IB Physics: The Laws and Applications of Radioactive Decay | IB物理：放射性衰变的规律与应用

📚 IB Physics: The Laws and Applications of Radioactive Decay | IB物理：放射性衰变的规律与应用

Radioactive decay is one of the most important topics in IB Physics because it connects quantum mechanics, nuclear structure, and practical technology. The behaviour of unstable nuclei follows a beautifully simple exponential law, yet the process itself is fundamentally random. This article introduces the key equations, decay modes, half-life calculations, and the real-world uses that you need for revision.

放射性衰变是IB物理中最重要的主题之一，因为它将量子力学、核结构和实际应用连接在一起。不稳定原子核的行为遵循一条简洁的指数规律，但这一过程本身在本质上是随机的。本文将介绍关键方程、衰变模式、半衰期计算以及备考所需的实际应用。

1. The Random and Spontaneous Nature of Decay | 衰变的随机性与自发性

Radioactive decay is a spontaneous process: an unstable nucleus emits radiation without any external trigger, because it is trying to reach a more stable energy state. It is also a random process: it is impossible to predict exactly which nucleus will decay next, or when it will decay.

放射性衰变是一个自发过程：不稳定的原子核无需外部触发就会发射辐射，因为它试图达到更稳定的能量状态。它同时也是一个随机过程：无法准确预测下一个会衰变的是哪一个原子核，也无法预测它何时衰变。

Although individual decays are unpredictable, the behaviour of a very large number of nuclei is statistically regular. This allows us to define a decay constant λ, which is the probability that a single nucleus will decay per unit time.

虽然单个衰变不可预测，但大量原子核的行为在统计上是规则的。因此我们可以定义衰变常数 λ，即单个原子核在单位时间内发生衰变的概率。

dN/dt = −λN

Here N is the number of undecayed nuclei remaining, and the negative sign shows that N decreases with time. The minus sign must not be forgotten when you write the differential form.

其中 N 是尚未衰变的原子核数，负号表示 N 随时间减少。在写微分形式时，一定不要遗漏负号。

2. Alpha, Beta and Gamma Emissions | α、β、γ 三种辐射

Alpha (α) radiation is a stream of helium-4 nuclei, written as ⁴₂He. When a nucleus emits an alpha particle, its mass number decreases by 4 and its atomic number decreases by 2.

α 辐射是氦-4原子核流，写作 ⁴₂He。当原子核发射一个α粒子时，其质量数减少4，原子序数减少2。

²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He

Beta-minus (β⁻) decay involves a neutron changing into a proton, an electron and an antineutrino. The mass number stays the same, but the atomic number increases by 1. For example, carbon-14 decays to nitrogen-14.

β⁻ 衰变涉及一个中子转变为质子，同时释放一个电子和一个反中微子。质量数不变，原子序数增加1。例如，碳-14衰变为氮-14。

¹⁴₆C → ¹⁴₇N + ⁰₋₁e + ν̄

Gamma (γ) radiation is high-energy electromagnetic radiation. It usually accompanies alpha or beta decay when the daughter nucleus is left in an excited state. Gamma emission changes neither the mass number nor the atomic number.

γ 辐射是高能电磁波。当子核处于激发态时，γ辐射通常伴随α或β衰变出现。γ发射既不改变质量数，也不改变原子序数。

Alpha particles are highly ionising but weakly penetrating. They can be stopped by a sheet of paper or a few centimetres of air.

α粒子电离能力很强，但穿透能力很弱，一张纸或几厘米空气就能阻挡。
Beta particles are less ionising than alpha particles but more penetrating. A few millimetres of aluminium are usually enough to stop them.

β粒子的电离能力比α粒子弱，但穿透能力更强，通常几毫米厚的铝就能阻挡。
Gamma rays are weakly ionising but highly penetrating. Dense materials such as lead or several centimetres of concrete are needed to reduce their intensity.

γ射线电离能力弱，但穿透能力很强，需要铅等致密材料或数厘米厚的混凝土来减弱其强度。

3. The Exponential Law of Decay | 指数衰变定律

Solving the differential equation dN/dt = −λN gives the exponential decay law for the number of undecayed nuclei:

求解微分方程 dN/dt = −λN 可得到未衰变原子核数的指数衰变定律：

N = N₀ e^(−λt)

Here N₀ is the initial number of nuclei, λ is the decay constant, and t is the elapsed time. The activity A is the number of decays per second, measured in becquerel (Bq), where 1 Bq = 1 s⁻¹. Since A = λN, the activity also decays exponentially:

其中 N₀ 是初始原子核数，λ 是衰变常数，t 是经过的时间。活度 A 是每秒发生的衰变次数，单位为贝克勒尔（Bq），1 Bq = 1 s⁻¹。由于 A = λN，活度也按指数规律衰变：

A = A₀ e^(−λt)

An exponential curve approaches zero but never actually reaches zero. In practice, after enough half-lives the remaining amount becomes so small that it is undetectable.

