📚 Nuclear Structure | 原子核结构
The nucleus lies at the heart of every atom, holding nearly all its mass and determining the element's identity. In IB Physics, understanding nuclear structure means exploring what the nucleus is made of, how its constituents interact, and why some combinations of protons and neutrons are stable while others decay. This article covers the composition, size, density, strong nuclear force, binding energy, and stability of nuclei, providing a self-contained revision guide aligned with the IB Physics syllabus.
原子核位于每个原子的中心，几乎承载了全部质量，并决定了元素的种类。在 IB 物理中，理解原子核结构意味着要探讨原子核由什么组成、其组成部分如何相互作用，以及为什么某些质子与中子的组合是稳定的，而另一些则会衰变。本文涵盖了原子核的组成、大小、密度、强核力、结合能以及稳定性，提供了一份与 IB 物理大纲相一致的完整复习指南。
1. What is the Atomic Nucleus? | 原子核是什么？
The atomic nucleus is an extremely small, dense region at the centre of an atom, consisting of protons and neutrons bound together. It carries a positive electric charge equal to the number of protons and accounts for more than 99.9% of the atom's mass, yet its radius is about 10 000 times smaller than that of the atom. The discovery of the nucleus by Rutherford's scattering experiment in 1911 overturned the earlier plum pudding model and opened the field of nuclear physics.
原子核是原子中心一个极小而致密的区域，由质子和中子结合而成。它带有相当于质子数的正电荷，承担了原子超过 99.9% 的质量，但其半径却比原子半径小约一万倍。1911 年卢瑟福散射实验发现了原子核，推翻了早期的枣糕模型，开启了核物理学领域。
In nuclear physics, the nucleus is treated as a quantum many-body system governed by the strong nuclear force. The study of nuclear structure aims to explain why certain combinations of nucleons form bound states and how those states can transform via radioactivity.
在核物理中，原子核被看作一个由强核力支配的量子多体系统。对核结构的研究旨在解释为什么某些核子组合能形成束缚态，以及这些态如何通过放射性发生转变。
2. Nucleons and Nuclear Notation | 核子与核素符号
mass unit u is defined as 1/12 of the mass of a carbon-12 atom, equal to 1.66 × 10⁻²⁷ kg.
质子和中子统称为核子。质子带正电荷 +e (1.60 × 10⁻¹⁹ C)，质量约为 1.0073 u；中子不带电，质量约为 1.0087 u。原子质量单位 u 定义为碳-12 原子质量的 1/12，等于 1.66 × 10⁻²⁷ kg。
A nuclide is represented by the notation:
ᴬZX
of nucleons), and the neutron number N = A - Z. For example, carbon-14 is written as ¹⁴₆C.
其中 X 是化学符号，Z 是原子序数（质子数），A 是质量数（核子总数），中子数 N = A - Z。例如，碳-14 写作 ¹⁴₆C。
3. Isotopes and Their Significance | 同位素及其意义
Isotopes are nuclides of the same element (same Z) that have different numbers of neutrons (different N and A). For instance, carbon has three naturally occurring isotopes: ¹²C, ¹³C and ¹⁴C, with 6, 7 and 8 neutrons respectively. Isotopes exhibit identical chemical properties because the electron configuration remains unchanged, but their nuclear properties – such as stability and mass – can differ dramatically.
同位素是同一元素（Z 相同）但中子数不同（N 和 A 不同）的核素。例如，碳有三种天然同位素：¹²C、¹³C 和 ¹⁴C，中子数分别为 6、7 和 8。同位素的化学性质相同，因为电子排布不变，但它们的核性质——如稳定性和质量——可能差异很大。
The existence of isotopes explains why atomic masses on the periodic table are not whole numbers: they represent the weighted average of all naturally occurring isotopes. Radioactive isotopes, such as ¹⁴C, are widely used in dating, medicine and tracer studies, making the concept of isotopes essential for both physics and applied science.
