📚 A-Level Physics: Nuclear Physics Key Points and Revision | A-Level 物理：核物理考点精讲
Nuclear physics digs into the heart of matter, exploring the structure of the atomic nucleus, the forces that hold it together, and the transformations that release enormous amounts of energy. From radioactivity and half-life to fission, fusion and E=mc², these concepts form a core part of the A-Level Physics syllabus and link directly to real-world applications such as nuclear power and medical imaging.
核物理深入物质的核心，探索原子核的结构、将其凝聚在一起的力以及释放巨大能量的核转变。从放射性、半衰期到裂变、聚变和 E=mc²，这些概念构成 A-Level 物理课程的核心部分，并与核电、医学成像等现实应用直接相连。
1. Structure of the Nucleus and Isotopes | 原子核结构与同位素
Every atom contains a tiny, dense nucleus made up of nucleons — positively charged protons and uncharged neutrons. The number of protons is the atomic number Z, which defines the element, while the total number of nucleons is the mass number A. A nuclide is written as AZX, for example 126C. Isotopes are atoms of the same element with the same Z but different A, such as carbon-12 and carbon-14. They have identical chemical properties but different nuclear stability.
每个原子都含有一个微小致密的原子核，由核子——带正电的质子和不带电的中子组成。质子数即原子序数 Z，决定了元素的种类，而核子总数是质量数 A。核素表示为 AZX，例如 126C。同位素是质子数相同但质量数不同的原子，比如碳‑12 和碳‑14。它们化学性质相同，但核稳定性不同。
2. The Strong Nuclear Force | 强核力
The nucleus does not fly apart despite electrostatic repulsion between protons because a powerful attractive force — the strong nuclear force — acts between all nucleons. This force is very short-range (∼10⁻¹⁵ m), attractive at separations around 1–3 fm, and repulsive at extremely small distances to prevent nucleon collapse. It is charge-independent, meaning it acts equally between proton-proton, neutron-neutron and proton-neutron pairs. In larger nuclei, extra neutrons are needed to add strong-force attraction without increasing electrostatic repulsion, which explains the neutron-to-proton ratio trend along the stability line.
尽管质子间存在静电斥力，原子核并未飞散，这是因为所有核子之间还存在一种强大的吸引力——强核力。该力为极短程力（约 10⁻¹⁵ m），在 1–3 fm 的间距上表现为引力，而在极近距离上表现为斥力，以防止核子塌缩。强核力与电荷无关，质‑质、中‑中和质‑中对间的作用强度相同。在较大的核中，需要额外的中子来增加强核力吸引而不增大静电斥力，这就解释了沿稳定线变化的中子‑质子比趋势。
3. Mass Defect and Binding Energy | 质量亏损与结合能
The mass of a nucleus is always less than the sum of the masses of its separate nucleons. This difference is called the mass defect (Δm). The missing mass has been converted into the binding energy that holds the nucleus together, according to E = Δmc². Binding energy is the work required to separate a nucleus into its individual protons and neutrons. It is usually expressed in MeV; 1 atomic mass unit (u) is equivalent to 931.5 MeV of energy. For example, the mass defect for helium-4 is about 0.0304 u, giving a total binding energy of roughly 28.3 MeV.
原子核的质量总是小于其各个独立核子的质量之和。这个差值称为质量亏损（Δm）。根据 E = Δmc²，亏损的质量已转化为将核子结合在一起的结合能。结合能是将一个原子核完全分离成独立质子和中子所需做的功，常以 MeV 表示；1 个原子质量单位 (u) 相当于 931.5 MeV 的能量。例如，氦‑4 的质量亏损约为 0.0304 u，对应的总结合能大约为 28.3 MeV。
4. Binding Energy per Nucleon and Stability | 每个核子的结合能与稳定性
Dividing the total binding energy by the number of nucleons gives the binding energy per nucleon. A graph of this quantity against mass number A shows a broad peak around iron-56, the most stable nucleus. For light nuclei, fusion dramatically increases binding energy per nucleon; for very heavy nuclei, fission increases it. A high binding energy per nucleon means a more tightly bound, lower energy state, so nuclei tend to move towards the iron peak through radioactive decay or nuclear reactions.