指数曲线会趋近于零，但永远不会真正等于零。实际上，经过足够多的半衰期后，剩余量会小到无法探测。

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

The half-life T½ is the time required for half of the original nuclei to decay, or equivalently, the time for the activity to fall to half of its initial value. The half-life is related to the decay constant by:

半衰期 T½ 是初始原子核数的一半发生衰变所需的时间，也等于活度降至初始值一半所需的时间。半衰期与衰变常数的关系为：

T½ = ln 2 / λ ≈ 0.693 / λ

For example, if a sample has a half-life of 5.0 years, its decay constant is λ = 0.693 / 5.0 = 0.139 yr⁻¹. After 15 years, which is three half-lives, only one-eighth of the original nuclei remain.

例如，如果某样品的半衰期为5.0年，则其衰变常数为 λ = 0.693 / 5.0 = 0.139 yr⁻¹。15年后，即3个半衰期后，只剩下原来原子核数的八分之一。

Number of half-lives n	Fraction remaining	Percentage remaining
0	1	100%
1	1/2	50%
2	1/4	25%
3	1/8	12.5%
10	1/1024	≈0.1%

A useful equivalent form is N = N₀(1/2)ⁿ, where n = t/T½ is the number of half-lives that have elapsed. This is often quicker than using the exponential form in simple problems.

一个有用的等价形式是 N = N₀(1/2)ⁿ，其中 n = t/T½ 是经过的半衰期个数。在简单问题中，这通常比使用指数形式更快。

5. Measuring Half-Life | 测量半衰期

For a short-half-life source, place a sample near a Geiger-Müller tube and record the count rate at regular time intervals. Before taking measurements, always measure and subtract the background radiation count.

对于短半衰期放射源，可将样品放在盖革-米勒管附近，每隔一定时间记录计数率。在测量之前，务必先测量本底辐射并扣除。

Because a detector does not catch every decay, the observed count rate is usually less than the true activity. However, as long as the detection efficiency is constant, the count rate is proportional to activity and follows the same exponential decay.

由于探测器无法捕捉到每一次衰变，观察到的计数率通常小于真实活度。但只要探测效率恒定，计数率就与活度成正比，并遵循相同的指数衰减规律。

For a long half-life, the decay over a short observation period is too small to measure directly. Instead, you can measure the current activity A and estimate the number N of nuclei from the mass of the sample, then use λ = A/N.

对于长半衰期样品，在较短观测时间内衰变量太小，无法直接测量。此时可以测量当前活度 A，并根据样品质量估算原子核数 N，然后利用 λ = A/N 计算。

Another graphical method is to plot ln A against t. The result is a straight line with slope −λ, so the decay constant can be found from the gradient.

另一种作图方法是将 ln A 对 t 作图，得到一条斜率为 −λ 的直线，从而求出衰变常数。

6. Decay Chains and Radioactive Equilibrium | 衰变链与放射性平衡

Many radioactive nuclei do not decay directly to a stable isotope. They form a decay chain, passing through a series of daughter nuclei until a stable end product is reached. The well-known uranium-238 series ends with stable lead-206.

许多放射性原子核并非直接衰变为稳定同位素，而是形成衰变链，经过一系列子核直到达到稳定的最终产物。著名的铀-238衰变链最终是稳定的铅-206。

In a decay chain, if the parent has a very long half-life, the activity of some shorter-lived daughters can become almost equal to the parent activity over time. This condition is called secular equilibrium, and it is important in mineral dating and environmental monitoring.

在衰变链中，如果母核半衰期很长，经过足够时间，某些短寿命子核的活度几乎等于母核活度。这种状态称为长期平衡，它在矿物定年和环境监测中非常重要。

For IB problems, you usually only consider a single decay step. But understanding chains explains why old uranium ores contain many different radioactive isotopes, not just uranium.

在IB题目中，通常只考虑单步衰变。但理解衰变链可以解释为什么古老的铀矿石中含有多种不同的放射性同位素，而不只是铀。

7. Radiocarbon Dating | 放射性碳定年法

Carbon-14 is continuously produced in the upper atmosphere by cosmic-ray neutrons reacting with nitrogen-14. Living organisms absorb carbon-14 as carbon dioxide through photosynthesis and the food chain, so the ratio of carbon-14 to carbon-12 in a living organism stays roughly constant.

碳-14由宇宙射线中的中子与大气上层氮-14反应不断产生。生物体通过光合作用和食物链以二氧化碳形式吸收碳-14，因此活生物体内碳-14与碳-12的比值大致保持不变。

When an organism dies, it stops exchanging carbon with the environment. The carbon-14 it contains decays with a half-life of about 5730 years, so its activity decreases. By comparing the remaining carbon-14 activity with that in a living sample, the time since death can be found:

当生物死亡后，它不再与外界交换碳。其体内的碳-14以约5730年的半衰期衰变，因此活度降低。通过比较剩余碳-14活度与活体样品中的活度，即可算出死亡至今的时间：

t = (1/λ) ln(A₀/A)

For older objects, carbon-14 dating is not practical because the remaining activity becomes too small. Instead, methods such as potassium-argon dating are used for geological samples, because potassium-40 has a half-life of about 1.25 billion years.

