📚 Randomness and Decay | 放射性随机性与衰变
Radioactive decay is a spontaneous and random process that cannot be controlled by external conditions such as temperature, pressure or chemical bonding. This topic explores how randomness is described statistically, how decay is modelled, and how half-life and activity are calculated.
放射性衰变是一种自发且随机的过程，不受温度、压力或化学键等外部条件控制。本主题将探讨如何用统计方法描述随机性，如何建立衰变模型，以及如何计算半衰期和活度。
1. The Random Nature of Radioactive Decay | 放射性衰变的随机性
Radioactive decay is a spontaneous process. It is impossible to predict which individual nucleus in a sample will decay next, or when a particular nucleus will decay.
放射性衰变是一种自发过程。无法预测样品中哪一个原子核会下一个衰变，也无法预测某个特定原子核何时会衰变。
However, for a large number of nuclei, the overall behaviour is governed by probability and is statistically predictable.
然而，对于大量原子核，整体行为由概率支配，并且具有统计可预测性。
This combination of individual randomness and collective statistical regularity is central to understanding decay.
这种个体随机性与集体统计规律性的结合，是理解衰变的核心。
2. Activity and the Decay Constant | 活度与衰变常量
Activity A is defined as the number of decays per unit time. The SI unit of activity is the becquerel (Bq), where 1 Bq = 1 decay per second.
活度 A 定义为单位时间内发生衰变的次数。活度的国际单位是贝克勒尔（Bq），其中 1 Bq = 每秒 1 次衰变。
The decay constant λ represents the probability that a single nucleus will decay per unit time. Its unit is s⁻¹.
衰变常量 λ 表示单个原子核在单位时间内发生衰变的概率。其单位为 s⁻¹。
For a sample containing N undecayed nuclei, the activity is given by:
对于含有 N 个未衰变原子核的样品，活度由下式给出：
A = λN
A larger decay constant means a higher probability of decay and therefore a more active sample for a given number of nuclei.
衰变常量越大，表示衰变概率越高，因此在原子核数量相同的情况下，样品的活度也越大。
3. The Decay Equation N = N₀e⁻λt | 衰变方程 N = N₀e⁻λt
Because decay is random, the number of undecayed nuclei decreases exponentially with time. If N₀ is the initial number of undecayed nuclei at t = 0, then after time t the number remaining is:
由于衰变是随机的，未衰变原子核的数量随时间呈指数减少。如果 N₀ 是 t = 0 时未衰变原子核的初始数量，那么经过时间 t 后剩余的数量为：
N = N₀e−λt
This equation follows from the differential equation dN/dt = −λN, which states that the rate of decay is proportional to the number of undecayed nuclei present.
该方程源自微分方程 dN/dt = −λN，它表明衰变速率与当前未衰变原子核的数量成正比。
The negative sign indicates that N decreases as time increases.
负号表示 N 随时间的增加而减少。
4. Half-Life | 半衰期
The half-life T½ is the average time taken for half of the unstable nuclei in a sample to decay, or equivalently, for the activity to fall to half of its initial value.
半衰期 T½ 是样品中一半不稳定原子核发生衰变所需的平均时间，等价地，也是活度降至其初始值一半所需的时间。
The half-life is related to the decay constant by:
半衰期与衰变常量的关系为：
T½ = ln 2 / λ = 0.693 / λ
A large decay constant gives a short half-life, meaning the isotope decays rapidly. A small decay constant gives a long half-life and a less active sample.
衰变常量越大，半衰期越短，意味着同位素衰变很快。衰变常量越小，半衰期越长，样品的活度也越低。
5. Exponential Decay Curve | 指数衰变曲线
A graph of the number of undecayed nuclei N against time t is a decreasing exponential curve. The curve never reaches zero, but approaches the time axis asymptotically.
未衰变原子核数量 N 随时间 t 变化的图像是一条递减的指数曲线。该曲线永远不会到达零，而是以时间轴为渐近线。
In each successive half-life interval, the number of undecayed nuclei falls by half. After one half-life, N = N₀/2; after two half-lives, N = N₀/4; after three half-lives, N = N₀/8.
在每一个连续的半衰期时间间隔内，未衰变原子核数量减少一半。经过一个半衰期后，N = N₀/2；经过两个半衰期后，N = N₀/4；经过三个半衰期后，N = N₀/8。
This constant-fraction decay is a key feature of an exponential process.
这种恒定比例衰变是指数过程的一个重要特征。
6. Activity Equation and Count Rate | 活度方程与计数率
Since activity A is proportional to N, the activity also decreases exponentially with time:
由于活度 A 与 N 成正比，活度也随时间呈指数下降：
A = A₀e−λt
In experiments, a Geiger-Muller tube measures a count rate C, which is directly proportional to the activity, provided the background count rate has been subtracted.
在实验中，盖革-米勒计数管测量计数率 C，只要已扣除背景计数率，计数率便与活度成正比。
Therefore the corrected count rate follows the same exponential law: C = C₀e−λt.
因此，修正后的计数率遵循相同的指数规律：C = C₀e−λt。
7. Statistical Fluctuations | 统计涨落
Because decay is random, repeated measurements of the same source over equal time intervals do not give exactly the same count. The counts show statistical fluctuations.
