Brueckner Theory in an Oscillator Basis. I. The Method of Reference Bethe-Goldstone Equations and Comparison of the Yale, Reid (Hard-Core), and Hamada-Johnston Interactions

Abstract

Arguments are given for preferring discrete localized excited single-particle states to plane waves in Brueckner theory for finite nuclei. Short-range internuclear repulsion requires transforming pair states to relative and cm. coordinates, in which only constant and harmonic potentials separate. This series of papers develops a form of Brueckner theory in which single-particle states are expanded in harmonicoscillator functions. This paper is limited to the single-oscillator-configuration (SOC) approximation. The reaction matrix is obtained in two steps: A reference matrix, involving the potential V s (ti) --C-f-JmcoVi 2 and Eden and Emery's rather good approximate Pauli operator, is obtained by solving radial Bethe-Goldstone equations. The off-diagonal elements of the tensor interaction are treated exactly in coupled equations. Second, essentially the exact SOC matrix is calculated; by making the Pauli and spectral corrections on the reference two-body rather than the relative matrix elements the exact SOC Pauli operator can be used, and energies of low-lying excited states can be varied (e.g., to satisfy self-consistency conditions). The Pauli corrections are much smaller than when the Pauli effect is omitted entirely in the reference matrix. By proper choice of C, the spectral corrections also may be reduced. Three interactions with hard cores are compared under the same conditions. The gap between occupied and excited states is greater than | the normal oscillator spacing na>. Occupied-state energies and pair-creation matrix elements of V R are made nearly self-consistent. The three interactions are practically equivalent, as demonstrated in a calculation of the energetics of 16 0. The most advantageous choice of excited-state spectrum remains the outstanding uncertainty, probably requiring careful evaluation of the Bethe three-body cluster for finite nuclei. ffiFM(V",4)=EA(0 = ET V^+FjPM^Vr+FiPM (1.1b) HL 2Mi J and its eigenstates are linear combinations of products of single-particle eigenstates of the single-particle Hamiltonian h. If all the particles are identical fermions, the IPM eigenstates are Slater determinants, and the antisymmetry is treated mds't conveniently in the 1264 M. Lax, Rev. Mod. Phys. 23, 287 (1951); K. M. Watson, Phys. Rev. 89, 575 (1953); N. C. Francis and K. M. Watson, ibid. 92, 291 (1953). 9 The fundamental ideas of Brueckner and his collaborators evolved in several papers published in 1954-55, attaining in K. A. Brueckner [Phys. Rev. 100, 36 (1955] a form which, except for the technical improvement of the particle-hole formalism, Ref. 7, has the basic structure used today. 10 The power of the Brueckner reaction-matrix approach was convincingly demonstrated in the detailed nuclear-matter calculations of K. A. Brueckner and J. L. Gammel, Phys. Rev. 109,1023. 11 H. A. Bethe, Phys. Rev. 138, B804 (1965); 158, 941 (1967). 12 Baird H. Brandow, Phys. Rev. 152, 863 (1966). BECKER, MACKELLAR, AND MORRIS

Toward ab initio density functional theory for nuclei

“…The diagrams that contribute in a finite system at first and second order in H 1 are shown in Fig. 3 [75] (those in the second row vanish in uniform matter by momentum conservation). A sample diagram contributing to Eq.…”

Goldstone-Brueckner perturbation theory extended in terms of mixed nonorthogonal Slater determinants

“…The choice of the p-p matrix elements of Γ α has been of considerable controversy but we will not go into more details here and refer to Ref. [18,19,20,21,22].…”

Two nucleon emission following bound pion absorption

“…The initial state correlation is taken into account by the use of the correlated wave function determined on the basis of the Brueckner theory [43] including also higher order effects in the definition of the single particle potential of Tripathi et al [43]. In our calculation we have employed for V the Reid soft core potential [44], the starting energy E s = -30 MeV and the oscillator range parameter b = 1.7678 fm.…”

“…The mutual scattering between two emitted nucleons is described by the plane wave expansion method [40,41] together with the T-matrix in momentum space [42]. The initial state correlations are treated by the Brueckner theory [43] using the Reid-soft core potential [44]. For the pion absorption operator, we take the non-relativistic reduction of the pseudovector interaction without rescattering.…”