📚 E.3 Radioactive decay | E.3 放射性衰变
In the microscopic world of nuclei, stability is a delicate balance. Many naturally occurring and man-made isotopes are unstable and undergo spontaneous transformations known as radioactive decay. This process, governed entirely by quantum mechanics and the weak and strong nuclear forces, has profound consequences for matter, energy, and even our understanding of time itself. In IB Physics Topic E.3, we explore the fundamental laws that describe how nuclei disintegrate, the types of radiation emitted, and how we can use these phenomena as clocks to probe history.
在微观的原子核世界里，稳定性是一种微妙的平衡。许多天然存在及人工合成的同位素并不稳定，它们会经历一种自发的转变，即放射性衰变。这一过程完全由量子力学以及弱核力和强核力支配，对物质、能量乃至我们对时间本身的理解都有着深远的影响。在IB 物理 E.3 章节中，我们将探索描述原子核蜕变的基本定律，释放的辐射种类，以及如何利用这些现象作为时钟去探寻历史。
1. The Nature of Radioactive Decay | 放射性衰变的本质
Radioactive decay is a completely spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. The decay occurs without any external influence and is impossible to predict for a single nucleus; we can only specify the probability that a given nucleus will decay in a certain time interval. This intrinsic randomness is a signature of quantum phenomena and distinguishes nuclear decay from deterministic classical processes.
放射性衰变是一个完全自发的过程，不稳定的原子核通过释放辐射而损失能量。衰变的发生不受任何外部影响，且对单个原子核而言是无法预测的；我们只能给出某个核在一段时间内发生衰变的概率。这种内禀的随机性是量子现象的典型特征，并将核衰变与决定论的经典过程区别开来。
When a nucleus decays, it transforms into a different nuclide, often emitting particles such as alpha particles, beta particles, or gamma photons. The parent nuclide disappears while a daughter nuclide appears, and the total energy is conserved, though some mass is converted into kinetic energy of the products or into photon energy.
当原子核衰变时，它会转变为不同的核素，常常发射α粒子、β粒子或γ光子等。母核消失，子核出现，同时总能量守恒，尽管有一部分质量转化为产物的动能或光子的能量。
2. Decay Constant and Activity | 衰变常数与活度
Although individual decays are random, a large collection of identical unstable nuclei exhibits a well-defined statistical behaviour. The probability per unit time that a nucleus will decay is called the decay constant, denoted by the symbol λ (lambda). Its SI unit is s⁻¹. A large λ means the isotope decays rapidly; a small λ indicates a slow decay rate.
尽管单个衰变是随机的，但大量相同的放射性核会表现出明确的统计行为。一个核在单位时间内发生衰变的概率称为衰变常数，用符号 λ 表示，其国际单位是 s⁻¹。λ 很大意味着该同位素衰变很快；λ 很小则表示衰变速率慢。
The activity = λN. The unit of activity is the becquerel (Bq), where 1 Bq equals one decay per second.
放射性样品的活度 A 是单位时间内发生的衰变次数。它与该时刻尚未衰变的核数目 N 成正比：A = λN。活度的单位是贝克勒尔 (Bq)，1 Bq 等于每秒一次衰变。
Activity is not a constant over time; as the number of parent nuclei decreases, the activity decreases in exactly the same exponential manner. Therefore, measuring activity provides a way to monitor the decay process without needing to count the actual number of nuclei.
活度并不是恒定的；随着母核数目的减少，活度也以完全相同的指数方式减小。因此，测量活度便提供了一种监测衰变过程的方法，而无需计数实际的核数目。
3. The Exponential Decay Law | 指数衰变定律
The fact that each nucleus has the same constant probability of decay per unit time leads directly to an exponential decrease in the number of parent nuclei N over time t. The governing equation is
N = N₀ exp(−λt)
where N₀ is the original number of undecayed nuclei at t = 0. This is called the exponential decay law and is one of the most important quantitative relationships in nuclear physics.
