev. E 92, 042146 (2015)
S. Sen, M. Long, Phys. Rev. B 46, 14617 (1992)
S. Sen, Phys. Rev. B 44, 7444 (1991)
S. Sen, Physica A 360, 304 (2006)
A.V. Mokshin, R.M. Yulmetyev, P. Hanggi, Phys. Rev. Lett. 95, 200601 (2005)
A. Wierling, Eur. Phys. J. B 85, 20571 (2012)
P.R.C. Guimaraes, J.A. Placak, O.F. de Alcantara Bonfim, J. Florencio, Phys. Rev. E 92, 042115 (2015)
M.E.S. Nunes, E.M. Silva, P.H.L. Martins, J.A. Placak, J. Florencil, Phys. Rev. E 98, 042124 (2018)
J. Florencio, Jr. M.H. Lee, Phys. Rev. A 31, 3231 (1985)
M.H. Lee, J. Florencio Jr. J. Hong, J. Phys. A 22, L331 (1989)
J. Kim, I. Sawada, Phys. Rev. E 61, R2172 (2000)
M.B. Yu, Eur. Phys. J. B 86, 57 (2013)
M.B. Yu, Eur. Phys. J. B 90, 87 (2017)
M.B. Yu, Physica A 398, 252 (2014)
M.B. Yu, Eur. Phys. J. B 91, 25 (2018)
M.B. Yu, Eur. Phys. J. B 92, 272 (2019)
L.M. Milne-Thomson, Jacobian Elliptic Functions and Theta Functions, inHandbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, Applied Mathematics Series 55, edited by M. Abramowitz & I. A. Stegun (National Bureau of Standards, Washington, D.C., 1964), Chap. 16, p. 575
M.H. Lee, Phys. Rev. Lett. 87, 250061 (2001)
Y.H. Shen, Z.Z. Liang, L.H. Xu, Q.Q. Cai, inPractical Handbook of Mathematics (Science Press, Beijing, 1999), p. 17
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Yu, M.B. A diatomic chain with an impurity in mass and Hooke constant. Eur. Phys. J. B 93, 152 (2020). https://doi.org/10.1140/epjb/e2020-10048-y
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2020-10048-y