Radioactive Decay Exam Essentials | 放射性衰变考点精讲

📚 Radioactive Decay Exam Essentials | 放射性衰变考点精讲

Radioactive decay is a random and spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. Understanding the types of decay, the mathematical laws governing activity, and real‑world applications such as carbon dating lies at the heart of IB and OCR A‑Level Physics. This revision guide walks you through every critical concept with paired bilingual explanations, worked examples, and the precise notation examiners expect.

放射性衰变是一种随机且自发的过程，不稳定的原子核通过发射辐射来释放能量。掌握衰变类型、支配活度的数学规律以及碳定年等实际应用，是 IB 和 OCR A‑Level 物理的核心。本考点精讲以中英双语对译的方式，梳理每一个关键概念，并提供例题和考官要求的准确符号。

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1. The Nature of Radioactive Decay | 放射性衰变的本质

Radioactive decay occurs when an unstable nucleus transforms into a more stable configuration. The process is spontaneous – it cannot be triggered by changes in temperature, pressure, or chemical environment – and it is random – we cannot predict which individual nucleus will decay at a given moment, only the probability per unit time.

当不稳定的原子核转变成更加稳定的结构时，就发生放射性衰变。该过程是自发的 —— 温度、压力或化学环境的变化无法触发它 —— 同时也是随机的 —— 我们无法预测某个特定核在某一时刻是否会衰变，只能知道单位时间内的衰变概率。

The emission of radiation carries away excess energy. The daughter nucleus may itself be radioactive, initiating a decay chain that continues until a stable isotope is reached. In all cases, charge, mass number, and energy–momentum are conserved in the reaction.

辐射的发射带走了多余的能量。子核本身也可能具有放射性，从而引发衰变链，直到抵达稳定同位素为止。在所有的反应中，电荷数、质量数以及能量–动量均守恒。

For exam credit, always state that the decay constant λ is the probability of decay per nucleus per second, and that the source of energy is the mass difference between parent and products according to E = mc².

考试拿分的关键在于：明确指出衰变常数 λ 是每个核每秒的衰变概率，并且衰变能量来源于母核与产物之间的质量差，经 E = mc² 转换而来。

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2. Types of Radiation: Alpha, Beta, Gamma | 辐射类型：α、β、γ

Three distinct forms of radiation are emitted from naturally occurring radioactive sources. Their properties differ markedly in terms of ionising power, penetrating ability, and behaviour in electric and magnetic fields.

天然放射源会发射出三种截然不同的辐射。它们的电离能力、穿透本领以及在电场和磁场中的偏转行为都存在显著差异。

Property | 性质	α (alpha)	β⁻ (beta-minus)	γ (gamma)
Nature | 本质	Helium nucleus, ⁴₂He	Electron, ⁰₋₁e	Photon (EM wave)
Charge | 电荷	+2e	–1e	0
Ionising ability | 电离本领	Strong	Moderate	Weak
Penetration | 穿透力	Stopped by paper or a few cm of air	Stopped by ~3 mm Al	Reduced by several cm of Pb, never fully stopped
Deflection in E/B fields | 电磁场偏转	Slight, opposite to β⁻	Large, opposite to α	None

Because α particles have the largest mass and charge, they produce the densest ionisation trails but are easily absorbed. β⁻ particles are lighter and faster, so they ionise less intensely but can penetrate further. γ photons interact only weakly with matter, requiring dense shielding for attenuation.

由于 α 粒子的质量和电荷最大，它们能产生最密集的电离径迹，但也最容易被吸收。β⁻ 粒子更轻、更快，因此电离强度较弱但穿透能力更强。γ 光子几乎不与物质相互作用，需要高密度屏蔽材料才能衰减。

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3. Alpha Decay | α衰变

In alpha decay, a heavy parent nucleus emits a ⁴₂He nucleus. The general equation is:

ᴬZX → ᴬ⁻⁴Z₋₂Y + ⁴₂He

在 α 衰变中，较重的母核发射一个 ⁴₂He 核。通式为：

ᴬZX → ᴬ⁻⁴Z₋₂Y + ⁴₂He

For example, uranium‑238 decays to thorium‑234:

²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He

例如，铀‑238 衰变成钍‑234：

²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He

Notice that mass number decreases by 4 and atomic number decreases by 2. The α particles are emitted with discrete kinetic energies, typically a few MeV, because the energy released is shared between the α particle and the recoiling daughter nucleus in a fixed ratio governed by conservation of momentum.

请注意，质量数减少了 4，原子序数减少了 2。α 粒子以分立的动能（典型值为几个 MeV）发射，因为释放的能量在 α 粒子和反冲子核之间按动量守恒规定的固定比例分配。

Alpha spectra are therefore line spectra, not continuous, providing evidence for discrete nuclear energy levels.