每个核在单位时间内具有恒定的衰变概率，这一事实直接导致母核数目 N 随时间 t 呈指数减少。其控制方程为
N = N₀ exp(−λt)
其中 N₀ 是 t = 0 时刻尚未衰变的核的初始数目。这称为指数衰变定律，是核物理中最重要的定量关系之一。
exp(−λt), and to the mass of the radioactive isotope in the sample. This universality makes the mathematics of decay very straightforward once the decay constant is known.
该定律可由微分方程 dN/dt = −λN 导出。同样的指数形式也适用于活度：A = A₀ exp(−λt)，以及样品中放射性同位素的质量。这种普适性使得一旦知道了衰变常数，衰变的数学处理就变得十分直观。
4. Half-Life | 半衰期
The half-life, T₁/₂, is the time required for half of the radioactive nuclei in a sample to decay. It is related to the decay constant by
T₁/₂ = ln 2 / λ
/2. After n half-lives, the fraction remaining is (1/2)ⁿ. Half-lives range from fractions of a second to billions of years, making this concept essential for comparing the stability of different isotopes.
半衰期 T₁/₂ 是指样品中一半的放射性核发生衰变所需要的时间。它与衰变常数的关系为
T₁/₂ = ln 2 / λ
经过一个半衰期后，N = N₀/2 且 A = A₀/2。经过 n 个半衰期后，剩余的份额为 (1/2)ⁿ。半衰期从几分之一秒到数十亿年不等，这使得该概念对于比较不同同位素的稳定性至关重要。
Experimentally, the half-life can be determined from the slope of a graph of ln N versus t, which is a straight line of gradient −λ. Alternatively, the decay curve can be analysed directly. The knowledge of half-lives underpins techniques such as radioactive dating and medical tracer protocols.
在实验中，半衰期可以通过绘制 ln N 对 t 图的斜率来确定，该图是一条斜率为 −λ 的直线。或者也可以直接分析衰变曲线。半衰期的知识为放射性定年和医用示踪剂方案等技术奠定了基础。
5. Alpha Decay | α 衰变
In alpha decay, an unstable nucleus emits an alpha particle, which is a helium-4 nucleus consisting of two protons and two neutrons. The general equation is
ᴬZX → ᴬ⁻⁴Z₋₂Y + ⁴₂He
where X is the parent nucleus and Y is the daughter. Alpha decay typically occurs in heavy nuclei with atomic number Z greater than 82, such as uranium and radium. The emitted alpha particle has a discrete kinetic energy, typically a few MeV, because the decay energy is shared in a fixed ratio between the daughter nucleus and the alpha particle.
在α 衰变中，不稳定的核发射一个 α 粒子，即由两个质子和两个中子组成的氦-4 核。一般方程为
ᴬZX → ᴬ⁻⁴Z₋₂Y + ⁴₂He
其中 X 是母核，Y 是子核。α 衰变通常发生在原子序数 Z 大于 82 的重核中，例如铀和镭。发射的 α 粒子具有分立的动能，通常为几个 MeV，因为衰变能量以固定的比例在子核和 α 粒子之间分配。
Alpha particles are highly ionising but have a very short range in matter; they can be stopped by a sheet of paper. Because of their ability to cause intense local ionisation, alpha emitters are hazardous if ingested, yet their external hazard is minimal.
α 粒子具有很强的电离能力，但在物质中的射程极短；一张纸便可以阻挡它们。由于 α 粒子能够引起强烈的局部电离，吸入或摄入 α 发射体是非常危险的，但它们的外部危害却极小。
6. Beta Decay and the Neutrino | β 衰变与中微子
Beta decay is the process by which a nucleus changes its proton-to-neutron ratio via the weak interaction. There are two common types: β⁻ decay, where a neutron converts into a proton with the emission of an electron and an antineutrino; and β⁺ decay, where a proton converts into a neutron with the emission of a positron and a neutrino.
