| Nuclear physics |
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High-energy processes |
Innuclear physics,atomic physics, andnuclear chemistry, thenuclear shell model utilizes thePauli exclusion principle to model the structure ofatomic nuclei in terms of energy levels.[1] The first shell model was proposed byDmitri Ivanenko (together with E. Gapon) in 1932. The model was developed in 1949 following independent work by several physicists, most notablyMaria Goeppert Mayer andJ. Hans D. Jensen, who received the 1963Nobel Prize in Physics for their contributions to this model, andEugene Wigner, who received the Nobel Prize alongside them for his earlier groundlaying work on the atomic nuclei.[2]
The nuclear shell model is partly analogous to theatomic shell model, which describes the arrangement ofelectrons in an atom, in that a filled shell results in better stability. When addingnucleons (protons andneutrons) to a nucleus, there are certain points where thebinding energy of the next nucleon is significantly less than the last one. This observation that there are specificmagic quantum numbers of nucleons (2, 8, 20, 28, 50, 82, and 126) 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-calledisland of stability. Some semi-magic numbers have been found, notablyZ = 40, which gives the nuclear shell filling for the various elements; 16 may also be a magic number.[3]
To get these numbers, the nuclear shell model starts with an average potential with a shape somewhere between thesquare well and theharmonic 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 athree-dimensional harmonic oscillator plus aspin–orbit interaction. A more realistic but complicated potential is known as theWoods–Saxon potential.
Consider athree-dimensional harmonic oscillator. This would give, for example, in the first three levels ("ℓ" is theangular momentum quantum number):
| leveln | ℓ | mℓ | ms |
|---|---|---|---|
| 0 | 0 | 0 | +1/2 |
| −1/2 | |||
| 1 | 1 | +1 | +1/2 |
| −1/2 | |||
| 0 | +1/2 | ||
| −1/2 | |||
| −1 | +1/2 | ||
| −1/2 | |||
| 2 | 0 | 0 | +1/2 |
| −1/2 | |||
| 2 | +2 | +1/2 | |
| −1/2 | |||
| +1 | +1/2 | ||
| −1/2 | |||
| 0 | +1/2 | ||
| −1/2 | |||
| −1 | +1/2 | ||
| −1/2 | |||
| −2 | +1/2 | ||
| −1/2 |
Nuclei are built by addingprotons andneutrons. 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 theperiodic 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 nuclearbinding 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 particular, the first six shells are:
where for everyℓ there are 2ℓ+1 different values ofml and 2 values ofms, giving a total of 4ℓ+2 states for every specific level.
These numbers are twice the values oftriangular numbers from the Pascal Triangle: 1, 3, 6, 10, 15, 21, ....
We next include aspin–orbit interaction. First, we have to describe the system by thequantum numbersj,mj andparity instead ofℓ,ml andms, as in thehydrogen–like atom. Since every even level includes only even values ofℓ, 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
where for everyj there are2j+1 different states from different values ofmj.
Due to the spin–orbit interaction, the energies of states of the same level but with differentj 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 casej =ℓ +s =ℓ +1/2. When is anti-parallel to (i.e. aligned oppositely), the interaction energy is negative, and in this casej=ℓ−s=ℓ−1/2. Furthermore, the strength of the interaction is roughly proportional toℓ.
For example, consider the states at level 4:
Theharmonic oscillator potential grows infinitely as the distance from the centerr goes to infinity. A more realistic potential, such as theWoods–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 theLaplace operator of theHamiltonian operator. Another main difference is that orbits with high average radii, such as those with highn or highℓ, will have a lower energy than in a harmonic oscillator potential. Both effects lead to a reduction in the energy levels of highℓ orbits.
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 highestj states have their energies shifted downwards, especially for highn (where the highestj is high). 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-highestj 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 highestj 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 byn, and the magic numbers are changed.
We may then suppose that the highestj states forn = 3 have an intermediate energy between the average energies ofn = 2 andn = 3, and suppose that the highestj states for largern (at least up ton = 7) have an energy closer to the average energy ofn−1. Then we get the following shells (see the figure)
and so on.
Note that the numbers of states after the 4th shell are doubled triangular numbersplus two. 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 (10 nucleons), leading to a new shell with 30 nucleons. 1g2d3s has 30 nucleons, and adding intruder 1h11/2 (12 nucleons) yields a new shell size of 42, and so on.
The magic numbers are then
and so on. This gives all the observed magic numbers and also predicts a new one (the so-calledisland of stability) 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).
Another way to predict magic (and semi-magic) numbers is by laying out the idealized filling order (with spin–orbit splitting but energy levels not overlapping). For consistency, s is split intoj =1/2 andj = −1/2 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.
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, ... .
This model also predicts or explains with some success other properties of nuclei, in particularspin andparity of nucleiground states, and to some extent theirexcited nuclear states as well. Take17
8O (oxygen-17) as an example: Its nucleus has eight protons filling the first three proton "shells", eight neutrons filling the first three neutron "shells", and one extra neutron. All protons in a complete proton shell have zero totalangular momentum, 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 4th shell, which is a d-shell (ℓ = 2), and sincep = (−1)ℓ, this gives the nucleus an overall parity of +1. This 4th d-shell has aj =5/2, thus the nucleus of17
8O is expected to have positive parity and total angular momentum5/2, which indeed it has.
