Shell model 50609 220589895 2008-06-20T16:21:58Z Cyfal 4637213 the shell model for electrons is, actually, somewhat more well-known... See also the discussion page. {{otheruses4|the shell model of nuclear physics|shell model for electrons (atomic physics)|electron configuration}} {{Refimprove|date=March 2008}} In [[nuclear physics]], the '''nuclear shell model''' is a [[nuclear model|model of]] the [[atomic nucleus]] which uses the [[Pauli exclusion principle|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 notably [[Eugene Paul Wigner]], [[Maria Goeppert-Mayer]] and [[J. Hans D. Jensen]], who shared the 1963 [[Nobel Prize in Physics]] for their contributions. The shell model is partly analogous to the [[Electron configuration|atomic shell model]] which describes the arrangement of [[electron]]s in an atom, in that a filled shell results in greater stability. When adding [[nucleon]]s ([[proton]]s or [[neutron]]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 number (physics)|magic number]]s 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. <!-- Image with unknown copyright status removed: [[Image:Shellmod.gif]] --> Nevertheless, the [[Magic number (physics)|magic number]]s of nucleons, as well as other properties, can be arrived at by approximating the model with a [[Quantum_harmonic_oscillator#Example:_three-dimensional_harmonic_oscillator|three-dimensional harmonic oscillator]] plus a [[spin-orbit interaction]]. A more realistic but also complicated potential is known as [[Woods Saxon potential]]. ==Deformed harmonic oscillator approximated model== Consider a [[Quantum_harmonic_oscillator#Example:_3D_isotropic_harmonic_oscillator|three-dimensional harmonic oscillator]]. This would give, for example, in the first two levels {| class="wikitable" ! level ''n'' !! ''l'' !! ''m<sub>l</sub>''!! m<sub>s</sub> |- | rowspan=2|0 || rowspan=2| 0 || rowspan=2| 0 || 1/2 |- | -1/2 |- | rowspan=6|1 || rowspan=6|1 || rowspan=2|1 || 1/2 |- | -1/2 |- | rowspan=2|0 || 1/2 |- | -1/2 |- | rowspan=2|-1 || 1/2 |- | -1/2 |} 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 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 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 [[Quantum_harmonic_oscillator#Example:_three-dimensional_harmonic_oscillator|three-dimensional harmonic oscillator]] the total [[degeneracy]] at level n is <math>{(n+1)(n+2)\over 2}</math>. Due to the [[Spin (physics)|spin]], the degeneracy is doubled and is <math>(n+1)(n+2)</math>. :Thus the magic numbers would be :<math>{\sum_{n=0}}^k (n+1)(n+2) = {(k+1)(k+2)(k+3)\over 3}</math> :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 ''m<sub>l</sub>'' and 2 values of ''m<sub>s</sub>'', 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_number#Quantum_numbers_with_spin-orbit_interaction|quantum numbers]] ''j'', ''m<sub>j</sub>'' and [[Parity (physics)|parity]] instead of ''l'', ''m<sub>l</sub>'' and ''m<sub>s</sub>'', as in the [[Hydrogen-like_atom#Including_spin-orbit_interaction|Hydrogen-like atom]]. Since every even level includes only even values of ''l'', it includes only states of even (positive) [[Parity (physics)|parity]]; Similarly every odd level includes only states of odd (negative) [[Parity (physics)|parity]]. Thus we can ignore [[Parity (physics)|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 ''m<sub>j</sub>''. 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 <math>\vec{s}</math> is parallel to <math>\vec{l}</math>, the interaction energy is negative; and in this case ''j'' = ''l'' + ''s'' = ''l'' + 1/2. When <math>\vec{s}</math> is anti-parallel to <math>\vec{l}</math> (i.e. aligned oppositely), the interaction energy is positive, and in this case ''j'' = ''l'' - ''s'' = ''l'' - 1/2. Furthermore, the [[Spin-orbit_interaction#Evaluating|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 negative [[spin-orbit interaction]] energy. * The 8 states with ''j'' = 7/2 came from ''l'' = 4 and ''s'' anti-parallel to ''l''. Thus they have a positive [[spin-orbit interaction]] energy. * The 6 states with ''j'' = 5/2 came from ''l'' = 2 and ''s'' parallel to ''l''. Thus they have a negative [[spin-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 positive [[spin-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 zero [[spin-orbit interaction]] energy. ===Deforming the potential=== The [[harmonic oscillator]] [[potential]] <math>V(r) = \mu \omega^2 r^2 /2</math> grows infinitely as the distance from the center ''r'' goes to infinity. A more realistic potential, such as [[Woods Saxon potential]], would approach a constant at this limit. One main consequence is that the average radius of [[nucleon]]s orbits would be larger in a realistic potential; This leads to a reduced term <math>\hbar^2 l(l+1)/ 2m r^2</math> in the [[Laplacian]] in the [[Hamiltonian]]. 