The discovery of the neutron by James Chadwick in 1932 finally made the nature and structure of the atomic nucleus clear. It consisted of a collection of two non-identical particles, proton and neutron, of almost the same mass, with the neutron having no charge, bound together by the strongest known force in nature! When the size of the nucleus, around 10^-13 cm, is compared to the size of the atom itself, about 10^-8 cm, it was realized that typical nucleon kinetic and binding energies are of the order of MeV, compared to eV for atomic electrons. Thus processes in which the strong nuclear force does work liberate a million times the kinetic energy per unit mass, compared to chemical processes in which the electromagnetic force does work.  Because of the extreme density of the nucleus, it was assumed that the nucleons formed a chaotic system of strongly interacting particles, and that a detailed description of nuclear states might be practically impossible. But at the end of WW 2, physicists were able to turn to a detailed study of the excited states of various nuclei, and an astonishing discovery quickly followed. Just as the state of an electron in a complex atom can be understood as an interaction of the single electron with a charge distribution generated by all the other electrons, plus the nucleus, the state of a single nucleon in the nucleus can be understood to a large degree as due to the interaction of that nucleon with a matter and charge distribution created by all the other nucleons... this concept is usually known as the “independent particle model.” The discovery was made independently by Maria Goeppert Mayer and Hans Jensen, and earned the Nobel Prize in 1963. Their first calculations were done with our old pal, the harmonic oscillator potential.

Consider a collision between two nuclei, A and a, which results in a rearrangement of nucleons. The nuclear force can do either positive or negative work during the process, leading to new nuclei B + b. If the total mass of B + b is less than the mass of A + a, the result is that B + b have more kinetic energy than the original A + a, by amount Q, the so-called Q-value. If the Q-value is negative, the process is obviously forbidden by conservation of energy, unless the original kinetic energy of A + a is great enough to make up the difference.
Rutherford and his collaborators observed the first nuclear reaction, ⁴He + ¹⁴N leading to p + ¹⁷O, in 1917.

The independent particle model (nucleons in a central potential generated by all the other nucleons, plus a spin-orbit interaction) gave a beautiful explanation of the observed fact that some specific numbers of protons or neutrons were “magic,” resulting in a nucleus that is unusually stable and tightly bound. In other words, that specific number of nucleons fills a shell, creating an inert core.  Again note how similar this is to the atomic case, where filled electron shells create atoms that don't participate in chemical reactions, the so-called noble gases, like helium and argon!

More generally, it was noticed early on that nuclei with even numbers of both protons and neutrons were unusually stable. If a nucleus has an odd number of either protons or neutrons, it is much less likely to be stable, and if it has an odd number of both protons and neutrons it is quite unlikely to be stable at all. This suggested that protons and neutrons in nuclei form stable S = 0 pairs, and that it takes considerable energy to break a pair.  These pairs behave like bosons, a phenomenon also seen as causing superconductivity in some metals (electrons bond into so-called Cooper pairs)!

It is a property of all unstable quantum states that the probability of a decay of the state, per unit time, is independent of time. This leads to a simple differential equation, dN/dt = −N/τ, where τ is the average lifetime of the state. The solution is by inspection N(t) = N(0)e^−t/τ.  Since essentially the entire population of planet earth is mathematically illiterate, and would never have heard of Napier's number e, the basis of the natural logarithms, it is more common to write the solution in terms of powers of 2, as N(t) = N(0)(1/2)^t/T_1/2, where the half-life T_1/2 = ㏑2 × τ is the time for half a sample of material to undergo the decay.

Heavy nuclei are often unstable to the penetration of tightly bound, paired clusters of 2 protons and 2 neutrons (the nucleus of the He atom!) through the Coulomb barrier that ordinarily helps to keep nucleons inside the region of the strong nuclear potential energy. Because the probability of barrier penetration depends exponentially on the energy of the particle hitting the barrier, lifetimes for this “alpha decay” cover an enormous range from billions of years to very tiny fractions of a second, such as 10⁻⁶ seconds!

The most common type of nuclear decay occurs when a nucleus has too many protons or too many neutrons to be, according to the mass formula, the most tightly bound system with that value of A. An excess neutron is unstable to the weak interaction, one of the four fundamental forces of nature. The weak interaction can change a neutron into a proton, an electron, and an anti-electron-neutrino. Since the neutrino has no charge, and little or no mass, the only particle that is normally observed being emitted is the electron, which before it was recognized was called a β⁻ ray.  However, when the energy is favorable to the process, an excess proton can be changed into a neutron, a positron (anti-electron) and an electron neutrino.  Before the positron was recognized, it was called a β⁺ ray.

What is actually going on in these decays is that either an up quark is changing into a down quark, by emitting a weak boson, or a down quark is changing into an up quark, by emitting a weak boson. The boson then creates a particle-antiparticle pair.

The final type of nuclear decay, gamma or γ decay, occurs because, after alpha or beta decay, the resulting nucleus is almost never in its ground state, but rather in one of its excited states. So the excited state or states decay to the ground state by emission of a photon. Because the spacing of the energy levels is of order MeV, the photon energies are of order MeV, a million times the kinetic energy of a photon of visible light.

Suppose you have a system of identical radioactive nuclei, with half lives of one day. So after one day, only 50% of the original nuclei remain. Suppose after 6 years, you have kept careful track of nuclei which have not yet decayed. What is the probability that one of these nuclei, which has not decayed in 6 years, decays in the next day? Answer: 50%! The probability of decay per unit time is a constant!

Example: find a rock that contains Ar but not K . This tells you the natural abundance of Ar. Now you can date a rock that contains both K and Ar, because the excess of Ar in that rock is due to decay of K, and the amount of  the excess gives you the time since the rock formed.