Decay Graphs and Equation Analysis | 衰变图像与方程分析

📚 Decay Graphs and Equation Analysis | 衰变图像与方程分析

Radioactive decay is a random, spontaneous process in which unstable nuclei emit radiation to become more stable. The mathematics of decay allows physicists to predict how the number of radioactive nuclei, the activity, and the count rate change with time. In this article, we will explore the fundamental decay equation, interpret decay graphs, and learn how to analyse them for A-Level Physics.

放射性衰变是一个随机、自发的过程，不稳定的原子核通过发射辐射而变得更加稳定。衰变的数学描述使物理学家能够预测放射性核数目、活度和计数率随时间的变化。在本文中，我们将深入探讨基本衰变方程、解读衰变图像，并学习如何为 A-Level 物理考试分析这些图像。

1. The Law of Radioactive Decay | 放射性衰变定律

The law of radioactive decay states that the rate of decay of a radioactive isotope is directly proportional to the number of undecayed nuclei present at that instant. This can be written as:

放射性衰变定律指出：放射性同位素的衰变速率与当前时刻未衰变的原子核数目成正比。可写为：

dN/dt = −λN

Here, N is the number of undecayed nuclei, t is time, λ is the decay constant, and the negative sign indicates that N decreases with time. The decay constant λ represents the probability of decay per unit time, and it is characteristic of each radioactive isotope.

其中 N 是未衰变核的数目，t 是时间，λ 是衰变常数，负号表示 N 随时间减少。衰变常数 λ 表示单位时间内每个核发生衰变的概率，是每种放射性同位素的特征量。

2. The Decay Equation and Its Solution | 衰变方程及其解

The differential equation dN/dt = −λN is solved by separating variables and integrating:

微分方程 dN/dt = −λN 可通过分离变量并积分求解：

∫(1/N) dN = −∫λ dt

Integrating gives ln N = −λt + C. If N = N₀ at t = 0, then C = ln N₀. Therefore:

积分后得 ln N = −λt + C。若在 t = 0 时 N = N₀，则 C = ln N₀。因此：

N = N₀ e^−λt

This exponential decay equation is the foundation of all decay graph analysis. It shows that the number of undecayed nuclei falls exponentially, never reaching zero in theory.

这个指数衰变方程是所有衰变图像分析的基础。它表明未衰变核的数目按指数方式减少，理论上永远不会减少到零。

3. Half-Life and Decay Constant | 半衰期与衰变常数

The half-life T½ is defined as the time taken for the number of undecayed nuclei (or the activity) to reduce to half its initial value. From the decay equation:

半衰期 T½ 定义为未衰变核数目（或活度）减少到初始值一半所需的时间。由衰变方程：

½ N₀ = N₀ e^−λT½

Taking natural logs gives:

取自然对数得到：

λT½ = ln 2, so T½ = ln 2 / λ

Alternatively, λ = ln 2 / T½. This relationship is essential for converting between half-life and decay constant in calculations.

反过来，λ = ln 2 / T½。这个关系对于在计算中联系半衰期和衰变常数至关重要。

4. The Decay Curve: N Against t | 衰变曲线：N-t 图

The graph of N against t for a radioactive isotope shows an exponential decay curve. It starts at N₀, decreases rapidly at first, and then approaches zero asymptotically. The half-life can be read directly from the graph by finding the time at which N has fallen to half the initial value.

放射性同位素的 N-t 图像是一条指数衰减曲线。它从 N₀ 开始，初期下降很快，随后逐渐趋近于零。通过找到 N 降至初始值一半的时刻，可以直接从图像上读取半衰期。

Key features of the decay curve:

衰变曲线的主要特征：

The gradient is always negative and its magnitude decreases with time.

斜率始终为负，且斜率大小随时间减小。
The curve never reaches zero; it only approaches the time axis asymptotically.

曲线永远不会达到零，只是渐近地接近时间轴。
After each half-life, the remaining fraction is ½, ¼, ⅛, 1/16, etc.

每经过一个半衰期，剩余比例依次为 ½、¼、⅛、1/16……

5. Linearised Graph: ln N Against t | 线性化图像：ln N-t 图

Taking the natural logarithm of both sides of N = N₀ e^−λt gives:

对 N = N₀ e^−λt 两边取自然对数，可得：

ln N = ln N₀ − λt

This is the equation of a straight line. A graph of ln N against t has a gradient of −λ and a y-intercept of ln N₀. Using this linear form is often more accurate than reading an exponential curve because it uses all data points and makes systematic errors easier to spot.

这是一条直线的方程。以 ln N 对 t 作图，得到斜率为 −λ、纵截距为 ln N₀ 的直线。使用线性形式通常比直接读取指数曲线更准确，因为它利用了所有数据点，并且更容易发现系统误差。

6. Activity and Count Rate | 活度与计数率

The activity A of a radioactive source is defined as the number of decays per second. It is proportional to N, so it also decays exponentially:

放射性源的活度 A 定义为每秒发生的衰变次数。它与 N 成正比，因此也按指数方式衰变：

A = A₀ e^−λt, where A = λN

The count rate R measured by a detector is usually proportional to the activity, assuming a fixed geometry and no dead-time losses. Hence count rate also follows the same exponential law. In experiments, the background radiation must be subtracted from the measured count rate before plotting the decay graph.

