Binding Energy and Nuclear Stability | 结合能与原子核稳定性

📚 Binding Energy and Nuclear Stability | 结合能与原子核稳定性

In nuclear physics, the concept of binding energy is essential for understanding why some nuclei are stable while others undergo radioactive decay. This article explains mass defect, binding energy, the binding energy per nucleon curve, and how these ideas relate to nuclear fission and fusion.

在核物理中，结合能的概念对于理解为什么有些原子核稳定而另一些会发生放射性衰变至关重要。本文将解释质量亏损、结合能、每核子结合能曲线，以及这些概念如何与核裂变和核聚变相关联。

1. The Strong Nuclear Force | 强核力

Protons inside a nucleus repel each other electrically. If only the electromagnetic force acted, no nucleus heavier than hydrogen could exist. A much stronger attractive force, called the strong nuclear force, binds protons and neutrons together. This force is short-range, acting only over distances of about 1–3 femtometres (1 fm = 10⁻¹⁵ m), and it is attractive at typical nuclear separations.

原子核内的质子之间相互电排斥。如果只存在电磁力，那么比氢更重的原子核都不可能存在。一种强得多的吸引力，称为强核力，将质子和中子束缚在一起。这种力是短程力，仅在约 1–3 飞米（1 fm = 10⁻¹⁵ m）的距离内起作用，并且在典型核间距下表现为吸引力。

At extremely short distances (< 0.5 fm), the strong force becomes repulsive, preventing the nucleus from collapsing. This balance between attraction and repulsion determines nuclear sizes and stability.

在极短距离（< 0.5 fm）下，强核力变为排斥力，防止原子核崩溃。这种吸引与排斥之间的平衡决定了原子核的大小和稳定性。

2. Mass Defect | 质量亏损

The total mass of a stable nucleus is always less than the sum of the masses of its individual protons and neutrons. This difference is called the mass defect, Δm. For example, for helium-4 (2 protons and 2 neutrons), the measured mass is 4.002602 u, while the sum of individual particles is 2 × 1.007276 u + 2 × 1.008665 u = 4.031882 u. The mass defect is therefore 0.029280 u.

一个稳定原子核的总质量总是小于其组成质子和中子各自质量之和。这个差值称为质量亏损 Δm。例如，对于氦-4（2个质子和2个中子），测得质量是 4.002602 u，而单个粒子质量之和为 2 × 1.007276 u + 2 × 1.008665 u = 4.031882 u。因此质量亏损为 0.029280 u。

Δm = (Z × mₚ + N × mₙ) − M_nucleus

In this equation, Z is the number of protons, N is the number of neutrons, mₚ is the proton mass, mₙ is the neutron mass, and M_nucleus is the actual mass of the nucleus.

在这个公式中，Z 是质子数，N 是中子数，mₚ 是质子质量，mₙ 是中子质量，M_nucleus 是原子核的实际质量。

3. Binding Energy | 结合能

According to Einstein’s mass–energy equivalence, the mass defect corresponds to an energy release when the nucleus forms from separate nucleons. This energy is called the binding energy, E_b. It is the minimum energy required to completely separate a nucleus into its individual protons and neutrons.

根据爱因斯坦的质能等价关系，质量亏损对应着原子核由独立核子形成时释放的能量。这个能量称为结合能 E_b。它是将一个原子核完全拆分成独立质子和中子所需的最小能量。

E_b = Δm c²

Using the helium-4 example, Δm = 0.029280 u. Since 1 u = 1.66054 × 10⁻²⁷ kg, Δm = 4.862 × 10⁻²⁹ kg. Then E_b = 4.862 × 10⁻²⁹ × (3.00 × 10⁸)² = 4.37 × 10⁻¹² J, which is about 27.3 MeV.

以氦-4为例，Δm = 0.029280 u。由于 1 u = 1.66054 × 10⁻²⁷ kg，Δm = 4.862 × 10⁻²⁹ kg。于是 E_b = 4.862 × 10⁻²⁹ × (3.00 × 10⁸)² = 4.37 × 10⁻¹² J，约 27.3 MeV。

In nuclear physics, energies are often quoted in electronvolts. 1 eV = 1.60 × 10⁻¹⁹ J, and 1 MeV = 10⁶ eV. The conversion factor is approximately 1 u = 931.5 MeV/c².

