9.2 Binding Energy and Nuclear Forces

Binding energy of a nucleus

Concept and calculation of binding energy

A nucleus weighs less than the sum of its individual protons and neutrons. That "missing" mass has been converted into energy that holds the nucleus together. This is binding energy, and it tells you how tightly a nucleus is bound.

You calculate binding energy using Einstein's mass-energy equivalence:

$BE = (\Delta m)c^2$

where $\Delta m$ is the mass defect (more on that below) and $c$ is the speed of light.

Binding energy per nucleon ($BE/A$, where $A$ is the mass number) is the real measure of nuclear stability. A higher $BE/A$ means each nucleon is more tightly bound, so the nucleus is harder to break apart.

The $BE/A$ curve is one of the most important graphs in nuclear physics:

• It rises steeply for light nuclei (hydrogen, helium, lithium).
• It peaks near iron-56 at roughly 8.8 MeV per nucleon, making iron-56 one of the most stable nuclei.
• It gradually decreases for heavier nuclei beyond iron.

This shape directly explains why fusion of light nuclei and fission of heavy nuclei both release energy: in each case, the products move closer to the iron peak, gaining binding energy per nucleon.

The semi-empirical mass formula (SEMF) provides an approximate way to calculate binding energies across the chart of nuclides by modeling the nucleus as a liquid drop with volume, surface, Coulomb, asymmetry, and pairing terms.

Applications and implications of binding energy

• Fission: Heavy nuclei (like uranium-235) split into mid-mass fragments with higher $BE/A$, releasing the difference as energy.
• Fusion: Light nuclei (like hydrogen isotopes) combine into heavier products with higher $BE/A$, also releasing energy.
• Binding energy differences determine the Q-value of a nuclear reaction, which tells you how much energy is released or absorbed.
• Stars forge elements up to iron through fusion because that's where the $BE/A$ curve peaks. Elements heavier than iron require energy input and are primarily produced in supernovae.
• The relative abundances of elements in the universe reflect these binding energy trends: iron-group elements are disproportionately common.

Mass defect and nuclear stability

Understanding mass defect

Mass defect ($\Delta m$) is the difference between the total mass of the separated nucleons and the actual mass of the nucleus:

$\Delta m = [Z \cdot m_p + N \cdot m_n] - m_{\text{nucleus}}$

where $Z$ is the number of protons, $N$ is the number of neutrons, $m_p$ is the proton mass, and $m_n$ is the neutron mass.

For any stable nucleus, $\Delta m$ is always positive. The nucleus has less mass than its parts because some mass has been converted into binding energy.

Mass defect is typically expressed in atomic mass units (u), where 1 u = 931.5 MeV/$c^2$. This conversion factor lets you move between mass and energy units quickly. For example, if $\Delta m = 0.03$ u, the binding energy is $0.03 \times 931.5 \approx 27.9$ MeV.

Precise nuclear masses are measured using mass spectrometry, which is how we get the data to calculate mass defects experimentally.

Relationship between mass defect and nuclear stability

A larger mass defect per nucleon generally means a more stable nucleus, because more mass has been converted into binding energy per nucleon.

• The variation of mass defect across the periodic table mirrors the $BE/A$ curve: nuclei near iron have the largest mass defect per nucleon.
• When a nuclear reaction occurs, the total mass of the products differs from the total mass of the reactants. That mass difference becomes the reaction's energy output (or input). This is how you calculate Q-values: $Q = (\Delta m_{\text{reactants}} - \Delta m_{\text{products}})c^2$.
• Nuclei with unusually low mass defect per nucleon for their mass number tend to be unstable and undergo radioactive decay toward more stable configurations.
• Magic numbers (2, 8, 20, 28, 50, 82, 126) correspond to closed nuclear shells where nuclei have especially large binding energies and mass defects, similar to how noble gases have closed electron shells.

Strong nuclear force and electrostatic repulsion

Characteristics of the strong nuclear force

The strong nuclear force is what keeps nuclei from flying apart. Protons are all positively charged, so electrostatic (Coulomb) repulsion pushes them apart. The strong force overcomes this repulsion, but only at very short distances.

Key properties:

• At nuclear distances (~1 fm), the strong force is roughly 100 times stronger than the electromagnetic force.
• It is charge-independent: the strong force between two protons, two neutrons, or a proton and a neutron is essentially the same.
• Its range is limited to about 1–2 femtometers ($10^{-15}$ m), roughly the size of a nucleon. Beyond this distance, it drops off rapidly.
• At the fundamental level, the strong force binds quarks within protons and neutrons via exchange of gluons (massless particles carrying color charge). The force between nucleons is a residual effect of this quark-level interaction, sometimes called the nuclear force or residual strong force.
• Color confinement means quarks are never found isolated; they always form color-neutral combinations (like the three quarks in a proton or neutron).
• At extremely short ranges (less than ~0.5 fm), the strong force actually becomes repulsive, creating a hard core that prevents nucleons from overlapping completely.

Balance between strong force and electrostatic repulsion

Nuclear stability comes from the competition between these two forces:

• The strong force is short-range and attractive (with a repulsive core at very short range). The Coulomb force is long-range and repulsive between protons.
• In light nuclei, every nucleon is close enough to interact with every other nucleon via the strong force, so stability is easy to achieve.
• In heavier nuclei, the strong force only acts between nearest neighbors (because of its short range), but the Coulomb repulsion acts between all proton pairs across the entire nucleus. This is why heavy nuclei become increasingly unstable.
• Nuclear saturation refers to the fact that each nucleon only interacts strongly with its nearest neighbors. This is why nuclear density stays roughly constant regardless of nucleus size (about $2.3 \times 10^{17}$ kg/m³).
• To compensate for growing Coulomb repulsion, stable heavy nuclei need a higher neutron-to-proton ratio. Light stable nuclei have $N/Z \approx 1$, while heavy stable nuclei like lead-208 have $N/Z \approx 1.5$. Extra neutrons add strong-force attraction without adding Coulomb repulsion.
• The Coulomb barrier is the energy a charged particle must have to get close enough for the strong force to take over. This barrier is why nuclear fusion requires extremely high temperatures (millions of kelvin) to occur, as in stellar cores.

Strong vs. weak nuclear forces

These are two of the four fundamental forces, and they play very different roles inside the nucleus.

Properties of the strong nuclear force

• Binds quarks into hadrons (protons, neutrons) and binds nucleons into nuclei
• Mediated by gluons (massless, carry color charge)
• Range: ~1–2 fm for the residual nuclear force
• Charge-independent (acts equally on protons and neutrons)
• Strongest of all four fundamental forces at nuclear distances
• Exhibits color confinement

Characteristics of the weak nuclear force

• Responsible for beta decay and other flavor-changing processes
• Mediated by massive W and Z bosons ($m_W \approx 80$ GeV/$c^2$, $m_Z \approx 91$ GeV/$c^2$)
• Extremely short range: ~$10^{-18}$ m (about 1000 times shorter than the strong force range). The large mass of the W and Z bosons is what limits the range so severely.
• Strength is roughly $10^{-6}$ times that of the strong force
• Can change quark flavors: for example, in beta-minus decay, a down quark in a neutron changes to an up quark, converting the neutron into a proton while emitting an electron and an antineutrino
• Plays a critical role in stellar nucleosynthesis (e.g., the proton-proton chain in the Sun depends on a weak-force step) and in neutrino interactions with matter