含义Meaning
由离子间库仑吸引与短程排斥计算离子晶体的晶格能（气态离子形成 1 mol 晶体释放的能量）。
Computes the lattice energy of an ionic crystal (energy released when gaseous ions form 1 mol of solid) from Coulomb attraction and short-range repulsion.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(M\) | 马德隆常数Madelung constant | 1 |
| \(z_\pm\) | 离子电荷数（|z|）ionic charges | 1 |
| \(r_0\) | 平衡阴阳离子间距equilibrium cation–anion distance | m |
| \(n\) | 玻恩指数（5–12，取决于电子构型）Born exponent (5–12) | 1 |
| \(\rho\) | 排斥范围参数 ≈ 0.345 Årepulsion range parameter ≈ 0.345 Å | m |
推导Derivation
一对离子的库仑能 \(z_+z_-e^2/(4\pi\varepsilon_0r)\)；对整个晶格求和得到马德隆常数 M（只依赖结构）。
The Coulomb energy of one pair summed over the lattice gives the Madelung constant M (structure only).
电子云重叠引起的排斥写作 \(B/r^n\)。每摩尔能量 \(U(r) = -\frac{N_AMz^2e^2}{4\pi\varepsilon_0 r} + \frac{N_AB}{r^n}\)。
Repulsion from overlapping electron clouds: \(B/r^n\).
平衡时 \(dU/dr = 0\)，解出 \(B = \frac{Mz^2e^2r_0^{\,n-1}}{4\pi\varepsilon_0 n}\)。
At equilibrium \(dU/dr = 0\), fixing B.
代回即得 \(U_L = -\frac{N_AMz^2e^2}{4\pi\varepsilon_0r_0}(1 - 1/n)\)。排斥项只修正约 10%。
Substituting back gives the result; repulsion corrects by only ~10 %.
n 由压缩率测定，或按离子构型取值：He 5、Ne 7、Ar 9、Kr 10、Xe 12（阴阳离子取平均）。
n from compressibility, or by configuration: He 5, Ne 7, Ar 9, Kr 10, Xe 12 (average for the pair).
适用条件与局限Assumptions & limitations
纯离子模型；对极化强、共价成分大的晶体（AgI、CuCl）计算值明显小于玻恩–哈伯循环实验值。
Purely ionic; underestimates for polarisable or covalent solids (AgI, CuCl).
忽略范德华色散能与零点能（约 1–2%）。
Neglects dispersion and zero-point energy (~1–2 %).
例题Worked example
NaCl：M = 1.74756，r₀ = 2.814 Å，n = 8：U_L = −755 kJ/mol（玻恩–哈伯实验值 −787 kJ/mol）。
NaCl: M = 1.74756, r₀ = 2.814 Å, n = 8: U_L = −755 kJ/mol (Born–Haber −787).