IB Physics: Energy Level Transitions and Radioactive Decay | IB物理：能级跃迁与放射性衰变

📚 IB Physics: Energy Level Transitions and Radioactive Decay | IB物理：能级跃迁与放射性衰变

Energy level transitions and radioactive decay are two cornerstone topics in IB Physics. They reveal the quantum nature of matter and the fundamental instability of certain nuclei. This article will guide you through the essential concepts, formulas, and exam-focused insights you need to master these topics.

能级跃迁与放射性衰变是IB物理中的两大基石课题。它们揭示了物质的量子本质以及某些原子核的基本不稳定性。本文将带你梳理核心概念、关键公式以及考试重点，帮助你全面掌握这两部分内容。

1. The Bohr Model of the Atom | 玻尔原子模型

The Bohr model describes electrons orbiting the nucleus in discrete energy levels. Electrons can only occupy specific orbits with quantized energy values, denoted by the principal quantum number n (n = 1, 2, 3, …). The ground state (n = 1) has the lowest energy, while higher values of n correspond to excited states.

玻尔模型描述了电子绕核运动于分立的能级之上。电子只能占据具有量子化能量值的特定轨道，这些能量值由主量子数n（n = 1, 2, 3, …）表示。基态（n = 1）能量最低，而n值越大对应的激发态能量越高。

The energy of an electron in a hydrogen atom is given by:

氢原子中电子的能量表达式为：

Eₙ = -13.6 eV / n²

Here, the negative sign indicates that the electron is bound to the nucleus. As n increases, the energy becomes less negative, meaning the electron is less tightly bound.

这里的负号表示电子被束缚在原子核周围。随着n增大，能量负值减小，说明电子受到的束缚变弱。

2. Photon Emission and Absorption | 光子的发射与吸收

When an electron transitions from a higher energy level (Eᵢ) to a lower energy level (E𝒻), it emits a photon with energy equal to the difference between the two levels. Conversely, a photon can be absorbed to excite an electron from a lower to a higher level.

当电子从高能级（Eᵢ）跃迁到低能级（E𝒻）时，会发射一个光子，其能量等于两个能级之差。反之，吸收一个光子可以将电子从低能级激发到高能级。

The energy of the emitted or absorbed photon is:

发射或吸收的光子能量为：

ΔE = Eᵢ – E𝒻 = hf = hc / λ

Where h is Planck’s constant (6.63 × 10⁻³⁴ J·s), f is the frequency of the photon, c is the speed of light, and λ is the wavelength.

其中h是普朗克常量（6.63 × 10⁻³⁴ J·s），f是光子频率，c是光速，λ是波长。

Key exam tip: Always ensure that the energy difference exactly matches the photon energy. If the photon energy is too large or too small, absorption cannot occur.

考试提示：务必确保光子的能量与能级差精确匹配。如果光子能量偏大或偏小，吸收过程就不会发生。

3. The Rydberg Formula and Hydrogen Spectrum | 里德伯公式与氢原子光谱

The Rydberg formula predicts the wavelengths of spectral lines for hydrogen:

里德伯公式可以预测氢原子光谱线的波长：

1/λ = R_H (1/n₁² – 1/n₂²)

Here, R_H is the Rydberg constant (1.097 × 10⁷ m⁻¹), n₁ and n₂ are positive integers with n₂ > n₁. The Lyman series (n₁ = 1) lies in the ultraviolet region, the Balmer series (n₁ = 2) in the visible region, and the Paschen series (n₁ = 3) in the infrared region.

其中R_H是里德伯常量（1.097 × 10⁷ m⁻¹），n₁和n₂为正整数且n₂ > n₁。莱曼系（n₁ = 1）位于紫外区，巴尔末系（n₁ = 2）位于可见光区，帕邢系（n₁ = 3）位于红外区。

Let us consider a worked example. What is the wavelength of the first line in the Balmer series for hydrogen?

我们来看一个计算示例。求氢原子巴尔末系第一条谱线的波长是多少？

For the first line of the Balmer series, n₁ = 2 and n₂ = 3:

对于巴尔末系第一条谱线，n₁ = 2，n₂ = 3：

1/λ = (1.097 × 10⁷ m⁻¹)(1/2² – 1/3²) = (1.097 × 10⁷)(1/4 – 1/9)

1/λ = (1.097 × 10⁷)(5/36) = 1.524 × 10⁶ m⁻¹, thus λ = 6.56 × 10⁻⁷ m = 656 nm. This is the well-known red H-alpha line.

1/λ = (1.097 × 10⁷)(5/36) = 1.524 × 10⁶ m⁻¹，因此λ = 6.56 × 10⁻⁷ m = 656 nm，这就是著名的H-α红线。

4. Emission and Absorption Spectra | 发射光谱与吸收光谱

An emission spectrum is produced when excited atoms return to lower energy levels, emitting photons at discrete wavelengths. These appear as bright lines on a dark background.

发射光谱是受激原子回到低能级时产生的，表现为暗背景上的明亮谱线，对应特定波长的光子。

An absorption spectrum occurs when white light passes through a cool gas. Atoms absorb photons at specific energies, leaving dark lines in the otherwise continuous spectrum. This is how astronomers determine the composition of distant stars.

