KAHIBARO
Table of Contents
The special role of the proton-neutron pair
Inside a nucleus, protons and neutrons are bound together by the strong nuclear force. In this chapter, we focus on one specific part of that picture, the interaction between a proton and a neutron. This is the simplest mixed pair of nucleons, and it shows several important features of nuclear forces very clearly.
A proton and a neutron are both nucleons. They have nearly the same mass, but only the proton carries electric charge. Because the neutron is neutral, a proton-neutron pair avoids the electric repulsion that exists between two protons. This makes the proton-neutron interaction especially important for understanding how nuclei are held together.
Why proton-neutron attraction is important
If nuclei contained only protons, the electric repulsion between them would push them apart. Neutrons help stabilize nuclei because they contribute strong attraction without adding electric repulsion. The attraction between a proton and a neutron is therefore one of the main reasons stable nuclei can exist.
In many nuclei, proton-neutron attraction is stronger, in an effective sense, than proton-proton or neutron-neutron attraction in comparable situations. This does not mean the force is completely different in origin, but it does mean that the proton-neutron pair has a particularly favorable binding behavior.
A proton and a neutron attract each other through the strong nuclear force, and unlike two protons, they do not repel electrically.
The deuteron as the simplest example
The clearest example of proton-neutron interaction is the deuteron, which is the nucleus of deuterium. It contains exactly one proton and one neutron. It is the simplest bound nucleus made of more than one nucleon.
The existence of the deuteron shows that a proton and a neutron can form a bound state. By contrast, two protons alone do not form a stable bound nucleus, and two neutrons alone do not form a stable bound nucleus under ordinary conditions. This makes the proton-neutron system special.
The deuteron is only weakly bound compared with many heavier nuclei, which tells us that the nuclear force is strong enough to bind the pair, but only when the conditions are right.
The deuteron is a bound proton-neutron system. It is the simplest direct evidence that the proton-neutron interaction can produce nuclear binding.
Spin dependence of the interaction
The proton-neutron interaction depends on how the spins of the two nucleons are arranged. Each nucleon has spin $1/2$, so when a proton and a neutron combine, their spins can be aligned or anti-aligned.
There are two basic total-spin possibilities. One is the singlet state with total spin $S = 0$, and the other is the triplet state with total spin $S = 1$. The bound deuteron exists in the triplet state, not in the singlet state. This tells us that the proton-neutron force is more attractive in the triplet configuration.
This is one of the most important lessons from the proton-neutron system. Nuclear forces are not just functions of distance. They also depend on quantum properties such as spin.
| Spin arrangement | Total spin | Bound state? |
|---|---|---|
| Anti-aligned spins | $S = 0$ | No stable bound proton-neutron state |
| Aligned combination | $S = 1$ | Yes, deuteron exists |
The proton-neutron interaction is spin dependent. It is sufficiently attractive to bind the pair in the $S = 1$ state, but not in the $S = 0$ state.
Short range behavior
The proton-neutron force acts only over very short distances, of the order of a few femtometers. At intermediate nuclear distances, it is attractive. At very small separations, it becomes strongly repulsive. This short-range repulsion prevents nucleons from collapsing into each other.
So the interaction has two key parts. There is an attractive region that allows binding, and a repulsive core that keeps nucleons apart at extremely small distances.
A simple sketch of the potential energy as a function of separation helps show this idea.
A useful qualitative picture is that the proton-neutron potential has a strong repulsive core at very small $r$ and an attractive well at slightly larger $r$.
Charge independence and charge symmetry
The strong interaction between nucleons is almost independent of electric charge. This means that, to a good approximation, the strong force between proton-proton, neutron-neutron, and proton-neutron pairs is similar once electromagnetic effects are removed.
However, the proton-neutron system is not exactly the same as the others. The proton has charge and the neutron does not, and there are also small intrinsic differences. Still, nuclear physics often treats protons and neutrons as two states of a more general particle, the nucleon.
This approximate similarity is described using the idea of isospin. A full treatment belongs elsewhere, but here it is enough to know that the proton-neutron interaction reflects an almost charge-independent strong force.
Exchange picture
A common physical picture is that the force between a proton and a neutron arises from the exchange of mesons, especially pions. This does not mean the nucleons are literally touching. Instead, the interaction can be modeled as being mediated by exchanged particles.
This helps explain why the force is short ranged. A heavier exchanged particle leads to a shorter interaction range than a massless one. Since pions have mass, the force they mediate does not extend far outside the nucleus.
A rough relation for the range is
$$
R \sim \frac{\hbar}{m_\pi c}
$$
where $m_\pi$ is the pion mass.
The short range of the proton-neutron interaction can be understood qualitatively through meson exchange, especially pion exchange.
Tensor character of the force
The proton-neutron interaction is not purely central. A central force depends only on distance. The real nuclear force also has a tensor component, which depends on the relative orientation of spins and position.
This tensor part is especially important in the deuteron. It mixes different orbital behaviors and helps explain why the deuteron is not perfectly spherically symmetric. In simple terms, the force depends not only on how far apart the proton and neutron are, but also on how their spins are oriented relative to the line joining them.
For beginners, the key point is that proton-neutron interaction has more structure than an ordinary simple attraction.
What proton-neutron interaction tells us about nuclei
The proton-neutron pair gives direct information about the nature of nuclear binding. From it, we learn that nuclear forces are short ranged, strongly attractive at the right distance, repulsive at very short distance, and dependent on spin.
We also learn why neutrons are so important in nuclei. They increase strong binding without increasing Coulomb repulsion. This is why, especially in medium and heavy nuclei, neutrons are essential for stability.
Summary relations and facts
Some of the most useful facts about proton-neutron interaction can be collected in one place.
| Feature | Proton-neutron system |
|---|---|
| Electric interaction | No Coulomb repulsion between the pair |
| Strong interaction | Attractive at nuclear distances |
| Very short distance behavior | Strong repulsive core |
| Bound state | Yes, the deuteron |
| Spin dependence | Stronger attraction in the triplet state |
| Range | About a few femtometers |
| Theoretical picture | Often modeled by meson exchange |
Key conclusions for the proton-neutron interaction are: it is short ranged, attractive enough to bind a deuteron, spin dependent, and free from proton-proton Coulomb repulsion.