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Physical Review

Numerical Study of the 8-32-Particle Eigenstates of the Pairing Hamiltonian

R. W. Richardson

Published Jan 21, 1966• 200 citations
Publisher

Abstract

Exact eigenstates of the pairing-force Hamiltonian have been calculated for $2N$ particles in $2N$ equispaced, doubly degenerate single-particle levels for $N=4, 8, 12, \mathrm{and} 16$. The states calculated are the ground state and the first two excited seniority-zero and seniority-two states. These states were calculated for values of the interaction strength that give pairing energies that are typical of nuclear systems. Numerical results are given for the energies and occupation probabilities of these states and are compared with the corresponding results of the Bardeen-Cooper-Schrieffer (BCS) theory. It is shown that the BCS theory is a worse approximation for the larger values of $N$ than it is for the smaller values, with errors as high as 140% (20%) for the excitation energies of the seniority-zero (-two) states. The methods used in the calculation of these states are outlined.

DOI
10.1103/physrev.141.949

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A Microcanonical Comparison of Richardsons Equations With The BCS Mean Field Approximation

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The exact solution of the reduced pairing Hamiltonian is compared to the BCS mean field approximation for low particle numbers in finite systems. A combination of two different methods for solving the Richardson equations is presented. One of the methods originally intended to be solved analytically is instead adapted to a numerical scheme. The comparison includes the pairing gap parameter, as approximated by the three-point formula in the exact solution, and a microcanonical investigation of systems with levels of varying degeneracies. The gaps as calculated within the two approaches is found to deviate more and more from each other with increasing pairing strength. However testing the three-point-formula with the BCS approximation suggests that this might be due to the formula itself. Furthermore the BCS approximation, when re-evaluating the gap for each excitation, does a very good job at reproducing the energy spectra as obtained for the Richardson solution of the highly excited states where the pairing effects are small. However for the low lying excitations the assessment is not so good. When keeping the gap constant for all excitations the excitation spectra of the BCS approximation broaden a lot more than that of the exact method. This procedure gives unreasonable results for the systems and low particle numbers considered.