吸收光谱则是白光穿过低温气体时产生的。原子吸收特定能量的光子，在连续光谱上留下暗线。天文学家正是利用这一原理来测定遥远恒星的成分。

The absorption lines of a gas occur at exactly the same wavelengths as its emission lines, a direct consequence of the quantized energy levels.

气体的吸收谱线与其发射谱线的波长完全相同，这是能级量子化的直接结果。

5. Introduction to Radioactive Decay | 放射性衰变概述

Radioactive decay is a spontaneous, random process in which an unstable nucleus loses energy by emitting radiation. It is governed by the laws of quantum mechanics and cannot be influenced by any chemical or physical means.

放射性衰变是不稳定原子核通过发射辐射来释放能量的自发随机过程。它由量子力学规律支配，不受任何化学或物理手段影响。

There are three main types of radiation produced during decay: alpha (α) particles, beta (β) particles, and gamma (γ) rays.

衰变过程中会产生三种主要辐射类型：α粒子、β粒子和γ射线。

Radiation Type	Nature	Symbol	Charge	Ionizing Power
Alpha	Helium nucleus	²⁴He or α	+2	High
Beta-minus	Fast electron	₋₁⁰e or β⁻	-1	Medium
Gamma	High-energy photon	γ	0	Low

Alpha particles are the most massive and carry the highest ionizing power, but are easily stopped by a sheet of paper. Beta particles are lighter and can penetrate a few millimeters of aluminum. Gamma rays are highly penetrating and require several centimeters of lead or meters of concrete to be significantly attenuated.

α粒子质量最大，电离能力最强，但穿透力最弱，一张纸即可阻挡。β粒子质量较轻，能穿透数毫米厚的铝板。γ射线穿透力极强，需要数厘米厚的铅或数米厚的混凝土才能有效衰减。

6. Alpha and Beta Decay Equations | α衰变与β衰变方程

In alpha decay, a nucleus loses two protons and two neutrons. The general equation is:

在α衰变中，原子核损失两个质子和两个中子。一般方程为：

^A_Z X → ^(A-4)_(Z-2) Y + ⁴₂He

For example, uranium-238 decays into thorium-234:

例如，铀-238衰变为钍-234：

²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He

In beta-minus decay, a neutron converts into a proton, emitting an electron and an antineutrino:

在β⁻衰变中，一个中子转化为质子，同时发射一个电子和一个反中微子：

^A_Z X → ^A_(Z+1) Y + ₋₁⁰e + ν̄

For example, carbon-14 decays into nitrogen-14:

例如，碳-14衰变为氮-14：

¹⁴₆C → ¹⁴₇N + ₋₁⁰e + ν̄

Notice that in beta decay, the mass number A remains unchanged, but the atomic number Z increases by one.

注意在β衰变中，质量数A不变，但原子序数Z增加1。

7. The Exponential Decay Law | 指数衰变定律

The number of undecayed nuclei in a radioactive sample decreases exponentially with time:

放射性样品中未衰变核的数量随时间呈指数衰减：

N = N₀e^(-λt)

Where N₀ is the initial number of nuclei, N is the number remaining after time t, and λ is the decay constant, measured in s⁻¹. The decay constant represents the probability per unit time that a nucleus will decay.

其中N₀是初始核数，N是经过时间t后剩余的核数，λ是衰变常量（单位s⁻¹）。衰变常量表示一个原子核在单位时间内发生衰变的概率。

The activity A of a sample is the rate of decay:

样品的活度A表示单位时间内发生衰变的次数：

A = A₀e^(-λt) = λN

Activity is measured in becquerels (Bq), where 1 Bq = 1 decay per second.

活度的单位是贝克勒尔（Bq），1 Bq = 1次衰变每秒。

8. Half-Life and Decay Constant | 半衰期与衰变常量

The half-life (T₁/₂) is the time required for half of the radioactive nuclei in a sample to decay. It is related to the decay constant by:

半衰期（T₁/₂）是指样品中一半放射性原子核发生衰变所需的时间。它与衰变常量的关系为：

T₁/₂ = ln 2 / λ ≈ 0.693 / λ

After n half-lives, the fraction of nuclei remaining is (1/2)ⁿ = 2⁻ⁿ. For example, after one half-life, 50% remains; after two half-lives, 25% remains; after three half-lives, only 12.5% remains.

经过n个半衰期后，剩余核素比例为(1/2)ⁿ = 2⁻ⁿ。例如，经过一个半衰期剩余50%，两个半衰期剩余25%，三个半衰期仅剩12.5%。

Worked example: A sample initially contains 8.0 × 10²⁰ radioactive nuclei with a half-life of 6.0 hours. How many nuclei remain after 24 hours?

计算示例：某样品初始含有8.0 × 10²⁰个放射性原子核，半衰期为6.0小时。24小时后还剩多少个原子核？

24 hours = 4 half-lives. N = N₀(1/2)⁴ = (8.0 × 10²⁰)(1/16) = 5.0 × 10¹⁹ nuclei.

