📚 Decay Constant and Half-Life | 衰变常数与半衰期
In the CIE A-Level Physics syllabus, radioactive decay is described statistically by two central quantities: the decay constant λ and the half-life T½. Together they allow us to predict the behaviour of a very large number of unstable nuclei, even though the decay of any individual nucleus is unpredictable.
在 CIE A-Level 物理考纲中，放射性衰变通过两个核心物理量进行统计描述：衰变常数 λ 和半衰期 T½。尽管单个原子核的衰变无法预测，但把二者结合起来，我们就能预测大量不稳定原子核的整体行为。
This article restates the original definitions, then develops the exponential decay equation, the relationship between λ and T½, worked numerical examples, graph analysis, the carbon-dating application, and the classic exam traps that lose marks.
本文在重述原有定义的基础上，进一步展开指数衰变方程、λ 与 T½ 的关系、计算例题、图像分析、碳定年应用，以及最容易失分的经典考点陷阱。
1. Radioactive Decay as a Random Process | 放射性衰变作为随机过程
Radioactive decay is a random and spontaneous process. We cannot predict which nucleus will decay next, nor when any particular nucleus will decay. Nuclei do not "age": a nucleus that has survived for a thousand half-lives is no more likely to decay in the next second than a freshly formed one.
放射性衰变是一个随机且自发的过程。我们无法预测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衰变。原子核不会"变老"：一个已经存活了一千个半衰期的原子核，在下一秒发生衰变的概率并不比刚生成的原子核更大。
The decay rate of a sample depends only on the number of undecayed nuclei present at that instant, not on temperature, pressure, chemical state or the history of the sample. Because the process is random, repeated measurements of count rate over short time intervals show fluctuations about an average value.
样品的衰变速率只取决于该时刻尚未衰变的原子核数量，而与温度、压强、化学状态或样品的历史无关。由于过程具有随机性，在短时间间隔内重复测量计数率时，结果会在平均值附近涨落。
When a very large number of nuclei is present, these fluctuations become relatively small and the average decay rate is statistically stable. It is this stability that allows us to define a fixed probability of decay per unit time for each nucleus — the decay constant.
当存在大量原子核时，这种涨落相对变小，平均衰变速率在统计上变得稳定。正是这种稳定性，使我们能够为每个原子核定义单位时间内固定的衰变概率——即衰变常数。
2. Defining the Decay Constant | 定义衰变常数
The decay constant λ is defined as the probability that an individual nucleus will decay per unit time. Its SI unit is s⁻¹ (per second). It is a property of the nuclide itself, not of the size or mass of the sample being studied.
衰变常数 λ 定义为单个原子核在单位时间内发生衰变的概率。其国际单位是 s⁻¹（每秒）。它是核素本身的属性，与研究样品的大小或质量无关。
If λ = 0.01 s⁻¹, each nucleus has a 1% perature, pressure, chemical bonding or physical state, because decay is a nuclear process governed by the weak or strong interaction, not by the electron cloud.
如果 λ = 0.01 s⁻¹，则每个原子核在任意一秒的间隔内有 1% 的概率发生衰变。λ 越大，衰变过程越快，半衰期越短。λ 不随温度、压强、化学键合或物态而改变，因为衰变是核过程，由弱相互作用或强相互作用支配，与核外电子云无关。
This is a key difference from chemical reactions, whose rates are strongly temperature dependent. A radioactive nuclide locked inside a compound decays at exactly the same rate as the pure element.
这是它与化学反应的关键区别——化学反应速率强烈依赖温度；而放射性核素即使存在于化合物中，其衰变速率也与纯元素完全相同。
| Quantity 物理量 | Symbol 符号 | Definition 定义 | SI unit 单位 |
| Decay constant 衰变常数 | λ | probability of decay per nucleus per unit time 每个原子核单位时间内衰变的概率 | s⁻¹ |
| Half-life 半衰期 | T½ | time for half the undecayed nuclei to decay 未衰变核数减半所需时间 | s |
| Activity 活度 | A | number of decays per second 每秒衰变次数 | Bq |
3. The Exponential Decay Equation | 指数衰变方程
Because the number of decays per second is proportional to the number of undecayed nuclei present, the rate of change of N is given by the decay rate equation. The minus sign shows that N decreases with time.
由于每秒衰变次数与当前未衰变原子核数量成正比，N 的变化率由衰变速率方程给出。负号表示 N 随时间减少。
dN/dt = −λN
question on this topic. Here N₀ is the number of undecayed nuclei at t = 0, and e = 2.718… is the base of natural logarithms.
对该关系式积分即可得到指数衰变方程，它是 CIE 这一考点几乎所有题目的核心公式。其中 N₀ 是 t = 0 时的未衰变核数，e = 2.718… 是自然对数的底。
N = N₀e−λt
Because activity A is proportional to N, and so are mass of undecayed nuclide and the background-corrected count rate, the same exponential form applies to all of them. This is why one equation covers so many different exam questions.
