📚 Mastering Combined-195: The Dice Analogue for Radioactive Decay (Edexcel A-Level) | 掌握 Combined-195：放射性衰变的骰子模拟（Edexcel A-Level）
In the Edexcel A-Level Physics course, the practical coded 'Combined-195' challenges students to model the statistical nature of radioactive decay using a simple yet powerful tool: ordinary dice. By repeatedly throwing a large set of dice and removing those that show a predetermined outcome, you replicate the spontaneous, random disintegration of atomic nuclei. This core practical not only reinforces key concepts such as half-life and decay constant but also develops essential skills in data collection, graphical analysis, and evaluating experimental uncertainties. Through this article, you will gain a thorough understanding of the procedure, the underlying physics, and how to maximise your marks in both practical assessments and examination questions related to Combined-195.
在 Edexcel A-Level 物理课程中，编号为 'Combined-195' 的实验要求学生利用一种简单而强大的工具——普通骰子，来模拟放射性衰变的统计规律。通过反复投掷大量骰子并移除那些出现特定结果的骰子，你能够重现原子核自发、随机分裂的过程。这个核心实验不仅巩固了半衰期和衰变常数等关键概念，还培养了你在数据收集、图形分析以及评估实验误差方面的重要技能。通过本文，你将深入了解操作步骤、背后的物理原理，以及如何在 Combined-195 相关的实验考核和考试题目中取得高分。
1. Introduction to Combined-195 | Combined-195 简介
radioactive sample, each unstable nucleus has a fixed probability of decaying per unit time, but the exact moment of disintegration is unpredictable. The dice analogue captures this behaviour: each die represents a radioactive nucleus, and each throw represents a unit time interval. Obtaining a specific face—typically a '6'—signifies that a nucleus has decayed. Over successive throws, the number of 'surviving' dice decreases exponentially, mirroring the decay law N = N₀ × e−λt or, in discrete terms, N = N₀ × (½)t/T₁/₂. This experiment allows you to determine the half-life from a graph of remaining dice against throw number, and to discuss the limitations of the model.
Combined-195 是 Edexcel 指定的核心实验，它利用骰子模拟来探究放射性衰变的随机性。在真实的放射性样品中，每个不稳定原子核在单位时间内都有固定的衰变概率，但具体的衰变时刻不可预测。骰子类比捕捉到了这一行为：每个骰子代表一个放射性原子核，每次投掷代表一个单位时间间隔。掷出某一特定点数——通常为 '6'——表示原子核发生了衰变。经过连续多次投掷，'存活' 骰子的数量呈指数下降，这反映了衰变规律 N = N₀ × e−λt，或者用离散形式表达为 N = N₀ × (½)t/T₁/₂。该实验使你能够从剩余骰子数量对投掷次数的图上求出半衰期，并讨论模型的局限性。
2. The Concept of Radioactive Decay | 放射性衰变的概念
ured in becquerels (Bq), is the number of disintegrations per second. For a given isotope, the probability per unit time that a nucleus will decay is constant and is characterised by the decay constant λ (unit s⁻¹). The fundamental equation linking the number of undecayed nuclei N at time t to the initial number N₀ is dN/dt = −λN. Solving this differential equation yields the exponential decay law: N = N₀ e−λt nships are central to understanding both real radioactive samples and the dice simulation.
放射性衰变是不稳定原子核通过辐射释放能量的一种自发过程。放射源的活度 A 以贝克勒尔（Bq）为单位，表示每秒发生衰变的次数。对于某一特定同位素，每个原子核在单位时间内发生衰变的概率是恒定的，并以衰变常数 λ（单位为 s⁻¹）来表征。联系 t 时刻未衰变原子核数 N 与初始数目 N₀ 的基本方程是 dN/dt = −λN。求解该微分方程即得到指数衰变规律：N = N₀ e−λt。半衰期 T₁/₂ 定义为一半的原始原子核发生衰变所需的时间，它与 λ 的关系为 T₁/₂ = ln 2 / λ ≈ 0.693 / λ。这些关系对于理解真实的放射性样品和骰子模拟都至关重要。
3. Why Use a Dice Analogue? | 为什么使用骰子类比？
s and often require dedicated radioactive source management. A dice simulation eliminates radiation hazards entirely while preserving the core statistical principles. Each die thrown has a one-in-six (≈16.7%) chance of showing a '6' ar decay. Moreover, the experiment can be completed quickly, repeated easily, and manipulated to explore how changing the 'decay probability' life. This makes Combined-195 a safe, cost-effective and pedagogically rich investigation.
