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arXiv:nucl-th/0003019 · Physical Review C

Odd-even staggering of binding energies as a consequence of pairing and mean-field effects

J. Dobaczewski, Piotr Magierski, Witold Nazarewicz, W. Satuła, Z. Szymański

Published Jan 19, 2001• 112 citations
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Abstract

Odd-even staggering of binding energies is studied in finite fermion systems with pairing correlations. We discuss contributions of the pairing and mean field to the staggering, and we construct the binding-energy indicators which measure the magnitude of pairing correlations and the effective single-particle spacings in a given system. The analysis is based on studying several exactly solvable many-body Hamiltonians as well as on the analytical formulas that can be applied in the weak and strong pairing limits.

DOI
10.1103/physrevc.63.024308

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Odd-even staggering of binding energies is studied in finite fermion systems with pairing correlations. We discuss contributions of the pairing and mean field to the staggering, and we construct the binding-energy indicators which measure the magnitude of pairing correlations and the effective single-particle spacings in a given system. The analysis is based on studying several exactly solvable many-body Hamiltonians as well as on the analytical formulas that can be applied in the weak and strong pairing limits.
Odd-even statistical staggering in a Lipkin-like few fermions model has been recently encountered. Of course, staggering in nuclear binding energies is a well established fact. Similar effects are detected in other finite fermion systems as well, as for example, ultra small metallic grains and metal clusters. We work in this effort with the above-mentioned Lipkin-like, two-level fermion model and show that statistical staggering effects can be detailedly explained by recourse to a straightforward analysis of the associated energy-spectra.
Pairing gaps for fermionic atoms in harmonic-oscillator traps are calculated for a wide range of interaction strengths and particle numbers, and compared to pairing in nuclei. Especially systems where the pairing gap exceeds the single-level spacing but is smaller than the shell splitting $\ensuremath{\Elzxh}\ensuremath{\omega}$ are studied, which applies to most trapped Fermi atomic systems as well as to finite nuclei. When solving the gap equation for a large trap with such multilevel pairing, one finds that the matrix elements between nearby harmonic-oscillator levels and the quasi particle energies lead to a double logarithm of the gap, and a pronounced shell structure at magic numbers. It is argued that neutron and proton pairing in nuclei belongs to the class of multilevel pairing, that their shell structure follows naturally, and that the gaps scale as $\ensuremath{\sim}{A}^{\ensuremath{-}1/3}$---all in qualitative agreement with odd-even staggering of nuclear binding energies. Pairing in large systems is related to that in the bulk limit. For large nuclei the neutron and proton superfluid gaps approach the asymptotic value in infinite nuclear matter: $\ensuremath{\Delta}\ensuremath{\simeq}1.1\mathrm{MeV}.$
We study a trapped system of fermions with an attractive zero-range two-body interaction using the Shell-Model Monte Carlo method. The method provides {\em ab initio} results in the low NNN limit where mean-field theory is not applicable. The energy and pairing properties are presented as functions of interaction strength, particle number, and temperature. In the interesting region where typical matrix elements of the two-body interaction are comparable to the level spacing of the trap we find large odd-even effects and signatures of shell structure. As a function of temperature, we observe the disappearance of these effects as in a phase transition.
Different steps leading to the new functional for pairing based on natural orbitals and occupancies proposed earlier [D. Lacroix and G. Hupin, Phys. Rev. B 82, 144509 (2010)] are carefully analyzed. Properties of quasiparticle states projected onto good particle numbers are first reviewed. These properties are used to (i) prove the existence of such a functional, (ii) provide an explicit functional through a 1/N1/N1/N expansion starting from the BCS approach, and (iii) give a compact form of the functional summing up all orders in the expansion. The functional is benchmarked in the case of the picket-fence pairing Hamiltonian where even and odd systems are studied, using the blocking technique, at various particle numbers and coupling strengths, with uniform and random single-particle level spacing. In all cases, very good agreement is found, with a deviation of $<$1% compared to the exact energy.
The lowest seniority-zero eigenstates of an exactly solvable multilevel pairing Hamiltonian for a finite Fermi system are examined at different pairing regimes. After briefly reviewing the form of the eigenstates in the Richardson formalism, we discuss a different representation of these states in terms of the collective pairs resulting from the diagonalization of the Hamiltonian in a space of two degenerate time-reversed fermions. We perform a two-fold analysis by working both in the fermionic space of these collective pairs and in a space of corresponding elementary bosons. On the fermionic side, we monitor the variations which occur, with increasing the pairing strength, in the structure of both these collective pairs and the lowest eigenstates. On the bosonic side, after reviewing a fermion-boson mapping procedure, we construct exact images of the fermion eigenstates and study their wave function. The analysis allows a close examination of the phenomenon of pair condensation in a finite Fermi system and gives new insights into the evolution of the lowest (seniority-zero) excited states of a pairing Hamiltonian from the unperturbed regime up to a strongly interacting one.
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 the 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 the 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 Sn isotopes with valence neutrons in the 50–82 neutron shell. The 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.