Decay Constant and Half-Life | 衰变常数与半衰期

📚 Decay Constant and Half-Life | 衰变常数与半衰期

Radioactive decay is a random and spontaneous process in which an unstable nucleus transforms into a more stable configuration by emitting radiation. In A-Level physics, the behaviour of radioactive substances is described mathematically using two key quantities: the decay constant λ and the half-life t½. These two quantities are intimately connected, and together they allow us to predict how the number of radioactive nuclei in a sample changes with time.

放射性衰变是不稳定核通过发射辐射而转变为更稳定组态的随机自发过程。在A-Level物理中，我们通过两个关键量来描述放射性物质的行为：衰变常数λ和半衰期t½。这两个量密切相关，它们使我们能够预测样品中放射性核数目随时间的变化。

1. Randomness and Statistical Nature of Decay | 衰变的随机性与统计规律性

It is impossible to predict which individual nucleus will decay next or exactly when it will decay. Decay is governed entirely by probability. However, when a very large number of identical radioactive nuclei are present, the overall decay rate shows a definite and predictable statistical pattern.

我们无法预测下一个发生衰变的是哪一个核，也无法预测它将在何时衰变。衰变完全由概率支配。然而，当存在大量相同的放射性核时，整体的衰变速率会表现出明确且可预测的统计规律。

This statistical law is reflected in the decay constant, which represents the probability per unit time that a single nucleus will decay. For example, a decay constant of 0.02 s⁻¹ means that each nucleus has a 2% chance of decaying in every second, assuming independent events.

这一统计规律通过衰变常数体现出来，它表示单个核在单位时间内发生衰变的概率。例如，衰变常数为0.02 s⁻¹意味着在假设事件相互独立的情况下，每个核在每一秒内有2%的概率发生衰变。

2. The Decay Constant λ | 衰变常数λ

The decay constant λ is defined as the probability per unit time of a nucleus decaying. For a sample containing N radioactive nuclei, the rate of decrease of N is proportional to N. This can be written as:

衰变常数λ定义为核在单位时间内发生衰变的概率。对于含有N个放射性核的样品，N的减少速率与N成正比。这可以写为：

dN/dt = −λN

Here, dN/dt is the instantaneous rate of change of the number of undecayed nuclei. The negative sign shows that N is decreasing. The decay constant λ is a positive constant that is characteristic of the particular radioactive nuclide under consideration.

其中dN/dt是未衰变核数目N的瞬时变化率。负号表示N在减少。衰变常数λ是正数，它是所讨论的特定放射性核素的特征量。

Importantly, λ is independent of environmental conditions such as temperature, pressure, or chemical state. It depends only on the nature of the nucleus. This makes radioactive decay a very reliable physical process.

重要的是，λ不依赖于环境条件，如温度、压力或化学状态。它只取决于原子核本身的性质。这使得放射性衰变成为一种非常可靠的物理过程。

3. The Exponential Decay Law | 指数衰减定律

The differential equation dN/dt = −λN can be solved to give the exponential decay law. If N₀ is the initial number of nuclei at time t = 0, then the number N remaining after time t is:

微分方程 dN/dt = −λN 可以求解，得到指数衰减定律。若N₀是t = 0时的初始核数目，则时间t后剩余的核数目N为：

N = N₀e^−λt

This equation shows that the number of undecayed nuclei decreases exponentially with time. It is a continuous, smooth curve that asymptotically approaches zero but never actually reaches zero in a finite time.

这个方程表明未衰变核的数目随时间呈指数衰减。它是一条连续光滑的曲线，渐近地接近零，但在有限时间内永远不会实际达到零。

Taking the natural logarithm of both sides gives ln(N/N₀) = −λt, or equivalently ln N = ln N₀ − λt. Thus a graph of ln N against time t produces a straight line with gradient −λ and intercept ln N₀. This linearised form is extremely useful in experiments.

对等式两边取自然对数，可得 ln(N/N₀) = −λt，或等价地 ln N = ln N₀ − λt。因此，以ln N对时间t作图会得到一条直线，斜率为−λ，截距为ln N₀。这种线性化形式在实验中非常有用。

4. Half-Life t½ | 半衰期t½

The half-life of a radioactive nuclide is defined as the time taken for the number of undecayed nuclei to decrease to half of its original value. It is denoted by t½ and is often more intuitive to use than the decay constant.

放射性核素的半衰期定义为未衰变核的数目减少到原来一半所需的时间。用t½表示，它通常比衰变常数更直观。

For example, if a sample initially contains 1000 nuclei and t½ = 2 hours, then after 2 hours 500 nuclei remain, after 4 hours 250 remain, after 6 hours 125 remain, and so on. In general, after n half-lives, the remaining fraction is (1/2)ⁿ.

