A two-level solvable model involving competing pairing interactions

A model is considered consisting of nucleons moving in two non-degenerate l-shells and interacting through two pairing residual interactions with (S, T) = (1, 0) and (0, 1). These, together with the single particle hamiltonian induce mutually destructive correlations, giving rise to various collective pictures that can be discussed as representing a two-dimensional space of phases. The model is solved exactly using an O(8) ⊗ O(8) group theoretical classification scheme. The transfer of correlated pairs and quartets is also discussed.     

Class of exactly solvable pairing models

We present three classes of exactly solvable models for fermion and boson systems, based on the pairing interaction. These models are solvable in any dimension. As an example we show the first results for fermions interacting with repulsive pairing forces in a two-dimensional square lattice. In spite of the repulsive pairing force the exact results show attractive pair correlations.     

Properties of proton-rich nuclei in a three-body model

Using a three-body model and realistic two-body potentials, we investigate the properties of the nuclei ¹⁸Ne and ²⁸S near the proton dripline. We figure out the two-proton separation energies, occupation of the valence protons, root-mean-square radii of matter and the valence protons. Besides, the spatial correlation densities are displayed to reflect the correlation between the two valence protons. The first excited 0⁺ state of ¹⁸Ne is most likely to be a halo state according to our calculation. Turning off the Coulomb interactions among the three-body systems, we get the two-neutron separation energies and configuration of the valence neutrons of their corresponding mirror nuclei. The results indicate that the three-body model is proper to describe some proton-rich nuclei and can be used to deduce reliable information.     

A schematic model for monopole and quadrupole pairing in nuclei

A schematic Hamiltonian with a pairing interaction plus a quadrupole-quadrupole interaction between nucleons is presented. It is shown that all the states of the fermion system can be classified according to the number of nucleons u not coupled to coherent monopole or quadrupole pairs. The states with u = 0 are shown to have a one-to-one correspondence to the states of the interacting boson model. The spectra for these states are derived analytically for various limits of the pairing strength and the quadrupole strength. Analytical forms for the matrix elements of operators are derived for these limits. The operators in fermion space are mapped onto boson operators. The matrix elements of operators in the fermion space are shown to be equal to matrix elements of the boson operators multiplied by analytical Pauli factors which are state dependent. The two-nucleon transfer strength is calculated in two limits and is compared to experimental values.     

Semi-microscopic model for the effective pairing interaction in atomic nuclei

A model has been proposed in which the ab initio effective pairing interaction in atomic nuclei is supplemented by a small phenomenological term containing one parameter universal for all intermediate-mass and heavy nuclei. The neutron and proton pairing gaps have been calculated for several chains of semimagic nuclei; these calculations demonstrate the applicability of this model.     

Disappearance of pairing correlations in a rotating nucleus and the role of particle-number projection discussed within a solvable model

The disappearance of pairing correlations in a rotating nucleus and the role of the particlenumber projection technique are discussed using a simple solvable two-level model. Exact solutions are compared to approximate ones obtained with the independent-quasiparticle method both with and without particle-number projection. Analytical formulas for particle-number projected quantities are obtained. It is shown that pairing correlations tend to disappear in a nucleus which rotates fast enough. It is also argued that the pairing parameter Δ which appears in the particle-number projected wave function does not provide an adequate measure of the pairing correlations in nuclei. The pair-transfer reaction cross section is shown to be related closely to the pairing correlation energy and it is suggested that it can be used as a criterion for the existence of the superfluidity in rotating nuclei.     

Nucleon pairing in atomic nuclei

The nucleon pairing effect that is analyzed in the present paper is one of the striking manifestations of nuclear dynamics. Nucleon pairing for different chains of nuclei dependent upon the number of protons or neutrons in the nucleus allows one to explain the emergence of a great number of positive-parity states, which form a ground state multiplet, in even-even nuclei in the excitation energy range E* < 4 MeV. The interaction of paired nucleons with vibrational and rotational degrees of freedom of a nucleus produces a wide variety of excitation spectra of positive-parity states in even-even nuclei.     

Dual origin of pairing in nuclei

The pairing correlations of the nucleus ¹²⁰Sn are calculated by solving the Nambu–Gor’kov equations, including medium polarization effects resulting from the interweaving of quasiparticles, spin and density vibrations, taking into account, within the framework of nuclear field theory (NFT), processes leading to self-energy and vertex corrections and to the induced pairing interaction. From these results one can not only demonstrate the inevitability of the dual origin of pairing in nuclei, but also extract information which can be used at profit to quantitatively disentangle the contributions to the pairing gap Δ arising from the bare and from the induced pairing interaction. The first is the strong ¹ S ₀ short-range NN potential resulting from meson exchange between nucleons moving in time reversal states within an energy range of hundreds of MeV from the Fermi energy. The second results from the exchange of vibrational modes between nucleons moving within few MeV from the Fermi energy. Short- (v _p ^bare) and long-range (v _p ^ind) pairing interactions contribute essentially equally to nuclear Cooper pair stability. That is to the breaking of gauge invariance in open-shell superfluid nuclei and thus to the order parameter, namely to the ground state expectation value of the pair creation operator. In other words, to the emergent property of generalized rigidity in gauge space, and associated rotational bands and Cooper pair tunneling between members of these bands.     

