A-Level物理 核衰变 半衰期 核反应

A-Level Physics: Nuclear Physics ： Radioactive Decay, Half-Life, and Nuclear Reactions

1. Introduction to Nuclear Physics

Nuclear physics is the branch of physics that studies the atomic nucleus, its constituents, and the forces that govern its behaviour. Unlike chemistry, which deals with electron interactions, nuclear physics focuses on the protons and neutrons (collectively called nucleons) bound together by the strong nuclear force. Understanding the nucleus is essential for explaining radioactivity, nuclear energy, and many applications in medicine and industry. The nucleus is characterised by its atomic number Z (number of protons), mass number A (total nucleons), and neutron number N = A − Z.

核物理学是研究原子核及其组成部分和相互作用力的物理学分支。与化学研究电子相互作用不同，核物理关注的是由强核力结合在一起的中子和质子（统称核子）。理解原子核是解释放射性、核能以及医学和工业中许多应用的基础。原子核由原子序数Z（质子数）、质量数A（总核子数）和中子数N = A − Z来表征。

2. Types of Radioactive Decay

Radioactive decay occurs when an unstable nucleus spontaneously transforms into a more stable configuration by emitting particles and/or energy. There are four main types of decay covered in the A-Level syllabus: alpha decay, beta-minus decay, beta-plus decay, and gamma emission. Each decay mode is governed by conservation laws: conservation of mass-energy, conservation of charge, and conservation of nucleon number must all be satisfied.

放射性衰变发生在不稳定的原子核通过发射粒子和/或能量自发转变为更稳定构型时。A-Level大纲涵盖四种主要衰变类型：α衰变、β⁻衰变、β⁺衰变和γ辐射。每种衰变模式都受守恒定律支配：质能守恒、电荷守恒和核子数守恒必须全部满足。

3. Alpha Decay (α)

Alpha decay occurs in heavy nuclei with too many protons and neutrons, typically those with A > 210. The nucleus emits an alpha particle, which is a helium-4 nucleus consisting of 2 protons and 2 neutrons. This reduces the parent nucleus’s mass number by 4 and its atomic number by 2. A classic example is the decay of uranium-238 into thorium-234: ^238_92U → ^234_90Th + ^4_2He. Alpha particles have a relatively low penetration ability ： they can be stopped by a sheet of paper or a few centimetres of air ： but they are highly ionising due to their +2 charge and large mass.

α衰变发生在质子和中子过多的重核中，通常是A > 210的核素。原子核发射一个α粒子，即由2个质子和2个中子组成的氦-4核。这使母核的质量数减少4，原子序数减少2。经典例子是铀-238衰变为钍-234：^238_92U → ^234_90Th + ^4_2He。α粒子穿透能力较低：一张纸或几厘米空气即可阻挡，但由于其+2电荷和大质量，电离能力很强。

4. Beta-Minus Decay (β⁻)

Beta-minus decay occurs in neutron-rich nuclei. A neutron inside the nucleus transforms into a proton, emitting an electron (the beta particle) and an electron antineutrino. The atomic number Z increases by 1 while the mass number A remains unchanged. The general equation is: ^A_ZX → ^A_{Z+1}Y + e⁻ + ν̄ₑ. A common example is carbon-14 dating: ^14_6C → ^14_7N + e⁻ + ν̄ₑ. Beta particles are more penetrating than alpha particles ： they can pass through a few millimetres of aluminium ： but are less ionising. The neutrino carries away some of the energy, explaining the continuous energy spectrum of beta particles that puzzled early physicists.

β⁻衰变发生在中子过多的核中。核内一个中子转变为质子，发射出一个电子（β粒子）和一个反电子中微子。原子序数Z增加1，而质量数A不变。通用方程为：^A_ZX → ^A_{Z+1}Y + e⁻ + ν̄ₑ。常见例子是碳-14测年法：^14_6C → ^14_7N + e⁻ + ν̄ₑ。β粒子比α粒子穿透力更强：可穿过几毫米铝片，但电离能力较弱。中微子带走部分能量，这解释了早期物理学家所困惑的β粒子连续能谱。

5. Beta-Plus Decay (β⁺) and Electron Capture

Beta-plus decay occurs in proton-rich nuclei where a proton transforms into a neutron, emitting a positron (the antiparticle of the electron) and an electron neutrino. The general equation is: ^A_ZX → ^A_{Z-1}Y + e⁺ + νₑ. An alternative process for proton-rich nuclei is electron capture, where the nucleus captures an inner-shell electron, combining it with a proton to form a neutron and emitting only an electron neutrino. Both processes reduce Z by 1 while keeping A constant. Beta-plus emitters are used in medical imaging, particularly in PET (Positron Emission Tomography) scans where the emitted positron annihilates with an electron to produce two gamma photons travelling in opposite directions.

