📚 The Mathematics of Radioactive Decay | 放射性衰变的数学
Radioactive decay is a fundamental concept in physics, but its mathematical description is elegantly simple and forms a key topic in CIE A-Level Mathematics. By modelling decay as a continuous process driven by a first-order differential equation, we unlock powerful tools for predicting the number of undecayed nuclei, calculating half-life, and analysing exponential decay graphs. This article unpacks the mathematics behind radioactive decay, from the differential equation through to worked examples and common pitfalls, providing a complete revision resource tailored to the CIE syllabus.
放射性衰变是物理学中的一个基本概念，但它的数学描述却简洁优美，是 CIE A-Level 数学的重要课题。通过将衰变过程建模为一个由一阶微分方程驱动的连续过程，我们能够利用强大的数学工具预测未衰变原子核的数量、计算半衰期并分析指数衰减图像。本文深入剖析放射性衰变背后的数学原理，从微分方程建立到典型例题和常见错误，为 CIE 考试提供一份完整的复习资源。
1. The Nature of Radioactive Decay | 放射性衰变的本质
Radioactive decay is a random process at the level of individual nuclei; we cannot predict exactly when a particular nucleus will decay. However, for a large collection of identical unstable nuclei, the overall decay behaviour follows a precise, predictable pattern. The rate at which nuclei decay is proportional to the number of nuclei present at that instant. This is the fundamental assumption that underpins the mathematical model.
对于单个原子核而言，放射性衰变是一个随机过程，我们无法精确预测某一个原子核何时会发生衰变。然而，对于大量相同的、不稳定的原子核来说，整体的衰变行为遵循一个精确且可预测的规律。原子核的衰变速率与当时存在的未衰变原子核数量成正比。这是支撑整个数学模型的基本假设。
In the context of CIE Mathematics, this proportionality leads directly to a differential equation. Understanding this link between physical behaviour and mathematical structure is essential for tackling related exam questions on differential equations and exponential models.
在 CIE 数学的框架下，这种正比关系直接引出一个微分方程。理解物理行为与数学结构之间的这种联系，对于处理有关微分方程和指数模型的考试题目至关重要。
2. Modelling with a Differential Equation | 用微分方程建模
Let N represent the number of undecayed radioactive nuclei at time t. The activity, or rate of decay, is the number of decays per unit time, given by dN/dt. Since N decreases as time progresses, dN/dt is negative. The proportionality to N is expressed as:
设 N 表示在时间 t 时未衰变的放射性原子核数量。活度，即衰变率，是单位时间内的衰变次数，用 dN/dt 表示。由于 N 随着时间推移而减少，dN/dt 为负值。与 N 的正比关系表达为：
dN/dt = -λN
Here, λ (lambda) is a positive constant called the decay constant. It is characteristic of the particular radioactive isotope and has units of time-1, such as s-1 or yr-1. The negative sign ensures that dN/dt is negative, reflecting the decrease in N.
这里，λ（拉姆达）是一个正的常数，称为衰变常数。它是特定放射性同位素的特征常数，单位是时间-1，例如秒-1或年-1。方程中的负号保证了 dN/dt 为负值，体现出 N 在不断减少。
This differential equation is separable and of a type frequently examined in CIE Pure Mathematics 3. Its form tells us that the rate of change of N is directly proportional to N itself, a hallmark of exponential decay.
该微分方程是可分离变量的，属于 CIE Pure Mathematics 3 中经常考查的类型。方程的形式告诉我们，N 的变化率与 N 本身成正比，这是指数衰减的标志。
3. Solving the Differential Equation | 解微分方程
To solve dN/dt = -λN, we separate the variables N and t:
为了求解 dN/dt = -λN，我们分离变量 N 和 t：
(1/N) dN = -λ dt
Integrating both sides gives:
对两边积分得到：
∫ (1/N) dN = ∫ -λ dt
ln|N| = -λt + C
Since N > 0, we can drop the absolute value. Exponentiating both sides yields the general solution:
因为 N > 0，我们可以去掉绝对值符号。对两边取指数，得到通解：
N = e-λt + C = eC e-λt
Let the initial number of nuclei at t = 0 be N0. Substituting t = 0 gives N0 = eC. Thus, the particular solution is the familiar exponential decay law:
设 t = 0 时的初始原子核数为 N0。代入 t = 0 得到 N0 = eC。因此，特解就是我们所熟悉的指数衰减定律：
N = N0 e-λt
This equation is central to any discussion of radioactive decay. It shows that the number of undecayed nuclei decreases exponentially over time.
