Radioactive Decay and Half-Life: The Caesium-137 Case Study | 放射性衰变与半衰期：铯-137案例解析

📚 Radioactive Decay and Half-Life: The Caesium-137 Case Study | 放射性衰变与半衰期：铯-137案例解析

Radioactive decay is a random, spontaneous process in which an unstable atomic nucleus emits ionising radiation to move towards a more stable arrangement. In the Edexcel A-Level Combined Science and Physics specifications, caesium-137 is a classic case study because it undergoes beta-minus decay followed by gamma emission, and its half-life of about 30 years makes it significant in medicine, industry and environmental monitoring. This article explains the nuclear processes, the mathematics of half-life, graphical methods and common exam techniques.

放射性衰变是一个随机、自发的过程，不稳定的原子核通过发射电离辐射变得更加稳定。在爱德思 A-Level 综合科学与物理考试中，铯-137是一个经典案例，因为它先发生β⁻衰变，再释放γ射线，约30年的半衰期使它在医学、工业和环境监测中具有重要意义。本文将讲解相关核过程、半衰期的数学计算、图像方法以及常见考试技巧。

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1. Nuclear Stability and the Origin of Radioactivity | 核稳定性与放射性的起源

The stability of an atomic nucleus depends on the balance between the strong nuclear force, which attracts nucleons, and the electrostatic repulsion between protons. A key indicator of stability is the neutron-to-proton ratio, often written as n/p. For light nuclei, this ratio is close to 1:1, but for heavier nuclei, a higher ratio is required because more neutrons are needed to dilute the repulsive forces between positively charged protons. Caesium-137 has 55 protons and 82 neutrons, giving an n/p ratio of about 1.49. This neutron-rich condition makes the nucleus unstable.

原子核的稳定性取决于核子之间的强核力与质子之间静电排斥力的平衡。稳定性的一个重要指标是中子-质子比，通常写作 n/p。对于轻核，这一比例接近1:1；但对于重核，需要更高的中子比例来削弱质子间的正电排斥。铯-137有55个质子和82个中子，n/p 约为1.49。这种中子过多的状态使原子核不稳定。

Radioactive decay is both spontaneous and random. Spontaneous means the decay is not affected by external conditions such as temperature, pressure or chemical bonding. Random means it is impossible to predict exactly which nucleus will decay at a given moment, although the overall behaviour of a large sample follows a precise statistical law. Edexcel exam questions often ask students to distinguish between these two terms.

放射性衰变既是自发的，也是随机的。自发意味着衰变不受温度、压力或化学键等外部条件的影响。随机意味着无法准确预测哪一个原子核会在某一时刻发生衰变，但大量原子核组成的样品整体行为遵循精确的统计规律。爱德思考试常要求学生区分这两个术语。

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2. Beta-Minus Decay Mechanism | β⁻ 衰变机制

Beta-minus decay occurs in neutron-rich nuclei. Inside the nucleus, a neutron is converted into a proton through the weak nuclear interaction. This increases the atomic number by 1 while the mass number remains unchanged. The process also produces a high-speed electron, called a beta particle, and an electron antineutrino. The antineutrino carries away some energy and momentum, which explains why beta particles are emitted with a continuous spectrum of kinetic energies.

β⁻衰变发生在中子过多的原子核中。在原子核内部，一个中子通过弱核相互作用转变为一个质子。这使原子序数增加1，而质量数保持不变。同时产生一个高速电子，称为β粒子，以及一个反电子中微子。反中微子带走部分能量和动量，这解释了为什么β粒子发射时具有连续动能谱。

n → p + e⁻ + ν̅ₑ

¹³⁷₅₅Cs → ¹³⁷₅₆Ba + β⁻ + ν̅ₑ

In this nuclear equation, the total mass number is conserved: 137 = 137 + 0. The total charge is also conserved: 55 = 56 − 1. This is a useful check in exam questions. The decay converts caesium-137 into barium-137, but the barium nucleus may be left in an excited state, which is discussed in the next section.

在这个核方程中，总质量数守恒：137 = 137 + 0。总电荷也守恒：55 = 56 − 1。这是考试中有效的检查方法。该衰变将铯-137转变为钡-137，但钡原子核可能处于激发态，下一节将对此进行讨论。

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3. Gamma Emission and Metastable Barium-137 | γ 射线发射与亚稳态钡-137

When caesium-137 undergoes beta-minus decay, the daughter nucleus is often formed in an excited, metastable state written as Ba-137m. Metastable means the nucleus has excess energy but survives for a short time before releasing it. Barium-137m drops to its stable ground state by emitting a gamma photon with an energy of about 0.662 MeV. Gamma emission does not change the mass number or atomic number because the photon is a massless, uncharged form of electromagnetic radiation.

