Thomas Fermi Research Articles

Thomas–Fermi profile of a fast rotating Bose–Einstein condensate

We study the minimizers of a magnetic 2D non-linear Schr\"odinger energy functional in a quadratic trapping potential, describing a rotating Bose-Einstein condensate. We derive an effective Thomas-Fermi-like model in the rapidly rotating limit where the centrifugal force compensates the confinement, and available states are restricted to the lowest Landau level. The coupling constant of the effective Thomas-Fermi functional is linked to the emergence of vortex lattices (the Abrikosov problem). We define it via a low density expansion of the energy of the corresponding homogeneous gas in the thermodynamic limit.

Parametric triggering of vortices in toroidally trapped rotating Bose–Einstein condensates

We study the creation of vortices by triggering the rotating Bose–Einstein condensates in a toroidal trap with trap parameters such as laser beam waist and Gaussian potential depth. By numerically solving the time-dependent Gross–Pitaevskii equation in two dimensions, we observe a change in vortex structure and a considerable increase in the number of vortices when the waist of the irradiated laser beam is in consonance with the area of the condensate as we vary the Gaussian potential depth. By computing the root mean square radius of the condensate, we confirm the variation in the number of vortices generated as a function of the ratio between the root-mean-square radius of the condensate and the laser beam waist. In particular, the number of hidden vortices reaches the maximum value when the above ratio is close to the value 0.7. We find the variation in the number of vortices is rapid for deeper Gaussian potentials, and we conclude that the larger beam waist and deeper Gaussian potentials generate more vortices. Further, we calculate the number of vortices using the Feynman rule with Thomas Fermi approximation and compare them with the numerical results. We also observe that the critical rotation frequency decreases with an increase in depth of Gaussian potential.

Surface energy coefficient of an N2LO Skyrme energy functional: A semiclassical extended Thomas-Fermi approach

We generalize to N2LO Skyrme functionals the semi-classical approach of Grammaticos and Voros in order to calculate the Extended Thomas Fermi expressions of the new densities and currents appearing at the N2LO level. Within a one dimensional symmetric semi infinite nuclear matter model and using a simple Fermi-like density profile, we obtain an easy-to-use formula for the surface energy including the contributions of the central, density-dependent and spin-orbit terms up to $\hbar^2$. Such a formula can be easily used as a first attempt to constrain the surface properties of new N2LO Skyrme parametrizations. The N2LO parametrization tested in this paper is shown to exhibit a shift (compared to a full Hartree-Fock calculation) which is quantitatively similar to the one obtained with the traditional Skyrme parametrizations.

Pasta Phases in Neutron Star Mantle: Extended Thomas–Fermi vs. Compressible Liquid Drop Approaches

Nuclear pasta phases in the neutron stars mantle can affect the mechanical and transport properties of superdense matter, thus playing an important role in the dynamics and evolution of neutron stars. In this paper, we compare results obtained by the Extended Thomas–Fermi (ETF) method with the compressible liquid drop model (CLDM), based on the thermodynamically consistent description of the surface properties calculated for the two-phase plane interface and the same energy-density functional (for numerical illustration, we applied the Skyrme-type functional SLy4). Our ETF calculations found that pasta phases in cylindrical form cover a significant crustal region (both normal and inverse phases, aka spaghetti and bucatini are presented). Meanwhile, within the applied CLDM framework, which includes the thermodynamically required effect of neutron adsorption on the cluster’s surface but neglects curvature corrections, only the spaghetti phase was found to be energetically favorable in the small density range prior to crust–core transition. On the other hand, the recent CLDM of Dinh Thi et al., 2021, which, on the contrary, accounts for curvature term but neglects neutron adsorption, predicts pasta phase onset in better agreement with the ETF. This fact highlights the importance of the curvature effects and allows counting on the potential validity of the CLDMs as a convenient, transparent and accurate tool for investigation of the pasta-phase properties.

