含义Meaning
每个放射性核在单位时间内衰变的概率 λ 恒定，与温度、压强、化学状态几乎无关，所以未衰变核数随时间指数下降。
Each radioactive nucleus has a constant decay probability λ per unit time, nearly independent of temperature, pressure and chemistry, so the number remaining decays exponentially.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(N\) | 未衰变核数number of undecayed nuclei | 1 |
| \(\lambda\) | 衰变常数decay constant | s⁻¹ |
| \(\tau = 1/\lambda\) | 平均寿命mean lifetime | s |
推导Derivation
大量核中，dt 内衰变数正比于现有核数：\(dN = -\lambda N\,dt\)（一级动力学）。
In dt the number decaying is proportional to N: \(dN = -\lambda N\,dt\) (first order).
分离变量积分 \(\int dN/N = -\lambda\int dt\) 得 \(\ln(N/N_0) = -\lambda t\)。
Integrate: \(\ln(N/N_0) = -\lambda t\).
平均寿命 \(\tau = \int_0^\infty t\,\lambda e^{-\lambda t}dt = 1/\lambda\)。
Mean lifetime \(\tau = 1/\lambda\).
适用条件与局限Assumptions & limitations
统计规律，只对大量原子核成立；单个核何时衰变不可预测，计数服从泊松分布（相对误差 \(1/\sqrt N\)）。
A statistical law for large numbers; counts follow Poisson statistics (relative error \(1/\sqrt N\)).
电子俘获和内转换的速率会因化学环境改变而有极小变化（约 0.1%，如 ⁷Be）。
Electron-capture rates shift slightly (~0.1 %, e.g. ⁷Be) with chemical environment.