📚 Decay Graphs and Equations | 衰变图与方程
Radioactive decay is a spontaneous and random process in which an unstable nucleus emits particles or electromagnetic radiation to become more stable. This process follows precise mathematical laws that govern how the number of undecayed nuclei changes with time. Understanding decay graphs and the associated exponential equations is essential for A-Level Physics, as they allow us to model the behaviour of radioactive samples, determine half-lives, and apply these ideas to carbon dating and medical uses.
放射性衰变是一种自发的、随机的过程，不稳定的原子核通过释放粒子或电磁辐射变得更稳定。这一过程遵循精确的数学规律，描述了未衰变核的数量如何随时间变化。理解衰变图以及相关的指数方程对于 A-Level 物理至关重要，因为我们可以借此模拟放射性样品的行为，测定半衰期，并将其应用于碳定年和医疗用途。
1. Introduction to Radioactive Decay | 放射性衰变简介
Radioactive decay is fundamentally governed by quantum mechanics, meaning we cannot predict when any single nucleus will decay. However, for a large number of identical nuclei, the rate of decay is proportional to the number of nuclei present. This leads to an exponential decrease in the population of parent nuclei over time. The randomness is often illustrated by the fact that decay events follow a Poisson distribution at the single-event level, yet the overall activity decays smoothly.
放射性衰变从根本上受量子力学支配，这意味着我们无法预言某个特定的核何时会衰变。不过，对于大量相同的核，其衰变速率与当前存在的核数量成正比。这导致母核数量随时间呈指数减少。这种随机性常体现在单个事件服从泊松分布，但总体活度却平稳地下降。
2. The Exponential Decay Law | 指数衰变定律
ime t = 0, the number N remaining after time t is given by the exponential decay equation:
N = N₀ e–λt
where λ (lambda) is the decay constant, a quantity characteristic of each radioactive isotope. This equation tells us that the number of parent nuclei decays exponentially: initially the decay is rapid because many nuclei are present, and it slows down as fewer remain.
如果我们从 t = 0 时刻有 N₀ 个未衰变核开始，经过时间 t 后剩余的核数量 N 由指数衰变方程给出：
N = N₀ e–λt
式中 λ（拉姆达）是衰变常数，是每种放射性同位素的特征量。该方程告诉我们，母核数量按指数规律减少：初始时衰变很快，因为现存核很多，而随着剩余数目减少，衰变慢下来。
3. Decay Constant (λ) | 衰变常数 (λ)
The decay constant λ has units of inverse time (e.g. s⁻¹, min⁻¹, yr⁻¹). It represents the probability per unit time that a single nucleus will decay. A large λ means the isotope is highly unstable and decays quickly. In an A-Level problem, you may be given λ or be asked to determine it from a half-life. Always remember that λ is constant for a given nuclide and does not change with temperature, pressure, or chemical bonding.
衰变常数 λ 的单位是时间的倒数（如 s⁻¹, min⁻¹, yr⁻¹）。它表示单个核在单位时间内发生衰变的概率。λ 很大意味着同位素极不稳定，衰变很快。在 A-Level 题目中，可能会直接给出 λ 或要求从半衰期算出 λ。务必记住，对于给定的核素，λ 是常数，不随温度、压强或化学键改变。
4. Half-Life (T½) | 半衰期
ial value. After one half-life, N = N₀/2; after two half-lives, N = N₀/4; after n half-lives, N = N₀ (½)ⁿ. Half-life is a very intuitive way to express the rate of decay and is particularly useful when reading decay graphs.
半衰期 T½ 指的是样品中一半的核发生衰变——或者说未衰变核数目降至初始值一半——所需的时间。经过一个半衰期，N = N₀/2；经过两个半衰期，N = N₀/4；经过 n 个半衰期，N = N₀ (½)ⁿ。半衰期是表达衰变速率的直观方式，在解读衰变图时特别有用。
5. Deriving T½ = ln2/λ | 推导 T½ = ln2/λ
Starting from N = N₀ e–λt, set N = N₀/2 at t = T½:
N₀/2 = N₀ e–λ T½
Cancel N₀ and take the natural logarithm of both sides:
ln(½) = –λ T½
Since ln(½) = –ln2, we obtain:
T½ = ln2 / λ ≈ 0.693 / λ
This relation is fundamental and appears in almost every radioactive decay calculation. It shows that the half-life is inversely proportional to the decay constant: a short half-life corresponds to a large λ.
从 N = N₀ e–λt 出发，设 t = T½ 时 N = N₀/2：
N₀/2 = N₀ e–λ T½
消去 N₀，两边取自然对数：
ln(½) = –λ T½
因为 ln(½) = –ln2，得到：
T½ = ln2 / λ ≈ 0.693 / λ
这一关系是基础，几乎出现在每一道放射性衰变计算题中。它表明半衰期与衰变常数成反比：半衰期越短，λ 越大。
6. Activity and the Decay Equation | 活度与衰变方程
y per second. Activity is directly proportional to the number of undecayed nuclei present:
A = λ N
Since N follows exponential decay, activity also decays exponentially:
A = A₀ e–λt
where A₀ is the initial activity. A common exam task is to read activity from a graph or to calculate the time needed for the activity to drop to a particular level.