由于活度 A 与 N 成正比，未衰变核素的质量以及扣除本底后的计数率也都与 N 成正比，因此同一指数形式对它们全部适用。这正是同一个方程能够覆盖如此多不同考题的原因。
A = A₀e−λt = λN
4. Where the Decay Equation Comes From | 衰变方程的来源
The derivation is short and worth understanding, because it explains why λ has units of s⁻¹ and why the decay curve is exponential rather than linear. Start from the statement that the activity is proportional to the number of remaining nuclei.
推导过程很短，值得理解，因为它解释了为什么 λ 的单位是 s⁻¹，以及为什么衰变曲线是指数型而不是直线。起点是"活度与剩余核数成正比"这一陈述。
(number N) gives a natural logarithm, which is then inverted to give the exponential form.
将变量 N 与 t 分离，并在 t = 0（N = N₀）到时刻 t（核数为 N）之间积分，得到自然对数形式，再取反对数即得指数形式。
∫ (1/N) dN = ∫ −λ dt → ln N = −λt + c
N = N₀e−λt
examinable because it is the equation of the straight-line graph used to measure λ experimentally.
CIE 通常要求你会使用并解释这个方程，而不是完整复现积分过程；但中间结果 ln N = ln N₀ − λt 是可考的，因为它正是实验测定 λ 时所用直线的方程。
5. Defining Half-Life | 定义半衰期
The half-life T½ of a radioactive nuclide is the average time taken for half of the undecayed nuclei in a sample to decay. Equivalently, it is the time taken for the activity of a sample to fall to half of its initial value, or for the count rate (corrected for background) to halve.
放射性核素的半衰期 T½ 是指样品中一半未衰变原子核发生衰变所需的平均时间。等价地说，它也是样品活度降为初值一半所需的时间，或（扣除本底后的）计数率减半所需的时间。
The half-life of a given nuclide is constant: it does not depend on the initial number of nuclei, on temperature or on chemical form. After one half-life, 50% of the original nuclei remain; after two, 25%; after three, 12.5%. After n half-lives the surviving fraction is 2⁻ⁿ.
给定核素的半衰期是恒定的：它与初始核数、温度或化学形态无关。经过一个半衰期，剩余原始核数的 50%；两个后剩 25%；三个后剩 12.5%。经过 n 个半衰期后，剩余比例为 2⁻ⁿ。
N = N₀ / 2ⁿ where n = t / T½
Half-lives span an enormous range, from about 10⁻²² s for extremely unstable nuclei to more than 10⁹ years for nuclides such as uranium-238. This is why some isotopes are used in medicine within hours of production, while others form the long-lived background radiation of rocks.
半衰期的跨度极大，从极不稳定核素的约 10⁻²² s，到铀-238 这类超过 10⁹ 年的核素。这正是为什么有些同位素必须在制成后数小时内用于医学，而另一些则构成了岩石中长期存在的本底辐射。
6. Linking Decay Constant and Half-Life | 衰变常数与半衰期的关系
riginal nuclei remain, into the exponential decay equation, then take natural logarithms of both sides.
这两个物理量描述的是同一物理过程，因此必然相互关联。把 t = T½（此时恰好剩下一半原始核数）代入指数衰变方程，再对两边取自然对数。
N₀/2 = N₀e−λT½ → ½ = e−λT½
ln ½ = −λT½ → λT½ = ln 2 = 0.693
Rearranging gives the two forms you must be able to quote and use without hesitation. Note that λ and T½ are inversely proportional: doubling λ halves the half-life.
整理后得到两个必须能够脱口而出并熟练使用的形式。注意 λ 与 T½ 成反比：λ 加倍，半衰期减半。
T½ = ln 2 / λ = 0.693 / λ and λ = ln 2 / T½ = 0.693 / T½
ly appears in extension reading. It is the average time a nucleus survives, and it is always longer than the half-life.
与之相关的量是平均寿命 τ = 1/λ = T½ / ln 2 ≈ 1.44 T½，在拓展阅读中偶尔出现。它表示原子核存活的平均时间，总是大于半衰期。
7. Worked Example: From Half-Life to Decay Constant | 例题：由半衰期求衰变常数
to λ = 0.693 / T½. Always convert the half-life into seconds first if λ is required in s⁻¹ — this is the single most common source of lost marks.
某医用同位素的半衰期为 20.0 分钟。先换算为秒，再代入 λ = 0.693 / T½。若要求 λ 的单位是 s⁻¹，必须先把半衰期换算成秒——这是最常见的失分点。
T½ = 20.0 × 60 = 1.20 × 10³ s
λ = 0.693 / (1.20 × 10³) = 5.78 × 10⁻⁴ s⁻¹
Reversing the calculation: a nuclide with λ = 4.2 × 10⁻⁹ s⁻¹ has T½ = 0.693 / (4.2
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