📚 Decay Constant and Half-Life | 衰变常数与半衰期
Radioactive decay is a fundamental process by which unstable atomic nuclei lose energy by emitting radiation. Understanding the rate of decay is essential for predicting the behaviour of radioactive sources, and two key quantities used to describe this rate are the decay constant (λ) and the half-life (T½). This article explores their definitions, the mathematical relationship between them, and their practical applications in A-Level Physics.
放射性衰变是不稳定原子核通过发射辐射而损失能量的基本过程。理解衰变的速率对于预测放射源的行为至关重要，而用于描述这一速率的两个关键量是衰变常数 (λ) 和半衰期 (T½)。本文探讨它们的定义、两者之间的数学关系及其在 A-Level 物理中的实际应用。
1. Introduction to Radioactive Decay | 放射性衰变简介
Many atomic nuclei are unstable; over time, they spontaneously transform into more stable nuclei while emitting alpha particles, beta particles, or gamma rays. This transformation is called radioactive decay. The decay of a given nucleus is independent of its age and of the behaviour of other nuclei.
许多原子核是不稳定的；随着时间的推移，它们会自发地转变为更稳定的核，同时发射出 α 粒子、β 粒子或 γ 射线。这种转变称为放射性衰变。某个特定核的衰变与其存在时间无关，也与其他核的行为无关。
Because the process is statistical in nature, we cannot predict exactly when an individual nucleus will decay. However, we can model the average decay rate for a large sample of nuclei using the decay constant and the half-life, which provide a reliable measure of the sample's activity over time.
由于该过程本质上是统计性的，我们无法准确预测某个特定核何时会发生衰变。但我们可以利用衰变常数和半衰期对大量原子核样品的平均衰变速率进行建模，从而可靠地衡量样品随时间变化的活度。
2. The Random Nature of Decay | 衰变的随机性
Radioactive decay is a random process at the level of a single nucleus. The moment a particular unstable nucleus decays cannot be foretold. Each nucleus has a constant probability of decaying per unit time, regardless of how long it has already existed.
在单个原子核层面上，放射性衰变是一个随机过程。一个特定不稳定核发生衰变的时刻是无法预知的。每个核在单位时间内发生衰变的概率是恒定的，与它已经存在了多长时间无关。
This randomness means that observations of radioactive emissions follow a Poisson distribution, but for large numbers of nuclei the decay can be described smoothly by an exponential law. The decay constant λ represents that fixed probability of decay per unit time for one nucleus.
这种随机性意味着对放射性发射的观测遵循泊松分布，但当原子核数量很大时，衰变可以用一个平滑的指数规律来描述。衰变常数 λ 表示一个核在单位时间内衰变的恒定概率。
3. Defining the Decay Constant (λ) | 衰变常数 (λ) 的定义
The decay constant, symbolised by the Greek letter lambda (λ), is defined as the probability per unit time that a given radioactive nucleus will decay. Its SI unit is s⁻¹ or a⁻¹, depending on the timescale used. A larger decay constant indicates a more rapid decay process.
衰变常数，用希腊字母 λ 表示，定义为一个特定放射性核在单位时间内发生衰变的概率。其 SI 单位是 s⁻¹ 或 a⁻¹，取决于所使用的时间尺度。衰变常数越大，表示衰变过程越快。
For a sample containing N undecayed nuclei, the instantaneous rate of decay — called the activity A — is proportional to N. This is expressed by the fundamental differential equation:
对于含有 N 个未衰变核的样品，衰变的瞬时速率——称为活度 A——与 N 成正比。这由基本的微分方程表示：
A = –dN/dt = λN
The negative sign indicates that the number of undecayed nuclei decreases with time. Solving this equation leads to the exponential decay law.
负号表明未衰变核的数量随时间减少。求解该方程即可得到指数衰变定律。
4. Exponential Decay Law | 指数衰变定律
By integrating the equation dN/dt = – the exponential decay formula for the number of radioactive nuclei remaining after time t:
通过对 dN/dt = –λN 进行积分，我们得到经过时间 t 后剩余放射性核数量的指数衰变公式：
N = N0 e–λt
Here N is the number of undecayed nuclei present at time t, N0 l logarithms (approximately 2.718). The same form applies to the mass of a radioactive sample and to the activity, since both are proportional to N.
