A-Level Physics: Unit 5 Jan22 Practical Investigation – Radioactive Decay and Half-Life Measurement | A-Level 物理：单元5 Jan22 实验探究

📚 A-Level Physics: Unit 5 Jan22 Practical Investigation – Radioactive Decay and Half-Life Measurement | A-Level 物理：单元5 Jan22 实验探究 – 放射性衰变与半衰期测定

This article breaks down the key skills and analysis needed for the experimental investigation question from the January 2022 Unit 5 paper. We use a classic experiment – measuring the half-life of a short-lived radioactive source – to illustrate data collection, graphical analysis, uncertainty evaluation, and exam-ready technique.

本文详细拆解 2022 年 1 月单元 5 试卷中的实验探究题型。我们以经典实验——测量短半衰期放射源的半衰期——为例，展示数据采集、图形分析、不确定度评估以及应试技巧。

1. Experiment Overview | 实验概述

The experiment aims to determine the half-life of a radioactive isotope such as protactinium-234 (Pa-234) or radon-220 (Rn-220) by recording the decay count rate over time and correcting for background radiation.

本实验旨在通过记录放射性同位素（如镤-234 或氡-220）的衰变计数率随时间的变化，并修正本底辐射，测定其半衰期。

A Geiger-Müller (GM) tube connected to a counter or data logger is used to measure the number of decays per unit time. A solvent extraction method may be employed to separate the short-lived daughter nuclide, giving a rapidly changing count rate.

使用连接至计数器或数据记录仪的盖革-米勒（GM）计数管测量单位时间内的衰变次数。可采用溶剂萃取法分离出短寿命子体核素，从而获得快速变化的计数率。

The investigation ties directly into the ‘Physics from Creation to Collapse’ unit, testing knowledge of radioactivity, exponential decay, and practical skills such as timing, reducing random errors, and plotting ln graphs.

该探究直接关联“从创生到坍缩的物理”单元，考察放射性、指数衰减以及计时、减少随机误差和绘制自然对数图等实操技能。

2. Background Theory | 理论背景

Radioactive decay follows the exponential law: N = N₀ e^(−λt), where N is the number of undecayed nuclei at time t, N₀ is the initial number, and λ is the decay constant.

放射性衰变遵循指数规律：N = N₀ e^(−λt)，其中 N 为时刻 t 未衰变的原子核数，N₀ 为初始原子核数，λ 为衰变常数。

Since the activity A (count rate corrected for background) is proportional to N, we can write A = A₀ e^(−λt). Taking the natural logarithm gives ln A = ln A₀ − λt, which is a straight line with gradient −λ.

由于活度 A（经本底修正的计数率）与 N 成正比，可写成 A = A₀ e^(−λt)。两边取自然对数得 ln A = ln A₀ − λt，这是一条斜率为 −λ 的直线。

The half-life t₁/₂ is related to the decay constant by t₁/₂ = ln 2 / λ. Thus, from the gradient of a ln A–t graph, both λ and t₁/₂ can be calculated.

半衰期 t₁/₂ 与衰变常数的关系为 t₁/₂ = ln 2 / λ。因此，由 ln A–t 图的斜率可同时求出 λ 和 t₁/₂。

Background radiation contributes a roughly constant count rate that must be subtracted from the measured counts to obtain the true count rate from the source.

本底辐射提供了近似恒定的计数率，为了得到放射源的真实计数率，必须从测量值中扣除本底。

3. Apparatus and Setup | 仪器与装置

GM tube and scalar counter or data logger – to measure counts per fixed time interval.

GM 计数管和定标器或数据记录仪 – 用于测量固定时间间隔内的计数。
Short-lived radioactive source (e.g. Pa-234 generated from a uranium salt in an organic solvent, or Rn-220 from a thorium generator bottle).

短半衰期放射源（例如，由铀盐在有机溶剂中生成的 Pa-234，或来自钍发生器瓶的 Rn-220）。
Stopwatch – to record time from the moment of extraction or from a reference event.

秒表 – 从萃取时刻或参考事件开始计时。
Retort stand, clamp, and holder – to fix the GM tube at a fixed distance from the source (e.g. 5 cm).

铁架台、夹子和支架 – 将 GM 计数管固定在距源一定距离处（如 5 cm）。
Lead shielding – optional, to reduce background or protect from scattered radiation.

铅屏蔽 – 可选，用于降低本底或防护散射辐射。

4. Data Collection Procedure | 数据采集流程

First, measure the background count rate by taking several counts (e.g. for 300 s) without the source present, then calculate the average background counts per second, B.

