Nuclear Shell Model Calculations Research Articles

A new technique for describing the electronic states of atoms and molecules — The vector method

Whitehead's numerical approach to nuclear shell-model calculations has been successfully applied to ground and low-lying excited states of atoms and molecules. This paper gives a description of the computational techniques and results for LiH and H 2O. Although presently somewhat slower than standard configuration interaction techniques, additional research should improve this situation.

Hidden configurations and effective interactions: A comparison of three different ways to construct renormalized hamiltonians for truncated shell-model calculations

We discuss in general some criteria and methods for constructing effective Hamiltonians. Then three different methods are compared for constructing an effective Hamiltonian to be used in nuclear shell-model calculations for A = 17–20, allowing ( A-16) active nucleons in the d 5 2 , s 1 2 vector space. For all three methods, the aim is to obtain a d 5 2 , s 1 2 model which will simulate the results of a given full d 5 2 , s 1 2 , d 3 2 model. The three methods for finding the effective Hamiltonian are. 1. (a) conventional low-order perturbation theory; 2. (b) a projection technique, in which we construct a Hamiltonian whose eigenvalues excactly match a selected subset of d 5 2 , s 1 2 , d 3 2 eigenvalues, and whose eigenvectors excatly match the projections of d 5 2 , s 1 2 , d 3 2 eigenvectors on the d 5 2 , s 1 2 space; and 3. (c) least-square fit to selected d 5 2 , s 1 2 , d 3 2 energies. For all three methods, we first restrict the effective Hamiltonian to a linear combination of 1-body and 2-body operators. Then for the perturbation and projection techniques, we also calculate the 3-body-operator terms in the effective Hamiltonian. When the effective Hamiltonians are limited to 1-body and 2-body terms, the leastsquare method yields the best overall fit to the low-lying spectrum of d 5 2 , s 1 2 , d 3 2

On the application of the Lanczos algorithm for the matrix eigenproblem to other than nuclear shell model calculations

An application of the Lanczos algorithm for the tridiagonalization of a finite matrix to shell model calculations has been recently suggested. Here we study the possibility to extend this method to cases where states lying in the continuum are included. First, we treat a model where a numerical value to the ground state energy is given in advance but the potential and the nature of the system are arbitrary. By means of this model we show that the extension is possible. Next, we demonstrate a practical application of the method in treating a two-body system. The results are promising.

A numerical approach to nuclear shell-model calculations

An application of the Lanczos algorithm for the matrix eigenproblem to nuclear shell-model calculations is described. The method has many advantages over conventional shell-model techniques, one of the most important being a very much slower growth of computation time with increasing dimensionality of the shell-model basis.

Configuration mixing and mysterious second zero states

A simple schematic model is used to highlight the difficulty of obtaining a low lying excited state in doubly closed shell nuclei having the same quantum numbers as the ground state in a nuclear shell-model configuration mixing calculation.

The Delphi-speakeasy system. I. Overall description

DELPHI is a modular computer program designed to aid scientists in their day-to-day research. Originally intended to serve the field of nuclear shell-model studies, it has since become a generalized program for theoretical physics. A user-oriented language called SPEAKEASY,included to make the system easily used, is itself becoming a powerful research tool. This paper, the first in a series, is intended to provide a general description of the program. A brief description of nuclear shell-model calculations is included.

Center-of-mass localization in self-bound many-body systems

The problem of self-localization in bound many-body systems is examined, with emphasis on the attempt to calculate density distributions relative to the center-of-mass and thereby examine properties of the surface. Such distributions ϱ(r) can be formally calculated in several ways; in terms of off-diagonal matrix elements of the ordinary density, in terms of new field amplitudes with non-local commutation relations, or in terms of diagonal elements of the ordinary density evaluated in states of an extended (translational symmetry-broken) Hamiltonian. The first two methods appear rather intractable for practical calculations, whereas the third formulation allows a simple exact solution for oscillator forces and is capable of extension to other interactions of interest. In view of the use to which oscillator states have been put in nuclear shell model calculations, we note here some of the difficulties inherent in an oscillator model calculation of the density ϱ(r) for large nuclei.

