Blog

📚 Analysis of Experimental Data in CIE A-Level Physics | CIE A-Level 物理实验数据分析

In CIE A-Level Physics, data analysis is a core skill assessed especially in Paper 5 (Planning, Analysis and Evaluation). It is not just about plugging numbers into formulas; it requires you to identify and quantify errors, choose the most informative graph, extract values from gradients and intercepts, and judge whether your data actually support the expected physical relationship.

在 CIE A-Level 物理中，数据分析是一项核心技能，尤其在 Paper 5（实验规划、分析与评估）中考查。它不仅仅是把数字代入公式，还要求你识别并量化误差、选择最有信息量的图像、从斜率和截距中提取数值，并判断数据是否真正支持预期的物理关系。

1. Types of Data and Errors | 数据类型与误差

Raw data are the readings you record directly from instruments. Processed data are quantities calculated from raw readings, such as averages, squares, or reciprocals. Errors are usually divided into random errors and systematic errors.

原始数据是你直接从仪器记录下来的读数。处理数据是由原始读数计算出来的量，例如平均值、平方或倒数。误差通常分为随机误差和系统误差。

Random errors cause readings to scatter unpredictably about the true value. They can be reduced by taking repeated measurements and using the mean. Systematic errors cause all readings to be shifted in the same direction, often due to poor calibration or a zero error, and they cannot be reduced by repetition.

随机误差使读数在真值附近不可预测地散布。可以通过多次测量并取平均值来减小。系统误差使所有读数朝同一方向偏移，通常由校准不良或零点误差引起，重复测量无法减小系统误差。

2. Recording Data with Absolute Uncertainties | 记录数据与绝对不确定度

Every measured value should be written with an absolute uncertainty. For a single reading on a digital instrument, the absolute uncertainty is at least ± the smallest scale division, or the manufacturer’s stated accuracy if larger. For an analogue instrument, it is usually half the smallest scale division.

每个测量值都应写出绝对不确定度。对于数字仪器的单次读数，绝对不确定度至少为最小分度值，或制造商标称的准确度（如果更大）。对于模拟仪器，通常取最小分度值的一半。

For repeated readings, the best estimate is the mean, and a simple estimate of the absolute uncertainty is half the range: (maximum − minimum) ÷ 2. You should record this as value ± uncertainty with a consistent unit.

对于重复读数，最佳估计值是平均值，绝对不确定度的一个简单估计是半极差：（最大值 − 最小值）÷ 2。记录时应写成 数值 ± 不确定度，并保持单位一致。

3. Percentage Uncertainties and Propagation | 百分比不确定度与传递

The percentage uncertainty is found from:

百分比不确定度由下式得出：

percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%

百分比不确定度 = (绝对不确定度 ÷ 测量值) × 100%

When two quantities are added or subtracted, absolute uncertainties add in quadrature or as a simple sum for a conservative estimate. The simple sum rule is often accepted at A-Level:

当两个量相加或相减时，绝对不确定度可以按平方根合成，或采用更保守的简单相加。A-Level 中通常接受简单相加规则：

更多咨询请联系16621398022（同微信）

📚 Decay graphs and equations | 衰变图与方程

Radioactive decay is the spontaneous disintegration of unstable nuclei, and CIE A Level Physics expects you to read, sketch, and interpret decay graphs and equations. This article explains the random nature of decay, the exponential law, half-life, logarithmic graphs, background corrections, and carbon dating.

放射性衰变是不稳定原子核的自发解体，CIE A Level 物理要求你能够读取、绘制并解释衰变图与方程。本文讲解衰变的随机性、指数规律、半衰期、对数图、本底修正以及碳定年。

1. Radioactive decay as a random process | 放射性衰变作为随机过程

Decay is random and spontaneous. You cannot predict which particular nucleus will decay next or when it will decay. However, for a large number of identical nuclei, the overall decay rate is statistically predictable.

衰变是随机且自发的。你无法预测哪个特定原子核将在何时衰变。但对于大量同种原子核，整体衰变率在统计上是可以预测的。

External conditions such as temperature, pressure, and chemical bonding do not change the decay constant of a given nuclide. This is because decay is governed by the weak and strong nuclear interactions inside the nucleus.

温度、压强和化学键等外部条件不能改变给定核素的衰变常数。这是因为衰变由原子核内部的弱相互作用和强相互作用支配。

In an experiment, counts fluctuate around a smooth exponential trend. The fluctuations arise from the random timing of individual decays and are more obvious for small count rates.

在实验中，计数围绕平滑的指数趋势波动。波动源于单个衰变发生时刻的随机性，在计数率较小时更明显。

2. Decay constant and activity | 衰变常数与活度

The activity A of a radioactive source is the number of decays per unit time. It is measured in becquerels (Bq), where 1 Bq = 1 decay per second.

放射源的活度 A 是单位时间内发生的衰变次数。它以贝克勒尔（Bq）为单位，1 Bq = 每秒 1 次衰变。

The decay constant λ is the probability per unit time that a given nucleus will decay. Its SI unit is s⁻¹.

衰变常数 λ 是给定原子核在单位时间内发生衰变的概率。其 SI 单位是 s⁻¹。

A = λN

If a sample contains N undecayed nuclei, its activity is directly proportional to N. The greater the number of parent nuclei, the more decays occur per second.

如果样品含有 N 个未衰变原子核，其活度与 N 成正比。母核数目越多，每秒发生的衰变就越多。

• N is the number of undecayed parent nuclei.
• A is the activity, measured in Bq.
• λ is the decay constant, measured in s⁻¹.

其中 N 是未衰变母核数目，A 是活度，单位为 Bq，λ 是衰变常数，单位为 s⁻¹。

3. The exponential decay equation | 指数衰变方程

Since activity is proportional to N, the rate of change of N is proportional to N itself. This gives the differential equation dN/dt = −λN.

由于活度与 N 成正比，N 的变化率也与 N 本身成正比。这给出微分方程 dN/dt = −λN。

N = N₀e^−λt

The solution is an exponential decay. N₀ is the initial number of undecayed nuclei at t = 0. The same form applies to activity and to mass of the parent nuclide.

其解为指数衰减。N₀ 是 t = 0 时未衰变原子核的初始数目。同样的形式适用于活度和母核素的质量。

A = A₀e^−λt

At any time t, the fraction remaining is e^−λt. When t is one half-life, this fraction equals 1/2.

在任意时刻 t，剩余的分数为 e^−λt。当 t 等于一个半衰期时，该分数等于 1/2。

The exponential law assumes a large number of nuclei. For very small samples, statistical deviations from the smooth curve become significant.

指数规律假设原子核数目很大。对于非常小的样品，与平滑曲线的统计偏差会变得显著。

4. Half-life and its equation | 半衰期及其方程

The half-life T½ is the time taken for the number of undecayed parent nuclei to decrease to half of its initial value. The same time is also the time for the activity to halve.

半衰期 T½ 是未衰变母核数目减少到初始值一半所需的时间。同样也是活度减半所需的时间。

T½ = ln 2 ÷ λ = 0.693 ÷ λ

This equation is independent of the initial number N₀. A useful rearrangement is λ = ln 2 ÷ T½.

该方程与初始数目 N₀ 无关。一个有用的变形是 λ = ln 2 ÷ T½。

After n half-lives, the remaining fraction is (1/2)ⁿ. For example, after 3 half-lives, 1/8 of the original nuclei remain.

经过 n 个半衰期后，剩余分数为 (1/2)ⁿ。例如，经过 3 个半衰期后，原始核的 1/8 保留下来。

The table above summarises how the remaining fraction and activity change with whole-number multiples of the half-life. It provides a quick calculation shortcut in exam questions.

上表总结剩余分数和活度随半衰期倍数的变化，是考试中快速计算的依据。

5. Reading N-t and A-t graphs | 读 N-t 与 A-t 图

A graph of N against t is a decreasing exponential curve. It starts at N₀, approaches the time axis asymptotically, and

Published by TutorHao | A-Level Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 Echo Sounding: Principles, Calculations and Applications | 回声测深：原理、计算与应用

Echo sounding is a method of using reflected sound waves to determine the depth of water or the distance to an object. A pulse of sound is emitted, travels through a medium, reflects from a boundary, and returns to a detector. By measuring the time interval between emission and reception and knowing the speed of sound, the distance can be calculated. This technique underpins SONAR, medical ultrasound and industrial testing, and is a standard application of wave reflection in the CIE A-Level Physics syllabus.

回声测深是一种利用反射声波来测定水深或物体距离的方法。发射一个声脉冲，声波在介质中传播，遇到边界反射后被探测器接收。通过测量发射与接收之间的时间间隔，并已知声速，就可以计算距离。该技术是声呐、医学超声和工业检测的基础，也是 CIE A-Level 物理大纲中波的反射的标准应用。

1. What Is Echo Sounding? | 什么是回声测深？

Echo sounding is a distance-measuring technique that uses the reflection of sound. A transmitter sends a short pulse of sound towards a boundary, such as the seabed. The pulse travels through the medium, reflects at the boundary, and returns to a receiver. The total time between transmission and reception is measured, and the distance to the boundary is calculated from the known speed of sound.

回声测深是一种利用声音反射来测量距离的技术。发射器向边界（例如海底）发送一个短声脉冲。脉冲在介质中传播，在边界处反射，然后返回接收器。测量发射和接收之间的总时间，并根据已知声速计算到边界的距离。

In CIE A-Level Physics, echo sounding appears under wave properties and applications of sound. It is a direct application of the wave equation and reflection. The method is sometimes called echo ranging or echo location because it locates a boundary by timing the return of an echo.

