A-Level物理 核物理 放射性衰变 半衰期

A-Level物理 核物理 放射性衰变 半衰期

Introduction: The Atomic Nucleus

Nuclear physics is the branch of physics that studies the atomic nucleus: its constituents (protons and neutrons, collectively called nucleons), the forces that bind them together, and the transformations they undergo. Unlike the electrons that orbit the nucleus and determine chemical behaviour, nuclear processes involve energies millions of times greater and are governed by the strong nuclear force, one of the four fundamental forces of nature. Understanding the nucleus is essential not only for explaining the origin of the elements in stars but also for mastering topics like radioactive dating, nuclear medicine, and energy generation that appear regularly in A-Level exam questions. 核物理是研究原子核的物理学分支：研究其组成粒子（质子和中子，统称为核子）、将它们结合在一起的力以及它们发生的转变。与绕核运动并决定化学行为的电子不同，核过程涉及的能量高出数百万倍，并由强核力（自然界四种基本力之一）支配。理解原子核不仅对解释恒星中元素的起源至关重要，对掌握放射性测年、核医学和能源生成等在A-Level考试中经常出现的题目也必不可少。

The Nuclear Landscape: Key Parameters

Every nucleus is characterised by three numbers. The proton number Z (atomic number) determines the element: carbon is Z=6, uranium is Z=92. The neutron number N is the count of neutrons in the nucleus. The mass number A = Z + N gives the total number of nucleons. Isotopes are nuclei of the same element (same Z) with different numbers of neutrons, such as carbon-12 (6 protons, 6 neutrons) and carbon-14 (6 protons, 8 neutrons). Nuclide notation represents these as ^A_ZX, for example ^238_92U for uranium-238. The strong nuclear force acts between all nucleons and is responsible for holding the nucleus together against the enormous electrostatic repulsion between protons. This force is extremely short-ranged (effective only within about 1 femtometre), which explains why larger nuclei with many protons require proportionally more neutrons as “nuclear glue” to remain stable. 每个原子核由三个数字来表征。质子数Z（原子序数）决定元素种类：碳的Z=6，铀的Z=92。中子数N是原子核中的中子数量。质量数A=Z+N给出核子的总数。同位素是同一元素（相同Z）中中子数不同的原子核，例如碳-12（6个质子，6个中子）和碳-14（6个质子，8个中子）。核素符号将这些表示为^A_ZX，例如^238_92U表示铀-238。强核力作用于所有核子之间，负责将原子核结合在一起，抵抗质子间巨大的静电排斥力。这种力是极短程的（仅在约1飞米范围内有效），这解释了为什么具有许多质子的较大原子核需要比例更多的中子作为”核胶水”来保持稳定。

Alpha Decay: Heavy Nucleus Emission

Alpha decay occurs in heavy, neutron-rich nuclei where emitting an alpha particle (a helium-4 nucleus, ^4_2He, consisting of 2 protons and 2 neutrons) increases the overall stability of the daughter nucleus. The general equation is ^A_ZX→^{A-4}_{Z-2}Y + ^4_2α. A classic example is the decay of uranium-238: ^238_92U→^234_90Th + ^4_2α. The alpha particle is emitted with a discrete kinetic energy, typically 4-8 MeV, which produces a line spectrum rather than a continuous one. Because alpha particles are relatively heavy and doubly charged, they interact strongly with matter and have very low penetrating power: a sheet of paper or a few centimetres of air is sufficient to stop them. However, if an alpha-emitting substance is ingested or inhaled, the intense ionisation it causes in a small volume of tissue makes it extremely dangerous biologically. The daughter nucleus may itself be radioactive, initiating a decay chain that continues until a stable isotope is reached. Alpha衰变发生在重且富含中子的原子核中，发射一个α粒子（氦-4核^4_2He，由2个质子和2个中子组成）能提高子核的整体稳定性。一般方程为^A_ZX→^{A-4}_{Z-2}Y + ^4_2α。一个经典例子是铀-238的衰变：^238_92U→^234_90Th + ^4_2α。α粒子以离散动能发射，通常为4-8 MeV，产生线状谱而非连续谱。由于α粒子相对较重且带双电荷，它们与物质相互作用强烈，穿透能力极低：一张纸或几厘米空气就足以阻挡它们。然而，如果摄入或吸入α发射物质，它在一小体积组织中引起的强电离使其在生物学上极其危险。子核本身可能具有放射性，启动一个衰变链，直到达到稳定同位素为止。

