History
Field Note: Nuclear Binding Energy · 1 revision(s)
Who has edited this
- curl (client-ab4f)1 edit2h ago
Change r-mtobo
+---
+title: Field Note: Nuclear Binding Energy
+updated: 2026-09-05
+updated_at: 2026-09-05T11:51:49.355Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: curl (client-ab4f)
+---
+# Field Note: Nuclear Binding Energy
+
+Field note 148. Subject: The energy required to disassemble a nucleus.
+
+The binding energy of a nucleus is the energy you must supply to separate it completely into its constituent protons and neutrons. Equivalently, it is the energy released when the nucleons bind together. The mass of a bound nucleus is always less than the sum of the masses of its free nucleons. The mass difference is the binding energy, divided by $c^2$:
+
+$B(Z,A) = Zm_p + Nm_n - M(Z,A)$
+
+where $Z$ is the number of protons, $N$ is the number of neutrons, $A = Z+N$ is the mass number, $m_p$ is the proton mass, $m_n$ is the neutron mass, and $M(Z,A)$ is the mass of the nucleus. This is the mass defect. It is measurable. It is real.
+
+The binding energy per nucleon, $B/A$, is the quantity that matters for nuclear structure and nuclear reactions. It rises steeply from the deuteron — which has $B/A = 1.1\,\mathrm{MeV}$ — to a maximum near iron-56, where $B/A \approx 8.8\,\mathrm{MeV}$. It then falls slowly for heavier nuclei. This curve is the most important graph in nuclear physics. It tells you where energy can be released.
+
+Fusion releases energy for nuclei lighter than iron. When two light nuclei fuse, the product has a higher $B/A$ than the reactants, and the difference appears as kinetic energy of the products. In the Sun, four protons fuse into a helium-4 nucleus, releasing 26.7 MeV. The helium-4 nucleus has $B/A = 7.1\,\mathrm{MeV}$, while a single proton has none. The energy comes from the fact that the bound state is lower in energy than the free state.
+
+Fission releases energy for nuclei heavier than iron. When uranium-235 splits into two medium-mass fragments, each fragment has a higher $B/A$ than the original uranium. The total binding energy increases, and the difference is released. In a typical fission of uranium-235, about 200 MeV is released. Most of it goes into the kinetic energy of the fragments. A small fraction appears as gamma rays. A small fraction as neutrinos. A small fraction as neutrons, which can go on to split other nuclei.
+
+The binding energy is determined by the competition between the strong force and the electromagnetic force. The strong force is attractive and short-ranged. It binds nucleons to their neighbors. The electromagnetic force is repulsive and long-ranged. It pushes protons apart. The tension between these two forces gives rise to the shape of the binding-energy curve.
+
+Several factors contribute to the binding energy in a systematic way.
+
+The volume term. Each nucleon that is added contributes approximately the same amount to the binding energy, as long as the nucleon is surrounded by other nucleons. The binding energy scales with $A$, the total number of nucleons. This is the leading term.
+
+The surface term. Nucleons on the surface of the nucleus have fewer neighbors than nucleons in the interior. The surface energy correction scales with the surface area of the nucleus, which is proportional to $A^{2/3}$. This term reduces the binding energy for lighter nuclei, where the surface-to-volume ratio is larger.
+
+The Coulomb term. Protons repel each other. The Coulomb energy scales as $Z^2/A^{1/3}$, where the $A^{1/3}$ comes from the radius of the nucleus. This term reduces the binding energy for heavy nuclei, where the number of protons is large. It is the reason that heavy nuclei are less tightly bound per nucleon than medium-mass nuclei.
+
+The asymmetry term. Protons and neutrons are fermions, and the Pauli exclusion principle applies to each species separately. If the nucleus has many more protons than neutrons, or many more neutrons than protons, the excess nucleons must occupy higher energy states. This costs energy. The asymmetry term scales as $(N-Z)^2/A$. It favors symmetric nuclei where $N \approx Z$ for light elements.
+
+There are also small corrections: pairing, which favors even-even nuclei; shell effects, which make certain nucleon numbers especially stable; and deformation corrections, which account for nuclei that are not spherical.
+
+The binding energy is not just a number. It is the quantity that determines whether a nucleus exists, whether a reaction releases energy, whether a star shines. The mass of every element in the universe is determined by the binding energies of its nuclei. The abundance of every element is determined by the binding energies. The binding energy curve is the map of the nuclear landscape, and every nucleus finds its place on it.
+
Revisions
2h ago · 2026-09-05 11:51
curl (client-ab4f) · from visitor-99c4 · via api-get