探测器的计数率 R 通常与活度成正比（假设几何条件固定且无死时间损失）。因此计数率也遵循相同的指数规律。在实验中，绘制衰变图像之前必须从测量到的计数率中减去本底辐射。

7. Writing Nuclear Decay Equations | 核衰变方程的书写

Analysing decay graphs often goes hand-in-hand with understanding the underlying decay equations. For α decay, a nucleus emits an α particle (⁴₂He), so the mass number decreases by 4 and the atomic number decreases by 2:

分析衰变图像往往需要同时理解其背后的衰变方程。对于 α 衰变，原子核放出 α 粒子（⁴₂He），因此质量数减少 4，原子序数减少 2：

²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He

For β⁻ decay, a neutron converts into a proton, emitting an electron and an antineutrino. The mass number stays the same, but the atomic number increases by 1:

对于 β⁻ 衰变，一个中子转化为质子，同时放出一个电子和一个反中微子。质量数不变，原子序数增加 1：

¹⁴₆C → ¹⁴₇N + ⁰₋₁e + ν̄

γ decay often accompanies α or β decay; it involves emission of a high-energy photon with no change in mass or atomic number. Balancing both mass number and atomic number is essential when writing any nuclear equation.

γ 衰变常伴随 α 或 β 衰变发生；它放出高能光子，质量数和原子序数均不改变。书写任何核方程时，保持质量数和原子序数守恒都是关键。

8. Analysing Decay Graphs to Find Half-Life | 从图像求半衰期

There are two common graphical methods to determine half-life from experimental data:

从实验数据确定半衰期有两种常用的图像方法：

Method 1: Direct reading from the N-t curve. Find the time when N equals N₀/2, then again when it equals N₀/4, etc. The average of these time intervals gives T½.

方法一：从 N-t 曲线上直接读取。找到 N 等于 N₀/2 的时刻，再找到 N 等于 N₀/4 的时刻，依此类推。这些时间间隔的平均值即为 T½。
Method 2: Plot ln N against t. The gradient is −λ, so T½ = ln 2 / λ. This method is more reliable because all data are used and the linear fit averages out random errors.

方法二：绘制 ln N-t 图。斜率为 −λ，因此 T½ = ln 2 / λ。这种方法更可靠，因为它使用了所有数据点，并且线性拟合可以平均随机误差。

When the background radiation is significant, the measured count rate C includes background B. The corrected count rate is C − B, and this corrected value should be used in the exponential analysis.

当本底辐射显著时，测量计数率 C 包含本底 B。校正后的计数率为 C − B，应使用该校正值进行指数分析。

9. Typical Exam Question Walk-Through | 典型考题分析

Consider a sample of a radioactive isotope with an initial activity of 800 Bq. After 120 s, the activity falls to 100 Bq. Find the half-life.

考虑一个放射性样品，初始活度为 800 Bq。经过 120 s 后，活度降至 100 Bq。求其半衰期。

Since 800 → 400 → 200 → 100, three half-lives have passed in 120 s. Therefore T½ = 120 / 3 = 40 s.

由于 800 → 400 → 200 → 100，120 s 内经过了三个半衰期。因此 T½ = 120 / 3 = 40 s。

Alternatively, use the decay equation: 100 = 800 e^−λ×120. Taking logs gives λ = (ln 8)/120 = 0.01733 s⁻¹, and T½ = ln 2 / λ ≈ 40.0 s.

另一种方法：使用衰变方程 100 = 800 e^−λ×120。取对数得 λ = (ln 8)/120 = 0.01733 s⁻¹，因此 T½ = ln 2 / λ ≈ 40.0 s。

In an exam, always state whether you are using the “fractional reduction” method or the exponential equation method, and show your working clearly.

考试中，务必说明你使用的是“比例减半”方法还是指数方程方法，并清晰展示计算过程。

10. Common Mistakes and Tips | 常见错误与提示

Students often make the following mistakes in decay graph analysis:

学生在衰变图像分析中常犯以下错误：

Forgetting to subtract background radiation from count-rate data before plotting.

绘制图像前忘记从计数率数据中减去本底辐射。
Using N₀/2, N₀/3, N₀/4 as equally spaced fractions; remember that decay is exponential, so equal time intervals correspond to equal ratios, not equal differences.

错误地将 N₀/2、N₀/3、N₀/4 当作等间隔的分数；记住衰变是指数式的，相等时间间隔对应相等比例，而不是相等差值。
Misreading the gradient of the ln N graph; the gradient is −λ, not λ.

读错 ln N 图像斜率；斜率为 −λ，而不是 λ。
Confusing half-life with mean life. The mean life τ = 1/λ, while T½ = ln 2 / λ.

混淆半衰期与平均寿命。平均寿命 τ = 1/λ，而 T½ = ln 2 / λ。

11. Summary | 总结

The decay equation N = N₀ e^−λt and its linear form ln N = ln N₀ − λt are powerful tools for analysing radioactive decay. By understanding the shape of the decay curve, using linearisation to determine λ, and correctly writing decay equations, you can solve a wide range of exam problems with confidence.

衰变方程 N = N₀ e^−λt 及其线性形式 ln N = ln N₀ − λt 是分析放射性衰变的强大工具。通过理解衰变曲线的形状、使用线性化方法确定 λ，并正确书写衰变方程，你可以自信地解决各类考试问题。

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Decay Graphs and Equation Analysis | 衰变图像与方程分析