📚 Decay graphs and equations | 衰变图与方程
Radioactive decay is the spontaneous disintegration of unstable nuclei, and CIE A Level Physics expects you to read, sketch, and interpret decay graphs and equations. This article explains the random nature of decay, the exponential law, half-life, logarithmic graphs, background corrections, and carbon dating.
放射性衰变是不稳定原子核的自发解体，CIE A Level 物理要求你能够读取、绘制并解释衰变图与方程。本文讲解衰变的随机性、指数规律、半衰期、对数图、本底修正以及碳定年。
1. Radioactive decay as a random process | 放射性衰变作为随机过程
Decay is random and spontaneous. You cannot predict which particular nucleus will decay next or when it will decay. However, for a large number of identical nuclei, the overall decay rate is statistically predictable.
衰变是随机且自发的。你无法预测哪个特定原子核将在何时衰变。但对于大量同种原子核，整体衰变率在统计上是可以预测的。
External conditions such as temperature, pressure, and chemical bonding do not change the decay constant of a given nuclide. This is because decay is governed by the weak and strong nuclear interactions inside the nucleus.
温度、压强和化学键等外部条件不能改变给定核素的衰变常数。这是因为衰变由原子核内部的弱相互作用和强相互作用支配。
In an experiment, counts fluctuate around a smooth exponential trend. The fluctuations arise from the random timing of individual decays and are more obvious for small count rates.
在实验中，计数围绕平滑的指数趋势波动。波动源于单个衰变发生时刻的随机性，在计数率较小时更明显。
2. Decay constant and activity | 衰变常数与活度
The activity A of a radioactive source is the number of decays per unit time. It is measured in becquerels (Bq), where 1 Bq = 1 decay per second.
放射源的活度 A 是单位时间内发生的衰变次数。它以贝克勒尔（Bq）为单位，1 Bq = 每秒 1 次衰变。
The decay constant λ is the probability per unit time that a given nucleus will decay. Its SI unit is s⁻¹.
衰变常数 λ 是给定原子核在单位时间内发生衰变的概率。其 SI 单位是 s⁻¹。
A = λN
If a sample contains N undecayed nuclei, its activity is directly proportional to N. The greater the number of parent nuclei, the more decays occur per second.
如果样品含有 N 个未衰变原子核，其活度与 N 成正比。母核数目越多，每秒发生的衰变就越多。
- N is the number of undecayed parent nuclei.
- A is the activity, measured in Bq.
- λ is the decay constant, measured in s⁻¹.
其中 N 是未衰变母核数目，A 是活度，单位为 Bq，λ 是衰变常数，单位为 s⁻¹。
3. The exponential decay equation | 指数衰变方程
Since activity is proportional to N, the rate of change of N is proportional to N itself. This gives the differential equation dN/dt = −λN.
由于活度与 N 成正比，N 的变化率也与 N 本身成正比。这给出微分方程 dN/dt = −λN。
N = N₀e−λt
The solution is an exponential decay. N₀ is the initial number of undecayed nuclei at t = 0. The same form applies to activity and to mass of the parent nuclide.
其解为指数衰减。N₀ 是 t = 0 时未衰变原子核的初始数目。同样的形式适用于活度和母核素的质量。
A = A₀e−λt
At any time t, the fraction remaining is e−λt. When t is one half-life, this fraction equals 1/2.
在任意时刻 t，剩余的分数为 e−λt。当 t 等于一个半衰期时，该分数等于 1/2。
The exponential law assumes a large number of nuclei. For very small samples, statistical deviations from the smooth curve become significant.
指数规律假设原子核数目很大。对于非常小的样品，与平滑曲线的统计偏差会变得显著。
4. Half-life and its equation | 半衰期及其方程
The half-life T½ is the time taken for the number of undecayed parent nuclei to decrease to half of its initial value. The same time is also the time for the activity to halve.
半衰期 T½ 是未衰变母核数目减少到初始值一半所需的时间。同样也是活度减半所需的时间。
T½ = ln 2 ÷ λ = 0.693 ÷ λ
This equation is independent of the initial number N₀. A useful rearrangement is λ = ln 2 ÷ T½.
该方程与初始数目 N₀ 无关。一个有用的变形是 λ = ln 2 ÷ T½。
After n half-lives, the remaining fraction is (1/2)n. For example, after 3 half-lives, 1/8 of the original nuclei remain.
经过 n 个半衰期后，剩余分数为 (1/2)n。例如，经过 3 个半衰期后，原始核的 1/8 保留下来。
| Elapsed time | Fraction remaining | Activity |
|---|---|---|
| 0 | 1 | A₀ |
| T½ | 1/2 | A₀/2 |
| 2T½ | 1/4 | A₀/4 |
| 3T½ | 1/8 | A₀/8 |
| 4T½ | 1/16 | A₀/16 |
The table above summarises how the remaining fraction and activity change with whole-number multiples of the half-life. It provides a quick calculation shortcut in exam questions.
上表总结剩余分数和活度随半衰期倍数的变化，是考试中快速计算的依据。
5. Reading N-t and A-t graphs | 读 N-t 与 A-t 图
A graph of N against t is a decreasing exponential curve. It starts at N₀, approaches the time axis asymptotically, and
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