The idea of mass defect is one of the clearest ways to understand why nuclei are bound together.
If you imagine taking a nucleus apart into separate protons and neutrons, the total mass of those free nucleons is
slightly larger than the mass of the bound nucleus. That “missing” mass is not lost; it appears as
binding energy, according to Einstein’s relation \(E = mc^2\).
Mass defect
For a nuclide with atomic number \(Z\), mass number \(A\), and neutron number \(N = A-Z\), the mass defect is the
difference between the separated-particle mass and the measured bound mass. There are two common ways to write the
formula, depending on whether the measured input is an atomic mass or a nuclear mass.
If the measured value is an atomic mass:
\[
\begin{aligned}
\Delta m
&= Z\,m_{\mathrm{H}} + N\,m_n - m_{\mathrm{atom}}.
\end{aligned}
\]
If the measured value is a nuclear mass:
\[
\begin{aligned}
\Delta m
&= Z\,m_p + N\,m_n - m_{\mathrm{nucleus}}.
\end{aligned}
\]
The atomic-mass version is often more convenient in practice because many tables list atomic masses rather than
bare nuclear masses. Using hydrogen-atom mass \(m_{\mathrm{H}}\) automatically accounts for the electrons on both
sides in a way that largely cancels them.
Binding energy
Once the mass defect is known, the total binding energy follows immediately:
Binding energy from mass defect.
\[
\begin{aligned}
E_b &= \Delta m\,c^2.
\end{aligned}
\]
In nuclear physics, it is extremely common to use the conversion
\[
\begin{aligned}
1\ \mathrm{u}\,c^2 &\approx 931.494\ \mathrm{MeV},
\end{aligned}
\]
so that the binding energy becomes
\[
\begin{aligned}
E_b &\approx 931.494\,\Delta m\ \mathrm{MeV}.
\end{aligned}
\]
This is why even a very small mass defect in atomic-mass units corresponds to a very large nuclear energy.
Binding energy per nucleon
To compare different nuclei fairly, one usually divides the total binding energy by the mass number:
Binding energy per nucleon.
\[
\begin{aligned}
\frac{E_b}{A} &= \frac{E_b}{A}.
\end{aligned}
\]
This quantity tells you, on average, how strongly each nucleon is bound inside the nucleus. A larger
\(E_b/A\) usually means a more stable nucleus.
Sample calculation: helium-4
A classic example is helium-4, with \(Z=2\), \(A=4\), and measured atomic mass approximately
\(4.00260\ \mathrm{u}\). Then the neutron number is
\[
\begin{aligned}
N &= A - Z = 4 - 2 = 2.
\end{aligned}
\]
Using the atomic-mass form of the defect,
\[
\begin{aligned}
\Delta m
&= 2\,m_{\mathrm{H}} + 2\,m_n - 4.00260.
\end{aligned}
\]
Numerically, this gives a mass defect of about
\[
\begin{aligned}
\Delta m &\approx 0.0304\ \mathrm{u}.
\end{aligned}
\]
Multiplying by \(931.494\ \mathrm{MeV/u}\) gives
\[
\begin{aligned}
E_b &\approx 28.3\ \mathrm{MeV}.
\end{aligned}
\]
Dividing by \(A=4\) gives the binding energy per nucleon:
\[
\begin{aligned}
\frac{E_b}{A} &\approx 7.07\ \mathrm{MeV/nucleon}.
\end{aligned}
\]
This is a very useful benchmark value and matches the scale expected for a tightly bound light nucleus.
Why the binding-energy-per-nucleon curve matters
If you plot \(E_b/A\) versus mass number \(A\), you get one of the most important curves in nuclear physics.
The curve rises rapidly for light nuclei, reaches a broad maximum around the iron and nickel region, and then
slowly decreases for very heavy nuclei. This pattern explains two key facts:
| Region |
Trend in \(E_b/A\) |
Physical implication |
| Light nuclei |
Increasing |
Fusion can release energy by moving toward more tightly bound nuclei |
| Middle-mass nuclei |
Near maximum |
These are among the most stable nuclei |
| Very heavy nuclei |
Gradually decreasing |
Fission can release energy by splitting into more stable fragments |
That is why both nuclear fusion and nuclear fission can produce energy, even though they move nuclei in opposite
directions on the chart.
Physical meaning of the mass defect
A positive mass defect means the bound nucleus has lower mass than its separated nucleons. This lower mass corresponds
to a lower total energy state, which is exactly what we mean by the nucleus being bound. To pull the nucleus apart,
you must put that binding energy back in.
If an input produces a negative mass defect, it usually means one of three things: the wrong mass type was entered,
the measured mass value is inconsistent, or the data belong to a different nuclide than the one specified by \(Z\) and
\(A\).
Advanced note
For more advanced nuclear-physics work, one often compares the measured binding energy with the
semi-empirical mass formula, which models contributions from volume, surface, Coulomb, asymmetry,
and pairing terms. That approach helps explain the shape of the binding-energy curve in a more systematic way.
Still, the direct mass-defect method remains the most transparent first calculation and the most fundamental physical
definition of nuclear binding energy.
| Concept |
Main relation |
Meaning |
| Neutron number |
\(N = A-Z\) |
Number of neutrons in the nucleus |
| Mass defect |
\(\Delta m = m_{\mathrm{free}} - m_{\mathrm{bound}}\) |
Missing mass due to nuclear binding |
| Atomic-mass form |
\(\Delta m = Zm_{\mathrm{H}} + Nm_n - m_{\mathrm{atom}}\) |
Convenient formula when atomic masses are tabulated |
| Nuclear-mass form |
\(\Delta m = Zm_p + Nm_n - m_{\mathrm{nucleus}}\) |
Uses bare proton and neutron masses |
| Binding energy |
\(E_b = \Delta m c^2\) |
Total energy required to separate the nucleus completely |
| Practical conversion |
\(E_b \approx 931.494\,\Delta m\ \mathrm{MeV}\) |
Standard nuclear-physics unit conversion |
| Binding energy per nucleon |
\(E_b/A\) |
Average binding strength per nucleon |