📚 A2 Physics: Nuclear Physics Exam Focus | A2物理：核物理考点精讲
Nuclear physics at A2 level explores the very core of matter — the nucleus. It ties together fundamental concepts of mass, energy and the forces that govern stability, radioactivity and nuclear reactions. This guide revisits every essential point, pairing clear explanations with worked examples and structured comparisons, ensuring you can approach exam problems on binding energy, decay equations, half-life calculations and nuclear processes with complete confidence.
A2阶段的核物理深入物质的核心——原子核。它把质量、能量以及支配原子核稳定性、放射性和核反应的基本概念串联在一起。这份考点精讲涵盖了每一个核心要点，用清晰的解释配合实例和结构化对比，让你在面对结合能、衰变方程、半衰期计算和核反应过程等考题时胸有成竹。
1. The Structure of the Nucleus | 原子核的结构
All atomic nuclei consist of nucleons — positively charged protons and electrically neutral neutrons. The number of protons defines the atomic (proton) number Z, while the total number of nucleons is the mass number A. Neutron number N = A − Z. Despite the repulsive electric force between protons, nuclei are stable due to the strong nuclear force, which acts attractively between all nucleons at very short ranges (~1 fm).
所有原子核都由核子（带正电的质子和电中性的中子）组成。质子数即原子序数 Z，核子总数称为质量数 A。中子数 N = A − Z。虽然质子间存在静电斥力，但由于强核力在极小距离（约1飞米）内对所有核子起吸引作用，原子核得以保持稳定。
- Radius dependence: Nuclear radius R ≈ r₀ A¹⁄³, where r₀ ≈ 1.2 fm. This demonstrates that nuclear volume scales with mass number A.
- 半径关系：核半径 R ≈ r₀ A¹⁄³，r₀ ≈ 1.2 fm。这表明原子核的体积与质量数 A 成正比。
- Density: Nuclear density is roughly constant (≈ 2.3×10¹⁷ kg m⁻³), independent of A, showing that nucleons are packed uniformly.
- 密度：核密度近似为常数（≈ 2.3×10¹⁷ kg m⁻³），与 A 无关，说明核子紧密均匀堆积。
2. Isotopes and Nuclide Notation | 同位素与核素符号
Atoms with the same Z but different N are called isotopes. They share identical chemical properties but have different nuclear masses and stabilities. Nuclide notation is written as ᴬX or X-A, for example ²³⁸U or uranium-238 means Z=92, A=238, N=146.
质子数相同而中子数不同的原子互称同位素。同位素化学性质相同，但核质量和稳定性不同。核素符号写成 ᴬX 或 X-A，如 ²³⁸U（铀-238）表示 Z=92，A=238，N=146。
The standard form: ²³⁸₉₂U places Z as a left subscript and A as a left superscript. This notation is essential for balancing nuclear equations.
标准格式 ²³⁸₉₂U 中 Z 为左下角标，A 为左上角标。这种符号对配平核反应方程至关重要。
3. Nuclear Forces and Stability | 核力与稳定性
Stability arises from the balance between the attractive strong nuclear force and the repulsive electrostatic (Coulomb) force. The strong force is charge-independent, short-range (acts only up to ~3 fm) and saturates — each nucleon interacts only with its nearest neighbours. As Z increases, more neutrons are needed to ‘dilute’ the Coulomb repulsion among protons, resulting in an N/Z ratio that rises from ~1 in light nuclei to ~1.5 in heavy nuclei like lead.
稳定性取决于强核吸引力与静电（库仑）斥力之间的平衡。强核力与电荷无关，作用距离极短（约 ≤ 3 fm），且具有饱和性——每个核子只与最近邻的核子相互作用。随着 Z 增大，需要用更多的中子来“稀释”质子间的库仑斥力，因此 N/Z 比从轻核的约 1 上升到铅等重核的约 1.5。
Nuclei that lie outside the stability band on an N–Z plot are likely radioactive. Very heavy nuclei (Z > 83) are unstable, often undergoing alpha decay.
在 N–Z 图上位于稳定带之外的核素很可能具有放射性。极重的核（Z > 83）不稳定，常发生 α 衰变。
4. Mass Defect and Binding Energy | 质量亏损与结合能
The measured mass of a nucleus is always less than the sum of the masses of its individual protons and neutrons. This difference is the mass defect Δm. Einstein’s mass–energy relation links this defect to the binding energy E = Δm c², which is the energy required to separate a nucleus into its constituent nucleons.
原子核的实际质量总是小于组成它的各个质子和中子质量之和。这个差值就是质量亏损 Δm。根据爱因斯坦质能方程，结合能 E = Δm c²，它是将原子核拆散为独立核子所需的能量。
In calculations, atomic masses are usually given in unified atomic mass units (u), where 1 u = 1.661×10⁻²⁷ kg. Energy equivalent: 1 u = 931.5 MeV (or use 1 u c² = 931.5 MeV). For precise exam work, remember Δm (in u) × 931.5 ≈ binding energy in MeV.
