Main Findings of Daehnick's Review Article on V_eff

Introduction

Daehnick's 1983 review article [DAE83] extended and critically assessed methods of extracting empirical effective interactions from data on single-nucleon transfer. The main equations used to extract V_eff from nuclear transfer data are given in the Equations Box. The general method was to construct the effective Hamiltonian by inserting measured energies E and spectroscopic amplitudes A into equ. (1), and then subtract the single particle energies to get matrix elements of V_eff by means of equ. (5). The energies and spectroscopic amplitudes were obtained by analyzing experiments on single-nucleon transfer reactions.

Results

Following the lead of Schiffer and True [SCH76] and Molinari et al. [MOL75], Daehnick was interested in some surprisingly simple universal features exhibited by the empirically extracted matrix elements of V_eff. For a given configuration, say a particle-particle configuration j₁j₂, the total angular momentum can take any value allowed by the rules of angular momentum coupling. This gives a set of states referred to as a multiplet. The average of the matrix elements of V_eff over all the states in the multiplet is:

Only the monopole (i.e. angle independent) part of V_eff contributes to v. Daehnick found that the monopole part of the effective interaction scales with mass number A according to v ∝ A^-0.75, and that this scaling is almost independent of the quantum numbers of the multiplet. Recognition of the A-dependence of the effective interaction was of some significance, because it had not been taken into account in the classic Argonne work of Cohen and Kurath [COH65] on p-shell nuclei ranging from A=8 to A=16.

For a wide range of nuclei, matrix elements of V_effwere found to exhibit universal behavior when scaled by v and plotted against q₁₂, the semi-classical angle between the vectors j₁ and j₂. This is defined by

The cases J = 0 and j₁ = 1/2 and/or j₂ = 1/2 were exceptions to the universal behavior, largely because semi classical approximations do not apply to them.

The scaled diagonal matrix elements were well approximated by as few as 6 universal functions of q₁₂, which can be thought of as arising from a delta force V_d, a quadrupole force V_Q (representing quadrupole-phonon core polarization), and a monopole force V_M of the forms

Fractionation and Centroids

Reassuringly, deviations from the universal behavior described were found to be appreciably reduced when the appropriate centroid was used for each energy, as specified in eq. (3) of the Equations Box, in place of the energy of the single dominant state. In fact there seemed to be no confirmed experimental disagreements with the trends. The importance of using centroids is illustrated by Fig. 1, which is based on data on the j²= (9/2)² configuration of two neutron holes in ⁸⁶Sr presented by PC Li et al. [LI85] When centroids are appropriately used to analyze multiplets near poorly-closed shells, the extracted V_eff agrees much better with the universal trends.

Comparison with Theoretical Results

Many authors have made theoretical calculations of matrix elements of V_eff by starting from realistic free nucleon-nucleon potentials. These realistic potentials reproduce the known properties of the deuteron and the experimental phase shifts for nucleon-nucleon scattering. The calculations of V_eff make use of the Brueckner G matrix, following developments of methods pioneered by Kuo and Brown [KUO66]. In the main the theoretical results conform well to the universal trends obtained from the experimental effective-interaction matrix elements. The scatter among the different theoretical results provides some measure of the "theoretical error bars".

Two-nucleon states should be classified by the isospin quantum number T (which is conserved by the strong interaction) as well as by the angular momentum quantum numbers. For two neutrons or two protons, T = 1, but for a neutron and a proton, T can be either 1 or 0. T = 1 corresponds to space-spin antisymmetry, T = 0 to space-spin symmetry of the wave function. A notable finding of Daehnick's review article is that the theoretical and empirical results agree well for T = 1 matrix elements, but deviate systematically for T = 0 matrix elements, showing too little attraction especially for low J values. This discrepancy remains an open problem and a challenge to theorists.

The striking dependence of the deviations on T is illustrated in Fig. 2.The scaled matrix element

is plotted against the semi classical angle θ₁₂ for multiplets corresponding to a variety of j₁j₂ = j² configurations in nuclei with mass numbers 18, 40, 48, and 210. The solid curves represent simple universal fits to the data; the dashed curves show the full regions over which the experimental matrix elements have been found to scatter. The theoretical points labeled "KUO from G matrix" systematically show too little attraction in the T = 0 case, but agree quite well with experiment in the T = 1 case.

Uncertainties

Of particular interest in this work was the careful attention given to the uncertainties of the analysis. Absolute cross sections are hard to measure accurately. Moreover, spectroscopic factors must be obtained by comparison with cross sections computed by the DWBA method, which depend quite sensitively on the optical potentials used. Fortunately there are "sum rules" which can serve as a check on the validity of the spectroscopic factors extracted. Thus, for transfer of a nucleon to the target,

One should avoid renormalizing extracted spectroscopic factors merely to satisfy this sum rule, unless one can be certain that all the spectroscopic strength has been seen experimentally. A set of perfectly correct spectroscopic factors may seem to be too small, when perhaps the left-hand side of the sum rule is less than 1 only because a number of states have not been seen, due to each having small spectroscopic strength, though the total spectroscopic strength of the unseen states may be appreciable.

In ways analogous to the treatment of two-particle ("p-p") states, two-hole ("h-h") states can be studied by pickup reactions on one-hole targets, and particle-hole ("p-h") states can be studied by pickup reactions on one-particle states or by stripping reactions on one-hole states. The relation between p-p and h-h matrix elements is trivial, while p-p and p-h matrix elements are related via the Pandya transformation [PAN56] which involves angular momentum recoupling coefficients. In some cases, the Pandya transformation allows the same effective interaction matrix elements to be obtained in two independent ways. Daehnick showed that missing spectroscopic strength produces errors in p-p matrix elements obtained from stripping on one-particle states oppositely directed to those in p-p matrix elements obtained from p-h matrix elements via the Pandya transformation. This allowed him to estimate some useful upper and lower "bounds" on the effective-interaction matrix elements.

In cases where the core nucleus is not a good closed-shell nucleus, the extraction of the single-particle energies e(j) becomes ambiguous. Daehnick used single-particle centroid energies to restore consistency as far as possible. These corrections sometimes influenced the extracted effective-interaction matrix elements by amounts of order 300 keV.