Validity of the compressibility equation and Kirkwood-Buff theory for crystalline matter

Volume integrals over the radial pair-distribution function, so-called Kirkwood-Buff integrals (KBIs), play a central role in the theory of solutions by linking structural with thermodynamic information. The simplest example is the compressibility equation, a fundamental relation in statistical mechanics of fluids. Until now, KBI theory could not be applied to crystals because the integrals strongly diverge when computed in the standard way. We solve the divergence problem and generalize KBI theory to crystalline matter by using the recently proposed finite-volume theory. For crystals with harmonic interaction, we derive an analytic expression for the peak shape of the pair-distribution function at finite temperature. From this we demonstrate that the compressibility equation holds exactly in harmonic crystals.

Figure 1

PDF $g\left(r\right)$ and correlation function ${\mathrm{\Gamma }}^{-1}\left(L\right)$, Eq. (4), of a perfect fcc lattice. $r$ is the pair distance and $L$ is the upper integration bound. The nearest-neighbor distance $d$ is taken as unit length. (a) PDF as histogram plot with bin size $\mathrm{\Delta }r=0.05$. (b) ${\mathrm{\Gamma }}_0^{-1}\left(L\right)$ computed with running KBI, Eq. (7). (c) ${\mathrm{\Gamma }}^{-1}\left(L\right)$ computed with finite-volume KBI, Eq. (8). The inset shows ${\mathrm{\Gamma }}^{-1}\left(L\right)$ on the full $y$ scale. The blue lines are a guide to the eye. They are $y=1$ in (a), $y=\pm 10L$ in (b), $y=0.33/L$ in (c).

Figure 2

PDF $g\left(r\right)$ and fluctuation function ${\mathrm{\Gamma }}^{-1}\left(L\right)$ at finite temperature for the fcc crystal in the harmonic approximation. The topmost (green) curve is the PDF for $T/T_0=0.02$, shifted up by 1 for clarity. All other curves are obtained with $T/T_0=0.1$. ${\mathrm{\Gamma }}^{-1}$ (red) and ${\mathrm{\Gamma }}_0^{-1}$ (black) correspond to running and finite-volume KBI, respectively. Note that ${\mathrm{\Gamma }}_0^{-1}$ is multiplied by a factor of 0.05.

Figure 3

$L\to \infty$ convergence of the fluctuation function ${\mathrm{\Gamma }}^{-1}\left(L\right)$ of the harmonic fcc crystal obtained with finite-volume KBI. The reduced temperature $T/T_0$ is indicated on each curve. The straight lines are linear fits in the range $1/L<0.075$.

Figure 4

Effect of peak shape on fluctuation function ${\mathrm{\Gamma }}^{-1}\left(L\right)$ of an fcc crystal at $T/T_0=0.1$, obtained with finite-volume KBI. (a) Simple Gaussian shape. (b) Exact shape and width [Eqs. (14) and (16)]. (c) Exact shape but constant width $[\beta =0$ in Eq. (14)].