- Shell model
In
nuclear physics , the nuclear shell model is a model of theatomic nucleus which uses the Pauli principle to describe the structure of the nucleus in terms of energy levels. The model was developed in 1949 following independent work by several physicists, most notablyEugene Paul Wigner ,Maria Goeppert-Mayer andJ. Hans D. Jensen , who shared the 1963Nobel Prize in Physics for their contributions.The shell model is partly analogous to the atomic shell model which describes the arrangement of
electron s in an atom, in that a filled shell results in greater stability. When addingnucleon s (proton s orneutron s) 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 certain magic numbers of nucleons: 2, 8, 20, 28, 50, 82, 126 which are more tightly bound than the next higher number, is the origin of the shell model.Note that the shells exist for both protons and neutrons individually, so that we can speak of "magic nuclei" where one nucleon type is at a magic number, and "doubly magic 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. This has a relevant role in the search of the so-called
island of stability . Besides, there have been found some semimagic numbers, noticeably Z=40.In order to get these numbers, the nuclear shell model starts from an average potential with a shape something between the square well and the harmonic oscillator. To this potential a relativistic spin orbit term is added. Even so, the total perturbation does not coincide with experiment, and an empirical spin orbit coupling, named Nilsson Term, must be added with at least two or three different values of its coupling constant, depending on the nuclei being studied.
Nevertheless, the magic numbers of nucleons, as well as other properties, can be arrived at by approximating the model with a plus a
spin-orbit interaction . A more realistic but also complicated potential is known asWoods Saxon potential .Deformed harmonic oscillator approximated model
Consider a . This would give, for example, in the first two levels
We can imagine ourselves building a nucleus by adding protons and neutrons. These will always fill the lowest available level. Thus the first two protons fill level zero, the next six protons fill level one, and so on. As with
electron s in theperiodic 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 binding energy than other nuclei with a similar total number of protons. All this is true for neutrons as well.This means that the magic numbers are expected to be those in which all occupied shells are full. We see that for the first two numbers we get 2 (level 0 full) and 8 (levels 0 and 1 full), in accord with experiment. However the full set of magic numbers does not turn out correctly. These can be computed as follows:
:In a the total
degeneracy at level n is . Due to the spin, the degeneracy is doubled and is .:Thus the magic numbers would be::for 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.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 2"l"+1 different values of "ml" and 2 values of "ms", giving a total of 4"l"+2 states for every specific level.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 (positive) parity; Similarly every odd level includes only states of odd (negative) parity. Thus we can ignore parity in counting states. The first six shells, described by the new quantum numbers, are
* level 0 ("n"=0): 2 states ("j" = 1/2). Even parity.
* level 1 ("n"=1): 4 states ("j" = 3/2) + 2 states ("j" = 1/2) = 6. Odd parity.
* level 2 ("n"=2): 6 states ("j" = 5/2) + 4 states ("j" = 3/2) + 2 states ("j" = 1/2) = 12. Even parity.
* level 3 ("n"=3): 8 states ("j" = 7/2) + 6 states ("j" = 5/2) + 4 states ("j" = 3/2) + 2 states ("j" = 1/2) = 20. Odd parity.
* level 4 ("n"=4): 10 states ("j" = 9/2) + 8 states ("j" = 7/2) + 6 states ("j" = 5/2) + 4 states ("j" = 3/2) + 2 states ("j" = 1/2) = 30. Even parity.
* level 5 ("n"=5): 12 states ("j" = 11/2) + 10 states ("j" = 9/2) + 8 states ("j" = 7/2) + 6 states ("j" = 5/2) + 4 states ("j" = 3/2) + 2 states ("j" = 1/2) = 42. Odd parity.where for every "j" there are 2"j"+1 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 negative; and in this case "j" = "l" + "s" = "l" + 1/2. When is anti-parallel to (i.e. aligned oppositely), the interaction energy is positive, and in this case "j" = "l" - "s" = "l" - 1/2. Furthermore, the strength of the interaction is roughly proportional to "l".For example, consider the states at level 4:
* The 10 states with "j" = 9/2 come from "l" = 4 and "s" parallel to "l". Thus they have a negativespin-orbit interaction energy.
* The 8 states with "j" = 7/2 came from "l" = 4 and "s" anti-parallel to "l". Thus they have a positivespin-orbit interaction energy.
* The 6 states with "j" = 5/2 came from "l" = 2 and "s" parallel to "l". Thus they have a negativespin-orbit interaction energy. However its magnitude is half compared to the states with "j" = 9/2.
* The 4 states with "j" = 3/2 came from "l" = 2 and "s" anti-parallel to "l". Thus they have a positivespin-orbit interaction energy. However its magnitude is half compared to the states with "j" = 7/2.
* The 2 states with "j" = 1/2 came from "l" = 0 and thus have zerospin-orbit interaction energy.Deforming the potential
The
harmonic oscillator potential grows infinitely as the distance from the center "r" goes to infinity. A more realistic potential, such asWoods Saxon potential , would approach a constant at this limit. One main consequence is that the average radius ofnucleon s orbits would be larger in a realistic potential; This leads to a reduced term in theLaplacian in theHamiltonian . 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" (where the highest "j" is high). This is both due to the negativespin-orbit interaction energy and to the reduction in energy resulting from deforming the potential to 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 to be 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" (at least up to "n" = 7) have an energy closer to the average energy of "n"-1. Then we get the following shells (see the figure)
* 1st Shell: 2 states ("n" = 0, "j" = 1/2).