在 CIE A-Level 物理中，回声测深出现在波的性质和声音应用部分。它是波动方程和反射的直接应用。该方法有时被称为回声测距或回声定位，因为它通过测量回声返回的时间来确定边界的位置。

2. The Wave Physics Behind an Echo | 回声背后的波动物理学

Sound is a longitudinal mechanical wave, so it requires a medium and travels as a series of compressions and rarefactions. When a sound wave meets a boundary between two materials with different acoustic properties, part of the wave energy is reflected and part is transmitted. The reflected part forms the echo.

声音是一种纵波，属于机械波，因此需要介质，并以一系列疏密波的形式传播。当声波遇到声学性质不同的两种材料之间的边界时，部分波能会被反射，部分会透射。反射的部分形成回声。

The strength of the echo depends on the acoustic impedance mismatch between the two media. A large difference, such as water-to-rock or water-to-air, produces a strong reflection. A small difference produces a weak reflection. Echo sounding relies on detecting this reflected pulse.

回声的强度取决于两种介质的声阻抗差异。差异越大（例如水与岩石或水与空气），反射越强。差异越小

Published by TutorHao | A-Level Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 Using Ultrasound in Medicine | 医学超声的应用

Ultrasound is high-frequency sound above the human hearing range, normally above 20 kHz. In medicine, frequencies from about 1 MHz to 15 MHz are used for both diagnosis and therapy. Because ultrasound can travel through soft tissue and reflect from internal boundaries, it produces images without using ionising radiation.

超声是高于人类听觉范围的高频声波，通常指频率超过 20 kHz 的声波。在医学中，常使用约 1 MHz 到 15 MHz 的超声波进行诊断和治疗。由于超声波能够穿过软组织并在内部界面发生反射，它可以在不使用电离辐射的情况下生成图像。

1. Basic Properties of Ultrasound | 超声波的基本性质

Ultrasound is a longitudinal mechanical wave, so it requires a medium and cannot travel through a vacuum. In medical imaging, pulses of ultrasound are sent into the body and echoes are detected from boundaries between different tissues.

超声是一种纵波机械波，因此它需要介质传播，不能在真空中传播。在医学成像中，超声波脉冲被送入人体，并由不同组织之间的边界反射产生回波，这些回波被检测出来。

In soft tissue, the speed of sound is approximately 1540 m s⁻¹. The wavelength λ is related to frequency f and speed v by the wave equation:

在软组织中，声速约为 1540 m s⁻¹。波长 λ 与频率 f 和速度 v 之间满足波动方程：

v = f × λ

For a typical diagnostic frequency of 1 MHz, λ = v ÷ f = 1540 ÷ 1.0 × 10⁶ ≈ 1.54 mm. Higher frequencies give shorter wavelengths, which can resolve smaller structures.

对于典型的诊断频率 1 MHz，λ = v ÷ f = 1540 ÷ 1.0 × 10⁶ ≈ 1.54 mm。频率越高，波长越短，能够分辨更小的结构。

2. Generating Ultrasound: The Piezoelectric Effect | 超声波的产生：压电效应

Ultrasound transducers use the piezoelectric effect. A piezoelectric crystal such as lead zirconate titanate (PZT) changes shape when a potential difference is applied across it. An alternating voltage makes the crystal vibrate and emit ultrasound at the same frequency.

超声换能器利用压电效应。诸如锆钛酸铅（PZT）之类的压电晶体在两端施加电势差时会发生形变。施加交变电压会使晶体振动，并以相同频率发射超声波。

The same crystal also acts as a receiver: when reflected ultrasound echoes strike it, the changing pressure induces an alternating potential difference across the crystal. This signal is then processed to build an image.

同一块晶体也可用作接收器：当反射的超声回波撞击晶体时，变化的压力会在晶体两端感应出交变电势差。该信号随后经处理生成图像。

In A-level work, you should recall that the emitted frequency equals the driving frequency of the alternating voltage, and that the crystal is cut to a suitable thickness to resonate at the desired frequency.

在 A-level 阶段，需要记住发射频率等于交变电压的驱动频率，并且晶体被切割成适当厚度，以便在所需频率下共振。

3. Frequency, Wavelength and Resolution | 频率、波长和分辨率

Axial resolution is limited by wavelength, so higher frequencies give better detail. For example, at 10 MHz the wavelength in soft tissue is about 0.154 mm, allowing structures smaller than a millimetre to be distinguished.

轴向分辨率受波长限制，因此频率越高，细节越好。例如，在 10 MHz 下，软组织中的波长约为 0.154 mm，能够分辨小于一毫米的结构。

However, higher frequency ultrasound is attenuated more rapidly. This means there is a trade-off between resolution and penetration depth. Obstetric scans often use 2-5 MHz to reach deep tissues, while scans of the eye or skin may use 15 MHz or more because these structures are close to the surface.

然而，频率越高的超声波衰减越快。这意味着分辨率和穿透深度之间存在权衡。产科扫描通常使用 2-5 MHz 以到达深部组织，而眼睛或皮肤扫描可能使用 15 MHz 或更高频率，因为这些结构靠近体表。

4. Acoustic Impedance and Intensity Reflection Coefficient | 声阻抗和强度反射系数

When ultrasound meets a boundary between two media, the amount of reflection depends on the difference in acoustic impedance Z. Acoustic impedance is defined as the product of density ρ and speed c of sound in the medium:

当超声波遇到两种介质的界面时，反射量取决于声阻抗 Z 的差异。声阻抗定义为介质密度 ρ 与声速 c 的乘积：

Z = ρ × c

The SI unit of acoustic impedance is kg m⁻² s⁻¹, sometimes written as rayl. For normal incidence, the intensity reflection coefficient R is given by:

声阻抗的国际单位是 kg m⁻² s⁻¹，有时写作 rayl。对于垂直入射，强度反射系数 R 由下式给出：

R = [(Z₂ − Z₁) ÷ (Z₂ + Z₁)]²

Here Z₁ and Z₂ are the acoustic impedances of the two media. If the impedances are equal, R = 0 and there is total transmission; if they are very different, R is close to 1 and almost all ultrasound is reflected.

其中 Z₁ 和 Z₂ 是两种介质的声阻抗。如果阻抗相等，R = 0，超声完全透射；如果阻抗差异很大，R 接近 1，几乎全部超声波被反射。

For example, at an air-soft tissue boundary, Z₁ ≈ 4.3 × 10² kg m⁻² s⁻¹ and Z₂ ≈ 1.63 × 10⁶ kg m⁻² s⁻¹. The reflection coefficient is approximately ((1.63 × 10⁶ − 430) ÷ (1.63 × 10⁶ + 430))² ≈ 0.999, so almost all ultrasound is reflected. That is why a coupling gel is essential.

例如，在空气-软组织界面，Z₁ ≈ 4.3 × 10² kg m⁻² s⁻¹，Z₂ ≈ 1.63 × 10⁶ kg m⁻² s⁻¹。反射系数约为 ((1.63 × 10⁶ − 430) ÷ (1.63 × 10⁶ + 430))² ≈ 0.999，因此几乎全部超声波被反射。这就是必须使用耦合剂的原因。

5. Impedance Matching with Coupling Gel | 使用耦合剂进行阻抗匹配

Air has a very low acoustic impedance compared with skin, so an air gap between the transducer and the body would cause strong reflection. A water-based coupling gel is placed between the transducer and the skin. Its impedance is close to that of skin and soft tissue, so R is small and most ultrasound enters the body.

空气的声阻抗与皮肤相比非常低，因此探头与身体之间的空气间隙会引起强烈反射。水性耦合剂被涂在探头和皮肤之间。其阻抗接近皮肤和软组织，因此 R 很小，大部分超声波能够进入体内。

This is an example of impedance matching. Without the gel, the reflected intensity would be so high that very little signal would penetrate or return, making imaging impossible.

这是阻抗匹配的一个例子。没有耦合剂时，反射强度会非常高，几乎很少有信号穿透或返回，使成像无法进行。

Typical values of acoustic impedance are shown below.

下表列出了一些典型声阻抗值。

6. Attenuation in Tissue | 组织中的衰减

As ultrasound travels through tissue, its intensity decreases because energy is absorbed and scattered. This reduction in intensity is called attenuation. Attenuation increases with distance and with frequency, roughly exponentially in homogeneous tissue.

当超声波穿过组织时，其强度因能量被吸收和散射而降低。这种强度下降称为衰减。衰减随距离和频率增加，在均匀组织中大致呈指数规律。

A higher frequency gives better resolution but also attenuates more quickly, so it can only image shallow structures. Lower frequencies are chosen when deep penetration is needed, such as in abdominal or obstetric imaging.

频率越高，分辨率越好，但衰减也越快，因此只能对浅表结构成像。当需要深部穿透时，例如腹部或产科成像，会选择较低频率。

Attenuation is often specified in decibels per centimetre per megahertz. For soft tissue, a useful approximation is about 1 dB cm⁻¹ MHz⁻¹. This means a 5 MHz pulse loses intensity about five times as quickly per centimetre as a 1 MHz pulse.

衰减通常以每厘米每兆赫的分贝数来表示。对于软组织，一个有用的近似值约为 1 dB cm⁻¹ MHz⁻¹。这意味着 5 MHz 脉冲每厘米的强度损失大约是 1 MHz 脉冲的五倍。

7. A-scan and B-scan Imaging | A 型扫描和 B 型扫描成像

In an A-scan, the transducer sends a short pulse into the body and records the amplitude of echoes as a function of time. The time delay gives the depth of each reflecting boundary because the pulse travels at a known speed. A-scans are used to measure distances, for example the length of the eye before cataract surgery.

