Radioactive Decay | 放射性衰变

📚 Radioactive Decay | 放射性衰变

Radioactive decay is the spontaneous and random disintegration of unstable atomic nuclei, resulting in the emission of particles and/or electromagnetic radiation. This process transforms the original parent nucleus into a more stable daughter nucleus, often after a series of decays. Understanding radioactive decay is crucial for grasping nuclear physics, half-life calculations, and applications ranging from medical imaging to radioactive dating.

放射性衰变是不稳定原子核自发且随机地发生蜕变，并发射粒子与／或电磁辐射的过程。这一过程会将母核转变为更稳定的子核，通常需经历一系列衰变。理解放射性衰变对于掌握核物理、半衰期计算以及从医学影像到放射性测年的各类应用至关重要。

1. Fundamentals of Radioactive Decay | 放射性衰变基础

Radioactive decay occurs because certain combinations of protons and neutrons are energetically unstable. An unstable nucleus will seek a lower energy state by emitting radiation. This emission is independent of physical conditions such as temperature and pressure, and the rate of decay is governed purely by the nuclear structure.

放射性衰变的发生是因为某些质子与中子的组合在能量上不稳定。不稳定的原子核会通过发射辐射来寻求更低的能态。这种发射与温度、压力等物理条件无关，衰变速率完全由核结构决定。

Stability depends on the neutron-to-proton (n/p) ratio. For light nuclei (Z ≤ 20), stable nuclides have an n/p ratio close to 1. For heavier nuclei, the stable ratio increases to around 1.5 because more neutrons are needed to counteract the electrostatic repulsion of protons via the strong nuclear force. Nuclei lying outside the ‘band of stability’ will undergo decay to approach stability.

稳定性取决于中子与质子数之比（n/p 比）。对于轻核（Z ≤ 20），稳定核素的 n/p 比接近 1。对于更重的核，稳定的 n/p 比会增至约 1.5，因为需要更多的中子通过强核力来抵消质子间的静电排斥。处于“稳定带”之外的原子核将发生衰变以趋向稳定。

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2. Types of Radiation Emitted | 发射的辐射类型

There are three main types of radiation emitted during radioactive decay: alpha (α) particles, beta (β) particles, and gamma (γ) rays. Each has distinct properties in terms of charge, mass, penetrating power, and ionising ability.

放射性衰变中发射的辐射主要有三种类型：α 粒子、β 粒子和 γ 射线。它们在电荷、质量、穿透力和电离能力方面各具特点。

Property | 属性	Alpha (α) | α 粒子	Beta (β) | β 粒子	Gamma (γ) | γ 射线
Nature | 本质	Helium-4 nucleus (²⁴₂He) | 氦-4 核 (²⁴₂He)	Electron (β⁻) or positron (β⁺) | 电子 (β⁻) 或正电子 (β⁺)	Electromagnetic wave | 电磁波
Charge | 电荷	+2e | +2e	−1e (β⁻) or +1e (β⁺) | −1e (β⁻) 或 +1e (β⁺)	0 | 0
Mass (u) | 质量	~4 | 约 4	~1/1840 | 约 1/1840	0 | 0
Penetration | 穿透力	Stopped by paper/skin | 被纸/皮肤阻挡	Stopped by ~3 mm Al | 被约 3 mm 铝阻挡	Reduced by several cm of lead | 被数厘米铅减弱
Ionising ability | 电离能力	Very high | 非常高	Moderate | 中等	Low | 低

Alpha particles have the highest ionising power and thus lose energy quickly, making them easily stopped. Beta particles are more penetrating but less ionising. Gamma rays are highly penetrating and require dense materials for shielding.