Microscopic-macroscopic level densities for low excitation energies

Level density ρ(E,Q) is derived within the micro-macroscopic approximation (MMA) for a system of strongly interacting Fermi particles with the energy E and additional integrals of motion Q, in line with several topics of the universal and fruitful activity of A. S. Davydov. Within the extended Thomas Fermi and semiclassical periodic orbit theory beyond the Fermi-gas saddle-point method, we obtain ρ∝Iν(S)/Sν, where Iν (S) is the modified Bessel function of the entropy S. For small shell-structure contribution, one finds ν = κ/2 + 1, where κ is the number of additional integrals of motion. This integer number is a dimension of Q, Q = {N, Z, …} for the case of two-component atomic nuclei, where N and Z are the numbers of neutrons and protons, respectively. For much larger shell structure contributions, one obtains ν = κ /2 + 2. The MMA level density ρ reaches the well-known Fermi gas asymptote for large excitation energies and the finite micro-canonical combinatoric limit for low excitation energies. The additional integrals of motion can also be the projection of the angular momentum of a nuclear system for nuclear rotations of deformed nuclei, number of excitons for collective dynamics, and so on. Fitting the MMA total level density ρ(E, Q) for a set of the integrals of motion Q = {N, Z}, to experimental data on a long nuclear isotope chain for low excitation energies, one obtains the results for the inverse level-density parameter K, which differs significantly from those of neutron resonances due to shell, isotopic asymmetry, and pairing effects.

On Bose–Einstein condensates in the Thomas–Fermi regime

We study a system of N trapped bosons in the Thomas–Fermi regime with an interacting pair potential of the form g_N N^{3beta -1} V(N^beta x) , for some beta in (0,1/3) and g_N diverging as N rightarrow infty . We prove that there is complete Bose–Einstein condensation at the level of the ground state and, furthermore, that, if beta in (0,1/6) , condensation is preserved by the time evolution.

Hot dense nuclear matter with the Thomas-Fermi approximation

We analyze the thermal effects on the EOS of dense nuclear matter, using a statistical approach where a semi-classical mean-field (MF) model based on the Thomas-Fermi (TF) approximation is introduced. Within this density-functional-based theory, the equation of state (EOS) of hot dense nuclear matter for different compositions of baryonic matter such as symmetric nuclear matter (SNM), pure neutron matter (PNM), and neutrino-free (neutrino-trapped) matter in β-equilibrium is investigated, while the contribution of photons is taken into account for only the β-equilibrated cases. The Landau effective mass, by which the thermal effects on dense matter can be judged, is also extracted from the cold single-particle (SP) potential of neutrons in phase space for both SNM and PNM. We show that the characteristic behavior of the nuclear symmetry free energy (SFE), especially at supra-saturation densities, has a clear relevance to the general influence of the temperature on the EOS. Based on the Γ-law approximation, a simple parametrization for the thermal properties of the EOS is provided by the thermal index Γth, while the presence of trapped neutrinos in β-equilibrated matter gives rise to significantly more consistency with the original astrophysical range 1.5≤Γth≤2. Our EOSs are then employed in an isentropic scheme to investigate the properties of a hot compact object under the assumption of the conserved baryonic mass, while complying with the current observational constraints on the structure of neutron stars (NSs). Within the present model, we cannot predict the possibility that the lighter component of GW190814 [1] is a massive cold neutron star (NS).

A class of traversable wormholes in the Starobinsky-like [formula omitted] gravity with anisotropic dark matter

This paper studies generalized Ellis–Bronikov (E–B) traversable wormholes in f(R) extended gravity. We assume that these objects are supported by anisotropic dark matter (DM) according to the most often used phenomenological models, with specific profile densities, namely those of Navarro–Frenk–White (NFW), Thomas–Fermi (T–F), and Pseudo-isothermal (PI). Initially, we obtain the field equations in a general scenario of f(R) theories in the metric formalism regarding the static and spherically symmetric Morris–Thorne spacetime. Then we detail a f(R) model that obeys a power-law, which includes the Starobinky modified gravity as a particular case. Following, we analyze the energy conditions which do not depend on the DM models (Null and Weak Energy Conditions—NEC and WEC) and those which are model-dependent (Strong and Dominant Energy Conditions SEC and DEC). Finally, we compare some E–B wormhole solutions and the analyzed DM models, discussing the feasibility of the wormhole-dark matter system in different scenarios.

Equation of State for Bismuth at High Energy Densities

The purpose of this work is to describe the thermodynamic properties of bismuth in a broad scope of mechanical and thermal effects. A model of the equation of state in a closed form of the functional relationship between pressure, specific volume, and specific internal energy is developed. A new expression is proposed for the internal energy of a zero-temperature isotherm in a wide range of compression ratios, which has asymptotics to the Thomas–Fermi model with corrections. Based on the new model, an equation of state for bismuth in the region of body-centered cubic solid and liquid phases is constructed. The results of calculating the thermodynamic characteristics of these condensed phases with the new EOS are compared with the available experimental data for this metal in waves of shock compression and isentropic expansion. The parameters of shock waves in air obtained earlier by unloading shock-compressed bismuth samples are reconsidered. The newly developed equation of state can be used in modeling various processes in this material at high energy densities.

