📚 Radioactive Decay and Half-Life | 放射性衰变与半衰期
Radioactive decay is a fundamental process by which unstable atomic nuclei lose energy by emitting radiation. This natural, spontaneous transformation is central to understanding nuclear physics, from carbon dating to medical isotopes. In this article, we explore the random nature of decay, derive the exponential decay law, and examine key concepts such as the decay constant, activity, and half-life. Mastering these ideas is essential for Edexcel A-level Physics and provides a gateway to interpreting nuclear phenomena accurately.
放射性衰变是不稳定原子核通过释放辐射而损失能量的基本过程。这种自发的、自然的转变对于理解从碳定年到医学同位素的核物理学至关重要。本文将探讨衰变的随机性，推导指数衰变定律，并深入分析衰变常数、活度和半衰期等关键概念。掌握这些思想对 Edexcel A-level 物理至关重要，也为准确解释核现象打开了大门。
1. The Random Nature of Decay | 衰变的随机性
Radioactive decay is a completely random process at the level of individual nuclei. It is impossible to predict exactly when a particular nucleus will disintegrate. Instead, we can only state the probability that a given nucleus will decay in a unit time interval. This inherent randomness is similar to throwing dice or flipping coins, but it does not prevent us from making very precise predictions when large numbers of nuclei are involved.
放射性衰变在单个原子核的层面上是完全随机的。我们无法准确预言某个特定原子核何时会分裂，只能指出它在单位时间间隔内发生衰变的概率。这种内在的随机性类似于掷骰子或抛硬币，但当涉及大量原子核时，我们依然可以做出非常精确的预测。
The random nature is also evidenced by the fact that the decay rate is unaffected by external conditions such as temperature, pressure, or chemical bonding. This distinguishes nuclear decay from chemical reactions and confirms that it originates from the forces within the nucleus itself.
衰变不受温度、压力或化学键等外部条件的影响，这也证实了其随机性。这一点将核衰变与化学反应区分开来，并肯定了它源于原子核内部的力。
2. Decay Constant and Activity | 衰变常数与活度
the timescale. A large λ means the nucleus is highly unstable and decays quickly; a small λ indicates a long-lived isotope. The activity A of a sample is defined as the number of disintegrations per second and is measured in becquerels (Bq), where 1 Bq = 1 decay per second.
单位时间内某个特定原子核衰变的概率称为衰变常数，符号为 λ。它的单位可以是 s⁻¹、min⁻¹ 或 yr⁻¹。λ 大意味着原子核极不稳定，衰变迅速；λ 小则表明同位素寿命长。样品的活度 A 定义为每秒钟发生的衰变次数，单位是贝克勒尔 (Bq)，1 Bq = 1 次衰变/秒。
s over time, activity also falls exponentially. This relationship forms the foundation for deriving the exponential decay law.
从数学上看，活度直接正比于样品中当时尚未衰变的原子核数 N：A = λN。由于 N 随时间减少，活度也呈指数下降。这一关系是推导指数衰变定律的基础。
3. Exponential Decay Law | 指数衰变定律
The number of undecayed nuclei N remaining in a sample after a time t is given by the exponential equation:
经过时间 t 后，样品中尚未衰变的原子核数 N 由指数方程给出：
N = N₀ e−λt
is the base of natural logarithms. This law arises because the rate of decay dN/dt is proportional to N itself, leading to a differential equation whose solution is the exponential function.
式中 N₀ 是 t = 0 时的初始原子核数，λ 是衰变常数，e 是自然对数的底。该定律的根源在于衰变速率 dN/dt 与 N 本身成正比，从而得到一个微分方程，其解即为指数函数。
to the mass of a radioactive isotope and to its activity. Therefore, we can write A = A₀ e−λt or m = m₀ e−λt, making it a very versatile expression in calculations.