β⁺衰变发生在质子过多的核中，一个质子转变为中子，发射出一个正电子（电子的反粒子）和一个电子中微子。通用方程为：^A_ZX → ^A_{Z-1}Y + e⁺ + νₑ。对于富质子核，另一种过程是电子俘获：原子核俘获一个内层电子，与质子结合形成中子，仅发射一个电子中微子。两种过程都使Z减少1而A不变。β⁺发射体用于医学成像，特别是PET（正电子发射断层扫描），其中发射的正电子与电子湮灭产生两个沿相反方向运动的光子。

6. Gamma Decay (γ)

Gamma decay is the emission of high-energy electromagnetic radiation from an excited nucleus. Unlike alpha and beta decay, gamma emission does not change the atomic number or mass number of the nucleus. After a nucleus undergoes alpha or beta decay, the daughter nucleus is often left in an excited state. It de-excites by emitting a gamma photon, carrying away the excess energy. The general notation is: ^A_ZX* → ^A_ZX + γ, where the asterisk denotes the excited state. Gamma rays are extremely penetrating ： several centimetres of lead or metres of concrete are required for effective shielding ： but they are the least ionising of the three types. Gamma sources such as cobalt-60 are used in radiotherapy for cancer treatment and in industrial radiography.

γ衰变是激发态原子核发射高能电磁辐射的过程。与α和β衰变不同，γ辐射不改变原子核的原子序数或质量数。原子核经历α或β衰变后，子核通常处于激发态。它通过发射γ光子退激，带走多余能量。通用记法为：^A_ZX* → ^A_ZX + γ，其中星号表示激发态。γ射线穿透性极强：需要几厘米铅或数米混凝土才能有效屏蔽，但电离能力最弱。钴-60等γ源用于癌症放疗和工业射线照相。

7. Half-Life and Decay Constant

The half-life T₁/₂ of a radioactive isotope is the time taken for half the nuclei in a sample to decay. It is related to the decay constant λ by the equation T₁/₂ = ln(2) / λ ≈ 0.693 / λ. The decay constant λ is the probability per unit time that a given nucleus will decay. Radioactive decay follows an exponential law: N = N₀e^{−λt}, where N is the number of undecayed nuclei at time t, and N₀ is the initial number. The activity A of a source, measured in becquerels (Bq), is defined as the number of decays per second: A = λN. Activity also decays exponentially: A = A₀e^{−λt}.

放射性同位素的半衰期T₁/₂是样本中半数原子核衰变所需的时间。它与衰变常数λ的关系为T₁/₂ = ln(2) / λ ≈ 0.693 / λ。衰变常数λ是单位时间内给定原子核衰变的概率。放射性衰变遵循指数规律：N = N₀e^{−λt}，其中N为时间t时未衰变的原子核数，N₀为初始数量。源的活度A以贝克勒尔（Bq）为单位，定义为每秒衰变次数：A = λN。活度也呈指数衰减：A = A₀e^{−λt}。

8. Half-Life Calculations (Worked Examples)

Example 1: A sample of iodine-131 has an initial activity of 800 Bq and a half-life of 8 days. Find the activity after 24 days. Solution: After 8 days, activity = 400 Bq; after 16 days, activity = 200 Bq; after 24 days, activity = 100 Bq. Alternatively, using the formula A = A₀e^{−λt}: λ = ln(2)/8 = 0.0866 per day; A = 800 × e^{−0.0866 × 24} = 800 × e^{−2.078} = 800 × 0.125 = 100 Bq. Example 2: Carbon-14 has a half-life of 5730 years. An archaeological sample has 25% of the original carbon-14 remaining. Find its age. Solution: 25% remaining means 2 half-lives have passed (100% = 50% = 25%), so age = 2 × 5730 = 11460 years.

例1：碘-131样本初始活度为800 Bq，半衰期为8天。求24天后的活度。解答：8天后，活度 = 400 Bq；16天后，活度 = 200 Bq；24天后，活度 = 100 Bq。或用公式A = A₀e^{−λt}：λ = ln(2)/8 = 0.0866/天；A = 800 × e^{−0.0866 × 24} = 800 × e^{−2.078} = 800 × 0.125 = 100 Bq。例2：碳-14半衰期为5730年。考古样本中碳-14残留量为原始的25%。求其年代。解答：25%残留意味着经过了2个半衰期（100% = 50% = 25%），因此年代 = 2 × 5730 = 11460年。

9. Nuclear Fission

Nuclear fission is the splitting of a heavy nucleus into two lighter fragments, accompanied by the release of several neutrons and a large amount of energy. The most common fissionable isotope is uranium-235. When a ^235U nucleus absorbs a thermal (slow) neutron, it becomes unstable and splits into two daughter nuclei, typically barium-141 and krypton-92, plus three neutrons: ^235_92U + ^1_0n → ^141_56Ba + ^92_36Kr + 3^1_0n. The energy released in fission appears as kinetic energy of the fission fragments and neutrons, and as gamma radiation. Fission is the basis of nuclear power reactors and atomic bombs. In a reactor, the chain reaction is controlled using control rods (typically boron or cadmium) that absorb excess neutrons, maintaining a steady rate of fission.

