Brueckner Reaction Matrix Research Articles

Dependence of off-energy-shell Brueckner reaction matrix elements for finite nuclei on the shape of the repulsive core

Off-energy-shell Brueckner reaction matrix elements for finite nuclei, which are required in any three or four body cluster calculation, are obtained for realistic hard- and soft-core interactions. Significant differences are found between the two interactions.

A=18Nuclear Reaction Matrix

The Brueckner reaction matrix for the $A=18$ nuclei is obtained in an accurate fashion from a nonlocal separable potential of the Tabakin type, but possessing a much stronger repulsive core. This potential, as well as those of Tabakin and Yamaguchi, is used in comparative studies of the reaction matrix. Both harmonic-oscillator and plane-wave intermediate states are employed. The results are compared extensively to the work by other authors on the local Hamada-Johnston potential. In the plane-wave treatment, the effect of replacing the kinetic-energy operator by a form (suggested by Baranger) more compatible with the exclusion principle is estimated.

Accurate calculation of the reaction matrix in light nuclei and in nuclear matter

A method for calculating the Brueckner reaction matrix in the oscillator representation, which was outlined in an earlier letter, is explained in detail. The basic equations are derived and solved numerically. It is shown how the t-matrix elements thus obtained can be used to calculate certain higher-order diagrams for the energy and r.m.s. radius; and numerical results are presented. Finally, the method is applied to infinite nuclear matter, and some numerical results are given for the Reid hard-core and soft-core potentials.

Binding-energy calculations for finite nuclei

A method for computing nuclear binding energies without making recourse to the full Brueckner reaction matrix is discussed. The view presented is that with the aid of the Moszkowski-Scott technique ordinary Rayleigh-Schrödinger theory may suffice in handling repulsive core potentials provided approximate self-consistency is maintained. Our arguments are supported by results obtained in applying the method to 16O using two simple hard-core potentials.

Nonsingular Formulation of the Brueckner Approximation for an Infinite Fermi System

In the approximation neglecting any but single pair correlations, singularities of the reaction matrix render Brueckner's integral for the average energy per particle a singular integral. Several attempts to overcome this difficulty have been unsuccessful. By considering the infinite Fermi system to be a limit of finite systems, it is shown that the correct result merely involves replacing Brueckner's ordinary integral over diagonal reaction matrix elements by a principal value integral. In a finite system the level shift of a Bethe-Goldstone state differs from the diagonal reaction matrix element by a normalization factor which does not approach unity uniformly in the integration variable as the volume becomes infinite. In the neighborhood of a singularity the expression for the two-particle energy shift takes the form $\frac{\mathrm{cy}}{({y}^{2}+c{\mathcal{U}}^{\ensuremath{-}1})}$, where $y$ is the unperturbed energy measured from the singularity, $c$ is the square of a matrix element, and $\mathcal{U}$ is the quantization volume. Hence as $\mathcal{U}\ensuremath{\rightarrow}\ensuremath{\infty}$ the sum over the energy $y$ indeed approaches a principal value integral. An alternative derivation, employing a modified reaction matrix for which there is no difference between level shift and matrix element, leads to the same result. The general derivations are preceded by a soluble example.The connection of the Brueckner approximation with a phase-shift approximation for low-density systems is discussed. Some corrections to the higher order terms in existing derivations of the "separation method" expansion of the Brueckner reaction matrix are given.

On the existence of solutions of the Brueckner equations for a many-fermion system

The Brueckner reaction matrix for a many-fermion system is frequently introduced as the formal “sum” of an infinite series which is a part of the complete many-body perturbation series. It is shown that, when the system is infinite, for any “well-behaved” potential, there are always some unperturbed states for which the reaction matrix series diverges. Conditions for the existence of solutions of the integral equations for the reaction matrix are discussed and it is shown that “Cooper-type” singularities arise for potentials v( r) with not too strongly bound states provided ∫ 0 ∞ v( r) sin 2 k f rd r is not positive (ℏ k F being the Fermi momentum). Explicit calcutations show that there are other types of reaction matrix singularities for nuclear matter but not for liquid helium three. The results provide a potential foundation for Belyaev's calculations of nuclear spectra and suggest that there should be no energy gap in the spectrum of liquid helium three.

Un développement du potentiel de Gibbs d'un système quantique composé d'un grand nombre de particules III—La contribution des collisions binaires

Starting from an expansion derived in a previous work, we study the contribution to the Gibbs potential of the two-body dynamical correlations, taking into account the statistical correlations. Such a contribution is of interest for low density systems at low temperature. In the zero density limit, it reduces to the Beth-Uhlenbeck expression for the second virial coefficient. For a system of fermions in the zero temperature limit, it yields the contribution of the Brueckner reaction matrix to the ground state energy, plus, under certain conditions, additional terms of the form exp(β|Δ|), where the Δ are the binding energies of “bound states” of the type first discussed by L. Cooper. Finally, we study the wave function of two particles immersed in a medium (defined by its temperature and chemical potential). It satisfies an equation generalizing the Bethe-Goldstone equation for an arbitrary temperature.