📚 Half-Life and Radioactive Decay | 半衰期与放射性衰变
In IGCSE Edexcel Science, Unit 7.5 introduces the concept of half-life, a crucial idea for understanding radioactive decay. This article explains what half-life means, how to calculate it, and how it is applied in real-world contexts such as carbon dating and medical imaging.
在 IGCSE Edexcel 科学课程中，第 7.5 单元介绍了半衰期的概念，这是理解放射性衰变的关键。本文将解释半衰期的含义、如何计算它，以及它如何在碳定年和医学成像等现实场景中应用。
1. What is Radioactive Decay? | 什么是放射性衰变？
Radioactive decay happens when an unstable atomic nucleus releases energy or particles in order to become more stable. The process may involve the emission of an alpha particle (α), a beta particle (β) or gamma radiation (γ). Each type of emission changes the nucleus in a different way.
放射性衰变发生在不稳定的原子核通过释放能量或粒子变得更稳定的过程中。该过程可能涉及发射 α 粒子、β 粒子或 γ 辐射。每种发射都以不同方式改变原子核。
In alpha decay the nucleus loses two protons and two neutrons, so the mass number falls by 4 and the atomic number falls by 2. For example, a uranium-238 nucleus decays by emitting an alpha particle to form thorium-234:
在 α 衰变中，原子核失去两个质子和两个中子，因此质量数减少 4，原子序数减少 2。例如，铀-238 原子核通过发射一个 α 粒子衰变为钍-234：
²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He
Beta decay is different. A neutron inside the nucleus turns into a proton and a fast-moving electron is emitted, so the mass number stays the same while the atomic number increases by 1. Carbon-14 decays in exactly this way:
β 衰变则不同。原子核内的一个中子转变为一个质子，同时发射出一个高速电子，因此质量数保持不变，而原子序数增加 1。碳-14 正是以这种方式衰变：
¹⁴₆C → ¹⁴₇N + ⁰₋₁e
Gamma emission on its own does not change the mass number or the atomic number; it simply carries away surplus energy from an excited nucleus. In all three cases the decay is spontaneous, and it cannot be started, speeded up or slowed down by heating, cooling, pressure or chemical reactions.
单一发射 γ 射线不会改变质量数或原子序数；它只是把多余的能量从激发态的原子核中带走。在这三种情况下，衰变都是自发的，不能通过加热、冷却、加压或化学反应来开启、加速或减慢。
2. The Random Nature of Decay | 衰变的随机性
Radioactive decay is a random process. This means it is impossible to predict exactly when a particular nucleus will decay, or which nucleus in a sample will decay next. Two identical nuclei sitting side by side may decay seconds apart or centuries apart.
放射性衰变是一个随机过程。这意味着无法准确预测某个特定原子核何时衰变，也无法预测样品中下一个衰变的是哪个原子核。两个紧挨着、完全相同的原子核，可能相隔几秒衰变，也可能相隔几个世纪才衰变。
A useful analogy is throwing a large number of dice. You cannot say when any single die will show a six, but if you throw a thousand dice, roughly one sixth of them will show a six every time. In the same way, the fate of one nucleus is unpredictable, but the behaviour of a large collection of nuclei is highly predictable.
一个有用的类比是掷大量骰子。你无法说出某一颗骰子何时会掷出六点，但如果你掷一千颗骰子，每一次大约会有六分之一掷出六点。同样，单个原子核的命运不可预测，但大量原子核的整体行为却高度可预测。
Because decay is random, the number of decays per second is proportional to the number of unstable nuclei still present. As the sample decays, there are fewer unstable nuclei left, so fewer decays happen each second. This is why activity falls in a smooth, predictable pattern rather than in a straight line.
正因为衰变是随机的，每秒衰变的次数与仍然存在的不稳定原子核数量成正比。随着样品衰变，剩下的不稳定原子核越来越少，因此每秒发生的衰变也越来越少。这就是为什么活度呈现出平滑而可预测的下降模式，而不是直线下降。
3. Defining Half-Life | 半衰期的定义
The half-life (t₁/₂) of a radioactive isotope is the average time taken for half the nuclei in a given sample to decay, or equivalently, the time for the activity (or count rate) of the sample to fall to half its initial value. Every radioactive isotope has a fixed half-life, ranging from fractions of a second to billions of years.
放射性同位素的半衰期 (t₁/₂) 是指给定样品中一半的原子核发生衰变所需的平均时间，或等价地说，样品的活度（或计数率）下降至初始值一半所需的时间。每种放射性同位素都有固定的半衰期，范围从不到一秒到数十亿年不等。
Note the phrase "half the nuclei in a given sample". Half-life is a statistical property: it says nothing about one individual nucleus, only about a very large population of them. It is also independent of the amount of the substance, so 2 g and 2 tonnes of the same isotope have exactly the same half-life.
