📚 E Nuclear and Quantum Physics | E 原子核与量子物理学
Modern physics rests on two revolutionary pillars: the study of the atomic nucleus and the strange rules that govern the microscopic world. IB Physics Topic E – Nuclear and Quantum Physics – takes you from the heart of the atom, through radioactive decays and nuclear energy, to the photon, the photoelectric effect, and wave‑particle duality. Understanding these concepts not only explains how stars shine and why some elements are stable, but also underpins technologies from medical imaging to quantum computing.
现代物理建立在两大革命性支柱之上：对原子核的研究以及支配微观世界的奇特规律。IB 物理主题 E——原子核与量子物理学——将带你从原子的核心出发，穿过放射性衰变与核能，到达光子、光电效应和波粒二象性。理解这些概念不仅能解释恒星为何发光、为何有些元素是稳定的，也是从医学成像到量子计算等技术的基础。
1. Atomic Structure and the Nucleus | 原子结构与原子核
nes the chemical element; the total number of nucleons A = Z + N (where N is the number of neutrons) defines the isotope. Nuclear radii are of order 10⁻¹⁵ m, roughly 10 000 times smaller than the atomic radius. Experiments show that nuclear radius R is approximately R = r₀ A1/3, with r₀ ≈ 1.2 × 10⁻¹⁵ m, implying that nuclear matter has a roughly constant density.
所有普通物质都由原子组成，每个原子有一个微小的中心原子核，核外有电子围绕。原子核包含质子和中子，统称为核子。质子带一个基本正电荷 +e，而中子不带电。质子数 Z 决定了化学元素；核子总数 A = Z + N（N 为中子数）决定了同位素。原子核半径的数量级为 10⁻¹⁵ m，比原子半径小约 10 000 倍。实验表明，核半径 R 近似为 R = r₀ A1/3，其中 r₀ ≈ 1.2 × 10⁻¹⁵ m，这意味着核物质具有几乎恒定的密度。
Nuclear forces are now understood in terms of the strong interaction. On the femtometre scale, the strong force binds nucleons together, overcoming the enormous electrostatic repulsion between protons. This force is very short‑range and saturates: each nucleon only interacts with its immediate neighbours.
如今核力用强相互作用来解释。在飞米尺度上，强力将核子束缚在一起，克服了质子间巨大的静电排斥力。这种力作用距离极短且具有饱和性：每个核子只与最邻近的核子发生作用。
2. Isotopes and Nuclear Stability | 同位素与核稳定性
r stable nuclei require a neutron excess to dilute the Coulomb repulsion. The stability curve on an N versus Z plot shows a band of stability. Nuclei that lie away from this band are radioactive and will undergo transmutation towards stability. The binding energy per nucleon is the key indicator of how tightly a nucleus is held together.
并非所有 Z 和 N 的组合都能产生一个稳定的原子核。轻核倾向于拥有大致相等的质子数和中子数（Z ≈ N），而较重的稳定核则需要中子过剩来稀释库仑排斥力。在 N-Z 图上，稳定核分布在一个稳定带内。偏离该带的原子核具有放射性，会通过嬗变向稳定方向转化。每个核子的平均结合能是衡量一个核束缚松紧程度的关键指标。
arate it into its constituent protons and neutrons. It is given by ΔE = (Z mₚ + N mₙ − mₙᵤ꜀ₗₑᵤₛ) c², where mₚ, mₙ and mₙᵤ꜀ cleus respectively. This energy is equivalent to the mass defect Δm multiplied by c². Iron‑56 sits near the peak of the binding‑energy‑per‑nucleon curve, making it one of the most stable nuclei.
