Nuclear Physics: Radioactive Decay and Half-Life Calculations — 核物理：放射性衰变与半衰期计算

Introduction to Nuclear Physics — 核物理导论

核物理是物理学中研究原子核的结构、性质和变化规律的分支学科。在 A-Level 物理课程中，核物理是一个核心模块，涵盖放射性衰变、半衰期计算、核反应以及核能在现代科技中的应用。对于 AQA 考试局的考生而言，掌握放射性衰变的数学模型和半衰期概念是取得高分的关键，因为这部分内容频繁出现在 AS 和 A2 试卷中，既考查理解力也考查计算能力。

Nuclear physics is the branch of physics that studies the structure, properties, and behavior of atomic nuclei. In the A-Level Physics curriculum, nuclear physics forms a core module covering radioactive decay, half-life calculations, nuclear reactions, and the applications of nuclear energy in modern technology. For AQA exam board candidates, mastering the mathematical model of radioactive decay and the concept of half-life is essential for achieving high marks, as this content appears frequently in both AS and A2 papers, testing both understanding and calculation skills.

The Structure of the Atomic Nucleus — 原子核的结构

原子核由质子和中子组成，两者统称为核子。质子带正电荷，中子不带电荷。核素通常用符号 ^A_ZX 表示，其中 A 为质量数（质子数 + 中子数），Z 为原子序数（质子数），X 为元素符号。例如，碳-14 表示为 ¹⁴₆C，具有 6 个质子和 8 个中子。理解核素符号对于放射性衰变方程的书写至关重要，因为衰变过程中质量数和原子序数必须守恒。

The atomic nucleus consists of protons and neutrons, collectively called nucleons. Protons carry a positive charge, while neutrons are electrically neutral. A nuclide is typically represented by the symbol ^A_ZX, where A is the mass number (protons + neutrons), Z is the atomic number (protons), and X is the element symbol. For example, carbon-14 is written as ¹⁴₆C, with 6 protons and 8 neutrons. Understanding nuclide notation is crucial for writing radioactive decay equations, as both mass number and atomic number must be conserved during decay processes.

在稳定核中，核力（强力）克服了质子之间的库仑斥力，将核子束缚在一起。然而，当核内中子与质子的比例偏离稳定带（stability band）时，核就会变得不稳定，从而发生放射性衰变。较轻的元素在质子数与中子数接近 1:1 时最稳定，而较重的元素则需要更多的中子来提供额外的核力以抵消更大的库仑斥力。

In stable nuclei, the strong nuclear force overcomes the Coulomb repulsion between protons, binding the nucleons together. However, when the neutron-to-proton ratio deviates from the stability band, the nucleus becomes unstable and undergoes radioactive decay. Lighter elements are most stable when the proton-to-neutron ratio is close to 1:1, while heavier elements require more neutrons to provide additional nuclear force to counteract the greater Coulomb repulsion.

Types of Radioactive Decay — 放射性衰变的类型

Alpha Decay — Alpha 衰变

Alpha 衰变发生在重核（通常 A > 200）中，当库仑斥力超过核力时，原子核会发射一个由 2 个质子和 2 个中子组成的 Alpha 粒子（即氦-4 核，⁴₂He）。衰变后，母核的质量数减少 4，原子序数减少 2，子核在周期表中向左移动两格。例如，镭-226 的 Alpha 衰变产生氡-222：²²⁶₈₈Ra -> ²²²₈₆Rn + ⁴₂He。

Alpha decay occurs in heavy nuclei (typically A > 200) when the Coulomb repulsion overcomes the nuclear force, causing the nucleus to emit an alpha particle consisting of 2 protons and 2 neutrons (a helium-4 nucleus, ⁴₂He). After decay, the parent nucleus loses 4 in mass number and 2 in atomic number, with the daughter nucleus shifting two places to the left in the periodic table. For example, the alpha decay of radium-226 produces radon-222: ²²⁶₈₈Ra -> ²²²₈₆Rn + ⁴₂He.

Alpha 粒子的穿透能力最弱，可以被一张纸或几厘米的空气阻挡。然而，其电离能力最强，一旦进入体内（如吸入或摄入），会对生物组织造成严重损伤。AQA 考试中常要求考生比较三种衰变类型在穿透能力和电离能力上的差异。

Alpha particles have the weakest penetrating power and can be stopped by a sheet of paper or a few centimeters of air. However, they have the strongest ionizing ability and can cause severe damage to biological tissue if they enter the body through inhalation or ingestion. AQA exams frequently ask candidates to compare the three decay types in terms of penetrating power and ionizing ability.

