Inelastic electron scattering from nuclear matter

Stationarity of the energy expectation value as required in the variational approach can in most cases of practical interest be formulated with reference to a unitary ‘variational group’ and its associated Lie algebra. In terms of a basis of this Lie algebra, Brillouin (hypervirial) conditions and a Hessean are defined. The formulation of stationary perturbation theory (as well as of multiple perturbation theory) in the Lie-algebraic framework is straightforward. If the operatorY, which describes the perturbation of the wave function, is expandable in the Lie algebra of the variational group, a Hellmann-Feynman theorem holds (as a special case of Wigner's [2n+1] rule) and the first order operatorY1 can be obtained from minimization of a Lie-algebraic Hylleraas functional. Under the same condition for two perturbations, Dalgarno's exchange theorem holds. An analysis of the spectrum the Hessean leads to a generalization of the RPA method for any chosen variational group. Any variational group automatically generates a model excitation spectrum. Sometimes one wants to formulate the variational approach in terms of two or more (independent and noncommuting) variational groups. An example is coupled-MC-SCF theory. One must specify the order in which operators of the two groups act, but otherwise there is not much change with respect to the case of a single group. Time-dependent stationary perturbation theory, based on Frenkel's stationarity principle, is possible on similar lines. Singularities related to an indefinite phase, which plague traditional time dependent perturbation theory are automatically avoided. In the framework of stationary perturbation theory the dipole length and dipole velocity formulas for a transition element are equivalent. For a time-dependent Hamiltonian there is no unique definition of the corresponding energy. There are various possibilities to define a ‘pseudo-energy’. One of these definitions is consistent with a special form of a time-dependent Hellmann-Feynman theorem. For a perturbation periodic in time so-called Floquet states exist and stationary time-dependent perturbation theory starting from a stationary state of the unperturbed problem automatically leads to these. For Floquet states a genuine stationarity condition can be derived, that is based on the concept of a generalized Hilbert space and that does not suffer from the shortcomings of Frenkel's principle. The perturbation formalism for these is surprisingly close to that of the time-independent theory. For degenerate and near-degenerate states a quasidegenerate generalization of stationary perturbation theory is possible.

The phase diagram of nuclear and quark matter at high baryon density

Time-dependent perturbation theory in quantum mechanics is divergent at long times when the perturbation induces a resonance between two eigenstates of the unperturbed Hamiltonian. Divergences in perturbation theory are also common in quantum field theory and in critical phenomena. The renormalization group (RG) was designed to deal with these divergences. In the last two decades, this procedure has been extended to dynamical systems where the perturbation theory diverges in the long-time limit. In this article, we first review the connection between RG in the context of field theory and RG in the context of dynamical systems. We then show that the long-time divergence in the resonant situation in the time-dependent perturbation theory in quantum mechanics can be removed by using a RG-aided calculational scheme.

Nuclear matter calculations using separable potentials

Realistic density dependent CDM3Yn versions of the M3Y interaction have been used in an extended Hartree-Fock (HF) calculation of nuclear matter (NM), with the nucleon single-particle potential determined from the total NM energy based on the Hugenholtz-van Hove theorem that gives rise naturally to a rearrangement term (RT). Using the RT of the single-nucleon potential obtained exactly at different NM densities, the density- and energy dependence of the CDM3Yn interactions was modified to account properly for both the RT and observed energy dependence of the nucleon optical potential. Based on a local density approximation, the double-folding model of the nucleus-nucleus optical potential has been extended to take into account consistently the rearrangement effect and energy dependence of the nuclear mean-field potential, using the modified CDM3Yn interactions. The extended double-folding model was applied to study the elastic $^{12}$C+$^{12}$C and $^{16}$O+$^{12}$C scattering at the refractive energies, where the Airy structure of the nuclear rainbow has been well established. The RT was found to affect significantly the real nucleus-nucleus optical potential at small internuclear distances, giving the potential strength close to that implied by the realistic optical model description of the Airy oscillation.

Heavy baryons in compact stars

Generalized Hartree-Fock Overhauser orbitals, corresponding to a (zero pressure) periodic structure of $\ensuremath{\alpha}$ particles, are shown to have lower energy than homogeneous nuclear matter with a Skyrme interaction, at subnuclear densities.NUCLEAR STRUCTURE Nuclear matter, Hartree-Fock approximation, cluster models.