同样的指数形式也适用于放射性同位素的质量和活度，因此可以写作 A = A₀ e−λt 或 m = m₀ e−λt，这在计算中是一个非常通用的表达式。
4. Derivation of the Decay Equation | 衰变方程的推导
Starting from the proportionality between activity and the number of nuclei, we can write the rate of change of N as:
从活度与原子核数的比例关系出发，我们可以写出 N 的变化率：
dN/dt = −λN
The negative sign indicates that N decreases with time. Separating variables and integrating gives:
负号表示 N 随时间减少。分离变量并积分可得：
∫ (1/N) dN = −λ ∫ dt → ln N = −λt + C
(N / N₀) = −λt. Taking the exponential of both sides gives the familiar decay law N = N₀ e−λt. This derivation is frequently examined at A-level and demonstrates how calculus links probability with measurable quantities.
当 t = 0 时 N = N₀，所以 C = ln N₀。代入后得到 ln(N / N₀) = −λt。两边取指数即得熟知的衰变定律 N = N₀ e−λt。这一推导过程在 A-level 考试中常见，它展示了微积分如何将概率与可测量量联系起来。
5. Understanding Half-Life | 理解半衰期
e is a characteristic property of each isotope and ranges from fractions of a second to billions of years, making it extremely useful for dating and identification.
放射性同位素的半衰期 T₁/₂ 是指原有原子核数衰变一半所需的时间。换句话说，当 t = T₁/₂ 时，N = N₀ / 2。半衰期是每种同位素的特征性质，其范围从几分之一秒到数十亿年，这使得它在年代测定和同位素识别中极其有用。
Because the activity also halves in one half-life, the half-life can be determined directly from an activity–time graph. For a pure isotope, the time for the count rate to drop to half its initial value (after background correction) equals T₁/₂. This simple relationship is often tested using graphical data.
由于活度在一个半衰期内也会减半，因此可直接从活度-时间图上确定半衰期。对于纯同位素，计数率降至初始值一半所需的时间（经背景修正后）即为 T₁/₂。这一简单关系经常通过图线数据来考查。
6. Relationship between Half-Life and Decay Constant | 半衰期与衰变常数的关系
By substituting N = N₀ / 2 and t = T₁/₂ into N = N₀ e−λt, we obtain:
将 N = N₀ / 2 和 t = T₁/₂ 代入 N = N₀ e−λt，得到：
1/2 = e−λ T₁/₂ → ln(1/2) = −λ T₁/₂
Since ln(1/2) = −ln 2, this simplifies to the widely used formula:
因为 ln(1/2) = −ln 2，上式简化为广泛使用的公式：
T₁/₂ = ln 2 / λ ≈ 0.693 / λ
This inverse relationship means that isotopes with large decay constants have short half-lives, and vice versa. It also provides a straightforward method to calculate λ from a measured half-life, or to predict how long a sample must be stored before its activity falls to a safe level.
这一反比关系表明，衰变常数大的同位素半衰期短，反之亦然。它也提供了一种直接的方法：从测得的半衰期计算 λ，或预测样品需要存放多久其活度才能降至安全水平。
7. Graphical Analysis of Decay | 衰变的图形分析
Plotting N, A, or mass against time yields an exponential decay curve. By taking natural logarithms, the exponential equation can be linearised:
将 N、A 或质量对时间作图会得到一条指数衰变曲线。通过取自然对数，指数方程可以线性化：
ln N = ln N₀ − λt
Thus, a graph of ln N (or ln A) against t gives a straight line with gradient −λ and y-intercept ln N₀. This is extremely useful for determining λ and T₁/₂ from experimental data, particularly when the half-life is not obvious from the raw decay curve.
因此，ln N（或 ln A）对 t 的关系图为一条直线，其斜率为 −λ，y 截距为 ln N₀。这非常有利于通过实验数据确定 λ 和 T₁/₂，特别是当原始衰变曲线上无法明显看出半衰期时。
In practice, students are frequently asked to use such graphs to find the decay constant or to verify the exponential nature. Remember that any intercept can yield the original activity or number of nuclei, and the half-life can be read from the graph using the time interval for N to halve directly.
