双语化学辞典Chem Lexicon
本科Undergraduate 离子晶体与晶格能 Ionic solids & lattice energy

半径比规则Radius ratio rule

\[\frac{r_+}{r_-}:\ \ 0.225\ (\mathrm{CN\,4})\ \ 0.414\ (\mathrm{CN\,6})\ \ 0.732\ (\mathrm{CN\,8})\ \ 1\ (\mathrm{CN\,12})\]
临界半径比的几何推导结果critical ratios from geometry
\[\left(\frac{r_+}{r_-}\right)_{\min,\,\mathrm{CN6}} = \sqrt2 - 1 = 0.414,\quad \left(\frac{r_+}{r_-}\right)_{\min,\,\mathrm{CN8}} = \sqrt3 - 1 = 0.732,\quad \left(\frac{r_+}{r_-}\right)_{\min,\,\mathrm{CN4}} = \sqrt{\tfrac32} - 1 = 0.225\]

含义Meaning

离子晶体中阳离子的配位数由阴阳离子半径比决定：阳离子必须与周围阴离子接触，而阴离子之间不重叠。

The cation’s coordination number is set by the radius ratio: the cation must touch its anions without the anions overlapping.

符号说明Symbols

符号Symbol含义 Meaning单位 Unit
\(r_+, r_-\)阳、阴离子半径cation and anion radiipm
\(\mathrm{CN}\)配位数coordination number1

推导Derivation

  1. 八面体（CN 6）：四个阴离子在正方形顶点相切，对角线 \(2(r_+ + r_-) = 2\sqrt2\,r_-\)，故 \(r_+/r_- = \sqrt2 - 1\)。

    Octahedral: four anions touching on a square, diagonal \(2(r_+ + r_-) = 2\sqrt2\,r_-\).

  2. 立方（CN 8）：体对角线 \(2(r_+ + r_-) = \sqrt3\cdot 2r_-\)，故 \(\sqrt3 - 1\)。

    Cubic: body diagonal gives \(\sqrt3 - 1\).

  3. 四面体（CN 4）：阳离子在立方体中心、阴离子在相间四个顶点；阴离子沿面对角线相切 \(\sqrt2 a = 2r_-\)，阴阳离子距离为半个体对角线 \(\tfrac{\sqrt3}{2}a = r_+ + r_-\)，得 \(\sqrt{3/2} - 1 = 0.225\)。

    Tetrahedral: anions on alternate cube corners touch along the face diagonal (\(\sqrt2 a = 2r_-\)); the cation–anion distance is half the body diagonal, giving \(\sqrt{3/2} - 1\).

  4. 半径比低于临界值时阳离子“晃动”，结构不稳定，转为较低配位数。

    Below the critical ratio the cation ‘rattles’ and a lower CN is adopted.

适用条件与局限Assumptions & limitations

  • 硬球模型；预测正确率只有约一半（离子半径非固定值、共价性、极化）。

    Hard-sphere model; right only about half the time (radii vary, covalency, polarisation).

  • 例：LiI 比值 0.35 预测 CN 4，实际为 NaCl 型（CN 6）。

    E.g. LiI (0.35) predicts CN 4 but is rock salt.