这里 N 是时间 t 时存在的未衰变核的数量，N0 是 t = 0 时未衰变核的初始数量，e 是自然对数的底（约等于 2.718）。相同的形式也适用于放射性样品的质量和活度，因为它们都与 N 成正比。
Thus the activity A at any time t is given by:
因此任意时间 t 的活度 A 由下式给出：
A = A0 e–λt
where A0 = λN0 is the initial activity. These exponential curves all have the same shape, characterised by the decay constant λ.
其中 A0 = λN0 为初始活度。这些指数曲线都具有相同的形状，其特征由衰变常数 λ 决定。
5. Deriving the Exponential Equation | 推导指数方程
onstant, it is useful to see the integration from the rate equation. Starting with dN/dt = –λN, we separate variables:
为了充分理解衰变常数的含义，看一下从速率方程进行积分会很有帮助。从 dN/dt = –λN 开始，分离变量：
dN/N = –λ dt
Integrating both sides from t = 0 to t = t, with N = N0 at t = 0 and N = N at t = t, gives:
从 t = 0 到 t = t 积分，t = 0 时 N = N0，t = t 时 N = N，得到：
∫N0N dN/N = –λ ∫0t dt
[ln N]N0N = –λt
ln N – ln N0 = –λt
ln(N/N0) = –λt
Exponentiating both sides yields the expected form N = N0 e–λt. This derivation highlights that λ appears in the exponent, controlling how rapidly the number of nuclei declines.
两边取指数即得到预期的形式 N = N0 e–λt。这一推导强调 λ 出现在指数中，控制着核数量的衰减速度。
6. Understanding Half-Life (T½) | 理解半衰期 (T½)
The half-life, denoted by T½ (or sometimes t1/2), is the time required for half of the radioactive nuclei initially present in a sample to decay. Equivalently, it is the time taken for the number of undecayed nuclei, the mass of the sample, or the activity to fall to one half of its original value.
半衰期，记为 T½（或有时用 t1/2），是指样品中最初存在的放射性核有一半发生衰变所需的时间。等效地，它也是未衰变核的数量、样品质量或活度下降到其初始值一半所需的时间。
ile radon-220 has a half-life of only 55.6 seconds. The half-life is independent of the initial number of nuclei; it depends solely on the decay constant λ.
半衰期是比较不同同位素稳定性的一种简便方法。例如，铀-238 的半衰期约为 4.5 × 10⁹ 年，而氡-220 的半衰期仅为 55.6 秒。半衰期与初始核的数量无关；它仅取决于衰变常数 λ。
7. Relating Decay Constant and Half-Life | 衰变常数与半衰期的关系
matical relationship links the decay constant and the half-life. By definition, when t = T½, the number of undecayed nuclei is N = N0/2. Substituting these values into the decay equation gives:
衰变常数和半衰期之间存在直接的数学关系。根据定义，当 t = T½ 时，未衰变核的数量为 N = N0/2。将这些值代入衰变方程得到：
N0/2 = N0 e–λ T½
Dividing both sides by N0 and taking natural logarithms:
两边除以 N0 并取自然对数：
½ = e–λ T½
ln(½) = –λ T½
–ln 2 = –λ T½
λ T½ = ln 2
Hence, the half-life is given by:
因此，半衰期由下式给出：
T½ = ln 2 / λ ≈ 0.693 / λ
This equation shows the inverse relationship: a larger decay constant means a shorter half-life, and vice versa. It is one of the most important results in radioactivity, allowing you to calculate one quantity if the other is known.
该方程显示了反比关系：衰变常数越大，半衰期越短，反之亦然。这是放射性领域最重要的结论之一，可以让你在已知其中一个量的情况下计算另一个量。
8. Graphical Determination of λ and T½ | 用图解法确定 λ 和 T½
The exponential decay of a radioactive source can be verified and analysed using graphs. A plot of N against t gives a smooth curve that halves in height over every half-life interval. To determine the half-life from such a curve, simply read the time interval over which the count rate or activity falls from a chosen value to half that value; repeat for several different starting points to show constancy.
