衰变链与放射平衡 Decay chains and radioactive equilibrium — 公式推导、符号与局限

\[N_2(t) = \frac{\lambda_1}{\lambda_2 - \lambda_1}N_1(0)\left(e^{-\lambda_1t} - e^{-\lambda_2t}\right) + N_2(0)e^{-\lambda_2t}\]

两种平衡secular and transient equilibrium

\[\lambda_1 \ll \lambda_2:\ A_1 = A_2\ (\text{长期平衡});\qquad \lambda_1 < \lambda_2:\ \frac{A_2}{A_1} = \frac{\lambda_2}{\lambda_2 - \lambda_1}\ (\text{暂时平衡})\]\[\lambda_1 \ll \lambda_2:\ A_1 = A_2\ (\text{secular eq.});\qquad \lambda_1 < \lambda_2:\ \frac{A_2}{A_1} = \frac{\lambda_2}{\lambda_2 - \lambda_1}\ (\text{transient eq.})\]

含义Meaning

母体 1 衰变为放射性子体 2（贝特曼方程）。母体寿命远长于子体时，经过数个子体半衰期后两者活度相等（长期平衡）——天然铀系、钍系中各子体活度都相同。

A parent decays to a radioactive daughter (Bateman equations). When the parent lives much longer, after several daughter half-lives their activities become equal (secular equilibrium) — as for every member of the natural U and Th series.

符号说明Symbols

符号Symbol	含义 Meaning	单位 Unit
\(\lambda_1, \lambda_2\)	母体、子体衰变常数parent and daughter decay constants	s⁻¹
\(A_i = \lambda_iN_i\)	活度activities	Bq

推导Derivation

\(dN_1/dt = -\lambda_1N_1\)，\(dN_2/dt = \lambda_1N_1 - \lambda_2N_2\)（与连续反应 A→B→C 相同）。
Same equations as consecutive first-order reactions A → B → C.
代入 \(N_1 = N_1(0)e^{-\lambda_1t}\)，用积分因子 \(e^{\lambda_2t}\) 解线性方程即得。
Solve the linear equation with integrating factor \(e^{\lambda_2t}\).
λ₁ ≪ λ₂、t ≫ 1/λ₂ 时 \(e^{-\lambda_2t} \to 0\)，\(\lambda_2N_2 \approx \lambda_1N_1\)。
For λ₁ ≪ λ₂ and long times, \(\lambda_2N_2 \approx \lambda_1N_1\).

适用条件与局限Assumptions & limitations

若 λ₁ > λ₂（子体更长寿），不出现平衡，母体先耗尽。
If the daughter lives longer, no equilibrium forms.
开放体系（如氡气逸出）会破坏平衡。
Open systems (e.g. radon escape) break equilibrium.

例题Worked example

⁹⁹Mo（66 h）→ ⁹⁹ᵐTc（6.0 h）：医用锝发生器处于暂时平衡，每天“挤奶”一次取 Tc。

⁹⁹Mo (66 h) → ⁹⁹ᵐTc (6.0 h): medical generators run in transient equilibrium and are ‘milked’ daily.