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=== [[Image:Shells.png|thumb|right|Low-lying energy levels in a single-particle shell model with an oscillator potential (with a small negative <math>\boldsymbol{l}^2</math> term) without spin-orbit (left) and with spin-orbin (right) interaction. The number to the right of a level indicates its degeneracy, (''2j+1''). The boxed integers indicate the 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 negative [[spin-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 number (physics)|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 number (physics)|magic numbers]] are then * &nbsp;&nbsp;2 * &nbsp;&nbsp;8 = 2+6 * &nbsp;20 = 2+6+12 * &nbsp;28 = 2+6+12+8 * &nbsp;50 = 2+6+12+8+22 * &nbsp;82 = 2+6+12+8+22+32 * 126 = 2+6+12+8+22+32+44 * 184 = 2+6+12+8+22+32+44+58 and so on. This gives all the observed [[Magic number (physics)|magic numbers]], and also predicts a [[Island of stability|new one]], at the value of 184 (for protons, the [[Magic number (physics)|magic number]] 126 has not been observed yet, and more complicated [[Island of stability|theoretical considerations]] predict the [[Magic number (physics)|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 (physics)|spin]] and [[Parity (physics)|parity]] of nuclei [[ground state]]s, and to some extent their [[excited state]]s as well. Take [[Isotopes of oxygen|<sup>17</sup><sub>8</sub>O<sub>9</sub>]] 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 total [[angular 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 (physics)|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 (physics)|spin]] (i.e. [[angular momentum]]) of the nucleus, as well as its [[Parity (physics)|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 [[Isotopes of oxygen|<sup>17</sup><sub>8</sub>O<sub>9</sub>]] is expected to have positive parity and spin 5/2, which indeed it has. For nuclei farther from the [[Magic number (physics)|magic numbers]] one must add the assumption that due to the relation between the [[strong nuclear force]] and [[angular momentum]], [[proton]]s or [[neutron]]s with the same ''n'' tend to form pairs of opposite [[angular momentum|angular momenta]]. Therefore a nucleus with an even number of protons and an even number of neutrons has 0 [[Spin (physics)|spin]] and positive [[Parity (physics)|parity]]. A nucleus with an even number of protons and an odd number of neutrons (or vice versa) has the [[Parity (physics)|parity]] of the last neutron (or proton), and the [[Spin (physics)|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 (physics)|parity]] of both the last neutron and the last proton. The nucleus parity will be a product of theirs, while the nucleus [[Spin (physics)|spin]] will be one of the possible results of the [[Angular_momentum#Addition_of_quantized_angular_momenta|sum]] of their [[total angular momentum|total angular momenta]] (with other possible results being [[excited 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). For [[nucleon]] pairs, however, it is often energetically favorable 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 the [[strong 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#Magnetic_and_electric_multipoles|deuterium]]. Therefore one gets several possible answers for the [[nuclear magnetic moment]], one for each possible combined ''l'' and ''s'' state, and the real state of the nucleus is a [[superposition]] 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 its [[ground state]] has a definite [[Parity (physics)|parity]], so its matter density (<math>\psi^2</math>, where <math>\psi</math> is the [[wavefunction]]) is always invariant under [[Parity (physics)|parity]]. This is usually the situations with the [[Dipole#Atomic_dipoles|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#Magnetic_and_electric_multipoles|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] [[Category:Nuclear physics]] [[de:Schalenmodell]] [[fr:Modèle en couches]] [[ja:シェルモデル]] [[hu:Héjmodell]] [[pl:Model powłokowy]] [[ru:Теория оболочечного строения ядра]]