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Nuclear collisions from 0.3 to 2 GeV/nucleon are studied in a microscopic theory based on Vlasov's self-consistent mean field and Uehling-Uhlenbeck's two-body collision term which respects the Pauli principle. The theory explains simultaneously the observed collective flow and the pion multiplicity and gives their dependence on the nuclear equation of state.
The novel momentum analysis technique introduced by Danielewicz and Odyniec can be used to detect and exhibit collective flow in the light system Ar(1800 MeV/nucleon) + KCl where the usual kinetic energy flow analysis fails. The microscopic Vlasov-Uehling-Uhlenbeck theory which includes the nuclear mean field, two-body collisions, and Pauli blocking is used to study this phenomenon. The resulting transverse momentum transfers turn out to be quite sensitive to the nuclear equation of state. From a comparison with experimental data, evidence is presented for a rather stiff nuclear equation of state. The cascade model is unable to describe the data.
The role of nonequilibrium and quantal effects in fast nucleus-nucleus collisions is studied via the Vlasov-Uehling-Uhlenbeck theory which includes the nuclear mean field dynamics, two-body collisions, and Pauli blocking. The intranuclear cascade model, where the dynamics is governed by independent NN collisions, and the Vlasov equation, where the nuclear mean field determines the collision dynamics, are also studied as reference cases. The Vlasov equation (no collision term) yields single particle distribution functions which–after the reaction–are only slightly modified in momentum space; even in central collisions, transparency is predicted. This is in agreement with the predictions of the quantal time-dependent Hartree-Fock method. In contrast, large momentum transfer is obtained when the Uehling-Uhlenbeck collision term is incorporated; then the final momentum distribution is nearly spherically symmetric in the center of mass and a well-equilibrated nuclear system is formed: the nuclei stop each other; the translational kinetic energy is transformed into randomized microscopic motion. The Vlasov-Uehling-Uhlenbeck theory is supplemented with a phase space coalescence model of fragment formation. Calculated proton spectra compare well with recent data for Ar(42, 92, and 137 MeV/nucleon) + Ca. Also the total yields of medium mass fragments are well reproduced in the present approach. The mean field dynamics without two-body collisions, on the other hand, exhibits forward peaked proton distributions, in contrast to the data. The cascade approach underpredicts the yields of low energy protons by more than an order of magnitude.
The influence of fluctuations of the shape degree of freedom in collisions of deformed nuclei with energies between 0.8 and 2.1 GeV/nucleon is analyzed on the basis of an intranuclear cascade simulation for the strongly deformed systems 46Ti+ 46Ti and 166Er+ 166Er. While there is a considerable sensitivity of the global event variables to the orientation for polarized beams and targets, this dependence disappears in the average over all orientations for impact parameter selected and integrated events. The dependence of the nuclear stopping and thermalization on the size of the system under consideration and on the bombarding energy is also investigated.
We study the recent claim that the intranuclear cascade model exhibits collective sidewards flow. 4000 intranuclear cascade simulations of the reaction Nb(400 MeV/nucleon)+Nb are performed employing bound and unbound versions of the Cugnon cascade. We show that instability of the target and projectile nuclei in the unbound cascade produces substantial spurious sidewards flow angles, for spectators as well as for participants. Once the nuclear binding is included, the peak of the flow angle distributions for the participants alone is reduced from 35° to 17°. The theoretical ‘‘data’’ are subjected to the experimental multiplicity and efficiency cuts of the plastic ball 4π electronic spectrometer system. The flow angular distributions obtained from the bound cascade—with spectators and participants subjected to the plastic ball filter—are forward peaked, in contrast to the plastic ball data. We discuss the uncertainties encountered with the application of the experimental efficiency and multiplicity filter. The influence of the Pauli principle on the flow is also discussed. The lack of flow effects in the cascade model clearly reflects the absence of the nuclear compression energy that can cause substantially larger collective sidewards motion—there is too little intrinsic pressure built up in the cascade model.
Time dependent dirac equation with relativistic mean field dynamics applied to heavy ion scattering
(1986)
We treat the relativistic propagation of nucleons coupled to scalar- and vector-meson fields in a mean-field approximation. The time-dependent Dirac and mean-meson-field equations are solved numerically in three dimensions. Collisions of 16O(300, 600, and 1200 MeV/nucleon) + 16O are studied for various impact parameters. The results are compared to other recent theoretical approaches. The calculations predict spallation, large transverse-momentum transfer, and positive-angle sidewards flow, in qualitative agreement with the data in this energy regime.
Studying Walecka's mean-field theory we find that one can reproduce the observed binding energy and density of nuclear matter within experimental precision in an area characterized by a line in the coupling-constant plane. A part of this line defines systems which exhibit a phase transition around Tc~200 MeV for zero baryon density. The rest corresponds to such systems where the phase transition is absent; in that case a peak appears in the specific heat around T~200 MeV. We interpret these results as indicating that the hadron phase of nuclear matter alone indicates the occurrence of an abrupt change in the bulk properties around ρV~0 and T~200 MeV.
We present a theoretical description of nuclear collisions which consists of a three-dimensional fluid-dynamical model, a chemical equilibrium breakup calculation for local light fragment (i.e., p, n, d, t, 3He, and 4He) production, and a final thermal evaporation of these particles. The light fragment cross sections and some properties of the heavy target residues are calculated for the asymmetric system Ne+U at 400 MeV/N. The results of the model calculations are compared with recent experimental data. Several observable signatures of the collective hydrodynamical processes are consistent with the present data. An event-by-event analysis of the flow patterns of the various clusters is proposed which can yield deeper insight into the collision dynamics.
Intranuclear cascade calculations and fluid dynamical predictions of the kinetic energy flow are compared for collisions of 40Ca + 40Ca and 238U + 238U. The aspect ratio, R13, as obtained from the global analysis, is independent of the bombarding energy for the intranuclear cascade model. Fluid dynamics, on the other hand, predicts a dramatic increase of R13 at medium energies Elab≲200 MeV/nucleon. In fact, R13(Elab) directly reflects the incompressibility of the nuclear matter and can be used to extract the nuclear equation of stat at high densities. Distortions of the flow tensor due to few nucleon scattering are analyzed. Possible procedures to remove this background from experimental data are discussed.
In this paper, I tackle the question whether one share - one vote should become a European law rule. I examine, first of all, the economic theory concerning one share - one vote and its optimality, and the law and economics literature on dual class recapitalizations and other deviations from one share - one vote. I also consider the agency costs of deviations from one share - one vote and examine whether they justify regulation. I subsequently analyze the rules implementing the one share - one vote standard in the US and Europe. In particular, I analyze the self-regulatory rules of US exchanges, the relevant provisions of the European Takeover Directive (including the well known break-through rule), and the European Court of Justice's position as to golden shares (which also are deviations from the one share - one vote standard). I conclude that one share - one vote is not justified by economic efficiency, as also confirmed by comparative law. Also the European breakthrough rule, which ultimately strikes down all deviations from one share - one vote, does not appear to be well grounded. Only transparency rules appear to be justified at EU level as disclosure of ownership and voting structures serves a pricing and governance function, while harmonisation of the relevant rules reduces transaction costs in integrated markets.