原子核的结合能是将它分离成组成它的质子和中子所需的能量，由 ΔE = (Z mₚ + N mₙ − mₙᵤ꜀ₗₑᵤₛ) c² 给出，其中 mₚ、mₙ 和 mₙᵤ꜀ₗₑᵤₛ 分别是质子、中子和原子核的静止质量。这一能量等价于质量亏损 Δm 乘以 c²。铁‑56 位于平均结合能曲线的峰值附近，是最稳定的核素之一。
3. Radioactive Decay | 放射性衰变
Unstable nuclei transform spontaneously through radioactive decay, emitting particles and energy. The three principal modes are alpha (α) decay, beta minus (β⁻) decay, beta plus (β⁺) decay and gamma (γ) emission. In α‑decay, a heavy nucleus ejects a helium‑4 nucleus, reducing Z by 2 and A by 4. In β⁻ decay, a neutron turns into a proton with the emission of an electron and an antineutrino, increasing Z by 1. In β⁺ decay, a proton converts into a neutron, emitting a positron and a neutrino, decreasing Z by 1. Gamma emission usually follows an alpha or beta decay when the daughter nucleus is left in an excited state; the nucleus releases a high‑energy photon to reach its ground state.
不稳定的原子核会通过放射性衰变自发地转化，发射粒子和能量。三种主要模式是 α 衰变、β⁻ 衰变、β⁺ 衰变和 γ 辐射。在 α 衰变中，重核射出一个氦‑4 核，Z 减少 2，A 减少 4。在 β⁻ 衰变中，一个中子转变成质子，同时发射一个电子和一个反中微子，Z 增加 1。在 β⁺ 衰变中，一个质子转变成中子，发射一个正电子和一个中微子，Z 减少 1。γ 辐射通常跟在 α 或 β 衰变之后，当子核处于激发态时，核释放高能光子以回到基态。
ay per unit time. The activity A = −dN/dt = λN, where N is the number of undecayed nuclei.
放射性衰变在单个核的层面上是随机的，但大量核组成的样品服从精确的统计规律。衰变常数 λ 给出单位时间内衰变的概率。活度 A = −dN/dt = λN，其中 N 为尚未衰变的核的数目。
4. Half‑life and Activity | 半衰期与活度
The half‑life T1/2 is the time taken for half the radioactive nuclei in a sample to decay. It is related to the decay constant by T1/2 = ln 2 / λ. The exponential decay law can be written as N = N₀ e−λt or as A = A₀ e−λt. Because activity is proportional to the number of parent nuclei, both follow the same behaviour. Half‑life values range from fractions of a second to billions of years, making some isotopes useful as geological clocks.
半衰期 T1/2 是样品中一半放射性原子核发生衰变所需的时间，它与衰变常数的关系为 T1/2 = ln 2 / λ。指数衰减规律可写为 N = N₀ e−λt 或 A = A₀ e−λt。由于活度与母核数目成正比，两者遵循相同的变化规律。半衰期短的只有几分之一秒，长的可达数十亿年，这使某些同位素成为地质年代计时钟。
equations above to solve problems on remaining fraction, age determination and background‑ s. Remember that activity is measured in becquerels (Bq), where 1 Bq = 1 decay per second.
在 IB 物理中，你需要用上述方程求解剩余比例、年代测定以及经本底校正的计数率等问题。请记住，活度以贝克勒尔（Bq）为单位，1 Bq 相当于每秒一次衰变。
5. Nuclear Reactions and Energy | 核反应与能量
Any process that rearranges nucleons can release or absorb energy. In a nuclear reaction, the total energy released, the Q‑value, is given by the mass difference between reactants and products multiplied by c². If the total mass of the products is less than that of the reactants, the reaction is exothermic (Q > 0); otherwise it is endothermic. The energy released in nuclear reactions is typically millions of times larger than that of chemical reactions because nuclear binding forces are far stronger.
任何重新排布核子的过程都可能释放或吸收能量。在核反应中，释放的总能量（Q 值）等于反应物与生成物的质量差乘以 c²。如果生成物的总质量小于反应物的总质量，反应为放热反应（Q > 0）；反之则为吸热反应。核反应释放的能量通常比化学反应大数百万倍，因为核束缚力比化学键强得多。
The binding energy per nucleon curve shows that both very light and very heavy nuclei are less tightly bound than mid‑mass nuclei like iron. This is why fusion of light nuclei and fission of heavy nuclei both liberate energy: both processes move towards a higher binding energy per nucleon.