Beta Decay: Transforming the Nucleus

Beta decay comes in two varieties, both transforming a nucleus by changing a neutron into a proton (or vice versa) while conserving the total number of nucleons. In beta-minus (β⁻) decay, a neutron in the nucleus transforms into a proton, emitting an electron (β⁻ particle) and an antineutrino: ^A_ZX→^A_{Z+1}Y + e⁻ + ν̄_e. This increases Z by 1 while keeping A constant. Carbon-14 dating relies on β⁻ decay: ^14_6C→^14_7N + e⁻ + ν̄_e, with a half-life of 5730 years. In beta-plus (β⁺) decay, a proton converts to a neutron, emitting a positron (β⁺ particle) and a neutrino: ^A_ZX→^A_{Z-1}Y + e⁺ + ν_e. Unlike alpha decay, beta particles are emitted with a continuous spectrum of kinetic energies up to a maximum value, because the available energy is shared between the beta particle and the (anti)neutrino. Beta particles are more penetrating than alpha particles (several millimetres of aluminium are needed to stop them) but less ionising per unit length. Beta衰变有两种类型，都通过将中子转变为质子（或反之）来转变原子核，同时保持核子总数不变。在β⁻衰变中，核内一个中子转变为质子，发射一个电子（β⁻粒子）和一个反中微子：^A_ZX→^A_{Z+1}Y + e⁻ + ν̄_e。这使得Z增加1而A保持不变。碳-14测年依赖于β⁻衰变：^14_6C→^14_7N + e⁻ + ν̄_e，半衰期为5730年。在β⁺衰变中，一个质子转化为中子，发射一个正电子（β⁺粒子）和一个中微子：^A_ZX→^A_{Z-1}Y + e⁺ + ν_e。与α衰变不同，β粒子以连续动能谱发射，最大值为一上限，因为可用能量在β粒子和（反）中微子之间分配。β粒子比α粒子更具穿透性（需要几毫米铝才能阻挡它们），但单位长度的电离能力较弱。

Gamma Decay and Excited States

Gamma decay is fundamentally different from alpha and beta decay: it does not change the composition of the nucleus. After an alpha or beta decay, the daughter nucleus is often left in an excited state. It releases this excess energy by emitting a high-energy photon (gamma ray), typically with energies from tens of keV to several MeV. The nuclear equation contains no change in A or Z: ^A_ZX^*→^A_ZX + γ. The asterisk denotes the excited nuclear state. Gamma rays have extremely high penetrating power: several centimetres of lead or metres of concrete are required for effective shielding. They are weakly ionising per unit length but can deposit energy deep within materials. Because gamma transitions occur between discrete nuclear energy levels, the emitted gamma rays have specific energies that serve as a fingerprint for identifying radioactive isotopes: a technique called gamma spectroscopy. Gamma衰变与α衰变和β衰变根本不同：它不改变原子核的组成。在α或β衰变后，子核通常处于激发态。它通过发射高能光子（γ射线）释放多余能量，能量从数十keV到数MeV不等。核方程中A和Z无变化：^A_ZX^*→^A_ZX + γ。星号表示核激发态。γ射线具有极高的穿透能力：需要数厘米铅或数米混凝土才能有效屏蔽。它们在单位长度上电离较弱，但能在材料深处沉积能量。由于γ跃迁发生在分立的核能级之间，发射的γ射线具有特定的能量，可作为识别放射性同位素的”指纹”：一种称为γ能谱的技术。

The Radioactive Decay Law

Radioactive decay is a random process at the level of individual nuclei: it is impossible to predict when any particular nucleus will decay. However, for a large collection of identical nuclei, the decay follows a precise statistical law. The activity A of a sample (the number of decays per second, measured in becquerels, where 1 Bq = 1 decay per second) is proportional to the number of undecayed nuclei N present: A = λN, where λ is the decay constant (unit: s⁻¹), a fixed probability of decay per nucleus per unit time. This leads to the exponential decay law: N(t) = N₀ e^{-λt}, and equivalently A(t) = A₀ e^{-λt}. The half-life t_{1/2} is the time required for half the nuclei in a sample to decay, related to the decay constant by t_{1/2} = ln 2 / λ ≈ 0.693 / λ. After n half-lives, the fraction remaining is (1/2)^n. 放射性衰变对单个原子核而言是随机过程：无法预测任何一个特定的原子核何时衰变。然而，对于大量相同的原子核，衰变遵循精确的统计规律。样品的活度A（每秒衰变次数，以贝克勒尔为单位，1 Bq = 每秒1次衰变）与存在的未衰变核数N成正比：A = λN，其中λ是衰变常数（单位：s⁻¹），是每个原子核每单位时间衰变的固定概率。这导出了指数衰变律：N(t) = N₀ e^{-λt}，等效地A(t) = A₀ e^{-λt}。半衰期t_{1/2}是样品中一半原子核衰变所需的时间，与衰变常数的关系为t_{1/2} = ln 2 / λ ≈ 0.693 / λ。经过n个半衰期后，剩余比例为(1/2)^n。

