In nuclear physics, atomic physics, and nuclear chemistry, the nuclear shell model is a model of the atomic nucleus that uses the Pauli exclusion principle to describe the structure of nuclei in terms of energy levels. The first shell model was proposed by Dmitri Ivanenko (together with E. Gapon) in 1932. The model was developed in 1949 following independent work by several physicists, most notably Maria Goeppert Mayer and J. Hans D. Jensen, who shared half of the 1963 Nobel Prize in Physics for their contributions.
The nuclear shell model is partly analogous to the atomic shell model, which describes the arrangement of electrons in an atom, in that a filled shell results in better stability. When adding nucleons (protons and neutrons) to a nucleus, there are certain points where the binding energy of the next nucleon is significantly less than the last one. This observation, that there are specific magic quantum numbers of nucleons (2, 8, 20, 28, 50, 82, 126) which are more tightly bound than the following higher number, is the origin of the shell model.
The shells for protons and neutrons are independent of each other. Therefore, there can exist both "magic nuclei", in which one nucleon type or the other is at a magic number, and "doubly magic quantum nuclei", where both are. Due to some variations in orbital filling, the upper magic numbers are 126 and, speculatively, 184 for neutrons, but only 114 for protons, playing a role in the search for the so-called island of stability. Some semi-magic numbers have been found, notably Z = 40, which gives the nuclear shell filling for the various elements; 16 may also be a magic number.
In order to get these numbers, the nuclear shell model starts from an average potential with a shape somewhere between the square well and the harmonic oscillator. To this potential, a spin orbit term is added. Even so, the total perturbation does not coincide with experiment, and an empirical spin orbit coupling must be added with at least two or three different values of its coupling constant, depending on the nuclei being studied.
The magic numbers of nuclei, as well as other properties, can be arrived at by approximating the model with a three-dimensional harmonic oscillator plus a spin–orbit interaction. A more realistic but also complicated potential is known as the Woods–Saxon potential.
Modified harmonic oscillator model
Consider a three-dimensional harmonic oscillator. This would give, for example, in the first three levels ("ℓ" is the angular momentum quantum number):
Nuclei are built by adding protons and neutrons. These will always fill the lowest available level, with the first two protons filling level zero, the next six protons filling level one, and so on. As with electrons in the periodic table, protons in the outermost shell will be relatively loosely bound to the nucleus if there are only few protons in that shell, because they are farthest from the center of the nucleus. Therefore, nuclei which have a full outer proton shell will have a higher nuclear binding energy than other nuclei with a similar total number of protons. The same is true for neutrons.
This means that the magic numbers are expected to be those in which all occupied shells are full. In accordance with the experiment, we get 2 (level 0 full) and 8 (levels 0 and 1 full) for the first two numbers. However, the full set of magic numbers does not turn out correctly. These can be computed as follows:
In a three-dimensional harmonic oscillator the total degeneracy of states at level n is . Due to the spin, the degeneracy is doubled and is . Thus, the magic numbers would befor all integer k. This gives the following magic numbers: 2, 8, 20, 40, 70, 112, ..., which agree with experiment only in the first three entries. These numbers are twice the tetrahedral numbers (1, 4, 10, 20, 35, 56, ...) from the Pascal Triangle.
In particular, the first six shells are: level 0: 2 states (ℓ = 0) = 2. level 1: 6 states (ℓ = 1) = 6. level 2: 2 states (ℓ = 0) + 10 states (ℓ = 2) = 12. level 3: 6 states (ℓ = 1) + 14 states (ℓ = 3) = 20. level 4: 2 states (ℓ = 0) + 10 states (ℓ = 2) + 18 states (ℓ = 4) = 30. level 5: 6 states (ℓ = 1) + 14 states (ℓ = 3) + 22 states (ℓ = 5) = 42. where for every ℓ there are 2ℓ+1 different values of ml and 2 values of ms, giving a total of 4ℓ+2 states for every specific level.
These numbers are twice the values of triangular numbers from the Pascal Triangle: 1, 3, 6, 10, 15, 21, ....