Mathematical Modelling of Radioactive Decay | 放射性衰变的数学建模

📚 Mathematical Modelling of Radioactive Decay | 放射性衰变的数学建模

Radioactive decay is a natural and spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. This process is inherently random at the level of individual atoms, yet it follows a precise mathematical pattern when observed across a large population of nuclei. Understanding this mathematical framework is essential for CIE A-Level Physics students, as it connects probability, calculus, and experimental measurement in a single elegant model.

放射性衰变是不稳定的原子核通过发射辐射而损失能量的自然自发过程。在单个原子层面，这一过程本质上是随机的，但当观察大量原子核时，它遵循精确的数学规律。理解这一数学框架对CIE A-Level物理学生至关重要，因为它将概率、微积分和实验测量统一在一个优雅的模型中。

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1. The Nature of Unstable Nuclei | 不稳定原子核的性质

An atom consists of protons and neutrons bound together in the nucleus. When the balance between protons and neutrons is unfavorable, or when the nucleus is too large, the nucleus becomes unstable. Such nuclei undergo radioactive decay to reach a more stable configuration. The three most common types of decay are alpha (α), beta (β), and gamma (γ) emission, each involving different changes to the nucleus.

原子由质子和中子组成，它们结合在原子核中。当质子和中子之间的比例失衡，或原子核过大时，原子核变得不稳定。这样的原子核发生放射性衰变以达到更稳定的构型。最常见的三种衰变类型是α衰变、β衰变和γ衰变，每种衰变都会引起原子核的不同变化。

For A-Level purposes, what matters most is not the detailed nuclear transformations, but the mathematical law that governs the rate at which these transformations occur. This law applies universally to all radioactive substances, regardless of the type of decay involved.

就A-Level考试而言，最重要的不是核转变的细节，而是支配这些转变速率的数学规律。这个规律普遍适用于所有放射性物质，与衰变类型无关。

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2. The Decay Constant λ | 衰变常数 λ

The decay constant, denoted by the Greek letter lambda (λ), represents the probability per unit time that a given nucleus will decay. Its unit is s⁻¹ (per second). Crucially, λ is a constant for a particular isotope under fixed conditions — it does not change with temperature, pressure, or chemical state.

衰变常数用希腊字母λ表示，它代表给定原子核在单位时间内发生衰变的概率。其单位是s⁻¹（每秒）。关键在于，λ在固定条件下对特定同位素是一个常数——它不随温度、压力或化学状态而变化。

If we have a sample containing N radioactive nuclei, the number of decays per second, denoted by ΔN/Δt, is directly proportional to N. This gives us the fundamental relationship: activity A = λN.

如果我们有一个含有N个放射性原子核的样品，每秒的衰变数用ΔN/Δt表示，它与N成正比。由此得出基本关系：活度 A = λN。

The activity A is measured in becquerels (Bq), where 1 Bq equals one decay per second. The minus sign in the differential equation indicates that N decreases over time.

活度A以贝克勒尔（Bq）为单位，1 Bq等于每秒一次衰变。微分方程中的负号表示N随时间减少。

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3. Deriving the Exponential Decay Law | 推导指数衰变定律

The rate of decay is proportional to the number of undecayed nuclei present. Mathematically, we write this as dN/dt = −λN. This is a first-order differential equation, and it can be solved by separating the variables.

衰变速率与尚未衰变的原子核数目成正比。数学上写作 dN/dt = −λN。这是一阶微分方程，可以通过分离变量法求解。

dN/N = −λ dt

Integrating both sides, with limits from N₀ (initial number) to N (number at time t), gives:

对两边积分，从N₀（初始数目）到N（t时刻的数目）：

ln(N/N₀) = −λt

Exponentiating both sides yields the famous exponential decay law:

两边取指数得到著名的指数衰变定律：

N = N₀ e⁻λᵗ

This equation tells us that the number of undecayed nuclei decreases exponentially with time. The same relationship applies to activity, since A = λN, giving A = A₀ e⁻λᵗ.

该方程告诉我们，未衰变原子核的数量随时间呈指数减少。同样的关系适用于活度，因为A = λN，所以A = A₀ e⁻λᵗ。

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4. Half-Life and Its Relationship to λ | 半衰期及其与λ的关系

The half-life (T₁/₂) is defined as the time taken for half of the radioactive nuclei in a sample to decay. After one half-life, N = N₀/2. Substituting this into the decay equation:

半衰期（T₁/₂）定义为样品中一半放射性原子核发生衰变所需的时间。经过一个半衰期后，N = N₀/2。将其代入衰变方程：

N₀/2 = N₀ e⁻λT₁/₂

Dividing through by N₀ and taking natural logarithms:

两边除以N₀并取自然对数：

T₁/₂ = ln2 / λ

Since ln2 ≈ 0.693, we obtain the practical formula T₁/₂ = 0.693/λ. This inverse relationship is a key result: a large decay constant corresponds to a short half-life, meaning a rapidly decaying substance.

