IB Physics: Radioactive Decay Key Points | IB 物理：放射性衰变考点精讲

📚 IB Physics: Radioactive Decay Key Points | IB 物理：放射性衰变考点精讲

Radioactive decay is a spontaneous and random process in which an unstable atomic nucleus loses energy by emitting radiation. In the IB Physics syllabus, this topic forms the bridge between nuclear structure and the practical applications of radioactivity in medicine, energy, and dating. Understanding the types of decay, the mathematical description of exponential decay, and the concepts of half-life and activity are essential for both Standard and Higher Level students.

放射性衰变是一种自发且随机的过程，不稳定的原子核通过放出辐射来损失能量。在 IB 物理大纲中，这一主题是连接核结构与放射性在医学、能源和测年等实际应用之间的桥梁。理解衰变类型、指数衰减的数学描述、半衰期和活度等概念，对于标准水平和高水平学生都至关重要。

1. Nuclear Stability and the Reason for Decay | 核稳定性与衰变的原因

The stability of a nucleus depends on the balance between the strong nuclear force, which holds nucleons together, and the electrostatic repulsion between protons. For light nuclei (Z ≤ 20), the neutron-to-proton ratio (N/Z) close to 1 is stable. As Z increases, more neutrons are needed to counteract the growing Coulomb repulsion, so the stable N/Z ratio rises to about 1.5 for heavy nuclei. Nuclei that lie outside this band of stability undergo radioactive decay to move toward a more stable configuration.

原子核的稳定性取决于核子间强核力与质子间静电斥力之间的平衡。对于轻核（Z ≤ 20），中子‑质子比（N/Z）接近 1 时稳定。随着 Z 增大，需要更多的中子来抵消不断增大的库仑斥力，因此重核的稳定 N/Z 值上升到约 1.5。位于稳定带之外的原子核会通过放射性衰变趋向更稳定的结构。

2. Types of Radioactive Decay | 放射性衰变的类型

The IB syllabus requires knowledge of four main decay modes: alpha (α) decay, beta minus (β⁻) decay, beta plus (β⁺) decay, and gamma (γ) emission. Alpha decay occurs in heavy nuclei with too many nucleons; an alpha particle (⁴₂He nucleus) is ejected, reducing the mass number by 4 and the atomic number by 2. Beta minus decay involves the conversion of a neutron into a proton, emitting an electron (β⁻) and an antineutrino. This occurs in neutron-rich nuclei, increasing Z by 1 while A remains unchanged. Beta plus decay is the emission of a positron (β⁺) and a neutrino when a proton changes into a neutron; it occurs in proton-rich nuclei and decreases Z by 1. Gamma decay usually accompanies alpha or beta decay: the daughter nucleus releases excess energy as a high-energy photon without altering A or Z.

IB 大纲要求掌握四种主要衰变模式：α 衰变、β⁻ 衰变、β⁺ 衰变和 γ 辐射。α 衰变发生在核子过多的重核中；放出一个 α 粒子（⁴₂He 核），质量数减少 4，原子序数减少 2。β⁻ 衰变是中子转变为质子，放出一个电子（β⁻）和一个反中微子，发生在中子过剩的核中，Z 增加 1，A 不变。β⁺ 衰变是质子转变为中子时放出一个正电子（β⁺）和一个中微子，发生在质子过剩的核中，Z 减少 1。γ 衰变通常伴随 α 或 β 衰变：子核以高能光子的形式释放多余能量，不改变 A 或 Z。

3. Decay Equations and Conservation Laws | 衰变方程与守恒定律

When writing decay equations, the total mass number (A) and total atomic number (Z) must be conserved. For alpha decay: ^A_ZX → ^A‑4_Z‑2Y + ⁴₂He. For beta minus: ^A_ZX → ^A_Z+1Y + e⁻ + ν̅. For beta plus: ^A_ZX → ^A_Z‑1Y + e⁺ + ν. Charge, nucleon number, and lepton number are all conserved. In beta decay, the antineutrino or neutrino carries away some energy and momentum, explaining the continuous energy spectrum of beta particles.

