Strong Nuclear Force and Nuclear Stability

The strong nuclear force is the short-range attraction between nucleons that holds the nucleus together against the electrostatic repulsion of the protons; whether a nucleus is stable depends on its neutron-to-proton ratio, on even or odd numbers of nucleons, and on magic numbers. In brief: light stable nuclei have N/Z close to 1 (carbon-12 has 6 protons and 6 neutrons), heavier ones need more neutrons (lead-208 has N/Z = 1.54), and all nuclides with more than 83 protons are radioactive.

There are about 250 stable nuclides, and well over 3000 known radioactive ones, and the difference between them is set by the balance of nuclear forces. Understanding stability makes it possible to predict how a given nucleus will decay. This article describes the strong force, the belt of stability, the patterns in stable nuclei, the magic numbers and the rules that predict the mode of decay, with data tables and worked examples.

What is the strong nuclear force?

Protons repel each other by the Coulomb force. At a distance of 1 fm, the repulsion between two protons is about 230 N, and yet nuclei exist. The reason is the strong nuclear force, an attraction between all nucleons (protons and neutrons alike) that is much stronger than the electrostatic force at short distances, but has a very short range of about 1–3 fm. At larger distances it falls off to zero, while the Coulomb repulsion decreases slowly (as 1/r²) and acts between all the protons in the nucleus. Each nucleon is attracted only by its close neighbors, whereas each proton is repelled by all the other protons.

For this reason a nucleus with many protons needs additional neutrons, which add attraction without adding repulsion. But too many neutrons also reduce stability, since the neutrons then fill higher energy levels. Nuclei with more than 83 protons cannot be stabilized, whatever the number of neutrons. The strong force is now understood as a residual effect of the force between quarks, which is carried by gluons, described by quantum chromodynamics.

What is the belt of stability?

If the number of neutrons N is plotted against the number of protons Z for all known nuclides, the stable nuclei lie in a narrow band, called the belt (or band) of stability, and the radioactive nuclei lie around it. For light elements (up to calcium, Z = 20) the stable nuclei have N ≈ Z. For heavier elements the band curves upward, and N/Z rises to about 1.5 for lead. All nuclei with Z > 83 are unstable. Technetium (Z = 43) and promethium (Z = 61) have no stable isotopes, although they lie in the middle of the periodic table.

Neutron-to-proton ratios of stable nuclei

Why do even numbers matter?

Nucleons, like electrons, have spin and tend to form pairs, so nuclei with even numbers of protons and neutrons are more stable than those with odd numbers. The count of stable nuclides shows this:

This is the reason for the Oddo–Harkins rule: elements with even atomic numbers are more abundant in the universe than their odd neighbors.

What are magic numbers?

Nuclei with 2, 8, 20, 28, 50, 82 or 126 protons or neutrons (the magic numbers) are particularly stable, just as atoms with filled electron shells (noble gases) are. This is explained by the nuclear shell model, in which nucleons fill energy levels in the potential well of the nucleus. A nucleus with a magic number of both protons and neutrons is “doubly magic”: ⁴He (2, 2), ¹⁶O (8, 8), ⁴⁰Ca (20, 20) and ²⁰⁸Pb (82, 126) are examples. Tin (Z = 50) has ten stable isotopes, the most of any element, and calcium-48 (20, 28) is a rare doubly magic isotope that is neutron-rich.

How do you predict the mode of decay?

Worked examples

Example 1: carbon-14

¹⁴C has Z = 6 and N = 8, so N/Z = 1.33. The stable carbon isotopes (¹²C, ¹³C) have N/Z of 1.00 and 1.17. ¹⁴C has too many neutrons, so it decays by beta-minus emission: ¹⁴₆C → ¹⁴₇N + e⁻ + ν̄ (half-life 5730 years).

Example 2: carbon-11

¹¹C has Z = 6 and N = 5, so N/Z = 0.83: too few neutrons. It decays by positron emission: ¹¹₆C → ¹¹₅B + e⁺ + ν (half-life 20.4 min), which is a nuclide that is used in positron emission tomography.

Example 3: uranium-238

Z = 92 is above 83, so the nuclide is unstable, and it decays by alpha emission: ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He. The half-life is 4.47 × 10⁹ years, so it still exists on Earth.

Example 4: is a nuclide likely to be stable?

Compare ⁴⁰Ca (20, 20), ⁴⁰K (19, 21) and ⁴⁰Ar (18, 22). ⁴⁰Ca is even-even and doubly magic, so it is stable. ⁴⁰K is odd-odd and unstable, with a half-life of 1.25 × 10⁹ years, and decays by beta-minus to ⁴⁰Ca (about 89 %) or electron capture to ⁴⁰Ar (about 11 %). ⁴⁰Ar is even-even and is stable. The nuclide ⁴⁰K is in the human body and contributes to the natural radioactivity.

Example 5: the electrostatic repulsion

The Coulomb force between two protons at 1.0 fm is F = kq²/r² = (8.99 × 10⁹)(1.602 × 10⁻¹⁹)² / (1.0 × 10⁻¹⁵)² = 2.3 × 10² N. The strong force at this distance is of the order of 10⁴ N, which is about 100 times larger.

How was nuclear stability understood?

Hideki Yukawa proposed in 1935 that the nuclear force is carried by a particle with a mass between those of the electron and proton, which was later identified with the pion, and received the Nobel Prize in Physics in 1949. The liquid-drop model of Carl von Weizsäcker (1935) and Hans Bethe gave a semi-empirical formula for the binding energy. The shell model with magic numbers was developed in 1949 by Maria Goeppert Mayer and independently by J. Hans D. Jensen, who shared the Nobel Prize in Physics in 1963.

Which mistakes are common?

• Thinking that gravity holds the nucleus together: it is the strong force.
• Assuming that N/Z = 1 is stable for all nuclei: heavier nuclei need more neutrons.
• Ignoring odd–even effects: even–even nuclei are the most stable.
• Mixing up the decay for neutron-rich and neutron-poor nuclei: neutron-rich ones emit β⁻.
• Forgetting that stable does not mean abundant: some stable isotopes are extremely rare.