这个方程是所有放射性衰变讨论的核心。它表明未衰变原子核的数量随时间呈指数级减少。
4. The Decay Constant λ | 衰变常数 λ
The decay constant λ determines the probability per unit time that a given nucleus will decay. A larger λ means a more rapid decay process. In CIE exam questions, λ might be given directly, or you may be asked to calculate it from the half-life or from a set of data. Being comfortable manipulating λ in equations is vital.
衰变常数 λ 决定了单位时间内某个原子核发生衰变的概率。λ 越大，表示衰变过程越快。在 CIE 考试题中，λ 可能直接给出，也可能要求你根据半衰期或一组数据将它计算出来。熟练地在方程中处理 λ 至关重要。
From the solution N = N0 e-λt, after one time constant (t = 1/λ), the number of nuclei falls to N = N0 e-1 ≈ 0.368 N0. This gives a tangible sense of what λ represents: it is the reciprocal of the time taken for the population to reduce to about 37% of its initial value.
从解 N = N0 e-λt 可知，经过一个时间常数（t = 1/λ）后，原子核数量下降为 N = N0 e-1 ≈ 0.368 N0。这让我们能够直观地理解 λ 的含义：它是原子核数量下降到初始值约 37% 所需时间的倒数。
The units of λ always harbour a trap: if time is measured in seconds, λ is in s-1; if in years, yr-1. Always check unit consistency, especially when using formulas involving half-life or mean lifetime.
λ 的单位常常暗藏陷阱：如果时间以秒为单位，λ 的单位就是 s-1；如果以年为单位，则为 yr-1。尤其在涉及半衰期或平均寿命的公式中，务必检查单位的一致性。
5. Half-Life and its Calculation | 半衰期及其计算
The half-life, denoted T½, is the time required for half of the radioactive nuclei initially present to decay, i.e. for N to fall to N0/2. Substituting into the decay equation:
半衰期，记作 T½，是指最初存在的放射性原子核有一半发生衰变所需的时间，即 N 下降到 N0/2。代入衰变方程：
N0/2 = N0 e-λT½
Dividing both sides by N0 and taking natural logarithms yields:
两边同除以 N0 并取自然对数：
ln(1/2) = -λT½
-ln 2 = -λT½
T½ = ln 2 / λ
This elegant formula shows that half-life is inversely proportional to the decay constant. A short half-life corresponds to a large λ and a highly unstable isotope. It is independent of the initial quantity N0, which is a hallmark of first-order processes.
这个简练的公式表明，半衰期与衰变常数成反比。半衰期短意味着 λ 大，同位素高度不稳定。半衰期与初始数量 N0 无关，这是一级过程的一个标志性特征。
In CIE questions, you may be given T½ also need to find the fraction remaining after a certain number of half-lives using N = N0 (1/2)n, where n = t / T½. Both approaches should be at your fingertips.
在 CIE 考题中，可能会给出 T½ 让你求 λ，反之亦然。你也可能需要根据经过的半衰期个数，利用 N = N0 (1/2)n（其中 n = t / T½）来求剩余的份额。这两种方法都应该熟练掌握。
6. Mean Lifetime | 平均寿命
resents the average time a nucleus exists before decaying. For a process governed by N = N0 e-λt, the mean lifetime is the reciprocal of the decay constant:
另一个有用的物理量是原子核的平均寿命 τ（塔乌）。它表示原子核在衰变前存在的时间的平均值。对于由 N = N0 e-λt 支配的过程，平均寿命是衰变常数的倒数：
τ = 1 / λ
Deriving this requires integration of t multiplied by the probability density function, but at CIE level you are usually expected to know and apply the result. The relationship T½ = τ ln 2 links the half-life and mean lifetime neatly.