由于衰变是随机的，在相同时间间隔内对同一放射源进行重复测量，得到的计数并不完全相同。计数会表现出统计涨落。
For a count N recorded in a given interval, the standard uncertainty is approximately √N. The fractional uncertainty is therefore 1/√N.
对于在给定时间间隔内记录到的计数 N，标准不确定度约为 √N。因此相对不确定度为 1/√N。
Recording a larger total count reduces the fractional uncertainty and improves the precision of the measurement. To achieve this, longer counting times or a stronger source are used.
记录更大的总计数可以降低相对不确定度，从而改善测量精度。为此，可使用更长的计数时间或更强的放射源。
8. Background Radiation | 背景辐射
Background radiation is always present due to cosmic rays, naturally occurring radioactive materials in rocks and soil, radon gas, and medical or industrial sources.
背景辐射始终存在，来源包括宇宙射线、岩石和土壤中的天然放射性物质、氡气以及医疗或工业源。
When measuring the activity of a source, the background count rate must be measured separately and subtracted from the total count rate to obtain the corrected count rate due to the source alone.
在测量放射源的活度时，必须单独测量背景计数率，并从总计数率中减去，以获得仅由该放射源产生的修正计数率。
Background radiation is itself random, so its contribution also fluctuates and should be measured over a sufficiently long time to reduce uncertainty.
背景辐射本身也是随机的，因此其贡献也会发生涨落，应在足够长的时间内测量以减小不确定度。
9. Measuring Half-Life | 半衰期的测量
To measure the half-life of a radioactive isotope, a detector such as a Geiger-Muller tube is used to record the count rate at regular time intervals.
为测量放射性同位素的半衰期，可使用盖革-米勒计数管等探测器，在固定的时间间隔记录计数率。
The background count rate is first measured and subtracted. The corrected count rate is then plotted against time to obtain an exponential decay curve, from which the half-life can be read directly.
首先测量并扣除背景计数率。然后绘制修正计数率随时间的变化曲线，得到指数衰变曲线，从曲线上可以直接读出半衰期。
Alternatively, a graph of ln C against t is plotted. Since ln C = ln C₀ − λt, the graph is a straight line with gradient −λ. The half-life is then found from T½ = ln 2 / λ.
另一种方法是绘制 ln C 对 t 的图像。由于 ln C = ln C₀ − λt，该图像是一条斜率为 −λ 的直线。然后利用 T½ = ln 2 / λ 求出半衰期。
10. Applications: Radioactive Dating | 应用：放射性定年
Carbon-14 dating is used to estimate the age of organic remains such as wood, cloth or bone. Living organisms continually exchange carbon with the atmosphere, so the ratio of carbon-14 to carbon-12 remains roughly constant while alive.
碳-14 定年法用于估算木材、织物或骨骼等有机遗骸的年龄。生物体在存活期间不断与大气交换碳，因此碳-14 与碳-12 的比率在存活期间大致保持恒定。
After death, carbon-14 decays with a half-life of about 5730 years and is no longer replaced. By measuring the remaining activity or the ratio of carbon-14 to carbon-12, the time since death can be calculated using N = N₀e−λt.
生物体死亡后，碳-14 以约 5730 年的半衰期衰变，且不再得到补充。通过测量剩余的活度或碳-14 与碳-12 的比率，可以利用 N = N₀e−λt 计算死亡至今的时间。
This method relies on the assumption that the atmospheric carbon-14 ratio has been approximately constant over time. Calibration with other dating methods helps to improve accuracy.
该方法依赖于大气中碳-14 比率随时间大致恒定的假设。利用其他定年方法进行校准有助于提高准确性。
11. Safety and Randomness | 安全与随机性
The random nature of decay means that the biological effects of ionising radiation are probabilistic. Even low doses carry a small but non-zero risk of cellular damage or mutation.
衰变的随机性意味着电离辐射的生物效应是概率性的。即使低剂量也存在微小但非零的细胞损伤或突变风险。
Safety measures therefore aim to minimise exposure. The three main principles are to keep the time of exposure short, increase distance from the source, and use shielding such as lead.
因此，安全措施旨在尽量减少照射。三个主要原则是：缩短照射时间、增大与放射源的距离，以及使用铅等屏蔽材料。
Sources should be handled with tongs and stored in lead-lined containers when not in use. Experiments should be planned so that the total count collected is high enough for precision while keeping exposure low.
放射源应使用镊子操作，不用时应存放在铅衬容器中。实验设计应使收集到的总计数足够高以保证精度，同时保持低照射量。
12. Key Equations Summary | 关键公式总结
The following table summarises the main relationships used in the randomness and decay topic.
下表总结了随机性与衰变主题中使用的主要关系式。
| Equation | Meaning | 中文含义 |
|---|---|---|
| A = λN | Activity equals decay constant times number of undecayed nuclei | 活度 = 衰变常量 × 未衰变核数 |
| N = N₀e−λt | Number of undecayed nuclei after time t | 时间 t 后未衰变核数 |
| A = A₀e−λt | Activity after time t | 时间 t 后的活度 |
| T½ = ln 2 / λ | Half-life in terms of decay constant | 用衰变常量表示的半衰期 |
These equations all assume that the decay process is random and that the number of nuclei is large enough for statistical treatment to be valid.
这些方程均假设衰变过程是随机的，并且原子核数量足够大，统计处理可以成立。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导