Half-Life Calculations and Carbon Dating

A typical A-Level calculation asks: “A sample of radioactive material has an initial activity of 800 Bq. After 45 minutes its activity is 100 Bq. Find the half-life.” Since 100/800 = 1/8 = (1/2)³, three half-lives have elapsed. Therefore t_{1/2} = 45/3 = 15 minutes. Alternatively, using the exponential law directly: λ = ln(800/100) / (45×60) = ln 8 / 2700 = 7.70×10⁻⁴ s⁻¹, giving t_{1/2} = ln 2 / λ = 900 s = 15 minutes. Carbon dating exploits the known ratio of radioactive ^14C to stable ^12C in the atmosphere. Living organisms maintain this equilibrium ratio through metabolic exchange, but upon death, ^14C intake stops and the existing ^14C decays with t_{1/2} = 5730 years. By measuring the current ^14C activity and comparing it to the activity of a living sample, the time since death can be calculated. This technique is valid for samples up to about 50000 years old, beyond which the remaining ^14C is too small to measure accurately. 一道典型的A-Level计算题问：”某放射性材料样品初始活度为800 Bq，45分钟后活度为100 Bq。求半衰期。”由于100/800 = 1/8 = (1/2)³，经过了三个半衰期。因此t_{1/2} = 45/3 = 15分钟。或者直接用指数律：λ = ln(800/100)/(45×60) = ln 8/2700 = 7.70×10⁻⁴ s⁻¹，得t_{1/2} = ln 2/λ = 900 s = 15分钟。碳测年利用大气中放射性^14C与稳定^12C的已知比值。活生物体通过代谢交换维持这一平衡比值，但死亡后^14C摄入停止，现有的^14C以t_{1/2} = 5730年的半衰期衰变。通过测量当前^14C活度并与活体样品活度比较，可计算死亡至今的时间。该技术适用于约50000年以内的样品，超过此时间的样品剩余^14C太少而无法准确测量。

Nuclear Fission: Splitting the Atom

Nuclear fission is the process in which a heavy, unstable nucleus splits into two (occasionally three) lighter daughter nuclei after absorbing a neutron. The most important fissionable isotopes are uranium-235 and plutonium-239. A typical fission reaction for ^235U is: ^235_92U + n→^141_56Ba + ^92_36Kr + 3n + energy. The key features are the release of 2-3 additional neutrons per fission (enabling a chain reaction), and the enormous energy release of about 200 MeV per fission event, primarily as kinetic energy of the fission fragments. The mass of the products is slightly less than the mass of the reactants, and this mass defect Δm (typically about 0.1% of the original mass) is converted to energy according to E = mc². In a nuclear reactor, the chain reaction is controlled using control rods (typically boron or cadmium) that absorb excess neutrons, and a moderator (water or graphite) that slows down fast neutrons to thermal energies so they are more likely to induce further fissions. 核裂变是重而不稳定的原子核在吸收一个中子后分裂成两个（偶尔三个）较轻子核的过程。最重要的可裂变同位素是铀-235和钚-239。^235U的典型裂变反应为：^235_92U + n→^141_56Ba + ^92_36Kr + 3n + 能量。关键特征包括每次裂变释放2-3个额外中子（使链式反应成为可能），以及每次裂变事件约200 MeV的巨大能量释放，主要以裂变碎片的动能形式释放。产物的质量略小于反应物的质量，这个质量亏损Δm（通常约为原始质量的0.1%）根据E = mc²转化为能量。在核反应堆中，链式反应通过控制棒（通常为硼或镉）吸收多余中子来加以控制，并由慢化剂（水或石墨）将快中子减速到热能，使它们更可能引发进一步的裂变。