The rules for the ordering of the nucleus shells are similar toHund's Rules of the atomic shells, however, unlike its use in atomic physics, the completion of a shell is not signified by reaching the nextn, as such the shell model cannot accurately predict the order of excited nuclei states, though it is very successful in predicting the ground states. The order of the first few terms are listed as follows: 1s, 1p3/2, 1p1/2, 1d5/2, 2s, 1d3/2... For further clarification on the notation refer to the article on the Russell–Saundersterm symbol.
For nuclei farther from themagic quantum numbers one must add the assumption that due to the relation between thestrong nuclear force and total angular momentum,protons orneutrons with the samen tend to form pairs of opposite angular momentum. 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 the total 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 thesum of their angular momenta (with other possible results being excited states 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 from a harmonic oscillator potential to a more realistic one). For nucleon pairs, however, it is often energetically favourable to be at high angular momentum, even if its energy level for a single nucleon would be higher. This is due to the relation between angular momentum and thestrong nuclear force.
Thenuclear magnetic moment of neutrons and protons is partly predicted by this simple version of the shell model. The magnetic moment is calculated throughj,ℓ ands of the "last" nucleon, but nuclei are not in states of well-definedℓ ands. Furthermore, forodd-odd nuclei, one has to consider the two "last" nucleons, as indeuterium. Therefore, one gets several possible answers for the nuclear magnetic moment, one for each possible combinedℓ ands 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.
Theelectric dipole of a nucleus is always zero, because itsground state has a definite parity. The matter density (ψ2, whereψ is thewavefunction) is always invariant under parity. This is usually the situation with theatomic electric dipole.
Higher electric and magneticmultipole moments cannot be predicted by this simple version of the shell model for reasons similar to those in the case ofdeuterium.
For nuclei having two or more valence nucleons (i.e. nucleons outside a closed shell), a residual two-body interaction must be added. This residual term comes from the part of the inter-nucleon interaction not included in the approximative average potential. Through this inclusion, different shell configurations are mixed, and the energy degeneracy of states corresponding to the same configuration is broken.[5][6]
These residual interactions are incorporated through shell model calculations in a truncated model space (or valence space). This space is spanned by a basis of many-particle states where only single-particle states in the model space are active. The Schrödinger equation is solved on this basis, using an effective Hamiltonian specifically suited for the model space. This Hamiltonian is different from the one of free nucleons as, among other things, it has to compensate for excluded configurations.[6]
One can do away with the average potential approximation entirely by extending the model space to the previously inert core and treating all single-particle states up to the model space truncation as active. This forms the basis of theno-core shell model, which is anab initio method. It is necessary to include athree-body interaction in such calculations to achieve agreement with experiments.[7]
In 1953 the first experimental examples were found of rotational bands in nuclei, with their energy levels following the same J(J+1) pattern of energies as in rotating molecules. Quantum mechanically, it is impossible to have a collective rotation of a sphere, so this implied that the shape of these nuclei was non-spherical. In principle, these rotational states could have been described as coherent superpositions of particle-hole excitations in the basis consisting of single-particle states of the spherical potential. But in reality, the description of these states in this manner is intractable, due to a large number of valence particles—and this intractability was even greater in the 1950s when computing power was extremely rudimentary. For these reasons,Aage Bohr,Ben Mottelson, andSven Gösta Nilsson constructed models in which the potential was deformed into an ellipsoidal shape. The first successful model of this type is now known as theNilsson model. It is essentially the harmonic oscillator model described in this article, but with anisotropy added, so the oscillator frequencies along the three Cartesian axes are not all the same. Typically the shape is a prolate ellipsoid, with the axis of symmetry taken to be z. Because the potential is not spherically symmetric, the single-particle states are not states of good angular momentum J. However, a Lagrange multiplier, known as a "cranking" term, can be added to the Hamiltonian. Usually the angular frequency vector ω is taken to be perpendicular to the symmetry axis, although tilted-axis cranking can also be considered. Filling the single-particle states up to the Fermi level produces states whose expected angular momentum along the cranking axis is the desired value.
Igal Talmi developed a method to obtain the information from experimental data and use it to calculate and predict energies which have not been measured. This method has been successfully used by many nuclear physicists and has led to a deeper understanding of nuclear structure. The theory which gives a good description of these properties was developed. This description turned out to furnish the shell model basis of the elegant and successfulinteracting boson model.
A model derived from the nuclear shell model is the alpha particle model developed byHenry Margenau,Edward Teller, J. K. Pering,T. H. Skyrme, also sometimes called theSkyrme model.[8][9] Note, however, that the Skyrme model is usually taken to be a model of the nucleon itself, as a "cloud" of mesons (pions), rather than as a model of the nucleus as a "cloud" of alpha particles.
