Thomas-Fermi Approximation Research Articles

On the ground state of one-dimensional quantum droplets for large chemical potentials

Abstract In the present work we revisit the problem of the quantum droplet in atomic Bose–Einstein condensates with an eye towards describing its ground state in the large density, so-called Thomas–Fermi (TF) limit. We consider the problem as being separable into 3 distinct regions: an inner one, where the TF approximation is valid, a sharp transition region where the density abruptly drops towards the (vanishing) background value and an outer region which asymptotes to the background value. We analyze the spatial extent of each of these regions, and develop a systematic effective description of the rapid intermediate transition region. Accordingly, we derive a uniformly valid description of the ground state that is found to accurately match our numerical computations. As an additional application of our considerations, we show that this formulation allows for an analytical approximation of excited states such as the (trapped) dark soliton in the large density limit.

Carrier transport simulations in twisted bilayer and turbostratic multilayer graphene systems

The effects of inserting a twisted bilayer graphene (tBLG) between turbostratic graphene layers and a SiO2 substrate on the transport properties are investigated, to explore the possibility that using a tBLG could be an effective way to screen the potential fluctuations due to the impurities on the substrate. The Fermi velocity in a tBLG changes from that of the pristine graphene depending on the twist angle. In the present study, a parameter α is introduced, which is defined as the ratio of the Fermi velocity in the tBLG to that of pristine graphene, and the transport properties are calculated as a function of α. The self-consistent calculation of the Poisson equation with the Thomas–Fermi approximation is performed for multilayer graphene systems consisting of a tBLG and turbostratic graphene layers, and the potential profiles are incorporated into a Monte Carlo simulator to calculate the drift velocities and mobilities. It is shown that the transport properties of the whole system strongly depend on the parameter α and the number of layers in the system.

Two-dimensional quantum droplets in binary quadrupolar condensates

We study the stability and characteristics of two-dimensional (2D) quasi-isotropic quantum droplets (QDs) of fundamental and vortex types, formed by binary Bose–Einstein condensate with magnetic quadrupole–quadrupole interactions (MQQIs). The magnetic quadrupoles are built as pairs of dipoles and antidipoles polarized along the x-axis. The MQQIs are induced by applying an external magnetic field that varies along the x-axis. The system is modeled by the Gross–Pitaevskii equations including the MQQIs and Lee-Huang-Yang correction to the mean-field approximation. Stable 2D fundamental QDs and quasi-isotropic vortex QDs with topological charges S⩽4 are produced by means of the imaginary-time-integration method for configurations with the quadrupoles polarized parallel to the system’s two-dimensional plane. Effects of the norm and MQQI strength on the QDs are studied in detail. Some results, including an accurate prediction of the effective area, chemical potential, and peak density of QDs, are obtained in an analytical form by means of the Thomas-Fermi approximation. Collisions between moving QDs are studied by means of systematic simulations.

Alpha-like correlations in Ne, comparison of quartetting wave function and THSR approaches

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The Effect of Global and Local Chemical Reactivity Descriptors in the Determination of Properties of Transition Metal Clusters

Recently, research on material properties has got lots of attention because of their promising technological applications. In this paper, the author’s focus is on the global and local chemical reactivity descriptors of some transition metal clusters. In addition, the author also include findings of chemical properties with reactivity descriptors. Furthermore, according to Thomas–Fermi approximation and from the exact formulation of Density Functional Theory by Hohenberg and Kohn’s theorem, the author introduce electronegativity and the theory of hardness and softness for further investigation of chemical properties of the clusters. As a part of investigations, the author also introduce the Fukui functions with an emphasis for the determinations of chemical reactivity descriptors. All the results are obtained by theoretical investigation of electronegativity), Chemical potential (), chemical hardness (), and polarizability () of the given transition metal clusters (Cr and Mn). From the results, the author observed that the ionization potential of the clusters increases with decreasing cluster size as the clusters become more stable and tend towards chemical inertness. In conclusion, determination of the reactivity descriptors of the transition metal clusters show results that are in line with previous experimental and theoretical findings.

Alpha-like correlations in 20Ne: Comparison of quartetting wave function and THSR approaches

20Ne can be considered as a double-magic 16O core nucleus surrounded by four nucleons, the constituents of an α-like quartet. Similar to other nuclei (212Po, 104Ti, etc.) with a quartet on top of a double-magic core nucleus, significant α-like correlations are expected. The quartetting wave function (QWF) approach predicts a large α-like cluster contribution near the surface of the nuclei. The Tohsaki-Horiuchi-Schuck-Röpke (THSR) approach describes α-like clustering in nuclear systems. The results of the QWF approach in the Thomas-Fermi and shell-model approximation are compared with THSR calculations for the container model.