同位素的存在解释了为什么元素周期表上的原子量不是整数：它们代表所有天然同位素的加权平均值。放射性同位素（如 ¹⁴C）广泛用于年代测定、医学和示踪研究，使得同位素概念对物理学和应用科学都至关重要。
4. The Size of the Nucleus | 原子核的大小
Nuclear radii can be estimated by electron scattering or from the diffraction of high-energy particles. Experiments show that the radius R of a nucleus increases with the cube root of its mass number A. This relationship is expressed as:
R = R₀ A1/3
where R₀ ≈ 1.2 fm (1 femtometre = 10⁻¹⁵ m). The dependence R ∝ A1/3 implies that nuclear volume is proportional to the number of nucleons, suggesting that nucleons behave rather like incompressible spheres packed together with a roughly constant spacing.
其中 R₀ ≈ 1.2 fm（1 飞米 = 10⁻¹⁵ m）。关系 R ∝ A1/3 表明核体积与核子数成正比，这意味着核子的行为类似于不可压缩的球体，以大致恒定的间距紧密堆积。
For example, a uranium-238 nucleus (A = 238) has a radius of about 1.2 × 2381/3 e radii of only a few femtometres, highlighting the extraordinary compactness of nuclear matter.
例如，铀-238 核（A = 238）的半径约为 1.2 × 2381/3 ≈ 7.4 fm。即使最重的原子核，半径也只有几飞米，突显了核物质的非凡致密性。
5. Nuclear Density | 核密度
Since nuclear radius follows R = R₀ A1/3 (4/3)π R₀³ A. The mass of a nucleus is approximately A × u, so the density ρ is given by:
ρ = mass / volume ≈ (A × u) / [(4/3)π R₀³ A] = u / [(4/3)π R₀³]
constant across all nuclei. Plugging in u = 1.66 × 10⁻²⁷ kg and R₀ = 1.2 × 10⁻¹⁵ m gives:
ρ ≈ 2.3 × 10¹⁷ kg m⁻³
A 被消去，表明所有原子核的密度大致恒定。代入 u = 1.66 × 10⁻²⁷ kg 和 R₀ = 1.2 × 10⁻¹⁵ m，得到 ρ ≈ 2.3 × 10¹⁷ kg m⁻³。
This density is enormously high – a matchbox filled with pure nuclear matter would have a mass of about 2.5 billion tonnes. The constant density reinforces the picture of nucleons packed uniformly, with the strong nuclear force providing the necessary binding against electrostatic repulsion among protons.
这一密度高得惊人——一个装满纯核物质的火柴盒，质量约为 25 亿吨。恒定的密度进一步证实了核子均匀堆积的图像，其中强核力提供了克服质子间静电排斥所需的结合。
6. The Strong Nuclear Force | 强核力
Within a nucleus, protons experience a large Coulomb repulsion due to their positive charges. If this were the only interaction, nuclei would instantly fly apart. The strong nuclear force overcomes this repulsion and binds nucleons together. Its key properties are:
- Short range: It acts only over distances of about 1–3 fm. Beyond this, the force becomes negligible.
- Attractive at nucleon separations: It is powerfully attractive at ∼1 fm, preventing collapse at shorter distances by becoming repulsive.
- Charge independent: The strong force between two protons, two neutrons, or a proton and a neutron is essentially the same.
在原子核内部，质子因其正电荷而承受巨大的库仑排斥力。如果只有这一相互作用，原子核会瞬间解体。强核力克服了这种排斥力，将核子结合在一起。其主要特性包括：短程性（仅在 1–3 fm 距离内起作用，超出后微不足道）、核子间距约 1 fm 时为强吸引力（而在更短距离上变为排斥，防止塌缩）、以及电荷无关性（质子-质子、中子-中子、质子-中子之间的强力基本相同）。
Because the strong force is short-ranged, each nucleon interacts significantly only with its immediate neighbours. In large nuclei, surface nucleons have fewer neighbours and are less tightly bound, a feature explained by the liquid drop model. The electromagnetic repulsion, which is long-ranged, acts between all proton pairs and grows with Z², eventually limiting the size of stable nuclei.