在 A 型扫描中，探头向体内发射短脉冲，并记录回波振幅随时间的变化。由于脉冲以已知速度传播，时间延迟给出了每个反射界面的深度。A 型扫描用于测量距离，例如白内障手术前测量眼球的长度。

In a B-scan, the transducer is moved or an array of transducers is used. The echo amplitudes are converted into brightness levels on a two-dimensional image. B-scan is the standard real-time ultrasound imaging mode used for observing a fetus, the heart, the liver and other organs.

在 B 型扫描中，探头移动或使用换能器阵列。回波振幅被转换为二维图像上的亮度等级。B 型扫描

Published by TutorHao | A-Level Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

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

In CIE A-Level Physics, radioactive decay is described statistically by the decay constant λ and the half-life T½. These two quantities give us powerful ways to predict the behaviour of a large number of unstable nuclei, even though the decay of any individual nucleus is random.

在 CIE A-Level 物理中，放射性衰变通过衰变常数 λ 和半衰期 T½ 进行统计描述。尽管单个原子核的衰变是随机的，这两个物理量让我们能够预测大量不稳定原子核的整体行为。

1. Radioactive Decay as a Random Process | 放射性衰变作为随机过程

Radioactive decay is a random and spontaneous process. We cannot predict which nucleus will decay next or when a particular nucleus will decay. The decay rate depends only on the number of undecayed nuclei present.

放射性衰变是一个随机且自发的过程。我们无法预测哪一个原子核会下一个衰变，也无法预测某个原子核何时会衰变。衰变速率仅取决于当前未衰变原子核的数量。

Because the process is random, experiments show fluctuations in count rate over short time intervals. However, when a very large number of nuclei are present, the average decay rate becomes statistically stable.

由于过程具有随机性，实验在短时间间隔内计数率会出现涨落。但当存在大量原子核时，平均衰变速率在统计上变得稳定。

This statistical behaviour allows us to define a fixed probability of decay per unit time for each nucleus — this probability is the decay constant.

这种统计行为使我们可以为每个原子核定义单位时间内固定的衰变概率——这个概率就是衰变常数。

2. Defining the Decay Constant | 定义衰变常数

The decay constant λ is defined as the probability that an individual nucleus will decay per unit time. Its SI unit is s⁻¹.

衰变常数 λ 定义为单个原子核在单位时间内发生衰变的概率。其国际单位是 s⁻¹。

If λ = 0.01 s⁻¹, each nucleus has a 1% chance of decaying in any one-second interval. A larger λ means a faster decay process.

如果 λ = 0.01 s⁻¹，则每个原子核在任意一秒间隔内有 1% 的概率发生衰变。λ 越大，衰变过程越快。

The decay constant is characteristic of a particular radioisotope and is not affected by temperature, pressure, or chemical bonding. This is a key difference from chemical reactions.

衰变常数是特定放射性同位素的特征量

Published by TutorHao | A-Level Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 Mass and Energy | 质量与能量

In CIE A Level Physics, the relationship between mass and energy is captured by Einstein’s famous equation E = mc². This principle is essential for understanding nuclear reactions, radioactive decay, and particle interactions. It tells us that mass can be converted into energy and energy can contribute to mass.

在 CIE A Level 物理中，质量与能量的关系由爱因斯坦著名方程 E = mc² 描述。这一原理对理解核反应、放射性衰变和粒子相互作用至关重要。它表明质量可以转化为能量，能量也可以表现为质量。

1. The Mass-Energy Equivalence Principle | 质能等价原理

Einstein’s special theory of relativity shows that mass and energy are equivalent. The total energy E of a particle has a contribution from its rest mass m₀: E₀ = m₀c². Here c is the speed of light in vacuum, c = 3.00 × 10⁸ m s⁻¹. Since c² is huge, a small mass corresponds to a very large energy.

爱因斯坦的狭义相对论指出质量与能量是等价的。粒子的总能量 E 包含来自其静止质量 m₀ 的贡献：E₀ = m₀c²。其中 c 是真空中的光速，c = 3.00 × 10⁸ m s⁻¹。由于 c² 极大，很小的质量就对应非常大的能量。

E₀ = m₀c²

The equation is not just a conversion factor; it states that mass is a form of energy. In nuclear physics, we often deal with rest mass energies because changes in mass are measurable.

该方程不仅仅是一个换算因子，它表明质量是能量的一种形式。在核物理中，我们经常处理静止质量能量，因为质量的变化是可测量的。

2. Rest Energy and the Electronvolt | 静能与电子伏特

The rest energy of a particle is the energy stored in its mass when the particle is at rest. For an electron, mₑ = 9.11 × 10⁻³¹ kg, so E₀ = mₑc² ≈ 8.19 × 10⁻¹⁴ J. This is more conveniently expressed in electronvolts: 1 eV = 1.60 × 10⁻¹⁹ J, so the electron rest energy is about 0.511 MeV.

粒子的静能是粒子静止时储存在其质量中的能量。对电子而言，mₑ = 9.11 × 10⁻³¹ kg，因此 E₀ = mₑc² ≈ 8.19 × 10⁻¹⁴ J。用电子伏特表示更方便：1 eV = 1.60 × 10⁻¹⁹ J，因此电子的静能约为 0.511 MeV。

In A Level calculations, you should be able to convert between joules and electronvolts. The proton and neutron rest energies are about 938 MeV and 940 MeV respectively.

在 A Level 计算中，你应能在焦耳和电子伏特之间转换。质子和中子的静能分别约为 938 MeV 和 940 MeV。

3. Atomic Mass Unit and Energy Equivalence | 原子质量单位与能量当量

Nuclear masses are often given in atomic mass units, u. By definition, 1 u is one twelfth of the mass of a carbon-12 atom, and 1 u = 1.66 × 10⁻²⁷ kg. Using E = mc², 1 u is equivalent to 931.5 MeV. Therefore a mass difference of 1 u corresponds to an energy of 931.5 MeV.

核质量通常以原子质量单位 u 表示。根据定义，1 u 是碳-12 原子质量的十二分之一，且 1 u = 1.66 × 10⁻²⁷ kg。利用 E = mc²，1 u 等价于 931.5 MeV。因此，1 u 的质量差对应 931.5 MeV 的能量。

1 u c² = 931.5 MeV

This conversion is extremely useful in nuclear calculations because mass differences are typically of the order of 10⁻³ u to 10⁻¹ u, giving energies in MeV.

这一换算在核计算中极其有用，因为质量差通常在 10⁻³ u 到 10⁻¹ u 量级，给出的能量以 MeV 为单位。

4. Mass Defect in Nuclei | 原子核的质量亏损

The mass of a nucleus is always less than the sum of the masses of its individual protons and neutrons. For a nucleus with Z protons and N neutrons, the mass defect Δm is:

原子核的质量总是小于其各个质子和中子质量之和。对于含有 Z 个质子和 N 个中子的原子核，质量亏损 Δm 为：

Δm = Z mₚ + N mₙ − m_nucleus

This missing mass has been converted into binding energy. It is important to use nuclear masses consistently. In CIE exams, data may give atomic masses including electrons; you should account for electron masses when required, although often differences cancel.

这些“消失”的质量转化为结合能。一致地使用核质量非常重要。在 CIE 考试中，数据可能给出包含电子的原子质量；需要时应计入电子质量，尽管差异通常会抵消。

5. Binding Energy and Nuclear Stability | 结合能与核稳定性

Binding energy E_b is the energy required to separate a nucleus into its individual nucleons. It is related to the mass defect by:

结合能 E_b 是将原子核拆分为单个核子所需的能量。它与质量亏损的关系为：

E_b = Δm c²

A larger binding energy means the nucleons are more tightly bound. However, total binding energy alone does not determine stability. A heavier nucleus has more nucleons, so it naturally has a larger total binding energy. Stability is better measured by binding energy per nucleon.

结合能越大，核子结合得越紧密。但仅凭总结合能不能决定稳定性。较重的原子核有更多核子，因此总结合能自然更大。稳定性更好地用比结合能（每个核子的结合能）来衡量。

6. Binding Energy per Nucleon Curve | 比结合能曲线

The binding energy per nucleon E_b/A is obtained by dividing the total binding energy by the nucleon number A. It rises rapidly for light nuclei, reaches a maximum of about 8.8 MeV per nucleon near iron-56, and then decreases slowly for heavier nuclei.

比结合能 E_b/A 由总结合能除以核子数 A 得到。它对于轻核迅速上升，在铁-56 附近达到约每核子 8.8 MeV 的最大值，然后对于更重的核缓慢下降。

This curve explains why energy can be released by both fission of heavy nuclei and fusion of light nuclei. Both processes move products toward the more stable middle region of the curve.

这条曲线解释了为什么重核裂变和轻核聚变都能释放能量。两种过程都使产物向曲线中部更稳定的区域移动。

7. Nuclear Fission and Fusion | 核裂变与核聚变

In nuclear fission, a heavy nucleus such as uranium-235 splits into lighter fragments, releasing energy because the products have a higher binding energy per nucleon. A typical fission reaction is:

在核裂变中，铀-235 等重核分裂成较轻的碎片，由于产物的比结合能更高而释放能量。一个典型的裂变反应为：

U-235 + n → Ba-141 + Kr-92 + 3 n + energy

The total mass of products is smaller than reactants, and the energy released is ΔE = Δm c².

产物的总质量小于反应物，释放的能量为 ΔE = Δm c²。

In nuclear fusion, light nuclei such as deuterium and tritium combine to form helium-4. The mass of the helium nucleus is less than the total mass of the reactants, so energy is released. Fusion powers the Sun.