α 粒子具有最强的电离能力，因此能量损失快，容易被阻挡。β 粒子穿透力更强，但电离能力较弱。γ 射线穿透力极强，需要高密度材料进行屏蔽。

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3. Alpha Decay | α 衰变

Alpha decay occurs predominantly in heavy nuclei (Z > 82) where the strong nuclear force struggles to overcome the electrostatic repulsion of many protons. The nucleus emits an alpha particle, reducing its mass number by 4 and its atomic number by 2. A general alpha decay equation can be written as:

α 衰变主要发生在重核（Z > 82）中，此时强核力难以克服众多质子间的静电排斥。原子核发射一个 α 粒子，使其质量数减少 4，原子序数减少 2。一般的 α 衰变方程可写为：

ᴬᵣX → ᴬ⁻⁴ᵣ₋₂Y + ⁴²He

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

例如，铀-238 衰变为钍-234：

²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He

The discrete energy of the emitted alpha particles gives rise to a line spectrum, as the daughter nucleus and the alpha particle share the released energy in a fixed ratio determined by conservation of momentum.

发射的 α 粒子具有离散的能量，产生线状谱，因为子核和 α 粒子按动量守恒所确定的固定比例分配释放的能量。

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4. Beta Minus (β⁻) Decay | β⁻ 衰变

Beta minus decay occurs in neutron-rich nuclei. A neutron is transformed into a proton, an electron (β⁻ particle), and an antineutrino (ν̄ₑ). This increases the atomic number by 1 while the mass number remains unchanged. The general equation is:

β⁻ 衰变发生在中子过剩的核中。一个中子转变为一个质子、一个电子（β⁻ 粒子）和一个反中微子（ν̄ₑ）。这使得原子序数增加 1，而质量数保持不变。通用方程为：

n → p + e⁻ + ν̄ₑ

For a nucleus, the decay is represented as:

对于原子核，衰变表示为：

ᴬᵣX → ᴬᵣ₊₁Y + β⁻ + ν̄ₑ

An example is the decay of carbon-14:

一个例子是碳-14 的衰变：

¹⁴₆C → ¹⁴₇N + e⁻ + ν̄ₑ

The emitted beta particles have a continuous energy spectrum up to a maximum value because the decay energy is shared among three particles. The existence of the neutrino was proposed to account for the ‘missing’ energy and momentum.

发射的 β 粒子具有连续能谱，直至一最大值，因为衰变能在三个粒子间分配。中微子的提出正是为了解释“消失”的能量和动量。

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5. Beta Plus (β⁺) Decay and Electron Capture | β⁺ 衰变与电子俘获

In proton-rich nuclei, beta plus decay may occur. A proton converts into a neutron, a positron (β⁺), and a neutrino (νₑ). The atomic number decreases by 1, while the mass number stays the same.

在质子过剩的核中，可能发生 β⁺ 衰变。一个质子转变为一个中子、一个正电子（β⁺）和一个中微子（νₑ）。原子序数减少 1，质量数不变。

p → n + e⁺ + νₑ

An alternative for proton-rich nuclides is electron capture, where an inner orbital electron is captured by a proton to form a neutron and a neutrino:

质子过剩核素的另一种衰变方式是电子俘获，即内层轨道电子被质子俘获，形成一个中子和一个中微子：

p + e⁻ → n + νₑ

Both processes shift the n/p ratio towards stability. Positron emission also yields a continuous energy spectrum, and the subsequent annihilation of the positron with an electron produces two 511 keV gamma photons.

这两种过程都使 n/p 比趋向稳定。正电子发射同样产生连续能谱，随后正电子与电子湮灭会产生两个 511 keV 的 γ 光子。

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6. Gamma Decay | γ 衰变

Gamma decay accompanies other decay modes when the daughter nucleus is left in an excited state. No transmutation occurs — the atomic number and mass number remain unchanged. The excited nucleus simply releases the excess energy as a high-energy photon (gamma ray).

γ 衰变伴随其他衰变模式，当子核处于激发态时发生。不发生核转变——原子序数和质量数均保持不变。激发态的核直接以高能光子（γ 射线）的形式释放多余能量。

ᴺᵣX* → ᴺᵣX + γ

For instance, in the beta decay of cobalt-60, the excited nickel-60 releases gamma rays:

例如，在钴-60 的 β 衰变中，激发态的镍-60 释放 γ 射线：

⁶⁰₂₇Co → ⁶⁰₂₈Ni* + β⁻ + ν̄ₑ → ⁶⁰₂₈Ni + γ

Gamma emission provides a powerful tool for medical imaging (e.g., technetium-99m) and industrial radiography.