由于强核力是短程的，每个核子仅与近邻核子发生显著相互作用。在大核中，表面核子的近邻数较少，束缚得较松，这一特点可由液滴模型解释。长程的电磁排斥力作用于所有质子对之间，并随 Z² 增长，最终限制了稳定核的大小。
7. Mass Defect and Binding Energy | 质量亏损与结合能
The mass of a nucleus is always less than the sum of the masses of its individual nucleons. This difference is called the mass defect Δm:
Δm = Zmₚ + Nmₙ - Mₙᵤcₗₑᵤₛ
are the proton and neutron masses, and M is the actual nuclear mass. According to Einstein's mass-energy equivalence E = mc², the mass defect corresponds to the binding energy Eb that holds the nucleus together:
Eb = Δm c²
原子核的实际质量总是小于其各个核子质量之和，这一差值称为质量亏损 Δm。根据爱因斯坦质能方程 E = mc²，质量亏损对应于将核子结合在一起的结合能 Eb。
eadily convert a mass defect in atomic mass units to binding energy. For instance, the mass defect of helium-4 (⁴He) is about 0.0304 u, yielding a binding energy of approximately 28.3 MeV.
结合能通常以 MeV 表示。利用 1 u = 931.5 MeV/c²，可以轻松地将以原子质量单位表示的质量亏损转换为结合能。例如，氦-4 (⁴He) 的质量亏损约为 0.0304 u，对应的结合能约为 28.3 MeV。
8. Binding Energy per Nucleon Curve | 平均结合能曲线
A more revealing quantity is the binding energy per nucleon, Eb/A. A plot of Eb 8.8 MeV/nucleon, and a gradual decline for heavier nuclei. This curve has profound implications:
- Fusion: Fusing light nuclei (e.g. hydrogen to helium) releases energy because the product has a higher Eb/A.
- Fission: Splitting heavy nuclei (e.g. uranium) releases energy because the fragments lie closer to the peak of the curve.
更有意义的量是平均结合能 Eb/A。Eb/A 对质量数 A 的图显示：轻核区快速上升，在 A ≈ 56（铁、镍）附近达到约 8.8 MeV/核子的宽峰，之后重核区缓慢下降。这条曲线具有深远意义：轻核聚变（如氢变为氦）释放能量，因为产物的 Eb/A 更高；重核裂变（如铀）也释放能量，因为裂变碎片更靠近曲线的峰值。
(A = 56, maximum Eb/A ≈ 8.8 MeV)
The peak at iron explains why stellar cores ultimately become iron; beyond that, energy cannot be extracted by fusion. This curve is a cornerstone of nuclear astrophysics and nuclear energy applications.
曲线在铁处的峰值解释了为什么恒星核心最终会变为铁；超过该点后，聚变无法再提取能量。这条曲线是核天体物理学和核能应用的基石。
9. The Stability of Nuclei | 原子核的稳定性
ncreases, the Coulomb repulsion grows, and additional neutrons are needed to provide extra strong-force binding without adding electrostatic repulsion. Therefore, the neutron-to-proton ratio N/Z rises from about 1 to around 1.5 for very heavy stable nuclei such as ²⁰⁸Pb.
原子核的稳定性取决于吸引性的强核力与排斥性的库仑力之间的平衡。对于轻核（Z ≤ 20），稳定核的质子与中子数量大致相等（N ≈ Z）。随着 Z 的增加，库仑排斥力增强，需要更多的中子来提供额外的强核力结合，而不增加静电排斥。因此，中子-质子比 N/Z 从大约 1 升高到极重稳定核（如 ²⁰⁸Pb）的约 1.5。
xist; all isotopes of elements heavier than lead are radioactive, ultimately decaying via alpha or beta emission.
位于稳定带之上的核富含中子，倾向于发生 β⁻ 衰变，将中子转化为质子。位于稳定带之下的核富含质子，可能发生 β⁺ 衰变或电子俘获。超过 Z = 82 后，没有稳定核存在；所有比铅重的元素的同位素都具有放射性，最终通过 α 或 β 衰变发生转变。
10. Radioactivity and Unstable Nuclei | 放射性与不稳定核
Radioactivity is a direct consequence of nuclear instability. Unstable nuclei emit particles or electromagnetic radiation to move toward a more stable configuration. The principal decay modes are:
- Alpha (α) decay: emission of a ⁴He nucleus, reducing A by 4 and Z by 2.
