📚 Discrete Energy and Radioactivity | 离散能量与放射性
In modern physics, the concepts of discrete energy levels and radioactivity revolutionised our understanding of matter and energy. At the atomic scale, electrons exist only in specific, quantised energy states, giving rise to unique line spectra for each element. Meanwhile, unstable atomic nuclei undergo spontaneous transformations, emitting radiation in the form of alpha, beta and gamma rays. These phenomena form the basis of nuclear physics, medical imaging, power generation, and our understanding of the universe at its most fundamental level. This revision guide brings together the definitions, equations, worked reasoning and exam technique you need to master the topic.
在现代物理学中，离散能级和放射性的概念彻底改变了我们对物质和能量的认知。在原子尺度上，电子只存在于特定的、量子化的能态中，从而产生每种元素独一无二的线状光谱。同时，不稳定的原子核会自发发生转变，以α、β和γ射线的形式发射辐射。这些现象构成了核物理、医学成像、能源发电以及我们对宇宙最基本层面理解的基础。本复习指南把关键定义、公式、推理过程与应试技巧系统地整理在一起。
1. Atomic Energy Levels and Discrete Spectra | 原子能级与离散光谱
Electrons in an atom occupy specific energy levels, and they cannot have energies between these levels. This quantisation means that the energy of an atom is discrete, not continuous: only certain allowed values exist. The lowest allowed level is called the ground state, and any level above it is an excited state, which is unstable and short-lived.
原子中的电子占据特定的能级，它们不能拥有这些能级之间的能量。这种量子化意味着原子的能量是离散的，而非连续的：只有某些特定的允许值存在。最低的允许能级称为基态，其上的任何能级都称为激发态，激发态不稳定且寿命极短。
When an electron moves from a higher energy level to a lower one, it emits a photon whose energy equals the difference between the two levels. Conversely, an electron can jump to a higher level by absorbing a photon of exactly the right energy. The transition energy is therefore fixed by the structure of the atom itself, not by anything external.
当电子从较高能级跃迁到较低能级时，它会发射一个光子，其能量等于这两个能级之差。反过来，电子也可以通过吸收一个能量恰好合适的光子跃迁到更高能级。因此跃迁能量由原子自身的结构决定，而不取决于任何外部因素。
ΔE = E₂ − E₁ = hf = hc ÷ λ
0⁻³⁴ J s), f is the frequency of the photon in hertz, and λ is its wavelength in metres. Because each element has a different set of energy levels, each element produces a different set of photon energies, and therefore a different spectrum.
其中 h 为普朗克常数（6.63 × 10⁻³⁴ J s），f 为光子频率（单位赫兹），λ 为波长（单位米）。由于每种元素的能级组合不同，每种元素产生的光子能量组合也不同，因而光谱也各不相同。
2. Photon Energy and the Photon Model | 光子能量与光子模型
A photon is a quantum of electromagnetic radiation. Its energy is directly proportional to frequency and inversely proportional to wavelength. This single relationship links the wave and particle descriptions of light and is used in almost every line-spectrum calculation.
光子是电磁辐射的量子。它的能量与频率成正比，与波长成反比。这一关系把光的波动描述与粒子描述联系起来，几乎出现在所有线状光谱的计算中。
E = hf = hc ÷ λ h = 6.63 × 10⁻³⁴ J s c = 3.00 × 10⁸ m s⁻¹
Atomic energies are usually quoted in electronvolts because the joule is far too large a unit at this scale. One electronvolt is the kinetic energy gained by an electron accelerated through a potential difference of one volt.
原子尺度的能量通常用电子伏特表示，因为焦耳在这个尺度上实在太大了。1 电子伏特等于一个电子经过 1 伏特电势差加速后所获得的动能。
1 eV = 1.60 × 10⁻¹⁹ J
= 1.63 × 10⁻¹⁸ ÷ 6.63 × 10⁻³⁴ ≈ 2.46 × 10¹⁵ Hz, and λ = c ÷ f ≈ 122 nm, which lies in the ultraviolet region. This is the Lyman-alpha line.
