Momentum-dependent Interaction Research Articles

Order parameter for pairing systems.

We compare the static approximation to the Hubbard-Stratonovich representation of the partition function with the order parameter representation based on the Landau theory of phase transitions and we find that the two expressions for the partition function are very similar if we choose the expectation value of the potential for the order parameter of the system. This choice for the order parameter has certain advantages above choosing the BCS energy gap for the order parameter in that it generalizes naturally for momentum-dependent pairing interactions and that it remains well defined in exact treatments. In a simple ${\mathit{i}}_{13/2}$-model calculation different choices for the order parameter are compared with the exact grand canonical and with the static path integral results.

Evidence for dx2-y2 pairing from nuclear-magnetic-resonance experiments in the superconducting state of YBa2Cu3O7.

We calculate the electron spin susceptibility for the superconducting state of ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7}$ using a band structure with nearest- and next-nearest-neighbor hopping, a momentum-dependent spin-spin interaction, and a superconducting gap with ${\mathit{d}}_{\mathit{x}}^{2}$-${\mathit{y}}^{2}$ symmetry. Our calculated nuclear magnetic relaxation rates and Knight shift agree favorably with experiment. Our work provides further evidence for a ${\mathit{d}}_{\mathit{x}}^{2}$-${\mathit{y}}^{2}$ pairing state and demonstrates that antiferromagnetic correlations persist in the superconducting state.

Thermal properties of nuclear matter with a momentum-dependent effective interaction.

We study the effect of the momentum dependence on the thermal properties of excited dilute nuclear matter in the Boltzmann approximation. The optical potential, mean field, [ital k] mass, pressure, and incompressibility are calculated. We improve on an earlier approximate treatment and find, in most cases, relatively small quantitative differences in the behavior of the calculated observables. An exception is the [ital k] mass, which acquires a temperature dependence in the improved approximation. Phase diagrams for nuclear matter are presented and the transition of hadronic matter to quark-gluon plasma is studied.

Probing the nuclear matter equation of state with neutrons from nucleus-nucleus collisions

We report the first neutron triple-differential cross-sections measured in multiplicity-selected heavy-ion collisions. We examine the maximum azimuthal anisotropy ratio of these neutron cross-sections from 400 AMeV Nb-Nb and Au-Au collisions as a probe of the nuclear equation of state. Experiment reveals that this ratio does not depend on the mass of the colliding nuclei, contrary to the predictions of the Quantum Molecular Dynamics (QMD) transport theory. We compare the Nb-Nb data also with theoretical calculations done within the Boltzmann-Uehling-Uhlenbeck framework with both momentum-dependent and momentum-independent interactions. We find that the experiment is sensitive to the size of the nuclear incompressibility modulus K in the nuclear matter equation of state, and that the data demand a K value compatible with state-of-the-art nuclear matter calculations.

Relativistic potentials for a counterstreaming nuclear-matter scenario with a covariant momentum-dependent interaction

We calculate relativistic potentials for a counterstreaming nuclear-matter scenario, employing a covariant effective interaction with momentum-dependent potentials. In contrast to the σω mean-field model, this interaction is formulated consistent with the observed momentum dependence of the optical potential in nucleon-nucleus collisions. We compare parametrizations of this new interaction to the nonlinear σω model used so far in relativistic transport theories. Finally, we present an approximate method to calculate the momentum-dependent potentials in actual simulations of heavy-ion collisions and compare to the exact solution.

Nuclear equation of state with momentum-dependent interactions.

The nuclear-matter equation of state is studied with a momentum-dependent effective interaction. The incompressibility and the zero-temperature optical potential are calculated analytically. The momentum distribution of nucleons is examined and the effect of the momentum dependence on pion-production rates is illustrated. The phase transition to a quark-gluon plasma, described by a bag-model equation of state, is studied in an approximation suitable for infinite nuclear matter.

Inversion of collective matter flow and equation of state

The multidetector array Mur + Tonneau has been used to perform a 4π detection of charged particles and fragments emitted in reactions at energies ranging from 25 to 95 MeV/u. The collective transverse momentum in the reaction plane (sidewards flow parameter) is observed to strongly vary as a function of impact parameter and incident energy. The measured values have to be corrected for the effects due to the error on the reaction plane determination and to the detector limitations. Those limitations caused by the method itself are more important than those caused by the detector. The flow value changes from negative values (negative scattering at low energies) to positive values (compression, at high energies). The inversion (or balance) energy is in the range 50–100 MeV/u, depending on the system and impact parameter value. Comparisons between experimental data and theoretical values are shown for the system 40Ar+ 27Al. These studies (BUU, Landau-Vlasov, QMD) show that the flow, and especially the inversion energy, is sensitive to the nucleon-nucleon cross section σNN in medium. The data indicate that a momentum dependent effective interaction should be used. Then, the flow value is not much sensitive to the incompressibility modulus K, but possibly in peripheral collisions. Additionnal observables should be used. The azimuthal distribution of mid-rapidity particles indicates a rotation-like behaviour of the interaction region.

