Pairing theory of the symmetry energy
- Fjordtoften 17, DK-4700 Naestved (Denmark)
A model is investigated that displays a picture of the symmetry energy as an energy of rotation in isospace of a Cooper pair condensate, briefly 'superfluid isorotation'. The Hamiltonian is isobarically invariant and has a one- and a two-nucleon term, where the two-nucleon interaction is composed of an isovector pairing force and an interaction of isospins. It is analyzed in the Hartree-Bogolyubov plus random-phase approximation (RPA). The Hartree-Bogolyubov energy minus Lagrangian multiplier terms proportional to the number of valence nucleons and the z component of the isospin is shown to be locally minimized by a product of neutron and proton Bardeen-Cooper-Schrieffer states. The equations of the RPA can be reduced to independent equations for two-neutron, two-proton, and neutron-proton quasiparticle pairs. In each of these spaces, they have a Nambu-Goldstone solution due to the global gauge invariance and isobaric invariance of the Hamiltonian. Except for the Nambu-Goldstone solutions, the RPA solutions are independent of the strength of the isospin interaction. If, in one space, the pertinent single-nucleon spectrum has a particle-hole symmetry, the RPA solutions are twofold degenerate except for the Nambu-Goldstone solution and one more solution. In an idealized case of infinitely many equidistant single-nucleon levels, the one-nucleon term in the Hamiltonian and the isospin interaction contribute terms in the symmetry energy quadratic in the isospin T. The pairing force and the two-neutron and two-proton RPA correlation energies do not contribute. The contribution of the neutron-proton correlation energy is dominated by the Nambu-Goldstone solution, which gives a linear term that makes the total symmetry energy proportional to T(T+1). The rest of this contribution is negative and can be written as the difference of two terms of the form {radical}((aT){sup 2}+b{sup 2})-b. Observations reported from Skyrme force calculations are discussed in the light of these results. Calculations with deformed Woods-Saxon single-nucleon levels give results similar to those of the idealized case. In calculations for the mass numbers A=56 and A=100 with spherical Woods-Saxon levels, the promotion of nucleons across magic gaps in the single-nucleon spectrum and the onset of superfluidity with the departure from magicity give rise to large linear terms in the symmetry energy. The calculations with Woods-Saxon single-nucleon levels reproduce surprisingly well the empirical symmetry energy. An experimental signature of superfluid isorotation is discussed.
- OSTI ID:
- 21293926
- Journal Information:
- Physical Review. C, Nuclear Physics, Vol. 80, Issue 4; Other Information: DOI: 10.1103/PhysRevC.80.044313; (c) 2009 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); ISSN 0556-2813
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
BCS THEORY
COMPUTERIZED SIMULATION
CONDENSATES
COOPER PAIRS
ELECTRON CORRELATION
GAUGE INVARIANCE
HAMILTONIANS
HARTREE-FOCK-BOGOLYUBOV THEORY
ISOSPIN
ISOVECTORS
LAGRANGIAN FUNCTION
MASS
MATHEMATICAL SOLUTIONS
NEUTRONS
PROTONS
RANDOM PHASE APPROXIMATION
SKYRME POTENTIAL
SPECTRA
SPHERICAL CONFIGURATION
SUPERFLUIDITY
SYMMETRY
VALENCE
WOODS-SAXON POTENTIAL