计算中，原子质量通常以原子质量单位 u 给出，1 u = 1.661×10⁻²⁷ kg。能量当量：1 u = 931.5 MeV（即 1 u c² = 931.5 MeV）。考试中请牢记：Δm (u) × 931.5 ≈ 结合能 (MeV)。
E_b = Δm c²
5. Binding Energy per Nucleon | 平均结合能
Dividing the total binding energy by the mass number A gives the binding energy per nucleon, a direct measure of nuclear stability. A graph of binding energy per nucleon against A peaks around iron-56 (~8.8 MeV per nucleon), indicating maximum stability. Lighter nuclei can release energy by fusion towards iron; heavier nuclei release energy by fission towards iron.
将总结合能除以质量数 A，得到平均结合能（单个核子的结合能），它直接衡量原子核的稳定性。平均结合能对 A 的曲线在铁-56 附近达到峰值（约 8.8 MeV/核子），说明此处最稳定。比铁轻的核可通过聚变向铁方向释放能量；比铁重的核则通过裂变向铁方向释放能量。
Examiners often ask you to interpret this curve: steep rise for A < 20, broad maximum, gentle decrease for heavy nuclei. This explains why both fission of heavy nuclei and fusion of very light nuclei are exothermic.
考官常要求解释该曲线：A < 20 时急剧上升，之后出现宽阔的峰值，重核区缓慢下降。这解释了为什么重核裂变和轻核聚变都能释放能量。
6. Radioactive Decay | 放射性衰变
Radioactive decay is a spontaneous and random process where an unstable nucleus emits radiation to become more stable. The process is unaffected by external conditions such as temperature or pressure. Key measurable quantities include activity A (the number of decays per second) measured in becquerels (Bq), and the decay constant λ, which is the probability of a given nucleus decaying per unit time.
放射性衰变是一个自发、随机的过程，不稳定的原子核通过发射辐射趋向稳定。衰变不受温度、压强等外部条件影响。关键可测量包括活度 A（每秒衰变次数，单位贝克勒尔 Bq）和衰变常数 λ（单个核在单位时间内发生衰变的概率）。
Activity equation: A = λN, where N is the number of undecayed nuclei present.
活度方程：A = λN，其中 N 为未衰变核的个数。
7. Decay Modes: Alpha, Beta, Gamma | 衰变模式：α、β、γ
Three primary types of radiation are emitted: alpha (α), beta (β⁻ or β⁺), and gamma (γ). Their properties are summarised below.
放射线主要分为三种：α（阿尔法）、β（贝塔，β⁻ 或 β⁺）和 γ（伽马）。其特性总结如下表。
| Property | Alpha (α) | Beta-minus (β⁻) | Gamma (γ) |
|---|---|---|---|
| Nature | Helium nucleus ⁴₂He | Fast electron e⁻ | Electromagnetic photon |
| Charge | +2e | −e (or +e for β⁺) | 0 |
| Penetration | Stopped by paper/skin | Stopped by ~3 mm Al | Reduced by thick Pb/concrete |
| Ionisation | Strong | Moderate | Weak |
In α decay, A decreases by 4 and Z by 2. In β⁻ decay, a neutron converts to a proton emitting an electron and an antineutrino: A stays constant, Z increases by 1. β⁺ decay reduces Z by 1. Gamma emission often accompanies other decays, releasing excess energy with no change in A or Z.
α 衰变中，A 减 4，Z 减 2。β⁻ 衰变中，一个中子转变为质子并发射电子和反中微子：A 不变，Z 增 1。β⁺ 衰变中 Z 减 1。γ 射线常伴随其他衰变，释放多余能量，A 和 Z 不变。
Example: ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He (alpha); ¹⁴₆C → ¹⁴₇N + e⁻ + antineutrino (beta-minus).
示例：²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He（α 衰变）；¹⁴₆C → ¹⁴₇N + e⁻ + 反中微子（β⁻ 衰变）。
8. Activity and the Decay Law | 活度与衰变定律
The activity A of a radioactive source decreases exponentially: A = A₀ exp(−λt), where A₀ is the initial activity. Equivalently, the number of undecayed nuclei follows N = N₀ exp(−λt). This exponential model arises because the probability of decay per unit time is constant.
放射性源的活度 A 随时间指数衰减：A = A₀ exp(−λt)，A₀ 为初始活度。同样，未衰变核的数目遵循 N = N₀ exp(−λt)。由于单位时间衰变概率恒定，衰变规律呈指数形式。
Taking natural logs gives ln N = ln N₀ − λt, which is a straight line of gradient −λ when ln N is plotted against t. This is a frequent exam data-analysis task.
