Relativistic Hartree Approximation Research Articles

Vacuum fluctuations in spherical nuclei

The effect of the vacuum fluctuation corrections to the ground state properties of spherical nuclei is studied. The model used combines the relativistic Hartree approximation forA valence nucleons and the local density approximation for infinite number of Dirac sea nucleons. The results are compared with predictions of the Walecka mean-field theory (MFT) and the nonlinear model with scalar meson self couplings. The vacuum fluctuation corrections alone appeared to be unable to improve the MFT description of nuclei.

Meson exchange in nuclei and the EMC effect

Assuming the NN interaction in nuclei to proceed via meson exchange only, we calculate the nuclear structure function within the Walecka model which directly follows from the OBE mechanism in relativistic Hartree approximation. The meson part of the structure function is found to be concentrated at extremely small values of the scaling variable ( x≲0.05), being very large in this region since the characteristic momenta of nuclear form factors are of order R −1, R is the nuclear radius. At the same time the nucleon part of the structure function is reduced because of the final-state interaction with the vector meson field of nucleus. As a result the ratio. R( x) = F 2A(N)( x)/ F 2N( x)) is the order of 100 at x<0.05 and about 0.64 at moderate x-values, where F 2A(N)( x) is the nuclear structure function related to one nucleon. This result is not altered when both the exchange corrections to the Walecka model and the many-meson exchange are taken into account. Thus, the exchange by vector meson, which is inherent to the Walecka model, is found to be incompatible with the observed magnitude of the EMC effect.

Finite-density effective sigma meson mass in chiral models.

Properties of chirally-invariant models of nuclear matter are calculated with the relativistic Hartree approximation. Our emphasis is on the behavior of the effective \ensuremath{\sigma} meson mass. We find that the effective \ensuremath{\sigma} mass does not scale with the effective nucleon mass, which in these calculations is proportional to the expectation value of the \ensuremath{\sigma} field, i.e., the chiral order parameter. These results suggest that a decrease in the effective \ensuremath{\sigma} mass with increasing nucleon density is not a generic feature of chiral models. We also find that the incompressibility of nuclear matter is lower for those models with higher effective \ensuremath{\sigma} masses.

Effects of the Dirac sea on the Coulomb sum value for electron scattering

Polarization effects of the Dirac sea on the Coulomb sum values for electron scattering are investigated within the framework of the random-phase approximation (RPA) in the relativistic σ-ω model. Divergence due to NN̄ excitations in the correlation functions is renormalized using a method of dimensional regularization. Particle-hole correlations reduce the sum values of the relativistic Hartree approximation by 15≈20%, while effects of the Dirac sea reduce them further by about 10≈20%. The quenching phenomena of the Coulomb sum values observed in medium heavy nuclei are rather well explained in the renormalized relativistic RPA.

A new method to calculate magnetic moments in relativistic mean field theories

Abstract We show that the magnetic moment problem in relativistic mean field theories can be solved entirely within the relativistic Hartree approximation if it is used in the proper way: the wave functions of odd-mass nuclei are no longer time reversal invariant and therefore baryonic currents cannot be neglected in the mean field approximation. If they are taken into account they produce a non-vanishing vector part of the ω-field, which changes the nuclear wavefunction and the resulting magnetic moments. For the isoscalar part of magnetic moments in LS closed-shell nuclei plus or minus one nucleon we find excellent agreement with the Schmidt values.

Quasielastic electron scattering on 12C and 40Ca in the relativistic Hartree approximation Relativistic corrections to the response functions

The quenching of the longitudinal response functions observed in high-energy electron scattering is studied within a self-consistent relativistic Hartree approximation in the σ-ω model. The σ-ω model yields a small nucleon effective mass which reduces the response functions around the peak position through the Perey effect and the relativistic nuclear current. This reduction, however, is partially cancelled by effects from the relativistic kinematics. Consequently, the reduction is not enough to explain the quenching phenomena of the longitudinal response functions around the peak. The relativistic Coulomb sum values, on the other hand, are little reduced, compared with those of nonrelativistic models, failing to reproduce experimental data. In contrast to the longitudinal ones, the transverse response functions around the peak are enhanced by relativistic effects on the nuclear current. Nevertheless, the present model underestimates experimental values at high momentum transfer.

Vacuum polarization effects on meson propagators

The stability of uniform nuclear matter in the relativistic Hartree approximation to the Walecka model is investigated by calculating the meson propagators in the one-loop approximation. In this renormalizable model, Dirac nucleons interact with neutral scalar and vector mesons. At the one-loop level, nuclear matter is found to be unstable to short-wavelength modes because of vacuum polarization effects. This occurs at momenta greater than twice the free-nucleon mass. The onset of instability is pushed to larger momenta with increasing baryon density as the nucleon effective mass decreases. The implications of short-range correlations and vertex corrections are discussed.

Relativistic description of deformed rare earth nuclei

Ground state properties of heavy deformed nuclei in the rare earth region are described in the framework of the relativistic Hartree approximation. Both linear and non-linear parameter sets of the lagrangian are used. The non-linear sets reproduce the experimentally observed binding energies, the charge radii and the quadrupole moments, with the same quality as non-relativistic density dependent mean field calculations with Skyrme forces.