请注意 "给定样品中一半的原子核" 这句话。半衰期是一个统计性质：它对单个原子核毫无意义，只对极大量的原子核群体成立。半衰期也与物质的多少无关，因此 2 g 和 2 吨同种同位素的半衰期完全相同。
The half-life is not affected by temperature, pressure, chemical bonding or physical state. Whether carbon-14 is inside a living tree, a lump of charcoal or a diamond, its half-life is still 5730 years. This is one reason radioactive isotopes are so useful as natural clocks and tracers.
半衰期不受温度、压力、化学键合或物态的影响。无论碳-14 是在活着的树木里、一块木炭里还是一颗钻石里，它的半衰期仍然是 5730 年。这正是放射性同位素作为天然时钟和示踪剂如此有用的原因之一。
4. Common Isotopes and Their Half-Lives | 常见同位素及其半衰期
The table below lists three common isotopes and their half-lives, along with typical uses. Notice how the half-life determines what an isotope is good for: very long half-lives suit geological dating and long-life devices, while short half-lives suit medical work where the activity must disappear quickly.
下表列出了三种常见的同位素及其半衰期和典型用途。请注意半衰期如何决定一种同位素的适用场景：极长的半衰期适合地质定年和长寿命装置，而较短的半衰期适合活度需要快速消失的医学工作。
| Isotope | Half-life | Typical Use |
|---|---|---|
| Carbon-14 (¹⁴C) | 5730 years | Archaeological dating |
| Iodine-131 (¹³¹I) | 8 days | Thyroid cancer treatment |
| Americium-241 (²⁴¹Am) | 432 years | Smoke detectors |
Radioactive isotopes span an enormous range. Some, such as polonium-214, have half-lives of less than a thousandth of a second, while uranium-238 has a half-life of about 4.5 billion years, which is why it is still present in rocks formed when the Earth was young.
放射性同位素的半衰期跨度极大。有些如钋-214，半衰期不到千分之一秒，而铀-238 的半衰期约为 45 亿年，这正是它至今仍存在于地球形成初期的岩石中的原因。
5. Simple Half-Life Calculations | 简单的半衰期计算
fraction of the sample remaining as (½)ⁿ. The remaining mass or number of nuclei is given by initial amount × (½)ⁿ.
进行简单的半衰期计算时，首先确定已经过的半衰期个数 n：n = 总时间 ÷ 半衰期 (t₁/₂)。然后计算剩余样品的分数为 (½)ⁿ。剩余质量或原子核数目由下式给出：初始量 × (½)ⁿ。
sp; remaining fraction = (½)ⁿ remaining amount = original amount × (½)ⁿ
80 g of a radioactive isotope with a half-life of 5 years. How much remains after 20 years? Here n = 20 ÷ 5 = 4 half-lives, so the remaining mass is 80 × (½)⁴ = 80 × 0.0625 = 5 g.
例题 1。某样品含有 80 g 某放射性同位素，半衰期为 5 年。20 年后还剩多少？此处 n = 20 ÷ 5 = 4 个半衰期，因此剩余质量为 80 × (½)⁴ = 80 × 0.0625 = 5 g。
life is 3 hours. How long does it take for the count rate to fall to 100 counts per minute? three times takes 800 → 400 → 200 → 100, so three half-lives have passed: 3 × 3 = 9 hours.
例题 2。某放射源的计数率为每分钟 800 次，半衰期为 3 小时。计数率降到每分钟 100 次需要多长时间？连续三次减半得到 800 → 400 → 200 → 100，即经过了 3 个半衰期：3 × 3 = 9 小时。
ready decayed is 1 − (½)ⁿ. After three half-lives, (½)³ = 0.125 remains, so 0.875, or 87.5% ecayed. Always check that your remaining fraction plus your decayed fraction adds up to 1.
已经衰变的分数为 1 − (½)ⁿ。经过 3 个半衰期后，剩余的分数 (½)³ = 0.125，因此已有 0.875，即 87.5% 的原始原子核发生了衰变。请始终检查剩余分数与已衰变分数之和是否等于 1。
6. Using Decay Curves and Graphs | 使用衰变曲线和图像
A decay curve is a graph of activity or count rate (y-axis) against time (x-axis). Because decay is a random process but happens at a rate proportional to the number of nuclei left, the curve is a smooth exponential decay curve that gets flatter and flatter but never quite reaches zero.
衰变曲线是活度或计数率（y 轴）对时间（x 轴）的图像。由于衰变是随机过程，但发生速率与剩余原子核数量成正比，因此曲线是一条平滑的指数衰减曲线，越来越平缓，但永远不会完全降为零。
To read a half-life from a curve, start at the initial value on the y-axis, go across to the curve, and then down to the time axis. That gives the first half-life. Repeat from half the initial value to get the second half-life, and from a quarter of the initial value for the third.
