# How are nuclei stable?

We all know that the density of the nucleus is very high.

Nuclei are made up of protons and neutrons, and while protons have the same charge, they are closely packed in a nucleus. How does the repulsion between protons not break apart the nucleus?

Since the gravitational force between two protons is negligible there must be another force holding the nucleus together. This is the strong nuclear force, which as the name suggests is extremely strong but it is also extremely short range and so it's effects are only felt on the scale of nuclei and baryons. As you can see in the graph, if two protons approach each other they repel each other due to electrostatic repulsion but once they get within about $\mathrm{3~fm}$ of each other the strong nuclear force begins to become significant and at about $\mathrm{2~fm}$ it starts to outweigh the electrostatic repulsion and bind the two protons together.

It is also interesting to note that at even shorter ranges the strong nuclear force is repulsive and so there is an equilibrium position, indicated on the graph, where the strong nuclear attraction and the electrostatic repulsion are equal in magnitude and so the net force on the protons is zero.

Note that the graph shown is for two protons and so in nuclei containing more than two protons the numbers will be different but the principle is the same.

• The thing is that once the 2 protons come within 2 fm it starts to outweigh the electrostatic repulsion.. BUT what is responsible for the 2 protons to come near to each other by over coming electrostatic repulsions??? – vamsi Jan 27 '15 at 13:23
• @vamsi two protons only overcome the electrostatic repulsion barrier in a high energy nuclear fusion process, for example in the sun or another star. Even in the core of a star, this can only happen due to quantum tunneling through the barrier. hyperphysics.phy-astr.gsu.edu/hbase/astro/procyc.html – DavePhD Jan 27 '15 at 14:39
• Then what about bon's answer?? – vamsi Jan 27 '15 at 15:19
• what are you confused about? – bon Jan 27 '15 at 18:09
• @bon two protons overcome electrostatic repulsions only when they are at a distance less than or equal to 2fm.what is the source that is bringing 2 protons closer to each other so that there will be nuclear force between them?How this is happening in all nuclie? – vamsi Jan 28 '15 at 7:59

Protons and neutrons in a nucleus are constantly emitting and absorbing little particles. When one nucleon emits a little particle called a "meson" that another nucleon absorbs, a strong force between the two nucleons results. This is called (strangely enough) the strong nuclear force1, and it is strong enough to counteract the powerful electrostatic repulsion of the protons in a nucleus.

Note: The modern explanation is richer and more complex than this. Since the 70's these "mesons" have been understood to be made up of smaller particles called quarks and gluons, which are also the building blocks of nucleons. In fact the strong force doesn't just bind the nucleus together; it holds the quarks in protons and neutrons together, too. The contemporary explanation of the strong interaction is in the realm of quantum chromodynamics, and really beyond the scope of chemistry. You can read more about it here.

Production and destruction of the messenger mesons violates the law of conservation of mass and energy! However, if the messenger particle has a very short lifetime, and so exists only for a very short time within a very small space, the particle can exist within the limitations set by the uncertainty principle. Particles like this are called virtual particles.

Nuclei are very small (on the order of femtometers in radius), so the range of the strong nuclear force must be very small. You can make a back-of-the-envelope estimate of the range as follows.

The uncertainty principle says that you can’t exactly determine the position and momentum of a very small particle simultaneously:

$$\sigma_x \, \sigma_p\ge \frac{\hbar}{2}$$

where $\sigma_x$ and $\sigma_p$ are the uncertainties in the position and momentum of a particle, and $\hbar$ is the reduced Planck constant. Given that the mass of the meson that mediates the strong force is $m \approx 2.4\times 10^{−28}$ kg, and the uncertainty in the velocity can’t be any larger than the speed of light $c$, you can compute $m c$ as a bound on $\sigma_p$.

You can then estimate $\sigma_x$, which should be related to the maximum distance that a meson can exist from the nucleon that generated it without violating the uncertainty principle, or the universal speed limit:

$$\sigma_x \approx \frac{\hbar}{2 m c} = 0.7\ \text{fm}$$

That’s the right order of magnitude for the range of the strong nuclear force; it's also about the distance between nucleons in the nucleus, and about 6/10ths of the classical radius of a proton.

1. There is a different nuclear force called the weak nuclear force which plays a role in fission and radioactive decay, so calling it just "nuclear force" might be a little ambiguous.

Coulomb's electrostatic repulsion between two protons is extremely large as compared to the gravitational force of attraction between them. Therefore, as you understand correctly, if Coulomb's repulsive and gravitational attractive forces are the only forces operating inside the nucleus, it cannot be stable. The stability of nucleus has been attributed to the existence of a third type of force inside the nucleus called Nuclear force.

Reference: Modern's abc of Physics by Satish K.Gupta, Part 2, Class 12.