含义Meaning
转动能级布居是简并度（2J+1，随 J 增大）与玻尔兹曼因子（随 J 减小）之积，所以存在一个布居最大的 J，决定了转动谱和振–转谱 P、R 支的强度包络。
Populations are degeneracy (2J + 1, rising) times Boltzmann factor (falling), so one J is most populated; this sets the intensity envelope of rotational and P/R-branch spectra.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(N_J\) | 能级 J 上的分子数population of level J | 1 |
| \(J_\text{max}\) | 布居最大的能级most populated level | 1 |
推导Derivation
玻尔兹曼分布：\(N_J \propto g_J e^{-E_J/k_\mathrm{B}T}\)，\(g_J = 2J+1\)。
Boltzmann: \(N_J \propto g_Je^{-E_J/k_\mathrm{B}T}\) with \(g_J = 2J+1\).
把 J 当作连续变量求导 \(dN_J/dJ = 0\)：\(2 = (2J+1)^2hcB/k_\mathrm{B}T\)。
Treat J as continuous and set \(dN_J/dJ = 0\).
解得 \(J_\text{max}\)，取最接近的整数。
Solve and round to the nearest integer.
适用条件与局限Assumptions & limitations
同核双原子分子需考虑核自旋统计（如 ¹⁶O₂ 只有奇数 J；H₂ 正、仲之比 3:1）。
Homonuclear diatomics need nuclear-spin statistics (¹⁶O₂ odd J only; ortho:para H₂ 3:1).
例题Worked example
HCl（B = 10.6 cm⁻¹）在 300 K：J_max ≈ 3；CO（B = 1.93）在 300 K：J_max ≈ 7。
HCl at 300 K: J_max ≈ 3; CO: J_max ≈ 7.