要从曲线上读取半衰期，先从 y 轴上的初始值出发，水平移动到曲线，再垂直向下到时间轴，这样就得到第一个半衰期。接着从初始值的一半重复这一步骤得到第二个半衰期，再从初始值的四分之一得到第三个半衰期。
You should find that these successive time intervals are equal. This constant interval is the half-life. In an exam, always show the horizontal and vertical guide lines on the graph, and quote the half-life with the correct unit, such as seconds, days or years.
你会发现这些连续的时间间隔是相等的。这个恒定间隔就是半衰期。在考试中，一定要在图上画出水平和垂直的辅助线，并用正确的单位写出半衰期，例如秒、天或年。
Real count-rate data are slightly "noisy" because decay is random: the points will not sit perfectly on the smooth curve. A common exam question asks why the points scatter a little, and the answer is simply the random nature of radioactive decay.
真实计数率数据会略显 "波动"，因为衰变是随机的：数据点不会完美地落在平滑曲线上。考试中常见的问题会问为什么数据点略有离散，答案就是放射性衰变的随机性。
7. Working Backwards: Finding Ages and Half-Lives | 反向计算：求年龄与半衰期
Sometimes you are given the remaining fraction and asked for the time, or given the time and asked for the half-life. The method is the same, just rearranged. Start by expressing the remaining amount as a fraction or percentage of the original amount, then keep halving until you reach it.
有时题目给出剩余分数求时间，或者给出时间求半衰期。方法相同，只是公式重新排列。先把剩余量表示为原始量的分数或百分比，然后不断减半，直到得到该数值。
Suppose a sample retains 25% of its original activity. Halving once gives 50%, halving twice gives 25% . If the isotope has a half-life of 5000 years, the sample must be about 10 000 years old.
假设某样品保留原始活度的 25%。减半一次得到 50%，减半两次得到 25%，因此经过了 n = 2 个半衰期。若该同位素半衰期为 5000 年，则样品大约有 10 000 年的历史。
The reverse also works. If a sample is known to be 15 000 years old and only 12.5% hen (½)ⁿ = 0.125, so n = 3 half-lives. The half-life is therefore 15 000 ÷ 3 = 5000 years.
反向计算同样可行。如果已知某样品有 15 000 年历史，且只剩原始活度的 12.5%，则 (½)ⁿ = 0.125，因此 n = 3 个半衰期。所以半衰期为 15 000 ÷ 3 = 5000 年。
A quick check: the percentages 50%, 25%, 12.5%, 6.25% and 3.125% correspond to 1, 2, 3, 4 and 5 half-lives. Memorising this short sequence makes many exam questions much faster to solve.
一个小技巧：50%、25%、12.5%、6.25% 和 3.125% 分别对应 1、2、3、4 和 5 个半衰期。记住这个短序列可以让许多考题的求解快得多。
8. Carbon Dating | 碳定年
Carbon-14 is produced in the upper atmosphere when cosmic rays collide with nitrogen atoms. Living plants take in carbon-14 through photosynthesis, and animals take it in by eating plants or other animals, so every living thing has a roughly constant proportion of carbon-14 in its tissues.
碳-14 是在高层大气中宇宙射线与氮原子碰撞时产生的。活着的植物通过光合作用吸收碳-14，动物则通过吃植物或其他动物摄入，因此每种生物的组织中都含有大致恒定比例的碳-14。
When an organism dies, it stops taking in new carbon. The carbon-14 inside it is no longer replaced, so it begins to decay with a half-life of 5730 years, while the stable carbon-12 remains unchanged. By measuring the ratio of carbon-14 to carbon-12, scientists can estimate how long ago the organism died.
当生物死亡后，它不再摄入新的碳。体内的碳-14 不再得到补充，于是开始以 5730 年的半衰期衰变，而稳定的碳-12 保持不变。通过测量碳-14 与碳-12 的比例，科学家可以估计该生物死亡距今有多久。
0 = 11 460 years old. Carbon dating is reliable for objects up to roughly 50 000 years old; beyond that, too little carbon-14 remains to be measured accurately.
例如，如果一块木头中碳-14 的含量是活木头的四分之一，那么已经过了两个半衰期，因此这块木头大约有 2 × 5730 = 11 460 年的历史。碳定年法对大约 5 万年以内的物体是可靠的；超过这个年限，剩余的碳-14 太少，无法准确测量。
The method assumes that the proportion of carbon-14 in the atmosphere has stayed roughly constant over time and that the sample has not been contaminated. These assumptions are exactly why carbon dating has an uncertainty range attached to every result.
该方法假设大气中碳-14 的比例在漫长时间里大致保持恒定，并且样品没有被污染。正是这些假设，使每个碳定年结果都带有不确定范围。
9. Medical Imaging and Treatment | 医学成像与治疗
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