Negative energy dust acoustic waves evolution in a dense magnetized quantum Thomas–Fermi plasma

Propagation of nonlinear waves in the magnetized quantum Thomas–Fermi dense plasma is analyzed. The Zakharov–Kuznetsov–Burgers equation is derived by using the theory of reductive perturbation. The exact solution contains both solitary and shock terms. Also, it is shown that rarefactive waves propagate in most cases. Both the associated electric field and the wave energy have been derived. The effects of dust and electrons temperature, dust density, magnetic field magnitude, and direction besides the effect of the kinematic viscosity on the amplitude, width, and energy of the formed waves are discussed. It is shown that the negative energy wave is formed and its value is enhanced due to the increase of the kinematic viscosity and the ambient magnetic field which lead to an increase in the instability. The present results are helpful in controlling the stabilization of confined Thomas–Fermi dense magnetoplasma that are found in white dwarfs and in the high-intensity laser-solid matter interaction experiments.

Description of covalent bond in terms of generalized charges: potential and dissociation energy of homonuclear compound

The theory of generalized charges is an asymptotic approximation of quantum mechanics for interatomic forces. It is based on the model of multicomponent electron gas, which extends the Thomas–Fermi model of inhomogeneous electron gas to pair electronic states. The present research takes in consideration the participation of the field of generalized charges in interatomic bonds. Herein, the definition of generalized charges (GC) by means of the overlap integral of the wave functions of valence electrons is given, and properties of GC are found out. Equations are derived for the potential and length of a homopolar bond, and dependence of these parameters on the generalized charges is shown. The dissociation energy of a diatomic molecule is expressed in terms of the nuclear charges, the function and multiplicity of the covalent bond, the angular momentum, and the vibrational energy of the binding electrons. The parametrization rules are substantiated and an a priori calculation of the dissociation energy of diatomic homonuclear molecules is carried out against the electronic configuration of the bond electrons and the nuclear charge. The calculation results for homonuclear compounds of the elements of the first four rows of the periodic table are shown to be of satisfactory accuracy.

Neutron star crust properties: comparison between the compressible liquid-drop model and the extended Thomas-Fermi approach

We present a detailed analysis of three models predicting the properties of non-uniform matter in the crust of neutron stars: the compressible liquid-drop model, the fourth order Extended Thomas Fermi (ETF) method, and ETF plus Strutinsky integral (ETFSI) correction. The former treats the nuclear clusters as uniform hard spheres, the second takes into account the density distribution which can be different for neutrons and protons, and the last one includes the proton shell effects within the Strutinsky approach. The purpose of this work is to understand the importance of the improvements in the nuclear modeling and to analyze the quantities which are the most sensitive to them. We find that thermodynamic quantities such as pressure, energy and chemical potential, as well as the electron fraction, are in very good agreement among the three models. This confirms previous results where we have shown that the improvement in the finite-size description of the nuclear clusters has a small impact on these quantities, since they are mainly constrained by the bulk properties. The refinements in the finite-size modeling are shown to impact mostly the composition of the nuclear clusters (Z cl, N cl) in an ordering which ranks according to the leptodermous expansion. This analysis is performed considering both the r-cluster and the e-cluster representations. The proton shell effects are shown to stabilize Z cl, which consequently impacts the neutron number N cl as well.

ELECTRODE: An electrochemistry package for atomistic simulations.

Constant potential methods (CPMs) enable computationally efficient simulations of the solid-liquid interface at conducting electrodes in molecular dynamics. They have been successfully used, for example, to realistically model the behavior of ionic liquids or water-in-salt electrolytes in supercapacitors and batteries. CPMs model conductive electrodes by updating charges of individual electrode atoms according to the applied electric potential and the (time-dependent) local electrolyte structure. Here, we present a feature-rich CPM implementation, called ELECTRODE, for the Large-scale Atomic/Molecular Massively Parallel Simulator, which includes a constrained charge method and a thermo-potentiostat. The ELECTRODE package also contains a finite-field approach, multiple corrections for nonperiodic boundary conditions of the particle-particle particle-mesh solver, and a Thomas-Fermi model for using nonideal metals as electrodes. We demonstrate the capabilities of this implementation for a parallel-plate electrical double-layer capacitor, for which we have investigated the charging times with the different implemented methods and found an interesting relationship between water and ionic dipole relaxations. To prove the validity of the one-dimensional correction for the long-range electrostatics, we estimated the vacuum capacitance of two coaxial carbon nanotubes and compared it to structureless cylinders, for which an analytical expression exists. In summary, the ELECTRODE package enables efficient electrochemical simulations using state-of-the-art methods, allowing one to simulate even heterogeneous electrodes. Moreover, it allows unveiling more rigorously how electrode curvature affects the capacitance with the one-dimensional correction.