Transverse momenta, nuclear equation of state, and momentum-dependent interactions in heavy-ion collisions.

We investigate the generation of transverse momentum in high-energy heavy-ion collisions and its relation to the nuclear equation of state. We find that streamer chamber data can be fitted by Boltzmann-Uehling-Uhlenbeck calculations with a momentum-dependent potential that closely models realistic nuclear matter interactions and yields an equation of state with {ital K}=215 MeV.

Multifragmentation, fragment flow, and the nuclear equation of state.

The quantum molecular dynamic method is used to study multifragmentation and fragment flow and their dependence on in-medium cross sections, momentum dependent interactions, and the nuclear equation of state, for collisions of $^{197}\mathrm{Au}$${+}^{197}$Au and $^{93}\mathrm{Nb}$${+}^{93}$Nb in the bombarding energy regime from 100 to 800A MeV. Time and impact parameter dependence of the fragment formation and their implications for the conjectured liquid-vapor phase transition are investigated. We find that the inclusive fragment mass distribution is independent of the equation of state and exhibits a power-law behavior Y(A)\ensuremath{\sim}${A}^{\mathrm{\ensuremath{-}}\ensuremath{\tau}}$ with an exponent \ensuremath{\tau}\ensuremath{\approxeq}-2.3. True multifragmentation events are found in central collisions for energies ${E}_{\mathrm{lab}\mathrm{\ensuremath{\sim}}30--200}$ MeV/nucleon. The associated light fragment (d,t,\ensuremath{\alpha}) to proton ratios increase with the multiplicity of charged particles and decrease with energy, in agreement with recent experiments. The calculated absolute charged particle multiplicities, the multiplicities of intermediate mass (Ag4) fragments, and their respective rapidity distributions do compare well with recent 4\ensuremath{\pi} data, but are quite insensitive to the equation of state.On the other hand, these quantities depend sensitively on the nucleon-nucleon scattering cross section, and can be used to determine \ensuremath{\sigma} experimentally. The transverse momentum flow of the complex fragments increases with the stiffness of the equation of state. Reduced (in-medium) n-n scattering cross sections reduce the fragment flow. Momentum dependent interactions increase the fragment flow. It is shown that the measured fragment flow at 200A MeV can be reproduced in the model. We find that also the increase of the ${p}_{x}$/A values with the fragment mass is in agreement with experiments. The calculated fragment flow is too small as compared to the plastic ball data, if a soft equation of state with in-medium corrections (momentum dependent interactions plus reduced cross sections) is employed. An alternative, most intriguing resolution of the puzzle about the stiffness of the equation of state could be an increase of the scattering cross sections due to precritical scattering in the vicinity of a phase transition.

Quantum molecular dynamics a microscopic model from UNILAC to CERN energies

We demonstrate that the microscopic QMD approach is useful to study heavy ion collisions from fusion fussion phenomena to the quest for signals of the quark gluon plasma. We discuss the possibilities and difficulties to determine the nuclear equation of state from heavy ion collisions. We investigate the influence of momentum dependent interactions and of in medium corrections to the nucleon-nucleon cross sections in the framework of the QMD model. The model is extended to low energies by including a Pauli potential in the nucleon-nucleon interaction. We show that it is possible to extract information on the effective cross sections from the experimental rapidity distributions of the fragments. We also investigate the transverse momentum of complex fragments with and without in medium corrections. The experimental data yield evidence for a stiff equation of state. A covariant extension of the QMD model is presented, which is applied to very high energy (10…200 AGeV) heavy ion collisions. Particle production and decay of heavy resonances are included. Predictions of the stopping power at AGS and SPS are presented. The importance of secondary scattering and nuclear stopping up to the highest energies is demonstrated. This is particularly important for the recently observed enhancement of strangeness production, which was proposed as a signal for QGP formation.

Computational model for nuclear reaction studies: Quasiparticle dynamics.

A computation model for nuclear reaction studies which is based upon a momentum dependent interaction between quasiparticles is extended to include collisions between the nucleons. The model is first used to examine the time scales involved in fragment formation at intermediate energies, both as a function of bombarding energy and impact parameter. The distribution of excitation energies and angular momenta of the reaction products is also investigated since the model possesses stable ground states. Several reactions are chosen as examples: Ar+C at 12A MeV, and Ca+Ca at 25A, 50A, and 100A MeV. The time scale for changes in mass distribution, etc., is discussed and the need for an afterburner scheme is established.

The momentum-dependent Tptensor interaction in intermediate-energy deuteron scattering

The role of the spin dependence associated with Pauli-principle-induced break-up is studied quantitatively in the case of intermediate-energy deuteron elastic scattering from a 58Ni target. The magnitude of the momentum-dependent Tp tensor interaction, calculated using realistic theoretical models of the nucleon-nucleon interaction, is shown to pass through a local maximum in the region of 400 MeV incident deuteron energy. Comparison of numerical calculations with the available experimental data at this energy shows that the Pauli mechanism is not responsible for the outstanding discrepancies between theory and experiment.