平均结合能曲线表明，很轻的核和很重的核都比铁这样的中等核束缚得松。这就是为什么轻核的聚变和重核的裂变都能释放能量：两种过程都趋向于更高的每个核子结合能。
6. Nuclear Fission | 核裂变
Fission is the splitting of a heavy nucleus into two lighter fragments of comparable mass, accompanied by the release of neutrons and a large amount of energy. The most familiar fission process involves the absorption of a slow (thermal) neutron by uranium‑235. The compound nucleus ²³⁶U is unstable and splits, for example:
¹₀n + ²³⁵₉₂U → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3¹₀n + energy
裂变是一个重核分裂成两个质量相当较轻碎块的过程，同时释放出中子和大量能量。最熟知的裂变过程是铀‑235 吸收一个慢（热）中子。复合核 ²³⁶U 不稳定并发生分裂，例如：
¹₀n + ²³⁵₉₂U → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3¹₀n + 能量
The two or three neutrons released can induce further fissions, sustaining a chain reaction. In a nuclear reactor, control rods absorb excess neutrons to keep the reaction rate steady. The energy released appears as kinetic energy of the fragments and neutrons, and is ultimately converted to electricity. Critical mass is the minimum mass of fissile material required to maintain a self‑sustaining chain reaction.
释放的两到三个中子可引发进一步的裂变，维持链式反应。在核反应堆中，控制棒吸收多余的中子，使反应速率保持稳定。释放的能量表现为碎片和中子的动能，并最终转化为电能。临界质量是维持自持链式反应所需的最低可裂变材料质量。
7. Nuclear Fusion | 核聚变
Fusion is the merging of light nuclei to form a heavier nucleus, again releasing energy because the product is more tightly bound. The Sun converts hydrogen into helium via the proton‑proton chain, releasing about 26.7 MeV per four protons consumed. Overall reaction:
4¹₁H → ⁴₂He + 2e⁺ + 2νe + 26.7 MeV
聚变是轻核合并形成一个较重核的过程，同样释放能量，因为产物束缚得更紧。太阳通过质子‑质子链将氢转化为氦，每消耗四个质子大约释放 26.7 MeV 的能量。总反应为：
4¹₁H → ⁴₂He + 2e⁺ + 2νe + 26.7 MeV
To initiate fusion, the positively charged nuclei must overcome their Coulomb barrier; this requires extremely high temperatures (tens of millions of kelvin) to give the nuclei enough kinetic energy. In stars, gravitational confinement provides the pressure and temperature needed, while on Earth magnetic confinement (tokamaks) and inertial confinement are being explored as routes to controlled fusion power.
要引发聚变，带正电的原子核必须克服其库仑势垒；这需要极高的温度（数千万开尔文），以使原子核获得足够的动能。在恒星中，引力约束提供了所需的压强和温度；而在地球上，磁约束（托卡马克）和惯性约束正被探索作为实现受控聚变能源的途径。
8. The Photoelectric Effect | 光电效应
The photoelectric effect is the emission of electrons from a metal surface when electromagnetic radiation of sufficiently high frequency falls on it. Classical wave theory could not explain the instantaneous emission, the existence of a threshold frequency f₀ below which no electrons are emitted, or the fact that the maximum kinetic energy of the photoelectrons depends only on frequency, not on intensity.
光电效应指的是当频率足够高的电磁辐射照射到金属表面时，会有电子发射出来的现象。经典波动理论无法解释瞬时发射、存在一个阈值频率 f₀（低于此频率时不发射电子）、以及光电子最大动能只与频率有关而与光强无关等事实。
Planck constant. An electron absorbs a single photon and requires a minimum energy φ, the work function of the metal, to escape. The photoelectric equation is:
h f = φ + Kmax
The stopping potential Vs is related to Kmax by e Vs = Kmax. Plotting Kmax against f yields a straight line whose gradient equals Planck's constant, and the intercept gives the work function.