Neutron star crust in Voigt approximation II: general formula for electron screening correction for effective shear modulus

ABSTRACT The main contribution to the effective shear modulus of neutron star crust can be calculated within Coulomb solid model and can be approximated by simple analytical expression for arbitrary (even multicomponent) composition. Here I consider correction associated with electron screening within Thomas–Fermi approximation. In particular, I demonstrate that for relativistic electrons (density ρ > 106 g cm−3) this correction can be estimated as $\delta \mu _\mathrm{eff}^\mathrm{V}= -9.4\times 10^{-4}\sum _Z n_Z Z^{7/3} e^2/a_\mathrm{e},$ where summation is taken over ion species, nZ is number density of ions with charge Ze, kTF is Thomas–Fermi screening wavenumber. Finally, ae = (4πne/3)−1/3 is electron sphere radius. Quasi-neutrality condition ne = ∑ZZnZ is assumed. This result holds true for arbitrary (even multicomponent and amorphous) matter and can be applied for neutron star crust and (dense) cores of white dwarfs. For example, the screening correction reduces shear modulus by ∼9 per cent for Z ∼ 40, which is typical for inner layers of neutron star crust.

Three-dimensional rogue waves and dust-acoustic dark soliton collisions in degenerate ultradense magnetoplasma in the presence of dust pressure anisotropy

A three-dimensional Thomas–Fermi dense anisotropic magnetized plasma having Fermi–Dirac distributed ions and electrons as well as classical fluid negative dust impurities is considered to analyze oblique modulational instability (MI) and head-on collisions among dust-acoustic dark solitons. The Chew–Golberger–Low description is employed to define the anisotropic dust pressure. The linear analysis is investigated. It is found that for larger wavelengths, the pressure anisotropy has a strong effect on the wave frequency. Following the multiscale reductive perturbation technique, a (3 + 1)-dimensional nonlinear Schrödinger equation is derived. Also, the MI criterion is identified, and the regions of (un)stable modulated waves are determined precisely. In addition to that, (un)stable domains of the modulated structures as well as the profile of the dust-acoustic rogue waves are found to be strongly affected by dust grain density, pressure anisotropy, and the strength of the magnetic field. In the stable regions, the face-to-face dark soliton collision and their phase shifts as well as their analytical trajectories are reported by applying the extended Poincare–Lighthill–Kuo method. Numerical analysis reveals that the phase shifts increase with dust concentration but decrease with dust pressure anisotropy. The present results may be applicable in exploring the nonlinear wave dynamics and solitary wave interactions in dense astrophysical plasmas especially to white dwarfs, interiors of the neutron stars, and magnet stars.

Propagation and energy of the dressed solitons in the Thomas–Fermi magnetoplasma

A theoretical investigation is presented for dust-acoustic (DA) waves in a collisionless Thomas–Fermi magnetoplasma. The plasma system consists of electrons, ions, and negatively charged dust grains, all existing in a quantizing magnetic field. The Korteweg–de Vries (KdV) and KdV type equations are derived by using the reductive perturbation method. The solutions of these evolved equations are obtained. The contribution of higher-order corrections to the DA is investigated. The electric field and the soliton energy were also derived. The K-dV and dressed soliton energies are depleted as the dust temperature and magnetic field increase. But they magnify as obliqueness increases. The present results are beneficial in understanding the waves propagating in Thomas–Fermi magnetoplasma that are applicable for high-intensity laser–solid matter interaction experiments and astrophysical compact objects such as white dwarfs.

Computational heuristics for solving nonlinear singular Thomas–Fermi equation with genetic exponential collocation algorithm

Computational heuristics for solving nonlinear singular Thomas–Fermi equation with genetic exponential collocation algorithm

DFT+U within the framework of linear combination of numerical atomic orbitals.

