Table of Contents
What Makes an Atomic Nucleus Stable?
Atomic nuclei consist of protons and neutrons (together called nucleons). Whether a nucleus is stable or unstable depends on a balance between attractive and repulsive forces and on how the nucleons are arranged.
Competing Forces in the Nucleus
Two fundamental interactions are especially important for nuclear stability:
- Coulomb (electrostatic) repulsion
Protons are positively charged and repel each other. The more protons there are, the stronger this repulsion becomes. - Strong nuclear force
A very strong but very short‑ranged attractive force that acts between all nucleons (proton–proton, proton–neutron, neutron–neutron). It is attractive at typical nucleon separations and overcomes Coulomb repulsion at short distances.
A nucleus is stable when, for its given number of protons, there are enough nucleons (especially neutrons) to provide sufficient strong‑force attraction to counteract proton–proton repulsion.
The Role of Neutrons: The Band of Stability
Neutrons add strong-force attraction without adding electric charge, so they help “glue” the nucleus together.
- Light stable nuclei (small atomic number $Z$) usually have roughly equal numbers of protons and neutrons: $N \approx Z$.
- Heavier stable nuclei require more neutrons than protons: $N > Z$. This extra neutron “excess” compensates for increased proton–proton repulsion.
If we plot the number of neutrons $N$ versus the number of protons $Z$ for known stable nuclei, the points lie in a relatively narrow region called the band (or valley) of stability.
- Nuclei on this band are stable (or at least very long‑lived).
- Nuclei above the band (too many neutrons) or below the band (too few neutrons) are unstable and undergo radioactive decay to move toward stability.
Mass Defect and Binding Energy
The mass of a nucleus is less than the sum of the masses of its free protons and neutrons. The “missing mass” is called the mass defect.
For a nucleus with $Z$ protons and $N$ neutrons and total mass $m_\text{nucleus}$:
$$
\Delta m
= \left(Z m_p + N m_n\right) - m_\text{nucleus}
$$
This mass defect corresponds to the binding energy $E_\text{B}$ that holds the nucleus together:
$$
E_\text{B} = \Delta m\, c^2
$$
(Here $c$ is the speed of light.)
A convenient quantity to compare different nuclei is the binding energy per nucleon:
$$
\frac{E_\text{B}}{A}, \quad A = Z + N
$$
- The higher the binding energy per nucleon, the more tightly bound and, generally, the more stable the nucleus.
- A plot of $E_\text{B}/A$ versus mass number $A$ shows a broad maximum near medium‑mass nuclei such as iron and nickel.
This has direct consequences for nuclear reactions (see below).
General Stability Trends
Some empirical patterns of nuclear stability are useful:
- Very light nuclei
Only certain combinations of $Z$ and $N$ are stable, e.g. $^1\text{H}$, $^2\text{H}$, $^4\text{He}$. - Medium‑mass nuclei
Nuclei with roughly $N \approx Z$ and reasonably high binding energy per nucleon are typically stable. - Heavy nuclei
For large $Z$, nuclei need significantly more neutrons ($N \gg Z$) but even then, above a certain size, the Coulomb repulsion becomes so strong that no combination is truly stable. The heaviest known elements are all radioactive.
Magic Numbers
Certain nucleon numbers correspond to unusually stable nuclei. These are called magic numbers. They are associated with “closed shells” of nucleons in the nucleus.
Typical magic proton or neutron numbers are:
$$
2,\ 8,\ 20,\ 28,\ 50,\ 82,\ 126
$$
Nuclei with a magic number of protons or neutrons are relatively stable; if both are magic, the nucleus is especially stable (so‑called “doubly magic” nuclei, e.g. $^{4}\text{He}$, $^{16}\text{O}$, $^{40}\text{Ca}$, $^{208}\text{Pb}$).
Radioactive Decay: Paths Toward Stability
Unstable nuclei transform spontaneously into more stable nuclei by emitting particles and/or radiation. This is called radioactive decay. The specific type of decay depends on how a nucleus deviates from the band of stability.
Beta Decay (Neutron–Proton Conversion)
Beta decays change the ratio of neutrons to protons.
- Beta minus decay ($\beta^-$)
Occurs in nuclei with too many neutrons (above the band of stability).
A neutron inside the nucleus converts to a proton, emitting an electron and an antineutrino:
$$
n \rightarrow p + e^- + \bar{\nu}_e
$$
In nuclear notation, for a nucleus $^{A}_{Z}\text{X}$:
$$
{}^{A}_{Z}\text{X} \rightarrow {}^{A}_{Z+1}\text{Y} + e^- + \bar{\nu}_e
$$
This moves the nucleus down toward the band of stability (decreasing $N/Z$).