Beta-Minus Decay — Beta- 衰变

Beta- 衰变发生在中子过剩的核中。核内的一个中子转变为质子，同时发射一个电子（Beta 粒子）和一个反电子中微子。衰变方程中，质量数保持不变，原子序数增加 1，子核在周期表中向右移动一格。经典例子是碳-14 衰变为氮-14：¹⁴₆C -> ¹⁴₇N + ⁰_-1e + v_e。在书写衰变方程时，必须同时标出反中微子，否则会丢分。

Beta-minus decay occurs in neutron-rich nuclei. A neutron in the nucleus transforms into a proton, simultaneously emitting an electron (beta particle) and an antineutrino. In the decay equation, the mass number remains unchanged, the atomic number increases by 1, and the daughter nucleus shifts one place to the right in the periodic table. The classic example is carbon-14 decaying to nitrogen-14: ¹⁴₆C -> ¹⁴₇N + ⁰_-1e + v_e. When writing decay equations, the antineutrino must be included, or marks will be lost.

Beta-Plus Decay — Beta+ 衰变

Beta+ 衰变发生在质子过剩的核中。核内的一个质子转变为中子，同时发射一个正电子（电子的反粒子）和一个电子中微子。原子序数减少 1，质量数不变。例如，碳-11 衰变为硼-11：¹¹₆C -> ¹¹₅B + ⁰₊₁e + v_e。Beta+ 衰变在 PET 扫描等医学成像技术中有重要应用。

Beta-plus decay occurs in proton-rich nuclei. A proton in the nucleus transforms into a neutron, simultaneously emitting a positron (the antiparticle of the electron) and an electron neutrino. The atomic number decreases by 1, while the mass number remains unchanged. For example, carbon-11 decays to boron-11: ¹¹₆C -> ¹¹₅B + ⁰₊₁e + v_e. Beta-plus decay has important applications in medical imaging techniques such as PET scanning.

Gamma Decay — Gamma 衰变

Gamma 衰变通常伴随 Alpha 或 Beta 衰变发生。当子核处于激发态时，会通过发射高能光子（Gamma 射线）回到基态。Gamma 衰变不改变质量数或原子序数，因此在核反应方程中通常可以不写，但在能量计算中必须考虑。Gamma 射线的穿透能力最强，需要几厘米厚的铅或几米厚的混凝土才能有效阻挡。

Gamma decay typically accompanies alpha or beta decay. When the daughter nucleus is in an excited state, it returns to the ground state by emitting high-energy photons (gamma rays). Gamma decay does not change the mass number or atomic number, so it is often omitted from nuclear reaction equations, but it must be considered in energy calculations. Gamma rays have the strongest penetrating power and require several centimeters of lead or several meters of concrete to be effectively blocked.

The Exponential Law of Radioactive Decay — 放射性衰变的指数规律

放射性衰变是一个随机过程。我们无法预测某一个特定的不稳定核何时会衰变，但对于大量核组成的样本，衰变速率遵循精确的统计规律。实验表明，单位时间内发生衰变的核的数目（衰变速率，也称活度 A）与当前尚未衰变的核的数目 N 成正比：A = lambda * N。其中 lambda 称为衰变常量，其单位是 s^{-1}，反映的是每个核在单位时间内发生衰变的概率。

Radioactive decay is a random process. We cannot predict when a specific unstable nucleus will decay, but for a sample consisting of a large number of nuclei, the decay rate follows a precise statistical law. Experiments show that the number of nuclei decaying per unit time (the decay rate, also called activity A) is proportional to the current number of undecayed nuclei N: A = lambda * N. Here, lambda is called the decay constant, with units of s^{-1}, representing the probability per unit time that any given nucleus will decay.

由微分方程 dN/dt = -lambda * N，通过积分可以得到放射性衰变的指数定律：N = N₀ * e^{-lambda t}。同样，活度也服从指数衰减规律：A = A₀ * e^{-lambda t}。这意味着无论起始数量是多少，每隔一个固定的时间段，剩余核的数量就会减少一半 – 这就是半衰期的物理本质。

From the differential equation dN/dt = -lambda * N, integration yields the exponential law of radioactive decay: N = N₀ * e^{-lambda t}. Similarly, activity also follows exponential decay: A = A₀ * e^{-lambda t}. This means that regardless of the starting quantity, the number of remaining nuclei halves after a fixed time interval – this is the physical essence of half-life.

