A Dirac description of ¹S0+³ S1-³ D1 pairing in nuclear matter

Abstract

We develop a Dirac-Hartree-Fock-Bogoliubov description of nuclear matter pairing in ¹S0 and ³S¹-³ D¹ channels. Here we investigate the density dependence ot the ¹S0 and ³S¹-³ D¹ pairing fields in asymmetric nuclear matter, using a Bonn meson-exchange interaction between Dirac nucleons. In this work, we present preliminary results.

A Dirac description of ¹S₀+³ S₁–³ D₁ pairing in nuclear matter

B. Funke Haas; B. V. Carlson; Tobias Frederico

Departamento de Física, Instituto Tecnológico de Aeronáutica 12228-901, São José dos Campos, SP, Brazil

ABSTRACT

We develop a Dirac-Hartree-Fock-Bogoliubov description of nuclear matter pairing in ¹S⁰ and ³S¹–³D¹ channels. Here we investigate the density dependence ot the ¹S⁰ and ³S¹–³D¹ pairing fields in asymmetric nuclear matter, using a Bonn meson-exchange interaction between Dirac nucleons. In this work, we present preliminary results.

1 Introduction

Nonrelativistic calculations of ¹S₀ + ³S₁–³D₁ in symmetric nuclear matter, using standard nucleon-nucleon interactions, yeald a ¹S₀ pairing gap of about the expected size, but an extremely large pairing gap, of the order of 10 MeV, in the ³S₁–³D₁ channel [1, 2, 3, 4, 5]. It has been suggested that relativistic effects substantially reduce this pairing gap [6], as is the case for ¹S₀ pairing at densities near saturation [7]. However, the size of the pairing gap has also been found to be related to the energy of the virtual/bound state in the vacuum of the channel under consideration [7]. This would imply a much larger pairing gap in the ³S₁–³D₁ channel, corresponding to the deuteron in the vacuum, than that in the ¹S₀ channel, which corresponds to the two-nucleon virtual state in the vacuum.

2 The Formalism

We take the hamiltonian form of the HFB equation to be

where

and

with

The index n denotes the 16 solutions to the HFB equation.

The self-consistency equations may be written as

and

where the index j refers to the different mesons exchanged and the indices a and b are their Lorentz/isospin indices. The normal and anomalous densities, g(q) and f(q) , respectively, are given by

with the sum over n running over the appropriate set of solutions of the HFB equation.

We take for the Dirac and isospin structure of the mean fields

and

where the rank two tensor Y₂() is given by

The densities can be decomposed similarly, where the component densities can be obtained with the appropriate traces,

The vertices and propagators of the mesons are given in Table 1, where we have defined

We will neglect retardation effects in the following so that the composite form-factor/reduced propagator will take the form

We will denote the remaining factors of the vertices as the bare vertices.

To obtain the reduced self-consistency equations, we substitute these decompositions in the unreduced equations, calculate and take traces. The equations for the components of ¹S₀ pairing field that result are uncoupled integral equations of the form

The equations for the l = 0 and l = 2 components of ³S₁–³D₁ pairing field reduce to coupled equations. To uncouple these further, we make an additional approximation - we replace the components of the mean field and pairing field by their (spherically symmetric) angular averages.

3 Results and Conclusions

In Figs. 1 and 2 we show the pairing gap for the pp and nn pairs as a function of asymmetry and density, calculated numerically for pure standard pairing in nuclear matter. In Figs. 3 to 5, we show pp, nn and quasi-deuteron pairings gaps for standard plus quasi-deuteron pairing. We have found that pairing in these two channels coexists in asymmetric nuclear matter. This mixed state is the ground state of nuclear matter in the region in which it exists.

Received on 7 October, 2003

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Publication Dates

• Publication in this collection
  26 Oct 2004
• Date of issue
  Sept 2004

History

• Accepted
  07 Oct 2003
• Received
  07 Oct 2003