An Improved Self-Consistent Phonon Approximation

Abstract

A self-consistent phonon approximation that includes odd-order anharmonic terms is found to give improvement over the usual lowest-order self-consistent phonon approximation when applied to computation of several properties of crystalline Ne and Ar. At low temperatures the improved theory joins smoothly onto the results of conventional perturbation theory.The failure of conventional lattice-dynamical perturbation theory when applied to ideal inert gas solids above -4/10 of their melting temperature (i.e., for rms amplitudes greater than -6%) has been pointed out recently. 1 At the same time the self-consistent phonon theory of Born and others has become well established. 2 The lowest-order form of this theory (hereafter called SC) has recently been applied to crystalline neon and argon by Gillis, Werthamer, and Koehler. 3 Since it omits odd derivatives of the interatomic potential (which are known to give substantial contributions at finite temperatures 1 ), the lowest-order theory is inadequate. It seems natural to investigate a self-consistent theory that includes odd-order anharmonic terms before investigating other problems such as hard-core effects and higher-order self-consistent theories.In the SC, frequencies oo^s 2 and wave vectors e a (qs) are obtained by iterating the following equations:^=e a (qs)D a^) e^s) 9 {<p (R.)) = (87r 3 detA)-1/2 /^3wexp(-|u-A.-1 .u)^ (R.+u), (A.) = 2MN~1TJ + {l-cos(q-R.)}e (qs)e 0 (qs)u+ /W 2 , v jaP ^q s L ^ 2 a 0^ qs' qs ' and the Helmholtz free energy is given by F^WEjW&j)) +E qs itt~l ln ( 2 sinh £^5 s >-i*% Gothic]Here, u+ =Hou+ (n+ +|), n+ ={exp(/3a>^ j-l}"" 1 , qs qs qs 2 ' qs L ^ qs J '(p(Rj) is the two-body potential, and /3 = #/&gT. These equations were applied to crystalline Ar and Ne by Gillis, Werthamer, and Koehler 3 with some disappointing results. For Ar they found C/,-26 J deg"" 1 mole" 1 at the triple point compared with the observed value 4 of 35 J deg"" 1 mole"" 1 , and wave velocities exhibited a temperature variation quite incompatible with the observed bulk modulus. 5 [Anticipating our results (Fig. 3) we will see the SC bulk modulus in reasonable agreement with experiment. Hence, Gillis, Werthamer, and Koehler's wave velocities are not those predicted by SC] An obvious omission from F$Q are terms involving odd derivatives of <p. With this in mind we introduce an improved self-consistent approximation (ISC) for the Helmholtz free energy, ^isc =^S C + AF, where *L = d#-V A^+ %

Thermodynamics and microstructure of vacancies in rare gas crystals at high temperature

“…Realization of this fact became a challenge that stimulated attempts to work out some methods for adequate self-consisting description of strongly anharmonic solids. At present, existing theoretical models [4][5][6][7] predict well enough thermal and elastic properties of the bulk RGC in a wide range of temperature and pressure, in agreement with the experimental data available.…”

Binary distribution functions of atoms of simple crystals

“…The ISC theory is known to give an excellent description of the thermodynamic properties of the quasiclassical anharmonic solids ͑Ar, Kr, and Xe͒, 11,20 though results for solid neon do not fit satisfactorily the observed data at high temperature. 21 Similarly to the ISC model, 11 we add a correction ⌬F 3 due to the cubic anharmonicity of atomic vibrations to the free energy ͑12͒ of the quasiharmonic crystal:…”

“…The SCP theory and its modifications were used successfully to evaluate the dynamical properties of both three-dimensional ͑3D͒ crystals [11][12][13][14][15] and plane adsorbed monolayers of atoms. 16 -18 Particularly, the improved self-consistent ͑ISC͒ model, which is the SCH approximation corrected for cubic anharmonicity, provides a satisfactory description of the phonon spectrum 13,19 and thermodynamic properties of the rare gas ͑RGC's͒ [11][12][13]20,21 crystals, except for temperatures near the melting point, where the iteration process shows poor convergence. This shortcoming is avoided within the effective potential theory, [22][23][24] developed during the past decade.…”

Anharmonicity in the surface and size effects in the specific heat of linear chains

“…Turning now to the anharmonic contribution cB,(50), we first note that various estimates [6] indicate that anharmonic perturbation theory cannot be expected to be valid for T* >0.3 to 0.5 for bulk crystals; for small crystals with surfaces this upper limit would be even smaller [7]. The vertical dashed line at T* = 0.3 in Fig.…”