硬晶体中发散直接关联的自由体积起源：来自精确一维模型的见解
Free-volume origin of diverging direct correlations in hard crystals: insights from an exact one-dimensional model
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中文总结 AI 辅助
研究三维硬球晶体中二阶直接关联函数，利用一维Percus泛函分离问题，通过高斯峰表示密度，分析自由体积与关联函数关系，解释其大小及晶格特定形式，数值解展示关联分布，提供最小自由体积解释。
中文摘要 AI 辅助
三维硬球晶体中的二阶直接关联函数比其液态对应物具有大得多的振幅（量级为$1/n_{\mathrm{vac}}$，其中$n_{\mathrm{vac}}$是空位浓度）以及明显更短程的更强结构化空间形式。我们使用硬棒的精确一维Percus泛函来分离两个潜在问题——为何关联大以及为何具有这样的形式。周期性晶体状密度由占据概率$q = 1 - n_{\mathrm{vac}}$的高斯峰表示。相关热力学量是局部插入自由体积$A(x)$。其最小值可写为$A_{\mathrm{min}}=\Delta_{\alpha a}+(1 - \Delta_{\alpha a})n_{\mathrm{vac}}$，其中$\Delta_{\alpha a}$是由有限局域化（由宽度参数$\alpha$表征）和晶格间距$a$导致的完全占据时的残余自由体积。因此，第一和第二直接关联包含奇异依赖性$c^{(1)}\sim\ln A_{\mathrm{min}}$和$c^{(2)}\sim - 1/A_{\mathrm{min}}$。观察到的$1/n_{\mathrm{vac}}$依赖性是空位主导极限$n_{\mathrm{vac}}\gg\Delta_{\alpha a}$，而非最一般结果。$c^{(2)}$的空间形式有单独的几何起源：硬棒权重函数选择排斥区间及其边界与小自由体积区域相交的构型。这产生了与底层周期性密度相关的平台、边缘和局部脊。非均匀Ornstein - Zernike方程的数值解展示了这些奇异直接关联如何在总关联和对关联中重新分布。该模型为晶体直接关联函数的大小和晶格特定形式提供了最小自由体积解释。
英文摘要
Second order direct correlation functions in three-dimensional hard-sphere crystals have much larger amplitudes (of the order of $1/n_{\mathrm{vac}}$ where $n_{\mathrm{vac}}$ is the vacancy concentration) and a more strongly structured spatial form of apparent shorter range than their liquid-state counterparts. We separate the two underlying questions---why the correlations are large and why they have the form they do---using the exact one-dimensional Percus functional for hard rods. A periodic crystal-like density is represented by Gaussian peaks with occupation probability $q=1-n_{\mathrm{vac}}$. The relevant thermodynamic quantity is the local insertion free volume $A(x)$. Its minimum can be written as $A_{\mathrm{min}} =Δ_{αa}+(1-Δ_{αa})n_{\mathrm{vac}}$, where $Δ_{αa}$ is the residual free volume at full occupation caused by finite localization (characterized by a width parameter $α$) and lattice spacing $a$. The first and second direct correlations therefore contain the singular dependences $c^{(1)}\sim\ln A_{\mathrm{min}} $ and $c^{(2)} \sim-1/A_{\mathrm{min}} $. The observed $1/n_{\mathrm{vac}}$ dependence is the vacancy-dominated limit $n_{\mathrm{vac}}\ggΔ_{αa}$, rather than the most general result. The spatial form of $c^{(2)}$ has a separate geometrical origin: the hard-rod weight functions select configurations in which exclusion intervals and their boundaries intersect regions of small free volume. This produces plateaus, edges, and localized ridges tied to the underlying periodic density. A numerical solution of the inhomogeneous Ornstein--Zernike equation shows how these singular direct correlations are redistributed in the total and pair correlations. The model provides a minimal free-volume explanation for both the magnitude and the lattice-specific form of crystalline direct correlation functions.