适合在学校实验室进行半衰期测量的真实放射源（例如镤-234 或氡-220）存在操作风险，通常需要专门的放射源管理。骰子模拟完全消除了辐射危害，同时保留了核心的统计原理。投掷一枚骰子出现 '6' 的概率为六分之一（≈16.7%），这类似于每次投掷中衰变概率为 p = 1/6。单个骰子的随机性反映了核衰变的量子力学不确定性。此外，实验可以快速完成、容易重复，而且可以通过改变 '衰变概率'（选择不同的目标面或使用不同面数的骰子）来探究半衰期如何变化。这使得 Combined-195 成为一种安全、经济且教学效果显著的研究方法。
4. Equipment and Setup | 设备和设置
To perform Combined-195, you will need a sufficiently large set of identical six-sided dice—typically 80 to 120 dice to ensure good statistics. A large tray or shallow box with raised edges prevents dice from escaping during throws. You will also require graph paper or a computer spreadsheet for recording and plotting data. Some versions of the practical use coloured stickers to mark the 'decayed' dice, but simply removing them works well. A ruler and a calculator with logarithmic functions are needed for later analysis. Ensure the workspace is clear and level so that dice can be thrown randomly without bias.
要进行 Combined-195 实验，你需要足够多且完全相同的六面骰子——通常 80 到 120 个以确保良好的统计效果。一个带高边的托盘或浅盒可以防止骰子在投掷时跳出。你还需要坐标纸或电脑电子表格来记录和绘制数据。有些版本的实验使用彩色贴纸来标记 '已衰变' 的骰子，但直接移除它们同样可行。后期的分析工作需要直尺和具有对数功能的计算器。请确保工作区域整洁且平坦，以便骰子能够无偏差地随机滚动。
5. Step-by-Step Procedure | 分步操作程序
Step 1 – Initial count: Count the total number of dice, N₀, and record it as the value for throw number 0. All are 'undecayed' at the start.
步骤1 – 初始计数： 数出骰子的总数 N₀，并将其记为投掷次数为 0 时的数值。实验开始时所有骰子均未 '衰变'。
Step 2 – First throw: Place all N₀ dice in the tray and shake or toss gently so that each die lands flat. Remove every die that shows a 6 (or your chosen decay face). Record the number of dice removed, and subtract from N₀ to find the number remaining, N₁, for throw 1.
步骤2 – 第一次投掷： 将所有 N₀ 个骰子放入托盘中，轻轻摇晃或抛掷，使每个骰子都能平整落下。移除每一个显示 6（或你选定的代表衰变的面）的骰子。记录移除的骰子数，并从 N₀ 中减去，得到投掷第 1 次的剩余数量 N₁。
Step 3 – Repeat throws: Take the remaining dice and throw them again. Remove all 6s, record the new remaining number. Repeat this process until the number of remaining dice becomes very small (fewer than 10) or until you have completed about 15 to 20 throws. For each throw number n, record the number of surviving dice Nn.
步骤3 – 重复投掷： 拿起剩余的骰子再次投掷。移除所有显示 6 的骰子，记录新的剩余数量。重复此过程，直到剩余骰子数变得非常少（少于 10 个）或者直到你完成了大约 15 至 20 次投掷。为每一次投掷编号 n，记录存活的骰子数 Nn。
6. Data Recording Table | 数据记录表
Organise your observations in a clear table. A typical layout is shown below, where N represents the number of dice remaining after each throw. The initial throw is numbered 0. If you use a spreadsheet, you can add a column for the natural logarithm of N to facilitate later analysis.
将你的观测结果整理到一个清晰的表格中。下表展示了一个典型的格式，其中 N 代表每次投掷后剩余的骰子数。初始投掷编号为 0。如果你使用电子表格，可以添加一列 N 的自然对数，以便后续分析。
| Throw number n | 投掷次数 n | Dice removed | 移除的骰子 | Dice remaining N | 剩余骰子 N | ln(N) |
|---|---|---|---|
| 0 | 0 | 100 | 4.605 |
| 1 | 17 | 83 | 4.419 |
| 2 | 14 | 69 | 4.234 |
| 3 | 12 | 57 | 4.043 |
| 4 | 10 | 47 | 3.850 |
| 5 | 8 | 39 | 3.664 |
| 6 | 6 | 33 | 3.497 |
| 7 | 6 | 27 | 3.296 |
| 8 | 4 | 23 | 3.135 |
| 9 | 4 | 19 | 2.944 |
| 10 | 3 | 16 | 2.773 |
Note: The exact numbers removed will vary due to the random nature of the throw. This table is only an example.