由于ln2 ≈ 0.693，我们得到实用公式T₁/₂ = 0.693/λ。这种反比关系是一个关键结果：大的衰变常数对应短的半衰期，意味着该物质衰变速度快。

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5. Graphical Representation | 图像表示

The exponential decay law produces a characteristic curve on a graph of N against t. The curve starts at N₀ and decreases rapidly at first, then levels off gradually, asymptotically approaching zero. The curve never actually reaches zero mathematically, which aligns with the probabilistic nature of individual decays.

指数衰变定律在N-t图上产生一条特征曲线。曲线从N₀开始，初始下降迅速，然后逐渐平缓，渐近趋向零。曲线在数学上永远不会真正达到零，这与单个原子衰变的概率性质一致。

However, if we plot ln(N) against t, we obtain a straight line with gradient −λ and vertical intercept ln(N₀). Alternatively, plotting ln(A) against t gives a straight line with gradient −λ and intercept ln(A₀). This linearisation is extremely useful for analysing experimental data and determining the decay constant.

然而，如果绘制ln(N)对t的图像，我们得到一条斜率为−λ、纵截距为ln(N₀)的直线。同样，绘制ln(A)对t的图像得到斜率为−λ、截距为ln(A₀)的直线。这种线性化处理对分析实验数据、确定衰变常数极其有用。

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6. Finding Half-Life from a Decay Curve | 从衰变曲线求半衰期

To determine the half-life from a decay graph, one can read the time at which the activity or number of nuclei falls to half its initial value. From this point, continue to measure successive half-way points to verify consistency. Alternatively, the half-life can be calculated from the gradient of a logarithmic plot, using the relation T₁/₂ = ln2/λ.

要从衰变图确定半衰期，可以读取活度或原子核数量降至初始值一半的时间。从该点继续测量后续的半值点以验证一致性。或者，可以使用关系式T₁/₂ = ln2/λ从对数图的斜率计算半衰期。

• Graphical method: read values directly from the decay curve

  图解法：直接从衰变曲线上读取数值

• Logarithmic method: measure the gradient of the ln(N) vs t plot

  对数法：测量ln(N)对t图像上的斜率

A common examination question provides a graph and asks students to estimate the half-life. The key skill is accurate reading of values and understanding that the fractional decrease is always 50% per half-life interval.

常见考题提供一幅图，要求学生估算半衰期。关键技能是准确读取数值，并理解每经过一个半衰期区间，减少的比例始终是50%。

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7. Carbon Dating: A Real-World Application | 碳定年法：实际应用

Carbon-14 (¹⁴C) is a radioactive isotope of carbon produced continuously in the upper atmosphere through cosmic ray interactions. Living organisms absorb carbon-14 through photosynthesis or the food chain, maintaining a constant ratio of ¹⁴C to ¹²C while alive. When an organism dies, it stops absorbing carbon, and the ¹⁴C present begins to decay with a half-life of 5730 years.

碳-14（¹⁴C）是碳的放射性同位素，通过宇宙射线与上层大气的相互作用不断产生。生物体通过光合作用或食物链吸收碳-14，在活着时维持恒定的¹⁴C与¹²C比例。当生物体死亡后，它停止吸收碳，体内的¹⁴C以5730年的半衰期开始衰变。

By measuring the current activity of ¹⁴C in a sample and comparing it to the expected initial activity, archaeologists can calculate the age of organic materials. This technique is effective for specimens up to about 50000 years old, as the activity becomes too low to measure accurately beyond this point.

通过测量样品中¹⁴C的当前活度并与预期初始活度比较，考古学家可以计算有机材料的年代。该技术对约5万年以内的标本有效，因为超过此范围后活度变得过低而无法准确测量。

The dating equation used is t = (1/λ)·ln(A₀/A), derived directly from the exponential decay law applied to activity.

定年使用的方程是 t = (1/λ)·ln(A₀/A)，直接从应用于活度的指数衰变定律推导而来。

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8. Worked Example: Half-Life Calculation | 例题：半衰期计算

Example: A radioactive isotope has a decay constant of 2.31 × 10⁻² year⁻¹. Calculate its half-life.