书写衰变方程时，总质量数（A）和总原子序数（Z）必须守恒。α 衰变：^A_ZX → ^A‑4_Z‑2Y + ⁴₂He。β⁻ 衰变：^A_ZX → ^A_Z+1Y + e⁻ + ν̅。β⁺ 衰变：^A_ZX → ^A_Z‑1Y + e⁺ + ν。电荷、核子数和轻子数均守恒。在 β 衰变中，反中微子或中微子带走了部分能量和动量，这解释了 β 粒子的连续能谱。

4. The Random and Spontaneous Nature | 随机性与自发性

Radioactive decay is spontaneous – it cannot be triggered by changes in temperature, pressure, or chemical bonding. It is also random – for a given nucleus, the exact time of decay cannot be predicted. However, for a large number of identical nuclei, a predictable statistical pattern emerges. This is modelled using the decay constant λ, which represents the probability per unit time that a single nucleus will decay.

放射性衰变是自发的——它不能被温度、压力或化学键变化触发。它也是随机的——对于某一个原子核，无法预言其确切的衰变时刻。然而，对于大量相同的原子核，会出现可预测的统计规律。这可以用衰变常数 λ 来建模，λ 表示单个原子核在单位时间内衰变的概率。

5. Exponential Decay Law | 指数衰减规律

The number of undecayed nuclei N at time t follows the equation:

N = N₀ e⁻^λt

where N₀ is the initial number of nuclei and λ is the decay constant. This relationship arises because the rate of decay is proportional to the number present: dN/dt = –λN. The same exponential form describes the mass of a radioactive sample m = m₀ e⁻^λt and the activity A = A₀ e⁻^λt. Graphical analysis of ln N versus t yields a straight line with slope –λ, which is a common IB data‑analysis task.

在时间 t 未衰变的核数目 N 遵循方程：N = N₀ e⁻^λt，其中 N₀ 为初始核数目，λ 为衰变常数。这一关系源于衰变率与现有核数目成正比：dN/dt = –λN。相同的指数形式也用于描述放射性样品的质量 m = m₀ e⁻^λt 和活度 A = A₀ e⁻^λt。对 ln N 与 t 作图会得到一条斜率为 –λ 的直线，这是 IB 常见的数据分析任务。

6. Half-Life and Decay Constant | 半衰期与衰变常数

The half-life T₁/₂ is the time taken for half of the radioactive nuclei in a sample to decay. It is related to the decay constant by:

T₁/₂ = ln 2 / λ

This relationship is derived by setting N = N₀/2 and solving for t. Note that λ has units of s⁻¹, so T₁/₂ is in seconds. Knowing the half-life allows calculation of the fraction remaining after a given time: fraction = (1/2)ⁿ, where n = t / T₁/₂. It is critical to recognise that after n half-lives, the activity and mass also decrease by the same factor (1/2)ⁿ.

半衰期 T₁/₂ 是样品中一半放射性原子核发生衰变所需的时间。它与衰变常数的关系为：T₁/₂ = ln 2 / λ。这个关系是通过令 N = N₀/2 代入求解 t 得到的。注意 λ 的单位是 s⁻¹，因此 T₁/₂ 的单位是秒。知道了半衰期，就可以计算经过任意时间后剩余的比例：剩余比例 = (1/2)ⁿ，其中 n = t / T₁/₂。必须认识到，经过 n 个半衰期后，活度和质量也按相同的因子 (1/2)ⁿ 减小。

7. Activity and Its Measurement | 活度及其测量

Activity A is defined as the number of decays per unit time, measured in becquerels (Bq), where 1 Bq = 1 decay per second. The instantaneous activity is A = λN. This leads to the exponential relationship A = A₀ e⁻^λt. In experiments, a Geiger‑Müller tube or a scintillation counter is used to record the count rate. Because the measured count rate includes background radiation, the true activity is obtained by subtracting the background count rate from the observed rate. IB questions often involve correcting for background counts and then plotting corrected count‑rate data to determine half‑life.

