S.S.M. Wong

Strength distributions and statistical spectroscopy. I. General theory

Abstract The strength distribution for an arbitrary excitation is given in terms of a double expansion, and its sum rules by single expansions, in polynomials defined by the initial and final energy spectra. In model spaces which are not too large, a rapid convergence, to within fluctuations, is assured by the action of a central limit theorem, as is shown in particular by considering the response of the system under infinitesimal deformations of the Hamiltonian. When larger spaces are decomposed into subspaces defined by a partitioning of the single-particle space a similar convergence results. At the same time, close contact is made with, and important corrections are found to, intuitive procedures which are often used for approximating strength distributions. The general features of the distribution are often easily understood in termsof a simple geometry made effective in the model space by the central limit theorem, and further features by exploiting the connection of this geometry to the unitary group of transformations in the single-particle space. Extensions are given for multipole strengths and sum rules, appropriate when the angular momenta (and isospins) are specified for the states involved in the transitions. Measures for the RMS fluctuations in the sum-rule quantities, and correlations between them, are given by combining the low-order-polynomial (statistically smoothed) strengths with an assumed Porter-Thomas distribution for the (high-order) strength fluctuations.

Strength distributions and statistical spectroscopy II. Shell-model comparisons

Abstract The theory of the preceding paper is applied to a number of M1, E2, and E4 electro-magnetic excitations in the (ds)6 space. Comparisons are made in detail with shell-model results for the pth energy-weighted sum rules, with p = 0, 1, 2 and starting states spanning the entire spectra, as well as with the exothermic-endothermic decomposition of the non-energy weighted sum rule, and the RMS fluctuations in, and correlations between, the sum-rule quantities. Further comparisons are made for the strength distributions themselves. In all cases the agreements are good, for the sum rules remarkably so, so that the statistical theory describes very well the essential features of the strength distribution. The only (partial) exception is with the usual low-lying quadrupole collectivity found microscopically for two of the starting states (for which most of the strength goes to a single final state) and predicted, though not in such detail, via a statistical calculation of the effective number of final states available for the quadrupole transitions. We are seeing here a real coexistence of collective and statistical phenomena. At higher excitations, where concentration of much of the strength into a single state is not to be expected, all the essential features should be statistically describable. As a result of the comparisons, we expect that the statistical theory, supplemented by further methods for the evaluation of the necessary input traces, should give an almost complete account of the essential features of the strength distributions, even in model spaces of arbitrarily large dimensionality.

“Exact” shell model calculations in A = 17 and A = 18 nuclei and effective operators in the (2s, 1d) shell

Abstract A shell model calculation in an enlarged space spanned by two particles in the (2s, 1d) shell and three-particle-one-hole 2 h ω excitations is performed using the “bare” G -matrix. An effective interaction for the (2s, 1d) shell is generated and its convergence properties are investigated. It is maintained that the series converges reasonably fast in G and that the singleparticle energies used in the calculation are instrumental in this respect. This conclusion is corroborated by the E2 effective charges and B (E2) strengths that we compute for A = 17 and A = 18 nuclei respectively.

Preliminary results on fixed-JT averaging method for many-particle spectroscopy

Abstract A practical method based on spectral distribution ideas is introduced to perform nuclear spectroscopic studies. Both the spin and the isospin are treated exactly. Extensive calculations of level densities and level positions are made in the sd shell and the results are compared with those of the exact shell model.

Polynomial expansions for excitation strengths

Abstract Rapidly convergent expansions for the strength function of an excitation in a many-particle model space are given in terms of orthogonal polynomials defined by the state or partial state densities of the system. Convergence is assured in a wide range of circumstances by the operation of a central limit theorem. Level-to-level fluctuations are in many cases small, their automatic elimination in the statistical smoothing generated by truncating the series then leading to only small errors.

A shell-model calculation of the mass 18 nuclei and the effective interaction in the sd-shell

Abstract A shell-model calculation of the energy levels of 18 O and 18 F is performed in the 2 particle and 3 particle-1 hole configuration space using the bare G -matrix elements of Kuo and Brown. The objective is to test the convergence of perturbative calculations of the effective interaction. The results are compared with those of first order perturbation theory and are found to be generally similar.

Calculations of the giant dipole resonance for sd-shell nuclei in the open-shell random phase approximation

Abstract The systematics of the giant dipole resonance have been calculated in the open-shell RPA for all the self-conjugate sd-shell nuclei, using (i) a phenomenological Rosenfeld interaction, (ii) Barret, Hewitt and McCarthy G -matrix elements and (iii) Kuo G -matrix elements. The excitations are based on shell model ground states for all nuclei except 28 Si for which a projected Hartree-Fock ground state was used.

Open-shell RPA calculations for the giant-dipole excitations of 12C

Abstract The RPA is extended to open-shell nuclei, in the equations of motion formalism, and applied to the J π = 1 − , T = 1 excitations of 12 C. It is found that a mixed configuration ground state results in a splitting of the giant photo-resonance.

Intermediate energy proton scattering from 40Ca, 90Zr and 208Pb

Abstract High momentum transfer measurements from TRIUMF are presented for proton elastic scattering from 40Ca, 90Zr and 208Pb. Differential cross sections and analyzing powers are given for proton kinetic energies of 200, 362 and 400 MeV out to a maximum momentum transfer of q≈4.8 fm−. The data are analyzed with a non-relativistic microscopic optical model to investigate its limits and to compare it with recent relativistic models.

The giant dipole excitations of 12C in the open shell tamm-dancoff approximation and in the intermediate-coupling shell model☆

Particle-hole and detailed shell model calculations have been performed for the giant dipole excitations of 12C. It is found that the particle-hole approximation gives the correct gross structure of the photo-resonance provided the particle-hole excitations are referred to the intermediate coupling ground state rather than to the unrealistic closed 1p32-shell.