Half-Life: Definition and Calculations — 半衰期：定义与计算

半衰期 T_{1/2} 定义为放射性核的数目（或活度）减少到初始值一半所需的时间。由指数衰变公式，当 N = N₀/2 时，有 N₀/2 = N₀ * e^{-lambda * T_{1/2}}，化简得 T_{1/2} = ln(2) / lambda 约等于 0.693 / lambda。这是 A-Level 物理中最基础也最重要的公式之一，必须熟记。

The half-life T_{1/2} is defined as the time required for the number of radioactive nuclei (or activity) to reduce to half of its initial value. From the exponential decay formula, when N = N₀/2, we have N₀/2 = N₀ * e^{-lambda * T_{1/2}}, which simplifies to T_{1/2} = ln(2) / lambda, approximately 0.693 / lambda. This is one of the most fundamental and important formulas in A-Level Physics and must be memorized.

不同放射性同位素的半衰期差异极大，从微秒级到数十亿年级不等。例如，钋-214 的半衰期仅为 164 微秒，而铀-238 的半衰期长达 44.7 亿年，与地球的年龄相当。AQA 考题中常见利用半衰期进行年代测定的应用，如碳-14 测年法用于考古学中测定有机物的年代（半衰期约 5730 年）。

The half-lives of different radioactive isotopes vary enormously, ranging from microseconds to billions of years. For example, polonium-214 has a half-life of just 164 microseconds, while uranium-238 has a half-life of 4.47 billion years, comparable to the age of the Earth. AQA exam questions frequently test applications of half-life for dating purposes, such as carbon-14 dating used in archaeology to determine the age of organic materials (half-life approximately 5730 years).

Exam-Style Calculation Problems — A-Level 考试计算题型

在 AQA 物理考试中，半衰期和衰变的计算题通常分为以下类型。一是「直接代入型」，给出初始活度和衰变常量，求某时刻的活度，直接使用 A = A₀ * e^{-lambda t} 即可。二是「半衰期反推型」，给出两次测量的活度数据及时间间隔，要求先计算衰变常量 lambda，再求半衰期。三是「分数型」，问经过多少个半衰期后剩余量为初始量的 1/8 或 1/16 等，这类题目利用 N = N₀*(1/2)^n 的关系更为便捷，其中 n 为经过的半衰期数。

In AQA Physics exams, half-life and decay calculation questions typically fall into the following types. The first is the “direct substitution” type: given the initial activity and decay constant, find the activity at a certain time – simply use A = A₀ * e^{-lambda t}. The second is the “reverse half-life” type: given two activity measurements at different times, calculate the decay constant lambda first, then the half-life. The third is the “fractional” type: asking after how many half-lives the remaining quantity is 1/8 or 1/16 of the initial amount. For these questions, using the relationship N = N₀*(1/2)^n is more convenient, where n is the number of half-lives elapsed.

典型例题：一种放射性样品的初始活度为 800 Bq（贝克勒尔），6 小时后活度降至 100 Bq。求半衰期。解题思路：利用 A = A₀ * (1/2)^n，代入得 100 = 800*(1/2)^n，即 (1/2)^n = 1/8，所以 n = 3。3 个半衰期对应 6 小时，因此 T_{1/2} = 2 小时。同时可以验证：800 -> 400 -> 200 -> 100，每步减半，符合结果。

Typical example: A radioactive sample has an initial activity of 800 Bq (becquerels). After 6 hours, the activity drops to 100 Bq. Find the half-life. Solution approach: Using A = A₀ * (1/2)^n, substitute to get 100 = 800*(1/2)^n, so (1/2)^n = 1/8, hence n = 3. Three half-lives correspond to 6 hours, therefore T_{1/2} = 2 hours. This can be verified: 800 -> 400 -> 200 -> 100, halving at each step, confirming the result.

