R.D. Lawson

The quasi-spin formalism and the dependence of nuclear matrix elements on particle number

Abstract It is shown that the quasi-spin formalism gives, in terms of simple vector-coupling coefficients, the explicit particle-number dependence of all matrix elements that enter a shell-model calculation for the configuration jn of identical nucleons. Reduction formulae are thus obtained which express matrix elements involving states of jn with seniority v in terms of the corresponding matrix elements in the configuration jv. Simple interpretations of various seniority selection rules for the matrix elements of energy and transition operators are given. The reduction formulae are generalized to mixed configurations of identical nucleons. These formal results are of considerable practical importance because seniority is a fairly good quantum number for systems of identical nucleons.

Shell-model study of the N = 49 isotones

Abstract The N = 49 isotones 88Y, 89Zr, 90Nb, 91Mo, 92Tc and 93Ru are studied in the framework of the shell model with valence nucleons occupying the 2 p 1 2 and 1 g 9 2 single-particle levels. The residual proton-neutron effective interaction is determined by a least-squares fit to the experimental energies of states in these nuclei with well established spins and parities. We examine the effects of requiring the effective interaction to conserve seniority and model-space isospin. It is found that the existing body of experimental data is consistent with conservation of T = 1 seniority but that the data call for small isospin mixing that can be traced to modelspace truncation effects. The resulting shell-model wave functions are used to calculate energy levels, single-nucleon spectroscopic factors and electromagnetic transition rates. These model wave functions also provide a basis for a qualitative discussion of the weak coupling multiplet structure that persists in the low-lying states of all of these nuclei. In general, all observed properties of yrast levels and analog states are well reproduced by our model as are the positions of many non-yrast levels. Within this model space unique 0+ anti-analog states emerge in 88Y, 90Nb and 92Tc, but the 9 2 + and 1 2 − anti-analogs are highly fragmented in the even-Z nuclei.

A SHELL-MODEL STUDY OF THE ISOTOPES OF O, F, AND Ne.

Abstract The nuclei 18,19,20 O, 18,19,20 F and 20 Ne are described in terms of the shell model within the configurations ( 1 d 5 2 , 2 s 1 2 ) n , n = 2–4 . The residual nucleon-nucleon interaction is parametrized in terms of its two-body matrix elements, which are varied until a best fit to the spectra is obtained. The resulting wave functions are used to compute static and dynamic nuclear properties. In general, the model fits the known experimental data on spectra, transition rates and static multipole moments quite satisfactorily. In particular, good agreement with experiment is obtained for the rotational spectrum of 20 Ne and for the lifetimes of several of the states of its ground-state rotational band. Fitting the level properties of 20 Ne, however, is found to impose on the interaction parameters few restrictions beyond those already implied by a successful treatment of the O and F spectra. Although 36 states are fitted with a 16-parameter potential, it is found that several of the two-body matrix elements are not very well determined. Those interaction parameters that are well determined are compared with the results of a recent reaction-matrix calculation. It is found that satisfactory over-all agreement with experiment can be obtained only if the 1 + excited state of 18 F at 1.7 MeV is omitted from the fitting procedure. Rough estimates suggest that a core-excited 1 + state should occur at about this energy. Thus although the shell model with effective interaction can absorb large amounts of configuration impurity, it cannot do so successfully when the degree of configuration impurity in one of the states under consideration differs radically from that in its neighbors.

Positive-parity states in mass-13 nuclei

Abstract The strength of coupling between the positive-parity nucleon and the C12 core is investigated. Strong coupling, as implied by using the Nilsson model to generate wave functions, contradicts experiment. A reasonable picture arises from weak coupling with a strength consistent with that derivable from summing the two-body interaction integrals between the positive-parity nucleon and the core.

A rotational model for Fe57

A model of Fe/sup 57/ is examined, in which the odd neutron is considered to move in the field of an axially symmetric rotor. The effect of band mixing is included. For a prolate deformation with sigma from 0.15 to 0.2, reasonable agreement with experiment is obtained for many of the properties of the states below 1 Mev in Fe/sup 57/. However, properties which depend on the intrinsic wave function of the 1/2- ground state are in serious disagreement with experiment. The predictions of the model are insensitive to reasonable changes in the moment of inentia of the rotor and in the parameters (other than tbe deformation) characterizing the shape of the Nilsson potential. (auth)

Energy dependence of the optical model potential for fast neutron scattering from cobalt

Abstract Differential elastic- and inelastic-scattering cross sections were measured from ∼ 1.5 to 10.0 MeV over the scattering-angle range ∼18° to 160°, with sufficient detail to define the energy-averaged behavior. Inelastic neutron groups were observed corresponding to measured excitation energies of: 1115 ± 29, 1212 ± 24, 1307 ± 24, 1503 ± 33, 1778 ± 40, 2112 ± 40, 2224 ± 35, 2423 ± 39, 2593 ± 41 and 2810 ± 67 keV. The experimental results were interpreted in terms of spherical-optical-statistical and coupled-channels models. A successful description of the differential elastic scattering below 10 MeV and the total cross section in the range 0–20 MeV was achieved using the spherical optical model with energy-dependent strengths and geometries. These energy dependencies are large below approximately 7.0 MeV, but become smaller and similar to those reported for “global” potentials at higher energies. This change in the energy dependence of the parameters, which occurs about 19 MeV above the Fermi energy, was also seen in the analysis of the 209 Bi data and probably marks the onset of the Fermi surface anomaly. Inelastic scattering to the levels below 1.8 MeV displays a forward peaked behavior. This non-statistical component is interpreted using the weak coupling model in which the f 7 2 proton hole is coupled to the 2 + state in 60 Ni. This vibrational characteristic provides an explanation of the unusual energy dependence and relatively small radius found for the imaginary optical model potential. In conjunction with the fact that cobalt is four neutrons away from the N = 28 closed shell, the coupling also provides an explanation for the large value of this potential. The real spherical optical-model potential derived from the neutron-scattering results was extrapolated to bound energies using the dispersion relationship and the method of moments. The resulting real-potential strength and radius peak at ∼ −10.0 MeV, whereas the real diffuseness is at a minimum at this energy. The extrapolated potential is ∼8% larger than that implied by reported particle-state energies, and ∼13% smaller than indicated by hole-state energies.

