Fock Model Research Articles

Investigation of high spin properties of $$^{55}$$Cr in deformed Hartree–Fock and angular momentum projection model

Investigation of high spin properties of $$^{55}$$Cr in deformed Hartree–Fock and angular momentum projection model

Existence of Minimizers for the Dirac–Fock Model of Crystals

Existence of Minimizers for the Dirac–Fock Model of Crystals

A rigorous justification of the Mittleman’s approach to the Dirac–Fock model

A rigorous justification of the Mittleman’s approach to the Dirac–Fock model

A Superunitary Fock Model of the Exceptional Lie Supergroup mathbb {D}(2, 1; alpha )

We construct a Fock model of the minimal representation of the exceptional Lie supergroup {mathbb {D}}(2,1; alpha ). Explicit expressions for the action are given by integrating to group level a Fock model of the Lie superalgebra D(2,1; alpha ) constructed earlier by the authors. It is also shown that the representation is superunitary in the sense of de Goursac–Michel.

Minimal bipartite dimers and higher genus Harnack curves

This paper completes the comprehensive study of the dimer model on infinite minimal graphs with Fock's weights [arXiv:1503.00289] initiated in [arXiv:2007.14699]: the latter article dealt with the elliptic case, i.e., models whose associated spectral curve is of genus one, while the present work applies to models of arbitrary genus. This provides a far-reaching extension of the genus zero results of [arXiv:math-ph/0202018, arXiv:math/0311062], from isoradial graphs with critical weights to minimal graphs with weights defining an arbitrary spectral data. For any minimal graph with Fock's weights, we give an explicit local expression for a two-parameter family of inverses of the associated Kasteleyn operator. In the periodic case, this allows us to prove local formulas for all ergodic Gibbs measures, thus providing an alternative description of the measures constructed in [arXiv:math-ph/0311005]. We also compute the corresponding slopes, exhibit an explicit parametrization of the spectral curve, identify the divisor of a vertex, and build on [arXiv:math/0311062, arXiv:1107.5588] to establish a correspondence between Fock's models on periodic minimal graphs and Harnack curves endowed with a standard divisor.

Development of various methods to the investigation of the spectral properties and collision dynamics of H‐like ions taking place in dense and hot plasma environments

AbstractThe correct understanding of the behavior of highly stripped ions immersed in a dense plasma environment is a hitherto challenge for theory and it is of importance in various applications. Here, we developed four kinds of novel theoretical models to the determination of the spectral properties and collision dynamics of H‐like ions in dense and hot plasmas by employing the analytical potential [Li et al., Phys. Plasmas 2019, 26, 033301] to describe the interactions among the charged particles: (i) The electron photon processes in plasma model within the relativistic scheme; (ii) The multiconfiguration Dirac–Fock model within the framework of relativistic self‐consistent iteration approximation; (iii) The generalized pseudospectral method within the relativistic framework; and (iv) An analytic formula based on the Hartree–Fock method and the irreducible tensor theory within the non‐relativistic regime, derived from a solution of Schrödinger equation. As a typical example, energy eigenvalues, radiative transition properties, spectral line shifts, and electron impact excitation and ionization cross sections of the selected N6+ and Ne9+ ions are determined for several temperature‐density cases, characteristic of inertial confinement fusion plasmas. A comparison of these results with each other and the results from earlier calculations as well as with the available experimental data is provided. Systematic trend is determined for all the properties under study concerning increased screening. The present study not only extends our understanding of the dense plasma shielding effects, but also opens the way for future investigations allowing accurate predictions of spectral properties of ions in dense and hot plasmas aimed at providing precision data for various practical applications.

Density Functional Theory for Two-Dimensional Homogeneous Materials

We study Density Functional Theory models for systems which are translationally invariant in some directions, such as a homogeneous 2-d slab in the 3-d space. We show how the different terms of the energy are modified and we derive reduced equations in the remaining directions. In the Thomas–Fermi model, we prove that there is perfect screening, and provide decay estimates for the electronic density away from the slab. In Kohn–Sham models, we prove that the Pauli principle is replaced by a penalization term in the energy. In the reduced Hartree–Fock model in particular, we prove that the resulting model is well-posed, and give some properties of the minimizer.

Impurity effects of the Λ hyperon in the hypernuclear systems and

Based on the beyond-mean-field Skyrme–Hartree–Fock model, impurity effects of the Λ hyperon in the hypernuclear systems Mg and Si are investigated, respectively. Four cases, in which the Λ hyperon occupies the single-particle orbitals [000], [110], [101] and [101], are focused. In each case, the potential energy surface and the energy curves projected on certain angular momenta are employed to show the influence of the Λ hyperon on the nuclear core. Beside the shrinkage effect that is induced by the Λ hyperon occupying the s Λ orbital, it is found that the Λ hyperon on the p Λ orbital, [110], drives the nuclear core toward a prolate shape, while the ones on the other two p Λ orbitals, [101] and [101], drive the nuclear core toward an oblate shape. The energy spectra and the corresponding intra-band E2 transition rates for the rotational bands are given as a prediction for future experiments.