例如，若样品最初含有1000个核，且t½ = 2小时，则2小时后剩余500个核，4小时后剩余250个，6小时后剩余125个，依此类推。一般来说，经过n个半衰期后，剩余比例为(1/2)ⁿ。

It is important to understand that the half-life is a constant for a given nuclide; it does not change with the age of the sample, the amount of material, or any external conditions. This constancy is the basis for many dating techniques.

重要的是要理解，半衰期对给定的核素是一个常数；它不随样品的年龄、物质的数量或任何外部条件而改变。这种恒定性是许多测年方法的基础。

5. Relationship Between λ and t½ | 衰变常数与半衰期的关系

Since the half-life occurs when N = N₀/2, we can substitute this into the exponential decay law:

由于当N = N₀/2时对应半衰期，我们可以将其代入指数衰减定律：

N₀/2 = N₀e^−λt½

Dividing by N₀ and taking the natural logarithm of both sides gives ln(1/2) = −λt½, so −ln2 = −λt½. Hence:

两边除以N₀并取自然对数，得到 ln(1/2) = −λt½，因此 −ln2 = −λt½。于是：

t½ = ln2 / λ ≈ 0.693 / λ

This relationship is fundamental. It shows that a short half-life corresponds to a large decay constant, meaning the nuclei decay rapidly. Conversely, a long half-life corresponds to a small decay constant.

这个关系非常重要。它表明短半衰期对应大的衰变常数，意味着核衰变得快。相反，长半衰期对应小的衰变常数。

6. Activity A | 活度A

The activity A of a radioactive sample is defined as the number of decays per unit time. From the decay law, the activity is given by:

放射性样品的活度A定义为每单位时间内发生衰变的次数。根据衰变定律，活度由下式给出：

A = |dN/dt| = λN

Since N = N₀e^−λt, the activity also follows the same exponential decay law:

由于N = N₀e^−λt，活度也遵循相同的指数衰减定律：

A = A₀e^−λt

Here A₀ is the initial activity. Activity is directly proportional to the number of undecayed nuclei, so measuring activity with a detector is a practical way to monitor the decay process.

这里A₀是初始活度。活度与未衰变核的数目成正比，因此用探测器测量活度是监测衰变过程的一种实用方法。

7. Units and Dimensions | 单位与量纲

The decay constant λ has the unit of reciprocal time, typically s⁻¹. The half-life t½ has the unit of time, for example seconds, hours, or years. The activity A is measured in becquerel (Bq), where 1 Bq is equal to one decay per second.

衰变常数λ的单位是时间的倒数，通常为s⁻¹。半衰期t½的单位是时间，例如秒、小时或年。活度A的单位是贝克勒尔（Bq），1 Bq等于每秒一次衰变。

For example, for carbon-14, t½ = 5730 years. The corresponding decay constant can be calculated as:

例如，对于碳-14，t½ = 5730年。相应的衰变常数可以计算为：

λ = ln2 / (5730 × 365 × 24 × 3600) ≈ 3.83 × 10⁻¹² s⁻¹

This very small value means that carbon-14 decays extremely slowly, which is why it is useful for dating ancient organic materials.

这个值非常小，意味着碳-14衰变极其缓慢，这正是它可用于测年古代有机材料的原因。

8. Experimental Determination of Half-Life | 半衰期的实验测定

To measure the half-life of a short-lived radioactive isotope, a detector such as a Geiger-Müller tube and counter is used to record the count rate (proportional to activity) over time. The background radiation count must first be subtracted from all measurements.

要测量短寿命放射性同位素的半衰期，通常使用盖革-米勒管和计数器记录随时间变化的计数率（与活度成正比）。必须先扣去所有测量中的本底辐射计数。

The net count rate is then plotted against time. For a pure sample, the decay curve is exponential. The half-life can be read directly from the graph as the time over which the count rate halves. Alternatively, plotting the natural logarithm of count rate against time gives a straight line of gradient −λ, from which t½ is obtained using t½ = ln2/λ.

然后绘制净计数率对时间的曲线。对于纯样品，衰变曲线是指数曲线。半衰期可直接从图中读出，即计数率减半所经过的时间。或者，绘制计数率的自然对数对时间的曲线，得到斜率为−λ的直线，再利用t½ = ln2/λ求得半衰期。

This method is reliable when the half-life is between several seconds and a few years. For very long half-lives, the change in activity over a reasonable measurement time is too small to be observed easily, and other methods are needed.

