Nuclear Structure, Binding Energy, and Mass Defect

The nucleus of an atom consists of protons and neutrons (nucleons) held together by the strong nuclear force; the mass of a nucleus is less than the sum of the masses of its separate nucleons, and this mass defect Δm corresponds to the binding energy Eb = Δm c2, the energy required to separate the nucleus into free nucleons. In brief: a nuclide is written AZX, with proton number Z, neutron number N and nucleon number A = Z + N. Isotopes have the same Z and different N. The binding energy per nucleon rises to a maximum of about 8.8 MeV near iron-56 and falls for heavier nuclei, so energy is released by fusing light nuclei and by splitting heavy nuclei. 1 u of mass is equivalent to 931.5 MeV.

The nucleus is about 100,000 times smaller than the atom but holds more than 99.9 percent of its mass. Understanding why nuclei are stable, and how much energy is stored in them, is the basis of nuclear power, nuclear medicine and the physics of stars. This guide describes the structure and size of the nucleus, the nuclear force, the mass defect and binding energy, compares nuclei in a table, works through numerical examples and corrects common mistakes.

What is the structure of the nucleus?

Rutherford’s scattering experiments (1911) showed that the positive charge and almost all the mass of an atom are concentrated in a tiny nucleus. James Chadwick discovered the neutron in 1932. The nucleus has Z protons (charge +e each) and N neutrons (no charge); A = Z + N. Nuclides with the same Z but different N are isotopes (for example, hydrogen-1, deuterium-2 and tritium-3). The mass of a proton is 1.00728 u and of a neutron 1.00866 u (1 u = 1.6605 × 10−27 kg). The nuclear radius is approximately R = r0A1/3, with r0 ≈ 1.2 fm (1 fm = 10−15 m): a nucleus has a volume proportional to A, so the density is about the same for all nuclei, 2.3 × 1017 kg/m3, about 1014 times the density of water. Light stable nuclei have N ≈ Z; heavy ones have more neutrons (N/Z up to about 1.5), to dilute the repulsion of the protons.

What holds the nucleus together?

Protons repel each other electrically with a force of tens to hundreds of newtons at nuclear distances, but the nucleus is stable because of the strong nuclear force between nucleons: it is attractive and much stronger than the electric force at distances of about 1 fm, but has a short range (about 2 to 3 fm), and falls to nearly zero beyond that. It is independent of charge (proton-proton, neutron-neutron and proton-neutron forces are similar), and becomes repulsive at very short distances (less than 0.7 fm), which prevents the collapse. Because of the short range, each nucleon interacts mainly with its nearest neighbors, so the binding energy per nucleon is roughly constant for medium and heavy nuclei (saturation), while the electric repulsion acts between all protons and grows as Z2. In large nuclei the repulsion becomes important, and nuclei beyond bismuth-209 (Z = 83) are unstable.

What are the mass defect and binding energy?

The mass of a stable nucleus is less than the total mass of its constituent protons and neutrons. The difference is the mass defect:

Δm = Zmp + (A − Z)mn − mnucleus.

By Einstein’s relation E = mc2, the mass that has “disappeared” corresponds to the energy released when the nucleus forms from free nucleons, the binding energy Eb = Δm c2. Equivalently, this is the energy that must be supplied to pull the nucleus apart. In practice, atomic masses (which include the electrons) are used: Δm = ZM(1H) + (A − Z)mn − M(atom), as the electron masses cancel. The conversion: 1 u = 931.5 MeV/c2. The binding energy per nucleon, Eb ÷ A, measures how tightly bound the nucleus is.

What does the binding energy curve show?

When Eb ÷ A is plotted against A, it rises steeply for light nuclei (deuterium 1.1 MeV, helium-4 7.07 MeV, carbon-12 7.68 MeV), has a broad maximum near A ≈ 56 (iron-56: 8.79 MeV; nickel-62 is slightly higher), and declines slowly to about 7.6 MeV for uranium-238. Peaks at helium-4, carbon-12 and oxygen-16 indicate particularly stable “alpha-particle” structures. Consequences: fusion of light nuclei moves up the curve, releasing energy (the Sun); fission of heavy nuclei into medium-mass fragments also moves toward the maximum and releases energy (nuclear reactors). Iron-group nuclei are the most stable, which is why fusion in stars ends at iron.