Initialization effects of nucleon profile on the π yields in heavy-ion collisions at medium energies

We study the problem of π production in heavy-ion collisions in the context of the isospin-dependent Boltzmann–Uehling–Uhlenbeck transport model. We generated nucleon densities using two different models, the Skyrme–Hartree–Fock (SHF) model and the configuration interaction shell model (SM). Indeed, inter-nucleon correlations are explicitly taken into account in SM, while they are averaged in the SHF model. As an application of our theoretical frameworks, we calculated the π − and π + yields in collisions of nuclei with A = 30–40 nucleons. We used different harmonic oscillator lengths b HO to generate the harmonic oscillator basis for SM in order to study both theoretical and experimental cases. It is found that the SM framework with b HO = 2.5 fm and SHF can be distinguished by the yield of π mesons, in this case the density distribution calculated by the SM produces more π in the collision. In comparison, SM with b HO = 2.0 fm is distinguished from SHF by the double π −/π + ratios with different large impact parameters, from which one can find that the double π −/π + ratios of SM change smoother and are less than those of SHF.

Experimental validation of theoretically predicted cut-off of L1–L3M4,5 transitions at Z = 50 through measurements of fluorescence and Coster–Kronig yields for Sn and Sb

In the present work, we have deduced the fluorescence (ω 1, ω 2, ω 3) and Coster–Kronig (CK)(f 12, f 13, f 23) yields for Sn (Z = 50) and Sb (Z = 51) from the L i (i = 1–3) sub-shell x-ray intensities measured using the energy tunable synchrotron radiation employing the selective photoionization method. For both the elements, yields have been obtained using two sets of theoretical photoionization cross sections based on the non-relativistic Hartree–Fock–Slater (HFS) model and the self-consistent Dirac–Hartree–Fock (DHF) model. In case of Sb, we have obtained a third set of measured yields also by using the experimental photoionization cross sections evaluated from independent measurements of the mass-attenuation coefficients. The experimental yields for Sb are reported for the first time by us. We have compared the present deduced fluorescence and CK yields with the Dirac–Hartree–Slater model based values, the semi-empirical values tabulated by Krause and the earlier reported values. In case of Sn, using the DHF and the HFS model based photoionization cross sections, two sets of present measured L1 sub-shell fluorescence yields (ω 1) are found to be 0.039 ± 0.007 and 0.036 ± 0.003, and the CK yields (f 13) are found to be 0.428 ± 0.107 and 0.405 ± 0.028, respectively. In case of Sb, using three sets of the photoionization cross sections (DHF, HFS and recent experimental values), the ω 1 values are measured to be 0.042 ± 0.007, 0.040 ± 0.004 and 0.047 ± 0.005, and the CK yields are measured to be 0.343 ± 0.085, 0.297 ± 0.021 and 0.247 ± 0.022, respectively. The comparison of these present measured yields with the theoretical values provided a reliable experimental evidence indicating cut-off of the intense L1–L3M4,5 CK transitions at Z = 50.

Neutron–proton interaction, bubble structure and neutron skin properties of Co in a Supernova

In the present paper, we investigated the neutron skin properties, central depletion effects, residual and average neutron–proton interactions for Co in a Type II Supernova 1987A. The neutron skin thicknesses performed by Skyrme–Hartree–Fock model for the isotopic chain of Co $$(17\le N\le 82)$$ are obtained with different methods to investigate the bulk and surface contributions to neutron skin thicknesses as a function of neutron number N and relative neutron excess I = (N-Z)/A. For this purpose, two-parameter Fermi (2pF) distributions for each nucleus are obtained by the proton and neutron density distributions. Since the neutron skin thickness demonstrates the dependency on the symmetry pressure of neutron matter and sensitive to the symmetry energy at the sub-saturation density, the neutron skin properties lead to the investigation on the formation and composition of the crusts in Supernova and neutron stars. For a proper understanding of surface effects of nuclei, the central depletion (or bubble structure) effects on neutron skin of nuclei in the Supernova are explained by calculating the bubble parameters b and b′. Furthermore, in order to illustrate accuracy of theoretical model used in the current work, the mass model approach is also analyzed by the calculating residual and average neutron–proton interactions, where the fluctuations in the mass model approaches have also been compared with the experimental and theoretical results in the literature. In order to evaluate more nuclear properties, we also analyzed the charge form factors and the single- particle energy levels for Co isotopes in the Supernova for providing nuclear data for electron-nucleus scattering experimental processes.