当半衰期在几秒到几年之间时，这种方法可靠。对于非常长的半衰期，在合理的测量时间内活度的变化太小，难以直接观察，需要采用其他方法。

9. Application: Radiometric Dating | 应用：放射性测年

The decay law allows scientists to determine the age of ancient objects by comparing the current activity with the initial activity. For carbon-14 dating, living organisms maintain a constant ratio of carbon-14 to carbon-12 because they continuously take in carbon from the atmosphere. Once the organism dies, the carbon-14 is no longer replenished and its activity declines exponentially.

衰变定律使科学家能够通过比较当前活度与初始活度来确定古代物体的年龄。对于碳-14测年，活生物体由于不断从大气中摄取碳，其碳-14与碳-12的比例保持恒定。一旦生物体死亡，碳-14不再得到补充，其活度便呈指数衰减。

If an ancient sample has an activity A and the initial activity (from a modern sample with the same mass) is A₀, then the age t is found from:

如果古代样品的活度为A，而初始活度（取同等质量现代样品的活度）为A₀，则年龄t可由下式求出：

t = (1/λ) ln(A₀/A)

Carbon-14 is suitable for dating organic materials up to about 50,000 years old, because after this time the remaining activity becomes very small and difficult to measure accurately. For older samples, isotopes with longer half-lives, such as potassium-40 or uranium-238, are used.

碳-14适用于测定约5万年以内的有机材料，因为超过这个时间后，剩余活度变得很小，难以准确测量。对于更老样品，则使用半衰期更长的同位素，如钾-40或铀-238。

10. Decay Chains and Effective Half-Life | 衰变链与有效半衰期

Some radioactive nuclei decay into daughter nuclei that are also radioactive, producing a decay chain. In such cases the overall decay is more complex because the daughter nuclei both form and decay simultaneously. The simple exponential law applies only to the first decay step unless the daughter is stable.

有些放射性核衰变成的子核也具有放射性，形成衰变链。在这种情况下，整体衰变更复杂，因为子核同时形成并衰变。简单的指数定律只适用于第一步衰变，除非子核是稳定的。

In medical applications involving radioactive substances inside the body, elimination can occur both by physical decay and by biological excretion. The effective decay constant is the sum of the physical decay constant λ_phys and the biological clearance constant λ_bio:

在涉及体内放射性物质的医学应用中，清除可以同时通过物理衰变和生物排出发生。有效衰变常数是物理衰变常数λ_phys和生物清除常数λ_bio之和：

λ_eff = λ_phys + λ_bio

The corresponding effective half-life is then shorter than either the physical or biological half-life alone. This concept is essential for calculating radiation doses in radiotherapy and diagnostics.

相应的有效半衰期比单独的物理半衰期或生物半衰期都要短。这一概念对于计算放疗和诊断中的辐射剂量至关重要。

11. Common Exam Points and Pitfalls | 常见考点与易错点

One of the most common errors is confusing the decay constant λ with the activity A. The decay constant is a probability per unit time, independent of sample size, while activity is the total number of decays per second and depends on both λ and N.

最常见的错误之一是混淆衰变常数λ与活度A。衰变常数是单位时间的概率，与样品大小无关；而活度是每秒总衰变次数，取决于λ和N两者。

Students must remember that the half-life of a given nuclide is invariant; it does not change with temperature, pressure, chemical state, or the age of the sample. It is also important to use the correct base for natural logarithms and to convert units of time consistently, especially when λ is in s⁻¹ and t½ is in years.

学生必须记住，给定核素的半衰期是不变的；它不随温度、压力、化学状态或样品年龄而改变。同样重要的是使用正确的自然对数底数，并统一时间单位，特别是当λ以s⁻¹为单位而t½以年为单位时。

When solving numerical problems, always quote the formula N = N₀e^−λt or A = A₀e^−λt, identify which quantities are given, and check that the argument of the exponential is dimensionless. Drawing a graph of lnN against t can often simplify the analysis and help avoid algebraic mistakes.

在解数值题时，一定要写出公式 N = N₀e^−λt 或 A = A₀e^−λt，确认哪些量是已知的，并检查指数中的量纲是否为1。绘制lnN对t的图通常能简化分析，避免代数错误。

Finally, remember that the exponential decay model applies to large numbers of nuclei. For a sample with only a few nuclei, statistical fluctuations become significant, and the smooth exponential curve is no longer a valid description.

最后，请记住指数衰变模型适用于大量核。对于只有少数核的样品，统计涨落变得显著，光滑的指数曲线不再适用。

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Decay Constant and Half-Life | 衰变常数与半衰期