- Beta-minus (β⁻) decay: a neutron changes into a proton with emission of an electron and an antineutrino.
- Beta-plus (β⁺) decay: a proton changes into a neutron with emission of a positron and a neutrino.
- Gamma (γ) decay: a nucleus in an excited state releases energy as a high-energy photon.
放射性是核不稳定的直接结果。不稳定的原子核通过发射粒子或电磁辐射，趋向更稳定的组态。主要衰变模式有：α 衰变（发射 ⁴He 核，A 减少 4，Z 减少 2）；β⁻ 衰变（中子转变为质子，并发射电子和反中微子）；β⁺ 衰变（质子转变为中子，并发射正电子和中微子）；γ 衰变（处于激发态的原子核以高能光子形式释放能量）。
The rate of decay is characterised by the decay constant λ and half-life T1/2. The exponential decay law N = N₀ e-λt governs the number of undecayed nuclei. Radioactive decay is a random and spontaneous process at the level of individual nuclei, with probabilities unaffected by external conditions such as temperature or pressure.
衰变速率由衰变常数 λ 和半衰期 T1/2 描述。指数衰变律 N = N₀ e-λt 控制了未衰变核的数目。放射性衰变对单个原子核而言是一个随机且自发的过程，其概率不受温度、压力等外界条件的影响。
11. Nuclear Models: Liquid Drop and Shell | 核模型：液滴模型与壳模型
Two models help explain nuclear behaviour: the liquid drop model and the nuclear shell model. The liquid drop model treats the nucleus as a drop of incompressible fluid of uniform density. It successfully accounts for the R ∝ A1/3 dependence and provides a semi-empirical mass formula that predicts binding energies using terms for volume, surface, Coulomb, asymmetry and pairing effects. This model explains fission and the general trend of the binding energy curve.
有两种模型有助于解释核行为：液滴模型和核壳模型。液滴模型将原子核视为一个密度均匀、不可压缩的液滴。它成功解释了 R ∝ A1/3 的关系，并通过体积能、表面能、库仑能、非对称能和配对能的项，提供了半经验质量公式，以预测结合能。该模型解释了裂变现象以及结合能曲线的总体趋势。
ean nuclear potential, analogous to electron shells in atoms. It explains the existence of 'magic numbers' ²⁰⁸Pb (Z=82, N=126) are doubly magic and have high binding energies and spherical shapes.
另一方面，壳模型认为核子在平均核势场中处于分立的能级（壳层）中运动，类似于原子中的电子壳层。它解释了“幻数”（2, 8, 20, 28, 50, 82, 126）的存在，在这些幻数处的原子核异常稳定。例如，⁴He（Z=2, N=2）和 ²⁰⁸Pb（Z=82, N=126）都是双幻核，具有高结合能和球形形状。
12. Nuclear Structure in the Bigger Picture | 更广阔图景中的核结构
Understanding nuclear structure is not only fundamental to physics but also underpins many applications: nuclear power, medical imaging and therapy, carbon dating, and the synthesis of elements in stars. The IB Physics syllabus connects these ideas to real-world contexts, for example by examining how binding energy determines the energy released in fission reactors or fusion in the Sun.
理解核结构不仅是物理学的基础，而且支撑着许多实际应用：核能、医学成像与治疗、碳年代测定以及恒星中元素的合成。IB 物理大纲将这些概念与现实世界联系起来，例如通过考察结合能如何决定裂变反应堆或太阳中聚变所释放的能量。
From the constant density of nuclear matter to the delicate balance of forces governing stability, the nucleus reveals a rich quantum world. Whether you are calculating the radius of a gold nucleus or predicting the decay route of an unstable isotope, the principles of nuclear structure provide a consistent and powerful framework.
从核物质的恒定密度到制约稳定性的精细力平衡，原子核揭示了一个丰富的量子世界。无论你是在计算金核的半径，还是预测不稳定同位素的衰变路径，核结构的原理都提供了一个一致且强大的框架。
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