Thermodynamical behaviour of nuclear matter with and without baryon density in a saturating chiral field theory

A saturating chiral field theory of nuclear matter has been used to investigate the phase transition in nucleon-antinucleon plasma at high temperatures and the liquid-gas phase transition in nuclear matter at low temperatures. Bulk properties are studied both in mean-field solution and in relativistic Hartree approximation. In addition, the influence of an explicit symmetry breaking term, added to the lagrangian to satisfy PCAC, is studied.

The Coulomb sum for electron scattering in the relativistic random phase approximation

The Coulomb sum value for electron scattering is estimated using the relativistic random phase approximation (RPA) in the field theory for nuclear matter. RPA correlations due to σ-, ω- and ϱ-meson exchanges reduce the sum value of the relativistic Hartree approximation by about 15–20%.

A relativistic quantum field theory of nuclear structure

A relativistic model of nuclear structure is presented based on renormalizable quantum field theories of mesons and baryons (quantum hadrodynamics). The relativistic Hartree approximation to the full field theory describes the nuclear ground state using four-component Dirac wave functions. With parameters determined from bulk nuclear properties, a nuclear shell structure is predicted. The ground state involves large Lorentz scalar and vector fields similar to the optical potentials used in calculations of mediumenergy nucleon-nucleus scattering. A consistent relativistic approach to electron-nucleus scattering is also discussed, and examples of elastic charge and magnetic scattering are illustrated.

Sudden change in the bulk properties of nucleon-antinucleon plasma at a certain temperature in the relativistic Hartree approximations.

The purpose of this paper is to study the bulk properties of nucleon-antinucleon plasma as a function of temperature in the relativistic Hartree approximation which takes into account the contribution of vacuum-fluctuation energy using both Walecka's model and the chirally symmetric \ensuremath{\sigma} model. This paper is an extension of previous work of Theis et al. based on Walecka's model and carried out in the mean-field-theory approximation. In our calculation the effective nucleon mass ${M}^{\mathrm{*}}$ can be both greater and less than the free-nucleon mass M. When we consider the ${M}^{\mathrm{*}}$of the solution, we find that there is a peak in the specific heat of the plasma divided by its limiting analytic Stefan-Boltzmann form around T=225 MeV in the case of Walecka's model. This fact may be interpreted by saying that the hadron phase of nuclear matter alone undergoes an abrupt change in the bulk properties around ${\ensuremath{\rho}}_{v}$\ensuremath{\sim}0 and T=225 MeV. We also find the surprising result that the energy density corresponding to the ${M}^{\mathrm{*}}$&gt;M branch of the solution is less than that of the ${M}^{\mathrm{*}}$of solution beyond a certain temperature. In the case of the \ensuremath{\sigma} model, the solution for nonzero ${M}^{\mathrm{*}}$not exist beyond a certain temperature.

Saturating chiral field theory for the study of nuclear matter in the relativistic Hartree approximation

The bulk properties of nuclear matter are studied by considering a chirally invariant lagrangian which contains an interaction term involving scalar and vector mesons, of the form (apart from a numerical factor) used by Boguta. The calculation is performed in the relativistic Hartree approximation which includes the baryon vacuum fluctuation correction.

Field-theoretical model for nuclear and neutron matter. I. Equation of state.

We analyze a field-theoretical Lagrangian model for the many-body problem of baryonic matter. The Lagrangian describes the baryonic interaction through the exchange of \ensuremath{\sigma}, \ensuremath{\pi}, \ensuremath{\omega}, and \ensuremath{\rho} mesons, and contains various models studied in the literature as particular cases. The model is solved in the relativistic Hartree approximation using the technique of the covariant Wigner function. The vacuum fluctuation effects are considered and renormalized through a counterterm procedure. The equation of state is obtained in this renormalized Hartree approximation at finite temperature and in the presence of homogeneous and isotropic meson condensates. In a first particular case (the Serot model) the parameters of the model are determined in order to explain the nuclear saturation and symmetry energy of symmetric nuclear matter at nuclear density and to account for the expected low-density behavior of dense matter after the neutron drip in a phenomenological way. In this case the incompressibility of nuclear matter at the nuclear density is 460 MeV. A second determination of the parameters of the general model leads to a \ensuremath{\sigma} model with an explicit symmetry-breaking term and is coupled to the \ensuremath{\omega} and \ensuremath{\rho} mesons in a renormalizable way. The equation of state thus obtained explains accurately all the properties of nuclear matter at nuclear density. In particular, we obtain a nuclear incompressibility of 225 MeV. The analysis of the resulting energy density as a function of the meson condensates shows that the presence of homogeneous and isotropic pion condensates or abnormal states of Lee-Wick type in neutron matter at T=0 is energetically forbidden in this approximation.