Graphical Analysis of Radioactive Decay — 放射性衰变的图像分析

AQA 考试非常重视图像分析能力。典型的活度-时间图是一个指数递减曲线。要从中提取半衰期，可以在 y 轴上选取任意一点（如初始活度的 75%），读取对应时间 t1，再找到活度为该值一半（37.5%）时对应的时间 t2，半衰期即为 t2 – t1。更精确的方法是对活度取自然对数，绘制 ln(A) 对 t 的图像：根据 ln(A) = ln(A₀) – lambda*t，这是一条斜率为 -lambda 的直线，从斜率可以直接求得衰变常量，进而计算半衰期。

AQA exams place great emphasis on graphical analysis skills. A typical activity-time graph is an exponential decay curve. To extract the half-life, pick any point on the y-axis (e.g., 75% of initial activity), read the corresponding time t1, then find the time t2 when the activity is half of that value (37.5%) – the half-life is t2 – t1. A more precise method is to take the natural logarithm of the activity and plot ln(A) against t: from ln(A) = ln(A₀) – lambda*t, this is a straight line with slope -lambda, from which the decay constant can be directly obtained and the half-life calculated.

Background Radiation and Corrections — 背景辐射与校正

在任何放射性测量实验中，探测器除了记录来自样品本身的辐射外，还会记录环境中的背景辐射。背景辐射来源于宇宙射线、地壳中的天然放射性核素（如氡气）以及人造辐射源。在精确的衰变实验中，必须在每次测量后减去背景计数率。AQA 实验题中常见的操作是：先在不放置放射源的情况下测量一段时间的背景计数，然后从每次样品测量结果中扣除该背景值。

In any radioactive measurement experiment, the detector records not only radiation from the sample itself but also background radiation from the environment. Background radiation originates from cosmic rays, naturally occurring radionuclides in the Earth’s crust (such as radon gas), and artificial sources. In precise decay experiments, the background count rate must be subtracted from each measurement. A common procedure in AQA practical questions is to first measure the background count over a period of time without the radioactive source present, then subtract this background value from each sample measurement.

Applications of Radioactive Isotopes — 放射性同位素的应用

放射性同位素在医学、工业和科学研究中有广泛的应用。在医学领域，碘-131 用于治疗甲状腺功能亢进和甲状腺癌，因为甲状腺会主动吸收碘。锝-99m（半衰期 6 小时）是最常用的医学成像示踪剂，其较短的半衰期意味着对患者的辐射剂量较低。在工业中，使用 Beta 源测量纸张、金属箔等材料的厚度；利用 Gamma 射线进行焊缝的无损检测。碳-14 测年法则彻底改变了考古学和地质学，使得测定数万年内有机遗骸的年代成为可能。

Radioactive isotopes have widespread applications in medicine, industry, and scientific research. In medicine, iodine-131 is used to treat hyperthyroidism and thyroid cancer because the thyroid gland actively absorbs iodine. Technetium-99m (half-life 6 hours) is the most commonly used medical imaging tracer – its short half-life means a lower radiation dose to patients. In industry, beta sources are used to measure the thickness of materials such as paper and metal foil, while gamma rays are used for non-destructive testing of welds. Carbon-14 dating has revolutionised archaeology and geology, making it possible to determine the age of organic remains up to tens of thousands of years old.

Key Equations Summary — 关键公式总结

以下是 AQA A-Level 物理核物理模块的核心公式，建议考生反复练习直到能够熟练运用：

The following are the core formulas for the AQA A-Level Physics nuclear physics module. Candidates are advised to practise them repeatedly until they can be applied proficiently:

1. 衰变速率（活度）：A = lambda * N

1. Decay rate (activity): A = lambda * N

2. 指数衰变定律：N = N₀ * e^{-lambda t}；A = A₀ * e^{-lambda t}

2. Exponential decay law: N = N₀ * e^{-lambda t}; A = A₀ * e^{-lambda t}

3. 半衰期与衰变常量的关系：T_{1/2} = ln(2) / lambda ≈ 0.693 / lambda

3. Relationship between half-life and decay constant: T_{1/2} = ln(2) / lambda, approximately 0.693 / lambda

4. 半衰期数 n 后的剩余量：N = N₀ * (1/2)^n

4. Remaining quantity after n half-lives: N = N₀ * (1/2)^n

5. 对数形式：ln(N) = ln(N₀) – lambda * t

5. Logarithmic form: ln(N) = ln(N₀) – lambda * t

Common Mistakes and Exam Tips — 常见错误与应试技巧

学生在核物理考试中常犯的错误包括：混淆质量数和原子序数在衰变方程中的变化规律；在 Beta 衰变方程中遗漏中微子或反中微子；忘记半衰期公式中自然对数的底为 e 而非 10；在对数图像分析中将斜率混淆为 -lambda 而不是 1/lambda。此外，计算活度时务必注意单位的统一：如果半衰期以年为单位，lambda 也必须转换为年^{-1}。

