转动常数与键长 Rotational constant and bond length — 公式推导、符号与局限

\[B = \frac{h}{8\pi^2 c\, I} = \frac{h}{8\pi^2 c\,\mu r_e^2},\qquad \tilde\nu_{J\to J+1} = 2B(J+1)\]

含义Meaning

由纯转动光谱（微波或远红外）谱线间隔 2B 求转动惯量，进而得到极其精确的键长。

From the 2B spacing of pure rotational (microwave or far-IR) lines one obtains the moment of inertia and hence very precise bond lengths.

符号Symbol	含义 Meaning	单位 Unit
\(B\)	转动常数rotational constant	cm⁻¹
\(I\)	转动惯量moment of inertia	kg·m²
\(r_e\)	平衡键长equilibrium bond length	m

刚性转子 \(E_J = hcBJ(J+1)\)，选择定则 ΔJ = +1（吸收），需有永久偶极矩。
Rigid-rotor levels with ΔJ = +1 and a permanent dipole.
\(\Delta E = hcB[(J+1)(J+2) - J(J+1)] = 2hcB(J+1)\)。
The transition energy is \(2hcB(J+1)\).
测出间隔 2B → \(I = h/(8\pi^2cB)\) → \(r_e = \sqrt{I/\mu}\)。
Spacing → I → \(r_e = \sqrt{I/\mu}\).

严格得到的是振动平均的 \(r_0\)（来自 \(B_0\)），与 \(r_e\) 有微小差别（\(B_v = B_e - \alpha_e(v + \tfrac12)\)）。
Measured \(B_0\) gives the vibrationally averaged \(r_0\), slightly different from \(r_e\).
离心畸变使高 J 谱线间隔略减小。
Centrifugal distortion narrows high-J spacings.

¹²C¹⁶O：B = 1.9313 cm⁻¹，μ = 6.856 u → r = 1.131 Å。

¹²C¹⁶O: B = 1.9313 cm⁻¹ → r = 1.131 Å.