Volume 112A, number 1,2 PHYSICS LETTERS 14 October 1985 zyxwvutsrq MELTING AND SPECTRA OF TWO-DIMENSIONAL CLASSICAL CRYSTALS Yu.E. LOZOVIK zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA International Laboratory of High M agnetic Fields und Low Temperature.% 53 529 W roctuaw , Poland V.M. FARZTDINOV, B. ABDULLAEV and S.A. KUCHEROV Institute of Spectroscopy, Academy of Sciences, 142092 Troitsk, Moscow region, USSR Received 23 May 1985; accepted for publication 5 August 1985 The anharmonicity-induced phonon softening and melting of 2D classical crystals (with interactions V(r) = l/r’. k = 1-12) are considered. The approximation taking account of skeleton (renormalized) diagrams of the first order on the modified melting parameter (MMP) is used. Melting points. temperature dependences and critical values of shear moduli and of MMP are in excellent agreement with physical and numerical experiments for not too high k. The existence of dimensionless melting parameters almost independent of the interaction law is established. Crystallization and melting of electrons on liquid helium [1,2] (with interaction V- I/r or V- l/r3 if the thin helium film is on the metal substrate [3]), of magnetic bubbles (V - l/r3), of colloidal system on surfaces [4] (V - l/r3), of adatom layers (where Vmay be approximated as I/- 1/r12 at high densi- ties [S]) and some other 2D systems [6] with power law interactions are widely investigated both experi- mentally [ 1,2,6] and by numerical simulations [2,4-71. The determination of the physical mecha- nism of melting, the order of the phase transition, the evaluation of the temperature dependences of the oscillation spectra and of some other physical quantities are of special interest in the theory of these systems. Our paper is devoted mainly to these questions. Besides we are interested in the existence of dimen- sionless parameters of melting which are almost in- dependent of the nature of the 2D crystals. We con- sider 2D classical *I crystals (2D CC) with interactions between particles V(r) = E(u/@, k = 1-12, 15. We take into account the phonon anharmonicities in the framework of the (3 t 4)-self-consistent approxima- tion ((3 t 4)-SCA) [8]. The (3 t 4)SCA corresponds *l The quantum case is considered in ref. [8]. -=-++ 4 zyxwvutsrqpon Q +u+ * h+b+e+* “: +OO+_+O CI Fjg. L (a) Dyson equation for the phonon Green function (PGF). Thin line is the harmonic PGF, thick one is the re- normalized PGF. (b) and(c) Skeleton diagrams of the first and second order on 7~. to the account of the skeleton diagrams (see fig. 1) (containing the renormalized phonon Green functions (PGF)) of the same, namely, the first order in the small modified melting parameter (MMP) rM = ((ui - ui+ 1)2>/a2, where Ui and IC~+~ are displacements in neighbouring lattice sites, a is the lattice constant and ( ) corresponds to the thermal average with the renormalized PGF. At temperatures T-% T, (T, is the melting temperature) the result of (3 t 4)-SCA agrees with the lowest order of the perturbation theory which corresponds to the replacement of the renormalized PGF by harmonic ones. This correct (at small 7’) limit is absent, e.g., in the self-consistent harmonic approximation (SCHA) which takes into 0.375-9601/85/$ 03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 61