Nuclear shell model


In nuclear physics, atomic physics, and nuclear chemistry, the nuclear shell model utilizes the Pauli exclusion principle to model the structure of atomic nuclei in terms of energy levels. The first shell model was proposed by Dmitri Ivanenko 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 received the 1963 Nobel Prize in Physics for their contributions to this model, and Eugene Wigner, who received the Nobel Prize alongside them for his earlier foundational work on atomic nuclei.
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 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 that 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 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.
To get these numbers, the nuclear shell model starts with 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 the 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 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 :
level nℓmℓms
000+
000−
11+1+
11+1−
110+
110−
11−1+
11−1−
200+
200−
22+2+
22+2−
22+1+
22+1−
220+
220−
22−1+
22−1−
22−2+
22−2−

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 a few protons in that shell because they are farthest from the center of the nucleus. Therefore, nuclei with 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 and 8 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 from the Pascal Triangle.
In particular, the first six shells are:
  • level 0: 2 states = 2.
  • level 1: 6 states = 6.
  • level 2: 2 states + 10 states = 12.
  • level 3: 6 states + 14 states = 20.
  • level 4: 2 states + 10 states + 18 states = 30.
  • level 5: 6 states + 14 states + 22 states = 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, ....

Including a spin-orbit interaction

We next include a spin–orbit interaction. First, we have to describe the system by the quantum numbers j, mj and parity instead of ℓ, ml and ms, as in the hydrogen–like atom. Since every even level includes only even values of ℓ, it includes only states of even parity. Similarly, every odd level includes only states of odd parity. Thus we can ignore parity in counting states. The first six shells, described by the new quantum numbers, are
  • level 0 : 2 states. Even parity.
  • level 1 : 2 states + 4 states = 6. Odd parity.
  • level 2 : 2 states + 4 states + 6 states = 12. Even parity.
  • level 3 : 2 states + 4 states + 6 states + 8 states = 20. Odd parity.
  • level 4 : 2 states + 4 states + 6 states + 8 states + 10 states = 30. Even parity.
  • level 5 : 2 states + 4 states + 6 states + 8 states + 10 states + 12 states = 42. Odd parity.
where for every j there are different states from different values of mj.
Due to the spin–orbit interaction, the energies of states of the same level but with different j will no longer be identical. This is because in the original quantum numbers, when is parallel to, the interaction energy is positive, and in this case j = ℓ + s = ℓ +. When is anti-parallel to , the interaction energy is negative, and in this case. Furthermore, the strength of the interaction is roughly proportional to ℓ.
For example, consider the states at level 4:
  • The 10 states with j = come from ℓ = 4 and s parallel to ℓ. Thus they have a positive spin–orbit interaction energy.
  • The 8 states with j = came from ℓ = 4 and s anti-parallel to ℓ. Thus they have a negative spin–orbit interaction energy.
  • The 6 states with j = came from ℓ = 2 and s parallel to ℓ. Thus they have a positive spin–orbit interaction energy. However, its magnitude is half compared to the states with j =.
  • The 4 states with j = came from ℓ = 2 and s anti-parallel to ℓ. Thus they have a negative spin–orbit interaction energy. However, its magnitude is half compared to the states with j =.
  • The 2 states with j = came from ℓ = 0 and thus have zero spin–orbit interaction energy.

    Changing the profile of the potential

The harmonic oscillator potential grows infinitely as the distance from the center r goes to infinity. A more realistic potential, such as the Woods–Saxon potential, would approach a constant at this limit. One main consequence is that the average radius of nucleons' orbits would be larger in a realistic potential. This leads to a reduced term in the Laplace operator of the Hamiltonian operator. Another main difference is that orbits with high average radii, such as those with high n or high ℓ, will have a lower energy than in a harmonic oscillator potential. Both effects lead to a reduction in the energy levels of high ℓ orbits.

Predicted magic numbers

Together with the spin–orbit interaction, and for appropriate magnitudes of both effects, one is led to the following qualitative picture: at all levels, the highest j states have their energies shifted downwards, especially for high n. This is both due to the negative spin–orbit interaction energy and to the reduction in energy resulting from deforming the potential into a more realistic one. The second-to-highest j states, on the contrary, have their energy shifted up by the first effect and down by the second effect, leading to a small overall shift. The shifts in the energy of the highest j states can thus bring the energy of states of one level closer to the energy of states of a lower level. The "shells" of the shell model are then no longer identical to the levels denoted by n, and the magic numbers are changed.
We may then suppose that the highest j states for n = 3 have an intermediate energy between the average energies of n = 2 and n = 3, and suppose that the highest j states for larger n have an energy closer to the average energy of. Then we get the following shells
  • 1st shell: 2 states.
  • 2nd shell: 6 states.
  • 3rd shell: 12 states.
  • 4th shell: 8 states.
  • 5th shell: 22 states.
  • 6th shell: 32 states.
  • 7th shell: 44 states.
  • 8th shell: 58 states.
and so on.
Note that the numbers of states after the 4th shell are doubled triangular numbers. Spin–orbit coupling causes so-called "intruder levels" to drop down from the next higher shell into the structure of the previous shell. The sizes of the intruders are such that the resulting shell sizes are themselves increased to the next higher doubled triangular numbers from those of the harmonic oscillator. For example, 1f2p has 20 nucleons, and spin–orbit coupling adds 1g9/2, leading to a new shell with 30 nucleons. 1g2d3s has 30 nucleons, and adding intruder 1h11/2 yields a new shell size of 42, and so on.
The magic numbers are then
  • 2
and so on. This gives all the observed magic numbers and also predicts a new one at the value of 184.
Another way to predict magic numbers is by laying out the idealized filling order. For consistency, s is split into and components with 2 and 0 members respectively. Taking the leftmost and rightmost total counts within sequences bounded by / here gives the magic and semi-magic numbers.
  • s/p > 2,2/6,8, so magic numbers 2,2/6,8
  • d:''s/f'':p > 14,18:20,20/28,34:38,40, so 14,20/28,40
  • g:''d:s''/h:''f:p'' > 50,58,64,68,70,70/82,92,100,106,110,112, so 50,70/82,112
  • i:''g:d'':s/''j:h'':f:''p'' > 126,138,148,156,162,166,168,168/184,198,210,220,228,234,238,240, so 126,168/184,240
The rightmost predicted magic numbers of each pair within the quartets bisected by / are double tetrahedral numbers from the Pascal Triangle: 2, 8, 20, 40, 70, 112, 168, 240 are 2x 1, 4, 10, 20, 35, 56, 84, 120, ..., and the leftmost members of the pairs differ from the rightmost by double triangular numbers: 2 − 2 = 0, 8 − 6 = 2, 20 − 14 = 6, 40 − 28 = 12, 70 − 50 = 20, 112 − 82 = 30, 168 − 126 = 42, 240 − 184 = 56, where 0, 2, 6, 12, 20, 30, 42, 56, ... are 2 × 0, 1, 3, 6, 10, 15, 21, 28, ... .