γ 射线的发射为医学影像（如锝-99m）和工业探伤提供了有力工具。

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7. Decay Law and the Decay Constant | 衰变定律与衰变常数

Radioactive decay is a random process, but for a large number of nuclei the average behaviour follows an exponential law. The decay constant λ (lambda) is the probability per unit time that a given nucleus will decay. Its unit is s⁻¹ (or min⁻¹, year⁻¹).

放射性衰变是随机过程，但对于大量核来说，平均行为遵循指数规律。衰变常数 λ 是单位时间内某个核发生衰变的概率，单位为 s⁻¹（或 min⁻¹、year⁻¹）。

If N is the number of undecayed nuclei present, the activity A (decays per second) is:

若 N 为当前尚未衰变的核数目，则活度 A（每秒衰变数）为：

A = λN

The fundamental differential equation governing decay is:

描述衰变的微分方程为：

dN/dt = −λN

Integration yields the exponential decay law:

积分后得到指数衰变律：

N(t) = N₀ e⁻ᴸᵗ

where N₀ is the number of nuclei at t = 0. The activity follows the same exponential form:

其中 N₀ 为 t = 0 时的核数目。活度遵循相同的指数形式：

A(t) = A₀ e⁻ᴸᵗ

The decay constant is unique to each radioisotope and links directly to the half-life.

每个放射性同位素都有其独特的衰变常数，并与半衰期直接关联。

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8. Half-Life and Its Significance | 半衰期及其意义

The half-life (T₁/₂) is the time taken for the number of undecayed nuclei (or the activity) to reduce to half of its initial value. From the exponential law, setting N = N₀/2 gives:

半衰期（T₁/₂）是未衰变核的数目（或活度）降至初始值一半所需的时间。由指数律，令 N = N₀/2 可得：

N₀/2 = N₀ e⁻ᴸᵗ(₁/₂) ⇒ ln(2) = λT₁/₂

Thus,

因此，

T₁/₂ = ln(2) / λ ≈ 0.693 / λ

Half-lives range from fractions of a second (e.g., polonium-214, T₁/₂ = 1.64×10⁻⁴ s) to billions of years (e.g., uranium-238, T₁/₂ = 4.47×10⁹ years). The half-life is a key parameter in determining the rate of decay and is used in radiometric dating.

半衰期从不到一秒（如钋-214，T₁/₂ = 1.64×10⁻⁴ 秒）到数十亿年（如铀-238，T₁/₂ = 4.47×10⁹ 年）不等。半衰期是决定衰变速率的关键参数，并用于放射性测年。

A short half-life means high activity per unit mass, while a long half-life means a sample remains radioactive for an extended period — important for waste management considerations.

短半衰期意味着单位质量的活度高，而长半衰期则意味着样品可长时间保持放射性——这对核废料管理至关重要。

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9. Solving Half-Life Problems | 半衰期问题求解

Problems involving half-life typically require the use of N = N₀ e⁻ᴸᵗ or the number of half-lives elapsed. If n = t / T₁/₂, then after n half-lives, the fraction remaining is (½)ⁿ.

涉及半衰期的问题通常需要运用 N = N₀ e⁻ᴸᵗ 或已过去的半衰期个数。若 n = t / T₁/₂，则经过 n 个半衰期后，剩余比例为 (½)ⁿ。

Example: A sample of iodine-131 (T₁/₂ = 8.0 days) has an initial activity of 400 Bq. What is its activity after 24 days?

示例：某碘-131 样品（T₁/₂ = 8.0 天）初始活度为 400 Bq。24 天后的活度是多少？

Number of half-lives: n = 24 / 8 = 3. Remaining fraction = (½)³ = ⅛. Activity = 400 × ⅛ = 50 Bq.

半衰期个数：n = 24 / 8 = 3。剩余比例 = (½)³ = ⅛。活度 = 400 × ⅛ = 50 Bq。

For problems where time is not an integer multiple of the half-life, use A = A₀ e⁻ᴸᵗ with λ = ln(2) / T₁/₂. Be careful with units: λ and t must have consistent time units.