在核物理中，能量常用电子伏特表示。1 eV = 1.60 × 10⁻¹⁹ J，1 MeV = 10⁶ eV。换算因子约为 1 u = 931.5 MeV/c²。

4. Binding Energy per Nucleon | 每核子结合能

The binding energy per nucleon is obtained by dividing the total binding energy by the mass number A. This quantity indicates how tightly bound the nucleons are, and it is a direct measure of nuclear stability. Higher binding energy per nucleon means greater stability.

每核子结合能等于总结合能除以质量数 A。该量表示核子结合的紧密程度，是核稳定性的直接度量。每核子结合能越高，原子核越稳定。

Binding energy per nucleon = E_b / A

For helium-4, the binding energy per nucleon is 27.3 MeV / 4 = 6.83 MeV. For iron-56, it is about 8.75 MeV, which is the highest among all nuclei.

对于氦-4，每核子结合能为 27.3 MeV / 4 = 6.83 MeV。对于铁-56，约为 8.75 MeV，是所有原子核中最高的。

5. The Binding Energy Curve | 结合能曲线

If we plot binding energy per nucleon against mass number A, we obtain a curve that rises steeply for light nuclei, peaks around A ≈ 56 (iron), then slowly decreases for heavier nuclei. This curve is fundamental to understanding energy release in nuclear reactions.

如果以每核子结合能对质量数 A 作图，可得一条曲线：轻核区域迅速上升，在 A ≈ 56（铁）附近达到峰值，然后对于更重的核缓慢下降。这条曲线是理解核反应中能量释放的基础。

Nucleus	A	Binding energy per nucleon (MeV)	每核子结合能 (MeV)
H-2	2	1.11	1.11
He-4	4	7.07	7.07
C-12	12	7.68	7.68
Fe-56	56	8.79	8.79
U-238	238	7.57	7.57

Light nuclei have low binding energy per nucleon because surface effects allow nucleons to be less tightly bound. Heavy nuclei have lower values because the Coulomb repulsion between many protons reduces the net attractive effect.

轻核的每核子结合能较低，因为表面效应使核子束缚得不够紧密。重核的值较低，是因为大量质子之间的库仑斥力削弱了净吸引作用。

6. Nuclear Stability and the Neutron–Proton Ratio | 核稳定性与中子–质子比

For stable nuclei, the ratio of neutrons to protons (N/Z) increases with atomic number. Light stable nuclei have N/Z ≈ 1, while heavy stable nuclei such as lead-208 have N/Z ≈ 1.54. The strong force is charge-independent, but Coulomb repulsion between protons grows with Z, so more neutrons are needed to provide extra attraction without adding repulsion.

对于稳定核，中子与质子之比 N/Z 随原子序数增大而增加。轻稳定核的 N/Z ≈ 1，而重稳定核如铅-208 的 N/Z ≈ 1.54。强核力与电荷无关，但质子间库仑斥力随 Z 增大而增强，因此需要更多中子来提供额外吸引而不增加排斥。

If a nucleus has too many or too few neutrons relative to the stable ratio, it will be unstable and undergo radioactive decay to move toward stability. For example, carbon-14 has N/Z = 8/6 = 1.33, which is too high, so it β⁻ decays to nitrogen-14.

如果原子核相对于稳定比值含有过多或过少的中子，它就不稳定，会通过放射性衰变趋向稳定。例如，碳-14 的 N/Z = 8/6 = 1.33，偏高，因此它通过 β⁻ 衰变转变为氮-14。

7. Fission and Fusion | 裂变与聚变

Nuclear fission occurs when a heavy nucleus (such as uranium-235) splits into two medium-sized nuclei. Since the products have higher binding energy per nucleon than the original nucleus, the total binding energy increases, and the mass defect releases energy. The energy released in fission can be calculated from the mass difference between the reactants and products.

核裂变是指重核（如铀-235）分裂成两个中等质量核的过程。由于产物比原始核的每核子结合能更高，总结合能增大，质量亏损释放能量。裂变释放的能量可通过反应物与产物之间的质量差来计算。

Nuclear fusion combines light nuclei (such as hydrogen isotopes) into a heavier nucleus. For light nuclei below iron-56 on the binding energy curve, fusion also increases binding energy per nucleon and releases energy. The Sun produces energy through fusion of hydrogen into helium.