Common mistakes students make in nuclear physics exams include: confusing the changes in mass number and atomic number in decay equations; omitting the neutrino or antineutrino in beta decay equations; forgetting that the base of the natural logarithm in the half-life formula is e, not 10; and confusing the slope in logarithmic graph analysis as -lambda rather than 1/lambda. Additionally, when calculating activity, always ensure unit consistency: if the half-life is in years, lambda must also be converted to year^{-1}.

在 AQA 考试中，单位转换是一个反复出现的考查点。学生需要熟练掌握从贝克勒尔（Bq，等同于 s^{-1}）到分钟^{-1}、小时^{-1}、年^{-1} 的转换，以及在衰变方程中正确使用科学记数法。例如，铀-238 的半衰期为 4.47 * 10^9 年，对应的 lambda 值约为 4.91 * 10^{-18} s^{-1} – 这种极小值的运算需要借助对数方法简化计算。

In AQA exams, unit conversion is a recurring point of assessment. Students need to be proficient in converting from becquerels (Bq, equivalent to s^{-1}) to min^{-1}, h^{-1}, yr^{-1}, and correctly using scientific notation in decay equations. For example, uranium-238 has a half-life of 4.47 * 10^9 years, corresponding to a lambda value of approximately 4.91 * 10^{-18} s^{-1} – calculations involving such extremely small values are simplified using logarithmic methods.

Nuclear Stability and the N-Z Curve — 核稳定性与 N-Z 曲线

核稳定性可以通过中子数 N 对质子数 Z 的曲线（N-Z 曲线）直观地表示。将所有已知的稳定核素绘制在 N-Z 坐标系中，可以观察到一条明显的稳定带。对于轻核（Z < 20），稳定核大致沿 N = Z 线分布。随着 Z 的增加，稳定带逐渐向 N > Z 的区域弯曲，这是因为需要更多的中子来提供核力以克服不断增大的库仑斥力。Z > 83（铋）之后，不存在任何稳定核素 – 所有核都不稳定，最终通过衰变链转变为稳定的铅同位素。

Nuclear stability can be visually represented by a plot of neutron number N against proton number Z – the N-Z curve. When all known stable nuclides are plotted in this coordinate system, a clear stability band is observed. For light nuclei (Z < 20), stable nuclei lie approximately along the N = Z line. As Z increases, the stability band gradually curves into the N > Z region because more neutrons are needed to provide nuclear force to overcome the growing Coulomb repulsion. Beyond Z > 83 (bismuth), no stable nuclides exist – all nuclei are unstable and ultimately decay through decay chains into stable lead isotopes.

位于稳定带上方的核素具有过多的中子，倾向于发生 Beta- 衰变，将中子转化为质子，从而向稳定带移动。位于稳定带下方的核素具有过多的质子，倾向于发生 Beta+ 衰变或电子俘获。而重核（A > 200）通常通过 Alpha 衰变减少核子总数，同时向稳定带靠拢。理解 N-Z 曲线不仅有助于预测衰变类型，也是 AQA 考试中常见的解释题素材。

Nuclides located above the stability band have an excess of neutrons and tend to undergo beta-minus decay, converting neutrons into protons to move towards the stability band. Nuclides located below the stability band have an excess of protons and tend to undergo beta-plus decay or electron capture. Heavy nuclei (A > 200) typically reduce their total nucleon count through alpha decay while moving towards the stability band. Understanding the N-Z curve not only helps predict decay types but also serves as common material for explanation questions in AQA exams.