We present a formulation and implementation of the density functional theory (DFT)+U method within the framework of linear combination of numerical atomic orbitals (NAO). Our implementation not only enables single-point total energy and electronic-structure calculations but also provides access to atomic forces and cell stresses, hence allowing for full structure relaxations of periodic systems. Furthermore, our implementation allows one to deal with non-collinear spin texture, with the spin-orbit coupling (SOC) effect treated self-consistently. The key aspect behind our implementation is a suitable definition of the correlated subspace when multiple atomic orbitals with the same angular momentum are used, and this is addressed via the "Mulliken charge projector" constructed in terms of the first (most localized) atomic orbital within the d/f angular momentum channel. The important Hubbard U and Hund J parameters can be estimated from a screened Coulomb potential of the Yukawa type, with the screening parameter either chosen semi-empirically or determined from the Thomas-Fermi screening model. Benchmark calculations are performed for four late transition metal monoxide bulk systems, i.e., MnO, FeO, CoO, and NiO, and for the 5d-electron compounds IrO2. For the former type of systems, we check the performance of our DFT+U implementation for calculating bandgaps, magnetic moments, electronic band structures, as well as forces and stresses; for the latter, the efficacy of our DFT+U+SOC implementation is assessed. Systematic comparisons with available experimental results, especially with the results from other implementation schemes, are carried out, which demonstrate the validity of our NAO-based DFT+U formalism and implementation.

Phenomenological potentials for the refractory metals Cr, Mo and W

Cohesion in the refractory metals Cr, Mo, and W is phenomenologically described in this work via a n-body energy functional with a set of physically motivated parameters that were optimized to reproduce selected experimental properties characteristic of perfect and defective crystals. The functional contains four terms accounting for the hard-core repulsion, the Thomas–Fermi kinetic energy repulsion and for contributions to the binding energy of s and d valence electrons. Lattice dynamics, molecular statics, and molecular dynamics calculations show that this model describes satisfactorily thermodynamic properties of the studied metals whereas, unlike other empirical approaches from the literature, predictions of phonon dispersion relations and of surface and point defect energetics reveal in fair good agreement with experiments. These results suggest that the present model is well adapted to large-scale simulations and whenever total energy calculations of thermodynamic properties are unfeasible.

Quantum Control of Nanoparticles at Low Temperature

In this work, quantum control of nanoparticles at low temperature s considered. It would be quite interesting for scientists and researchers to take the well known nanoparticles at the scale of 10􀀀9m(=1nm) as they are at proper low temperature. It is desired to create Bose-Einstein-Condensates (BEC) with the nanoparticles. In our former study (cf. [Wang, 2016]) of controlling nanoparticle, the density function theory (DFT) described by time-dependent Schr¨odinger equation had been utilized to apply control theory to nanoparticles at matter surface. In the framework of Thomas-Fermi (TF) model, nanoparticles governed quantum system had been considered in the complex Hilbert spaces. In this investigation, the factor of temperature is taken account into time dependent Schr¨odinger equation. First, physically, a lot of questions will be arising in here for the control purpose of BEC phenomena of low temperature. Such as, whether BEC can be created using nanoparticles? what can the nanoparticles do at low temperature? which differences have for the nanoparticles BEC than other size particles? how control would be proceeded for nanoparticles at magnetic-electric field? Second, theoretically, how to apply of quantum control theory to nanoparticles BEC in the framework of variational theory? Then, how to get the first hand theoretic results to do nanoparticles control at low temperature. Review current contributed work and literatures, the survey of control Bose-Einstein-Condenstates had been occurred at amount of areas and had obtained significant milestone results as atomic particles (e.g. 87Rb, 7Li, 23Na,52Cr, 39K) cooling to temperature below of BEC thresholds. Indeed, the behavior of nanoparticles at the room temperature had also been considered, and had already been created to Carbon (C) nanotube, nanowire, nanomotor, nanorod as advanced materials. Particularly, the focus point is nanoparticles at low temperature, what is happened? the exciting and attractive conclusion will be hoped in this paper. As a kind of prediction, this theoretical research for control of nanoparticles would be fairly interesting, the control theory could be applied to these sort of nanoparticles perfectly. Without lost of generality, the nanoparticles composed BEC will be much more useful and can be quickly utilized to real society in the world. It is good to connect these researches to chemical and physical laboratory, and to do further interdisciplinary work concerning the control of quantum system.