📚 Half-Life and Radioactive Decay | 半衰期与放射性衰变
In IGCSE Science (Edexcel), the concept of half-life is central to understanding radioactivity. It links the random nature of nuclear decay to measurable predictions, and it appears in almost every radioactivity exam question.
在爱德思 IGCSE 科学中，半衰期是理解放射性的核心概念。它将核衰变的随机性与可预测的测量联系了起来，也几乎出现在每一道与放射性相关的考试题中。
1. What Is Radioactive Decay? | 什么是放射性衰变？
Radioactive decay is a random process in which an unstable nucleus loses energy by emitting radiation. This may change the identity of the atom, turning one element into another.
放射性衰变是一个随机过程：不稳定的原子核通过释放辐射来失去能量。这个过程可能改变原子的身份，使一种元素变成另一种元素。
- Random: we cannot predict which nucleus will decay next.
- Spontaneous: it is not affected by temperature, pressure, or chemical state.
- Statistical: for a large number of nuclei, a reliable pattern emerges.
- 随机：我们无法预测下一个衰变的原子核是哪一个。
- 自发：温度、压强或化学状态都不会影响它。
- 统计性：对于大量原子核，会出现可靠的统计规律。
The most common forms of radiation are alpha (α), beta (β) and gamma (γ). Each has different ionising power and penetrating ability, but all can be detected using a Geiger-Müller tube.
最常见的辐射类型是 α、β 和 γ。它们的电离能力和穿透能力各不相同，但都可以通过盖革-米勒管进行探测。
2. The Meaning of Half-Life | 半衰期的含义
Half-life is defined as the time taken for half of the radioactive nuclei in a sample to decay.
半衰期的定义是：样品中一半的放射性原子核发生衰变所需的时间。
half-life = time for number of undecayed nuclei to fall to half its original value
半衰期 = 未衰变原子核数量降到原来一半所需的时间
Equivalently, it is the time taken for the count rate from a sample to decrease to half of its initial value.
等价地说，半衰期也是样品的计数率下降到初始值一半所需的时间。
The symbol used is often \( t_{1/2} \), but in this course we simply write half-life.
在课程中通常使用 \( t_{1/2} \) 表示半衰期，但这里我们直接写成 half-life。
3. Representing Decay with Graphs | 用图像表示衰变
A radioactive decay graph plots the number of undecayed nuclei (or count rate) against time. It is a smooth, decreasing curve that asymptotically approaches zero.
放射性衰变图以时间为横轴，以未衰变原子核数（或计数率）为纵轴。这是一条平滑下降的曲线，逐渐接近零但不会瞬间到达零。
- After one half-life, the value drops to 50%.
- After two half-lives, it drops to 25%.
- After three half-lives, it drops to 12.5%.
- After four half-lives, it drops to 6.25%.
- 经过一个半衰期，数值下降到 50%。
- 经过两个半衰期，下降到 25%。
- 经过三个半衰期，下降到 12.5%。
- 经过四个半衰期，下降到 6.25%。
To find the half-life from a graph, read the time for the count rate to halve. You can repeat this at several points to confirm the value.
要从图像中求半衰期，读取计数率减半所需的时间。可以取多个点重复验证该数值。
4. Half-Life and the Fraction Remaining | 半衰期与剩余比例
If \(n\) half-lives have passed, the fraction of the original nuclei remaining is:
如果经过了 \(n\) 个半衰期，剩余原子核占原始的比例为：
fraction remaining = (½)ⁿ
剩余比例 = (½)ⁿ
For example, after 3 half-lives:
例如，经过 3 个半衰期后：
(½)³ = ½ × ½ × ½ = 1/8
This simple pattern allows quick calculation of remaining mass or activity.
这个简单规律可以帮助我们快速计算剩余质量或活度。
5. Half-Life Calculations | 半衰期计算
The two most common calculation types are:
最常见的两种计算类型是：
| Type 1 | Find the remaining mass after a given time. |
| Type 2 | Find the half-life from decay data. |
| 第一种 | 已知时间，求剩余质量。 |
| 第二种 | 根据衰变数据求半衰期。 |
For Type 1, use the fraction formula and multiply by the original mass.
对于第一种类型，先用公式计算剩余比例，再乘以原始质量。
6. Worked Example | 例题解析
Example: A radioactive sample has an initial mass of 80 g and a half-life of 6 hours. What mass remains after 24 hours?
例题：某放射性样品初始质量为 80 g，半衰期为 6 小时。24 小时之后剩余质量是多少？
Step 1: Find the number of half-lives.
第一步：求半衰期的个数。
n = 24 ÷ 6 = 4
Step 2: Calculate the fraction remaining.
第二步：计算剩余比例。
fraction remaining = (½)⁴ = 1/16
Step 3: Multiply by the original mass.
第三步：乘以原始质量。
remaining mass = 80 × 1/16 = 5 g
So after 24 hours, only 5 g of the original radioactive nuclei remain.
因此 24 小时后，只剩下 5 g 原始放射性原子核。
For Type 2, you can use a table of count rate against time.
对于第二种类型，可以利用计数率随时间变化的表格。
7. Using a Decay Table | 利用衰变数据表
Consider the count rate of a sample measured every 10 minutes:
考虑一个样品每隔 10 分钟测量到的计数率：
| Time / min | 0 | 10 | 20 | 30 | 40 |
| Count rate / counts per minute | 1200 | 600 | 300 | 150 | 75 |
The count rate drops from 1200 to 600 in 10 minutes, from 600 to 300 in another 10 minutes, and so on. Therefore the half-life is 10 minutes.
