含义Meaning
自旋 1/2 的核在磁场 B₀ 中分裂为两个能级，吸收频率等于核磁矩进动的拉莫尔频率。能差极小，上下能级布居差只有约 10⁻⁵，所以 NMR 灵敏度低，需要强磁场。
A spin-½ nucleus in field B₀ has two levels; it absorbs at the Larmor precession frequency. The gap is tiny, so the population difference is ~10⁻⁵ — hence NMR’s low sensitivity and need for strong magnets.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\gamma\) | 旋磁比（¹H 2.675×10⁸ rad·s⁻¹·T⁻¹）gyromagnetic ratio (¹H 2.675×10⁸) | rad·s⁻¹·T⁻¹ |
| \(B_0\) | 外磁场applied field | T |
| \(\nu_0\) | 共振频率resonance frequency | Hz |
推导Derivation
核磁矩 \(\boldsymbol\mu = \gamma\hat{\mathbf I}\)，在磁场中能量 \(E = -\mu_zB_0 = -\gamma\hbar m_IB_0\)，\(m_I = \pm\tfrac12\)。
Nuclear moment \(\mu = \gamma I\); energy \(E = -\gamma\hbar m_IB_0\).
两能级差 \(\Delta E = \gamma\hbar B_0 = h\nu_0\)，得 \(\nu_0 = \gamma B_0/2\pi\)。
\(\Delta E = \gamma\hbar B_0 = h\nu_0\).
经典图像：磁矩受力矩绕 B₀ 进动，角频率 \(\omega_0 = \gamma B_0\)。
Classically the moment precesses at \(\omega_0 = \gamma B_0\).
布居差 \(\approx \gamma\hbar B_0/(2k_\mathrm{B}T)\)：¹H 在 9.4 T、298 K 约 3×10⁻⁵。
Population excess ≈ 3×10⁻⁵ for ¹H at 9.4 T and 298 K.
适用条件与局限Assumptions & limitations
化学环境使局部磁场 \(B_\text{loc} = B_0(1 - \sigma)\)，频率随之偏移（化学位移）。
Shielding makes \(B_\text{loc} = B_0(1-\sigma)\), shifting the frequency (chemical shift).
I > 1/2 的核有电四极矩，谱线常因快速弛豫而变宽。
Quadrupolar nuclei (I > ½) often give broad lines.
例题Worked example
¹H 在 9.4 T：ν₀ = 2.675×10⁸×9.4/2π = 400 MHz（“400 MHz 核磁”）；¹³C 约 100 MHz。
¹H at 9.4 T resonates at 400 MHz; ¹³C at ~100 MHz.