The nuclear force
What holds a nucleus together?
A nucleus packs many protons into a space a few femtometres across. Every pair of those protons repels. So why does the nucleus not fly apart?
Let us see how strong that repulsion is. Two protons, 1 fm apart. Coulomb’s law: F equals k e squared over r squared.
9 × 10⁹ times 1.6 × 10⁻¹⁹ squared, over 10⁻¹⁵ squared. That is about 230 newtons, on a single proton! Like the weight of a 23 kilogram bag, acting on a particle of 10⁻²⁷ kilogram.
Could gravity hold them? The gravitational pull between the two protons is G m squared over r squared: only about 1.9 × 10⁻³⁴ newton. About 10³⁶ times weaker than the repulsion. Gravity is hopeless.
And the binding energies we found are about 8 MeV per nucleon, far bigger than the few electron volts that bind electrons in atoms.
So there must be a new, strongly attractive force, of a totally different kind. It must overcome the Coulomb repulsion and bind both protons and neutrons into the tiny nuclear volume.
This is the nuclear force. It is much stronger than the Coulomb force, and far, far stronger than gravity.
It also acts between neutrons, which have no charge at all. Without it, a nucleus with more than one proton could not exist.
A strong, short-range force
Experiments from 1930 to 1950 revealed the main features of the nuclear force. The first we have seen: it is much stronger than the Coulomb force.
Second, it has a short range. Between two nucleons it falls quickly to zero once they are more than a few femtometres apart.
Here is the potential energy of a pair of nucleons against their separation r. Far apart, it is zero. As they come closer it dips down, to a deep minimum at r₀, about 0.8 fm.
Beyond r₀, the curve slopes upward as r increases, so the force pulls the nucleons together: it is attractive.
Closer than r₀, the curve shoots up steeply. Now the force is strongly repulsive: it pushes the nucleons apart. This keeps nucleons from being crushed into one another.
The short range explains the flat binding energy curve. A nucleon deep inside a big nucleus feels only its nearest neighbours. Nucleons farther away have no effect on it. This is saturation.
Third, the nuclear force is the same between neutron and neutron, proton and neutron, and proton and proton. It does not depend on electric charge.
Finally, unlike Coulomb’s law or Newton’s law of gravitation, there is no simple formula for the nuclear force. We describe it by its features and by curves like this one.
Compare it with the Coulomb repulsion between two protons, falling off slowly as 1 over r. Near 1 fm the nuclear force wins easily; a few fm away it has vanished, and only the Coulomb repulsion remains.
Try it: two nucleons
- Drag across the graph (or use the slider) to change the separation r. The arrows show the net force on each nucleon.
- Move r below 0.8 fm: the force turns strongly repulsive.
- For the p–p pair, compare the nuclear and Coulomb forces near 1 fm, and again at 4 fm. Which one wins at each?
- Switch to n–p and n–n: the nuclear curve does not change, but the Coulomb part disappears.
- The nuclear curve is a model with the shape of NCERT Fig. 13.2. There is no simple formula for the real force.
Too close
Two nucleons are 0.5 fm apart. What is the nuclear force between them?
- Strongly attractive, because the potential energy is negative
- Zero, because 0.5 fm is beyond the range of the force
- Strongly repulsive
- Attractive for n–p, repulsive for p–p
Answer
C. The potential energy has its minimum at r₀ ≈ 0.8 fm. At 0.5 fm the nucleons are closer than r₀, where the curve rises steeply as r decreases, so the force is strongly repulsive. The sign of the force depends on the slope of U(r), not on the sign of U. And the nuclear force is the same for n–p and p–p.
Coulomb energy of two protons
What is the Coulomb potential energy of two protons 2 fm apart? (k = 9 × 10⁹ N m² C⁻², e = 1.6 × 10⁻¹⁹ C, 1 MeV = 1.6 × 10⁻¹³ J)
- 2.88 MeV
- 0.72 MeV
- 1.44 MeV
- 1.15 × 10⁻¹³ MeV
Answer
B. U = ke²/r = 9 × 10⁹ × (1.6 × 10⁻¹⁹)² / (2 × 10⁻¹⁵) = 1.15 × 10⁻¹³ J = 0.72 MeV. Multiplying by 2 instead of dividing gives 2.88 MeV; forgetting to divide by 2 gives 1.44 MeV; 1.15 × 10⁻¹³ is the answer in joules, not MeV. Compare this with the nuclear potential energy, which is tens of MeV deep near 1 fm: that is why the nuclear force wins inside a nucleus.
Summary
- Coulomb repulsion of two protons:
F = ke²/r² ≈ 230 N at r = 1 fm - Minimum of nucleon–nucleon U(r):
r₀ ≈ 0.8 fm
- The nuclear force is a strong attraction that binds protons and neutrons; it is much stronger than the Coulomb force and far stronger than gravity.
- It is short-ranged: it falls to zero beyond a few fm. So each nucleon feels only its neighbours (saturation), and E_bn is nearly constant.
- It is attractive for r > r₀ ≈ 0.8 fm and strongly repulsive for r < r₀.
- It is about the same for n–n, n–p and p–p: it does not depend on charge. There is no simple formula for it.