Short range correlations are introduced using the unitary correlation method in a relativistic approach to the equation of state of the infinite nuclear matter in the framework of the Hartree-Fock approximation. It is shown that the correlations give rise to an extra node in the ground state wave function in the nucleons, contrary to what happens in nonrelativistic calculations with a hard core. The effect of the correlations on the ground state properties of nuclear and neutron matter is studied. The nucleon effective mass and equation of state (EOS) are very sensitive to short range correlations. In particular, if the pion contact term is neglected, a softening of the EOS is predicted. Correlations have also an important effect on the neutron matter EOS which presents no binding but only a very shallow minimum, contrary to the Walcka model.

Non-Hermitian time-dependent perturbation theory: Asymmetric transitions and transitionless interactions

Several problems encountered on the way to a microscopic derivation of covariant density functionals are discussed. The starting point are semi-microscopic functionals adjusted to ab-initio calculations in nuclear matter. Since it is impossible to derive in this way the strength of tensor terms in such functionals we propose a solution of the relativistic Brueckner- Hartree-Fock equations in finite nuclei and show first results.

Localized solutions of the BETHE—GOLDSTONE equation for two nucleons in nuclear matter are examined as a function of the center-of-mass momentum (c. m. m.) of the two nucleons. The equation depends upon the c. m. m. as parameter due to the dependence upon the c. m. m. of the projection operator appearing in the equation. An analytical solution of the equation is obtained for a non-local but separable potential, whereby a numerical solution is also obtained. An approximate solution for small c. m. m. is calculated for a square-well potential. In the range of the approximation the two analytical solutions agree exactly.

Off-shell effects in rank-two separable potentials

The tensor-optimized Fermi sphere (TOFS) theory is applied first for the study of the property of nuclear matter using the Argonne V4$^\prime$$NN$ potential. In the TOFS theory, the correlated nuclear matter wave function is taken to be a power-series type of the correlation function $F$, where $F$ can induce central, spin–isospin, tensor, etc. correlations. This expression has been ensured by a linked cluster expansion theorem established in the TOFS theory. We take into account the contributions from all the many-body terms arising from the product of the nuclear matter Hamiltonian $\mathcal{H}$ and $F$. The correlation function is optimally determined in the variation of the total energy of nuclear matter. It is found that the density dependence of the energy per particle in nuclear matter is reasonably reproduced up to the nuclear matter density $\rho \simeq 0.20$ fm$^{-3}$ in the present numerical calculation, in comparison with other methods such as the Brueckner–Hartree–Fock approach.

We present a fully self-consistent and relativistic calculation of the nucleon mean free path in nuclear matter and finite nuclei. Starting from the Bonn potential, the Dirac-Brueckner-Hartree-Fock results for nuclear matter are parametrized in terms of an effective \ensuremath{\sigma}-\ensuremath{\omega} Lagrangian suitable for relativistic density-dependent Hartree-Fock (RDHF) approximation. The nucleon mean free path in nuclear matter is derived from this effective Lagrangian taking diagrams up to the fourth order into account. For nucleon mean free path in finite nuclei, we make use of the density determined by the RDHF calculation in the local density approximation. Our microscopic results are in good agreement with the empirical data and the predictions of the Dirac phenomenology.

The nuclear potential, the transfer-induced dissipation, and the mass diffusion coefficient in heavy-ion collisions are investigated in a proximity formulation. An energy-dependent nuclear potential is calculated in the frozen wave function approximation using two slabs of symmetric nuclear matter, each described by Hartree-Fock single-particle wave functions. Corrections to the inertia parameter are also evaluated from this potential. The flux entering the window formula for the friction between two heavy ions is calculated in a simple barrier penetration model. The classically forbidden flux is found to make a significant contribution. The transfer flux arising from both the relative motion and finite temperature of the nuclei is calculated and the latter is used to estimate the mass diffusion coefficient. Using the mean trajectories from time-dependent Hartree-Fock calculations the charge variance is calculated for the reaction 84 Kr(712 MeV) + 209 Bi and is found to be in agreement with experiment.