核物理

统计规律 \[-\frac{\mathrm{d}N}{\mathrm{d}t} = \lambda N,\; N = N_0 \mathrm{e}^{-\lambda t}.\]

衰变常量 \(\lambda\)

一个核在单位时间内的衰变概率

半衰期

核因衰变减至半数所需时间 \[\frac{N_0}{2} = N_0 \mathrm{e}^{-\lambda T_{1/2}},\; T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}.\] \(t \rightarrow t +\mathrm{d}t\) 时段衰变核数 \[-\mathrm{d}N = \lambda N\mathrm{d}t = \lambda N_0 \mathrm{e}^{-\lambda t}\mathrm{d}t.\]

平均寿命

\[\tau = \frac{1}{N_0}\int\limits_0^{\infty}\lambda N_0 t \mathrm{e}^{-\lambda t}\mathrm{d}t = \frac{1}{\lambda} = \frac{T_{1/2}}{\ln 2}.\]

放射性活度 (衰变率)

样品单位时间内衰变的核数 \[R = \lambda N = R_0 \mathrm{e}^{-\lambda t}.\] 单位有 \(1\text{Bq} = 1\text{s}^{-1}, 1\text{Ci} = 3.7\times 10^{10} \text{Bq}.\)