取自然对数得 ln N = ln N₀ − λt，绘制 ln N 对 t 图像可得一条斜率为 −λ 的直线。这是常见的考试数据分析题。
9. Half-life and Exponential Decay | 半衰期与指数衰变
Half-life t½ is the time taken for the activity (or number of undecayed nuclei) to halve. It relates to the decay constant by t½ = ln 2 / λ ≈ 0.693 / λ. After n half-lives, the fraction remaining is (1/2)ⁿ.
半衰期 t½ 是活度（或未衰变核数）减半所需的时间。它与衰变常数的关系为 t½ = ln 2 / λ ≈ 0.693 / λ。经过 n 个半衰期后，剩余比例为 (1/2)ⁿ。
Exam problems will often provide a count rate or mass and ask for t½ using either the exponential formula or a graph. Always convert counts to the same background-corrected units.
考试题经常会给出计数率或质量，要求利用指数公式或图像求出 t½。切记将计数转换为扣除本底后的数值。
Worked idea: If initial activity is 800 Bq and after 30 minutes it is 100 Bq, then 800 → 400 → 200 → 100 takes three half-lives, so t½ = 10 minutes.
示例思路：若初始活度 800 Bq，30 分钟后为 100 Bq，则 800→400→200→100 经过 3 个半衰期，故 t½ = 10 min。
10. Nuclear Reactions: Fission and Fusion | 核反应：裂变与聚变
Induced nuclear fission occurs when a heavy nucleus (e.g. ²³⁵U or ²³⁹Pu) absorbs a slow neutron and splits into two smaller nuclei, releasing further neutrons and a huge amount of energy. A chain reaction is possible if emitted neutrons trigger further fission events. In a nuclear reactor, control rods absorb excess neutrons and a moderator slows them down.
诱导核裂变是重核（如 ²³⁵U 或 ²³⁹Pu）吸收慢中子后，分裂成两个较轻的核，同时释放出几个中子和巨大能量。如果释放的中子引起更多裂变，就会形成链式反应。在核反应堆中，控制棒吸收多余中子，慢化剂减缓中子速度。
Nuclear fusion combines light nuclei (e.g. deuterium and tritium) into a heavier one (helium), releasing energy because the binding energy per nucleon increases. Fusion requires extremely high temperatures and pressures to overcome Coulomb repulsion. Stars achieve this in their cores; on Earth, magnetic or inertial confinement is investigated.
核聚变将轻核（如氘和氚）结合成较重的核（氦），因平均结合能增大而释放能量。聚变需要极高的温度和压强以克服库仑斥力。恒星在其核心实现聚变；地球上的研究采用磁约束或惯性约束。
11. Energy Released in Nuclear Reactions | 核反应中的能量释放
Energy released in fission or fusion is calculated from the difference in total mass before and after the reaction: Q = (minitial − mfinal) c². Use atomic masses consistently. For example, in a typical fission fragment pair (e.g. ¹⁴¹Ba and ⁹²Kr from ²³⁵U), a mass decrease of ~0.2 u corresponds to roughly 180 MeV released.
裂变或聚变释放的能量由反应前后总质量差计算：Q = (m初始 − m最终) c²。需一致地使用原子质量。例如，一次典型裂变（²³⁵U → ¹⁴¹Ba + ⁹²Kr + 中子），质量减少约 0.2 u，相当于释放约 180 MeV 能量。
Exam calculations: Write the balanced equation, add initial masses, subtract final masses, multiply by 931.5 MeV/u. Also recognise that fusion’s energy yield per unit mass is far larger than fission’s.
考试计算：写出配平的方程，求初始总质量减去最终总质量，乘以 931.5 MeV/u。还要认识到，聚变单位质量释放的能量远大于裂变。
12. Practical Applications and Safety | 实际应用与安全
Radioisotopes are used in medicine (e.g. technetium-99m for imaging, iodine-131 for therapy), industry (tracing leaks, thickness gauging) and carbon dating. In carbon dating, the ratio of ¹⁴C to ¹²C in a dead sample follows the decay law to determine age, using t½ = 5730 years.
放射性同位素应用于医学（如锝-99m 用于成像，碘-131 用于治疗）、工业（泄漏示踪、厚度测量）和碳定年。碳定年中，死体样品中 ¹⁴C/¹²C 比值按衰变定律变化，利用半衰期 5730 年确定年代。
Handling radioactive materials requires safety measures: minimising exposure time, keeping distance, using shielding appropriate to the radiation type, and wearing protective clothing. Always point sources away from people and store securely.
操作放射性物质需采取安全措施：尽量缩短照射时间、保持距离、根据射线类型使用合适的屏蔽，并穿戴防护服。始终将放射源指向无人方向，并安全储存。
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