例题：氢原子中的一个电子跃迁回基态并释放 10.2 eV 的能量。换算得 E = 10.2 × 1.60 × 10⁻¹⁹ = 1.63 × 10⁻¹⁸ J，则 f = E ÷ h = 1.63 × 10⁻¹⁸ ÷ 6.63 × 10⁻³⁴ ≈ 2.46 × 10¹⁵ Hz，λ = c ÷ f ≈ 122 nm，位于紫外区，这正是莱曼α线。
3. Line Spectra as Evidence for Quantisation | 线状光谱作为量子化的证据
A hot, dense object such as a filament lamp emits a continuous spectrum containing all wavelengths. A low-pressure gas discharge tube, however, emits a line emission spectrum consisting of bright, sharp lines on a dark background. Each line corresponds to one specific electron transition.
炽热而致密的物体（如白炽灯丝）发出包含所有波长的连续光谱。但低压气体放电管发出的是线状发射光谱，表现为暗背景上一条条明亮锐利的谱线。每一条谱线都对应一次特定的电子跃迁。
If white light passes through a cool gas, photons with exactly the right energies are absorbed and re-emitted in all directions, so dark lines appear at the same wavelengths as the emission lines of that gas. This is an absorption spectrum, the basis of the Fraunhofer lines in the solar spectrum.
如果白光穿过冷气体，那些能量恰好合适的光子会被吸收并向各个方向再发射，于是在该气体发射线相同的波长处出现暗线，这就是吸收光谱，也是太阳光谱中夫琅禾费暗线的成因。
The hydrogen spectrum is the classic example and is organised into series according to the lower level involved:
氢光谱是经典范例，按跃迁所涉及的较低能级分为若干谱系：
- Lyman series (n = 1, ultraviolet) / 莱曼系（n = 1，紫外区）
- Balmer series (n = 2, visible; Hα at 656 nm, Hβ at 486 nm) / 巴尔末系（n = 2，可见光区；Hα 656 nm，Hβ 486 nm）
- Paschen series (n = 3, infrared) / 帕申系（n = 3，红外区）
Within each series the lines converge towards a series limit, beyond which a continuous spectrum appears because the electron has been completely removed from the atom. The existence of sharp lines, and of a limit, cannot be explained by any continuous model of atomic energy, and this is the strongest experimental evidence for discrete energy levels.
每个谱系内的谱线都向系限收敛，超过系限后出现连续谱，因为电子已被完全移出原子。锐利谱线的存在以及系限的存在，都无法用任何连续能量模型解释，这是离散能级最有力的实验证据。
4. Excitation, Ionisation and Energy Level Diagrams | 激发、电离与能级图
The ground state is the lowest energy state of an atom. Excitation is the process of raising an electron to a higher energy level within the atom, and the excitation energy is the energy required to do so. Ionisation is the complete removal of an electron from the atom, and the ionisation energy is the energy required to take an electron from the ground state to infinity.
基态是原子的最低能量状态。激发是把电子提升到原子内较高能级的过程，所需的能量称为激发能。电离则是把电子完全移出原子，把电子从基态移出到无穷远所需的能量称为电离能。
A free electron in a collision can transfer kinetic energy to a bound electron. If it carries more than the required ΔE, the excess simply remains as kinetic energy, so collisions can excite an atom with any energy above the threshold. A photon, by contrast, must carry exactly ΔE; if it does not, it passes straight through and no absorption occurs. This contrast is a favourite exam question.
自由电子在碰撞中可以把动能传递给束缚电子。如果它携带的能量大于所需的 ΔE，多余部分仍作为动能保留，因此碰撞可以在高于阈值能量的任意能量下激发原子。光子则不同，它必须携带恰好等于 ΔE 的能量；若不是，它就直接穿过，不产生吸收。这一对比是考试中的常见考点。
When drawing an energy level diagram, remember the conventions: horizontal lines at their correct relative positions, labelled with energy values or quantum numbers, and vertical arrows showing transitions. A downward arrow represents emission and an upward arrow represents absorption. The arrow length should be proportional to the photon energy.
绘制能级图时要遵守约定：用水平线按正确的相对位置表示能级，标注能量值或量子数，用竖直箭头表示跃迁。向下箭头表示发射，向上箭头表示吸收，箭头长度应与光子能量成正比。
5. Nuclear Structure and Stability | 原子核的结构与稳定性
A nuclide is written in the form ᴬ A − Z. Isotopes are nuclides of the same element with the same Z but different A, so they have identical chemical behaviour but different nuclear stability.