对于更古老的物体，碳-14定年法不再适用，因为剩余活度太小。此时可采用钾-氩定年法等地学方法，因为钾-40的半衰期约为12.5亿年。

8. Medical Applications | 医学应用

Radioactive tracers are used to diagnose medical conditions. A small amount of a radioactive isotope is injected into the body, and its progress is followed using a gamma camera. The isotope must have a short half-life so that the patient is not exposed to radiation for a long time.

放射性示踪剂用于医学诊断。将少量放射性同位素注入体内，然后用γ相机追踪其运动过程。同位素必须具有短半衰期，以免患者长时间受到辐射。

Technetium-99m has a half-life of about 6 hours and emits gamma rays ideal for imaging. It is widely used for bone, heart and brain scans.

锝-99m的半衰期约为6小时，发射适用于成像的γ射线，广泛用于骨骼、心脏和脑部扫描。
Iodine-131 is absorbed by the thyroid gland. It can be used to diagnose and treat thyroid conditions such as hyperthyroidism.

碘-131会被甲状腺吸收，可用于诊断和治疗甲状腺功能亢进等疾病。
In radiotherapy, focused beams of gamma radiation from sources such as cobalt-60 are used to destroy cancerous tissue. Rapidly dividing cells are especially sensitive to radiation.

在放射治疗中，来自钴-60等放射源的γ射线束被用来摧毁癌变组织。快速分裂的细胞对辐射尤其敏感。

9. Industrial and Archaeological Applications | 工业与考古应用

Smoke detectors contain a small americium-241 source that emits alpha particles. The alpha particles ionise the air in a detection chamber, creating a small electric current. If smoke enters the chamber, it absorbs the alpha particles and reduces the current, triggering the alarm.

烟雾报警器中含有少量镅-241α放射源。α粒子使探测室中的空气电离，产生微弱电流。当烟雾进入探测室时，会吸收α粒子并减弱电流，从而触发警报。

Thickness gauges use the absorption of beta or gamma radiation to measure the thickness of paper, plastic or metal sheets during production. As the material becomes thicker, more radiation is absorbed, allowing a detector to send a feedback signal to adjust the rollers.

厚度计利用β或γ辐射的吸收情况来测量纸张、塑料或金属板材在生产过程中的厚度。材料越厚，吸收的辐射越多，探测器可向轧辊发送反馈信号进行调节。

In archaeology and geology, the same exponential decay law is used to date samples. Radiocarbon dating is ideal for once-living materials up to about 50,000 years old, while uranium-lead and potassium-argon methods are used for much older rocks.

在考古学和地质学中，相同的指数衰变规律被用于样品定年。碳-14定年适用于约5万年以内的生物材料，而铀-铅法和钾-氩法适用于更古老的岩石。

10. Radiation Safety and Biological Effects | 辐射安全与生物效应

Ionising radiation can remove electrons from atoms and damage DNA. High doses can kill cells, while lower doses increase the long-term risk of cancer. Because alpha particles deposit their energy over a very short distance, they are especially damaging if ingested or inhaled.

电离辐射能使原子失去电子并损伤DNA。高剂量可杀死细胞，低剂量则增加长期患癌风险。由于α粒子在很短距离内释放能量，如果被摄入或吸入，其危害尤其大。

The absorbed dose D is defined as the energy absorbed per unit mass:

吸收剂量 D 的定义是单位质量吸收的能量：

D = E / m

The unit of absorbed dose is the gray (Gy), where 1 Gy = 1 J kg⁻¹. However, different types of radiation have different biological effects, so physicists also define the equivalent dose H = wR × D, measured in sieverts (Sv). For gamma and beta radiation, wR = 1; for alpha radiation, wR = 20.

吸收剂量的单位是戈瑞（Gy），1 Gy = 1 J kg⁻¹。但不同辐射的生物效应不同，因此物理学家还定义等效剂量 H = wR × D，单位为希沃特（Sv）。对于γ和β辐射，wR = 1；对于α辐射，wR = 20。

Practical safety measures include minimising time near a source, maximising distance, using appropriate shielding, and handling sources with tongs behind lead screens. Monitoring devices such as film badges are used by workers to track accumulated exposure.

实际安全措施包括缩短在放射源附近的时间、尽量加大距离、使用适当屏蔽，并使用长柄工具和铅屏操作。工作人员还会佩戴胶片剂量计等监测设备来记录累积照射量。

11. Common IB Exam Pitfalls and Tips | 常见IB考试误区与提示

Do not say that a particular nucleus ‘will decay after one half-life’. Half-life is a statistical concept; it applies to a large population of nuclei, not to a single nucleus.

不要说某个特定原子核“会在一个半衰期后衰变”。半衰期是统计概念，适用于大量原子核，而不是单个原子核。
Remember that λ is the probability of decay per nucleus per unit time. It is not the same as the activity, unless N = 1.

记住 λ 是每个原子核在单位时间内的衰变概率。除非 N = 1，否则它不等同于活度。

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IB Physics: The Laws and Applications of Radioactive Decay | IB物理：放射性衰变的规律与应用