若时间不是半衰期的整数倍，使用 A = A₀ e⁻ᴸᵗ，其中 λ = ln(2) / T₁/₂。注意单位：λ 与 t 必须使用一致的时间单位。

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10. Activity and the Becquerel | 活度与贝克勒尔

Activity (A) is the number of nuclear decays per unit time. The SI unit is the becquerel (Bq), where 1 Bq = 1 decay per second. An older unit is the curie (Ci): 1 Ci = 3.7×10¹⁰ Bq.

活度（A）是单位时间内的核衰变数。国际单位制单位为贝克勒尔（Bq），1 Bq = 1 次衰变/秒。旧单位为居里（Ci）：1 Ci = 3.7×10¹⁰ Bq。

Activity depends on the number of undecayed nuclei and the decay constant. Since N decreases exponentially, activity also decreases exponentially. In practice, the measured count rate (counts per second) is not exactly equal to the activity due to detector efficiency and background radiation, but it is proportional if corrections are applied.

活度取决于未衰变核的数目和衰变常数。由于 N 呈指数衰减，活度也指数衰减。实际测量中，计数率（每秒计数）并不完全等于活度，因为存在探测器效率和背景辐射的影响，但若进行修正，计数率与活度成正比。

Knowing the activity allows calculation of the number of radioactive atoms present: N = A / λ.

知道了活度，就可以计算存在的放射性原子数：N = A / λ。

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11. Background Radiation and Safety | 背景辐射与安全

We are constantly exposed to background radiation from cosmic rays, rocks, food, radon gas, and even our own bodies. Typical background count rates are around 0.5–1.0 counts per second, but they vary by location. When measuring radioactive sources, the background count must be subtracted to obtain the corrected count rate.

我们不断受到来自宇宙射线、岩石、食物、氡气乃至人体自身的背景辐射照射。典型背景计数率约为 0.5–1.0 次/秒，但随地点而异。测量放射源时，必须减去背景计数以得到修正后的计数率。

Handling radioactive sources requires strict safety protocols: minimise exposure time, maximise distance (inverse-square law applies to gamma), and use appropriate shielding. Alpha sources are hazardous only if ingested or inhaled, while gamma sources demand heavy shielding.

操作放射源需严格遵循安全规程：尽量缩短照射时间，增大距离（γ 射线的强度遵循平方反比定律），并使用适当的屏蔽材料。α 源仅在摄入或吸入时才具有危害性，而 γ 源则需要重屏蔽。

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12. Applications of Radioisotopes | 放射性同位素的应用

Radioisotopes are widely used in medicine, industry, and research. In medicine, iodine-125 and iridium-192 are used in brachytherapy to destroy cancer cells. Technetium-99m (T₁/₂ ≈ 6 h) is a gamma emitter used in diagnostic imaging; its short half-life minimises patient radiation dose.

放射性同位素广泛应用于医学、工业和研究领域。医学上，碘-125 和铱-192 用于近距离放射治疗以杀死癌细胞。锝-99m（T₁/₂ ≈ 6 小时）是用于诊断影像的 γ 发射体，其短半衰期可降低患者所受辐射剂量。

In industry, americium-241 emits alpha particles for smoke detectors, while cobalt-60 is used for sterilising medical equipment and food irradiation. Carbon-14 dating relies on the β⁻ decay of ¹⁴C (T₁/₂ = 5730 years) to determine the age of organic materials up to about 50 000 years.

工业上，镅-241 发射 α 粒子用于烟雾探测器，而钴-60 用于医疗器械消毒和食品辐照。碳-14 测年依靠 ¹⁴C 的 β⁻ 衰变（T₁/₂ = 5730 年）来测定有机材料的年代，可测定约五万年以内的样品。

Leak detection and thickness monitoring in manufacturing exploit the fact that radiation attenuation depends on material thickness and density, using beta or gamma sources.

制造业中的检漏和厚度监控利用辐射衰减与材料厚度和密度相关的特性，使用 β 或 γ 源。

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