放射源的指数衰变可以通过图形来验证和分析。N 对 t 的曲线是一条平滑曲线，每经过一个半衰期区间，高度便减半。要从这样的曲线确定半衰期，只需读取计数率或活度从选定值下降到其一半时所经历的时间间隔；对不同起始点重复该过程以显示其恒定性。
od uses the linear relationship obtained by taking natural logarithms. Starting from N = N0 e–λt:
一种更精确的方法是利用取自然对数后得到的线性关系。由 N = N0 e–λt 出发：
ln N = ln N0 – λt
If ln N is plotted against t, a straight line with gradient –λ and intercept ln N0 is produced. The half-life can then be found either from the gradient (T½ or directly from the graph by noting the time taken for ln N to decrease by ln 2 (≈0.693).
如果绘制 ln N 对 t 的图，会得到一条斜率为 –λ、截距为 ln N0 的直线。然后可以根据斜率 (T½ = ln 2 / λ) 或直接从图中观察 ln N 减少 ln 2 (≈0.693) 所需的时间来求出半衰期。
Because activity A is proportional to N, a plot of ln A against t also yields a straight line of the same gradient. This is particularly useful in experiments where a Geiger–Müller tube monitors the count rate.
由于活度 A 与 N 成正比，绘制 ln A 对 t 的图也会得到一条相同斜率的直线。这在用盖革-米勒管监测计数率的实验中特别有用。
9. Activity and the Decay Constant | 活度与衰变常数
q = 1 decay per second. Since A = λN, the activity at any instant can be calculated if the decay constant and the number of undecayed nuclei are known.
放射源的活度 A 是单位时间内发生的衰变数。在 SI 单位制中，活度以贝可勒尔 (Bq) 度量，1 Bq = 1 次衰变/秒。由于 A = λN，只要知道衰变常数和未衰变核的数量，就可以计算出任意时刻的活度。
Combining A = λN with the exponential law yields A = A0 e–λt, where A0 = λN0. This shows that activity decays with the same half-life as the number of nuclei. For example, if the activity of a sample is 800 Bq initially, after one half-life it will be 400 Bq, after two half-lives 200 Bq, and so on.
将 A = λN 与指数规律相结合可得 A = A0 e–λt，其中 A0 = λN0。这表明活度以与核数目相同的半衰期衰减。例如，若某样品的初始活度为 800 Bq，经过一个半衰期后变为 400 Bq，经过两个半衰期后变为 200 Bq，以此类推。
10. Carbon-14 Dating Application | 碳-14 定年法应用
One of the most celebrated applications of the decay constant and half-life concept is radiocarbon dating. Carbon-14 (¹⁴C) is a radioactive isotope with a half-life of about 5730 years. It is continuously produced in the atmosphere and incorporated into living organisms. When an organism dies, it stops exchanging carbon with the environment, and the ¹⁴C within its remains decays exponentially.
衰变常数和半衰期概念最著名的应用之一是放射性碳定年法。碳-14 (¹⁴C) 是一种放射性同位素，半衰期约为 5730 年。它在大气中不断产生，并被活着的生物体吸收。当生物死亡后，它停止与环境交换碳，其遗骸内的 ¹⁴C 便呈指数衰变。
iving organism), the age of the sample can be determined using the equation t = (1/λ) ln(N0/N). With T½ = 5730 years, the decay constant is:
通过测量样品中剩余的 ¹⁴C 活度，并将其与初始活度（假定与活体生物的活度相等）进行比较，可以利用公式 t = (1/λ) ln(N0/N) 确定样品的年代。已知 T½ = 5730 年，衰变常数为：
λ = ln 2 / 5730 ≈ 1.21 × 10⁻⁴ year⁻¹
This technique has revolutionised archaeology and geology, enabling the dating of artefacts and fossils up to about 50 000 years old.
