📚 E.3 Radioactive Decay | 放射性衰变
Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable one, accompanied by the emission of particles and/or electromagnetic radiation. This process is fundamental to nuclear physics and has far-reaching applications in medicine, energy generation, and archaeological dating. In this article we explore the nature of radioactivity, the different types of emissions, the mathematical laws governing decay, and the techniques used for detection. Higher Level (HL) extensions are included, such as the derivation of the exponential decay law from a differential equation and the use of logarithms to solve problems involving half-life.
放射性衰变是不稳定的原子核自发转变为更稳定核的过程，同时伴随着粒子和/或电磁辐射的发射。这一过程是核物理的基础，在医学、能源生产和考古年代测定等领域有着广泛的应用。本文将探讨放射性的本质、不同类型的辐射、支配衰变的数学定律，以及检测技术。文中包含了高水平 (HL) 扩展内容，例如从微分方程推导指数衰变定律，以及运用对数解决与半衰期相关的问题。
1. What is Radioactivity? | 什么是放射性？
Radioactivity was discovered by Henri Becquerel in 1896 when he observed that uranium salts emitted rays capable of fogging photographic plates. Further work by Marie and Pierre Curie revealed that the emission originates from the nucleus of the atom and is independent of chemical state. An unstable nucleus has an excess of energy or an imbalance between the number of protons and neutrons. To achieve stability it releases energy in the form of particles or electromagnetic waves.
放射性由亨利·贝克勒尔于 1896 年发现，他观察到铀盐能发射出使照相底片感光的射线。玛丽·居里和皮埃尔·居里的进一步研究表明，这些辐射源自原子核，并且与化学状态无关。不稳定的原子核具有过高的能量或质子与中子数失衡。为了达到稳定，它通过释放粒子或电磁波的形式释放能量。
2. Types of Radioactive Emissions | 放射性发射的类型
There are three main types of radiation emitted during radioactive decay: alpha (α) particles, beta (β) particles, and gamma (γ) rays. Alpha particles consist of two protons and two neutrons — identical to a helium-4 nucleus. Beta-minus (β⁻) decay involves the emission of an electron and an antineutrino when a neutron converts into a proton. Beta-plus (β⁺) decay, also called positron emission, occurs when a proton changes into a neutron, releasing a positron and a neutrino. Gamma decay usually accompanies alpha or beta decay and involves the emission of high-energy photons as the nucleus moves from an excited state to a lower energy state.
放射性衰变中主要发射三种类型的辐射：α 粒子、β 粒子和 γ 射线。α 粒子由两个质子和两个中子组成，与氦-4 核完全相同。β⁻ 衰变是在中子转变为质子时发射一个电子和一个反中微子。β⁺ 衰变又称正电子发射，发生在一个质子转变为中子时，释放出一个正电子和一个中微子。γ 衰变通常伴随 α 或 β 衰变发生，涉及原子核从激发态跃迁到较低能态时发射高能光子。
3. Properties of α, β, and γ Radiation | α、β 和 γ 辐射的性质
Alpha particles are relatively heavy and carry a +2 charge. They have low penetrating power — a sheet of paper or a few centimetres of air can stop them — but they are highly ionising because they interact strongly with matter. Beta particles are much lighter, carry a charge of –1 (or +1 for β⁺), and have greater penetrating ability, requiring a few millimetres of aluminium to be absorbed. Gamma rays are uncharged electromagnetic waves of very high frequency, possessing the highest penetrating power; they can be reduced in intensity but not completely stopped by several centimetres of lead or thick concrete. Their ionising ability is much lower than that of alpha or beta radiation.
α 粒子相对较重，带 +2 电荷。它们的穿透能力很弱——一张纸或几厘米空气就能阻挡——但因为与物质相互作用强，电离能力很强。β 粒子轻得多，带 –1（β⁺ 为 +1）电荷，穿透能力较强，需要几毫米的铝才能吸收。γ 射线是不带电的甚高频电磁波，穿透能力最强；几厘米铅板或厚混凝土可以降低其强度但无法完全阻挡。其电离能力远低于 α 或 β 辐射。
4. Random Nature of Decay | 衰变的随机性
Radioactive decay is a truly random process on the level of individual nuclei. We cannot predict when a particular unstable nucleus will decay; we can only state the probability that it will decay in a given time interval. This randomness means that for a large number of nuclei, the number of decays per unit time follows a statistical pattern described by the decay constant, but for a single nucleus the exact moment of decay is indeterminate. This inherent randomness distinguishes radioactive decay from many classical deterministic processes.
