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 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.

Characteristic spectrum of collective quantum oscillation of the nucleus as a whole

Characteristic spectrum of collective quantum rotation of the nucleus as a whole

All even-even nuclei have 0⁺ ground states. The excitations invariably follow a simple pattern which suggests a quantum vibration of the entire (spherical) nuclear shape, in other words a collective excitation of the entire nucleus... except for some heavy nuclei, where the excitation pattern is unmistakably rotational. These nuclei have a permanent equilibrium deformation, and the basic excitations are quantum rigid-rotor excitations. These two different types of nuclear states are obviously best understood as modes in which the entire nucleus as a whole participates... collective excitations. But attention quickly focused on the odd-A nuclei. These nuclei exhibited a simple single-particle spectrum, with a strong spin-orbit splitting. Furthermore, nuclei with Z or N equal to 2, 8, 20, 28, 50, 82 or 126 turned out to be extremely stable, that is, extremely deeply bound compared to neighboring nuclei. It was worth a Nobel prize to realize that the excitations of odd-A nuclei can be understood by picturing a single nucleon in a simple potential generated by all the other nucleons! This “independent particle model,” or Shell Model, quickly became one of the most useful of all tools in nuclear physics.

The effect of the spin-orbit term is to make the state with j = ℓ + (1/2) more tightly bound than the state with j = ℓ - (1/2).  The spin-orbit contribution is a surface term (derivative of the central potential) and is attractive.  One also needs to add a simple Coulomb potential if the individual particle is a proton. Solving the Schrödinger equation results in astonishing agreement with experiment.

Coupling particle pairs.

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RESIDUAL INTERACTIONS

When nuclear physicists talk about the shell model, they are not referring to the independent particle model, but rather that model augmented to include residual interactions. These interactions include particle-particle and particle-hole interactions, providing an extended Hamiltonian that has to be diagonalized in the IMP basis. In early calculations only a few particles and holes were allowed to interact. It is computationally impossible to allow all particles to interact, so even today it is necessary to choose carefully which particles and holes need to interact in order to understand the observed nuclear excitations.

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A very different approach, pioneered by theorists at Yale and at UT... by bosonizing the nuclear Hamiltonian, which is a very difficult mathematical transformation, but only has to be done once, the happy result is a Hamiltonian it is very simple to calculate with. The basic idea is to use nucleon pairs as a basis.

Tamura

Arima

A tale of two physicists, Taro Tamura (1923 - 1988) and Akito Arima (1930 - 2020). For reasons unknown to me they were always great rivals, and both got interested in a boson approach to nuclear structure calculations at about the same time. Tamura, characteristically, tried to do the most realistic possible calculations, even though the algebra was massively difficult and the computations even more so. Arima's approach was surprisingly simple, relying heavily on group theory and using many adjustable parameters. The competition might have continued indefinitely except that a heart attack killed Tamura at the age of 65. After that, the simplicity of (and heavy publicity in favor of) Arima's approach resulted in it being the only one remembered today.

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