Properties of nuclear matter in relativistic Brueckner–Hartree–Fock model with high-precision charge-dependent potentials

Properties of nuclear matter are investigated in the framework of relativistic Brueckner–Hartree–Fock model with the latest high-precision charge-dependent Bonn (pvCD-Bonn) potentials, where the coupling between pion and nucleon is adopted as a pseudovector form. These realistic pvCD-Bonn potentials are renormalized to effective nucleon–nucleon (NN) interactions, G matrices. They are obtained by solving the Blankenbecler-Sugar (BbS) equation in nuclear medium. Then, the saturation properties of symmetric nuclear matter are calculated with pvCD-Bonn A, B, C potentials. The energies per nucleon are around −10.72 MeV to −16.83 MeV at saturation densities, 0.139 fm−3 to 0.192 fm−3 with these three potentials, respectively. It clearly demonstrates that the pseudovector coupling between pion and nucleon can generate reasonable saturation properties comparing with pseudoscalar coupling. Furthermore, these saturation properties have strong correlations with the tensor components of NN potentials, i.e., the D-state probabilities of deuteron, PD to form a relativistic Coester band. The equations of state of pure neutron matter from pvCD-Bonn potentials are almost identical, since the prominent difference of pvCD Bonn potentials are the components of tensor force, which provides very weak contributions in the case of total isospin T = 1. In addition, the charge symmetry breaking (CSB) and charge independence breaking (CIB) effects are also discussed in nuclear matter from the partial wave contributions with these high-precision charge-dependent potentials. In general, the magnitudes of CSB from the differences between nn and pp potentials are about 0.05 MeV, while those of CIB are around 0.35 MeV from the differences between np and pp potentials.

Correlation Study for The Determination of pKa of A Number of Schiff Bases Derived from N-Formyl Pyridine using Quantum Mechanical Methods

Statistical analysis employing multiparametric regression analysis was used to correlate the pKa values of 16 substituted Schiff bases (imines) derived from heterogeneous carbonyl compounds with various descriptors as parameters. These parameters are based on quantum mechanical methods. Four sets of parameters are used for these treatments. Two of them were derived depending on the semiempirical calculation represented by Austin method1 (AM1) and parameterized method3 (PM3) .. The other two sets were chosen as Ab initio models, expressed by Hartree fock (HF) and MP2 calculation models performed at 6.311G (d,p) level of theory. The descriptors employed to test their impact on the variation of the experimental pKa(s) are of two kinds, electronic and geometrical including atomic charges (Muliken and lowding), length of the imine (C=N) and connected bonds. Other molecular characteristic are also used such as energies of the HOMO and LUMO orbitals, total energies (TE), kinetic energy (KE), torsion, Vender Waals interactions of the 1,4 type (VDW 1,4) and non 1,4 type (Non 1,4 VDW). The correlation between the pKa values of the studied compounds and the selected parameters are investigated. Depending on the obtained results, several sets of parameters were constructed. The best set is indicated by the values of correlation coefficient (R‌‌‌‌‌2) and standard error (SE). Comparison are carried out between the experimental and calculated pKa obtained from AM1, PM3, MP2, and HF methods. Models of HF showed better consistency among the experimental and the calculated pKa(s) of the studied compounds. Hypothetical compounds of similar structures but with different substituents not included in the regression analysis are used to test the success of the derived equations. Good results are obtained.

A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra osp(m,2|2n)

The minimal representation of a semisimple Lie group is a 'small' infinite-dimensional irreducible unitary representation. It is thought to correspond to the minimal nilpotent coadjoint orbit in Kirillov's orbit philosophy. The Segal-Bargmann transform is an intertwining integral transformation between two different models of the minimal representation for Hermitian Lie groups of tube type. In this paper we construct a Fock model for the minimal representation of the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2|2n)$. We also construct an integral transform which intertwines the Schr\"odinger model for the minimal representation of the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2|2n)$ with this new Fock model.

Screening in the Finite-Temperature Reduced Hartree–Fock Model

We prove the existence of solutions of the reduced Hartree–Fock equations at finite temperature for a periodic crystal with a small defect, and show the total screening of the defect charge by the electrons. We also show the convergence of the damped self-consistent field iteration using Kerker preconditioning to remove charge sloshing. As a crucial step of the proof, we define and study the properties of the dielectric operator.