实践中，学生常被要求利用这类图线求出衰变常数或验证指数规律。需记住，截距可给出原始活度或原子核数，而半衰期可直接从图上读取 N 减半所用的时间间隔来获得。
8. Applications: Carbon Dating | 应用：碳定年法
One of the most famous uses of radioactive decay is radiocarbon dating (carbon-14 dating). Living organisms constantly exchange carbon with the atmosphere, maintaining a tiny proportion of the radioactive isotope ¹⁴C. When the organism dies, this exchange stops and the ¹⁴C nuclei begin to decay with a half-life of approximately 5730 years.
放射性衰变最著名的应用之一便是放射性碳定年法（碳-14 定年）。活着的生物体不断与大气进行碳交换，从而维持极小比例的放射性同位素 ¹⁴C。当生物体死亡后，这种交换停止，¹⁴C 原子核开始衰变，其半衰期约为 5730 年。
By measuring the remaining activity or the number of ¹⁴C atoms in an archaeological sample and comparing it with the assumed initial concentration, the age of the sample can be calculated using the exponential decay law. Climate science, archaeology, and geology all rely on this principle, though careful calibration is needed to account for variations in ancient atmospheric ¹⁴C.
通过测量考古样品中残留的活度或 ¹⁴C 原子数，并与假定的初始浓度比较，便可利用指数衰变定律计算出样品的年代。气候科学、考古学和地质学都依赖这一原理，尽管需要仔细校正以计及古代大气中 ¹⁴C 浓度的变化。
9. Background Radiation and Correction | 背景辐射与修正
In any experiment to measure the decay of a radioactive source, the detected count rate includes contributions from background radiation — cosmic rays, naturally occurring radon, and terrestrial sources. To obtain a reliable activity value, the background count rate must be measured separately and subtracted from the total observed count rate.
在任何测量放射源衰变的实验中，探测器给出的计数率都包含背景辐射的贡献——宇宙射线、天然存在的氡气及地球上的辐射源。为获得可靠的活度值，必须单独测量背景计数率，并将其从总观测计数率中扣除。
Background correction is especially important when the source is weak or its activity has fallen to a level comparable to background. For half-life determinations, failure to correct for background can lead to an overestimation of half-life and a non-exponential graph at later times.
当放射源较弱或其活度已经衰减至与背景相当的水平时，背景修正确实非常重要。对于半衰期的测定，若未进行背景修正，可能导致高估半衰期，并使后期图线不再呈现指数形状。
10. Summary and Exam Tips | 总结与考试技巧
ities are the decay constant λ, activity A = λN, and half-life T₁/₂ = ln 2 / λ. The equations N = N₀ e−λt and its logarithmic form are fundamental to solving almost any decay problem.
放射性衰变将原子核尺度上的随机性与通过指数定律表现出来的高度可预测的宏观行为结合在一起。关键量包括衰变常数 λ、活度 A = λN 和半衰期 T₁/₂ = ln 2 / λ。方程 N = N₀ e−λt 及其对数形式是解决几乎所有衰变问题的基础。
In the exam, be ready to switch between the exponential and logarithmic forms, interpret straight-line graphs of ln(count rate) against time, and always perform background subtraction where necessary. Show derivations step by step and remember that the law applies equally to activity and mass. Understanding these principles will serve you exceptionally well in nuclear and particle physics topics.
在考试中，要准备好指数形式与对数形式的灵活转换，解释 ln(计数率) 随时间变化的直线图，并在必要时进行背景扣除。推导步骤要完整，并记住该定律同样适用于活度和质量。理解这些原理将使你在处理核与粒子物理题目时游刃有余。
Published by TutorHao | Physics Revision Series | aleveler.com
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