Dissipative Time Dependent Density Functional Theory

The simplest density functional theory due to Thomas, Fermi, Dirac and Weizsäcker is employed to describe the non-equilibrium thermodynamic evolution of an electron gas. The temperature effect is introduced via the Fermi-Dirac entropy, while the irreversible dynamics is described by a non-linear diffusion equation. A dissipative Kohn-Sham equation is also proposed, which improves the Thomas-Fermi-Weizsäcker kinetic functional.     

Kinetic energy functionals for molecular calculations

The orbitals and eigenvalues of a Kohn-Sham density functional theory calculation can be used to determine the kinetic potential, the functional derivative of the non-interacting kinetic energy. Thus, approproximate kinetic energy functionals can be systematically parameterized to improve their functional derivatives. Fitting procedures have been applied to various functional forms and the quality of the resulting functionals investigated using variationally optimized densities. The best functionals include the full correction of Weizsäcker and a modulation of the Thomas-Fermi p^5/3 term by a function of the distance from the nucleus. These atom-specific functionals predict virtually exact shell structure, and may be combined readily into a functional which supports molecular binding.     

A rigorous energy density functional approach

A rigorous approach for constructing of energy density functionals is developed using the local-scale point transformation method. The exact kinetic energy density τ[ρ] contains two terms: the original Weizsäcker term τ_w[ρ] and a Thomas-Fermi-like term τ_w[ρ]. An energy density functionalE[ρ] is built up within a Slater determinant approximation using the most general Skyrme-type forces. It is demonstrated that all Hartree-Fock nuclear ground state results are reproduced quite accurately and consequentlyE[ρ] includes the shell effects accounted mostly by the particular form of τ[ρ]. The approach presented is compared with both Extended Thomas-Fermi and Expectation Value methods. Further possible applications are briefly considered.     

Zur isotropen Turbulenz

On Isotropic Turbulence We demonstrate for the example of isotropic turbulence according to the statistical approaches by A. N. Komolgoroff and by W. Heisenberg and C. F. v. Weizsäcker that the mathematical “chaos” is a state with a high degree of order. This isotropic “chaos” vanishes in a finite time by energy dissipation (J. R. Mayer's fundamental experiment). According Einstein's formula for stochastic motions “chaos” becomes a static fluid with a higher temperature.     

Zero-temperature equation of state of metals in the statistical model with density gradient correction

The density gradient correction to kinetic energy (nine times smaller than the original von Weizsäcker correction) has been used within local density formalism to calculate the cold compression curve of metals. Numerical results are reported for Li, Be, Al and Cu. The modifications to the Thomas-Fermi-Dirac (TFD) results strongly depend, in the low compression range (?/?₀ ? 5), on the electron density of the material. The relative difference between the pressures in the present model and the TFD model regularly decreases with increasing compression, suggesting that TFD is reliable for ?/?₀ ? 50 in Li, ??₀ ? 20 in Be and Al and ?/?₀ ? 10 in Cu.     

A quantitative approach to ionic adsorption

Binding properties of multi-atomic systems with ionic bonds are calculated by using a recently developed method based on the density functional formalism. The charge density is obtained from a superposition of the respective atomic densities where the charge transfer between these atoms is chosen such that the total energy attains a minimum. The kinetic energy of the electrons can to a very good approximation be calculated by means of a modified Thomas-Fermi-v. Weizsäcker expression. The molecules NaCl and NaW and a NaW₄ cluster have been treated as model systems for ionic interaction. Moreover, we have computed the binding properties of a Na atom adsorbed on a W(100) surface. This particular problem is the primary subject of the present study. The calculations yield binding energies, binding distances, vibrational frequencies, and induced dipole moments.     

Nonexistence in Thomas-Fermi-Dirac-von Weizsäcker Theory with Small Nuclear Charges

We study the ionization problem in the Thomas-Fermi-Dirac-von Weizsäcker theory for atoms and molecules. We prove the nonexistence of minimizers for the energy functional when the number of electrons is large and the total nuclear charge is small. This nonexistence result also applies to external potentials decaying faster than the Coulomb potential. In the case of arbitrary nuclear charges, we obtain the nonexistence of stable minimizers and radial minimizers.     

