The binding energy curve and nuclear forces

Which nuclei are held most tightly?

In the last lesson we found $B/A = 7.07$ MeV for helium-4 and $7.98$ MeV for oxygen-16. If we find $B/A$ for every nucleus and plot it against the mass number $A$, we get one of the most useful graphs in physics.

Read the graph in four steps.

1. A steep rise. From deuterium ($1.1$ MeV) to helium-4 ($7.07$ MeV) the value climbs fast. A small nucleus has few neighbours for each nucleon.
2. Peaks at the "alpha" nuclei. Helium-4, carbon-12 and oxygen-16 stand a little above the curve. Each is made of whole alpha particles (helium-4 nuclei), which are very tightly bound units.
3. A broad plateau. For $A$ between about $30$ and $170$, $B/A$ stays near $8$ MeV or more. The top is about $8.8$ MeV near $A = 56$ to $62$.
4. A slow fall. Beyond the top, $B/A$ drops to $7.6$ MeV for uranium-238.

What NCERT shows, and the exact peak. NCERT puts the peak at $A = 56$, iron. Careful measurements put nickel-62 just above iron-56, $8.795$ MeV against $8.790$ MeV. Iron-56 is often called the most stable nucleus, but the true record belongs to nickel-62. The difference is tiny, and exams accept iron.

Why both fission and fusion release energy

A system moves to a state of more binding, and gives out the difference as energy. On the graph, the most tightly bound nuclei are at the top. So

• Fusion: two light nuclei join into one that is higher up the curve. Energy is released.
• Fission: one heavy nucleus splits into two that are higher up the curve. Energy is released.

Fission example. Take a nucleus with $A = 240$ and $B/A = 7.6$ MeV. It splits into two nuclei of $A = 120$, each with $B/A = 8.5$ MeV. The gain per nucleon is $0.9$ MeV, and all $240$ nucleons gain it:

This is close to the $200$ MeV that one real fission of uranium releases.

Fusion example. Two deuterium nuclei fuse into helium-4. The mass defect is $2 \times 2.014102\ \text{u} - 4.002603\ \text{u} = 0.025601\ \text{u}$. So $E = 0.025601\ \text{u} \times 931.494\ \text{MeV/u} = 23.85\ \text{MeV}$. That is $\dfrac{23.85\ \text{MeV}}{4} = 5.96$ MeV for each nucleon involved. In fission only $0.85$ MeV is released for each nucleon. Fusion gives about seven times more energy from the same mass of fuel.

Set the heavy nucleus to $A = 235$, $7.6$ MeV and the halves to $8.5$ MeV. Read the energy. Now lower the halves to $8.2$ MeV. How much energy do you lose?

The nuclear force

Why do protons stay together, when like charges repel? Two protons at $1$ fm push each other apart with a force of about $230$ N. Gravity is far too weak to hold them. A much stronger force must exist. It is the strong nuclear force, and it has these properties.

• It is attractive and very strong at a distance of about $1$ fm, far stronger than the electric repulsion there.
• It has a short range. It acts only up to a few femtometres. Beyond a few femtometres it is negligible.
• It is charge independent. The force between two protons, two neutrons or a proton and a neutron is nearly the same, once the effect of the electric charge is removed.
• It saturates. A nucleon attracts only its nearest neighbours, not all the others in the nucleus. This is why $B/A$ stays about the same and does not grow with $A$, and why $R^3 \propto A$.

The electric repulsion between the protons has a long range. Every proton repels every other proton. In a big nucleus this adds up. This is why $B/A$ falls for heavy nuclei and why heavy nuclei need more neutrons than protons. The extra neutrons add attraction and no repulsion. Light stable nuclei have $N \approx Z$. Iron-56 has $N/Z = 1.15$. Lead-208 has $N/Z = 1.54$. Uranium-238 has $N/Z = 1.59$. Above $Z = 82$ (lead) no nucleus is truly stable.

Why is $B/A$ low for very light nuclei? Nucleons on the surface have fewer neighbours, so they are less tightly held. In a small nucleus most nucleons are on the surface.

Using the curve in a calculation

Suppose a nucleus with $A$ nucleons and $B/A = b_1$ turns into products with the same total $A$ and $B/A = b_2$. The energy released is

For fusion, the total number of nucleons does not change, but there are fewer, bigger nuclei. For fission, there are more, smaller nuclei. In both cases, count the nucleons once and use $A$ of the whole system.

Practise with new numbers

Try it in Python

The program finds $B/A$ for seven nuclei and prints a bar for each. The tallest bar belongs to the most tightly bound nucleus in the list.

Common mistakes

• Saying only heavy nuclei release energy. Fusion of light nuclei releases energy too. Both move towards the top of the curve.
• Saying iron releases energy when it fuses. Iron is at the top. Fusion or fission of iron absorbs energy.
• Mixing up $B/A$ and $B$. The curve plots the energy per nucleon.
• Using the energy of one nucleus for all nucleons. In $E = A(b_2 - b_1)$, $A$ is the number of nucleons in the whole system.
• Saying the nuclear force is the electric force. It is a separate, much stronger force of short range.

Remember these

• The $B/A$ curve rises to about $8.8$ MeV near $A = 56$ to $62$ and falls to $7.6$ MeV at uranium.
• Fusion of light nuclei and fission of heavy nuclei both release energy, because both move up the curve.
• $E = A(b_2 - b_1)$. Fission gives about $200$ MeV. Deuterium to helium-4 gives $23.85$ MeV.
• The nuclear force is strong, attractive, short range, charge independent and saturating.
• Heavy nuclei need extra neutrons because the proton repulsion has a long range.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Two nuclei have $B/A$ of $5$ MeV and $8$ MeV. Which has the smaller mass for each nucleon?

2. A nucleus of $A = 100$ and $B/A = 8.2$ MeV is formed from nucleons far apart. How much energy is released?

3. Why do large stable nuclei have more neutrons than protons?

4. Why can the nuclear force not be gravity?

References

• NCERT Physics Part II (Class 12, rationalised edition), Chapter 13 Nuclei, on the binding energy curve and nuclear force.
• OpenStax University Physics Volume 3, Chapter 10 Nuclear Physics, on nuclear binding and nuclear forces.