24小时 = 4个半衰期。N = N₀(1/2)⁴ = (8.0 × 10²⁰)(1/16) = 5.0 × 10¹⁹个原子核。

9. Radiocarbon Dating | 碳-14测年法

Radiocarbon dating is a practical application of radioactive decay. Carbon-14 is continuously produced in the upper atmosphere by cosmic ray neutrons interacting with nitrogen-14:

碳-14测年法是放射性衰变的实际应用之一。碳-14在上层大气中由宇宙射线中子与氮-14相互作用而持续产生：

¹⁴₇N + ¹₀n → ¹⁴₆C + ¹₁H

Living organisms maintain a constant ratio of carbon-14 to carbon-12 through respiration and photosynthesis. When an organism dies, the intake of carbon stops, and the carbon-14 begins to decay with a half-life of 5,730 years. By measuring the remaining carbon-14 activity, scientists can estimate the age of ancient organic materials.

活体生物通过呼吸和光合作用维持体内碳-14与碳-12的恒定比例。当生物死亡后，碳的摄入停止，碳-14开始以5,730年的半衰期衰变。通过测量剩余的碳-14活度，科学家可以估算远古有机材料的年代。

Limitations: This method is only reliable for samples younger than about 60,000 years, and assumes that the atmospheric carbon-14 concentration has remained constant over time.

局限性：该方法仅对约60,000年以内的样品可靠，并且假设大气中碳-14的浓度在历史上保持恒定。

10. Nuclear Equations and Conservation Laws | 核反应方程与守恒定律

When writing and balancing nuclear equations, several conservation laws must be satisfied:

在书写和配平核反应方程时，必须满足以下守恒定律：

Conservation of mass number (A): Total A before the decay equals total A after.
质量数守恒：反应前后总质量数A相等。
Conservation of atomic number (Z): Total Z before the decay equals total Z after.
电荷数守恒：反应前后总原子序数Z相等。
Conservation of mass-energy (E = mc²): The total energy before and after the reaction is conserved.
质能守恒（E = mc²）：反应前后总能量守恒。

Momentum is also conserved in nuclear decays. In alpha decay, the daughter nucleus recoils in the opposite direction to the emitted alpha particle, which is why the alpha particle carries away most of the kinetic energy.

动量同样在核衰变中守恒。在α衰变中，子核沿α粒子发射的相反方向反冲，因此α粒子带走了大部分动能。

11. Background Radiation and Safety | 背景辐射与辐射安全

Background radiation is the low-level ionizing radiation that is always present in the environment. Sources include cosmic rays, radon gas from the ground, natural isotopes in rocks and food, and artificial sources such as medical X-rays and nuclear weapons tests.

背景辐射是环境中始终存在的低水平电离辐射。来源包括宇宙射线、地下的氡气、岩石和食物中的天然同位素，以及医疗X射线和核武器试验等人为来源。

Radiation safety principles are based on three pillars: time, distance, and shielding. Minimizing exposure time, maximizing distance from the source, and using appropriate shielding (e.g., lead for gamma rays) all reduce the radiation dose received.

辐射安全原则基于三大支柱：时间、距离和屏蔽。缩短受照时间、增大与辐射源的距离以及使用合适的屏蔽材料（如铅屏蔽γ射线），都能有效降低所受辐射剂量。

In IB Physics exams, you may be asked to calculate the energy released in a decay using the mass defect:

在IB物理考试中，你可能会被要求利用质量亏损计算衰变释放的能量：

E = Δmc²

Here, Δm is the difference between the initial mass and the sum of the final masses. This mass defect is converted into kinetic energy of the decay products and any emitted gamma radiation.

其中Δm是初始总质量与终态总质量之差。这质量亏损转化为衰变产物的动能以及可能发射的γ辐射的能量。

12. Energy Level Transitions and Gamma Emission | 能级跃迁与γ发射

Following alpha or beta decay, the daughter nucleus is often left in an excited state. Similar to electron transitions in atoms, the excited nucleus can transition to a lower, more stable energy level by emitting a gamma photon. The energy of the gamma photon equals the energy difference between the nuclear energy levels.

发生α或β衰变后，子核往往处于激发态。与原子中的电子跃迁类似，处于激发态的原子核可以通过发射γ光子跃迁到较低、更稳定的能级。γ光子的能量等于核能级之间的能量差。

This connection between atomic energy level transitions and nuclear gamma emission illustrates the unifying principle of quantum mechanics: energy is quantized at both the atomic and the nuclear scale.

原子能级跃迁与核γ发射之间的联系体现了量子力学的统一原理：能量在原子尺度和原子核尺度上都是量子化的。

Both processes obey the same fundamental relationship, E = hf. However, the energy differences between nuclear levels are typically millions of times larger than atomic levels, which is why gamma photons have much higher energies than visible light photons.

两个过程都遵守相同的基本关系E = hf。然而，核能级之间的能量差通常比原子能级大数百万倍，这就是为什么γ光子比可见光光子的能量高得多的原因。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IB Physics: Energy Level Transitions and Radioactive Decay | IB物理：能级跃迁与放射性衰变