R.R. Whitehead

Computational Methods for Shell-Model Calculations

The nuclear shell model has a very long history, during most of which as much effort has gone into the development of the formal and technical aspects of the model as into applying it to the description of nuclear properties. This has been partly due to the formidable difficulties attendant on what would seem at the start to be an idiotic pursuit — the description of a complicated many-body system by means of a model lacking any theoretical justification and using a Hamiltonian about which nothing was known. It was also due in part to the utterly fascinating and beautiful mathematical structures that were devised in order to make the calculations possible at all. The earliest shell-model calculations, those of Feenberg and Wigner (FW 37), used methods based on group theory (supermultiplet theory). Racah (Rac 43, 49) further developed the algebra of angular momentum and the method of fractional parentage to the point where they became usable in atomic shell-model calculations. This work was taken up and extended to the case of protons and neutrons by H. A. Jahn and his collaborators (Jah 50, 51; JvW 51) and by Flowers (Flo 52). The problem tackled by them was that of ascribing sufficient approximate quantum numbers or labels to distinguish the different states arising from the filling of a shell and to produce tables of the corresponding coefficients of fractional parentage for use in calculating matrix elements.

A shell-model investigation of the binding energies of some exotic isotopes of sodium and magnesium

Standard shell-model calculations of the binding energies of the neutron-rich isotopes of sodium and magnesium are in strong disagreement with the experimental values near N=20. The authors show that the discrepancy can be explained by allowing neutron excitations from the d3/2 shell into the f7/2 shell.

On the distribution of shell-model eigenvector components

Abstract We present results that show that the components of shell-model eigenvectors are not distributed like those of a randomly oriented vector. An argument based on the invariance properties of the ensemble of random matrices arising from two-body hamiltonians is used to suggest the correct form of the distribution. The agreement with distributions obtained in actual shell-model calculations is found to be excellent.

High-spin rotational states in 24Mg

Abstract The states of T = 0, J = 8, 9, 10, 11 and 12 in 24 Mg, calculated using the Preedom-Wildenthal interaction, have been studied with a view to assigning some of them to rotational bands. With the possible exception of a J = 9 member of the K = 2 excited band, there is no evidence that band structure persists beyond J = 8. The lowest J = 8 state has a very peculiar structure corresponding to a condensation of particles into the d 5 2 shell.

On the Lanczos method and the method of moments

The authors analyse some aspects of the numerically well-behaved Lanczos method of tri-diagonalising a matrix and the mathematically equivalent but numerically disastrous method of moments. It is shown how the elements of the Lanczos tri-diagonal matrix depend only on certain identifiable contributions to the moments, these contributions being those that do not duplicate any information about the matrix already contained in lower moments, all other contributions cancelling exactly.

Shell-model calculations in the sd shell. V. The structure of mass-23 nuclei

For pt.IV see ibid., vol.I, p.213, 1975. Untruncated shell-model calculations have been performed for even-parity low lying levels in 23Na, 23Mg and 23Ne, using the two-body interactions of Kuo and Preedom and Wildenthal. Excitation energies, transition rates and static moments have been calculated. When allowance is made for shifted bands in all spectra good agreement is achieved with both interactions for a wide range of data. The structure of the three nuclei is discussed in the light of these calculations.

Crossing of single-particle energy levels resulting from neutron excess in the sd shell

It has been shown that the large discrepancies between calculation and experiment in the binding energies of neutron-rich isotopes of sodium and magnesium can be removed by extending the model space to include the f7/2 shell. It is shown that the physical explanation of this result lies in the severe distortion of the underlying single-particle spectrum by the large neutron excess.

Shell-Model and protected Hartree-Fock calculations for 24Mg, 28Si and 32S

Abstract Shell-model and projected Hartree-Fock calculations of the spectra of 24 Si and 3 S are compared. The shell-model calculations were done in the untruncated (sd) n basis. The results show an increasing discrepancy between the two models from 24 Mg to 32 S. In particular odd- J states in 28 Si found in the shell-model calculation are absent in the Hartree-Fock spectrum.

Exact shell-model and projected Hartree-Fock calculations for 24Mg

Abstract The results of a projected Hartree-Fock calculation are compared with those of an exact shell-model calculation for states of J ⩾ 6 in the nucleus 24 Mg. The realistic Kuo interaction is used, and gives reasonable agreement with experiment.

Shell-model calculations in the sd shell. IX. Masses, spectra and beta decays of the mass-24 nuclei

For pt.VIII see ibid., vol.3, no.7, p.919 (1977). The authors present the results of a shell-model investigation of the masses, spectra, Coulomb energies and beta decays of the mass-24 nuclei. They have used the effective interaction of Chung and Wildenthal together with an empirically-determined Coulomb interaction. The ab initio use of a reasonably good Coulomb interaction reveals the presence of quite large errors in the calculated energies of certain states and hence of serious imperfections in the nuclear effective interaction. The calculated Coulomb energy differences in the mass-24 system have been used to improve the empirical Coulomb matrix elements.