Existence and uniqueness of positive solutions of semilinear elliptic equations with Coulomb potentials on ℝ3

We study the positive solutions of a semilinear equation with a Coulomb potential on ?³. We give a new uniqueness theorem for the positive radial solutions of such an equation and we apply these results to the Thomas-Fermi-Dirac-von Weizsäcker equation without electrostatic repulsion.     

The History and Impact of the CNO Cycles in Nuclear Astrophysics

The carbon cycle, or Bethe-Weizsäcker cycle, plays an important role in astrophysics as one of the most important energy sources for quiescent and explosive hydrogen burning in stars. This paper presents the intellectual and historical background of the idea of the correlation between stellar energy production and the synthesis of the chemical elements in stars on the example of this cycle. In particular, it addresses the contributions of Carl Friedrich von Weizsäcker and Hans Bethe, who provided the first predictions of the carbon cycle. Further, the experimental verification of the predicted process as it developed over the following decades is discussed, as well as the extension of the initial carbon cycle to the carbon-nitrogen-oxygen (CNO) multi-cycles and the hot CNO cycles. This development emerged from the detailed experimental studies of the associated nuclear reactions over more than seven decades. Finally, the impact of the experimental and theoretical results on our present understanding of hydrogen burning in different stellar environments is presented, as well as the impact on our understanding of the chemical evolution of our universe.     

A nuclear mass formula based onSU(4) symmetry

It is argued that mass anomalies at the N≈Z line are associated with SU(4) isospin-spin symmetry. Drawing on these arguments, a Weizsäcker-type nuclear mass formula is investigated which has the eigenvalue of the quadratic Casimir operator of SU(4) as a Wigner term. This SU(4)-based mass formula yields a better agreement than the one with the usual Wigner term |N—Z|/A. In addition, the SU(4) eigenvalue expression adequately replaces the usual pairing term of the Weizsäcker formula giving a lower overall rms deviation than the latter.     

A geometric model of space-time and matter

A geometric model of a spin one-half charged particle based on a modified version of Weyl's unified model of electromagnetism and gravity is presented. The model predicts a short-range force which we identify with the strong force. A nuclear mass formula similar to the Weizsäcker mass formula provides experimental verification of the geometric approach.     

Fragmentation of light nuclei by intermediate energy photons

New data on the fragmentation of carbon nuclei by photons with energies from 800 to 1500 MeV, obtained in the collaboration GRAAL, are presented. These data include the yields of heavier fragments than nucleons. Comparison of new results with literature data, obtained with real and virtual photons in reactions with electrons and relativistic ions (Coulomb dissociation) is done using a general approach in frame of the Weizsäcker–Williams model. Possible reasons for the observed differences between them are discussed.     

Mass formula based on SU(4)

The quality of the fit of experimental masses by a mass formula based on the two-body Casimir operator of SU(4) is tested and found to be at least as good as that of the Weizsäcker mass formulae, in spite of the fact that this formula is inherently less flexible. The physical basis for, and some ramifications of, this formula are discussed. A simple form for the shell corrections is then added in the formulae, leading to improved fits without modification of the above conclusions.     

Two-photon production of leptons at hadron colliders in semielastic and elastic cases

The mechanism of two-photon dilepton production is studied in the equivalent-photon (Weizsäcker–Williams) approximation. This approximation is shown to describe well experimental data from hadron accelerators. The respective total and differential cross sections were obtained for the LHC and for the Tevatron collider at various energies of colliding hadrons. The differential cross sections were studied versus the dilepton invariant mass, transverse momentum, and emission angle in the reference frame comoving with the center of mass of colliding hadrons. The cases of semielastic and inelastic collisions were examined.     