Gravitational collapse of Bose-Einstein condensate dark matter halos with logarithmic nonlinearity

"If dark matter is composed of massive bosons, a Bose-Einstein Condensation process must have occurred during the cosmological evolution. Therefore galactic dark matter may be in a form of a condensate, characterized by a strong self-interaction. One of the interesting forms of the self-interaction potential of the condensate dark matter is the logarithmic form. In the present work we investigate one of the astrophysical implications of the condensate dark matter with logarithmic self-interaction, namely, its gravitational collapse. To describe the condensate dark matter we use the Gross-Pitaevskii equation, and the Thomas-Fermi approximation. By using the hydrodynamic representation of the Gross-Pitaevskii equation we obtain the equation of state of the condensate, which has the form of the ideal gas equation of state, with the pressure proportional to the dark matter density. In the Thomas-Fermi approximation, the evolution equations of the condensate reduce to the classical continuity, and Euler equations of fluid dynamics. We obtain the equations of motion of the condensate radius in spherical symmetry, by assuming certain particular forms for the velocity and density of the condensate. The collapse time required for the formation of a stable macroscopic astrophysical object is obtained in an integral form, and explicit numerical estimations for the formation of astrophysical objects with masses ranging from 106M⊙ to 1012M⊙ are presented."

Interaction of one-dimensional quantum droplets with potential wells and barriers

We address static and dynamical properties of one-dimensional (1D) quantum droplets (QDs) under the action of local potentials in the form of narrow wells and barriers. The dynamics of QDs is governed by the 1D Gross–Pitaevskii equation including the mean-field cubic repulsive term and the beyond-mean-field attractive quadratic one. In the case of the well represented by the delta-function potential, three exact stable solutions are found for localized states pinned to the well. The Thomas–Fermi approximation for the well and the adiabatic approximation for the collision of the QD with the barrier are developed too. Collisions of incident QDs with the wells and barriers are analyzed in detail by means of systematic simulations. Outcomes, such as fission of the moving QD into transmitted, reflected, and trapped fragments, are identified in relevant parameter planes. In particular, a counter-intuitive effect of partial or full rebound of the incident QD from the potential well is studied in detail and qualitatively explained.

Dynamic dielectric function and phonon self-energy from electrons strongly correlated with acoustic phonons in 2D Dirac crystals

The unique structure of two-dimensional (2D) Dirac crystals, with electronic bands linear in the proximity of the Brillouin-zone boundary and the Fermi energy, creates anomalous situations where small Fermi-energy perturbations critically affect the electron-related lattice properties of the system. The Fermi-surface nesting (FSN) conditions determining such effects via electron–phonon interaction require accurate estimates of the crystal’s response function as a function of the phonon wavevector q for any values of temperature, as well as realistic hypotheses on the nature of the phonons involved. Numerous analytical estimates of for 2D Dirac crystals beyond the Thomas–Fermi approximation have been so far carried out only in terms of dielectric response function , for photon and optical-phonon perturbations, due to relative ease of incorporating a q-independent oscillation frequency in calculation. Models accounting for Dirac-electron interaction with acoustic phonons, for which ω is linear to q and is therefore dispersive, are essential to understand many critical crystal properties, including electrical and thermal transport. The lack of such models has often led to the assumption that the dielectric response function in these systems can be understood from free-electron behavior. Here, we show that, different from free-electron systems, calculated for acoustic phonons in 2D Dirac crystals using the Lindhard model, exhibits a cuspidal point at the FSN condition. Strong variability of persists also at finite temperatures, while tend to infinity in the dynamic case where the speed of sound is small, albeit non negligible, over the Dirac-electron Fermi velocity. The implications of our findings for electron-acoustic phonon interaction and transport properties such as the phonon line width derived from the phonon self-energy will also be discussed.