在核聚变中，氘和氚等轻核结合形成氦-4。氦核的质量小于反应物总质量，因此释放能量。聚变为太阳提供能量。

8. Annihilation and Pair Production | 湮灭与电子对产生

Mass–energy equivalence is also demonstrated in particle physics. When a particle and

Published by TutorHao | A-Level Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 Balanced Equations in Physics: Conservation and Nuclear Reactions | 物理中的平衡方程：守恒定律与核反应

In A-Level Physics, a balanced equation is a concise way to represent a nuclear process or particle interaction while showing that certain physical quantities are conserved on both sides of the arrow. Unlike chemical equations, which balance atoms by count, nuclear balanced equations focus on the conservation of mass number A, proton number Z, charge and, where relevant, lepton number.

在A-Level物理中，平衡方程是表示核过程或粒子相互作用的简洁方式，同时表明箭头两侧的某些物理量是守恒的。与化学方程式按原子数配平不同，核平衡方程关注质量数A、质子数Z、电荷以及相关情况下轻子数的守恒。

1. What is a Balanced Equation in Physics? | 物理中的平衡方程是什么？

A balanced equation in nuclear physics shows that the total mass number and total proton number before a reaction are exactly equal to the totals after the reaction. This does not mean mass is unchanged overall, because small mass differences appear as energy, but nucleon number and charge are strictly conserved in every allowed process.

核物理中的平衡方程表明，反应前的总质量数和总质子数与反应后的总数完全相等。这并不意味着总质量完全不变，因为微小的质量差会以能量形式出现，但核子数和电荷在每一个允许的过程中都是严格守恒的。

For example, in alpha decay the parent nucleus loses two protons and two neutrons, so the daughter nucleus has a mass number four units lower and a proton number two units lower. The emitted alpha particle carries away those nucleons.

例如，在α衰变中，母核失去两个质子和两个中子，因此子核的质量数减少4，质子数减少2。发射出的α粒子带走了这些核子。

2. Nuclear Notation: A and Z | 核符号：A 和 Z

Nuclear species are written as ²³⁵₉₂U, where the superscript A is the mass number and the subscript Z is the proton number. The mass number A equals the total number of protons plus neutrons, while Z identifies the element and gives the number of protons.

核素写作²³⁵₉₂U，其中上标A是质量数，下标Z是质子数。质量数A等于质子数与中子数之和，而Z确定了元素并给出质子数。

Because neutrons have no charge, they contribute to A but not to Z. This is why isotopes of the same element have the same Z but different A. Balancing a nuclear equation therefore means making the sums of A values and the sums of Z values equal on both sides.

由于中子不带电荷，它们对A有贡献但对Z没有贡献。这就是为什么同一元素的同位素具有相同的Z但不同的A。因此，配平核方程意味着使两侧的A值之和与Z值之和相等。

3. Alpha Decay Equations | α衰变方程

Alpha decay occurs when a heavy nucleus emits an alpha particle, which is a helium-4 nucleus written as ⁴₂He. The general pattern is that A decreases by 4 and Z decreases by 2.

α衰变发生在重核发射α粒子时，α粒子是氦-4核，写作⁴₂He。一般规律是A减少4，Z减少2。

²²⁶₈₈Ra → ²²²₈₆Rn + ⁴₂He

In this example, radium-226 decays to radon-222. The left side has A = 226 and Z = 88, while the right side has A = 222 + 4 = 226 and Z = 86 + 2 = 88. Both totals match, so the equation is balanced.

在这个例子中，镭-226衰变为氡-222。左侧A = 226，Z = 88，而右侧A = 222 + 4 = 226，Z = 86 + 2 = 88。两侧总数一致，因此方程是平衡的。

4. Beta Minus Decay Equations | β⁻衰变方程

Beta minus decay occurs when a neutron inside a nucleus changes into a proton, emitting an electron and an electron antineutrino. The emitted electron is written as ⁰₋₁e because its mass number is zero and its charge is −1.

β⁻衰变发生在原子核内的一个中子转变为质子时，同时发射一个电子和一个电子反中微子。发射出的电子写作⁰₋₁e，因为其质量数为零，电荷为−1。

¹⁴₆C → ¹⁴₇N + ⁰₋₁e + ν̅

Here the mass number stays at A = 14 on both sides. The proton number rises from 6 to 7 because a neutron has turned into a proton, and the electron’s Z = −1 balances the increase, giving 6 = 7 − 1. The antineutrino has A = 0 and Z = 0, so it does not affect these totals.

这里两侧的质量数都保持A = 14。质子数从6上升至7，因为一个中子变成了质子，而电子的Z = −1抵消了这一增加，得到6 = 7 − 1。反中微子的A = 0且Z = 0，因此它不影响这些总数。

5. Beta Plus Decay and Electron Capture | β⁺衰变与电子俘获

Beta plus decay involves a proton converting into a neutron, emitting a positron and a neutrino. The positron is written as ⁰₊₁e, with zero mass number and charge +1. In beta plus decay, A stays the same but Z decreases by 1.

β⁺衰变涉及一个质子转变为中子，发射一个正电子和一个中微子。正电子写作⁰₊₁e，其质量数为零，电荷为+1。在β⁺衰变中，A保持不变，但Z减少1。

¹¹₆C → ¹¹₅B + ⁰₊₁e + ν

Electron capture is another process that reduces Z by 1. An inner orbital electron is absorbed by the nucleus, combining with a proton to form a neutron and a neutrino.

电子俘获是另一种使Z减少1的过程。一个内层轨道电子被原子核吸收，与质子结合形成一个中子和一个中微子。

⁷₄Be + ⁰₋₁e → ⁷₃Li + ν

In both beta plus decay and electron capture, the total charge is conserved: the positron carries away positive charge, while the captured electron supplies negative charge to the nucleus.

在β⁺衰变和电子俘获中，总电荷都是守恒的：正电子带走正电荷，而被俘获的电子为原子核提供负电荷。

6. Gamma Emission and Excited States | γ发射与激发态

Gamma emission usually follows alpha or beta decay when the daughter nucleus is left in an excited state. The nucleus releases excess energy as a high-energy photon, written as ⁰₀γ or simply γ.

γ发射通常发生在α或β衰变之后，此时子核处于激发态。原子核以高能光子形式释放多余能量，写作⁰₀γ或简写为γ。

²³⁴₉₀Th* → ²³⁴₉₀Th + ⁰₀γ

Because the gamma photon has no mass and no charge, A and Z do not change during gamma emission. An asterisk is sometimes used to indicate an excited nucleus, but the balanced equation remains straightforward.

由于γ光子没有质量和电荷，因此在γ发射过程中A和Z不发生变化。有时用星号表示激发态核，但平衡方程仍然很简单。

7. Fission and Fusion Equations | 裂变与聚变方程

Nuclear fission occurs when a heavy nucleus such as uranium-235 absorbs a neutron and splits into two smaller nuclei, releasing more neutrons. The equation must still conserve A and Z on both sides.

核裂变发生在铀-235等重核吸收一个中子并分裂成两个较小原子核，同时释放更多中子时。方程两侧仍必须守恒A和Z。

²³⁵₉₂U + ¹₀n → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3¹₀n

Here the left side has A = 235 + 1 = 236 and Z = 92 + 0 = 92. The right side has A = 141 + 92 + 3 = 236 and Z = 56 + 36 + 0 = 92, so the fission equation is balanced.

这里左侧A = 235 + 1 = 236，Z = 92 + 0 = 92。右侧A = 141 + 92 + 3 = 236，Z = 56 + 36 + 0 = 92，因此裂变方程是平衡的。

Nuclear fusion is the joining of light nuclei. A common example is deuterium and tritium fusing to form helium-4 and a neutron.

核聚变是轻原子核的结合。一个常见例子是氘和氚聚变形成氦-4和一个中子。

²₁H + ³₁H → ⁴₂He + ¹₀n

The left side has A = 2 + 3 = 5 and Z = 1 + 1 = 2. The right side has A = 4 + 1 = 5 and Z = 2 + 0 = 2. This confirms the fusion equation is balanced.

左侧A = 2 + 3 = 5，Z = 1 + 1 = 2。右侧A = 4 + 1 = 5，Z = 2 + 0 = 2。这证实了聚变方程是平衡的。

8. Conservation Laws Behind Balancing | 配平背后的守恒定律

The most basic conservation laws tested in CIE A-Level Physics are conservation of mass number and conservation of proton number. These are effective summaries of bary

Published by TutorHao | A-Level Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 Medical Imaging: X-rays, Ultrasound, PET and MRI | 医学成像：X 射线、超声波、PET 与 MRI

Medical imaging uses different parts of the electromagnetic spectrum and sound waves to visualise internal body structures for diagnosis and treatment planning. This revision guide covers the core CIE A-Level Physics topics: X-ray production and attenuation, computed tomography, ultrasound, radionuclide imaging, PET and MRI, with the key physical principles and equations you need to know.

医学成像利用电磁波谱的不同部分和声波来显示体内结构，以辅助诊断和制定治疗方案。本复习指南涵盖 CIE A-Level 物理核心考点：X 射线产生与衰减、计算机断层扫描、超声波、放射性核素成像、PET 和 MRI，包括你需要掌握的关键物理原理和公式。

1. X-ray Production and the X-ray Spectrum | X 射线产生与能谱

In an X-ray tube, a heated filament emits electrons by thermionic emission. A high potential difference accelerates these electrons across a vacuum towards a tungsten or molybdenum target.