•Jan 1, 2016•0 citations
Exact eigenstates of the pairing-force Hamiltonian have been calculated for 2N2N2N particles in 2N2N2N equispaced, doubly degenerate single-particle levels for N=4,8,12,and16N=4, 8, 12, \mathrm{and} 16N=4,8,12,and16. The states calculated are the ground state and the first two excited seniority-zero and seniority-two states. These states were calculated for values of the interaction strength that give pairing energies that are typical of nuclear systems. Numerical results are given for the energies and occupation probabilities of these states and are compared with the corresponding results of the Bardeen-Cooper-Schrieffer (BCS) theory. It is shown that the BCS theory is a worse approximation for the larger values of NNN than it is for the smaller values, with errors as high as 140% (20%) for the excitation energies of the seniority-zero (-two) states. The methods used in the calculation of these states are outlined.
A previous study of the pairing interaction Hamiltonian by means of the equations of motion is extended. In the prior note, ground-state and single-quasiparticle properties were treated by an approximation method which was strictly particle conserving. According to the general theory, further improvement of the ground-state energies and single-particle occupation probabilities requires information about excited states of seniority zero and unity. A nonlinear, number-conserving random-phase approximation is developed for this purpose. Numerical calculations are carried out in a linearized version which is almost equivalent to the usual quasi-boson approximation. As part of the approximation, the conservation-of-particles condition is relaxed for the excited states. Results of the numerical study establish the consistency of the assumptions made and yield improved ground-state and single-particle properties. By comparison with the few models for which exact numerical results are available, the results of this paper are seen to be comparable with or superior to those of other improved approximation methods so far reported, especially so as the number of particles increases. Finally, it is pointed out that just as the results of the previous paper provide the input data for the present work, the results reported here can serve in like manner for more elaborate approximations, which are proposed.
The Euler-Lagrange equations corresponding to a Bardeen-Cooper-Schrieffer state that is an eigenstate of the number operator are derived and solved numerically for a $\ensuremath{\delta}$ interaction. The errors due to the nonconservation of particle number in the usual Bardeen-Cooper-Schrieffer theory are studied as a function of particle number, level density, and strength of the pairing interaction. A proof is given that for attractive pairing interactions the lowest energy solution corresponds always to real positive probability amplitudes ${v}_{\ensuremath{\nu}}$, ${u}_{\ensuremath{\nu}}$.
Equations for all the seniority-zero eigenstates of 2N2N2N nucleons in an arbitrary charge- and spin-independent potential well and interacting through charge- and spin-independent pairing forces are derived. These equations are solved exactly for a large number of states of this system. The interaction in this Hamiltonian is effective in the L=0L=0L=0 states of the two-nucleon system, and its strength is independent of the remaining quantum numbers of the two nucleons. We solve our equations exactly for those states whose wave functions are totally symmetric functions of the spin-isospin coordinates of the NNN L=0L=0L=0 pairs of nucleons in the state. The wave functions of these states factor into a spin-isospin-dependent part and a spatially dependent part. The spin-isospin-dependent part of one of these wave functions is an eigenvector of three tridiagonal matrices which insure that the state is a spin, isospin, and supermultiplet eigenstate, respectively. Explicit expressions are given for the eigenvalues of these three matrices in terms of the quantum numbers of the state. The spatially-dependent part of one of these wave functions is given explicitly in terms of NNN parameters which we call pair energies. These pair energies are shown to satisfy NNN coupled algebraic equations which depend parametrically upon the supermultiplet quantum numbers of the state. An expression for the occupation probabilities of the levels of the single-particle well is given. Throughout this work, an arbitrary splitting of the single-particle levels is treated exactly.
Two advances are reviewed in the application of pairing-force theory in the nuclear shell model. The first exploits a discovery that a wide range of two-nucleon interactions conserve seniority as a good quantum number. Asa consequence, the eigenstates of a hamiltonian with such an interaction belong to irreducible representations of a compact unitary-symplectic group. This makes it possible to extend the simply-solvable models with J = 0 pairing forces to a much richer set of models and still obtain states uniquely classified by their seniority and angular momentum quantum numbers. Moreover, many of the low-lying energy levels of such models can be obtained algebraically; in technical terms, the models are in some cases completely solvable and in other cases partially solvable by algebraic methods. The second advance exploits the discovery that, in a coherent state representation, states of good nucleon number can be projected analytically from BCS vacuum and excited quasiparticle states. This makes it possible to perform calculations in a number-projected BCS basis without losing much of the advantage of working in a quasiparticle scheme.
We study the excited states of zero seniority for various like-particle systems interacting by pairing forces and by general two-body interactions. We consider two types of excitations, generated from a ground state described by a pair condensate. One type is obtained by breaking a pair from the ground-state condensate and replacing it by ``excited'' collective pairs built on time-reversed single-particle orbits. The second type of zero-seniority excited states is described by a condensate of identical excited pairs. The structure of these excited states is analyzed for the picket fence model and for the valence neutrons of 108Sn^{108}\mathrm{Sn}108Sn. For a state-depending pairing interaction, the first type of excited states agree well with the J=0J=0J=0 states which are known in 108Sn^{108}\mathrm{Sn}108Sn. At the same time, these states can be also associated unambiguously with those exact states which are the closest in energy to the experimental levels. The states corresponding to the excited pair condensate (EPC) appear at low energies, around the energy of the second-excited state of the first type, and they do not have a simple correspondence with exact eigenstates. However, at a much higher excitation energy there is an exact state which is similar in structure to an EPC state. It is shown that this EPC state has the features of a giant pairing vibration.
The exact solution of the reduced pairing Hamiltonian is compared to the BCS mean field approximation for low particle numbers in finite systems. A combination of two different methods for solving the Richardson equations is presented. One of the methods originally intended to be solved analytically is instead adapted to a numerical scheme. The comparison includes the pairing gap parameter, as approximated by the three-point formula in the exact solution, and a microcanonical investigation of systems with levels of varying degeneracies. The gaps as calculated within the two approaches is found to deviate more and more from each other with increasing pairing strength. However testing the three-point-formula with the BCS approximation suggests that this might be due to the formula itself. Furthermore the BCS approximation, when re-evaluating the gap for each excitation, does a very good job at reproducing the energy spectra as obtained for the Richardson solution of the highly excited states where the pairing effects are small. However for the low lying excitations the assessment is not so good. When keeping the gap constant for all excitations the excitation spectra of the BCS approximation broaden a lot more than that of the exact method. This procedure gives unreasonable results for the systems and low particle numbers considered.