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arXiv:1001.2979 · Physics Letters B

Pair-vibrational states in the presence of neutron–proton pairing

Richard R. Chasman, P. Van Isacker

Published Jan 19, 2010• 4 citations
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Abstract

Pair vibrations are studied for a Hamiltonian with neutron–neutron, proton–proton and neutron–proton pairing. The spectrum is found to be rich in strongly correlated, low-lying excited states. Changing the ratio of diagonal to off-diagonal pairing matrix elements is found to have a large impact on the excited-state spectrum. The variational configuration interaction (VCI) method, used to calculate the excitation spectrum, is found to be in very good agreement with exact solutions for systems with large degeneracies having equal T=0 and T=1 pairing strengths.

DOI
10.1016/j.physletb.2010.01.032

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•Aug 16, 2016•15 citations
Pair vibrations are studied for a Hamiltonian with neutron–neutron, proton–proton and neutron–proton pairing. The spectrum is found to be rich in strongly correlated, low-lying excited states. Changing the ratio of diagonal to off-diagonal pairing matrix elements is found to have a large impact on the excited-state spectrum. The variational configuration interaction (VCI) method, used to calculate the excitation spectrum, is found to be in very good agreement with exact solutions for systems with large degeneracies having equal T=0 and T=1 pairing strengths.
Previous studies have shown that the ground state of systems of nucleons composed by an equal number of protons and neutrons interacting via proton-neutron pairing forces can be described accurately by a condensate of α-like quartets. Here we extend these studies to the low-lowing excited states of these systems and show that these states can be accurately described by breaking a quartet from the ground state condensate and replacing it with an “excited” quartet. This approach, which is analogous to the one-broken-pair approximation employed for like-particle pairing, is analyzed for various isovector and isovector-isoscalar pairing Hamiltonians which can be solved exactly by diagonalization.
A variational method is developed to determine configuration interaction wave functions. The method is straightforward and is applied to a pairing Hamiltonian with constant matrix elements, for which exact eigenvalues are available. Comparisons are made with the exact results. Calculations can be carried out to any desired degree of accuracy. The method is also applied to a Hamiltonian that has neutron-neutron, proton-proton, and neutron-proton pairing. No difficulties are found in extending the method to this Hamiltonian that has many collective modes. In practice, the method scales linearly with , where is the number of variational basis states.
A method is proposed for determining the ground state of the Hamiltonian $H=\ensuremath{\Sigma}\stackrel{}{k}{\ensuremath{\epsilon}}_{k}{N}_{k}\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{\mathrm{ijkl}}{V}_{\mathrm{ijkl}}{a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{k}{a}_{l}$when the residual interaction is such that $〈{a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{k}{a}_{l}〉={〈{N}_{i}{N}_{j}(1\ensuremath{-}{N}_{k})(1\ensuremath{-}{N}_{l})〉}^{\frac{1}{2}}{〈{N}_{k}{N}_{l}(1\ensuremath{-}{N}_{i})(1\ensuremath{-}{N}_{j})〉}^{\frac{1}{2}}$is a good approximation. The method is applied to pairing interactions involving one type of nucleon as well as to the generalized pairing problem containing neutron-neutron, neutron-proton, and proton-proton interactions.
The seniority-zero excited-state spectrum of the Hamiltonian $H=\ensuremath{\Sigma}\stackrel{}{k>0}{\ensuremath{\epsilon}}_{k}({{a}_{k}}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{k}+{{a}_{\ensuremath{-}k}}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{\ensuremath{-}k})\ensuremath{-}G \ensuremath{\Sigma}\stackrel{}{k,l>0}{{a}_{k}}^{\ifmmode\dagger\else\textdagger\fi{}}{{a}_{\ensuremath{-}k}}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{\ensuremath{-}l}{a}_{l}$ is studied in detail. Approximate excited-state wave functions of the form ${\ensuremath{\psi}}^{\ensuremath{\lambda}}=\ensuremath{\Sigma}\stackrel{}{\ensuremath{\alpha}}F(\ensuremath{\lambda},\ensuremath{\alpha})\ensuremath{\Sigma}\stackrel{}{i}D(\ensuremath{\alpha},i){{a}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}{{a}_{\ensuremath{-}i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Sigma}\stackrel{}{j>i}D(\ensuremath{\alpha},j){{a}_{j}}^{\ifmmode\dagger\else\textdagger\fi{}}{{a}_{\ensuremath{-}j}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\cdots}|0〉$ are developed. These solutions are compared with exact solutions of a small system and the agreement is quite good. A somewhat larger system is studied in order to see how the excited-state spectrum changes as the number of separable functions is increased. The method is applied to heavy nuclei and is in good agreement with observed 0+{0}^{+}0+ excited states for nuclei having 144 to 150 neutrons. A theoretical 0+{0}^{+}0+ spectrum is displayed for each even neutron system from 144 to 152 neutrons.
The exact solution of the proton-neutron isoscalar-isovector (T=0,1) pairing Hamiltonian with nondegenerate single-particle orbits and equal pairing strengths is presented for the first time. The Hamiltonian is a particular case of a family of integrable SO(8) Richardson-Gaudin models. The exact solution of the T=0,1 pairing Hamiltonian is reduced to a problem of 4 sets of coupled nonlinear equations that determine the spectral parameters of the complete set of eigenstates. The microscopic structure of individual eigenstates is analyzed in terms of evolution of the spectral parameters in the complex plane for a system of A=80 nucleons. The spectroscopic trends of the exact solutions are discussed in terms of generalized rotations in isospace.
The ground state correlations induced by a general pairing Hamiltonian in a finite system of like fermions are described in terms of four-body correlated structures (quartets). These are real superpositions of products of two pairs of particles in time-reversed states. Quartets are determined variationally through an iterative sequence of diagonalizations of the Hamiltonian in restricted model spaces and are, in principle, all distinct from one another. The ground state is represented as a product of quartets to which, depending on the number of particles (supposed to be even, in any case), an extra collective pair is added. The extra pair is also determined variationally. In case of pairing in a spherically symmetric mean field, both the quartets and the extra pair (if any) are characterized by a total angular momentum J=0. Realistic applications of the quartet formalism are carried out for the Sn isotopes with the valence neutrons in the 50-82 neutron shell. Exact ground state correlation energies, occupation numbers and pair transfer matrix elements are reproduced to a very high degree of precision. The formalism also lends itself to a straightforward and accurate description of the lowest seniority 0 and 2 excited states of the pairing Hamiltonian. A simplified representation of the ground state as a product of identical quartets is eventually discussed and found to improve considerably upon the more traditional particle-number projected-BCS approach.