Nuclear shell model

The shell model in nuclear physics is a model of the structure of atomic nuclei. It attributes this to quantum mechanical laws back, mainly due to the angular momentum quantization and the Pauli principle, and explains for example the successful magic numbers.


While the liquid-drop model the nucleus compares with a drop of water, whose behavior can be described mainly with classical mechanics, the shell model considers the individual nucleons and their motion in a potential field according to the rules of quantum mechanics, similar to the shell model for electrons in the atomic shell. Proton and neutron have as the electron spin quantum number 1/2. However, there are important differences from the atomic shell:

  • The nuclear core is comprised of two different types of particles,
  • There is no common potential of the center of force, but the field that acts on a single particle is produced from the remaining particles,
  • Between the nucleons act much stronger forces.

Well suited for describing a Woods - Saxon potential appears. Since this but can only handle numerical, one chooses to analytic treatment, for example, a similarly extending modified potential of a harmonic oscillator. Obtained as solutions of the Schrödinger equation discrete energy levels that can accommodate certain numbers of particles depending on the quantum numbers; they are - referred to as " shells " - based on the description of the atomic shell.

The level of the protons and neutrons are not the same, because the charge of the proton provides by mutual repulsion that the proton levels are somewhat higher than that of the neutron. Most nuclides ( up to about 80 protons), the distances of the levels to one another but for the protons and neutrons are approximately equal, the two -level diagrams are so displaced relative to one another substantially only. You can confirm this in mirror nuclei. This shift corresponds to the liquid-drop model of the Coulomb part.

The magic numbers

The number of identical particles, which can be located on a shell, is limited by the Pauli principle. The 1s shell, for example, is already fully occupied by two nucleons, and the next nucleon must occupy the 1p - shell with a correspondingly higher energy.

When all the protons or neutron shells are either completely filled or empty in a core, this is a particularly stable configuration, comparable to noble gases in the chemical; the special stability is evident in many properties and metrics. Such nuclei are called magic nuclei. The magic numbers observed in naturally occurring nuclides are 2, 8, 20, 28, 50, 82 and 126 nuclei in which protons and neutrons satisfy the condition called doubly magic.

From 28 upwards, these figures differ from the corresponding figures in the atomic shell. The reason is that in the nucleus acting stronger spin -orbit coupling. The energy gap, ie the energetically large distance between allowed states, is formed in a total of 28 particles of a type not full filling a shell, but by the spin -orbit coupling of 1f levels. f - level means as the electrons that the quantum number ( orbital angular momentum quantum number ) l = 3. With the spin quantum number s = ± 0.5 therefore arise as a possible total angular momentum j = l s, the values ​​3.5 and 2.5, where j = 3,5 energetically lower than j = 2,5. If all states are occupied with j = 3,5, we obtain a more stable level of the magic number 28

The spin -orbit coupling principle is similar to the fine structure splitting of the electron levels in the atom. However, in the nucleus enters the strong nuclear force in the place of the electromagnetic interaction; the splitting of the nucleon orbitals is thus both absolutely and relatively much greater than in shell electrons, and the state with j = 3,5 here is energetically more favorable ( lower) than the one with j = 2,5. To put it clearly: electrons "want" to minimize the total angular momentum j, nucleons "want" to maximize it.

Accordingly, the magic numbers 50, 82 and 126 are due to the spin-orbit coupling of the 1g, 1h and 1i orbitals.

Nuclear spin and parity

The shell model also predicts the nuclear spin and the positive or negative parity for the ground state of most nuclides correctly predicted. For example, all even-even nuclei have spin zero and positive parity ground state.


The shell model was postulated in 1949 by Maria Goeppert- Mayer and independently in the same year by J. Hans D. Jensen and employees. Goeppert- Mayer and Jensen received for the 1963 Nobel Prize in Physics. The fact that the shell model, despite the strong nucleon -nucleon force ( see above) is used sensibly, was only 1955 made ​​clear by Keith Brueckner and M. developed the approximate solutions for the many-body problem.