Properties of a local pairing model of high-Tc superconductivity

A simulation technique is used to study the properties of the “hole modulated hopping” model introduced by Hirsch. The superconducting order parameter, energy gap and pair size have been determined for a range of particle densities and temperatures in the neighbourhood of the superconducting phase transition. Results are consistent with the interpretation of the superconducting transition to be Bose-like at low hole densities and BCS-like at high hole densities, with a crossover near the T_c maximum in the T_c versus hole density curve. This behaviour is related to the existence of small non-overlapping pairs at low hole densities and large strongly interpenetrating pairs at hole densities above the T_c maximum.     

Algebraic solutions of an extended pairing model for well deformed nuclei

A Nilsson mean-field plus extended pairing interaction Hamiltonian with many-pair interaction terms is proposed. Eigenvalues of the extended pairing model are easy to obtain. Our investigation shows that one- and two-body interactions continue to dominate the dynamics for relatively small values of the pairing strength. As the strength of the pairing interaction grows, however, the three- and higher many-body interaction terms grow in importance. A numerical study of even-odd mass differences in the (154-171)Yb isotopes shows that the extended pairing model is applicable to well deformed nuclei.     

Nilsson mean-field plus the extended pairing model description of rare earth nuclei

The Nilsson mean-field plus the extended pairing model for well-deformed nuclei is applied to some representative rare earth examples. The binding energies, some low-lying pair-excited states and even-odd mass differences of Er, Yb and Hf isotopes are calculated systematically within the proton frozen-pair excitation limit. A comparison with experimental data for these nuclei shows that the results of the extended pairing model are better than those for the standard pairing model with the BCS approximation and the nearest-orbit pairing model.     

Anharmonic collective excitation in a solvable model

We apply the time-dependent variational principle, the nuclear field theory, and the boson expansion method to the Lipkin model to discuss anharmonicities of collective vibrational excitations. It is shown that all of these approaches lead to the same anharmonicity to leading order in the number of particles. Comparison with the exact solution of the Lipkin model shows that these theories reproduce quite well.     

New magic nuclei and neutron-proton pairing

Special features of new magic nuclei and their connection with the shell structure are considered. The mechanism of neutron-proton pairing is proposed as a basis for the formation of new magic nuclei. A law of nucleon pairing is introduced. Spin-parity values are explained for a number of odd-odd nuclei.     

Exactly solvable model of a spin glass

Sherrington and Kirkpatrick presented a solvable model of a spin glass. In the solution, they used a mathematically unwarranted procedure. In the present article, we show that the problem is exactly solved by starting with the virial expansion formula, and confirm the results of Sherrington and Kirkpatrick. The solution is obtained for the random Ising magnet in which the external field of each site and the exchange integral between each pair of sites are random variables. We obtain the exact thermodynamic properties for this system in the limit of n_w→∞, assuming that the exchange integrals of a spin with O(n_w) neighbours are $\text{O(n}{}_{\mathrm{w}}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$ and the average value of each is O(n_w^?1). The system is found to show the spin-glass state as well as the paramagnetic and the ferromagnetic state.     

Large amplitude collective motion in nuclei and metallic clusters — Applicability of adiabatic theory for a pairing model

A model Hamiltonian describing a two-level system with a crossing plus a pairing force is investigated using the technique of large-amplitude collective motion. The collective path, which is determined by the decoupling conditions, is found to be almost identical to the one in the Born-Oppenheimer approximation for the case of a strong pairing force. For the weak pairing case, the obtained path describes a diabatic dynamics of the system. Presented by T. Nakatsukasa at the International Conference on “Atomic Nuclei and Metallic Clusters”, Prague, September 1–5, 1997. This work is supported by EPSRC (UK).     

The nuclear scissors mode in a solvable model

The coupled dynamics of the scissors mode and the isovector giant quadrupole resonance is studied in a model with separable quadrupole–quadrupole residual interactions. The method of Wigner function moments is applied to derive the dynamical equations for angular momentum and quadrupole moment. Analytical expressions for energies, B(M1)- and B(E2)-values, sum rules and flow-patterns of both modes are found for arbitrary values of the deformation parameter. Some predictions for the case of superdeformation are given. The subtle nature of the phenomenon and its peculiarities are clarified.