首先，测量本底计数率：在无源条件下进行数次计数（如每次 300 s），计算平均本底计数率 B（每秒计数）。

Prepare the source and start the stopwatch immediately when the short-lived nuclide is formed. Record the total counts detected in successive equal time intervals, e.g. 10 s intervals, for at least 5–6 minutes.

准备好放射源，并在短寿命核素生成瞬间启动秒表。在连续相等的时间间隔（如每 10 s）内记录检测到的总计数，持续至少 5–6 分钟。

For each time interval, convert the total counts to count rate C = total counts / interval duration. Then calculate the corrected count rate A = C − B.

对每个时间间隔，将总计数转换为计数率 C = 总计数 / 间隔时长。然后计算经本底修正的计数率 A = C − B。

Take repeat readings where possible to reduce random error. If using a data logger, set the sampling time to match the expected rate of change.

在可能的情况下重复读数以减少随机误差。若使用数据记录仪，将采样时间设置为与预期变化速率相匹配。

5. Example Data Table | 示例数据表

Below is a typical structure for recording data. (This mirrors the layout often seen in A‑Level mark schemes.)

下面是一个典型的数据记录表格结构（与 A-Level 评分方案中常见布局一致）。

Time t / s	Total counts in 10 s	Count rate C / s⁻¹	Corrected A / s⁻¹	ln A
20	1540	154	153.5	5.034
30	980	98.0	97.5	4.579
40	620	62.0	61.5	4.119
50	390	39.0	38.5	3.650

Note: Background B = 0.5 s⁻¹ is subtracted from C. The time t is taken as the midpoint of the counting interval or the elapsed time.

注意：本底 B = 0.5 s⁻¹ 已从 C 中扣除。时间 t 取计数间隔中点或经过时间。

6. Graphical Analysis | 图形分析

Plot a graph of ln A (y‑axis) against time t (x‑axis). Use a sharp pencil and appropriate scales. Draw a best‑fit straight line through the plotted points.

绘制 ln A（纵轴）对时间 t（横轴）的图形。使用尖铅笔和合适的坐标尺度，通过数据点画出最佳拟合直线。

The gradient m of the line equals −λ. To calculate the gradient, choose two well‑separated points on the line (not necessarily data points) and use m = (ln A₂ − ln A₁) / (t₂ − t₁).

直线的斜率 m 等于 −λ。计算斜率时，在直线上选取两个相距较远的点（不一定是数据点），使用 m = (ln A₂ − ln A₁) / (t₂ − t₁)。

From the gradient, λ = −m. Then the half‑life is t₁/₂ = ln 2 / λ. For example, if the gradient is −0.035 s⁻¹, then λ = 0.035 s⁻¹ and t₁/₂ = 0.693 / 0.035 ≈ 19.8 s.

由斜率得 λ = −m。进而半衰期 t₁/₂ = ln 2 / λ。例如，若斜率为 −0.035 s⁻¹，则 λ = 0.035 s⁻¹，t₁/₂ = 0.693 / 0.035 ≈ 19.8 s。

Check for linearity – if the points deviate from a straight line at later times, it may indicate errors such as a changing background or inaccurate timing.

检查线性度——若在较晚时间数据点偏离直线，可能表明存在误差，如本底变化或计时不准确。

7. Uncertainty Estimation | 不确定度评估

In the January 2022 paper, candidates were expected to estimate the uncertainty in the half‑life using either the spread of repeat readings or the worst‑fit line method.

在 2022 年 1 月的试卷中，考生需使用重复读数的离散度或最劣拟合线法估算半衰期的不确定度。

Draw two additional lines: a steepest possible line and a shallowest possible line that are still reasonable fits to the error bars or data scatter. Their gradients, m_max and m_min, give the extreme values of λ.

另画两条线：一条是可能的最大斜率线，另一条是可能的最小斜率线，这两条线仍须与误差棒或数据散点合理拟合。它们的斜率 m_max 和 m_min 给出 λ 的极端值。

Calculate t₁/₂ for each extreme gradient: t₁/₂,min = ln 2 / (−m_max) and t₁/₂,max = ln 2 / (−m_min). The uncertainty Δt₁/₂ can be expressed as half the range: Δt₁/₂ = (t₁/₂,max − t₁/₂,min) / 2.