β衰变是原子核通过弱相互作用改变其质子与中子之比的过程。常见的两种类型是：β⁻ 衰变，中子转化为质子并发射一个电子和一个反中微子；β⁺ 衰变，质子转化为中子并发射一个正电子和一个中微子。
The β⁻ decay equation for carbon-14 is: ¹⁴₆C → ¹⁴₇N + e⁻ + ν̄ₑ . In β⁺ decay, an example is: ¹¹₆C → ¹¹₅B + e⁺ + νₑ. The symbol ν̄ d νₑ the electron neutrino, nearly massless particles that carry away energy and momentum.
碳-14 的 β⁻ 衰变方程为：¹⁴₆C → ¹⁴₇N + e⁻ + ν̄ₑ。β⁺ 衰变的一个例子是：¹¹₆C → ¹¹₅B + e⁺ + νₑ。符号 ν̄ₑ 代表电子反中微子，νₑ 代表电子中微子，它们是几乎无质量的粒子，带走能量和动量。
The beta particle spectrum is continuous, which historically implied the existence of the neutrino, as energy and momentum appeared not to be conserved in two-body decay. Wolfgang Pauli postulated the neutrino in 1930, and it was later confirmed experimentally. The antineutrino is essential for balancing lepton number in β⁻ decay.
β 粒子的能谱是连续的，这在历史上暗示了中微子的存在，因为在二体衰变中能量和动量似乎不守恒。沃尔夫冈·泡利于 1930 年假设了中微子，随后被实验证实。在 β⁻ 衰变中，反中微子对于平衡轻子数是必不可少的。
7. Gamma Decay and Isomeric Transitions | γ 衰变与同质异能跃迁
Gamma decay occurs when an excited nucleus releases its surplus energy by emitting a high-energy photon, usually following an alpha or beta decay that left the daughter in an excited state. The nucleons rearrange themselves into a lower energy configuration, and the emitted gamma ray has a well-defined energy equal to the difference between nuclear energy levels.
γ 衰变发生在激发态的原子核通过发射高能光子来释放多余能量时，通常紧随 α 或 β 衰变之后，因为衰变使子核处于激发态。核子重新排布到较低能级的组态，发射的 γ 射线具有明确的能量，等于核能级间的差值。
decay of metastable technetium-99m, widely used in medical imaging: ⁹⁹ᵐTc → ⁹⁹Tc + γ. The 'm' harge and has very high penetrating power; thick lead or concrete is needed for shielding.
一个例子是亚稳态锝-99m 的衰变，它广泛用于医学成像：⁹⁹ᵐTc → ⁹⁹Tc + γ。符号“m”表示一种具有可测半衰期的核同质异能素。γ 光子不带电，具有极高的穿透能力；需要厚铅板或混凝土进行屏蔽。
Gamma emission does not change the composition of the nucleus in terms of nucleon numbers, so the element and isotope remain the same before and after the transition. This contrasts with alpha and beta decays which transmute the element.
γ 放射并不改变核子数方面的核组成，因此在跃迁前后元素和同位素保持不变。这与能够嬗变元素的 α 和 β 衰变形成对比。
8. Radioactive Decay Series | 放射性衰变系列
Many heavy nuclei decay through a chain of successive transformations until a stable nuclide is reached. Such a sequence is called a radioactive decay series. There are four such series found in nature: the thorium series (4n), neptunium series (4n+1, now extinct except for traces produced artificially), uranium series (4n+2), and actinium series (4n+3).
许多重核通过一连串的连续衰变最终达到稳定核素，这样的序列称为放射性衰变系列。自然界中存在四个这样的系列：钍系 (4n)、镎系 (4n+1，现已绝迹，只有人工产生的微量存在)、铀系 (4n+2) 和锕系 (4n+3)。
Each series involves a mixture of alpha and beta decays, often with accompanying gamma radiation. For instance, uranium-238 decays in 14 steps to become stable lead-206. The intermediate decay products include radium and radon, which have environmental significance.