An Orbital‐Free Quantum Hypernetted Chain Model Based Perturbation Theory for Equation of State of Hydrogen and Helium in Warm Dense Regime

We present in this work, a thermodynamic perturbation theory for equation of state of hydrogen and helium in the warm dense regime. The system is modeled as a mixture of classical point ions and quantum electrons. A perturbation series for Helmholtz free energy and correlation functions of the ions and electrons as a function of density and temperature is proposed. Combining the classical thermodynamic perturbation theory and the orbitial‐free quantum hyper‐netted chain theory, a systematic procedure to obtain the terms of the perturbation series is developed. The ion‐ion correlations are treated within the hyper‐netted chain approximation and the ion‐electron correlations are treated within the Thomas‐Fermi‐Dirac‐Weizsäcker approximation. The method has been applied to obtain isotherms of hydrogen and helium in the warm dense regime. The isotherms are compared with available ab‐initio data and the results are analyzed. A good agreement with ab‐initio data has been observed for pressures greater than one Mbar. Advantages and limitations of the present method are discussed along with possible future improvements. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Quest for a Microscopic Nuclear Mass Formula

After stressing how well the purely macroscopic 1935 mass formula of von Weizsäcker works, we discuss the general problem of deriving nuclear masses from basic nucleonic interactions. We then describe the very recent Skyrme-Hartree-Fock-BCS mass formula of Goriely et al., the first and only to be entirely microscopic. We conclude by stressing how much more work has to be done before reliable extrapolations can be made from the mass data out to the highly neutron-rich part of the nuclear chart where the r-process of nucleosynthesis takes place.     

Mean-field vs. Monte Carlo equation of state for the expansion of a Fermi superfluid in the BCS-BEC crossover

The equation of state (EOS) of a Fermi superfluid is investigated in the BCS-BEC crossover at zero temperature. We discuss the EOS based on Monte Carlo (MC) data and asymptotic expansions and the EOS derived from the extended BCS (EBCS) mean-field theory. Then we introduce a time-dependent density functional, based on the bulk EOS and Landau’s superfluid hydrodynamics with a von Weizsäcker-type correction, to study the free expansion of the Fermi superfluid. We calculate the aspect ratio and the released energy of the expanding Fermi cloud showing that MC EOS and EBCS EOS are both compatible with the available experimental data of ⁶Li atoms. We find that the released energy satisfies and approximate analytical formula that is quite accurate in the BEC regime. For an anisotropic droplet, our numerical simulations show an initially faster reversal of anisotropy in the BCS regime, later suppressed by the BEC fluid.     

Charge oscillations and photoabsorption of the Thomas-Fermi-Dirac-Weizsäcker atom

Following the work of Bloch and Ball et al. the theory of linear response of a hydrodynamic Thomas-Fermi model is extended to a general density dependent energy functional. The theory is applied to the Thomas-Fermi-Dirac-Weizsäcker atom. Normal mode solutions and the atomic photoabsorption cross section are calculated.     

A survey of neutron and proton binding energies

Limiting values of last-nucleon binding energies are available from measured reaction energies of induced transformations satisfying the selection rule ΔA=0, ±1 (A=mass number) in about 140 cases. In 14 cases upper and lower limiting values are known separately, and these agree in each case within the limits of experimental error. Taking the other 126 values as ‘true’ (rather than limiting) values, and using information concerning total energies of β-disintegration, the number of last-nucleon binding energies which are reasonably well determined is increased to more than 600. These values are tabulated, and discussed in relation to the v. Weizsäcker mass formula. Apart from certain anomalies which are treated individually, the discussion brings out the effect of α-unit structure in light nuclei, diminishing in importance to become almost negligible beyond A=40, and yields mean values for nucleon-parity dependent terms in the range A～215. The recovery of ‘normal’ last-nucleon binding energy after completion of a closed shell is followed in detail over the range 126ˇ-Nˇ-136 (N=neutron number) and somewhat less closely over the range 82ˇ-Zˇ-92 (Z=proton number).