分别计算每个极端斜率对应的半衰期：t₁/₂,min = ln 2 / (−m_max) 和 t₁/₂,max = ln 2 / (−m_min)。不确定度 Δt₁/₂ 可表示为极差的一半：Δt₁/₂ = (t₁/₂,max − t₁/₂,min) / 2。

Alternatively, if a large number of repeats is available, calculate the standard deviation and use a percentage uncertainty. Always quote the final half‑life with its absolute uncertainty, e.g., t₁/₂ = 19.8 ± 0.6 s.

或者，若有大量重复数据，可计算标准差并使用百分比不确定度。最终半衰期务必附带绝对不确定度，例如 t₁/₂ = 19.8 ± 0.6 s。

8. Sources of Error and Improvements | 误差来源与改进

Background fluctuation: Radiation levels are not perfectly constant. Measure background for a longer time and repeat before and after the experiment to get a reliable mean.

本底波动：辐射水平并非完全恒定。延长本底测量时间，并在实验前后分别测量，以获得可靠的平均值。
Dead time of GM tube: At high count rates the tube may miss some decays. Keep the source at a distance to limit the initial count rate, or apply a dead‑time correction formula.

GM 计数管死时间：高计数率时计数管可能漏记部分衰变。使源保持一定距离以限制初始计数率，或应用死时间修正公式。
Timing uncertainty: Human reaction time in starting the stopwatch can shift the entire dataset. Use a data logger with automatic triggering or a video recording of the counter display to improve precision.

计时不确定度：启动秒表时的人为反应时间可使整个数据集偏移。使用带自动触发的数据记录仪，或对计数器显示屏进行视频录制以提高精度。
Sample variability: If the extraction process varies, the initial amount of short‑lived nuclide changes. Repeat the entire procedure several times and average the derived half‑lives.

样品差异性：若萃取过程存在差异，短寿命核素的初始量会改变。多次重复整个过程并对所得半衰期取平均。

9. Exam-Style Tips from Jan22 | 源自 Jan22 的应试技巧

The Jan22 Unit 5 paper required students to interpret a given ln(count rate) vs time graph, calculate the gradient, and determine the half‑life. Marks were specifically allocated for correctly recording data in a table with consistent significant figures and for stating the background correction.

2022 年 1 月单元 5 的试卷要求学生解读所给的 ln(计数率)-时间图，计算斜率并求半衰期。正确在表格中记录数据并保持有效数字一致、以及说明本底修正，均单独给分。

The question expected candidates to comment on the reliability of the result by considering the fit of the graph and any anomalous points. Always label axes with quantities and units, and draw the best‑fit line with equal distribution of points above and below.

试题期望考生通过考虑图形拟合程度及任何异常点来评述结果的可靠性。务必在坐标轴上标明物理量和单位，绘制最佳拟合线时应使线上方和线下方的数据点大致均匀分布。

For uncertainty, the mark scheme accepted both the worst‑fit gradient method and the direct use of half‑range from repeats. Whichever method you choose, show your working clearly and state the final result with the appropriate number of decimal places.

对于不确定度，评分方案既接受最劣拟合斜率法，也接受直接使用重复读数的极差法。无论选择何种方法，均需清晰展示计算过程，并以合适的小数位数表述最终结果。

In your conclusion, compare the experimental half‑life with the accepted value (if known) and suggest targeted improvements, not just generic ones like ‘be more careful’.

在结论中将实验半衰期与公认值（若已知）进行比较，并提出针对性的改进措施，而非只说“要更认真”之类的泛泛之谈。

10. Summary and Key Takeaways | 总结与核心要点

This practical investigation encapsulates the essential skills tested in A‑Level Physics Unit 5: collecting and tabulating data, plotting logarithmic graphs, extracting decay constants, and quantifying uncertainties.

该实验探究浓缩了 A-Level 物理单元 5 所测试的核心技能：采集和制表数据、绘制对数图、提取衰变常数以及量化不确定度。

Mastering the step‑by‑step analysis – from background subtraction to the final uncertainty statement – will not only prepare you for similar tasks on the Jan22 paper but also build solid foundations for university‑level laboratory work.

掌握从本底扣除到最终不确定度陈述的逐步分析，不仅能帮你应对 Jan22 试卷的类似任务，还能为大学阶段的实验课打下坚实基础。

Always remember that in radioactive decay experiments, the random nature of the process means that longer counting times or larger numbers of counts are essential to reduce statistical uncertainties.

请始终记住，在放射性衰变实验中，过程的随机性意味着必须延长计数时间或增大计数总量以减小统计不确定度。

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A-Level Physics: Unit 5 Jan22 Practical Investigation – Radioactive Decay and Half-Life Measurement | A-Level 物理：单元5 Jan22 实验探究 – 放射性衰变与半衰期测定