在 X 射线管中，加热灯丝通过热电子发射释放电子。高电势差使这些电子在真空中加速，撞向钨或钼靶。

Most of the electron kinetic energy is converted to thermal energy, but a small fraction produces X-ray photons by two mechanisms: bremsstrahlung (braking radiation) and characteristic radiation. Bremsstrahlung gives a continuous spectrum, while characteristic lines arise from electron transitions between inner atomic shells.

大部分电子动能转化为热能，但一小部分通过两种机制产生 X 射线光子：轫致辐射（制动辐射）和特征辐射。轫致辐射产生连续能谱，而特征谱线来自原子内壳层之间的电子跃迁。

λ_min = hc/eV

The minimum wavelength λ_min occurs when all of an electron’s kinetic energy is converted into one photon. Increasing the tube voltage V raises the maximum photon energy and shifts λ_min to a shorter wavelength. Increasing the tube current raises the intensity of the beam without changing λ_min.

最小波长 λ_min 出现在电子全部动能转化为单个光子的情况。增大管电压 V 会提高最大光子能量，使 λ_min 向更短波长方向移动。增大管电流会提高射线束强度，但不会改变 λ_min。

2. Attenuation of X-rays | X 射线衰减

When X-rays pass through matter, their intensity I decreases exponentially with thickness x according to the equation below, where μ is the linear attenuation coefficient of the material.

当 X 射线穿过物质时，其强度 I 随厚度 x 按以下公式指数衰减，其中 μ 为材料的线性衰减系数。

I = I₀ e^(−μx)

The half-value thickness x½ is the thickness that reduces the transmitted intensity to half its original value. It is related to μ by x½ = ln 2 / μ, and it is useful for comparing the penetrating ability of different materials.

半值厚度 x½ 是使透射强度降为原来一半的厚度。它与 μ 的关系为 x½ = ln 2 / μ，可用于比较不同材料的穿透能力。

Bone has a much larger μ than soft tissue at diagnostic X-ray energies because bone contains calcium with a higher effective atomic number. This difference produces strong contrast in an X-ray image.

在诊断用 X 射线能量下，骨的 μ 远大于软组织，因为骨含有有效原子序数更高的钙。这种差异会在 X 射线图像中产生强烈对比。

3. Conventional X-ray Imaging and Contrast | 常规 X 射线成像与对比度

A conventional X-ray image is a two-dimensional projection of the attenuation along each ray path. Dense structures such as bone absorb more radiation and appear white, while air-filled lungs appear dark.

常规 X 射线图像是沿每条射线路径衰减的二维投影。骨等致密结构吸收更多辐射而呈现白色，而充满空气的肺呈现黑色。

Soft tissues have similar attenuation coefficients, so contrast can be enhanced by administering a contrast medium such as barium sulfate or iodine. The contrast medium absorbs X-rays strongly and outlines the organ of interest.

软组织的衰减系数相近，因此可以通过服用硫酸钡或碘等造影剂来增强对比度。造影剂强烈吸收 X 射线，从而勾勒出目标器官的轮廓。

Collimation and lead shielding are used to reduce patient dose and scattered radiation. Image intensifiers or flat-panel detectors improve image quality while allowing lower exposure times.

使用准直器和铅屏蔽来减少患者剂量和散射辐射。影像增强器或平板探测器可提高图像质量，同时缩短曝光时间。

4. Computed Tomography (CT) Scanning | 计算机断层扫描（CT）

In CT, an X-ray tube and an array of detectors rotate around the patient. The attenuation of many thin beams is recorded at different angles as the patient moves slowly through the gantry.

在 CT 中，X 射线管和探测器阵列围绕患者旋转。随着患者缓慢移动通过机架，系统记录不同角度下许多细射线束的衰减。

A computer reconstructs a cross-sectional image of the slice using algorithms such as filtered back projection. Each pixel in the image represents the local linear attenuation coefficient of the tissue.

计算机利用滤波反投影等算法重建该层的横断面图像。图像中的每个像素代表组织的局部线性衰减系数。

CT gives much better soft-tissue contrast and three-dimensional information than a single projection radiograph, but the radiation dose is considerably higher. It is widely used for brain, chest and abdominal imaging.

CT 的软组织对比度和三维信息远优于单次投影 X 线摄影，但辐射剂量显著更高。它广泛用于脑、胸和腹部成像。

5. Ultrasound and Piezoelectric Transducer | 超声波与压电换能器

Ultrasound refers to longitudinal sound waves with frequencies above the human hearing range, typically 1–15 MHz for medical scans. Higher frequencies give better resolution but poorer penetration.

超声波是指频率超过人耳听觉范围的纵波，医学扫描通常使用 1–15 MHz。频率越高，分辨率越好，但穿透力越差。

A piezoelectric crystal such as lead zirconate titanate (PZT) changes shape when a voltage is applied, producing ultrasound pulses. When an echo returns, the crystal is compressed and generates a small voltage, so the same transducer acts as both transmitter and receiver.

锆钛酸铅（PZT）等压电晶体在施加电压时会改变形状，从而产生超声脉冲。当回波返回时，晶体受到压缩并产生小电压，因此同一换能器既充当发射器又充当接收器。

A water-based gel is applied to the skin to eliminate air gaps, because air has an acoustic impedance very different from skin and would reflect almost all the ultrasound.

皮肤上涂抹水性耦合凝胶以消除空气间隙，因为空气的声阻抗与皮肤差异极大，几乎会反射所有超声波。

6. Acoustic Impedance and Reflection at Boundaries | 声阻抗与界面反射

The acoustic impedance Z of a medium is defined as Z = ρc, where ρ is the density and c is the speed of sound. Its SI unit is kg m⁻² s⁻¹.

介质的声阻抗 Z 定义为 Z = ρc，其中 ρ 为密度，c 为声速。其 SI 单位是 kg m⁻² s⁻¹。

Z = ρc

At a boundary between two media, the fraction of ultrasound intensity reflected is given by R = (Z₂ − Z₁)² / (Z₂ + Z₁)². A large impedance mismatch causes a strong echo, while closely matched impedances allow most energy to be transmitted.

在两种介质的界面上，超声强度反射比由 R = (Z₂ − Z₁)² / (Z₂ + Z₁)² 给出。阻抗失配越大，回声越强；阻抗匹配越好，大部分能量越能透射。

R = (Z₂ − Z₁)² / (Z₂ + Z₁)²

For example, the impedance mismatch between air and skin is huge, so ultrasound cannot enter the body without coupling gel. Bone and gas bubbles also produce strong reflections that can shadow deeper tissue.

例如，空气与皮肤的阻抗失配极大，因此没有耦合凝胶时超声波无法进入人体。骨骼和气泡也会产生强反射，可能遮挡更深层组织。

7. A-scan and B-scan Ultrasound | A 扫描与 B 扫描超声

In an A-scan, the transducer sends a short

Published by TutorHao | A-Level Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 The Nature of Light – Waves or Particles? | 光的本性——波动还是粒子？

For centuries, physicists have debated whether light is a stream of particles or a wave motion. The modern answer is that light exhibits both wave-like and particle-like behaviour, depending on the experiment performed. This dual nature is central to quantum physics and is tested throughout the CIE A-Level syllabus, from interference and diffraction to the photoelectric effect.

几个世纪以来，物理学家一直在争论光究竟是粒子流还是波动。现代物理学给出的答案是：光既表现出波动性，又表现出粒子性，具体取决于所进行的实验。这种双重本性是量子物理的核心，也是 CIE A-Level 课程中从干涉、衍射到光电效应等内容的考查重点。

1. The Historical Debate: Newton vs Huygens | 历史争论：牛顿与惠更斯

Isaac Newton’s corpuscular theory proposed that light consists of tiny particles, or corpuscles, travelling in straight lines. This model naturally explained reflection and sharp shadows, but it struggled to account for refraction correctly and could not explain interference effects.

艾萨克·牛顿的微粒说认为，光由沿直线传播的微小粒子或微粒组成。该模型能自然地解释反射和清晰的影子，但在正确解释折射方面存在困难，也无法解释干涉现象。

Christiaan Huy

Published by TutorHao | A-Level Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 Binding Energy and Nuclear Stability | 结合能与核稳定性

Nuclear reactions release energies millions of times greater than typical chemical reactions. The fundamental reason is that a small amount of mass disappears when nucleons bind together, and this mass difference is converted into energy. In this article, we connect mass defect, binding energy, binding energy per nucleon and the stability curve, with a strong focus on CIE A-Level Physics requirements.

核反应释放的能量比一般化学反应高数百万倍。其根本原因在于核子结合时会有少量质量消失，这些质量差被转化为能量。本文将围绕质量亏损、结合能、比结合能以及核稳定性曲线展开，紧扣 CIE A-Level 物理考点。

1. Mass Defect and E = mc² | 质量亏损与质能方程

The mass defect Δm of a nucleus is defined as the difference between the total mass of its separate protons and neutrons and the mass of the nucleus itself. When nucleons join together, the bound system has slightly less mass than the sum of its parts.

原子核的质量亏损 Δm 定义为组成它的独立质子和中子总质量与原子核自身质量之差。当核子结合在一起时，形成的束缚系统质量略小于各部分质量之和。

This missing mass is not lost; it is converted into energy when the nucleus is formed. Einstein’s mass-energy relation gives the binding energy:

这些亏损的质量并没有消失，而是在原子核形成时转化为能量。爱因斯坦质能关系给出结合能：

E = Δm c²

In SI units, Δm is in kilograms, c = 3.00 × 10⁸ m s⁻¹, and E is measured in joules. For a single nucleus, this energy is extremely small in joules, so nuclear physicists often use MeV and atomic mass units.