爱因斯坦提出光由量子（现称光子）组成，每个光子携带能量 E = h f，其中 h 是普朗克常量。电子吸收一个光子，需要最小能量 φ（金属的逸出功）才能逸出。光电方程为：
h f = φ + Kmax
遏止电势 Vs 与 Kmax 的关系为 e Vs = Kmax。将 Kmax 对 f 作图可得一条直线，其斜率等于普朗克常量，截距给出逸出功。
9. Wave–Particle Duality and de Broglie Wavelength | 波粒二象性与德布罗意波长
wavelength λ = h / p. For electrons accelerated through a potential difference V, the non‑relativistic de Broglie wavelength is λ = h / √2 m e V.
光电效应展示了光的粒子性，但光也能表现出干涉和衍射等波动现象。这种二象性延伸到物质上：路易·德布罗意提出，任何动量为 p 的粒子都具有一个与之相关的波长 λ = h / p。对于经电势差 V 加速的电子，非相对论德布罗意波长为 λ = h / √2 m e V。
Davisson and Germer’s experiment, as well as the diffraction of electrons by a thin graphite film, confirmed that electrons can produce interference patterns, providing direct evidence for matter waves. The scale at which wave behaviour becomes significant is when the de Broglie wavelength is comparable to the size of obstacles or slits. This is why macroscopic objects do not exhibit observable wave properties.
戴维森和革末的实验以及电子通过薄石墨膜产生的衍射证实了电子可以产生干涉图样，为物质波提供了直接证据。当德布罗意波长与障碍物或狭缝尺寸相当时，波动行为才会变得显著。这就是宏观物体不表现出可观测波动性质的原因。
10. Atomic Spectra and Quantum Energy Levels | 原子光谱与量子能级
A gas of excited atoms emits light only at specific wavelengths, producing a line spectrum rather than a continuous rainbow. This is explained by the existence of discrete energy levels within the atom. When an electron makes a transition from a higher energy level Ei to a lower one Ef, a photon of energy h f = Ei − Ef is emitted. The hydrogen spectrum is described by the empirical Rydberg formula, but it was Bohr’s model that first linked this to quantised angular momentum.
受激原子组成的气体只在特定波长发射光，产生线状光谱而非连续彩虹。这可以用原子内部存在分立的能级来解释。当一个电子从高能级 Ei 向低能级 Ef 跃迁时，会放出一个能量为 h f = Ei − Ef 的光子。氢光谱由经验性的里德伯公式描述，但波尔模型首次将其与量子化的角动量联系起来。
The Bohr model postulates that electrons orbit the nucleus only in certain allowed radii and that angular momentum is integer multiples of h/2π. Although the model works well for hydrogen, it cannot explain multi‑electron atoms. The modern quantum‑mechanical picture replaces fixed orbits with probability clouds described by wavefunctions. Students should be able to sketch absorption and emission spectra and interpret them in terms of transitions between energy levels.
波尔模型假设电子只在特定允许半径的轨道上运行，且角动量为 h/2π 的整数倍。尽管该模型对氢原子很成功，却无法解释多电子原子。现代量子力学图像用波函数描述的概率云取代了固定轨道。学生应能画出吸收和发射光谱，并用能级之间的跃迁来解释它们。
≥ h/4π. This uncertainty is not due to measurement limitations but is intrinsic to nature.
海森堡不确定性原理是量子理论的另一块基石，它对某些物理量对（如位置和动量）能够被同时知晓的精确度施加了一个基本限制：Δx Δp ≥ h/4π。这种不确定性并非源于测量限制，而是自然界固有的。
Nuclear and quantum physics changes the way we view matter, energy and the universe. From unlocking the power of stars to enabling the electronics of the information age, these principles are central to IB Physics and to our technological future.
原子核和量子物理学改变了我们看待物质、能量和宇宙的方式。从释放恒星的能量到实现信息时代的电子技术，这些原理都是 IB 物理以及我们科技未来的核心。
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