放射源的活度 A 是指单位时间内发生的衰变次数，单位是贝克勒尔 (Bq)，1 Bq = 每秒衰变一次。活度与现存未衰变核数目成正比：
A = λ N
由于 N 按指数规律衰变，活度同样按指数规律衰变：
A = A₀ e–λt
其中 A₀ 是初始活度。考试中常见任务是读取活度-时间图像，或计算活度降至某一水平所需的时间。
7. N–t and A–t Decay Graphs | N–t 与 A–t 衰变图
A graph of the number of undecayed nuclei N against time t is a smooth curve that starts at N₀ and asymptotically approaches zero. The shape is identical to that of the activity–time graph, except the vertical axis is A instead of N. On such graphs, the half-life can be read directly: pick any value of N (or A), find the time taken for it to halve, and that interval is T½. The curve never crosses the horizontal axis, reflecting the theoretical nature of exponential decay.
未衰变核数目 N 随时间 t 变化的图像是一条光滑曲线，起点为 N₀，以渐近线方式趋近于零。活度–时间图像的形状与之完全相同，只不过纵轴换成 A。在这类图上可以直接读出半衰期：任选一个 N（或 A）值，找到其减半所用的时间，该时间间隔就是 T½。曲线永远不会穿过横轴，这体现了指数衰变的理论特征。
8. Using Graphs to Find Half-Life | 利用图像求半衰期
To obtain an accurate T½ from a graph, choose a clear initial value, such as N = 100 %, and note the time when N = 50 %. The difference in time is T½. Then repeat between 50 % and 25 % tial model and is often used to confirm that a sample contains only a single radioisotope.
为了从图像中得到准确的 T½，选择一个清晰的初始值，例如 N = 100 %，记下 N = 50 % 所对应的时间，二者之差即为 T½。然后在 50 % 与 25 % 之间重复此操作；若衰变确实服从指数规律，这些间隔应当相等。半衰期的恒定是对指数模型的一种检验，常用于确认样品中只含单一放射性同位素。
9. Semi-Logarithmic Plots | 半对数坐标图
Taking natural logs of N = N₀ e–λt gives:
ln N = ln N₀ – λ t
This is the equation of a straight line with gradient –λ and y-intercept ln N₀. Plotting ln N against t therefore yields a linear graph, which makes it much easier to determine λ and identify any deviations from pure exponential behaviour. In the laboratory, you might use such a plot to correct for background radiation or to confirm half-life values.
对 N = N₀ e–λt 两边取自然对数得到：
ln N = ln N₀ – λ t
这是一条斜率为 –λ、纵截距为 ln N₀ 的直线方程。因此，绘制 ln N 对 t 的图像会得到线性关系，从而更易于确定 λ 以及识别与纯粹指数行为的偏离。在实验中，你可以利用这样的图校正本底辐射或验证半衰期值。
10. Background Radiation Correction | 本底辐射校正
When measuring the count rate from a radioactive source, a detector will always record some background counts due to natural radiation from the environment and cosmic rays. The true activity of the source is obtained by subtracting the background count rate from the measured count rate. In decay graphs, failing to subtract background leads to a curve that does not fall to zero but rather approaches a constant background level. For accurate half-life determination, always correct the data before plotting.
在测量放射源的计数率时，探测器总会记录到一些由环境中的天然辐射和宇宙射线产生的本底计数。将实测计数率减去本底计数率，才能得到源的真实活度。在衰变图中，若不扣除本底，曲线就不会降至零，而是趋近于一个恒定的本底水平。为准确测定半衰期，务必在作图前对数据进行本底校正。
11. Radioactive Dating Applications | 放射性测年应用
tion of decay equations. Living organisms maintain a constant proportion of ¹⁴C (half-life ~ ¹⁴C decays exponentially. By measuring the current activity of a sample and using A = A₀ e–λt, the age can be estimated. Other dating methods use isotopes with much longer half-lives, such as uranium-lead dating for rocks. All rely on the same exponential decay principles.
碳-14 定年是衰变方程的经典应用。活着的生物通过与大气的碳交换维持恒定的 ¹⁴C 比例（半衰期约 5730 年）。生物死亡后，¹⁴C 便按指数规律衰变。通过测量样品当前的活度并利用 A = A₀ e–λt，就可以估算出年代。其他定年方法使用半衰期长得多的同位素，例如铀–铅法测定岩石年龄。所有这些方法都依赖同样的指数衰变原理。
12. Key Equations and Summary | 关键方程与总结
Mastering decay graphs and equations means being comfortable with the following:
- Exponential decay: N = N₀ e–λt, A = A₀ e–λt
- Activity link: A = λ N
- Half-life: T½ = ln2 / λ
- Linearised form: ln N = ln N₀ – λ t
- Graphical features: exponential curves, constant half-life on N–t or A–t graphs, background correction, semi-log straight-line plots
Always check that your calculated half-life remains constant over several intervals. Remember that λ is an intrinsic property, while activity depends on the sample size. With these tools, you can analyse any radioactive decay problem that appears in CIE A-Level Physics.
掌握衰变图与方程意味着要熟练运用以下内容：
- 指数衰变：N = N₀ e–λt, A = A₀ e–λt
- 活度关系：A = λ N
- 半衰期：T½ = ln2 / λ
- 线性化形式：ln N = ln N₀ – λ t
- 图形特征：指数曲线、N–t 或 A–t 图上恒定的半衰期、本底校正、半对数直线图
始终检查你算出的半衰期在多个区间内是否保持不变。记住 λ 是内禀属性，而活度取决于样品大小。凭借这些工具，你就能分析 CIE A-Level 物理中出现的任何放射性衰变问题。
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