核素写作 ᴬ_Z X 的形式，其中 X 为化学符号，Z 为质子数（原子序数），A 为核子数（质量数），中子数 N = A − Z。同位素是同一元素中 Z 相同而 A 不同的核素，因此化学性质相同，但核稳定性不同。
rt range, but has a very short range of only a few femtometres. Beyond that distance it falls to zero, which is why the electrostatic repulsion between protons dominates in large nuclei and why nuclei above about Z = 83 are all unstable.
原子核由强核力束缚在一起。强核力在所有核子之间都表现为吸引，在短程内远强于静电力，但作用范围极短，只有几飞米。超过这个距离它就降为零，这正是大核中质子间静电斥力占主导的原因，也是 Z 大于约 83 的原子核全部不稳定的原因。
For light, stable nuclei the neutron-to-proton ratio is close to 1. For heavy nuclei such as lead it rises to about 1.5, because extra neutrons add strong-force attraction without adding electrostatic repulsion. A nucleus with too many neutrons, or too few, lies outside the band of stability and will decay.
轻而稳定的原子核中，中子与质子之比接近 1；而在铅这样的重核中，该比值升至约 1.5，因为多余的中子只增加强核力的吸引，不增加静电斥力。中子过多或过少的原子核处在稳定带之外，会发生衰变。
6. Alpha, Beta and Gamma Radiation | α、β 与 γ 辐射
An unstable nucleus undergoes spontaneous transformation, emitting radiation in the form of alpha particles, beta particles or gamma rays. Spontaneous means the decay is not triggered by anything external, and random means we cannot predict which nucleus will decay next or exactly when it will happen.
不稳定的原子核会自发发生转变，以α粒子、β粒子或γ射线的形式发射辐射。自发意味着衰变不由任何外部因素触发，随机意味着我们无法预测下一个衰变的是哪个核、也无法预测准确时刻。
The three radiations differ greatly in charge, mass, penetrating power and ionising ability, and these differences are the key to identifying them in experiments.
这三种辐射在电荷、质量、穿透能力和电离能力上差别极大，而这些差异正是实验中鉴别它们的关键。
| Property / 性质 | Alpha α / α粒子 | Beta β⁻ / β粒子 | Gamma γ / γ射线 |
| Nature / 本质 | Helium nucleus ⁴₂He / 氦核 | Fast electron ⁰₋₁e / 高速电子 | Electromagnetic photon / 电磁光子 |
| Charge / 电荷 | +2e | −1e | 0 |
| Penetration / 穿透力 | Stopped by paper, a few cm of air / 纸或几厘米空气即可阻挡 | Stopped by a few mm of aluminium / 几毫米铝板即可阻挡 | Reduced by thick lead or concrete / 需厚铅或混凝土显著衰减 |
| Ionising power / 电离能力 | Strongest / 最强 | Moderate / 中等 | Weakest / 最弱 |
| Deflection in fields / 场中偏转 | Slight, opposite to β⁻ / 偏转小，方向与 β⁻ 相反 | Large / 偏转大 | None / 不偏转 |
A beta particle is not an orbital electron. It is created inside the nucleus when a neutron transforms into a proton, an electron and an antineutrino. A positron is emitted when a proton transforms into a neutron, a positron and a neutrino. Gamma emission usually follows alpha or beta decay, when a daughter nucleus left in an excited state releases energy as a photon.
β粒子并不是核外电子，而是在原子核内部由一个中子转变为一个质子、一个电子和一个反中微子时产生的。当一个质子转变为一个中子、一个正电子和一个中微子时，则会发射正电子。γ辐射通常发生在α或β衰变之后，此时处于激发态的daughter核（子核）以光子形式释放能量。
7. Nuclear Equations and Conservation Laws | 核方程与守恒定律
Every nuclear equation must conserve nucleon number and electric charge. In alpha decay the nucleon number falls by 4 and the proton number falls by 2. In beta-minus decay the proton number rises by 1 while the nucleon number is unchanged. In beta-plus decay the proton number falls by 1, and in gamma emission neither changes.
每一个核方程都必须满足核子数守恒与电荷守恒。α衰变中核子数减少 4、质子数减少 2；β⁻衰变中质子数增加 1 而核子数不变；β⁺衰变中质子数减少 1；γ发射中两者都不变。
²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He
¹⁴₆C → ¹⁴₇N + ⁰₋₁e + ν̄ₑ
²²₁₁Na → ²²₁₀Ne + ⁰₊₁e + νₑ
The neutrino was proposed
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