将总结合能除以核子数得到每个核子的平均结合能。该量随质量数 A 变化的曲线在铁‑56 附近呈现一个宽阔的峰顶，表明铁‑56 是最稳定的核。对于轻核，聚变可大幅提高每个核子的结合能；对于很重的核，裂变则能提高该值。每个核子结合能越高，表示原子核束缚越紧、能量越低，因此原子核会通过放射性衰变或核反应向铁峰移动。
5. Radioactive Decay: Alpha, Beta and Gamma | 放射性衰变：α、β 和 γ 辐射
Unstable nuclei spontaneously emit radiation to become more stable. The three main types are alpha (α), beta (β) and gamma (γ). Their properties are summarised below:
不稳定的原子核会自发辐射以变得更加稳定，主要类型有 α、β 和 γ 三种。它们的性质总结如下：
| Property | α (alpha) | β⁻ (beta-minus) | γ (gamma) |
|---|---|---|---|
| Nature | Helium nucleus 42He | Fast electron 0−1e | Electromagnetic wave |
| Ionising power | High | Medium | Low |
| Penetration | Stopped by paper / few cm of air | Stopped by ~3 mm aluminium | Reduced by several cm of lead or concrete |
| Deflection in electric / magnetic field | Deflected towards negative plate; small deflection due to large mass | Deflected towards positive plate; large deflection | No deflection |
Beta decay also involves an antineutrino (ν̄) to conserve energy and momentum. For beta-plus (β⁺) decay, a positron 0+1e and a neutrino are emitted.
β 衰变还伴随反中微子 (ν̄) 以保持能量和动量守恒。对于 β⁺ 衰变，会发射正电子 0+1e 和一个中微子。
6. Decay Equations and Conservation Laws | 衰变方程与守恒定律
Nuclear decay equations must obey conservation of nucleon number and charge. In alpha decay, the parent nucleus loses two protons and two neutrons, so A decreases by 4 and Z by 2. For example:
23892U → 23490Th + 42He
核衰变方程必须遵守核子数守恒和电荷守恒。在 α 衰变中，母核失去两个质子和两个中子，因此 A 减 4，Z 减 2。例如：
23892U → 23490Th + 42He
In β⁻ decay, a neutron changes into a proton, so A stays the same, Z increases by 1, and an electron plus an antineutrino are emitted:
146C → 147N + 0−1e + ν̄
在 β⁻ 衰变中，一个中子转变成质子，因此 A 不变，Z 增加 1，同时发射一个电子和一个反中微子：
146C → 147N + 0−1e + ν̄
Gamma emission usually follows alpha or beta decay when the daughter nucleus is left in an excited state; it involves no change in A or Z.
γ 辐射通常在子核处于激发态时伴随 α 或 β 衰变发生，不改变 A 和 Z。
7. Activity and the Decay Constant | 活度与衰变常量
The activity A of a radioactive sample is the number of decays per unit time. It is proportional to the number of undecayed nuclei N present:
A = −dN/dt = λN
放射性样品的活度 A 是单位时间内发生衰变的核数目。它与尚未衰变的核数目 N 成正比：
A = −dN/dt = λN
where λ is the decay constant, a probability of decay per nucleus per second. The SI unit of activity is the becquerel (Bq), where 1 Bq = 1 decay per second. A large λ corresponds to a rapidly decaying source.
其中 λ 是衰变常量，表示每个核每秒发生衰变的概率。活度的国际单位是贝克勒尔 (Bq)，1 Bq = 每秒 1 次衰变。λ 越大，源衰变得越快。
8. Half-Life and Exponential Decay | 半衰期与指数衰减
Radioactive decay is a random process described by an exponential law:
N = N₀ e−λt
放射性衰变是一个随机过程，遵循指数规律：
N = N₀ e−λt
The half-life T1/2 is the time taken for the number of undecayed nuclei (and the activity) to halve. Setting N = N₀/2 gives:
T1/2 = ln2 / λ
半衰期 T1/2 是未衰变核数（及活度）减少一半所需的时间。令 N = N₀/2 可得：
T1/2 = ln2 / λ
The same exponential form applies to activity and mass. A useful rule: after n half-lives, the remaining fraction is (1/2)ⁿ.