Binding Energy and Mass Defect — 结合能与质量亏损

原子核的质量总是小于其各组成核子质量之和，这个差值称为质量亏损。根据爱因斯坦质能方程 E = mc^2，质量亏损对应着将核子束缚在一起的结合能。结合能越大，核越稳定。将结合能除以核子数得到平均结合能（binding energy per nucleon），它反映了每个核子对核稳定性的平均贡献。铁-56 具有最大的平均结合能（约 8.8 MeV/核子），因此是最稳定的核素。

The mass of an atomic nucleus is always less than the sum of the masses of its constituent nucleons – this difference is called the mass defect. According to Einstein’s mass-energy equation E = mc^2, the mass defect corresponds to the binding energy that holds the nucleons together. The greater the binding energy, the more stable the nucleus. Dividing the binding energy by the number of nucleons gives the average binding energy (binding energy per nucleon), which reflects the average contribution of each nucleon to nuclear stability. Iron-56 has the highest average binding energy (approximately 8.8 MeV per nucleon), making it the most stable nuclide.

AQA 考试中要求考生能够从平均结合能曲线的形状推断出核能的释放途径。轻核通过聚变（fusion）结合能增大，释放能量 – 这就是太阳的能量来源。重核通过裂变（fission）分裂为中等质量核，同样释放能量 – 这是核电站的基本原理。平均结合能曲线在 A ~ 56 处达到峰值，意味着无论从轻核聚变还是重核裂变的路径靠近铁-56，都有能量释放。

AQA exams require candidates to infer energy release pathways from the shape of the average binding energy curve. Light nuclei release energy through fusion as their binding energy increases – this is the energy source of the Sun. Heavy nuclei release energy through fission into medium-mass nuclei – this is the basic principle of nuclear power stations. The average binding energy curve peaks around A ~ 56, meaning energy is released whether approaching iron-56 from lighter nuclei via fusion or from heavier nuclei via fission.

Nuclear Fission — 核裂变

核裂变是指一个重核（如铀-235）在中子轰击下分裂成两个中等质量的碎片，同时释放能量和2-3个中子的过程。这些释放的中子可以引发更多的裂变事件，形成链式反应。典型的裂变方程：²³⁵₉₂U + ¹₀n -> ⁹²₃₆Kr + ¹⁴¹₅₆Ba + 3¹₀n + 能量。每次裂变约释放 200 MeV 的能量，远大于化学反应的能量释放。

Nuclear fission is the process in which a heavy nucleus (such as uranium-235) splits into two medium-mass fragments upon neutron bombardment, simultaneously releasing energy and 2-3 neutrons. These released neutrons can trigger further fission events, creating a chain reaction. A typical fission equation: ²³⁵₉₂U + ¹₀n -> ⁹²₃₆Kr + ¹⁴¹₅₆Ba + 3¹₀n + energy. Each fission event releases approximately 200 MeV of energy, far exceeding the energy release of chemical reactions.

在核反应堆中，链式反应通过控制棒（吸收中子的硼或镉）和慢化剂（减速中子以增加裂变概率的水或石墨）进行精密调控。临界质量是维持自持链式反应所需的最小裂变材料质量。AQA 课程要求考生能够描述核反应堆的关键组件及其功能，并能够使用结合能数据计算裂变反应释放的能量。

In nuclear reactors, the chain reaction is precisely controlled using control rods (boron or cadmium, which absorb neutrons) and moderators (water or graphite, which slow down neutrons to increase fission probability). The critical mass is the minimum mass of fissile material required to sustain a self-sustaining chain reaction. The AQA syllabus requires candidates to describe the key components of a nuclear reactor and their functions, and to calculate the energy released in fission reactions using binding energy data.

Nuclear Fusion — 核聚变

核聚变是两个轻核在极高温度和压力下结合成一个较重核的过程，同时释放巨大能量。太阳内部的质子-质子链反应是自然界中最常见的聚变过程：四个质子最终融合成一个氦-4 核，释放约 26.7 MeV 的能量。要实现聚变，核必须克服它们之间的库仑斥力，这需要温度达到数千万到数亿开尔文的等离子体状态。

Nuclear fusion is the process in which two light nuclei combine under extremely high temperature and pressure to form a heavier nucleus, releasing enormous energy. The proton-proton chain reaction inside the Sun is the most common fusion process in nature: four protons ultimately fuse into one helium-4 nucleus, releasing approximately 26.7 MeV of energy. To achieve fusion, the nuclei must overcome the Coulomb repulsion between them, requiring plasma temperatures of tens to hundreds of millions of kelvins.

