含义Meaning
把化学键看作弹簧，伸缩振动波数随力常数（键强）增大而增大，随约化质量增大而减小。红外光谱官能团区位置的物理基础。
Treating a bond as a spring, the stretching wavenumber rises with force constant (bond strength) and falls with reduced mass — the physical basis of IR group frequencies.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\tilde\nu\) | 波数wavenumber | cm⁻¹ |
| \(k\) | 力常数（单键约 500，双键约 1000，三键约 1500）force constant (≈500 single, 1000 double, 1500 triple) | N·m⁻¹ |
| \(\mu\) | 约化质量reduced mass | kg |
| \(c\) | 光速speed of light | cm·s⁻¹ |
推导Derivation
经典谐振子 \(\mu\ddot x = -kx\) 的角频率 \(\omega = \sqrt{k/\mu}\)，频率 \(\nu = \omega/2\pi\)。
A classical oscillator has \(\omega = \sqrt{k/\mu}\), \(\nu = \omega/2\pi\).
量子谐振子能级间隔 \(h\nu\)，基频吸收 \(v = 0 \to 1\) 恰在此频率。
The quantum oscillator’s level spacing is hν, so the fundamental absorbs at this frequency.
换成波数 \(\tilde\nu = \nu/c\)。常用形式：\(\tilde\nu \approx 1303\sqrt{k/\mu}\)（k 用 mdyn/Å，μ 用原子质量单位）。
In wavenumbers \(\tilde\nu = \nu/c\); practically \(\tilde\nu \approx 1303\sqrt{k/\mu}\) with k in mdyn/Å and μ in u.
适用条件与局限Assumptions & limitations
谐振子近似；真实键有非谐性，基频略低于谐振频率。
Harmonic approximation; anharmonicity lowers the observed fundamental.
多原子分子振动是简正模式的耦合，不只是单个键的伸缩；指纹区耦合尤其严重。
Polyatomic vibrations are coupled normal modes, especially in the fingerprint region.
例题Worked example
C–H：k ≈ 500 N/m，μ = 12×1/13 u = 0.923 u → ν̃ ≈ 3030 cm⁻¹；C–D：μ = 1.71 u → ν̃ ≈ 2220 cm⁻¹。
C–H: k ≈ 500 N/m, μ = 0.923 u → ≈ 3030 cm⁻¹; C–D: μ = 1.71 u → ≈ 2220 cm⁻¹.