计数率在 10 分钟内从 1200 降到 600，再经过 10 分钟从 600 降到 300，依此类推。因此半衰期为 10 分钟。
Always check that the drop is exactly half; if there is background radiation, subtract it first.
务必确认下降是否恰好为一半；如果存在本底辐射，应先减去本底辐射。
8. Background Radiation | 本底辐射
Background radiation is the low-level radiation present all around us. Sources include cosmic rays, rocks, soil, food, and radon gas.
本底辐射是存在于我们周围的低水平辐射。来源包括宇宙射线、岩石、土壤、食物和氡气。
When measuring a radioactive source, you must first measure the background count rate and then subtract it from the total count rate.
在测量放射源时，必须先测量本底计数率，再从总计数率中减去它。
corrected count rate = total count rate − background count rate
校正计数率 = 总计数率 − 本底计数率
This corrected value is what should be used in half-life calculations.
半衰期计算中应使用的正是这个校正后的数值。
9. Radiocarbon Dating | 碳定年法
Carbon-14 is a radioactive isotope of carbon with a half-life of about 5730 years. It is constantly produced in the atmosphere and absorbed by living things.
碳-14 是碳的一种放射性同位素，半衰期约为 5730 年。它在大气中不断产生，并被生物体吸收。
When an organism dies, it stops absorbing carbon-14. The remaining carbon-14 then decays. By measuring the ratio of C-14 to stable carbon-12, scientists can estimate the age of ancient organic material.
生物体死亡后，不再吸收碳-14。剩余的碳-14 开始衰变。通过测量 C-14 与稳定碳-12 的比例，科学家可以估算古代有机物的年代。
Carbon dating is reliable for materials up to about 50,000 years old; for older materials, too little C-14 remains.
碳定年法适用于距今约 5 万年以内的材料；对于更古老的样品，残留的 C-14 太少，难以准确测量。
10. Medical and Industrial Uses | 医学与工业应用
In medicine, half-life is used to choose appropriate radioactive tracers. A tracer must last long enough for the scan to be completed, but short enough to disappear from the body quickly.
在医学中，半衰期用于选择适合的放射性示踪剂。示踪剂需要持续足够长的时间以完成扫描，但又要足够短，以便快速从体内消失。
For example, technetium-99m has a half-life of about 6 hours, making it ideal for medical imaging.
例如，锝-99m 的半衰期约为 6 小时，非常适合医学成像。
In industry, half-life helps control the thickness of materials. A source of beta radiation is placed on one side of a sheet, and a detector on the other. If the sheet becomes too thick, fewer beta particles pass through, and the detector signal weakens. This allows machines to adjust production automatically.
在工业中，半衰期可用于控制材料厚度。将 β 辐射源放在材料一侧，探测器放在另一侧。如果材料变厚，穿过的 β 粒子减少，探测器信号变弱，机器就会自动调整生产参数。
11. Common Misconceptions | 常见误区
Misconception 1: “After two half-lives, the sample has fully decayed.”
误区一：“经过两个半衰期后，样品就完全衰变完了。”
Correction: After two half-lives, 25% remains; after three, 12.5% remains. The sample never completely disappears in a finite time.
更正：经过两个半衰期后还剩 25%；三个后还剩 12.5%。样品在有限时间内永远不会完全消失。
Misconception 2: “Temperature changes affect the half-life.”
误区二：“温度变化会影响半衰期。”
Correction: Half-life is unaffected by physical or chemical conditions. It is a fixed property of each radioactive isotope.
更正：半衰期不受物理或化学条件影响，是每种放射性同位素的固定属性。
Misconception 3: “Count rate can tell us exactly how many atoms will decay next minute.”
误区三：“计数率可以告诉我们下一分钟具体有多少个原子会衰变。”
Correction: Decay is random. Count rate only gives an average trend for a large number of nuclei.
更正：衰变是随机的。计数率只给出大量原子核的平均趋势。
12. Practice Questions | 练习与总结
Practice 1: A source has a count rate of 800 counts per minute. Its half-life is 2 days. What will the count rate be after 6 days?
练习 1：某源计数率为每分钟 800 次，半衰期为 2 天。6 天后的计数率是多少？
Answer: 6 days = 3 half-lives; 800 → 400 → 200 → 100 counts per minute.
答案：6 天 = 3 个半衰期；800 → 400 → 200 → 100 次每分钟。
Practice 2: A sample contains 64 g of a radioactive isotope. After 30 minutes, only 8 g remains. Find the half-life.
练习 2：某样品含 64 g 放射性同位素。30 分钟后只剩 8 g。求半衰期。
Answer: 64 g → 32 g → 16 g → 8 g takes 3 half-lives. So 30 min = 3 × half-life; half-life = 10 minutes.
答案：64 g → 32 g → 16 g → 8 g 经过了 3 个半衰期。因此 30 分钟 = 3 × 半衰期；半衰期 = 10 分钟。
Summary: Half-life is a measure of how quickly a radioactive isotope decays. Use the fraction (½)ⁿ, subtract background radiation, and always read the graph carefully.
总结：半衰期是衡量放射性同位素衰变快慢的指标。使用比例 (½)ⁿ，减去本底辐射，并且仔细读图。
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