Comparison of Vlasov-Uehling-Uhlenbeck model with 4 pi heavy ion data.

Streamer chamber data for collisions of Ar + KCl and Ar + ${\mathrm{BaI}}_{2}$ at 1.2 GeV/nucleon are compared with microscopic model predictions based on the Vlasov-Uehling-Uhlenbeck equation, for various density-dependent nuclear equations of state. Multiplicity distributions and inclusive rapidity and transverse momentum spectra are in good agreement. Rapidity spectra show evidence of being useful in determining whether the model uses the correct cross sections for binary collisions in the nuclear medium, and whether momentum-dependent interactions are correctly incorporated. Sideward flow results do not favor the same nuclear stiffness parameter at all multiplicities.

Effective interactions and elementary excitations in nuclear matter

We present a polarization potential theory for nuclear matter. Starting from a realistic two-nucleon interaction, position space Pseudopotentials which describe effective nucleon-nucleon interactions are constructed in a manner similar to that used by Aldrich and Pines in the helium liquids. Important modifications which result from tensor forces and three-body interactions are incorporated; exchange and screening effects are included. The resulting momentum-dependent quasiparticle interactions are used to calculate higher-order Fermi liquid parameters, dynamic and static structure functions, and static polarizabilities. Comparisons are made with recent microscopic calculations.

Generalised non-equilibrium equation of state for nuclear matter with momentum-dependent interactions

A generalisation of the equation of state (EOS) for nuclear matter to non-equilibrium situations is presented using momentum-dependent interactions. This concept is important for the case of two interpenetrating nuclei in the early state of heavy-ion collisions. The streaming-through of the two nuclei enters directly into the generalised non-equilibrium EOS via an additional Lagrange multiplier in the grand partition function. The importance of this concept is demonstrated for momentum-dependent interactions (Skyrme forces and a relativistic meson field theory) in the independent-particle approximation.

IMPORTANCE OF MOMENTUM DEPENDENT INTERACTIONS FOR THE EXTRACTION OF THE NUCLEAR EQUATION OF STATE FROM HIGH ENERGY HEAVY ION COLLISIONS

We demonstrate that momentum dependent nuclear interactions (MDI) have a large effect on the dynamics and on the observables of high energy heavy ion collisions : Pion and kaon yields are strongly suppressed by MDI. This effect is much stronger than the supression due to a hard local potential. which introduces an ambiguity into the recent extraction of the nuclear equation of state from pion (and kaon) yields. Also the transverse rnomentum distributions exhibit a strong dependence on the MDI. The collective flow angles and the deuteron to proton ratios, on the other hand, are rather insensitive to the MDI. They may therefore be related directly to the nuclear equation of state at high density. Similtaneous measurements of these observables in heavy ion collisions can give clues on the nuclear equation of state at densities of interest for supernova collapse and neutron star stability.

Importance of momentum-dependent interactions for the extraction of the nuclear equation of state from high-energy heavy-ion collisions.

We demonstrate that momentum-dependent nuclear interactions (MDI) have a large effect on the dynamics and on the observables of high-energy heavy-ion collisions: A soft potential with MDI suppresses pion and kaon yields much more strongly than a local hard potential and results in transverse momenta intermediate between soft and hard local potentials. The collective-flow angles and the deuteron-to-proton ratios are rather insensitive to the MDI. Only simultaneous measurements of these observables can give clues on the nuclear equation of state at densities of interest for supernova collapse and neutron-star stability.

Heavy-ion collision theory with momentum-dependent interactions.

We examine the influence of momentum-dependent interactions on the momentum flow in 400 MeV/nucleon heavy ion collisions. Choosing the strength of the momentum dependence to produce an effective mass ${m}^{\mathrm{*}}$=0.7m at the Fermi surface, we find that the characteristics of a stiff equation of state can be obtained with a much softer compressibility.

Semiclassical quantization of relative skyrmion-skyrmion motion

The relative motion of the two-skyrmion system is quantized semiclassically. The reduced mass is calculated as a function of the relative distance and a large enhancement is seen at short distances. An effective momentum-dependent interaction is discussed quantitatively.

Random-phase-approximation calculations and residual interactions in the sigma tau channel.

We compare numerical results for renormalized spin-isospin matrix elements within the random phase approximation. Extended calculations are carried out for a pionlike excitation in a finite nucleus; the effects of contact versus momentum-dependent residual interactions, and of \ensuremath{\Delta}-h configurations, are demonstrated. We point out the implications of our results on nuclear structure calculations, where such effects are sometimes neglected. The role of the finite geometry of the system is crucial in determining the features of the spin-isospin responses. We emphasize that a crucial test of the theory can only be achieved through a direct comparison of the longitudinal and transverse channels.