放射性衰变在单个原子核层面是一个完全随机的过程。我们无法预测某个特定不稳定核何时会衰变；只能给出它在给定时间间隔内发生衰变的概率。这种随机性意味着，对于大量原子核，单位时间的衰变数目遵循由衰变常数描述的统计规律，但对单个原子核而言，衰变的确切时刻是不确定的。这种固有的随机性使放射性衰变有别于许多经典的确定性过程。
5. Decay Constant and Activity | 衰变常数和活度
The decay constant λ (lambda) is the probability per unit time that a given nucleus will decay. If a sample contains N undecayed nuclei, the activity A is defined as the number of decays per second: A = λN ecause the number of undecayed nuclei decreases over time, the activity also decreases exponentially. For a sample initially having N₀ nuclei, the activity at time t is given by A = A₀ e–λt, where A₀ = λN₀ is the initial activity.
衰变常数 λ（lambda）是单个原子核在单位时间内发生衰变的概率。若样品中含有 N 个未衰变的核，则活度 A 定义为每秒衰变数：A = λN。活度的单位是贝克勒尔 (Bq)，1 Bq = 每秒 1 次衰变。由于未衰变核的数目随时间减少，活度也呈指数下降。对于初始含有 N₀ 个核的样品，t 时刻的活度为 A = A₀ e–λt，其中 A₀ = λN₀ 为初始活度。
6. Exponential Decay Law | 指数衰变定律
The number N of undecayed nuclei in a sample obeys the exponential decay law: N = N₀ e–λt. This equation shows that the quantity of radioactive material decreases by the same factor over equal time intervals. If we take the natural logarithm of both sides, we obtain a linear relationship: ln N = ln N₀ – λt. Plotting ln N against t yields a straight line with a negative slope equal to –λ, a useful method for verifying the law and determining λ experimentally.
样品中未衰变核的数目 N 服从指数衰变定律：N = N₀ e–λt。该方程表明，放射性物质的量在相等的时间间隔内以相同的因子减少。如果对等式两边取自然对数，得到线性关系：ln N = ln N₀ – λt。以 ln N 对 t 作图会得到一条斜率为 –λ 的直线，这是验证该定律并通过实验确定 λ 的有效方法。
7. Half-life and Its Calculation | 半衰期及其计算
The half-life T½ the undecayed nuclei in a sample to decay. Substituting N = N₀/2 into the decay law gives N₀/2 = N₀ e–λT½, which simplifies to T½ = ln 2 / λ. Half-lives can range from fractions of a second to billions of years depending on the isotope. Because activity is proportional to N, the activity also halves after each half-life. Half-life is a convenient parameter for characterising radioactive materials and for calculating the amount remaining after a given time using the formula N = N₀ (½)t/T½.
半衰期 T½ 是样品中一半未衰变核发生衰变所需的时间。将 N = N₀/2 代入衰变定律得到 N₀/2 = N₀ e–λT½，化简后得 T½ = ln 2 / λ。半衰期因同位素而异，可从几分之一秒到数十亿年不等。由于活度与 N 成正比，活度同样每隔一个半衰期减半。半衰期是表征放射性物质的便捷参数，也用于通过公式 N = N₀ (½)t/T½ 计算经过给定时间后剩余的量。
8. (HL) Derivation of the Decay Law | (HL) 衰变定律的推导
The decay law can be derived from the fundamental assumption that the rate of decay is proportional to the number of undecayed nuclei present. Mathematically, this is expressed as dN/dt = –λN, where the negative sign indicates a decrease. Separating variables gives dN/N = –λ dt. Integrating both sides: ∫ dN/N = –λ ∫ dt, which yields ln N = –λt + C. At t = 0, N = N₀, so ln N₀ = C. Thus ln N – ln N₀ = –λt or ln(N/N₀) = –λt. Exponentiating both sides gives the familiar form N = N₀ e–λt. This derivation clarifies the origin of the exponential behaviour and is a key HL learning point.