Orbital-free approach for large-scale electrostatic simulations of quantum nanoelectronics devices

The route to reliable quantum nanoelectronic devices hinges on precise control of the electrostatic environment. For this reason, accurate methods for electrostatic simulations are essential in the design process. The most widespread methods for this purpose are the Thomas-Fermi (TF) approximation, which provides quick approximate results, and the Schrödinger-Poisson (SP) method, which better takes into account quantum mechanical effects. The mentioned methods suffer from relevant shortcomings: the TF method fails to take into account quantum confinement effects that are crucial in heterostructures, while the SP method suffers severe scalability problems. This paper outlines the application of an orbital-free approach inspired by density functional theory. By introducing gradient terms in the kinetic energy functional, our proposed method incorporates corrections to the electronic density due to quantum confinement while it preserves the scalability of a theory that can be expressed as a functional minimization problem. This method offers a new approach to addressing large-scale electrostatic simulations of quantum nanoelectronic devices.

Elastic properties of nuclear pasta in a fully three-dimensional geometry

Realistic estimations on the elastic properties of neutron star matter are carried out with a large strain (ε≲0.5) in the framework of relativistic-mean-field model with Thomas-Fermi approximation, where various crystalline configurations are considered in a fully three-dimensional geometry with reflection symmetry. Our calculation confirms the validity of assuming Coulomb crystals for the droplet phase above neutron drip density, which nonetheless does not work at large densities since the elastic constants are found to be decreasing after reaching their peaks. Similarly, the analytic formulae derived in the incompressible liquid-drop model give excellent description for the rod phase at small densities, which overestimates the elastic constants at larger densities. For slabs, due to the negligence on the variations of their thicknesses, the analytic formulae from liquid-drop model agree qualitatively but not quantitatively with our numerical estimations. By fitting to the numerical results, these analytic formulae are improved by introducing dampening factors. The impacts of nuclear symmetry energy are examined adopting two parameter sets, corresponding to the slope of symmetry energy L=41.34 and 89.39 MeV. Even with the uncertainties caused by the anisotropy in polycrystallines, the elastic properties of neutron star matter obtained with L=41.34 and 89.39 MeV are distinctively different, results in detectable differences in various neutron star activities.

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.

Sound Propagation in Cigar-Shaped Bose Liquids in the Thomas-Fermi Approximation: A Comparative Study between Gross-Pitaevskii and Logarithmic Models

A comparative study is conducted of the propagation of sound pulses in elongated Bose liquids and Bose-Einstein condensates in Gross-Pitaevskii and logarithmic models, by means of the Thomas-Fermi approximation. It is demonstrated that in the linear regime the propagation of small density fluctuations is essentially one-dimensional in both models, in the direction perpendicular to the cross section of a liquid’s lump. Under these approximations, it is demonstrated that the speed of sound scales as a square root of particle density in the case of the Gross-Pitaevskii liquid/condensate, but it is constant in a case of the homogeneous logarithmic liquid.

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).

THE EFFECT OF THE WAVE FUNCTION PARAMETERS ON THE ATOM LASER DENSITY

This paper studies the effect of the wave function parameters on the density of 87Rb atoms laser. The density equation has been derived through the use of Thomas-Fermi approximation, Green function and the stationary phase approximations. We investigated the change of density behavior with the effect of Rabi frequency, period of output coupling and detuning frequency. Consequently the number of released atoms can be controlled by determining the suitable parameters necessary for producing the best atom laser beam.

K-Essence Lagrangians of Polytropic and Logotropic Unified Dark Matter and Dark Energy Models

We determine the k-essence Lagrangian of a relativistic barotropic fluid. The equation of state of the fluid can be specified in different manners depending on whether the pressure is expressed in terms of the energy density (model I), the rest-mass density (model II), or the pseudo rest-mass density for a complex scalar field in the Thomas-Fermi approximation (model III). In the nonrelativistic limit, these three formulations coincide. In the relativistic regime, they lead to different models that we study exhaustively. We provide general results valid for an arbitrary equation of state and show how the different models are connected to each other. For illustration, we specifically consider polytropic and logotropic dark fluids that have been proposed as unified dark matter and dark energy models. We recover the Born-Infeld action of the Chaplygin gas in models I and III and obtain the explicit expression of the reduced action of the logotropic dark fluid in models II and III. We also derive the two-fluid representation of the Chaplygin and logotropic models. Our general formalism can be applied to many other situations such as Bose-Einstein condensates with a |φ|4 (or more general) self-interaction, dark matter superfluids, and mixed models.