核聚变是将轻核（如氢同位素）结合成更重的核。对于结合能曲线中铁-56以下的轻核，聚变同样提高每核子结合能并释放能量。太阳通过氢聚变为氦来产生能量。

Fission: heavy nucleus → two medium nuclei + energy + neutrons

裂变：重核 → 两个中核 + 能量 + 中子
Fusion: two light nuclei → one heavier nucleus + energy

聚变：两个轻核 → 一个更重的核 + 能量

8. Worked Example: Energy Released in Fusion | 例题：聚变释放的能量

Calculate the energy released when two deuterium nuclei (²H) fuse to form helium-4. Given masses: ²H = 2.014102 u, ⁴He = 4.002602 u.

计算两个氘核（²H）聚变形成氦-4 时释放的能量。已知质量：²H = 2.014102 u，⁴He = 4.002602 u。

Total mass before = 2 × 2.014102 u = 4.028204 u. Mass after = 4.002602 u. Mass defect = 4.028204 − 4.002602 = 0.025602 u.

反应前总质量 = 2 × 2.014102 u = 4.028204 u。反应后质量 = 4.002602 u。质量亏损 = 4.028204 − 4.002602 = 0.025602 u。

Since 1 u corresponds to 931.5 MeV, the energy released is 0.025602 × 931.5 = 23.85 MeV.

由于 1 u 对应 931.5 MeV，释放能量为 0.025602 × 931.5 = 23.85 MeV。

This example shows how a tiny mass loss can produce a large amount of energy. In A-Level questions, always convert masses to kg if you use E = mc², or use the u-to-MeV conversion factor directly.

这个例子表明微小的质量损失能够产生巨大能量。在 A-Level 考试中，若使用 E = mc² 需将质量换算为 kg，或直接使用 u 到 MeV 的换算因子。

9. Common Misconceptions and Exam Tips | 常见误区与考试提示

Misconception: “Larger mass number means greater stability.” Actually, stability is measured by binding energy per nucleon, not total binding energy. Iron-56 is more stable than uranium-238 even though its mass number is smaller.

误区：“质量数越大越稳定。”实际上，稳定性由每核子结合能衡量，而不是总结合能。铁-56 比铀-238 更稳定，尽管其质量数更小。
Misconception: “Mass is destroyed in nuclear reactions.” Mass is converted into energy. The total mass–energy is conserved.

误区：“核反应中质量被消灭了。”质量转化为能量。总质能守恒。
Always state that binding energy is the energy required to separate the nucleus into nucleons, not the energy that binds nucleons together (though it is equal in magnitude).

务必说明：结合能是将原子核拆分核子所需的能量，而不是束缚核子的能量（尽管数值相等）。
When drawing the binding energy curve, label the axes clearly: x-axis = mass number A, y-axis = binding energy per nucleon in MeV.

画结合能曲线时，清晰标注坐标轴：x 轴为质量数 A，y 轴为每核子结合能（单位 MeV）。

10. Conclusion | 总结

Binding energy and mass defect are central to nuclear physics. The binding energy per nucleon curve explains why fissile heavy nuclei and fusible light nuclei can release energy, and why iron is the most stable nucleus. Understanding these concepts allows us to predict nuclear stability and to calculate energy changes in nuclear reactions.

结合能与质量亏损是核物理的核心。每核子结合能曲线解释了为什么重核可裂变、轻核可聚变并释放能量，也解释了为什么铁是最稳定的原子核。理解这些概念使我们能够预测核稳定性，并计算核反应中的能量变化。

In CIE A-Level Physics, you should be able to calculate mass defect, convert between u and MeV, interpret the binding energy curve, and explain fission and fusion in terms of binding energy per nucleon. Practice these skills with past paper questions, and you will master this topic.

在 CIE A-Level 物理考试中，你应该能够计算质量亏损，在 u 与 MeV 之间换算，解读结合能曲线，并利用每核子结合能解释裂变与聚变。通过练习真题掌握这些技能，你就能完全掌握这一主题。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Binding Energy and Nuclear Stability | 结合能与原子核稳定性