Bethe-Goldstone Equation Research Articles

On-top pair-correlation function in the homogeneous electron liquid

The ladder theory, in which the Bethe-Goldstone equation for the effective potential between two scattering particles plays a central role, is well known for its satisfactory description of the short-range correlations in the homogeneous electron liquid. By solving exactly the Bethe-Goldstone equation in the limit of large transfer momentum between two scattering particles, we obtain accurate results for the on-top pair-correlation density $g(0)$, in both three dimensions and two dimensions. Furthermore, we prove, in general, the ladder theory satisfies the cusp condition for the pair-correlation density $g(r)$ at zero distance $r=0$.

Correlation between characteristic energies in non- s-wave pairing superconductors

By solution of the Bethe–Goldstone equation for the Cooper pairing problem, an approximate analytic relation is derived between coherence length ξ and the binding energy of the Cooper pair. This relation is then qualitatively confirmed by numerically solving the corresponding self-consistent gap equations, following the crossover from weak to strong coupling, in non- s-wave superconductors. The relation applies to non-conventional superconductors, and in particular to heavy Fermions and to high- T c cuprates. Utilizing in addition a phenomenological link between k B T c and a characteristic energy ε c=ℏ 2/2m ∗ξ 2 , with m ∗ the effective mass, major differences are exposed in the functional relation between k B T c and ε c for s-wave materials and for non-conventional superconductors. The relation between critical temperature and ε c thereby proposed correctly reflects the qualitative properties of heavy Fermion superconductors.

Reactions of Proton Halo Nuclei in a Relativistic Optical Potential

The reaction cross section, σR, of the proton halo nuclei 17Ne and 12N on Si is calculated using an optical potential derived from the solution of the Dirac–Brueckner–Bethe–Goldstone equation, starting from the one-boson-exchange-potential of Bonn. The nuclear densities are generated from self-consistent Hartree–Fock calculations using the recent Skyrme interaction SKRA. It is found that the enhancement in the reaction cross section found experimentally for the 17Ne + Si system in comparison to 15O + Si, where 15O has been considered as a core of 17Ne, is mainly due to the proton halo structure of 17Ne which increases the interaction, in the surface and tail regions. Glauber model calculations did not produce this enhancement in σR for proton halo nuclei.

Local-potential approximation for the Brueckner G matrix and problem of optimally choosing model subspace

The validity of the local-potential approximation, which was proposed previously for the singlet-pairing problem in semi-infinite nuclear matter, is investigated in the Bethe-Goldstone equation for the Brueckner G matrix. For this purpose, use is made of the method developed earlier for solving this equation for a planar slab of nuclear matter in the case of a separable form of NN interaction. The 1 S 0 singlet and the 3 S 1+3 D 1 triplet channel are considered. The complete two-particle Hilbert space is split into a model and the complementary subspace that are separated by an energy E 0. The two-particle propagator is calculated precisely in the first subspace, and the local-potential approximation is used in the second subspace. With an eye to subsequently employing the G matrix to calculate the Landau-Migdal amplitude, the total two-particle energy is fixed at E=2μ, where μ is the chemical potential of the system under consideration. Specific numerical calculations are performed at μ=−8 MeV. The applicability of the local-potential approximation is investigated versus the cutoff energy E 0. It is shown that, in either channel being considered, the accuracy of the local-potential approximation is rather high for E 0≥10 MeV.

Final state interaction in exclusive(e,e′NN)reactions

Contributions of nucleon-nucleon (NN) correlations, meson exchange currents and the residual final state interactions (FSI) on exclusive two-nucleon knock-out reactions induced by electron scattering are investigated. All contributions are derived from the same realistic meson exchange model for the NN interaction. Effects of correlations and FSI are determined in a consistent way by solving the NN scattering equation, the Bethe-Goldstone equation, for two nucleons in nuclear matter. One finds that the FSI re-scattering terms are non-negligible even if the two nucleons are emitted back to back.