活度 A 定义为每单位时间衰变的次数，单位是贝克勒尔（Bq），1 Bq = 每秒 1 次衰变。瞬时活度 A = λN。由此得出指数关系 A = A₀ e⁻^λt。实验中，使用盖革‑米勒管或闪烁计数器记录计数率。由于实测计数率包含本底辐射，真实的活度需要从观测值中减去本底计数率来得到。IB 考题常常要求进行本底修正，然后用修正后的计数率数据作图以确定半衰期。

8. Background Radiation and Corrections | 本底辐射与修正

Background radiation comes from cosmic rays, rocks (containing uranium, thorium), building materials, and even the human body. It must be measured before any experiment by running the detector with no radioactive source present, often for the same duration as the actual measurement. The corrected count rate is then: corrected rate = measured rate – background rate. The uncertainty in the corrected rate is calculated by adding the absolute uncertainties from the two measurements in quadrature, as both follow Poisson statistics.

本底辐射来自宇宙射线、岩石（含铀、钍）、建筑材料乃至人体。必须在实验前通过在没有放射源的情况下运行探测器来测量本底，通常测量时长与实际测量相同。修正后的计数率为：修正后计数率 = 实测计数率 – 本底计数率。修正后计数率的不确定度通过对两个测量的绝对不确定度进行平方和开根来计算，因为两者均遵循泊松统计。

9. Radioactive Dating – Carbon‑14 | 放射性测年——碳‑14

Carbon‑14 dating is a key application. Living organisms have a constant proportion of ¹⁴C to ¹²C because they constantly exchange carbon with the environment. After death, the ¹⁴C decays with T₁/₂ ≈ 5700 years. By measuring the remaining ¹⁴C activity per gram of carbon and comparing it with the activity in living tissue (about 0.23 Bq per gram), the time since death can be estimated. IB questions often require calculating the age using A = A₀ e⁻^λt or the half‑life method. Limitations include the need for calibration due to past variations in atmospheric ¹⁴C concentration and contamination of samples.

碳‑14 测年是一个关键应用。活着的生物体因与环境不断交换碳而保持 ¹⁴C 与 ¹²C 的恒定比例。生物死亡后，¹⁴C 以约 5700 年的半衰期衰减。通过测量每克碳中剩余的 ¹⁴C 活度，并与活体组织中的活度（约 0.23 Bq/g）比较，可以估算死亡时间。IB 考题常要求利用 A = A₀ e⁻^λt 或半衰期方法计算年龄。局限性包括因过去大气 ¹⁴C 浓度变化需要校准，以及样品污染问题。

10. Decay Chains and Secular Equilibrium (HL) | 衰变链与长期平衡（HL）

Many heavy nuclei undergo a series of decays before reaching a stable isotope. For instance, ²³⁸U decays through a chain that includes radium and radon, ending as ²⁰⁶Pb. In HL, students may encounter the concept of secular equilibrium: when a parent nuclide has a very long half‑life compared to its daughter, after several half‑lives of the daughter, the daughter’s activity becomes equal to that of the parent. This occurs because the rate at which the daughter is produced equals its own decay rate. Mathematically, A₂ ≈ A₁ if λ₁ ≪ λ₂ and sufficient time has passed.

许多重核在达到稳定同位素之前要经历一系列衰变。例如 ²³⁸U 通过包含镭和氡的链衰变，最终成为 ²⁰⁶Pb。在 HL 中，学生可能接触长期平衡的概念：当母核的半衰期远长于子核时，经过子核的几个半衰期后，子核的活度变得与母核相等。这是因为子核的生成速率等于其本身的衰变速率。数学上，若 λ₁ ≪ λ₂ 且经过足够时间，A₂ ≈ A₁。

Published by TutorHao | IB Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IB Physics: Radioactive Decay Key Points | IB 物理：放射性衰变 考点精讲