当铯-137发生β⁻衰变时，子核通常形成在激发态，即亚稳态，写作 Ba-137m。亚稳态意味着原子核具有过剩能量，但会在短时间内释放。钡-137m 通过发射能量约为 0.662 MeV 的γ光子回到稳定基态。γ发射不改变质量数和原子序数，因为光子是无质量、不带电的电磁辐射。

¹³⁷₅₅Cs → ¹³⁷ᵐ₅₆Ba + β⁻ + ν̅ₑ

¹³⁷ᵐ₅₆Ba → ¹³⁷₅₆Ba + γ

The presence of the gamma ray is highly significant for practical applications and safety. Beta particles from a sealed caesium-137 source may be stopped by relatively thin layers of material, but the gamma ray is highly penetrating. Therefore caesium-137 is used as a gamma source in industrial radiography, thickness gauges and medical radiotherapy equipment.

γ射线的存在对于实际应用和安全至关重要。密封铯-137源发出的β粒子可以被相对薄的材料阻挡，但γ射线具有很强的穿透力。因此铯-137被用作工业射线照相、厚度计和医疗放射治疗设备中的γ源。

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4. The Concept of Half-Life | 半衰期的概念

Half-life, often represented as t½, is the time taken for the number of undecayed nuclei in a sample to decrease by half. It is equivalent to the time taken for the activity of the sample to fall to half its initial value. For caesium-137, the half-life is about 30.17 years. This means a sample that starts with 1.00 × 10⁶ undecayed caesium-137 nuclei will contain about 5.0 × 10⁵ undecayed nuclei after 30.17 years, and about 2.5 × 10⁵ after another 30.17 years.

半衰期通常用 t½ 表示，是样品中未衰变原子核数量减少一半所需的时间。它也等于样品的放射性活度降至初始值一半所需的时间。铯-137的半衰期约为30.17年。这意味着一个初始含有 1.00 × 10⁶ 个未衰变铯-137原子核的样品，在30.17年后约有 5.0 × 10⁵ 个未衰变原子核，再经过30.17年后约为 2.5 × 10⁵ 个。

Half-life is a property of the particular isotope and is not affected by temperature, pressure or chemical state. This constant probability of decay per unit time is represented by the decay constant λ. A long half-life indicates a small decay constant and low activity per unit mass; a short half-life indicates a large decay constant and high activity per unit mass. Caesium-137 has a relatively long half-life, so it remains a radiological hazard for many decades if released into the environment.

半衰期是特定同位素的性质，不受温度、压力或化学状态影响。单位时间内衰变的恒定概率用衰变常数 λ 表示。半衰期长表示衰变常数小，单位质量活度低；半衰期短表示衰变常数大，单位质量活度高。铯-137的半衰期相对较长，因此如果释放到环境中，会在数十年内持续构成放射性危害。

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5. Mathematical Treatment of Radioactive Decay | 放射性衰变的数学处理

The activity A of a radioactive source is directly proportional to the number of undecayed nuclei N. This relationship is expressed using the decay constant λ:

放射源的活度 A 与未衰变原子核数 N 成正比。这一关系用衰变常数 λ 表示：

A = λN

The number of undecayed nuclei and the activity both decrease exponentially with time t. The exponential decay law can be written as:

未衰变原子核数量和活度都随时间 t 按指数规律减小。指数衰变定律可写作：

N = N₀e^(−λt) and A = A₀e^(−λt)

The relationship between half-life and decay constant is derived by setting N = N₀/2 and t = t½. Taking natural logarithms gives the essential equation needed in Edexcel calculations:

半衰期与衰变常数之间的关系通过令 N = N₀/2 和 t = t½ 推导。取自然对数后得到爱德思考试所需的公式：

t½ = ln 2 / λ = 0.693 / λ

For caesium-137, the decay constant in seconds can be calculated from its half-life of 30.17 years. First convert years to seconds: 30.17 × 365 × 24 × 3600 ≈ 9.52 × 10⁸ s. Then λ = 0.693 / 9.52 × 10⁸ s ≈ 7.28 × 10⁻¹⁰ s⁻¹. This very small decay constant indicates that any given caesium-137 nucleus has a low probability of decaying in one second.

对于铯-137，可以先将其半衰期30.17年换算为秒：30.17 × 365 × 24 × 3600 ≈ 9.52 × 10⁸ s。然后 λ = 0.693 / 9.52 × 10⁸ s ≈ 7.28 × 10⁻¹⁰ s⁻¹。这个非常小的衰变常数表明，任意一个铯-137原子核在一秒内衰变的概率很低。

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6. Activity and the Becquerel | 放射性活度与贝克勒尔

Activity is measured in becquerels, symbol Bq, where 1 Bq equals one decay per second. In exam calculations, students may be given activity in kBq, MBq or GBq and must convert to appropriate powers of ten. For example, 5.0 MBq is equal to 5.0 × 10

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