在国际单位制中，Δm 以千克为单位，c = 3.00 × 10⁸ m s⁻¹，能量 E 以焦耳为单位。对单个原子核而言，这个能量以焦耳表示非常小，因此核物理中常用 MeV 和原子质量单位。

Δm = (Z mₚ + N mₙ) − M

Here Z is the proton number, N is the neutron number, mₚ is the proton mass, mₙ is the neutron mass, and M is the mass of the assembled nucleus.

其中 Z 是质子数，N 是中子数，mₚ 是质子质量，mₙ 是中子质量，M 是形成后的原子核质量。

2. Atomic Mass Unit and Energy Equivalence | 原子质量单位与能量当量

In nuclear physics, masses are usually measured in unified atomic mass units, u. By definition, 1 u is exactly one twelfth of the mass of a neutral carbon-12 atom. In kilograms, this is approximately:

在核物理中，质量通常以统一原子质量单位 u 表示。根据定义，1 u 严格等于中性碳-12 原子质量的十二分之一。换算成千克约为：

1 u ≈ 1.661 × 10⁻²⁷ kg

Using E = mc², one atomic mass unit is equivalent to 931.5 MeV of energy. This conversion is essential in binding energy calculations.

利用 E = mc²，一个原子质量单位相当于 931.5 MeV 的能量。这个换算关系在结合能计算中至关重要。

1 u ≈ 931.5 MeV/c²

To convert a mass defect given in u into energy in MeV, we multiply by 931.5 MeV/c². To convert from MeV to joules, use 1 MeV = 1.602 × 10⁻¹³ J.

要将以 u 表示的质量亏损转换为以 MeV

Published by TutorHao | A-Level Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 Modelling with Particles and Waves | 粒子与波建模

In CIE A-Level Physics, the way we interpret light, electrons and other quantum objects depends on the model we choose. A particle model treats energy and momentum as localised bundles, while a wave model treats disturbances as spreading through space and showing interference. This article explains when each model works, the key experimental evidence and the mathematical links behind wave-particle duality.

在 CIE A-Level 物理中，我们如何解释光、电子等量子对象取决于所选模型。粒子模型把能量和动量视为局域束，而波动模型把扰动视为在空间中传播并产生干涉。本文解释两种模型各自适用的情境、关键实验证据以及波粒二象性背后的数学联系。

1. Why Models Matter in Physics | 为什么物理需要模型

A model in physics is a simplified representation used to predict observations. No single classical model is perfect for quantum objects; light sometimes behaves like a stream of photons, and electrons sometimes behave like waves.

物理模型是一种用于预测观测结果的简化表示。对量子物体而言，没有单一经典模型是完美的；光有时表现为光子流，电子有时表现为波。

Using the wrong model can lead to incorrect predictions. For example, Maxwell’s wave model explains interference but cannot explain the photoelectric effect without quantisation of light.

使用错误模型会导致错误预测。例如，麦克斯韦波动模型能解释干涉，但若不引入光的量子化，就无法解释光电效应。

2. The Particle Model of Light | 光的粒子模型

In the particle model, light consists of discrete packets of electromagnetic energy called photons. Each photon carries energy E = hf, where h is the Planck constant and f is the frequency.

在粒子模型中，光由离散的电磁能量包组成，称为光子。每个光子携带能量 E = hf，其中 h 是普朗克常量，f 是频率。

E = hf

The photon model was introduced by Einstein in 1905 to explain the photoelectric effect. It treats photons as having no rest mass but carrying momentum p = h/λ.

光子模型由爱因斯坦于 1905 年提出，用于解释光电效应。该模型认为光子没有静止质量，但具有动量 p = h/λ。

p = h / λ

3. The Wave Model of Light | 光的波动模型

The wave model describes light as a transverse electromagnetic wave. Electric and magnetic fields oscillate perpendicular to the direction of energy transfer.

波动模型将光描述为横电磁波。电场和磁场垂直于能量传递方向振荡。

Young’s double-slit experiment provides strong evidence for the wave nature of light. Monochromatic light passing through two slits creates bright and dark fringes due to constructive and destructive interference.

杨氏双缝实验为光的波动性提供了有力证据。单色光通过双缝后因相长干涉和相消干涉产生明暗条纹。

The fringe spacing Δx is given by:

条纹间距 Δx 由以下关系给出：

Δx = λD / d

where λ is the wavelength, D is the distance from slits to screen, and d is the slit separation.

其中 λ 是波长，D 是双缝到屏的距离，d 是双缝间距。

4. Photoelectric Effect: Evidence for Photons | 光电效应：光子的证据

When ultraviolet light shines on a clean metal surface, electrons are emitted. The wave model predicts that brighter light should give electrons more kinetic energy, but experiments show otherwise.

当紫外光照射洁净金属表面时，会发射电子。波动模型预测光越强，电子动能越大，但实验结果并非如此。

Key observations are: emission is instantaneous above a threshold frequency f₀; maximum electron kinetic energy depends only on frequency, not intensity; intensity affects only the number of emitted electrons.

关键观察是：高于阈值频率 f₀ 时发射立即发生；电子最大动能只取决于频率而非光强；光强只影响发射电子的数量。

Eₖ,ₘₐₓ = hf – Φ

Here Φ is the work function of the metal, the minimum energy needed to remove an electron. The threshold frequency is f₀ = Φ / h.

这里 Φ 是金属的功函数，即移除一个电子所需的最小能量。阈值频率为 f₀ = Φ / h。

5. Electron Diffraction: Evidence for Matter Waves | 电子衍射：物质波的证据

Electrons accelerated through a potential difference V gain kinetic energy Eₖ = eV. When they pass through a thin graphite film, a diffraction pattern of concentric rings appears on a fluorescent screen.

电子经电势差 V 加速后获得动能 Eₖ = eV。当它们穿过薄石墨膜时，荧光屏上出现同心环衍射图样。

This pattern is evidence that electrons behave as waves. Increasing the accelerating voltage reduces the ring spacing, showing that the wavelength decreases as electron speed increases.

这一图样证明电子表现出波动行为。提高加速电压会减小环间距，表明电子速度增加时波长减小。

The electron wavelength is given by:

电子波长由下式给出：

λ = h / p = h / √(2meV)

where m is the electron mass and e is the elementary charge.

其中 m 是电子质量，e 是元电荷。

6. De Broglie Wavelength | 德布罗意波长

Louis de Broglie proposed that all particles have a wavelength related to their momentum. This unified the particle and wave descriptions of matter.

德布罗意提出所有粒子都具有与其动量相关的波长。这统一了物质的粒子描述和波动描述。

The de Broglie wavelength is:

德布罗意波长为：

λ = h / p = h / mv

For macroscopic objects, the wavelength is extremely small, so wave behaviour is not observed. For electrons and other small particles, the wavelength can be comparable to atomic spacing, producing observable diffraction.

对于宏观物体，波长极小，因此观察不到波动行为。对于电子和其他小粒子，波长可与原子间距相当，从而产生可观察的衍射。

Example: An electron accelerated through 54 V has a de Broglie wavelength of about 1.67 × 10⁻¹⁰ m, similar to atomic spacing. This is why electron diffraction is useful for studying crystal structure.

例题：经 54 V 加速的电子德布罗意波长约为 1.67 × 10⁻¹⁰ m，与原子间距相近。这就是电子衍射可用于研究晶体结构的原因。

7. Wave-Particle Duality | 波粒二象性

Wave-particle duality states that quantum objects have both wave-like and particle-like properties. Neither classical model alone gives a complete description.

波粒二象性指出量子物体同时具有波动性和粒子性。单独使用任一经典模型都无法给出完整描述。

Light shows particle behaviour in the photoelectric effect and wave behaviour in interference. Electrons show particle behaviour in collisions and wave behaviour in diffraction.

光在光电效应中表现出粒子性，在干涉中表现出波动性。电子在碰撞中表现出粒子性，在衍射中表现出波动性。

Which property is observed depends on the experiment. A single quantum object interacts at a point like a particle, but the probability distribution of many events shows a wave-like interference pattern.

观察到哪种性质取决于实验。单个量子物体像一个粒子那样在一点发生相互作用，但大量事件的概率分布却显示出类波的干涉图样。

8. Atomic Spectra and Energy Quantisation | 原子光谱与能量量子化

Atoms emit or absorb light at discrete wavelengths, producing line spectra. These spectra can only be explained if electron energy levels are quantised.

原子以离散波长发射或吸收光，产生线光谱。只有电子能级是量子化的，才能解释这些光谱。

When an electron drops from a higher energy level E₂ to a lower level E₁, it emits a photon of energy:

当电子从较高能级 E₂ 跃迁到较低能级 E₁ 时，会发射一个光子，能量为：

hf = E₂ – E₁

The hydrogen spectrum is a classic example. The Balmer series lies in the visible region and corresponds to transitions ending at n = 2.

氢光谱是一个经典例子。巴耳末系位于可见光区，对应于终止在 n = 2 的跃迁。

This quantisation supports the particle model of light, as photons carry fixed energy differences, while the wavelengths of spectral lines link back to the wave model through c = fλ.

这种量子化支持了光的粒子模型，因为光子携带固定的能量差，而谱线波长又通过 c = fλ 与波动模型联系起来。

9. Probabilistic Interpretation of Waves | 波的概率诠释

In quantum physics, a particle’s wave function describes the probability amplitude for finding the particle at a location. The square of the wave function gives the probability density.

在量子物理中，粒子的波函数描述在某位置找到粒子的概率幅。波函数的平方给出概率密度。

This means a wave-like diffraction pattern does not show a particle spreading out; it shows the distribution of possible detection positions over many identical trials.

