KAHIBARO
Table of Contents
What binding energy means
Binding energy is the energy needed to completely separate a nucleus into its individual protons and neutrons. It is also the energy released when those nucleons come together to form the nucleus.
A nucleus is a bound system. That means its protons and neutrons are held together by the nuclear force. Because of this binding, the nucleus has less total mass than the sum of the masses of the free particles that make it up. That missing mass is related to energy.
If a nucleus is strongly bound, it takes a large amount of energy to pull it apart. If it is weakly bound, less energy is needed.
The binding energy of a nucleus is the energy required to separate it completely into free nucleons.
Equivalently, it is the energy released when the nucleus is formed from free nucleons.
Relation to mass defect
Binding energy is calculated from the mass defect. The mass defect is the difference between the mass of the separate nucleons and the actual mass of the nucleus.
If a nucleus contains $Z$ protons and $N$ neutrons, then the expected mass of the separated particles is
$$
m_{\text{separate}} = Z m_p + N m_n
$$
The mass defect is
$$
\Delta m = \left(Z m_p + N m_n\right) - m_{\text{nucleus}}
$$
The binding energy is then
$$
E_B = \Delta m c^2
$$
where $c$ is the speed of light.
The key formula for nuclear binding energy is
$$
E_B = \Delta m c^2
$$
A larger mass defect means a larger binding energy.
Why mass decreases when a nucleus forms
When nucleons join to form a nucleus, energy is released to the surroundings. Since energy and mass are equivalent, the system loses an amount of mass corresponding to that released energy. The final nucleus therefore has a smaller mass than the total mass of the free nucleons.
This does not mean mass disappears without reason. It means some of the original mass appears as released energy.
Interpreting binding energy physically
Binding energy tells us how tightly the nucleus is held together. A larger binding energy generally means a more stable nucleus against being broken apart into protons and neutrons.
For example, if nucleus A has greater binding energy than nucleus B, then more energy is needed to split nucleus A completely. So nucleus A is more strongly bound.
It is important to be precise. Binding energy refers to the whole nucleus. A large nucleus often has a larger total binding energy simply because it contains more nucleons. For comparing different nuclei, physicists often use binding energy per nucleon, which is treated in the next section of the course.
Example structure of the calculation
Suppose a nucleus has $Z$ protons and $N$ neutrons. The steps are always the same.
First, add the masses of the free nucleons:
$$
m_{\text{separate}} = Zm_p + Nm_n
$$
Next, subtract the actual nuclear mass:
$$
\Delta m = m_{\text{separate}} - m_{\text{nucleus}}
$$
Finally, convert mass defect into energy:
$$
E_B = \Delta m c^2
$$
In nuclear physics, energy is often expressed in electron volts, especially mega electron volts, MeV.
Units commonly used
Because nuclear masses are very small, atomic mass units are commonly used for mass, and MeV for energy.
A very useful conversion is
$$
1\ \text{u} \, c^2 \approx 931.5\ \text{MeV}
$$
So if the mass defect is known in atomic mass units, the binding energy can be found from
$$
E_B (\text{MeV}) \approx \Delta m (\text{u}) \times 931.5
$$
Useful nuclear conversion:
$$
1\ \text{u} \, c^2 \approx 931.5\ \text{MeV}
$$
Therefore,
$$
E_B (\text{MeV}) \approx \Delta m (\text{u}) \times 931.5
$$
Simple numerical example
Consider a nucleus made of 2 protons and 2 neutrons. Let the total mass of the separate nucleons be larger than the nuclear mass by
$$
\Delta m = 0.030\ \text{u}
$$
Then the binding energy is
$$
E_B = 0.030 \times 931.5\ \text{MeV}
$$
$$
E_B \approx 27.9\ \text{MeV}
$$
This means about $27.9\ \text{MeV}$ of energy would be needed to separate the nucleus completely, and the same amount would have been released when it formed.
Comparison of ideas
| Quantity | Meaning |
|---|---|
| Nuclear mass | Actual mass of the bound nucleus |
| Sum of separate nucleon masses | Mass of all protons and neutrons if free |
| Mass defect $\Delta m$ | Missing mass due to binding |
| Binding energy $E_B$ | Energy equivalent of that missing mass |
Energy picture
A bound nucleus has lower energy than the same nucleons when they are far apart. This lower energy state is why the nucleus can exist as a stable object. To take the nucleons apart, energy must be supplied from outside.
This idea is similar to an object sitting in a valley. To remove it from the valley, energy must be added.
Binding energy and nuclear reactions
Binding energy is central to nuclear reactions. If the products of a reaction have a greater total binding energy than the initial nuclei, energy can be released. If the products have less total binding energy, energy must be supplied.
This is why both fusion of light nuclei and fission of very heavy nuclei can release energy. The deeper comparison across nuclei belongs to the topic of binding energy per nucleon and the binding energy curve.
Common point of confusion
Students often confuse binding energy with the force holding the nucleus together. They are related, but not the same. Force describes interaction at a given situation. Binding energy describes the total energy needed to break the nucleus apart completely.
Another common confusion is with chemical binding energy. Nuclear binding energies are much larger than chemical energies.
Binding energy is not just any energy inside the nucleus.
It is specifically the energy required to separate the nucleus into free protons and neutrons.
Final summary
Binding energy measures how strongly a nucleus is held together. It comes from the mass defect through Einstein's relation,
$$
E_B = \Delta m c^2
$$
A nucleus with greater binding energy is more tightly bound. The idea of comparing nuclei using binding energy per nucleon is the natural next step.