Frequencies of Longitudinal Vibrations of a Two-Dimensional Wigner Crystal Coupled to Ripplons on the Surface of Liquid Helium

  • A. G. Eguiluz
  • A. A. Maradudin
  • R. J. Elliott

Abstract

We outline a first-principles derivation of the frequencies of the vibration modes of a two-dimensional Wigner crystal coupled to ripplons on a liquid helium surface, based on the use of thermodynamic Green1s functions. The phonon operators enter the interaction Hamiltonian through exp \((i\vec q.\vec u(\ell ))\) where \(\vec q\) is a two-dimensional wave vector and \(\vec u(\ell )\) is the two-dimensional displacement of an electron from its equilibrium lattice site. This exponential operator is retained unexpanded in our evaluation of the phonon self-energy. The phonon self-energy is obtained to second order in the (weak) phonon- ripplon interaction. Our result for the self-energy clearly displays the resonant nature of the coupling between the phonons and the ripplons. We show that the weights with which the ripplons enter the dispersion relation of the coupled modes are given in terms of an exponential whose argument originates from the difference in electron- displacement correlation functions given by \( < \vec q.\vec u({\ell _1};{t_1})\vec q.\vec u({\ell _2};{t_2}) > \, - \, < {(\vec q.\vec u)^2} > .\). Contact is made with the experimental results of Grimes and Adams.

Keywords

Hexa Helium Hexagonal 

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    What we actually did was compute u for three meshes defined by N1=N2=55, N1=71 and N1=N2=85, respectively. Here N1 and N2 are defined through the usual boundary conditions in two 2 2 dimensions. We set \( < {u^2} > /a_o^2 = \ell n(C(T)L/{a_o})\,and\,\ell nC\, = \,\ell n{C_\infty }\, + \,A/{N_1}\, + \,B/N_1^2.\,(Note\,that\,\ell n{C_\infty }\, = \,\ell nC\,for\,N\, = \infty ).\). The constants \(A,B\,and\,\ell n{C_\infty }\) are obtained from the computed values of \( < {u^2} > /a_o^2\). \((We\,used\,{n_s}\, = \,4.55\,x\,{10^8}c{m^{ - 2}}\,and\,T\, = \,0.42^\circ K)\).Google Scholar

Copyright information

© Plenum Press, New York 1983

Authors and Affiliations

  • A. G. Eguiluz
    • 1
  • A. A. Maradudin
    • 1
  • R. J. Elliott
    • 2
  1. 1.Department of PhysicsUniversity of California IrvineCaliforniaUSA
  2. 2.Department of Theoretical PhysicsUniversity of OxfordOxfordUK

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