推导这个结果需要对 t 乘以概率密度函数进行积分，但在 CIE 层次，你通常只要求知道并应用这个结果。关系式 T½ = τ ln 2 将半衰期与平均寿命简洁地联系了起来。
Mean lifetime is particularly useful when comparing the longevity of different isotopes. A longer τ indicates a more slowly decaying species.
在比较不同同位素的寿命时，平均寿命特别有用。τ 越长，表明该核素衰变得越慢。
7. Activity and the Decay Rate | 活度与衰变率
Activity A is defined as the magnitude of the rate of decay: A = |dN/dt| = λN. Since N follows exponential decay, activity also decays exponentially:
活度 A 定义为衰变率的大小：A = |dN/dt| = λN。由于 N 遵循指数衰减，活度也呈指数衰减：
A = λN0 e-λt = A0 e-λt
where A0 = λN0 is the initial activity. The SI unit of activity is the becquerel (Bq), defined as one decay per second. In exam problems, you may encounter data given as counts per minute or counts per second; always convert to consistent units.
其中 A0 = λN0 是初始活度。活度的国际单位是贝克勒尔（Bq），定义为每秒一次衰变。在考试题中，你可能会遇到以“每分钟计数”或“每秒计数”形式给出的数据；请始终转换为一致的单位。
Measuring activity is often more practical than counting undecayed nuclei. The exponential decay of activity provides an alternative route to determine λ and T½ from experimental data.
测量活度往往比计数未衰变原子核更为实际。活度的指数衰减为通过实验数据确定 λ 和 T½ 提供了另一条途径。
8. Exponential Decay Graph | 指数衰变图像
The graph of N against t (or A against t) is a decreasing exponential curve. It never reaches zero, but asymptotically approaches the t-axis. The curve is steeper at the start when N is large, because the rate of decay is proportional to N.
N 关于 t（或 A 关于 t）的图像是一条单调递减的指数曲线。它永远不会到达零，而是渐近地接近 t 轴。曲线在起始阶段（N 较大时）更陡，因为衰变速率与 N 成正比。
Key features to identify and label in exams: the initial value N0, the half-life on the time axis (where N = N0/2), and the smooth, ever-decreasing shape. Drawing tangents at various points can visually demonstrate that the gradient is proportional to the y-coordinate.
在考试中需要识别和标记的关键特征包括：初始值 N0、时间轴上对应半衰期的点（该处 N = N0/2），以及光滑且持续递减的形状。在曲线上不同点作切线，可以直观地展示斜率与 y 坐标成正比。
Understanding the geometric meaning of λ: λ is equal to the magnitude of the relative gradient, -(dN/dt)/N. This is a useful check when estimating λ from a given graph.
理解 λ 的几何意义：λ 等于相对梯度的绝对值，即 -(dN/dt)/N。这在从给定图像估算 λ 时是一个很有用的检验工具。
9. Logarithmic Transformation | 对数变换
Taking natural logarithms of the decay equation converts the exponential relationship into a linear one:
对衰变方程取自然对数，可以将指数关系转化为线性关系：
ln N = ln N0 - λt
This is of the form y = mx + c, with y = ln N, gradient m = -λ, and intercept c = ln N0. Plotting ln N against t yields a straight line, which is a powerful data analysis tool. If the plotted points lie on a straight line, the decay is confirmed to be exponential, and λ can be found from the gradient.
这个式子具有 y = mx + c 的形式，其中 y = ln N，斜率 m = -λ，截距 c = ln N0。画出 ln N 关于 t 的图像会得到一条直线，这是一种强大的数据分析工具。如果绘制出的数据点落在一条直线上，就能证实衰变是指数型的，并且可以由斜率求出 λ。
The same transformation applies to activity: ln A = ln A0 - λt. Many CIE questions supply a table of count rate versus time and ask you to process the data using logarithms to determine λ or T½. Mastering this technique is essential.