同样的指数形式适用于活度和质量。一个常用的经验法则：经过 n 个半衰期后，剩余的比例为 (1/2)ⁿ。
9. Nuclear Fission | 核裂变
Nuclear fission occurs when a heavy nucleus, such as uranium-235, captures a neutron and splits into two smaller daughter nuclei, releasing energy and two or three neutrons. A typical reaction is:
23592U + 10n → 14156Ba + 9236Kr + 3 10n + energy
核裂变是指重核（如铀‑235）俘获一个中子后分裂成两个较小的子核，同时释放能量和两到三个中子。一个典型的反应方程为：
23592U + 10n → 14156Ba + 9236Kr + 3 10n + 能量
The released neutrons can trigger further fissions, creating a chain reaction. In a nuclear reactor, control rods absorb excess neutrons to keep the reaction steady, and a moderator slows down neutrons to increase the probability of fission. The energy released comes from the increase in binding energy per nucleon; the fission fragments are more tightly bound than the original heavy nucleus.
释放出的中子可诱发更多的裂变，从而形成链式反应。在核反应堆中，控制棒吸收过剩的中子以维持反应平稳，慢化剂则减慢中子速度以提高裂变概率。释放的能量源于每个核子结合能的增加：裂变碎片的结合比原来的重核更紧密。
10. Nuclear Fusion | 核聚变
Nuclear fusion is the joining together of light nuclei to form a heavier nucleus, releasing energy because the product has a higher binding energy per nucleon. For example, in the Sun, hydrogen nuclei fuse into helium through the proton-proton chain. A simple representation of fusion is:
21H + 31H → 42He + 10n + energy
核聚变是轻核结合成一个较重核的过程，由于产物每个核子的结合能更高，因而释放能量。例如，在太阳中，氢核通过质子‑质子链反应聚变成氦核。聚变的一个简化表达式为：
21H + 31H → 42He + 10n + 能量
Fusion requires extremely high temperatures (millions of kelvin) to give nuclei enough kinetic energy to overcome the Coulomb barrier. On Earth, this is achieved in experimental reactors like tokamaks, but sustained, controlled fusion for power generation remains a challenge.
聚变需要极高的温度（数百万开尔文），以使核获得足够的动能来克服库仑势垒。在地球上，托卡马克等实验装置可以达到这些温度，但实现持续、可控的聚变发电仍是一项挑战。
11. Mass–Energy Equivalence: E = mc² | 质能等价：E = mc²
Einstein’s famous equation states that mass and energy are interchangeable. In nuclear reactions, the total mass of the products is slightly smaller than that of the reactants; this ‘lost’ mass Δm is converted into kinetic energy of the products. The energy released is given by:
E = Δm c²
爱因斯坦著名的方程指出质量与能量可以相互转化。在核反应中，产物的总质量略小于反应物的总质量；这个“亏损”的质量 Δm 转化为产物的动能。释放的能量由下式给出：
E = Δm c²
With 1 u = 1.661 × 10⁻²⁷ kg and c = 3.00 × 10⁸ m s⁻¹, it follows that 1 u is equivalent to 931.5 MeV. This conversion factor is vital for calculating energy changes in both fission and fusion.
因为 1 u = 1.661 × 10⁻²⁷ kg, c = 3.00 × 10⁸ m s⁻¹，可得 1 u 相当于 931.5 MeV。这个转换因子对于计算裂变和聚变中的能量变化至关重要。
12. Applications of Radioactivity: Carbon Dating | 放射性的应用：碳定年法
Radioactive isotopes have many practical uses. One key application is carbon-14 dating. Cosmic rays produce neutrons in the upper atmosphere that convert nitrogen-14 into radioactive carbon-14:
147N + 10n → 146C + 11H
放射性同位素有许多实际应用，其中一个重要应用是碳‑14 定年法。宇宙射线在高空大气中产生中子，将氮‑14 转化为放射性碳‑14：
147N + 10n → 146C + 11H
Living organisms constantly exchange carbon with their environment, maintaining a steady proportion of carbon-14 to carbon-12. When an organism dies, the intake stops and the carbon-14 decays with a half-life of about 5730 years. By measuring the remaining activity, the time since death can be estimated. This technique is widely used in archaeology and geology for organic materials up to around 50 000 years old.
活体生物与环境不断进行碳交换，使体内碳‑14 与碳‑12 的比例保持恒定。生物死亡后，碳的摄入停止，碳‑14 按照约 5730 年的半衰期衰变。通过测量剩余的活度，便可估算死亡以来的时间。该方法广泛应用于考古学和地质学，可测定距今约 5 万年以内的有机物质。
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