这意味着类波衍射图样并不表示粒子分散开来，而是显示在多次相同实验中可能被探测到的位置分布。

The probabilistic model is important for A-Level when understanding why individual electrons arrive at the screen one by one, yet build up an interference pattern over time.

概率模型在 A-Level 中很重要，可帮助理解为什么单个电子一个接一个到达屏幕，但随时间积累却能形成干涉图样。

Published by TutorHao | A-Level Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 Randomness and Decay | 放射性随机性与衰变

Radioactive decay is a spontaneous and random process that cannot be controlled by external conditions such as temperature, pressure or chemical bonding. This topic explores how randomness is described statistically, how decay is modelled, and how half-life and activity are calculated.

放射性衰变是一种自发且随机的过程，不受温度、压力或化学键等外部条件控制。本主题将探讨如何用统计方法描述随机性，如何建立衰变模型，以及如何计算半衰期和活度。

1. The Random Nature of Radioactive Decay | 放射性衰变的随机性

Radioactive decay is a spontaneous process. It is impossible to predict which individual nucleus in a sample will decay next, or when a particular nucleus will decay.

放射性衰变是一种自发过程。无法预测样品中哪一个原子核会下一个衰变，也无法预测某个特定原子核何时会衰变。

However, for a large number of nuclei, the overall behaviour is governed by probability and is statistically predictable.

然而，对于大量原子核，整体行为由概率支配，并且具有统计可预测性。

This combination of individual randomness and collective statistical regularity is central to understanding decay.

这种个体随机性与集体统计规律性的结合，是理解衰变的核心。

2. Activity and the Decay Constant | 活度与衰变常量

Activity A is defined as the number of decays per unit time. The SI unit of activity is the becquerel (Bq), where 1 Bq = 1 decay per second.

活度 A 定义为单位时间内发生衰变的次数。活度的国际单位是贝克勒尔（Bq），其中 1 Bq = 每秒 1 次衰变。

The decay constant λ represents the probability that a single nucleus will decay per unit time. Its unit is s⁻¹.

衰变常量 λ 表示单个原子核在单位时间内发生衰变的概率。其单位为 s⁻¹。

For a sample containing N undecayed nuclei, the activity is given by:

对于含有 N 个未衰变原子核的样品，活度由下式给出：

A = λN

A larger decay constant means a higher probability of decay and therefore a more active sample for a given number of nuclei.

衰变常量越大，表示衰变概率越高，因此在原子核数量相同的情况下，样品的活度也越大。

3. The Decay Equation N = N₀e⁻λt | 衰变方程 N = N₀e⁻λt

Because decay is random, the number of undecayed nuclei decreases exponentially with time. If N₀ is the initial number of undecayed nuclei at t = 0, then after time t the number remaining is:

由于衰变是随机的，未衰变原子核的数量随时间呈指数减少。如果 N₀ 是 t = 0 时未衰变原子核的初始数量，那么经过时间 t 后剩余的数量为：

N = N₀e^−λt

This equation follows from the differential equation dN/dt = −λN, which states that the rate of decay is proportional to the number of undecayed nuclei present.

该方程源自微分方程 dN/dt = −λN，它表明衰变速率与当前未衰变原子核的数量成正比。

The negative sign indicates that N decreases as time increases.

负号表示 N 随时间的增加而减少。

4. Half-Life | 半衰期

The half-life T½ is the average time taken for half of the unstable nuclei in a sample to decay, or equivalently, for the activity to fall to half of its initial value.

半衰期 T½ 是样品中一半不稳定原子核发生衰变所需的平均时间，等价地，也是活度降至其初始值一半所需的时间。

The half-life is related to the decay constant by:

半衰期与衰变常量的关系为：

T½ = ln 2 / λ = 0.693 / λ

A large decay constant gives a short half-life, meaning the isotope decays rapidly. A small decay constant gives a long half-life and a less active sample.

衰变常量越大，半衰期越短，意味着同位素衰变很快。衰变常量越小，半衰期越长，样品的活度也越低。

5. Exponential Decay Curve | 指数衰变曲线

A graph of the number of undecayed nuclei N against time t is a decreasing exponential curve. The curve never reaches zero, but approaches the time axis asymptotically.

未衰变原子核数量 N 随时间 t 变化的图像是一条递减的指数曲线。该曲线永远不会到达零，而是以时间轴为渐近线。

In each successive half-life interval, the number of undecayed nuclei falls by half. After one half-life, N = N₀/2; after two half-lives, N = N₀/4; after three half-lives, N = N₀/8.

在每一个连续的半衰期时间间隔内，未衰变原子核数量减少一半。经过一个半衰期后，N = N₀/2；经过两个半衰期后，N = N₀/4；经过三个半衰期后，N = N₀/8。

This constant-fraction decay is a key feature of an exponential process.

这种恒定比例衰变是指数过程的一个重要特征。

6. Activity Equation and Count Rate | 活度方程与计数率

Since activity A is proportional to N, the activity also decreases exponentially with time:

由于活度 A 与 N 成正比，活度也随时间呈指数下降：

A = A₀e^−λt

In experiments, a Geiger-Muller tube measures a count rate C, which is directly proportional to the activity, provided the background count rate has been subtracted.

在实验中，盖革-米勒计数管测量计数率 C，只要已扣除背景计数率，计数率便与活度成正比。

Therefore the corrected count rate follows the same exponential law: C = C₀e^−λt.

因此，修正后的计数率遵循相同的指数规律：C = C₀e^−λt。

7. Statistical Fluctuations | 统计涨落

Because decay is random, repeated measurements of the same source over equal time intervals do not give exactly the same count. The counts show statistical fluctuations.

由于衰变是随机的，在相同时间间隔内对同一放射源进行重复测量，得到的计数并不完全相同。计数会表现出统计涨落。

For a count N recorded in a given interval, the standard uncertainty is approximately √N. The fractional uncertainty is therefore 1/√N.

对于在给定时间间隔内记录到的计数 N，标准不确定度约为 √N。因此相对不确定度为 1/√N。

Recording a larger total count reduces the fractional uncertainty and improves the precision of the measurement. To achieve this, longer counting times or a stronger source are used.

记录更大的总计数可以降低相对不确定度，从而改善测量精度。为此，可使用更长的计数时间或更强的放射源。

8. Background Radiation | 背景辐射

Background radiation is always present due to cosmic rays, naturally occurring radioactive materials in rocks and soil, radon gas, and medical or industrial sources.

背景辐射始终存在，来源包括宇宙射线、岩石和土壤中的天然放射性物质、氡气以及医疗或工业源。

When measuring the activity of a source, the background count rate must be measured separately and subtracted from the total count rate to obtain the corrected count rate due to the source alone.

在测量放射源的活度时，必须单独测量背景计数率，并从总计数率中减去，以获得仅由该放射源产生的修正计数率。

Background radiation is itself random, so its contribution also fluctuates and should be measured over a sufficiently long time to reduce uncertainty.

背景辐射本身也是随机的，因此其贡献也会发生涨落，应在足够长的时间内测量以减小不确定度。

9. Measuring Half-Life | 半衰期的测量

To measure the half-life of a radioactive isotope, a detector such as a Geiger-Muller tube is used to record the count rate at regular time intervals.

为测量放射性同位素的半衰期，可使用盖革-米勒计数管等探测器，在固定的时间间隔记录计数率。

The background count rate is first measured and subtracted. The corrected count rate is then plotted against time to obtain an exponential decay curve, from which the half-life can be read directly.

首先测量并扣除背景计数率。然后绘制修正计数率随时间的变化曲线，得到指数衰变曲线，从曲线上可以直接读出半衰期。

Alternatively, a graph of ln C against t is plotted. Since ln C = ln C₀ − λt, the graph is a straight line with gradient −λ. The half-life is then found from T½ = ln 2 / λ.

另一种方法是绘制 ln C 对 t 的图像。由于 ln C = ln C₀ − λt，该图像是一条斜率为 −λ 的直线。然后利用 T½ = ln 2 / λ 求出半衰期。

10. Applications: Radioactive Dating | 应用：放射性定年

Carbon-14 dating is used to estimate the age of organic remains such as wood, cloth or bone. Living organisms continually exchange carbon with the atmosphere, so the ratio of carbon-14 to carbon-12 remains roughly constant while alive.

碳-14 定年法用于估算木材、织物或骨骼等有机遗骸的年龄。生物体在存活期间不断与大气交换碳，因此碳-14 与碳-12 的比率在存活期间大致保持恒定。

After death, carbon-14 decays with a half-life of about 5730 years and is no longer replaced. By measuring the remaining activity or the ratio of carbon-14 to carbon-12, the time since death can be calculated using N = N₀e^−λt.

生物体死亡后，碳-14 以约 5730 年的半衰期衰变，且不再得到补充。通过测量剩余的活度或碳-14 与碳-12 的比率，可以利用 N = N₀e^−λt 计算死亡至今的时间。

This method relies on the assumption that the atmospheric carbon-14 ratio has been approximately constant over time. Calibration with other dating methods helps to improve accuracy.

该方法依赖于大气中碳-14 比率随时间大致恒定的假设。利用其他定年方法进行校准有助于提高准确性。

11. Safety and Randomness | 安全与随机性

The random nature of decay means that the biological effects of ionising radiation are probabilistic. Even low doses carry a small but non-zero risk of cellular damage or mutation.

衰变的随机性意味着电离辐射的生物效应是概率性的。即使低剂量也存在微小但非零的细胞损伤或突变风险。

Safety measures therefore aim to minimise exposure. The three main principles are to keep the time of exposure short, increase distance from the source, and use shielding such as lead.