Properties of Neutron Stars Described by a Relativistic Ab Initio Model

Abstract Properties of neutron stars are investigated by an available relativistic ab initio method, the relativistic Brueckner–Hartree–Fock (RBHF) model, with the latest high-precision, relativistic charge-dependent potentials, pvCD-Bonn A, B, C. The neutron star matter is solved within the beta equilibrium and charge neutrality conditions in the framework of the RBHF model. Compared to the conventional treatment, where the chemical potential of leptons was approximately represented by the symmetry energy of nuclear matter, the equation of state of neutron star matter in the present self-consistent calculation with pvCD-Bonn B has a striking difference above the baryon number density n b = 0.55 fm−3. However, these differences influence the global properties of neutron stars only about 1% to 2%. Then, three two-body potentials pvCD-Bonn A, B, C, with different tensor components, are systematically applied in the RBHF model to calculate the properties of neutron stars. It is found that the maximum masses of neutron stars are around 2.21–2.30 M ⊙, and the corresponding radii are R = 11.18–11.72 km. The radii of a 1.4 M ⊙ neutron star are predicated as R 1.4 = 12.34–12.91 km, and their dimensionless tidal deformabilities are Λ1.4 = 485–626. Furthermore, the direct URCA process in neutron star cooling will happen from n b = 0.414 to 0.530 fm−3 with the proton fractions Y p = 0.136–0.138. All of the results obtained from the RBHF model only with two-body pvCD-Bonn potentials completely satisfy various constraints from recent astronomical observations of massive neutron stars, gravitational wave detection (GW170817), and simultaneous mass–radius measurement.

Asymmetry of the neutrino mean free path in partially spin polarized neutron matter within the Skyrme model

The asymmetry in the neutrino mean free path for the absorption reaction $\nu + n \rightarrow e^{-} + p$, is evaluated within hot neutron matter under a strong magnetic field. We consider densities in the range $0.05 \leq \rho \leq 0.4$ fm$^{-3}$, several temperatures up to 30 MeV and magnetic field strengths from $B=0$ up to $B=10^{18}$~G. Polarized neutron matter is described within the non--relativistic Hartree--Fock model using the LNS Skyrme interaction. The neutrino mean free path has a weak dependence with the temperature and in the strong magnetic field region, it decreases for growing values of it. This contrast with the scattering reaction $\nu + n \rightarrow \nu' + n'$, where the average mean free path is almost independent of the magnetic field and has a strong dependence with the temperature. We have evaluated the asymmetry from both the absorption and scattering reactions. Our results shows that the total asymmetry depends on the magnetic field intensity, the density and the temperature. For a density of $0.16$ fm$^{-3}$ and for a magnetic field strength of B=$10^{17}$~G, the asymmetry in the mean free path is found to be, $\sim 9\%$ and $\sim 3.4\%$ for temperatures of T$=15$ and $30$ MeV, respectively. While the same set of asymmetries for B=$10^{18}$~G, is $\sim 58\%$ and $\sim 48\%$.

Spectral line list of potential cosmochronological interest deduced from new calculations of radiative transition rates in singly ionized thorium (Th ii)

ABSTRACTIn this work, we report a list of 91 strong Th ii spectral lines in the visible wavelength region that could be used as cosmochronometers in future investigations to determine stellar ages from thorium radioactive decay. This list was established on the basis of a pseudo-relativistic Hartree–Fock model including core-polarization corrections (HFR+CPOL) allowing us to compute the corresponding radiative decay parameters, the latter being found to be in good overall agreement with the most accurate experimental data when available. Besides this semi-empirical approach, the fully relativistic ab initio multiconfiguration Dirac–Hartree–Fock (MCDHF) method was attempted, showing large discrepancies with both the present HFR+CPOL and the most recent experimental oscillator strengths, although the calculated wavelengths from theoretical energy levels agreed with observed spectral line wavelengths within 2.4 per cent.

Deformed K− nuclei in the Skyrme–Hartree–Fock approach

Abstract The properties of kaonic nuclei are studied using a two-dimensional Skyrme–Hartree–Fock model with a $KN$ Skyrme force. We focus in particular on the instability of the solutions for a too strong $KN$ interaction, which determines a maximum value of the kaon binding in this approach. We then analyze the change of the deformation properties of several core-deformed nuclei caused by the added kaon, and find a shrinking of the core and in some cases a complete loss of deformation.

A reduced Hartree–Fock model of slice-like defects in the Fermi sea

Studying the electronic structure of defects in materials is an important subject in condensed matter physics. From a mathematical point of view, nonlinear mean-field models of localized defects in insulators are well understood. We present here a mean-field model to study a particular instance of extended defects in metals. These extended defects typically correspond to taking out a slice of finite width in the three-dimensional homogeneous electron gas. We work in the framework of the reduced Hartree–Fock model with either Yukawa or Coulomb interactions. Using techniques developed in Frank et al (2011 Phys. Rev. Lett. 106 150402; Frank et al 2013 Duke Math. J. 162 435–95) to study local perturbations of the free-electron gas, we show that our model admits minimizers, and that Yukawa ground state energies and density matrices converge to ground state Coulomb energies and density matrices as the Yukawa parameter tends to zero. These minimizers are unique for Yukawa interactions, and are characterized by a self-consistent equation. We moreover present numerical simulations where we observe Friedel oscillations in the total electronic density.