A SKYRME PARAMETRIZATION BASED ON NUCLEAR MATTER BHF CALCULATIONS

Using a modified energy density functional of nuclear matter derived by solving the Bethe–Goldstone equation with a realistic nucleon–nucleon interaction and by including corrections due to relativistic and three-body effects, an effective Skyrme parameter set is derived. These corrections are found to be important in order to well describe the saturation properties of nuclear matter. The obtained Skyrme parameter set, which we denoted by SKRA, is found to better account for nuclear correlations and satisfactory describes finite nuclei, when used in the Skyrme–Hartree–Fock theory. The SKRA interaction can also be considered as an important step toward removing the ambiguities in the determination of Skyrme parameters.

Small excitonic complexes in a disk-shaped quantum dot

Confined excitonic complexes in two dimensions, consisting of N e electrons and N h holes, are studied by means of Bethe–Goldstone equations. Systems with up to 12 pairs, and asymmetric configurations with N e≠ N h are considered. The weak confinement regime gives indication of weak binding or even unbinding in the triexciton and the four-exciton system, and binding in the higher complexes.

Nucleon-nucleon correlations and two-nucleon currents in exclusive(e,e′NN)reactions

The contributions of short-range nucleon-nucleon $(\mathrm{NN})$ correlations, meson exchange current (MEC) terms, and the influence of $\ensuremath{\Delta}$ isobar excitations (isobaric currents) on exclusive two-nucleon knockout reactions induced by electron scattering are investigated. The nuclear structure functions are evaluated for nuclear matter. Realistic $\mathrm{NN}$ interactions derived in the framework of a one-boson-exchange model are employed to evaluate the effects of correlations and MEC in a consistent way. The correlations are determined by solving the Bethe-Goldstone equation. This yields significant contributions to the structure functions ${W}_{L}$ and ${W}_{T}$ of the ${(e,e}^{\ensuremath{'}}pn)$ and ${(e,e}^{\ensuremath{'}}pp)$ reactions. These contributions compete with MEC corrections originating from the $\ensuremath{\pi}$ and $\ensuremath{\rho}$ exchange terms of the same interaction. Special attention is paid to the so-called ``superparallel'' kinematics at momentum transfers which can be measured, e.g., at MAMI in Mainz.

Ground-state wave functions and energies of solids

It is possible to calculate the ground-state wave function and energy of a periodic solid by applying quantum chemical methods. The accuracy achieved is comparable with that for small molecules. The many-body problem of the correlated ground state is expressed in terms of cumulant scattering matrices. This provides a link to the method of increments which can also be derived from the Bethe–Goldstone equations. The theory is applied to calculate primarily the cohesive energy, but also other properties of group IV semiconductors, III–V compounds, and the ionic crystals MgO, CaO, and NiO. It is demonstrated that the scattering-matrix approach can be also applied to strongly correlated electron systems. As a first step in that direction a diamond lattice is considered when the bond lengths are stretched toward infinity. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 385–395, 2000

Erratum: Pauli exclusion operator and binding energy of nuclear matter [Phys. Rev. C 59, 2934 (1999)

Brueckner-Hartree-Fock calculations are performed for nuclear matter with an exact treatment of the Pauli exclusion operator in the Bethe-Goldstone equation. The differences in the calculated binding energy, compared to the angle-average approximation, which is commonly used, are non-negligible. These difference exhibits a specific density dependence, which shifts the calculated saturation point towards smaller densities. This effect is observed for various versions of modern models for the NN interaction.

Pauli exclusion operator and binding energy of nuclear matter

Brueckner-Hartree-Fock calculations are performed for nuclear matter with an exact treatment of the Pauli exclusion operator in the Bethe-Goldstone equation. The differences in the calculated binding energy, compared to the angle-average approximation, which is commonly used, are non-negligible. These differences exhibit a specific density dependence, which shifts the calculated saturation point towards smaller densities. This effect is observed for various versions of modern models for the nucleon-nucleon interaction.

ELASTIC SCATTERING OF 28Si–27Al IN A REALISTIC OPTICAL POTENTIAL

Within a realistic model for the optical potential between nuclei based on the solution of the Bethe-Goldstone equation with the Reid soft-core potential, the differential cross-sections of the 28 Si –27 Al system, recently measured at energies around the Coulomb barrier, are calculated. The experimental data are well-reproduced over the entire energy range without any adjustable parameters to fit the cross-section.

Two-nucleon spectral function of 16O at high momenta.