- Beta plus decay ($\beta^+$, positron emission)
Occurs in nuclei with too few neutrons (below the band of stability).
A proton converts to a neutron, emitting a positron and a neutrino:
$$
p \rightarrow n + e^+ + \nu_e
$$
Nuclear notation:
$$
{}^{A}_{Z}\text{X} \rightarrow {}^{A}_{Z-1}\text{Y} + e^+ + \nu_e
$$
This moves the nucleus up toward the band of stability (increasing $N/Z$).
- Electron capture (EC)
An alternative to positron emission for proton‑rich nuclei. The nucleus captures an inner‑shell electron:
$$
p + e^- \rightarrow n + \nu_e
$$
Nuclear notation:
$$
{}^{A}_{Z}\text{X} + e^- \rightarrow {}^{A}_{Z-1}\text{Y} + \nu_e
$$
The proton number decreases, again adjusting the $N/Z$ ratio.
Alpha Decay
Very heavy nuclei (large $Z$) often undergo alpha decay to reduce both proton and neutron numbers simultaneously.
An alpha particle is a helium‑4 nucleus:
$$
{}^{4}_{2}\text{He}
$$
In alpha decay:
$$
{}^{A}_{Z}\text{X} \rightarrow {}^{A-4}_{Z-2}\text{Y} + {}^{4}_{2}\text{He}
$$
This:
- Decreases $Z$ by 2 and $A$ by 4,
- Reduces Coulomb repulsion,
- Often brings heavy nuclei closer to regions of higher stability.
Gamma Emission
After a nuclear transformation (such as alpha or beta decay), the daughter nucleus may be created in an excited state with excess energy.
- This excess energy is released as gamma radiation ($\gamma$) without changing $Z$ or $A$:
$$
{}^{A}_{Z}\text{X}^* \rightarrow {}^{A}_{Z}\text{X} + \gamma
$$
Gamma emission adjusts the energy state of the nucleus rather than its composition.
Types of Nuclear Reactions Beyond Radioactive Decay
Radioactive decay is spontaneous. Nuclear reactions can also be induced by bombarding nuclei with particles such as neutrons, protons, or alpha particles, or by high‑energy photons.
General Form of a Nuclear Reaction
A nuclear reaction can be written as:
$$
a + A \rightarrow B + b
$$
- $A$ is the target nucleus,
- $a$ is the incoming particle,
- $B$ is the product (residual) nucleus,
- $b$ is an outgoing particle.
Mass number and charge are always conserved:
- Sum of mass numbers on left = sum on right,
- Sum of charges on left = sum on right.
Fission and Fusion: Connecting to Stability
The shape of the binding‑energy‑per‑nucleon curve explains why two very different types of nuclear reactions can release energy:
- Nuclear fission
Heavy nuclei (e.g. uranium, plutonium) split into two (or more) lighter nuclei plus neutrons.
The products are closer to the peak of the binding‑energy curve (around medium mass numbers), so the total binding energy increases and energy is released. - Nuclear fusion
Very light nuclei (e.g. hydrogen isotopes) combine to form heavier nuclei.
Again, products lie higher on the binding‑energy‑per‑nucleon curve than the reactants, releasing energy.
In both cases, the system moves toward nuclei that are more tightly bound and therefore more stable.
Energy Balance in Nuclear Reactions
Whether a nuclear reaction releases or requires energy can be determined from the mass difference between reactants and products.
- Consider a reaction
$$
\text{reactants} \rightarrow \text{products}
$$
- The mass difference is
$$
\Delta m = m_\text{reactants} - m_\text{products}
$$
- The corresponding energy change is
$$
Q = \Delta m \, c^2
$$
Interpretation:
- If $Q > 0$, the reaction is exothermic (energy is released).
- If $Q < 0$, the reaction is endothermic (energy must be supplied).
Because nuclear binding energies are large, even small mass differences correspond to large energy changes.
Summary: Stability and Nuclear Transformations
- Nuclear stability arises from a balance between strong nuclear attraction and Coulomb repulsion.
- Stable nuclei occupy a band of stability in the $N$–$Z$ diagram; neutron‑to‑proton ratio and “magic numbers” are key.
- Unstable nuclei undergo radioactive decay (alpha, beta, gamma, electron capture) to move toward greater stability.
- Induced nuclear reactions (including fission and fusion) change nuclei and can release large amounts of energy when they lead to products with higher binding energy per nucleon.