Binding energy per nucleon against mass number 0 50 100 150 200 250 0 2 4 6 8 10 Mass number A Eb / A (MeV) 2H 4He 12C 56Fe 238U
Binding energy per nucleon for the nuclides in the table (the line joins the points schematically). The curve climbs steeply for light nuclei, peaks near iron-56 at 8.79 MeV, and falls slowly for heavy nuclei such as uranium-238 (7.57 MeV). Fusion of light nuclei and fission of heavy nuclei both move toward the peak.

Binding energies compared

Binding energy of some nuclides
Nuclide A Total binding energy (MeV) Per nucleon (MeV)
Deuterium (2H)22.221.11
Helium-4428.37.07
Carbon-121292.27.68
Iron-56564928.79
Uranium-2382381,8027.57

Worked example: helium-4 and iron-56

  1. Mass defect of helium-4. Using atomic masses: M(1H) = 1.007825 u, mn = 1.008665 u, M(4He) = 4.002603 u. Δm = 2 × 1.007825 + 2 × 1.008665 − 4.002603 = 2.015650 + 2.017330 − 4.002603 = 0.030377 u.
  2. Binding energy: Eb = 0.030377 × 931.5 = 28.3 MeV, which is 7.07 MeV per nucleon. In joules: 28.3 × 1.602 × 10−13 = 4.53 × 10−12 J.
    ⁴He 7.07 MeV ⁵⁶Fe 8.79 MeV
  3. Iron-56 (Z = 26, M = 55.934936 u): Δm = 26 × 1.007825 + 30 × 1.008665 − 55.934936 = 26.203450 + 30.259950 − 55.934936 = 0.528464 u. Eb = 0.528464 × 931.5 = 492 MeV, 8.79 MeV per nucleon.
  4. Nuclear radius and density. For iron-56: R = 1.2 × 561/3 = 1.2 × 3.83 = 4.6 fm. Density = (56 × 1.66 × 10−27 kg) ÷ (4/3 π (4.6 × 10−15)3) = 9.3 × 10−26 ÷ 4.1 × 10−43 = 2.3 × 1017 kg/m3. A teaspoon of this matter would have a mass of about a billion tonnes.
  5. Energy scale. The 28.3 MeV for one helium nucleus corresponds to 28.3 × 1.602 × 10−13 J × 6.022 × 1023 ÷ 0.004 kg = 6.8 × 1014 J per kg of helium: about 23 million times the energy of burning 1 kg of coal (3 × 107 J/kg).

How was the nucleus discovered?

Ernest Rutherford, Hans Geiger and Ernest Marsden observed in 1909 that a few alpha particles are scattered at large angles by a thin gold foil, and Rutherford concluded in 1911 that the atom has a small, dense nucleus. Francis Aston’s mass spectrometer measurements (1919) showed that nuclear masses are not the sums of the nucleon masses, and Einstein’s mass-energy equivalence (1905) explained the difference. Chadwick found the neutron in 1932, Hideki Yukawa proposed in 1935 that the strong force is carried by mesons, and Maria Goeppert Mayer and Hans Jensen developed the nuclear shell model in 1949 (Nobel Prize 1963), explaining “magic numbers” of especially stable nuclei (2, 8, 20, 28, 50, 82, 126).

Which mistakes are common?

Mistake 1: Confusing binding energy with the energy needed to remove one nucleon

The binding energy is for the complete separation into all nucleons. The energy to remove a single nucleon (the separation energy) is different.

Mistake 2: Thinking that mass is destroyed

The mass difference is carried away as energy (radiation and kinetic energy) when the nucleus forms. The total mass-energy is conserved.

Mistake 3: Forgetting that more binding energy means more stable

Per nucleon, the highest is iron-56. Nuclei at the peak are the most stable and cannot release energy by fusion or fission.

Mistake 4: Mixing atomic and nuclear masses

Use atomic masses with hydrogen-atom masses, or nuclear masses with proton masses, but do not mix them.