* 2nd Shell: 6 states ("n" = 1, "j" = 1/2 or 3/2).
* 3rd shell: 12 states ("n" = 2, "j" = 1/2, 3/2 or 5/2).
* 4th shell: 8 states ("n" = 3, "j" = 7/2).
* 5th shell: 22 states ("n" = 3, "j" = 1/2, 3/2 or 5/2; "n" = 4, "j" = 9/2).
* 6th shell: 32 states ("n" = 4, "j" = 1/2, 3/2, 5/2 or 7/2; "n" = 5, "j" = 11/2).
* 7th shell: 44 states ("n" = 5, "j" = 1/2, 3/2, 5/2, 7/2 or 9/2; "n" = 6, "j" = 13/2).
* 8th shell: 58 states ("n" = 6, "j" = 1/2, 3/2, 5/2, 7/2, 9/2 or 11/2; "n" = 7, "j" = 15/2).and so on.The magic numbers are then
* 2
* 8 = 2+6
* 20 = 2+6+12
* 28 = 2+6+12+8
* 50 = 2+6+12+8+22
* 82 = 2+6+12+8+22+32
* 126 = 2+6+12+8+22+32+44
* 184 = 2+6+12+8+22+32+44+58and so on. This gives all the observed magic numbers, and also predicts a new one, at the value of 184 (for protons, the magic number 126 has not been observed yet, and more complicated theoretical considerations predict the magic number to be 114 instead).
Other properties of nuclei
This model also predicts or explains with some success other properties of nuclei, in particular spin and parity of nuclei
ground state s, and to some extent theirexcited state s as well. Take 178O9 as an example - its nucleus has eight protons filling the two first proton shells, eight neutrons filling the two first neutron shells, and one extra neutron. All protons in a complete proton shell have totalangular momentum zero, since their angular momenta cancel each other; The same is true for neutrons. All protons in the same level ("n") have the same parity (either +1 or -1), and since the parity of a pair of particles is the product of their parities, an even number of protons from the same level ("n") will have +1 parity. Thus the total angular momentum of the eight protons and the first eight neutrons is zero, and their total parity is +1. This means that the spin (i.e.angular momentum ) of the nucleus, as well as its parity, are fully determined by that of the ninth neutron. This one is in the first (i.e. lowest energy) state of the 3rd shell, and therefore have "n" = 2, giving it +1 parity, and "j" = 5/2. Thus the nucleus of 178O9 is expected to have positive parity and spin 5/2, which indeed it has.For nuclei farther from the magic numbers one must add the assumption that due to the relation between the
strong nuclear force andangular momentum ,proton s orneutron s with the same "n" tend to form pairs of opposite angular momenta. Therefore a nucleus with an even number of protons and an even number of neutrons has 0 spin and positive parity. A nucleus with an even number of protons and an odd number of neutrons (or vice versa) has the parity of the last neutron (or proton), and the spin equal to thetotal angular momentum of this neutron (or proton). By "last" we mean the properties coming from the highest energy level.In the case of a nucleus with an odd number of protons and an odd number of neutrons, one must consider the
total angular momentum and parity of both the last neutron and the last proton. The nucleus parity will be a product of theirs, while the nucleus spin will be one of the possible results of the sum of their total angular momenta (with other possible results beingexcited state s of the nucleus).The ordering of angular momentum levels within each shell is according to the principles described above - due to
spin-orbit interaction , with high angular momentum states having their energies shifted downwards due to the deformation of the potential (i.e. moving form a harmonic oscillator potential to a more realistic one). Fornucleon pairs, however, it is often energetically favorable to be at highangular momentum , even if its energy level for a singlenucleon would be higher. This is due to the relation betweenangular momentum and thestrong nuclear force .Nuclear magnetic moment is partly predicted by this simple version of the shell model. The magnetic moment is calculated through "j", "l" and "s" of the "last" nucleon, but nuclei are not in states of well defined "l" and "s". Furthermore, for odd-odd nuclei, one has to consider the two "last"nucleon s, as in deuterium. Therefore one gets several possible answers for thenuclear magnetic moment , one for each possible combined "l" and "s" state, and the real state of the nucleus is asuperposition of them. Thus the real (measured)nuclear magnetic moment is somewhere in between the possible answers.The
electric dipole of a nucleus is always zero, because itsground state has a definite parity, so its matter density (, where is thewavefunction ) is always invariant under parity. This is usually the situations with the atomic electric dipole as well.Higher electric and magnetic
multipole moments cannot be predicted by this simple version of the shell model, for the reasons similar to those in the case of the deuterium.See also
*
Interacting boson model
*Liquid drop model
*Nuclear structure External links
* [http://dftuz.unizar.es/~rivero/research/#0405076 The Lamb's Balance] , a proposed mechanism to substitute Nilsson terms.
* [http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/shell.html Nuclear Shell Model]
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