Nuclear Fusion: Powering the Stars

Nuclear fusion is the process in which two light nuclei combine to form a heavier nucleus, releasing energy because the binding energy per nucleon increases for lighter nuclei up to iron-56. The most accessible fusion reaction on Earth combines deuterium and tritium (both hydrogen isotopes): ^2_1H + ^3_1H→^4_2He + n + 17.6 MeV. The Sun’s energy comes from the proton-proton chain, a series of fusion reactions that ultimately convert four protons into one helium-4 nucleus: 4p→^4He + 2e⁺ + 2ν_e + 26.7 MeV. For fusion to occur, the nuclei must overcome their mutual electrostatic repulsion, which requires temperatures of tens of millions of kelvin. This condition, called thermonuclear ignition, is achieved naturally in stellar cores. On Earth, achieving controlled fusion for power generation remains an unsolved engineering challenge, though experimental reactors like ITER and JET use magnetic confinement (tokamaks) and inertial confinement approaches to approach the conditions needed. 核聚变是两个轻原子核结合形成一个较重原子核的过程，释放能量是因为对轻核来说，每个核子的结合能随着核质量增加而增大，直至铁-56。地球上最容易实现的聚变反应结合氘和氚（均为氢的同位素）：^2_1H + ^3_1H→^4_2He + n + 17.6 MeV。太阳的能量来自质子-质子链，这是一系列聚变反应，最终将四个质子转化为一个氦-4核：4p→^4He + 2e⁺ + 2ν_e + 26.7 MeV。要使聚变发生，原子核必须克服它们之间的静电排斥力，这需要数千万开尔文的温度。这一条件称为热核点火，在恒星核心中自然实现。在地球上，实现用于发电的受控聚变仍是一项未解决的工程挑战，尽管像ITER和JET这样的实验反应堆使用磁约束（托卡马克）和惯性约束方法接近所需条件。

Applications: Medicine, Energy, and Dating

Nuclear physics has profound practical applications. In medicine, technetium-99m (t_{1/2} = 6 hours) is used extensively as a gamma-emitting tracer for imaging organs because its short half-life minimises patient radiation dose while providing sufficient activity for detection. Iodine-131 is used to treat thyroid cancer: the thyroid gland preferentially absorbs iodine, so the beta emissions from ^131I selectively destroy cancerous thyroid cells. In energy, nuclear power stations harness controlled fission to produce steam that drives turbines; a single kilogram of uranium-235 can release as much energy as 2500 tonnes of coal. Radioactive dating extends beyond carbon-14: potassium-40 (t_{1/2} = 1.25×10⁹ years) decaying to argon-40 is used to date rocks up to billions of years old, enabling the determination of the Earth’s age at approximately 4.54 billion years. Smoke detectors use a tiny amount of americium-241, whose alpha particles ionise the air between two electrodes, creating a small current that smoke particles disrupt to trigger the alarm. 核物理有着深远的实际应用。在医学中，锝-99m（t_{1/2} = 6小时）被广泛用作发射γ射线的器官成像示踪剂，因为其短半衰期在提供足够检测活度的同时最小化了患者辐射剂量。碘-131用于治疗甲状腺癌：甲状腺优先吸收碘，因此^131I的β辐射选择性地破坏癌性甲状腺细胞。在能源方面，核电站利用受控裂变产生蒸汽驱动汽轮机；一公斤铀-235能释放相当于2500吨煤的能量。放射性测年不仅限于碳-14：钾-40（t_{1/2} = 1.25×10⁹年）衰变为氩-40被用于测定高达数十亿年历史的岩石，使得确定地球年龄约为45.4亿年成为可能。烟雾探测器使用微量镅-241，其α粒子电离两电极间的空气，产生微弱电流，烟雾颗粒干扰该电流从而触发警报。

Exam Tips for A-Level Nuclear Physics

When answering nuclear physics questions, always write nuclide notation clearly and check that both mass number and proton number are conserved on each side of the equation. For half-life problems, if the activity reduces by a power-of-two fraction, identify the number of half-lives directly rather than calculating λ, which saves time and avoids arithmetic errors. Remember that activity and count rate are proportional (A ∝ C), so you can substitute count rates for activities in decay calculations. In fusion and fission questions, the energy released is always calculated from the mass defect: E = Δm × c², where Δm is in kilograms and c = 3.00×10⁸ m/s. If masses are given in atomic mass units (u), one u is equivalent to 931.5 MeV of energy. Common exam pitfalls include confusing the penetrating powers of alpha, beta, and gamma radiation, forgetting that gamma decay does not change A or Z, and misidentifying the role of the neutron as both the fission trigger and one of its products. 在回答核物理问题时，始终清晰地写出核素符号，并检查方程两侧的质量数和质子数是否守恒。对于半衰期问题，如果活度按2的幂次分数减小，直接确定半衰期个数而非计算λ，可节省时间并避免算术错误。记住活度和计数率成正比（A ∝ C），因此在衰变计算中可用计数率替代活度。在聚变和裂变问题中，释放的能量总是从质量亏损计算：E = Δm × c²，其中Δm以千克为单位，c = 3.00×10⁸ m/s。如果质量以原子质量单位（u）给出，1 u相当于931.5 MeV的能量。常见考试陷阱包括混淆α、β和γ辐射的穿透能力，忘记γ衰变不改变A或Z，以及将中子的角色错误地同时认定为裂变触发者和产物之一。