A procedure for the calculation of the two-body spectral function of a finite nucleus is presented. This spectral function is used to calculate the longitudinal part of the $^{16}\mathrm{O}$(e,${\mathit{e}}^{\ensuremath{'}}$pp) cross section assuming plane waves for the outgoing nucleons. Short-range correlation effects are included in the pair-removal amplitudes by adding corresponding defect wave functions that are obtained from the solution of the Bethe-Goldstone equation in the finite nucleus. The associated G matrix is used as the effective interaction in a large but finite model space to calculate the pair-removal amplitudes in a random-phase approach. The resulting spectral functions exhibit clear differences between different realistic interactions in the momentum range 2--5 ${\mathrm{fm}}^{\mathrm{\ensuremath{-}}1}$ for the initial proton momenta. \textcopyright{} 1996 The American Physical Society.

Solution of the Bethe-Goldstone equation with an exact propagator.

The accuracy of the usual angle-averaging approximations involved in the solution of the nuclear matter Bethe-Goldstone equation is tested numerically in the case where self-energy effects are taken into account. It is found that the errors stemming from the additional angle averaging, which is needed to get rid of the ``energy denominator'' angular dependence, are much smaller than those already introduced by the Pauli operator angle averaging. These results apply to both the bound and scattering regimes. \textcopyright{} 1996 The American Physical Society.

Spectroscopic factors for nucleon knock-out fromO16at small missing energy

Spectroscopic factors for one-nucleon knock-out from $^{16}\mathrm{O}$ are calculated for states with low excitation energy in $^{15}\mathrm{N}$ with the Bonn-C potential. A method is proposed to deal with both short- and long-range correlations consistently. For this purpose a Green's function formalism is used and the self-energy in the Dyson equation is approximated as the sum of an energy-dependent Hartree-Fock (HF) term and dispersion and correlation terms of higher order in the G-matrix interaction. This G matrix is obtained by solving the Bethe-Goldstone equation with a Pauli operator which excludes just the model space treated in the subsequent calculation of the self-energy. The energy dependence of the HF energies induces an additional reduction of the spectroscopic factors for quasiparticle states close to the Fermi level by about 10%. Experimental data may signal the need of some further improvement in the treatment of intermediate- and long-range correlations. \textcopyright{} 1996 The American Physical Society.

Memory effects and virial corrections in nonequilibrium dense systems.

A dense interacting Bose or Fermi gas is considered. Within the Kadanoff-Baym real-time Green-function technique a generalized kinetic equation is derived in the nonequilibrium ladder approximation. As a Markovian limit the Boltzmann-Uehling-Uhlenbeck equation is recovered. The relevance of retardation effects in the kinetic equation of dense systems is shown. It turns out that the stationary solution of the generalized kinetic equation reproduces the second virial coefficient of the equation of state, including the Beth-Uhlenbeck quantum corrections. We establish a generalized kinetic equation describing correlation effects in a consistent approach to nonequilibrium for Bose and Fermi systems. In this way we give a generalization of the Beth-Uhlenbeck formula that applies to nonequilibrium and to finite systems. The derivation makes use of the time dependent T-matrix equation, which establishes a generalization of the Bethe-Goldstone equation. The correlated parts of the density and energy, which can be interpreted as virial corrections, are thermodynamically consistent. In this way global energy conservation is fulfilled.

Temperature-dependent mean field and its effect on heavy-ion reactions

Temperature-dependent mean field potentials of nucleons are obtained by solving the Bethe-Goldstone equation for a realistic force in nuclear matter at finite temperature. For a more efficient utilization of these potentials in studying the heavy-ion reactions using a transport theory, the density and temperature dependence of these potentials is parametrized in a Skyrme type form. These parametrized temperature-dependent potentials are implemented in quantum molecular dynamics. The temperature during the simulations is deduced using a hot Thomas-Fermi approach generalized for the case of two interpenetrating pieces of nuclear matter. First of all, we show that our formalism works well in the nuclear matter limit. In order to study the effect of temperature dependence in the mean-field potential in heavy-ion reactions, the reactions 40Ca+ 40Ca and 93Nb+ 93Nb are simulated using both a finite temperature-dependent potential and a temperature-independent (i.e. zero temperature) potential. Our detailed investigation shows that the temperature dependence of the mean field affects the heavy-ion reaction dynamics to a significant amount. These effects are stronger in case of heavier nuclei and are of the same order as the differences between the usual “soft” and “hard” equation of state. An analytical parametrization of the temperature dependence of the self-consistent field is given in a Skyrme type form.