这项技术彻底革新了考古学和地质学，使得对距今约五万年以前的文物和化石进行年代测定成为可能。
11. Worked Example | 计算示例
s an initial activity of 6.0 × 10⁵ Bq. The half-life of iodine-131 is 8.0 days. Calculate:
某碘-131 同位素样品初始活度为 6.0 × 10⁵ Bq。碘-131 的半衰期为 8.0 天。计算：
(a) the decay constant λ in day⁻¹ and s⁻¹;
(b) the activity after 24 days.
(a) 衰变常数 λ（以 day⁻¹ 和 s⁻¹ 为单位）；
(b) 24 天后的活度。
Solution | 解答
(a) Using λ = ln 2 / T½:
(a) 使用 λ = ln 2 / T½：
λ = 0.693 / 8.0 = 8.66 × 10⁻² day⁻¹
In SI units, converting 8.0 days to seconds: 8.0 days = 8.0 × 24 × 3600 s = 6.912 × 10⁵ s.
以 SI 单位表示，将 8.0 天换算为秒：8.0 天 = 8.0 × 24 × 3600 s = 6.912 × 10⁵ s。
λ = 0.693 / (6.912 × 10⁵) ≈ 1.00 × 10⁻⁶ s⁻¹
(b) The activity after time t is A = A0 e–λt. For t = 24 days, λt = 0.0866 × 24 = 2.0784.
(b) 时间 t 后的活度为 A = A0 e–λt。当 t = 24 天时，λt = 0.0866 × 24 = 2.0784。
A = 6.0 × 10⁵ × e–2.0784 ≈ 6.0 × 10⁵ × 0.125 = 7.5 × 10⁴ Bq
lved three times: 6.0 × 10⁵ → 3.0 × 10⁵ → 1.5 × 10⁵ → 7.5 × 10⁴ Bq, confirming the result.
注意 24 天恰好是 3 个半衰期，因此活度减半三次：6.0 × 10⁵ → 3.0 × 10⁵ → 1.5 × 10⁵ → 7.5 × 10⁴ Bq，验证了计算结果。
12. Summary and Key Points | 总结与要点
The decay constant λ is the probability of decay per unit time for a single nucleus, and the half-life T½ is the time for the number of undecayed nuclei (or activity) to halve. They are linked by the fundamental equation T½ = ln 2 / λ.
衰变常数 λ 是单个核在单位时间内的衰变概率，而半衰期 T½ 是未衰变核数量（或活度）减半所需的时间。它们通过基本方程 T½ = ln 2 / λ 联系在一起。
Radioactive decay follows an exponential law N = N0 e–λt, derived from the assumption that the rate of decay is proportional to the number of radioactive nuclei present. The same exponential form applies to activity and to the mass of the radioactive substance. The half-life is independent of the initial number of nuclei and is constant for a given isotope, making it a powerful tool for dating and for predicting the behaviour of radioactive materials in fields ranging from medicine to nuclear power.
放射性衰变遵循指数规律 N = N0 e–λt，该规律源于衰变速率与存在的放射性核数目成正比这一假设。相同的指数形式也适用于活度和放射性物质的质量。半衰期与初始核数目无关，对于给定的同位素是恒定的，这使其成为一种强大的工具，可用于年代测定以及预测从医学到核电等各个领域中放射性物质的行为。
Key formulas to remember:
要记住的关键公式：
- Activity: A = λN; A = A0 e–λt / 活度：A = λN; A = A0 e–λt
- Decay law: N = N0 e–λt / 衰变定律：N = N0 e–λt
- Half-life: T½ = ln 2 / λ / 半衰期：T½ = ln 2 / λ
- Linear form: ln N = ln N0 – λt / 线性形式：ln N = ln N0 – λt
Always ensure that the units of λ and time are consistent when performing calculations. With these concepts, you can confidently tackle any CIE A-Level question on decay constant and half-life.
在进行计算时，务必确保 λ 和时间的单位一致。掌握了这些概念，你就能自信地应对任何 CIE A-Level 中关于衰变常数和半衰期的问题。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图