Neutron scattering and the fermi surface anomaly in 51V

Abstract Differential neutron elastic- and inelastic-scattering cross sections of vanadium were measured from 4.5 to 10.0 MeV. These results were combined with previous 1.5 to 4.0 MeV scattering data from this laboratory, the 11.1 MeV elastic-scattering results obtained at Ohio University, and the reported neutron total cross sections to energies of ∼ 20 MeV, to form a data base which was interpreted in terms of the spherical optical-statistical model. A fit to the data was achieved by making both the strengths and geometries of the optical-model potential energy dependent. These energy dependencies were large below ∼ 6 MeV, but were smaller, and similar to those characteristic of global models, at higher energies. Using the dispersion relationship and the method of moments, the optical-model potential deduced from the 0 to 11.1 MeV neutron scattering data was extrapolated to higher energies and to the bound-state regime. This extrapolation led to predicted neutron total cross sections that are in good agreement with experimental values to at least 20 MeV. For negative energies the values of the volume-integral per nucleon of the real potential are in excellent agreement with those needed to reproduce the observed binding energies of particle and hole states and give clear evidence of the Fermi surface anomaly. It is argued that the use of a global optical model for interpreting low-energy data is suspect but is probably a reasonable approximation at higher energies.

On the energy dependence of the optical model of neutron scattering from niobium

Neutron differential-elastic-scattering cross sections of niobium were measured from 1.5 to 10.0 MeV at intervals of less than or equal to200 keV below 4.0 MeV, and of approx. =500 keV from 4.0 to 10.0 MeV. Ten to more than fifty differential-cross-section values were determined at each incident energy, distributed over the angular range approx. =20 to 160/sup 0/. The observed values were interpreted in the context of the spherical optical-statistical model. It was found that the volume integral of the real potential decreased with energy whereas the integral of the imaginary part increased. The energy dependence in both cases was consistent with a linear variation. There is a dispersion relationship between the real and imaginary potentials, and when this was used, in conjunction with the experimental imaginary potential, it was possible to predict the observed energy dependence of the real potential to a good degree of accuracy, thus supporting the consistency of the data and its analysis. The real-potential well depths needed to give the correct binding energies of the 2d/sub 5/2/, 3s/sub 1/2/, 2d/sub 3/2/ and 1g/sub 7/2/ particle states and of the 1g/sub 9/2/ hole state are in reasonable agreement with those given by a linear extrapolation ofmorexa0» the scattering potential. However, the well depths needed to give the observed binding of the 2p/sub 3/2/, 1f/sub 5/2/ and 2p/sub 1/2/ hole states are about 10% less than the extrapolated values. 40 refs., 5 figs.«xa0less

SHELL MODEL WITH REALISTIC FORCES: SPECTRA AND THE EFFECTS OF CORRELATIONS IN A = 6 NUCLEI.

Abstract Correlated wave functions that describe the motion of two nucleons outside an inert core are found by solving the coupled integro-differential equations describing the shell-model problem. The eigenvalues that emerge from the solution of this problem give the two-particle interaction energies relative to the closed shell, and the eigenfunctions contain all possible two-particle correlations that arise from exciting valence nucleons to states outside the Fermi sea. The theory is applied to 6 Li, and reasonable agreement with experiment is found when the Hamada-Johnston potential is used for the residual two-body force. The quantities compared with experiment are level scheme, E2 transition rates, M1 transition rates, and in 6 Be, the Coulomb energy of the two valence nucleons. It is argued that better agreement with experiment would emerge if the tensor component of the Hamada-Johnston potential were weakened. The variation of the effective-interaction matrix elements with both the single-particle level scheme and the oscillator constant ħω is examined. This approach is shown to be equivalent to a reformulation of the Bloch-Horowitz theory in which not only the “model-space” states but also all other states allowed by the exclusion principle are treated on an equal footing.

E2 transitions in nuclei

Abstract If one assumes a single major oscillator shell to describe the properties of low-lying nuclear states, shell-model calculations on single-closed-shell nuclei or nuclei with T z = 0 invariably predict the following result for doubly even nuclei: If one introduces an effective charge for the neutron and proton, the E2 transition rates between the lowest state with spin J i (denoted by J i (0) ) and the lowest state with spin J f (denoted by J f (0) ) are in good agreement with experiment. On the other hand, for ΔT = 0 the E2 transition rate between the state J i (0) and the state J f (k) is quite small when J i ≠ J f and k ≠ 0. (The superscript k denotes which of the many states with spin J f we refer to − k = 0 means the lowest one, k = 1 the second state etc.) In this note, it is shown that the lack of an E2 component in the gamma transition J i (0) → J f (k) ( J i ≠ J f , k ≠ 0) may be traced to an asymptotic selection rule which, although holding rigorously only when the nucleus has large deformation, persists to a high degree of approximation even when the nucleus is almost spherical. A strong E2 transition that violates this rule therefore indicates a form of core excitation different from that already included through the introduction of an effective charge. Thus, a measurement of B (E2) for such states provides a sensitive method of testing the “single major oscillator shell” assumption that goes into many shell-model calculations.