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A = 16 Theoretical
Because of the very large body of theoretical work that has been carried out for the A = 16, and the importance of the spherical shell model in this work, a general discussion of the shell model description of A = 16 nuclei is provided here. a
a We are very grateful to Dr. John Millener for providing these comments on the shell model for the A = 16 system.
The spherical shell-model provides a complete basis for the description of nuclear states. It is convenient to use harmonic oscillator single-particle wave functions since the coordinate transformations necessary to separate spurious center of mass states, to relate shell-model to cluster-model wave functions and so on can be made exactly. Configurations are classified by the number of oscillator quanta that they carry beyond the minimum allowed by the Pauli Principle as 0ℏω, 1ℏω, 2ℏω,... excitations. Non-spurious states of A = 16 in general involve admixtures of npnh configurations but the lowest excitations of each isospin can, with the exception of the Kπ = 0- band with the 16O 9.58 MeV 1- state as bandhead, be thought of as dominantly p-n(sd)n excitations. In fact, the lowest eigenstates of an nℏω calculation can usually be written economically in terms of product states of low-lying p-n and (sd)n eigenstates. In the simplest version of this weak-coupling model, one identifies the p-n and (sd)n eigenstates with the physical states of the relevant nuclei and takes the diagonal expectation value of Hp + Hsd from the known masses. The contribution from the cross-shell, or particle-hole, interaction can often be quite reliably estimated by using ph matrix elements extracted from the nominal 1p1h states of 16O or 16N.
The 2p2h states with T = 0 and T = 1 cannot, in general, be
described in terms of the simple weak-coupling model, although there
are examples to which such a description can be applied. Shell-model
calculations which use empirical interactions fitted to data on
1ℏω excitations in the mass region do, however, produce
2p2h T = 1 states in one-to-one correspondence with the lowest
positive-parity states of 16N (see 
16.5 [Table of Energy Levels for 16N] (in PDF or PS)). They also
produce T = 0 2p2h states starting at around 12 MeV in 16O.
In this case, the 2p2h states are interleaved with 4p4h states
which begin at lower energies. The lowest 2p2h T = 0 states can be
related in energy to the 14.82 MeV 6+ state which is strongly
populated by the addition of a stretched d5/22 pair in the
14N(α,
d)16O reaction. The lowest six 2p2h
T = 2 states can be very well described in this way.
Weak-coupling ideas can be extended to the lowest 3p3h and 4p4h states. Since the 3 and 4 particle (or hole) configurations are strongly configuration mixed in the jj-coupling scheme, the ph interaction is usually represented in the simple monopole form Eph = a + btp·th plus a small attractive Coulomb contribution. The ph interaction then gives a repulsive contribution of 9a and 16a → 3p3h and 4p4h configurations and separates the T = 0 and T = 1 3p3h states by b MeV. The empirical values of a and b are a ≈ 0.4 MeV and b ≈ 5 MeV, which put the 4p4h 0+ state and the 3p3h 1- states close to experimental candidates at 6.05, 12.44 and 17.28 MeV respectively, each of which is the lowest member of a band.
The weak-coupling states can be used as a basis for shell-model
calculations, but the elimination of spurious center-of-mass motion
is approximate even within an oscillator framework; orbits outside
the p(sd) space are needed and can be important components of
states of physical interest. If complete nℏω spaces are
used, the choice of basis can be one of computational convenience. A
more physical LS-coupled basis is obtained by classifying the states
according to the Wigner supermultiplet scheme
(SU4 
SU2 × SU2 symmetry [≈ f] in spin-isospin
space) and the SU3 symmetry (λ, μ) of
the harmonic
oscillator. States with the highest spatial symmetry [f] maximize
the number of spatially symmetric interacting pairs to take
advantage of the fact that the NN interaction is most strongly
attractive in the relative 0s state and weak or repulsive in
relative p states. These symmetries are broken mainly by the
one-body spin-orbit interaction. In np and nh calculations the
lowest states are dominated by the [f], (λ, μ)
configurations [n],(2n, 0) and [42 4-n], (0, n) respectively
(these symmetries are very good if the one-body spin-orbit
interaction is turned off). In npnh calculations, the lowest
states are dominated by the highest spatial symmetry allowed for
given isospin T and (2n, n) SU3 symmetry. These states are
identical to harmonic oscillator cluster-model states with 2n
quanta on the relative motion coordinate between the nh core and
the np cluster. States with a large parentage to the ground state
of the core should be seen strongly in the appropriate transfer
reaction.
In the above, a basic nℏω (mainly npnh) shell-model structure has been matched, through characteristic level properties and band structures, with experimental candidates. The mixing between shell-model configurations of different nℏω is of several distinct types.
First, there is direct mixing between low-lying states with different npnh structure; the p2 to (sd)2 mixing matrix elements (SU3 tensor character mainly (4, 2)) are not large (up to a few MeV) although the mixing can be large in cases of near degeneracy.
A second type of mixing is more easily understood by reference to cluster models in which an oscillator basis is used to expand the relative motion wave function. To get a realistic representation of the relative motion wave function for a loosely-bound state or an unbound resonance requires many oscillators up to high nℏω excitation. A related problem, which also involves the radial structure of the nucleus, occurs for the expansion of deformed states (of which cluster states are an example) in a spherical oscillator (shell-model) basis; e.g., deformed Hartree-Fock orbits may require an expansion in terms of many oscillator shells. It is difficult to accomodate this type of radial mixing in conventional shell-model calculations, but sympletic Sp(6, R) shell-models, in which the SU3 algebra is extended to include 1p1h 2ℏω monopole and quadrupole excitations, do include such mixing up to high nℏω.
A third type of mixing involves the coupling of npnh excitations to high-lying (n + 2)ℏω configurations via the strong (λ, μ) = (2, 0) component of the p2 → (sd)2 interaction. In the full (0 + 2 + 4)ℏω calculations, the large (30 - 45%) 2p2h admixtures in the ground state are mainly of the (2, 0) type, which are intimately related to the ground-state correlations of RPA theory, and lead to the enhancement (quenching) of ΔT = 0, ΔS = 0 (otherwise) exitations at low momentum transfer.
For most detailed structure questions, a shell-model calculation is required to include the relevant degrees of freedom. For example, (1990HA35) address two important problems with complete (0 + 2 + 4)ℏω and (1 + 3)ℏω model spaces. One is the rank-zero 16N(0-) → 16O(gs) β decay and the inverse μ capture which receive large two-body meson-exchange current contributions. The other is the distribution of M1 and Gamow-Teller strength based on the 16O ground state; this is a complicated problem which involves 2p2h,... admixtures in the ground state which break SU4 symmetry.
Many interesting structure problems remain. A detailed understanding of the shapes and magnitudes of inelastic form factors is lacking, particularly the shapes at momentum transfers beyond 2 fm-1. Even in the relatively simple case of M4 excitations, much studied via (e, e'), (p, p') and (π, π') reactions, a rather low value of the oscillator parameter b is required to describe the form factor. Also, the configuration mixing which splits the 4-; T = 0 strength into two major components and causes isospin mixing has not been satisfactorily described by a shell-model calculation. Similar interesting problems occur for isospin-mixed negative-parity states near 13 MeV excitation energy. It is worth noting that, to avoid some serious consistency problems, the large shell-model calculations have omitted orbits outside the p(sd) space except in as much to cleanly separate spurious center-of-mass states. A consistent treatment of 1p1h and 2p2h correlations in multi-ℏω shell-model spaces remains a challenging question.