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In
nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies the ...
,
atomic physics Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...
, and
nuclear chemistry Nuclear chemistry is the sub-field of chemistry dealing with radioactivity, nuclear processes, and transformations in the nuclei of atoms, such as nuclear transmutation and nuclear properties. It is the chemistry of radioactive elements such as ...
, the nuclear shell model is a
model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
of the
atomic nucleus The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron i ...
which uses the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
to describe the structure of the nucleus in terms of energy levels. The first shell model was proposed by Dmitri Ivanenko (together with E. Gapon) in 1932. The model was developed in 1949 following independent work by several physicists, most notably
Eugene Paul Wigner Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his con ...
,
Maria Goeppert Mayer Maria Goeppert Mayer (; June 28, 1906 – February 20, 1972) was a German-born American theoretical physicist, and Nobel laureate in Physics for proposing the nuclear shell model of the atomic nucleus. She was the second woman to win a Nobel Pr ...
and
J. Hans D. Jensen Johannes Hans Daniel Jensen (; 25 June 1907 – 11 February 1973) was a German nuclear physicist. During World War II, he worked on the German nuclear energy project, known as the Uranium Club, where he contributed to the separation of uranium is ...
, who shared the 1963
Nobel Prize in Physics ) , image = Nobel Prize.png , alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
for their contributions. The nuclear shell model is partly analogous to the atomic shell model, which describes the arrangement of
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
s in an atom in that filled shell results in better stability. When adding
nucleon In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number). Until the 1960s, nucleons were ...
s (
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
s or
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
s) to a nucleus, there are certain points where the
binding energy In physics and chemistry, binding energy is the smallest amount of energy required to remove a particle from a system of particles or to disassemble a system of particles into individual parts. In the former meaning the term is predominantly use ...
of the next nucleon is significantly less than the last one. This observation that there are specific magic quantum numbers of nucleons (2, 8, 20, 28, 50, 82, 126) which 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, "magic nuclei" exist in which one nucleon type or the other is at a magic number, and " doubly magic quantum 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, playing a role in the search for the so-called island of stability. Some semi-magic numbers have been found, notably ''Z'' =  40 giving nuclear shell filling for the various elements; 16 may also be a magic number. In order to get these numbers, the nuclear shell model starts from an average potential with a shape something between the
square well In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypo ...
and the harmonic oscillator. To this potential, a spin orbit term is added. Even so, the total perturbation does not coincide with 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 nucleons, as well as other properties, can be arrived at by approximating the model with a three-dimensional harmonic oscillator plus a
spin–orbit interaction In quantum physics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–or ...
. A more realistic but also complicated potential is known as
Woods–Saxon potential The Woods–Saxon potential is a mean field potential for the nucleons ( protons and neutrons) inside the atomic nucleus, which is used to describe approximately the forces applied on each nucleon, in the nuclear shell model for the structure of ...
.


Modified harmonic oscillator model

Consider a three-dimensional harmonic oscillator. This would give, for example, in the first three levels ("''ℓ''" is the angular momentum quantum number) Nuclei are built by adding
protons A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mas ...
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 The periodic table, also known as the periodic table of the (chemical) elements, is a rows and columns arrangement of the chemical elements. It is widely used in chemistry, physics, and other sciences, and is generally seen as an icon of ch ...
, 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 nuclear
binding energy In physics and chemistry, binding energy is the smallest amount of energy required to remove a particle from a system of particles or to disassemble a system of particles into individual parts. In the former meaning the term is predominantly use ...
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. 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 three-dimensional harmonic oscillator the total degeneracy of states at level ''n'' is . *Due to the spin, the degeneracy is doubled and is (n+1)(n+2). *Thus the magic numbers would be \sum_^k (n+1)(n+2) = \frac 3 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. These numbers are twice the
tetrahedral number A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The th tetrahedral number, , is the sum of the first triangular numbers, that is, ...
s (1, 4, 10, 20, 35, 56, ...) from the
Pascal Triangle In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ...
. 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. These numbers are twice the values of
triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
s from the Pascal Triangle: 1, 3, 6, 10, 15, 21, ....