因此，安全措施旨在尽量减少照射。三个主要原则是：缩短照射时间、增大与放射源的距离，以及使用铅等屏蔽材料。

Sources should be handled with tongs and stored in lead-lined containers when not in use. Experiments should be planned so that the total count collected is high enough for precision while keeping exposure low.

放射源应使用镊子操作，不用时应存放在铅衬容器中。实验设计应使收集到的总计数足够高以保证精度，同时保持低照射量。

12. Key Equations Summary | 关键公式总结

The following table summarises the main relationships used in the randomness and decay topic.

下表总结了随机性与衰变主题中使用的主要关系式。

These equations all assume that the decay process is random and that the number of nuclei is large enough for statistical treatment to be valid.

这些方程均假设衰变过程是随机的，并且原子核数量足够大，统计处理可以成立。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 Energy Released in Radioactive Decay | 放射性衰变释放的能量

Radioactive decay releases energy when an unstable nucleus transforms into a more stable configuration. This energy appears as kinetic energy of emitted particles and gamma photons. The source of the energy is a small loss of mass, described by Einstein’s mass-energy equivalence.

放射性衰变在不稳定原子核转变为更稳定组态时释放能量。这些能量表现为发射粒子和γ光子的动能。能量来源于微小的质量损失，由爱因斯坦的质能等价关系描述。

1. Mass-Energy Equivalence and Atomic Mass Unit | 质能等价与原子质量单位

Einstein’s relation E = mc² shows that mass can be converted into energy. In nuclear physics, masses are conveniently expressed in atomic mass units, u. One atomic mass unit is defined as 1/12 of the mass of a carbon-12 atom.

爱因斯坦关系 E = mc² 表明质量可以转化为能量。在核物理中，质量通常用原子质量单位 u 表示。一个原子质量单位定义为碳-12 原子质量的 1/12。

1 u = 1.6605 × 10⁻²⁷ kg

1 u c² = 931.5 MeV

This conversion factor is the key to all decay-energy calculations: a mass loss of 1 u releases 931.5 MeV of energy.

这个换算因子是所有衰变能计算的关键：质量亏损 1 u 释放 931.5 MeV 能量。

2. Binding Energy and Nuclear Stability | 结合能与核稳定性

The binding energy of a nucleus is the energy required to separate it into its individual protons and neutrons. The mass of a nucleus is always less than the sum of its separated nucleons; this difference is the mass defect. Multiplying the mass defect by c² gives the binding energy.

原子核的结合能是将它拆散为单个质子和中子所需的能量。原子核的质量总是小于其分离核子质量之和；这个差值就是质量亏损。将质量亏损乘以 c² 即得结合能。

Δm = Z m_p + N m_n − m_nucleus

BE = Δm c²

The binding energy per nucleon increases from light nuclei up to iron-56 and then decreases. A decay is energetically allowed if the products have greater binding energy per nucleon than the parent, so energy is released as the system becomes more tightly bound.

每个核子的结合能从轻核到铁-56 逐渐增大，之后又下降。如果产物的每个核子结合能大于母核，则衰变在能量上是允许的，系统变得更紧密时释放能量。

3. Mass Defect in Decay Processes | 衰变过程中的质量亏损

In any spontaneous decay, the total rest mass of the products is less than the rest mass of the parent. This mass

Published by TutorHao | A-Level Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 Line Spectra | 线光谱

When white light is dispersed by a prism or diffraction grating, it forms a continuous spectrum containing all visible wavelengths. However, light emitted or absorbed by individual atoms produces a very different pattern: a line spectrum consisting of discrete, well-defined wavelengths. Understanding line spectra is essential for explaining the quantised nature of atomic energy levels in A-Level Physics.

当白光通过棱镜或衍射光栅发生色散时，会形成包含所有可见波长的连续光谱。然而，单个原子发射或吸收的光会产生截然不同的图案：由分离且明确的波长组成的线状光谱。理解线光谱对于解释 A-Level 物理中原子能级的量子化本质至关重要。

1. Continuous vs Line Spectra | 连续光谱与线状光谱

A continuous spectrum contains all wavelengths over a given range, appearing as an unbroken band of colours. It is produced by hot solids, liquids, or dense gases under high pressure.

连续光谱在给定范围内包含所有波长，表现为一条不间断的色带。它由炽热的固体、液体或高压稠密气体产生。

A line spectrum contains only certain discrete wavelengths, showing either bright lines on a dark background or dark lines on a continuous background. Line spectra are produced by individual atoms in low-pressure gases.

线状光谱只包含特定分立的波长，在暗背景上显示亮线，或在连续背景上显示暗线。线状光谱由低压气体中的单个原子产生。

At A-Level, you must be able to distinguish between emission and absorption line spectra, and relate them to electronic transitions between quantised energy levels.

在 A-Level 课程中，你必须能够区分发射线光谱和吸收线光谱，并将它们与量子化能级之间的电子跃迁联系起来。

2. Emission Line Spectra | 发射线光谱

An emission line spectrum is produced when a hot, low-pressure gas emits light. The gas is excited by heating or by an electric discharge, giving its atoms extra energy.

当炽热的低压气体发光时，会产生发射线光谱。气体通过加热或放电被激发，使原子获得额外能量。

The spectrum consists of bright coloured lines at specific wavelengths on a dark background. Each line corresponds to a particular frequency emitted when an excited electron falls from a higher energy level to a lower one.

光谱由暗背景上特定波长的明亮彩色线条组成。每条线对应于激发态电子从高能级跃迁到低能级时发射的特定频率。

For example, a hydrogen discharge tube gives visible emission lines at 656 nm (red), 486 nm (blue-green), 434 nm (blue-violet) and 410 nm (violet).

例如，氢放电管发出的可见发射谱线分别为 656 nm（红）、486 nm（蓝绿）、434 nm（蓝紫）和 410 nm（紫）。

3. Absorption Line Spectra | 吸收线光谱

An absorption line spectrum is formed when white light passes through a cool gas. The gas absorbs only those photons whose energies exactly match the energy differences between its own atomic levels.

当白光穿过低温气体时，会形成吸收线光谱。气体只吸收能量与其自身原子能级差完全匹配的光子。

The observed spectrum is a continuous rainbow with dark lines missing at specific wavelengths. These dark lines occur at exactly the same wavelengths as the bright lines in the same element’s emission spectrum.

观察到的光谱是连续彩虹背景上在特定波长处出现暗线。这些暗线出现的波长与该元素发射光谱中亮线的波长完全一致。

This correspondence is strong evidence that atoms can only absorb or emit photons of certain discrete energies, supporting the idea of quantisation.

这种对应关系有力地证明了原子只能吸收或发射某些分立能量的光子，支持了量子化的观点。

4. The Bohr Model and Energy Levels | 玻尔模型与能级

Niels Bohr proposed that electrons in an atom occupy only certain allowed orbits, called energy levels or shells. The energy of each level is fixed, so the possible energies are quantised.

尼尔斯·玻尔提出，原子中的电子只能处于某些允许的轨道，称为能级或壳层。每个能级的能量是固定的，因此可能的能量是量子化的。

The lowest energy level is the ground state (n = 1). Higher levels are excited states (n = 2, 3, 4, …). For hydrogen, the ground state energy is −13.6 eV.

最低能级是基态（n = 1）。较高能级为激发态（n = 2, 3, 4, …）。对于氢原子，基态能量为 −13.6 eV。

The negative sign means the electron is bound to the nucleus. Zero energy corresponds to the electron being free from the atom, which occurs when ionisation has taken place.

负号表示电子被原子核束缚。零能量对应电子完全脱离原子，即发生电离。

5. Photon Energy and Transitions | 光子能量与跃迁

When an electron moves from a higher energy level E₂ to a lower energy level E₁, it emits a photon. The photon energy equals the difference between the two levels.

当电子从高能级 E₂ 跃迁到低能级 E₁ 时，会发射一个光子。光子能量等于两个能级之间的能量差。

ΔE = E₂ − E₁ = hf

Here ΔE is the energy difference in joules, h is the Planck constant (6.63 × 10⁻³⁴ J s), and f is the photon frequency in hertz.

其中 ΔE 是能量差（单位焦耳），h 是普朗克常数（6.63 × 10⁻³⁴ J s），f 是光子频率（单位赫兹）。

Since c = fλ, the wavelength of the emitted photon can be found from:

由于 c = fλ，发射光子的波长可由下式求得：

ΔE = hc/λ

Therefore λ = hc/ΔE. A small energy difference gives a photon with low energy and long wavelength; a large energy difference gives a photon with high energy and short wavelength.

因此 λ = hc/ΔE。能量差小，则光子能量低、波长长；能量差大，则光子能量高、波长短。

Before using this formula, convert energy level differences from electronvolts to joules using 1 eV = 1.60 × 10⁻¹⁹ J.

使用该公式前，需要将能级差从电子伏特转换为焦耳：1 eV = 1.60 × 10⁻¹⁹ J。

6. Hydrogen Spectral Series | 氢光谱线系

Hydrogen has the simplest line spectrum. Its spectral lines are grouped into series, each corresponding to transitions ending at a particular lower energy level n₁.

氢具有最简单的线状光谱。其谱线分为若干个线系，每个线系对应于终止在特定低能级 n₁ 的跃迁。

The Lyman series ends at n₁ = 1 and lies in the ultraviolet region. The Balmer series ends at n₁ = 2 and includes four visible lines.

莱曼线系终止于 n₁ = 1，位于紫外区。巴尔末线系终止于 n₁ = 2，包含四条可见谱线。