虽然受控核聚变作为清洁能源的潜力巨大，但在地球上实现持续的能量输出仍然面临巨大的技术和工程挑战。国际热核聚变实验堆（ITER）等项目正在探索磁约束和惯性约束两种主要技术路径。AQA 考试中，聚变通常以定性论述的形式出现，重点考查聚变相对于裂变的优势（燃料丰富、放射性废物较少）以及技术挑战（极高的温度和约束要求）。

Although controlled nuclear fusion has enormous potential as a clean energy source, achieving sustained energy output on Earth remains a formidable technological and engineering challenge. Projects such as the International Thermonuclear Experimental Reactor (ITER) are exploring two main technical approaches: magnetic confinement and inertial confinement. In AQA exams, fusion typically appears in qualitative discussion form, focusing on the advantages of fusion over fission (abundant fuel, less radioactive waste) and the technical challenges (extreme temperature and confinement requirements).

Practice Questions with Solutions — 练习题目与解析

Question 1: A sample of iodine-131 has an initial activity of 1200 Bq. The half-life of iodine-131 is 8 days. Calculate the activity after 32 days.

问题 1：碘-131 样品的初始活度为 1200 Bq，半衰期为 8 天。计算 32 天后的活度。

Solution: Number of half-lives: n = 32 / 8 = 4. Activity after 4 half-lives: A = A₀ * (1/2)^4 = 1200 * (1/16) = 75 Bq. Alternatively, using A = A₀ * e^{-lambda t}: lambda = ln(2) / 8 = 0.0866 day^{-1}, A = 1200 * e^{-0.0866 * 32} = 75 Bq.

解析：半衰期数 n = 32 / 8 = 4。4 个半衰期后活度：A = A₀ * (1/2)^4 = 1200 / 16 = 75 Bq。也可使用指数公式验证：lambda = ln(2) / 8 = 0.0866 天^{-1}，A = 1200 * e^{-0.0866 * 32} = 75 Bq。

Question 2: A radioactive source has an activity of 640 Bq at t = 0 and 160 Bq at t = 1.5 hours. Determine the half-life of the source.

问题 2：某放射源在 t = 0 时活度为 640 Bq，在 t = 1.5 小时时活度为 160 Bq。求该放射源的半衰期。

Solution: 640/160 = 4 = 2^2, so 2 half-lives have elapsed. Time for 2 half-lives = 1.5 hours, thus T_{1/2} = 1.5 / 2 = 0.75 hours = 45 minutes. Verification: 640 -> 320 -> 160, which takes 2 steps of 0.75 hours each.

解析：640 / 160 = 4 = 2^2，说明经过了 2 个半衰期。2 个半衰期对应 1.5 小时，因此 T_{1/2} = 1.5 / 2 = 0.75 小时 = 45 分钟。验证：640 -> 320 -> 160，每步 0.75 小时。

Summary and Revision Checklist — 总结与复习清单

总结核物理 A-Level 模块的核心知识：理解原子核的结构和核素符号；能够区分并书写 Alpha、Beta-、Beta+ 和 Gamma 衰变方程；掌握指数衰变定律 N = N₀ * e^{-lambda t} 及其应用；熟练运用半衰期公式 T_{1/2} = ln(2) / lambda 解决定量问题；能够通过 N-Z 曲线判断核稳定性并预测衰变类型；理解结合能、质量亏损以及裂变与聚变中的能量释放原理。

To summarise the core knowledge of the A-Level nuclear physics module: understand the structure of the nucleus and nuclide notation; be able to distinguish and write alpha, beta-minus, beta-plus, and gamma decay equations; master the exponential decay law N = N₀ * e^{-lambda t} and its applications; skillfully use the half-life formula T_{1/2} = ln(2) / lambda to solve quantitative problems; be able to judge nuclear stability and predict decay types using the N-Z curve; understand binding energy, mass defect, and the principles of energy release in fission and fusion.

建议考生在复习时重点关注历年 AQA 真题中的核物理计算题和解释题，特别是半衰期计算与图像分析的组合题型。熟练掌握对数运算和科学记数法是解题速度的关键。对于描述题，注意使用准确的物理术语，如”随机过程”、”指数衰减”、”链式反应”、”临界质量”等。

Candidates are advised to focus on nuclear physics calculation and explanation questions from past AQA papers during revision, particularly combined questions on half-life calculations and graphical analysis. Proficiency in logarithmic operations and scientific notation is key to solving problems quickly. For descriptive questions, use precise physics terminology such as “random process”, “exponential decay”, “chain reaction”, and “critical mass”.