核裂变是一个重核分裂为两个较轻碎片的过程，同时释放出数个中子和大量能量。最常见的可裂变同位素是铀-235。当一个^235U核吸收一个热（慢）中子后，变得不稳定并分裂为两个子核，通常是钡-141和氪-92，加上三个中子：^235_92U + ^1_0n → ^141_56Ba + ^92_36Kr + 3^1_0n。裂变释放的能量表现为裂变碎片和中子的动能以及γ辐射。裂变是核电站和原子弹的基础。在反应堆中，使用控制棒（通常是硼或镉）吸收多余中子来控制链式反应，维持稳定的裂变速率。

10. Nuclear Fusion

Nuclear fusion is the process in which two light nuclei combine to form a heavier nucleus, releasing energy. This is the energy source of stars, including our Sun. The most important fusion reaction in stars is the proton-proton chain, where four protons ultimately fuse to form helium-4: 4p → ^4_2He + 2e⁺ + 2νₑ + energy. Another key reaction is deuterium-tritium fusion: ^2_1H + ^3_1H → ^4_2He + ^1_0n + 17.6 MeV. For fusion to occur, the reacting nuclei must overcome their mutual electrostatic repulsion (the Coulomb barrier), which requires extremely high temperatures ： on the order of millions of kelvin ： to give the nuclei sufficient kinetic energy. This is why fusion is called a thermonuclear reaction. Controlled fusion on Earth remains an active area of research, with ITER being the flagship international project aiming to demonstrate net energy gain from fusion.

核聚变是两个轻核结合形成较重核并释放能量的过程。这是包括我们太阳在内的恒星的能量来源。恒星中最重要的聚变反应是质子-质子链式反应，其中四个质子最终聚变形成氦-4：4p → ^4_2He + 2e⁺ + 2νₑ + 能量。另一个关键反应是氘-氚聚变：^2_1H + ^3_1H → ^4_2He + ^1_0n + 17.6 MeV。要发生聚变，反应核必须克服彼此间的静电排斥（库仑势垒），这需要极高的温度：数百万开尔文量级，以使核子具有足够的动能。这就是聚变被称为热核反应的原因。地球上的受控聚变仍是活跃的研究领域，ITER是旨在证明聚变净能量增益的标志性国际项目。

11. Binding Energy and Mass Defect

The mass of a nucleus is always less than the sum of the masses of its individual nucleons. This difference is called the mass defect Δm, and the equivalent energy ： the binding energy ： is given by Einstein’s mass-energy relation: E = Δm c². Binding energy represents the energy required to disassemble a nucleus into its constituent protons and neutrons. The average binding energy per nucleon peaks around iron-56 (A ≈ 56), which is the most stable nucleus. This explains why energy is released in both fission (splitting heavy nuclei into lighter ones, moving up the curve toward iron) and fusion (combining light nuclei into heavier ones, also moving toward iron).

原子核的质量始终小于其各个核子质量之和。这个差值称为质量亏损Δm，其等效能量即结合能，由爱因斯坦质能关系给出：E = Δm c²。结合能代表将原子核拆解为其组成质子和中子所需的能量。每个核子的平均结合能在铁-56（A ≈ 56）附近达到峰值，铁-56是最稳定的核。这解释了为什么裂变和聚变都释放能量：裂变将重核分裂为较轻的核，向铁的曲线方向移动；聚变将轻核结合成较重的核，同样向铁的方向移动。

12. Exam Tips for A-Level Nuclear Physics

When answering exam questions on nuclear physics, always write balanced nuclear equations showing both mass number and atomic number on each side. Remember that the total mass number A and total atomic number Z must be conserved. For half-life calculations, practise using both the exponential formula N = N₀e^{−λt} and the step-by-step halving method ： examiners often test both approaches. Be meticulous with units: activity is in becquerels (Bq, decays per second), half-life in appropriate time units (seconds, days, or years), and energy often in MeV. When comparing alpha, beta, and gamma radiation, use a structured approach covering nature, charge, penetration, ionisation, and deflection in electric and magnetic fields.

在回答核物理考试问题时，始终写出平衡的核方程，两边都标明质量数和原子序数。记住总质量数A和总原子序数Z必须守恒。对于半衰期计算，练习使用指数公式N = N₀e^{−λt}和逐步减半法：考官通常两种方法都考。注意单位：活度以贝克勒尔（Bq，每秒衰变次数）为单位，半衰期使用适当的时间单位（秒、天或年），能量通常以MeV为单位。比较α、β和γ辐射时，使用结构化方法涵盖性质、电荷、穿透力、电离能力以及在电场和磁场中的偏转。