Four-body interaction in nuclear shell-model calculations

A four-body potential is used in shell-model calculations as an alternative possibility of describing cluster structures in nuclei. Volume- and surface-contact interactions are considered as simple examples. Application to20Ne shows good agreement in predicting the energy spectrum in the case of surface interaction, but fails in evaluatingB(E2) transitions, especially at high excitations, where two-body forces (here neglected) surely predominate.

What configuration mixing can be concealed in nuclear shell model calculations?

A modification of the pseudonium nuclei model shows that most types of configurations mixing cannot be concealed in shell model calculations. It is still possible to conceal configurations mixing in the ground states. This provides a clear indication as to the limited types of configurations mixing which can be concealed in shell model calculations.

Methods in Computational Physics. Advances in Research and Applications. Vol. 6. Nuclear Physics 1966

B. Alder, S. Fernbach and M. Rotenberg (Eds) New York: Academic Press. 1967 Pp. xiv + 303. Price £5 8s. Although this is primarily a volume for specialists in nuclear physics computation, it is sufficiently well larded with accounts of the relevance and results of the various methods to extend its appeal to a wider reading public. Chapters are provided on nuclear optical model calculations, numerical methods for the many-body theory of finite nuclei, application of the matrix Hartree–Fock method to problems in nuclear structure, variational calculations in few-body problems with the Monte Carlo method, automated nuclear shell model calculations and nucleon-nucleon phase shift analyses by x2 minimization.

Central and tensor potentials in nuclear shell model calculations

We present here the results of an attempt to determine an effective interaction for nuclear shell-model problems. A residual interaction with gaussian shape and both central and tensor parts was assumed. The ranges of these parts were fixed, but the strengths of the various spin-parity components were varied to obtain a good fit to a number of nuclear levels. The levels chosen for this fit were those in nuclei which consist of two particles or holes relative to the following closed shell cores: 16O, 40Ca, 88Sr, 208Pb. The attempt failed, in that no reasonable set of potential parameters could be found which worked reasonably well for all cases. The greatest success was obtained for those nuclei which have 208Pb as the core, although one of the worst cases of disagreement was 206Tl.

Spurious states arising from the centre-of-mass motion of a nucleus

A simple method is given for generating the spurious states which arise from the incorrect treatment of the centre-of-mass motion by nuclear shell model calculations. It is not necessary to know the spurious states in order to find their weight in a particular shell model state. It is also shown which jj configurations have no spurious states. Some examples are given.

Shell-Model Theory ofPb206

A detailed nuclear shell model calculation of the energy levels and gamma-ray transition rates in ${\mathrm{Pb}}^{206}$ is carried through. For pure singlet two-body forces with the same effective range and strength as for the low-energy two-body system, energy level agreement is good---for 13 known levels, the mean discrepancy between theory and experiment is 0.057 Mev. For singlet forces 75% as strong as for the low-energy two-body system, plus weak coupling to collective vibration, the energy level agreement is somewhat better---the mean discrepancy between theory and experiment is 0.035 Mev per level. In the latter theory, the strength of collective motion is determined from the known Coulomb excitation cross section in ${\mathrm{Pb}}^{206}$. In either theory, the calculated transition rates are in qualitative accord with experiment, but quantitative agreement is lacking.Electric quadrupole transition rates are shown to be describable in terms of a neutron effective charge. The effective charge is about $1.15e$ and the same in ${\mathrm{Pb}}^{206}$ as in ${\mathrm{Pb}}^{207}$. Other calculated quantities in good agreement with experiment are the relative cross sections to final $p$ states in the ${\mathrm{Pb}}^{206}(d, p){\mathrm{Pb}}^{207}$ reaction and the difference in binding energies of the last neutron in ${\mathrm{Pb}}^{206}$ and ${\mathrm{Pb}}^{207}$. The results are shown to be insensitive to substantial amounts of triplet-odd force, either attractive or repulsive.

Centre-of-mass effects in the nuclear shell-model

An examination is made of the possible errors that may arise in nuclear shell-model calculations by failing to separate out centre-of-mass motions from the internal motions of the system; a central oscillator potential is used to make possible a simple analysis. The most important effect is the appearance of spurious states that do not refer to the internal motion at all; these must be considered whenever the state is described by two or more unclosed shells (in the sense of L-S coupling), as, for example, in the excited states of 16 O.