The Mathematics of Radioactive Decay | 放射性衰变的数学

📚 The Mathematics of Radioactive Decay | 放射性衰变的数学

Radioactive decay provides one of the cleanest applications of exponential functions and differential equations in A-level Mathematics. The same mathematics is used in physics, archaeology, medicine and nuclear engineering, so it is an excellent cross-topic revision area.

放射性衰变是 A-level 数学中指数函数与微分方程最清晰的应用之一。同样的数学在物理、考古、医学和核工程中都有使用，因此它是一个很好的跨主题复习领域。

1. The Decay Model and Assumptions | 衰变模型与假设

Radioactive decay is modelled as a continuous process in which the rate of decay is proportional to the number of undecayed nuclei present.

放射性衰变被建模为一个连续过程，其中衰变速率与当前未衰变的原子核数成正比。

This assumption works well when the number of nuclei N is very large, so that random fluctuations become negligible.

当原子核数 N 非常大时，这个假设成立，随机涨落可以忽略不计。

We also assume that every nucleus has the same fixed probability of decaying per unit time, independent of external conditions.

我们还假设每个原子核在单位时间内都有相同的固定衰变概率，且与外部条件无关。

2. The Differential Equation dN/dt = -λN | 微分方程 dN/dt = -λN

Let N(t) be the number of undecayed nuclei at time t. The activity, or number of decays per unit time, is given by the rate of decrease of N.

设 N(t) 为 t 时刻未衰变原子核的数量。活度，即单位时间内的衰变次数，由 N 的减少速率给出。

The fundamental law states that this rate is proportional to N itself, which gives the first-order differential equation

基本定律指出该速率与 N 本身成正比，因此得一阶微分方程

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