Coarse-grained density and compressibility of nonideal crystals: General theory and an application to cluster crystals

J. M. Häring, C. Walz, G. Szamel, and M. Fuchs

Phys. Rev. B 92, 184103 – Published 9 November 2015

Abstract

The isothermal compressibility of a general crystal is analyzed within classical density functional theory. Our approach can be used for homogeneous and unstrained crystals containing an arbitrarily high density of local defects. We start by coarse-graining the microscopic particle density and then obtain the long-wavelength limits of the correlation functions of elasticity theory and the thermodynamic derivatives. We explicitly show that the long-wavelength limit of the microscopic density correlation function differs from the isothermal compressibility. We apply our theory to crystals consisting of soft particles which can multiply occupy lattice sites (“cluster crystals”). The multiple occupancy results in a strong local disorder over an extended range of temperatures. We determine the cluster crystals' isothermal compressibility, the fluctuations of the lattice occupation numbers and their correlation functions, and the dispersion relations. We also discuss their low-temperature phase diagram.

Received 29 July 2015
Revised 13 October 2015

DOI:https://doi.org/10.1103/PhysRevB.92.184103

Authors & Affiliations

J. M. Häring¹, C. Walz¹, G. Szamel², and M. Fuchs¹

¹Fachbereich für Physik, Universität Konstanz, 78457 Konstanz, Germany
²Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523, USA

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Vol. 92, Iss. 18 — 1 November 2015

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Images

Figure 1
Compressibilities of the GEM-4 system in units of ${[n_{0}^{2} ε σ^{3}]}^{- 1}$ versus the reduced thermodynamic variable $\frac{k_{B} T}{ε n_{0} σ^{3}}$ . While $κ$ is taken at fixed stress $h_{α β}, κ^{c}$ is taken at fixed stress $σ_{α β}$ , and $κ^{0}$ is the approximation neglecting the strain-density coupling introduced in Eq. (54). The approximation $κ \approx 1 / ν$ to neglect the strain-defect density coupling holds within the line thickness; see Fig. 4.
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Figure 2
The bulk modulus $B = \frac{1}{κ}$ in dimensionless units for three different temperatures versus $n_{0} σ^{3}$ . The three points are MC simulation results [16].
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Figure 3
Probability distribution functions for the occupation numbers in GEM-4 cluster crystals of fcc and bcc structure from MC simulations [17]. Gaussian distributions with the variances calculated from Eq. (80) and the mean value $n_{c}$ obtained through minimization of Eq. (78) (lines from left to right with increasing $n_{0} σ^{3}$ ) are compared with the MC data (symbols). Complete parameters are given in Table 1.
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Figure 4
Top panel: Phonon dispersion relation for a cluster crystal with fcc structure along four symmetry lines in the first Brillouin zone; the state at $k_{B} T / ε = 1.1$ and $n_{0} σ^{3} = 8.5$ is also included in Figs. 2 and 3 and Table 1. Bottom panel: The $q$ -dependent density correlation function from Eq. (64) and its dominating part for small $q$ given by $ν^{- 1} (q)$ (dashed blue line). Insets: The difference of both quantities $Δ (q) = 〈 δ n^{2} (q) 〉 σ^{3} / V - ν^{- 1} (q) k_{B} T n_{0}^{2}$ in a small range around $q = 0$ for the same symmetry lines. The different limits $Δ (q \to 0)$ depending on direction are apparent.
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Figure 5
Low-temperature phase diagram of the GEM-4 system as determined in MC simulations [19] (squares) and phonon theory [49] (crosses); red points connected by lines as guides to the eye indicate the coexistence regions. Pure fcc phases with integer site-occupations (denoted fccn with $n = 2, 3, ...$ ) survive only at extremely low temperatures. Mean-field DFT provides a good estimates of the critical temperatures for a reasonable numerical value of the occupation number variance, $\sqrt{〈 Δ n_{c}^{2} 〉} = 0.3$ (labeled blue line).
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Figure 6
The occupation number fluctuation $\sqrt{〈 Δ n_{c}^{2} 〉}$ (standard deviation) versus the temperature in units of $ε / k_{B}$ . The standard deviation is a function of temperature and density. It is plotted for several integer occupied states. The densities are chosen with the approximation $n_{c} / n_{0} \approx 2$ , which differs by about two percent from the optimal DFT-value. The red dotted line denotes the point of the curve at the critical temperature $k_{B} T_{c} / ε = 0.471$ which is obtained from $[N] p T$ simulations [20]; the blue line is the estimate of Fig. 5.
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