因此，α 粒子能谱是线状谱而非连续谱，这为原子核具有分立能级提供了证据。

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4. Beta-minus and Beta-plus Decay | β⁻与β⁺衰变

Beta decay involves the weak interaction and converts a neutron into a proton (β⁻) or a proton into a neutron (β⁺). In β⁻ decay, an electron and an anti‑electron‑neutrino are emitted:

n → p + e⁻ + ν̄ₑ

β 衰变涉及弱相互作用，将中子转变为质子（β⁻）或将质子转变为中子（β⁺）。在 β⁻ 衰变中，发射一个电子和一个反电子中微子：

n → p + e⁻ + ν̄ₑ

At the nuclide level: ¹⁴₆C → ¹⁴₇N + ⁰₋₁e + ν̄ₑ

¹⁴₆C → ¹⁴₇N + ⁰₋₁e + ν̄ₑ

在核素层面：碳‑14 变为氮‑14：

¹⁴₆C → ¹⁴₇N + ⁰₋₁e + ν̄ₑ

In β⁺ decay, a positron and an electron neutrino are released:

p → n + e⁺ + νₑ

在 β⁺ 衰变中，释放一个正电子和一个电子中微子：

p → n + e⁺ + νₑ

Example: ¹¹₆C → ¹¹₅B + ⁰₊₁e + νₑ

¹¹₆C → ¹¹₅B + ⁰₊₁e + νₑ

例如：碳‑11 衰变成硼‑11：

¹¹₆C → ¹¹₅B + ⁰₊₁e + νₑ

The key exam point is that β particles are emitted with a continuous spectrum of kinetic energies up to a maximum endpoint. The missing energy is carried away by the virtually undetectable neutrino, a hypothesis that confirmed the conservation of energy in nuclear processes.

考试的重点是，β 粒子以连续的动能谱发射，能量上限为某一端点值。被带走的“缺失能量”由几乎无法探测的中微子携带，这一假设证实了核反应中能量守恒的普遍性。

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5. Gamma Emission | γ辐射

Gamma photons are electromagnetic waves of very short wavelength (typically < 10⁻¹² m). They are usually emitted after an α or β decay has left the daughter nucleus in an excited state. The nucleus drops to a lower energy level, releasing the energy difference as a photon: E = hƒ = hc/λ.

γ 光子是波长极短（通常小于 10⁻¹² m）的电磁波。它们通常是在 α 或 β 衰变将子核留在激发态后发射的。子核跃迁到较低能级，将能量差以光子的形式释放：E = hƒ = hc/λ。

Since the nuclear energy levels are discrete, gamma spectra are line spectra, and the photon energies are characteristic of the particular nuclide. Gamma emission does not change the mass number or atomic number; it simply de‑excites the nucleus.

由于核的能级是分立的，γ 能谱是线状谱，光子能量是特定核素的特征。γ 辐射不会改变质量数或原子序数；仅仅使原子核退激。

An example: metastable technetium‑99m decays to technetium‑99 by emitting a 140 keV gamma photon, widely used in medical imaging.

一个实例：亚稳态的锝‑99m 通过发射一个 140 keV 的 γ 光子衰变到锝‑99，广泛用于医学成像。

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6. Decay Constant and Activity | 衰变常数与活度

The activity A of a radioactive sample is the number of nuclei that decay per unit time. It is proportional to the number of undecayed nuclei N present:

A = λN

放射性样品的活度 A 是单位时间内衰变的核的数目。它与样品中尚未衰变的核的数目 N 成正比：

A = λN

The constant of proportionality λ is the decay constant, measured in s⁻¹. It represents the probability of a given nucleus decaying in one second. Activity is measured in becquerels (Bq), where 1 Bq = 1 decay per second.

比例常量 λ 称为衰变常数，单位是 s⁻¹。它表示一个给定的核在一秒内发生衰变的概率。活度的单位是贝克勒尔（Bq），1 Bq = 每秒 1 次衰变。

Because the number of parent nuclei decreases with time, activity also decreases. The definition of λ ensures that over a short time Δt, the change in N is given by ΔN = –λN Δt, which leads to the exponential decay law.

由于母核数目随时间减少，活度也随时间减小。λ 的定义确保了在很短的时间 Δt 内，N 的变化量满足 ΔN = –λN Δt，由此可导出指数衰变规律。

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7. Exponential Decay Law | 指数衰变规律

Solving the differential equation dN/dt = –λN yields the exponential decay equation:

N = N₀ e⁽⁻λᵗ⁾

求解微分方程 dN/dt = –λN 得到指数衰变方程：

N = N₀ e⁽⁻λᵗ⁾

Here N₀ is the original number of radioactive nuclei at t = 0. Since activity is proportional to N, the same equation applies to activity:

A = A₀ e⁽⁻λᵗ⁾

此处 N₀ 是 t = 0 时放射性核的初始数目。因为活度与 N 成正比，同样的方程适用于活度：

A = A₀ e⁽⁻λᵗ⁾

When plotted on a log‑linear scale, ln A against t gives a straight line of slope –λ, allowing λ to be determined experimentally.

在单对数坐标中（ln A 对 t 作图），可以得到一条斜率为 –λ 的直线，从而通过实验测定 λ。

In IB and OCR exam questions, you may be given the decay constant and asked to find the fraction of nuclei remaining after a certain time, or vice‑versa. Always use natural logarithms and be careful with units of time.