G-matrix calculations in finite hypernuclei.

g-matrix calculations have been performed for $_{\mathrm{\ensuremath{\Lambda}}}^{5}\mathrm{He}$, $_{\mathrm{\ensuremath{\Lambda}}}^{17}\mathrm{O}$, and $_{\mathrm{\ensuremath{\Lambda}}}^{41}\mathrm{Ca}$ by solving the \ensuremath{\Lambda}-\ensuremath{\Sigma} coupled channels Bethe-Goldstone equation. The potentials employed were the Nijmegen model D and Nijmegen soft core. Calculations were performed with various choices for the Pauli-forbidden space and the energy denominator. The Nijmegen model D does a reasonable job of reproducing most experimental single-particle energies, but the calculated ground-state binding energies vary smoothly from being underbound in the $_{\mathrm{\ensuremath{\Lambda}}}^{5}\mathrm{He}$ to overbound in $_{\mathrm{\ensuremath{\Lambda}}}^{41}\mathrm{Ca}$. The Nijmegen soft core underbinds all single \ensuremath{\Lambda} states, but gives level spacings which agree with experimental values.

Neutron-proton pairing in light nuclei and two-neutrino double-beta decay

We have investigated neutron-proton (np) pairing in light nuclei (2p-lf shell) using a specialized Hartree-Fock-Bogoliubov (HFB) method. For the pairing interaction we used the constant Kisslinger-Sorensen interaction and the realistic G-matrix obtained by solving the Bethe-Goldstone equation with the Bonn one-boson-exchange potential. Our results show that there are at least two minima of the total energy corresponding to two solutions of the HFB equation. The lower solution shows large np pairing. This is confirmed by a recent empirical determination of the np pairing gap, which turns out to be surprisingly large in agreement with our theoretical result. If one renormalizes the Brueckner reaction matrix elements of the Bonn potential by a factor g np multiplying the neutron-proton particle-particle matrix elements to reproduce the experimental np gaps δ np, one obtains large T = 1 np pairing. Preliminary results indicate that the np pairing increases even more if particle-number projection is included. These effects are expected to affect strongly the double-beta decay. The 2 v double-beta decay in 20 48Ca 28 → 22 48Ti 26 is calculated. The Gamow-Teller matrix elements are reduced remarkably by the large np pairing compared to the results without np pairing. The sensitivity of the 2 tv double-beta decay transition on the particle-particle force strength parameter g pp is also reduced compared with the results without np pairing correlations.

THE QUARK MODEL, DEUTERON FORM FACTORS AND NUCLEAR MAGNETIC MOMENTS

The study of the deuteron electromagnetic form factors based on the quark cluster model is reviewed. The deuteron wave function is derived from a microscopic quark Hamiltonian with the help of the Resonating Group Method. One-pion and one-gluon exchange potentials are included in addition to a quadratic confinement potential. The photon is coupled directly to the quarks. Aside from the one-body impulse current, pion and gluon exchange currents are included on the quark level. Due to the Pauli principle on the quark level, new electromagnetic currents arise which are not present on the nucleon level. These currents, called quark exchange currents, describe processes in which a photon couples to a quark or a pair of quarks interacting via gluon or pion exchange and which are accompanied by a simultaneous quark interchange between the two threequark clusters (nucleons). They are small for low momentum transfers but appreciably influence the electromagnetic structure of the deuteron beyond a momentum transfer of q=5 fm−1. The discussion is extended to the magnetic moments of 15N, 17O and 39K by introducing the quark exchange currents as effective operators on the nucleon level. The quark exchange currents written in terms of nonlocal and spin-isospin dependent nuclear operators are effective only at short distances. They are evaluated with shell-model (harmonic oscillator) wave functions including the (short-range) Brueckner correlations. The Bethe-Goldstone equation is solved with our effective NN potential, which is derived from a microscopic quark Hamiltonian. The quark exchange currents shift the isovector magnetic moment of 39K by −20% from its Schmidt value.