Including a spin–orbit interaction

We next include a
spin–orbit interaction In quantum physics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–or ...
. First we have to describe the system by the
quantum numbers In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be k ...
''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'' = ). Even parity. * level 1 (''n'' = 1): 2 states (''j'' = ) + 4 states (''j'' = ) = 6. Odd parity. * level 2 (''n'' = 2): 2 states (''j'' = ) + 4 states (''j'' = ) + 6 states (''j'' = ) = 12. Even parity. * level 3 (''n'' = 3): 2 states (''j'' = ) + 4 states (''j'' = ) + 6 states (''j'' = ) + 8 states (''j'' = ) = 20. Odd parity. * level 4 (''n'' = 4): 2 states (''j'' = ) + 4 states (''j'' = ) + 6 states (''j'' = ) + 8 states (''j'' = ) + 10 states (''j'' = ) = 30. Even parity. * level 5 (''n'' = 5): 2 states (''j'' = ) + 4 states (''j'' = ) + 6 states (''j'' = ) + 8 states (''j'' = ) + 10 states (''j'' = ) + 12 states (''j'' = ) = 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 \scriptstyle \vec is parallel to \scriptstyle \vec, the interaction energy is positive; and in this case ''j'' = ''ℓ'' + ''s'' = ''ℓ'' + . When \scriptstyle \vec is anti-parallel to \scriptstyle \vec (i.e. aligned oppositely), 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 V(r) = \mu \omega^2 r^2 /2 grows infinitely as the distance from the center ''r'' goes to infinity. A more realistic potential, such as
Woods–Saxon potential The Woods–Saxon potential is a mean field potential for the nucleons ( protons and neutrons) inside the atomic nucleus, which is used to describe approximately the forces applied on each nucleon, in the nuclear shell model for the structure of ...
, 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 \scriptstyle \hbar^2 l(l+1)/ 2m r^2 in the Laplace operator of the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified 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'' (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 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 . Then we get the following shells (see the figure) * 1st shell: 2 states (''n'' = 0, ''j'' = ). * 2nd shell: 6 states (''n'' = 1, ''j'' = or ). * 3rd shell: 12 states (''n'' = 2, ''j'' = , or ). * 4th shell: 8 states (''n'' = 3, ''j'' = ). * 5th shell: 22 states (''n'' = 3, ''j'' = , or ; ''n'' = 4, ''j'' = ). * 6th shell: 32 states (''n'' = 4, ''j'' = , , or ; ''n'' = 5, ''j'' = ). * 7th shell: 44 states (''n'' = 5, ''j'' = , , , or ; ''n'' = 6, ''j'' = ). * 8th shell: 58 states (''n'' = 6, ''j'' = , , , , or ; ''n'' = 7, ''j'' = ). 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 very 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 addition of intruder 1h11/2 (12 nucleons) 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 (the so-called '' island 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 into j = 1⁄2 and j = -1⁄2 components with 2 and 0 members respectively. Taking leftmost and rightmost total counts within sequences marked bounded by / here gives the magic and semi-magic numbers. * ''s''(2,0)/p(4,2) > 2,2/6,8, so (semi)magic numbers 2,2/6,8 * ''d''(6,4):''s''(2,0)/''f''(8,6):''p''(4,2) > 14,18:20,20/28,34:38,40, so 14,20/28,40 * ''g''(10,8):''d''(6,4):''s''(2,0)/''h''(12,10):''f''(8,6):''p''(4,2) > 50,58,64,68,70,70/82,92,100,106,110,112, so 50,70/82,112 * ''i''(14,12):''g''(10,8):''d''(6,4):''s''(2,0)/''j''(16,14):''h''(12,10):''f''(8,6):''p''(4,2) > 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, ... .


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 The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
s, and to some extent their excited nuclear states as well. Take (
oxygen-17 Oxygen-17 (17O) is a low-abundance, natural, stable isotope of oxygen (0.0373% in seawater; approximately twice as abundant as deuterium). As the only stable isotope of oxygen possessing a nuclear spin (+5/2) and a favorable characteristic of fi ...
) as an example: Its nucleus has eight protons filling the three first proton "shells", eight neutrons filling the three first neutron "shells", and one extra neutron. All protons in a complete proton shell have zero total
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, 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 since ''p'' = (−1), this gives the nucleus an overall parity of +1. This 4th d-shell has a ''j'' = , thus the nucleus of is expected to have positive parity and total angular momentum , which indeed it has. The rules for the ordering of the nucleus shells are similar to
Hund's Rules In atomic physics, Hund's rules refers to a set of rules that German physicist Friedrich Hund formulated around 1927, which are used to determine the term symbol that corresponds to the ground state of a multi-electron atom. The first rule is e ...
of the atomic shells, however, unlike its use in atomic physics the completion of a shell is not signified by reaching the next ''n'', 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, 1p, 1p, 1d, 2s, 1d... For further clarification on the notation refer to the article on the RussellSaunders
term symbol In quantum mechanics, the term symbol is an abbreviated description of the (total) angular momentum quantum numbers in a multi-electron atom (however, even a single electron can be described by a term symbol). Each energy level of an atom with a giv ...
. For nuclei farther from the magic quantum numbers one must add the assumption that due to the relation between the
strong nuclear force The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
and total angular momentum,
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
s or
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
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 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 the sum 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 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 The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
.
Nuclear magnetic moment The nuclear magnetic moment is the magnetic moment of an atomic nucleus and arises from the spin of the protons and neutrons. It is mainly a magnetic dipole moment; the quadrupole moment does cause some small shifts in the hyperfine structure as ...
of neutron and proton 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" nucleons, as in
deuterium Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two Stable isotope ratio, stable isotopes of hydrogen (the other being Hydrogen atom, protium, or hydrogen-1). The atomic nucleus, nucleus of a deuterium ato ...
. 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 The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb- meter (C⋅m). T ...
of a nucleus is always zero, because its
ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
has a definite parity, so its matter density (''ψ'', where ''ψ'' is the
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
) is always invariant under parity. This is usually the situations with the atomic electric dipole as well. Higher electric and magnetic
multipole moments A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly ...
cannot be predicted by this simple version of the shell model, for the reasons similar to those in the case of
deuterium Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two Stable isotope ratio, stable isotopes of hydrogen (the other being Hydrogen atom, protium, or hydrogen-1). The atomic nucleus, nucleus of a deuterium ato ...
.