在 IB 和 OCR 考试题中，可能会给出衰变常数，要求计算经过一定时间后剩余的核的比例，或反过来求解时间。务必使用自然对数，并注意时间的单位。

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8. Half-life | 半衰期

The half‑life T½ is the time taken for half the radioactive nuclei in a sample to decay, or equivalently for the activity to fall to half its initial value. Substituting N = N₀/2 into the decay equation gives:

T½ = ln 2 / λ ≈ 0.693 / λ

半衰期 T½ 是样品中一半放射性核发生衰变所需的时间，等价于活度降低到初始值一半所用的时间。将 N = N₀/2 代入衰变方程可得：

T½ = ln 2 / λ ≈ 0.693 / λ

Half‑life is a characteristic property of a given isotope and is independent of the initial quantity. For example, the half‑life of carbon‑14 is approximately 5730 years, while that of uranium‑238 is 4.47 × 10⁹ years.

半衰期是给定同位素的特征属性，与初始数量无关。例如，碳‑14 的半衰期约为 5730 年，而铀‑238 的半衰期为 4.47 × 10⁹ 年。

In problem solving, if multiple half‑lives have elapsed, the fraction remaining is (½)ⁿ, where n = t / T½. This shortcut is useful for quick checks, but the full exponential equation must be used for non‑integer multiples.

在解题时，若经过了若干个半衰期，剩余比例为 (½)ⁿ，其中 n = t / T½。这种捷径用于快速验算很有效，但对于非整数倍的半衰期，必须使用完整的指数方程。

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9. Carbon-14 Dating | 碳‑14定年法

Radiocarbon dating relies on the continual production of ¹⁴C in the upper atmosphere and its subsequent uptake by living organisms. The reaction is ¹⁴N + n → ¹⁴C + p, and the ¹⁴C then decays via β⁻ with T½ = 5730 years.

放射性碳定年依赖于高层大气中 ¹⁴C 的持续生成及其被活体有机体的吸收。生成反应为 ¹⁴N + n → ¹⁴C + p，随后 ¹⁴C 通过 β⁻ 衰变，半衰期为 5730 年。

While a plant or animal is alive, it exchanges carbon with the environment and maintains an equilibrium level of ¹⁴C. When it dies, exchange stops and the ¹⁴C activity in the remains begins to decay exponentially. By measuring the current activity and comparing it with the activity of a living sample, the age can be calculated:

t = (T½ / ln 2) × ln (A₀ / A)

当动植物存活时，它们与环境进行碳交换，维持 ¹⁴C 的平衡水平。死亡后，交换停止，遗骸中的 ¹⁴C 活度开始按指数规律衰减。通过测量当前的活度并与活体样品的活度比较，即可计算年代：

t = (T½ / ln 2) × ln (A₀ / A)

It is assumed that the atmospheric ¹⁴C/¹²C ratio has remained constant, an assumption that requires calibration with tree‑ring data. The technique is reliable for ages up to about 50 000 years (roughly 8–9 half‑lives).

该方法假设大气中 ¹⁴C/¹²C 的比例保持恒定，这一假设需通过树木年轮数据进行校准。该技术对于大约 50 000 年以内的年代（相当于 8–9 个半衰期）是可靠的。

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10. Applications and Safety | 应用与安全

Radioactive sources are used in medicine (radiation therapy, tracers), industry (thickness gauges, crack detection), and research. Ionising radiation can kill living cells, so its ability to destroy cancerous tumours is exploited in radiotherapy, while carefully chosen γ emitters are used as tracers because they can be detected outside the body.

放射源用于医学（放射治疗、示踪剂）、工业（厚度计、裂纹检测）和研究。电离辐射能杀死活细胞，因此其破坏癌变肿瘤的能力被用于放射治疗；而精心选择的 γ 发射体可用作示踪剂，因为它们可以被体外探测器记录。

Safety measures follow the ALARA principle (As Low As Reasonably Achievable) and involve three main tactics: time – minimise exposure duration; distance – maximise separation from the source (intensity ∝ 1/r²); and shielding – use appropriate materials such as lead for γ, thick plastic for β. Never handle sources with bare hands; always use tongs and store sources in lead‑lined containers.

安全措施遵循 ALARA 原则（合理可行尽量低），并包含三个主要策略：时间—— 尽量缩短暴露时间；距离—— 尽量增大与源的距离（强度与 1/r² 成正比）；屏蔽—— 使用合适材料，如铅屏蔽 γ、厚塑料屏蔽 β。切勿徒手接触放射源；始终使用长柄钳，并在铅衬容器中储存放射源。

Candidates should be ready to compare the suitability of different isotopes for specific tasks based on half‑life and type of radiation. For example, a medical tracer requires a half‑life long enough to allow imaging but short enough to limit patient dose, and it must emit γ radiation that can be detected efficiently.

考生应能基于半衰期和辐射类型，比较不同同位素对特定任务的适用性。例如，医用示踪剂需要足够长的半衰期以完成成像，但又足够短以限制患者剂量，并且必须发射能够被高效探测的 γ 辐射。

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