例题：某放射性同位素的衰变常数为2.31 × 10⁻² 年⁻¹。计算其半衰期。

Solution: Using the formula T₁/₂ = ln2/λ, we substitute the given value:

解答：使用公式T₁/₂ = ln2/λ，代入给定数值：

T₁/₂ = 0.693 / (2.31 × 10⁻²) = 30.0 years

A second part of the question might ask how much of a 10 g sample remains after 90 years. Since 90 years equals three half-lives, the remaining mass is 10 g × (½)³ = 1.25 g.

问题的第二部分可能询问10 g样品经过90年后剩余多少。由于90年等于三个半衰期，剩余质量为10 g × (½)³ = 1.25 g。

This example illustrates two essential calculation types: finding the half-life from λ, and using integer multiples of half-life to calculate remaining quantities.

这个例题说明了两种基本计算类型：从λ求半衰期，以及利用半衰期的整数倍数计算剩余量。

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9. The Exponential Form: Alternative Expressions | 指数形式的其他表达式

While N = N₀ e⁻λᵗ is the most common form, alternative expressions are occasionally encountered. Writing λ = ln2/T₁/₂, the decay law becomes:

虽然N = N₀ e⁻λᵗ是最常见的形式，但偶尔会遇到其他表达式。将λ = ln2/T₁/₂代入，衰变定律变为：

N = N₀ (½)ᵗ/ᵀ₁/₂

This form is particularly intuitive: after t/T₁/₂ half-lives have elapsed, the factor (½) is applied that many times. Experts recommend being comfortable with both forms, as some questions may state one and ask for calculation using the other.

这种形式特别直观：经过t/T₁/₂个半衰期后，因子(½)被应用那么多次。建议熟练掌握两种形式，因为有些题目可能给出一种形式，要求使用另一种形式进行计算。

For example, if a sample has a half-life of 8 days, after 24 days the fraction remaining is (½)³ = ⅛. Expressing answers in fractions or decimals must be done correctly; both are acceptable in examinations.

例如，如果某样品半衰期为8天，经过24天后剩余比例为(½)³ = ⅛。用分数或小数表达答案都可以，考试中两种形式均被认可。

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10. Common Misconceptions and Exam Pitfalls | 常见误区与考试陷阱

A frequent misunderstanding is believing that radioactive decay can be affected by external conditions such as temperature or pressure. In reality, the decay constant is unaffected by these factors. Another common error involves confusing the terms activity and count rate, or misinterpreting the decay constant as the probability of decay over a full second rather than an instantaneous rate.

一个常见的误解是认为放射性衰变可以受温度或压力等外部条件的影响。实际上，衰变常数不受这些因素的影响。另一个常见错误涉及混淆活度和计数率这两个术语，或错误地将衰变常数理解为完整一秒内的衰变概率，而不是瞬时速率。

• Always use consistent units — convert time to seconds if λ is in s⁻¹

  始终保持单位一致——如果λ以s⁻¹为单位，需要将时间转换为秒

• Check whether a question asks for activity or number of undecayed nuclei

  检查题目问的是活度还是未衰变原子核的数目

• Distinguish between the decay constant λ and half-life T₁/₂ in calculations

  在计算中区分衰变常数λ和半衰期T₁/₂

Examiners frequently ask students to interpret the gradient of a logarithmic decay graph. A positive gradient value must be converted to a negative decay constant, and the half-life calculated accordingly.

考官经常让学生解读对数衰变图上的斜率。正的斜率值必须转换为负的衰变常数，并据此计算半衰期。

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11. Summary of Key Equations | 关键方程总结

The core relationships in radioactive decay mathematical modelling are listed in the table below. These equations form the foundation of nearly every examination question on this topic.

放射性衰变数学建模的核心关系如下表所示。这些方程构成了几乎所有相关考题的基础。

Relationship | 关系式	Formula | 公式
Definition of activity | 活度定义	A = −dN/dt = λN
Exponential decay | 指数衰变	N = N₀ e⁻λᵗ
Activity decay | 活度衰变	A = A₀ e⁻λᵗ
Half-life | 半衰期	T₁/₂ = ln2/λ
Alternative form | 替代形式	N = N₀ (½)ᵗ/ᵀ₁/₂
Dating equation | 定年方程	t = (1/λ)·ln(A₀/A)

Mastery of these equations, combined with accurate graphical interpretation, equips students to solve a wide range of CIE A-Level Physics problems on radioactive decay. Practice with past papers is strongly recommended to build confidence in applying these models under examination conditions.

熟练掌握这些方程，结合准确的图像解读能力，学生就能解决CIE A-Level物理中关于放射性衰变的各类问题。强烈建议通过练习历年真题，在考试条件下增强应用这些模型的信心。

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