同样的变换也适用于活度：ln A = ln A0 - λt。许多 CIE 考题会提供一个计数率随时间变化的数据表，要求你使用对数处理数据以确定 λ 或 T½。掌握这一技巧至关重要。
10. Worked Example | 典型例题
The following example illustrates the typical steps expected in a CIE exam. A sample of a radioactive isotope has an initial activity of 800 Bq. After 30 minutes, the activity has dropped to 200 Bq. Find the decay constant λ and the half-life T½.
以下例题展示了 CIE 考试中期望的典型解题步骤。某放射性同位素样品的初始活度为 800 Bq。30 分钟后，活度降为 200 Bq。求衰变常数 λ 和半衰期 T½。
Using A = A0 e-λt and substituting the given values:
使用 A = A0 e-λt 并代入给定数值：
200 = 800 e-λ(30)
Divide by 800: 0.25 = e-30λ. Take natural logs:
除以 800：0.25 = e-30λ。取自然对数：
ln(0.25) = -30λ
-1.3863 = -30λ ⇒ λ = 0.0462 min-1
Then T½ p; 15.0 minutes. This can be verified by noting that the activity has halved twice in 30 minutes (800 → 400 → 200), confirming a half-life of 15 min.
然后 T½ = ln 2 / λ ≈ 0.6931 / 0.0462 ≈ 15.0 分钟。可以通过注意到活度在 30 分钟内减半了两次（800 → 400 → 200）来验证，这证实半衰期确实是 15 分钟。
11. Common Mistakes | 常见错误
sign when integrating. Always check that your final expression for N decreases with time.
许多学生因可以避免的错误而失分。一个常见错误是未能正确处理 dN/dt = -λN 中的负号，导致积分时符号出错。务必检查你最终得到的 N 表达式随时间递减。
Another pitfall is confusing half-life with mean lifetime. Half-life is the time for N to halve, whereas mean lifetime is 1/λ. Using T½ = 1/λ is a serious error; the correct relation is T½ = ln 2 / λ.
另一个陷阱是将半衰期与平均寿命混淆。半衰期是 N 减半所需的时间，而平均寿命是 1/λ。错误地使用 T½ = 1/λ 是一个严重错误；正确的关系是 T½ = ln 2 / λ。
Unit inconsistencies also cause problems. If time is given in minutes but λ is required in s-1, a conversion is necessary. Additionally, when using the logarithmic method, ensure you use natural logs (ln) consistently, not log10, unless the question specifically instructs otherwise.
单位不一致也会引发问题。如果时间以分钟给出，但所求的 λ 需要以 s-1 为单位，就必须进行换算。此外，在使用对数方法时，确保始终使用自然对数（ln），而不是 log10，除非题目有特别说明。
12. Summary and Key Points | 总结与要点
oactive decay revolves around the differential equation dN/dt = -λN and its solution N = N0 e-λt. Key quantities include the decay constant λ, half-life T½ ivity A follows the same exponential law. Logarithmic transformation provides a linear method for data analysis. Mastery of these concepts, combined with careful unit handling and sign conventions, will equip you to tackle any CIE A-Level question on radioactive decay with confidence.
放射性衰变的数学核心是微分方程 dN/dt = -λN 及其解 N = N0 e-λt。关键物理量包括衰变常数 λ、半衰期 T½ = ln 2 / λ 和平均寿命 τ = 1/λ。活度 A 遵循同样的指数规律。对数变换为数据分析提供了一种线性化方法。扎实掌握这些概念，并注意单位处理和符号约定，将使你有信心应对任何 CIE A-Level 中关于放射性衰变的考题。
Always remember that this topic bridges pure mathematics and applied contexts. Practising past paper questions, particularly those requiring you to switch between exponential and logarithmic forms, is the best way to secure high marks.
请始终记住，这一课题联结了纯数学与应用背景。通过练习历年真题，尤其是那些要求你在指数形式和对数形式之间进行转换的题目，是确保获得高分的上佳途径。
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