Including residual interactions

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. 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 it among other things has to compensate for excluded configurations. 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 the no-core shell model, which is an ab initio method. It is necessary to include a three-body interaction in such calculations to achieve agreement with experiments.


Collective rotation and the deformed potential

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 nonspherical. 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 Aage Niels Bohr (; 19 June 1922 – 8 September 2009) was a Danish nuclear physicist who shared the Nobel Prize in Physics in 1975 with Ben Roy Mottelson and James Rainwater "for the discovery of the connection between collective motion and pa ...
,
Ben Mottelson Ben Roy Mottelson (9 July 1926 – 13 May 2022) was an American-Danish nuclear physicist. He won the 1975 Nobel Prize in Physics for his work on the non-spherical geometry of atomic nuclei. Early life Mottelson was born in Chicago, Illinois on ...
, and Sven Gösta Nilsson constructed models in which the potential was deformed into an ellipsoidal shape. The first successful model of this type is the one now known as the Nilsson model. It is essentially the harmonic oscillator model described in this article, but with anisotropy added, so that 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 -\omega\cdot J, 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 then produces states whose expected angular momentum along the cranking axis \langle J_x\rangle is the desired value.


Related models

Igal Talmi Igal Talmi (Hebrew: יגאל תלמי) (born January 31, 1925) is an Israeli nuclear physicist. Biography Igal Talmi was born in 1925 in Kiev, Ukraine, then part of the Soviet Union. His family immigrated to Mandate Palestine later that year an ...
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 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 successful
interacting boson model The interacting boson model (IBM) is a model in nuclear physics in which nucleons (protons or neutrons) pair up, essentially acting as a single particle with boson properties, with integral spin of 0, 2 or 4. It is sometimes known as the Interac ...
. A model derived from the nuclear shell model is the alpha particle model developed by
Henry Margenau Henry Margenau (April 30, 1901 – February 8, 1997) was a German-American physicist, and philosopher of science. Biography Early life Born in Bielefeld, Germany, Margenau obtained his bachelor's degree from Midland Lutheran College, Nebraska befo ...
,
Edward Teller Edward Teller ( hu, Teller Ede; January 15, 1908 – September 9, 2003) was a Hungarian-American theoretical physicist who is known colloquially as "the father of the hydrogen bomb" (see the Teller–Ulam design), although he did not care fo ...
, J. K. Pering, T. H. Skyrme, also sometimes called the
Skyrme model In particle theory, the skyrmion () is a topologically stable field configuration of a certain class of non-linear sigma models. It was originally proposed as a model of the nucleon by (and named after) Tony Skyrme in 1961. As a topological solito ...
. 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.


See also

*
Interacting boson model The interacting boson model (IBM) is a model in nuclear physics in which nucleons (protons or neutrons) pair up, essentially acting as a single particle with boson properties, with integral spin of 0, 2 or 4. It is sometimes known as the Interac ...
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Isomeric shift The isomeric shift (also called isomer shift) is the shift on atomic spectral lines and gamma spectral lines, which occurs as a consequence of replacement of one nuclear isomer by another. It is usually called isomeric shift on atomic spectral line ...
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Liquid drop model In nuclear physics, the semi-empirical mass formula (SEMF) (sometimes also called the Weizsäcker formula, Bethe–Weizsäcker formula, or Bethe–Weizsäcker mass formula to distinguish it from the Bethe–Weizsäcker process) is used to approxi ...
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Nuclear structure Understanding the structure of the atomic nucleus is one of the central challenges in nuclear physics. Models The liquid drop model The liquid drop model is one of the first models of nuclear structure, proposed by Carl Friedrich von Weizsäcke ...


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External links

* {{DEFAULTSORT:Nuclear Shell Model Nuclear physics German inventions