KAHIBARO
Table of Contents
What the Curve Shows
The binding energy curve is a graph that shows how tightly nuclei are held together. It usually plots the binding energy per nucleon, written as $BE/A$, against the mass number $A$, where $A$ is the total number of protons and neutrons in the nucleus.
This graph is important because it summarizes a huge amount of nuclear behavior in one picture. By looking at its shape, we can understand why energy can be released both by fission of heavy nuclei and by fusion of light nuclei.
A nucleon is either a proton or a neutron, so $BE/A$ tells us the average binding energy for each particle in the nucleus. A larger value of $BE/A$ means the nucleus is, on average, more tightly bound.
A nucleus with a larger binding energy per nucleon is generally more stable than a nucleus with a smaller binding energy per nucleon.
Reading the Axes
On the horizontal axis is the mass number $A$. Small values of $A$ correspond to light nuclei such as hydrogen and helium. Large values of $A$ correspond to heavy nuclei such as uranium.
On the vertical axis is the binding energy per nucleon, usually measured in MeV per nucleon.
If the graph rises, nuclei are becoming more tightly bound as $A$ increases. If the graph falls, the average binding becomes weaker for larger nuclei.
General Shape of the Curve
The binding-energy curve has a characteristic shape. It rises steeply for very light nuclei, reaches a broad maximum around nuclei in the iron and nickel region, and then slowly decreases for heavier nuclei.
This means that medium-mass nuclei are the most tightly bound. Very light nuclei and very heavy nuclei are less tightly bound by comparison.
A simplified summary is shown below.
| Region of the curve | Typical nuclei | Behavior of $BE/A$ | Meaning |
|---|---|---|---|
| Very light nuclei | H, D, He, Li | Rises quickly | Adding nucleons greatly increases binding |
| Medium nuclei | Fe, Ni region | Near maximum | Nuclei are most stable |
| Heavy nuclei | Pb, U | Slowly decreases | Average binding weakens somewhat |
Why the Curve Rises at First
For light nuclei, adding more nucleons allows the strong nuclear force to bind them more effectively. Since the strong force acts over a short range, each added nucleon can contribute significantly to binding when the nucleus is still small.
So from hydrogen up to medium nuclei, the binding energy per nucleon generally increases. This means the nucleus becomes more stable as it grows, at least in this range.
Why the Curve Peaks Near Iron
The peak of the curve is near iron and nickel, around $A \approx 56$ to $62$. These nuclei have some of the highest binding energies per nucleon, about $8$ to $9 \, \text{MeV}$ per nucleon.
This region represents the best balance between attractive nuclear forces and the repulsive electric force between protons. The strong nuclear force binds nearby nucleons together, but proton-proton repulsion grows as the number of protons increases. Around the iron region, the balance is especially favorable.
The most tightly bound nuclei lie near the iron and nickel region, where $BE/A$ is largest.
Why the Curve Falls for Heavy Nuclei
For very heavy nuclei, the binding energy per nucleon decreases slowly. The strong nuclear force is short-range, so each nucleon mainly interacts with nearby nucleons. But the electric repulsion between protons acts across the whole nucleus and grows as the nucleus gets larger.
As a result, very heavy nuclei are less tightly bound on average than medium-mass nuclei. This does not mean they have small total binding energy. In fact, heavy nuclei can have very large total binding energy, but when divided by the large number of nucleons, the binding per nucleon is lower.
Connection to Fusion
Fusion is the joining of light nuclei into a heavier nucleus. If the resulting nucleus lies higher on the binding-energy curve, then the binding energy per nucleon has increased. The final nucleus is more tightly bound, and energy is released.
For example, light nuclei such as hydrogen isotopes can fuse to form helium. Since helium has a larger binding energy per nucleon than the separate starting nuclei, the process releases energy.
Fusion releases energy when light nuclei combine to form a nucleus with larger binding energy per nucleon.
Connection to Fission
Fission is the splitting of a very heavy nucleus into smaller nuclei. If the products lie closer to the peak of the binding-energy curve, then they have larger binding energy per nucleon than the original heavy nucleus. The products are more tightly bound, and energy is released.
This is why nuclei such as uranium can release energy by splitting into medium-mass fragments.
Fission releases energy when a heavy nucleus splits into nuclei with larger binding energy per nucleon.
Why Both Fusion and Fission Can Release Energy
At first, it may seem strange that both combining nuclei and splitting nuclei can release energy. The curve explains this clearly.
Light nuclei are on the left side of the peak. Moving toward the peak means fusing them can increase $BE/A$.
Heavy nuclei are on the right side of the peak. Moving toward the peak means splitting them can increase $BE/A$.
So in both cases, energy is released when nuclear reactions move nuclei toward the peak region.
Energy Interpretation
If the total binding energy of the products is greater than the total binding energy of the reactants, the reaction releases energy. This energy comes from the difference in mass and appears as kinetic energy or radiation.
For a nucleus with mass number $A$, the total binding energy is
$$
BE = A \left(\frac{BE}{A}\right)
$$
This equation shows that the binding-energy curve gives average binding, while the actual total binding energy depends on both the average value and the number of nucleons.
A Sketch of the Curve
This drawing is only qualitative. The real curve has detailed local variations due to shell effects and differences between specific nuclei.
Important Consequences
The binding-energy curve helps explain several major facts of nuclear physics.
| Observation | Explanation from the curve |
|---|---|
| Light nuclei can power stars through fusion | Fusion moves nuclei upward toward greater $BE/A$ |
| Heavy nuclei such as uranium can release energy by fission | Fission products lie closer to the peak |
| Iron region nuclei are especially stable | They are near the maximum of $BE/A$ |
| Very heavy nuclei are less stable | Their $BE/A$ is lower than in the medium-mass region |
A Common Point of Confusion
Students often confuse total binding energy with binding energy per nucleon. Heavy nuclei usually have a very large total binding energy because they contain many nucleons. But they may still have a smaller binding energy per nucleon than medium nuclei.
For stability comparisons, the binding energy per nucleon is often the more useful quantity.
Do not confuse $BE$ with $BE/A$.
$BE$ is the total binding energy of the nucleus.
$BE/A$ is the average binding energy per nucleon, and it is the quantity shown in the binding-energy curve.
Final Picture
The binding-energy curve is one of the most useful graphs in nuclear physics. It shows that nuclei become more tightly bound as we move from very light nuclei toward the iron region, and less tightly bound as we move to very heavy nuclei. This single pattern explains why fusion of light nuclei and fission of heavy nuclei can both release energy, and why nuclei near iron are among the most stable in nature.