每个系列都包含 α 和 β 衰变的组合，通常伴有 γ 辐射。例如，铀-238 经过 14 步衰变变成稳定的铅-206。中间的衰变产物包括镭和氡，它们具有环境重要性。
Understanding these series is crucial for managing nuclear waste and for radiometric dating of geological materials. The secular equilibrium that can develop between parent and daughter activities is an important concept in applied nuclear physics.
理解这些系列对于核废料管理和地质材料的放射性测年是至关重要的。在母核与子核活度之间可能建立的长期平衡是应用核物理学中的一个重要概念。
9. Radiocarbon Dating | 放射性碳定年
Radiocarbon dating exploits the β⁻ decay of carbon-14 (¹⁴C, half‑ e of organic materials. Cosmic rays produce ¹⁴C in the atmosphere, where it combines with oxygen to form CO₂ and is incorporated into living organisms via photosynthesis. While an organism is alive, the ratio of ¹⁴C to stable ¹²C remains roughly constant, matching the atmospheric ratio.
放射性碳定年利用碳-14 (¹⁴C，半衰期约 5730 年) 的 β⁻ 衰变来确定有机材料的年龄。宇宙射线在大气中产生 ¹⁴C，它与氧结合形成 CO₂，并通过光合作用进入生物体。当生物体活着时，其体内 ¹⁴C 与稳定 ¹²C 的比例基本保持恒定，与大气中的比例一致。
When the organism dies, it stops exchanging carbon with the environment, and the ¹⁴C within its remains decays exponentially. By measuring the remaining ¹⁴C activity (or the ¹⁴C/¹²C ratio using accelerator mass spectrometry), the time elapsed since death can be calculated from the exponential decay formula, provided one knows the original activity.
当生物体死亡后，它停止与环境交换碳，其遗骸中的 ¹⁴C 便以指数方式衰变。通过测量剩余的 ¹⁴C 活度（或使用加速器质谱法测量 ¹⁴C/¹²C 的比例），只要知道初始活度，便可利用指数衰变公式计算出从死亡以来经过的时间。
Radiocarbon dating is valid for ages up to about 50,000 years. Corrections are needed for variations in atmospheric ¹⁴C over time, which are calibrated using tree rings and other archives. This technique has revolutionised archaeology, palaeontology, and climate science.
放射性碳定年对于约 5 万年以内的年龄有效。需对大气中 ¹⁴C 随时间的变化进行校正，这些校正是利用树木年轮及其他档案进行校准的。这一技术彻底改变了考古学、古生物学和气候科学。
10. Summary and Conceptual Connections | 总结与概念联系
The law of radioactive decay, with its characteristic exponential form, brings together statistical physics, quantum mechanics, and conservation principles. Decay constants and half-lives are immutable fingerprints of unstable isotopes, unaffected by temperature, pressure, or chemical environment.
放射性衰变定律以其特有的指数形式将统计物理、量子力学和守恒原理结合在一起。衰变常数和半衰期是不稳定同位素不可改变的特征指纹，不受温度、压力或化学环境的影响。
The three principal emissions—alpha, beta, and gamma—illustrate the different ways a nucleus can shed excess energy and move towards stability, each governed by distinct interactions. The requirement for the neutrino in beta decay highlights how fundamental conservation laws (energy, momentum, and lepton number) guide the development of particle physics.
三种主要的辐射——α、β 和 γ——说明了原子核释放多余能量并趋于稳定的不同方式，每种方式都由不同的相互作用所支配。β 衰变中中微子的必要性突出显示了基本的守恒定律（能量、动量和轻子数）如何指导粒子物理学的发展。
These ideas are not only indispensable for IB Physics examinations but also provide the intellectual foundation for nuclear energy, medical imaging, and dating the Earth's history. Mastery of the exponential decay mathematics and an appreciation of the underlying randomness will serve students well in many areas of science.
这些概念不仅是 IB 物理考试中不可或缺的内容，也为核能、医学成像和测年地球历史提供了智力基础。掌握指数衰变的数学并理解其背后的随机性将使学生在许多科学领域中受益匪浅。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图