Single-particle potential in a relativistic Hartree-Fock mean field approximation

A relativistic Hartree-Fock mean field approximation is investigated in a model in which the nucléon field interacts with scalar and vector meson fields. The Hartree-Fock potential felt by individual nucléons enters in a relativistic Dirac single-particle equation. It is shown that in the case of symmetric nuclear matter one can always find a potential which is fully equivalent to the most general mean field and which is only the sum of a Lorentz scalar, of one component of a Lorentz tensor and of the fourth component of a Lorentz vector. A non-relativistic potential is derived which yields exactly the same single-particle energies and elastic scattering phase shifts as the relativistic Hartree-Fock potential. Analytical results are presented in the case of nuclear matter. A local density approximation is constructed which enables one to consider finite nuclei. The input parameters of the model can be chosen in such a way that the empirical saturation properties of nuclear matter are well reproduced. Good agreement is obtained between the calculated non-relativistic potential and the empirical value of the real part of the optical-model potential at low and at intermediate energy. At intermediate energy, the wine-bottle bottom shape which had previously been found for the potential in the framework of the relativistic Hartree approximation is maintained when the Fock contribution is included.

Optical-model potential in a relativistic quantum field model

The average nucleon-nucleus potential at low and medium energy is investigated in the framework of a relativistic quantum field model. Using the same input parameters as Brockmann in his recent study of nuclear ground states, we calculate the self-consistent relativistic Hartree potential at positive energy in the case of infinite nuclear matter and of $^{16}\mathrm{O}$ and $^{40}\mathrm{Ca}$. This potential is the sum of a scalar operator and of the fourth component of a vector operator. We construct its Schr\odinger-equivalent potential by eliminating the small component of the Dirac spinor. The central part of this Schr\odinger-equivalent potential is in fair agreement with empirical values at low and intermediate energy. Particular attention is paid to the intermediate energy domain, in which the calculated potential is repulsive in the nuclear interior and attractive at the nuclear surface. This is in keeping with some empirical evidence and is similar to results found in the framework of the nonrelativistic Brueckner-Hartree-Fock approximation. The spin-orbit potential of the relativistic Hartree model is also in good agreement with empirical values.NUCLEAR REACTIONS Calculated average nuclear field of nuclear matter, $^{16}\mathrm{O}$ and $^{40}\mathrm{Ca}$ at positive energy from relativistic Hartree approximation.

Relativistic fermion gas interacting through a scalar field. I. Hartree approximation

A high-density fermion gas interacting through a scalar field is studied in the relativistic Hartree approximation, as a model for superhigh-density astrophysical systems. The Hartree quasiparticle states are the superposition of positive- and negative-energy states and can be constructed by a Bogoliubov-type canonical transformation. The resulting effective mass is strongly field-dependent, and approaches 0 as the density increases into the ultrarelativistic domain. The ensuing equation of state exhibits a phase transition and a bound state in the intermediate-density range, but becomes perfect-gas-like at high densities. There is no collapse, and stronger coupling enhances the perfect-gas-like behavior. An alternate picture is provided by considering the fermion gas in the background of zero-momentum bosons. Numerical comparisons with existing scalar-meson coupling data are provided, with emphasis on regions of neutron-star densities.

Solid-State and Nuclear Results from a Measurement of the Pressure Dependence of the Energy of the Resonance Gamma Ray ofAu197

Metallic gold has been subjected to a sequence of pressures in the range from 0 to 70.6 kbar at a temperature of 4.2\ifmmode^\circ\else\textdegree\fi{}K, and the change of the energy of the $^{197}\mathrm{Au}$ resonance $\ensuremath{\gamma}$ ray with pressure has been measured. The measurements were made by the M\ossbauer method. The result for $\frac{d{E}_{\ensuremath{\gamma}}}{\mathrm{dP}}$ is in good agreement with an earlier prediction for this quantity based on a correlation of the M\ossbauer isomer shift with residual electrical resistance for a series of dilute gold alloys. We have used the Wigner-Seitz approximation to describe the gold $6s$ band, and this wave function has been calculated in a relativistic Hartree approximation. Within this treatment we find that the gold nucleus increases in size when excited, with $\frac{\ensuremath{\Delta}r}{r}\ensuremath{\cong}+1.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$. A brief comment is made on the bearing which these results have on the charge and screening of gold atoms in dilute gold alloys.

Equilibrium and Linear Response of a Classical Scalar Plasma

The properties of a classical many-particle system interacting through a scalar field (a scalar plasma) are studied. A relativistic Hartree approximation is applied to equilibrium to evaluate thermodynamic characteristics such as the energy, entropy, and equation of state. A gas-liquid phase transition appears at moderate densities. There is no collapse for an attractive interaction: The system can accomodate any density. Stability against small perturbation and the behavior of collective excitations are analyzed through the linear-response function.

Atomic structures and electron scattering in the relativistic Hartree approximation

The problems of atomic structure and electron scattering from heavy atoms are investigated using the relativistic Hartree approximation. The results obtained from the self-consistent field calculation of structure indicate the necessity for including relativistic effects for heavy atoms. Previously reported scattering calculations, also made using the Hartree approximation, have been extended and it is shown that the approximation ceases to be valid for an incident energy below a few hundred eV.