注意：由于投掷的随机性，移除的精确数目会有所变化。此表仅为示例。
7. Graphical Analysis: Plotting the Decay Curve | 图形分析：绘制衰变曲线
cal axis against the throw number n on the horizontal axis. Use a smooth curve of best fit— ecreasing exponential. From your graph, identify the half-life in terms of throws. To do this, draw a horizontal line from N₀/2 on the y-axis across to the curve, then drop a vertical line down to the x-axis. The throw number at that intersection is the half-life throw count, n₁/₂. In a perfect 1/6 decay probability simulation, theory predicts n₁/₂ = ln 2 / ln(6/5) ≈ 3.8 throws. Your experimental value should be close to this.
绘制剩余骰子数 N（纵轴）对投掷次数 n（横轴）的图形。请使用平滑的最佳拟合曲线——不要简单地将各点用直线连接，因为数据具有统计性，曲线更能代表预期的指数趋势。典型的曲线形状是逐渐下降的指数曲线。从图中找出以投掷次数表示的半衰期。做法如下：从纵轴上 N₀/2 处画一条水平线与曲线相交，然后从交点向下画一条垂直线至横轴。该交点所对应的投掷次数即为半衰期投掷次数 n₁/₂。在衰变概率为 1/6 的理想模拟中，理论预测 n₁/₂ = ln 2 / ln(6/5) ≈ 3.8 次投掷。你的实验值应接近该值。
8. Determining Half-Life from the Graph | 从图确定半衰期
Besides the direct half-life point, you can verify the constancy of the half-life by checking whether the number of dice halves again after a further n₁/₂ throws. Starting from N₀/2, drop by another factor of two and confirm that the throw interval is roughly the same. This demonstrates the characteristic property of exponential decay: equal time intervals produce equal fractional decreases. Any slight variation in the interval indicates statistical fluctuations, which are expected and provide a good discussion point for error analysis. Record your measured half-life with its absolute uncertainty, estimated from the width of the lines you drew on the graph.
除了直接读取半衰期点外，你还可以通过检查再经过 n₁/₂ 次投掷后骰子数是否再次减半来验证半衰期的恒定性。从 N₀/2 出发，再次减少一半，确认两次所需的投掷间隔大致相同。这展示了指数衰变的一个特征性质：相同的时间间隔产生相同的分数递减。若间隔有微小变化，则表明存在统计涨落，这在意料之中，并为误差分析提供了一个很好的讨论点。记录你测量到的半衰期及其绝对不确定度，该不确定度可根据你在图形上所画线条的宽度来估算。
9. Calculating Half-Life Using Logarithms | 使用对数计算半衰期
exponential relationship in logarithmic form. The dice decay model predicts N = N₀ × (5/6)n c + m x, where the slope m = ln(5/6). If you plot ln N against n, the gradient of the line should be negative, and its magnitude can be used to find the half-life. From the decay constant definition, λ = −ln(5/6) per throw, so the half-life in throws is T₁/₂ = ln 2 / λ = ln 2 / (−ln(5/6)). Your own gradient from the ln N vs n graph should match −ln(5/6) approximately. Use the formula T₁/₂ = ln 2 / |gradient| to calculate your experimental half-life and compare it with the graphical value.
若要采用更具分析性的方法，你可以运用对数形式的指数关系。骰子衰变模型预测 N = N₀ × (5/6)n，因为每个骰子每次投掷有 5/6 的概率存活。取自然对数可得 ln N = ln N₀ + n × ln(5/6)。这是一个形如 y = c + m x 的线性方程，其中斜率 m = ln(5/6)。如果你绘制 ln N 对 n 的图形，该直线的斜率应为负值，且其大小可用来求出半衰期。根据衰变常数的定义，λ = −ln(5/6) 每次投掷，因此以投掷次数表示的半衰期为 T₁/₂ = ln 2 / λ = ln 2 / (−ln(5/6))。你从 ln N 对 n 图中得到的梯度应近似等于 −ln(5/6)。利用公式 T₁/₂ = ln 2 / |梯度| 计算你的实验半衰期，并将其与从曲线读出的值进行比较。
10. Sources of Error and Improvements | 误差来源和改进
The main source of uncertainty in Combined-195 is the statistical fluctuation inherent in a finite number of dice. A smaller set of dice, say 50, will produce larger relative variations from the expected exponential curve than a set of 200. To improve the accuracy, use as many dice as practically possible and consider combining class data to increase the 'sample size'. Another potential error is non-random throwing: dice that are not shaken thoroughly may show systematic bias, for example if they are simply dropped without rolling. Ensure every throw gives each die an equal chance of showing any face. Parallax error when reading graph values and drawing lines can be reduced by using sharp pencils, a transparent ruler, and taking several measurements. Finally, always record the actual number
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