结合能Nuclear binding energy

原子核的质量总是小于组成它的核子质量之和，差值 Δm 对应结合能 E_B = Δm·c²（1 u 相当于 931.5 MeV）。比结合能曲线从氢迅速上升，在 ⁵⁶Fe、⁶²Ni 附近达到最大（约 8.8 MeV/核子），之后缓慢下降至铀的约 7.6 MeV/核子。因此轻核聚变和重核裂变都会释放能量。⁴He 的比结合能（7.07 MeV）明显高于邻近核素，解释了 α 粒子的特殊稳定性。

A nucleus always weighs less than its separate nucleons; the difference Δm corresponds to binding energy E_B = Δm·c², with 1 u equivalent to 931.5 MeV. Binding energy per nucleon rises steeply from hydrogen, peaks near ⁵⁶Fe and ⁶²Ni at about 8.8 MeV, and falls slowly to about 7.6 MeV for uranium, so both fusion of light nuclei and fission of heavy ones release energy. ⁴He's value (7.07 MeV) stands well above its neighbours, explaining the special stability of the alpha particle.

1935 年魏茨泽克提出半经验质量公式（贝特-魏茨泽克公式），把结合能分解为体积项、表面项、库仑项、不对称项和对能项，基于玻尔等人的液滴模型。1949 年迈耶和延森提出核壳层模型，解释了质子或中子数为 2、8、20、28、50、82、126 的“幻数”核特别稳定，两人获 1963 年诺贝尔物理学奖。

In 1935 von Weizsäcker proposed the semi-empirical mass formula (the Bethe–Weizsäcker formula), splitting binding energy into volume, surface, Coulomb, asymmetry and pairing terms, based on the liquid-drop model of Bohr and others. In 1949 Mayer and Jensen proposed the nuclear shell model, explaining why nuclei with 2, 8, 20, 28, 50, 82 or 126 protons or neutrons, the magic numbers, are especially stable; they shared the 1963 Nobel Prize in Physics.

结合能曲线解释了核能的来源、恒星中元素合成在铁附近终止的原因，以及比铁重的元素主要由超新星爆发和中子星并合中的中子俘获过程（r 过程、s 过程）产生。核结合能比化学键能大约百万倍，这就是核燃料能量密度远高于化学燃料的原因。

The binding-energy curve explains the source of nuclear energy, why stellar nucleosynthesis stops near iron, and why heavier elements form mainly through neutron capture (the r- and s-processes) in supernovae and neutron-star mergers. Nuclear binding energies are about a million times larger than chemical bond energies, which is why nuclear fuel is so much denser in energy than chemical fuel.