衰变定律可从“衰变速率与现存未衰变核数成正比”这一基本假设推导出来。数学上表示为 dN/dt = –λN，其中负号表示数目减少。分离变量得 dN/N = –λ dt。两边积分：∫ dN/N = –λ ∫ dt，得到 ln N = –λt + C。当 t = 0 时，N = N₀，因此 ln N₀ = C。于是 ln N – ln N₀ = –λt 或 ln(N/N₀) = –λt。取指数得熟悉的形式 N = N₀ e–λt。该推导阐明了指数行为的起源，是 HL 的关键学习内容。
9. Background Radiation | 背景辐射
Background radiation is the ionising radiation that is always present in our environment from natural and artificial sources. Natural sources include cosmic rays from space, terrestrial radionuclides in rocks and soil (such as uranium and thorium), and internal radionuclides like carbon‑14 found in living organisms. Artificial sources include medical X‑rays, nuclear waste, and fallout from past nuclear testing. When measuring the activity of a radioactive sample, the background count rate must be subtracted to obtain the true net count rate from the source alone.
背景辐射是环境中始终存在的电离辐射，来自天然和人工源。天然源包括宇宙射线、岩石和土壤中的陆地放射性核素（如铀和钍），以及生物体内的碳-14 等内源性核素。人工源包括医用 X 射线、核废料以及过去核试验的沉降物。在测量放射性样品的活度时，必须扣除背景计数率，才能得到仅来自放射源的净计数率。
10. Detection of Radiation | 辐射的探测
One of the most common detectors is the Geiger–Müller (GM) tube. It consists of a cylindrical metal cathode and a thin central anode wire, filled with a low‑pressure inert gas. When ionising radiation enters the tube, it creates ion pairs, which are accelerated by the high voltage and cause an electron avalanche, producing a short current pulse that is counted. Scintillation counters use a material that emits tiny flashes of light when struck by radiation; these flashes are amplified by a photomultiplier tube and counted. Other detectors include cloud chambers that make tracks visible and semiconductor detectors that offer high energy resolution.
最常见的探测器之一是盖革-米勒 (GM) 计数管。它由一个圆柱形金属阴极和一根细中心阳极丝组成，内部充有低压惰性气体。当电离辐射进入计数管，会产生离子对，在高电压下加速并引发电子雪崩，产生一个短暂的电流脉冲被计数。闪烁计数器使用一种在辐射照射下发出微小闪光的材料；这些闪光由光电倍增管放大并计数。其他探测器包括能显示径迹的云室和提供高能量分辨率的半导体探测器。
11. Applications of Radioisotopes | 放射性同位素的应用
Radioactive isotopes have numerous practical uses. In medicine, iodine‑131 is used to treat thyroid disorders, and technetium‑99m serves as a tracer in diagnostic imaging. In industry, gamma radiography helps inspect welds and metal castings, while beta emitters are used for thickness control of paper or plastic films. In archaeology and geology, carbon‑14 dating relies on the known half‑life of 14C (5730 years) to determine the age of organic materials up to about 50 000 years. Gamma radiation from cobalt‑60 is used to sterilise medical equipment and preserve food by killing bacteria.
放射性同位素有许多实际用途。在医学中，碘-131 用于治疗甲状腺疾病，锝-99m 用作诊断成像中的示踪剂。在工业中，γ 射线照相术有助于检查焊缝和金属铸件，而 β 放射源用于纸张或塑料薄膜的厚度控制。在考古学和地质学中，碳-14 测年利用已知的 14C 半衰期（5730 年）来确定年龄最高约 5 万年的有机物质年代。来自钴-60 的 γ 辐射用于对医疗设备进行消毒，并通过杀灭细菌来保存食品。
12. Safety and Summary | 安全与总结
Handling radioactive sources requires careful safety measures: minimising exposure time, maximising distance, and using appropriate shielding. Alpha sources are safe outside the body but extremely dangerous if ingested; beta sources require light shielding; gamma sources demand heavy shielding. The fundamental principles of radioactive decay — its random nature, exponential decrease, and characteristic half‑life — are central to nuclear physics. Understanding these concepts, including the HL derivation, allows us to predict the behaviour of radioactive materials and to harness their properties responsibly in science and society.
处理放射源需要严格的安全措施：尽量缩短暴露时间、增大距离和使用适当的屏蔽。α 放射源在体外是安全的，但若摄入则极其危险；β 放射源需要轻屏蔽；γ 放射源则需要重屏蔽。放射性衰变的基本原理——其随机性、指数衰减和特征的半衰期——是核物理的核心。理解这些概念，包括 HL 推导，使我们能够预测放射性物质的行为，并在科学和社会中负责任地利用它们的特性。
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