Nuclear pasta structures at high temperatures

We investigate nuclear pasta structures at high temperatures in the framework of relativistic mean field model with Thomas-Fermi approximation. Typical pasta structures (droplet, rod, slab, tube, and bubble) are obtained, which form various crystalline configurations. The properties of those nuclear pastas are examined in a three-dimensional geometry with reflection symmetry, where the optimum lattice constants are fixed by reproducing the droplet/bubble density that minimizes the free energy adopting spherical or cylindrical approximations for Wigner-Seitz cells. It is found that different crystalline structures can evolve into each other via volume conserving deformations. For fixed densities and temperatures, the differences of the free energies per baryon of nuclear pasta in various shapes and lattice structures are typically on the order of tens of keV, suggesting the possible coexistence of those structures. As temperature increases, the thermodynamic fluctuations are expected to disrupt the long-range ordering in nuclear pasta structures. We then estimate the critical conditions for nuclear pasta to become disordered and behave like liquid, which are found to be sensitive to the densities, temperatures, proton fractions, and nuclear shapes. If we further increase temperature, eventually the nonuniform structures of nuclear pasta become unstable and are converted into uniform nuclear matter. The phase diagrams of nuclear matter are then estimated, which should be useful for understanding the evolutions of neutron stars, supernova dynamics, and binary neutron star mergers.

Unified neutron star EOSs and neutron star structures in RMF models

In the framework of the Thomas-Fermi approximation, we systematically study the EOSs and microscopic structures of neutron star matter in a vast density range with n b ≈ 10−10-2 fm−3, where various covariant density functionals are adopted, i.e., those with nonlinear self couplings (NL3, PK1, TM1, GM1, MTVTC) and density-dependent couplings (DD-LZ1, DDME-X, PKDD, DD-ME2, DD2, TW99). It is found that the EOSs generally coincide with each other at n b ≲ 10−4 fm−3 and 0.1 fm−3 ≲ n b ≲ 0.3 fm−3, while in other density regions they are sensitive to the effective interactions between nucleons. By adopting functionals with a larger slope of symmetry energy L, the curvature parameter K sym and neutron drip density generally increases, while the droplet size, proton number of nucleus, core-crust transition density, and onset density of non-spherical nuclei, decrease. All functionals predict neutron stars with maximum masses exceeding the two-solar-mass limit, while those of DD2, DD-LZ1, DD-ME2, and DDME-X predict optimum neutron star radii according to the observational constraints. Nevertheless, the corresponding skewness coefficients J are much larger than expected, while only the functionals MTVTC and TW99 meet the start-of-art constraints on J. More accurate measurements on the radius of PSR J0740 + 6620 and the maximum mass of neutron stars are thus essential to identify the functional that satisfies all constraints from nuclear physics and astrophysical observations. Approximate linear correlations between neutron stars’ radii at M = 1.4M ⊙ and 2M ⊙, the slope L and curvature parameter K sym of symmetry energy are observed as well, which are mainly attributed to the curvature-slope correlations in the functionals adopted here. The results presented here are applicable for investigations of the structures and evolutions of compact stars in a unified manner.

Interface effects of quark matter: Light-quark nuggets and compact stars

The interface effects of quark matter play important roles in the properties of compact stars and small nuggets, such as strangelets and nonstrange quark matter ($ud\mathrm{QM}$) nuggets. By introducing a density derivative term to the Lagrangian density and adopting Thomas-Fermi approximation, we find it is possible to reproduce the results obtained by solving Dirac equations. Adopting certain parameter sets, the energy per baryon of $ud\mathrm{QM}$ nuggets decreases with baryon number $A$ and become more stable than nuclei at $A\ensuremath{\gtrsim}300$. The effects of quark matter symmetry energy are examined, where $ud\mathrm{QM}$ nuggets at $A\ensuremath{\approx}1000$ can be more stable than others if large symmetry energy is adopted. In such cases, larger $ud\mathrm{QM}$ nuggets will decay via fission and the surface of a $ud\mathrm{QM}$ star will fragment into a crust made of $ud\mathrm{QM}$ nuggets and electrons, which resembles the cases of a strange star's crust. The corresponding microscopic structures are then investigated adopting spherical and cylindrical approximations for the Wigner-Seitz cells, where the droplet phase is found to be the most stable configuration with $ud\mathrm{QM}$ stars' crusts and $ud\mathrm{QM}$ dwarfs made of $ud\mathrm{QM}$ nuggets ($A\ensuremath{\approx}1000$) and electrons. For the cases considered here, the crust thickness of $ud\mathrm{QM}$ stars is typically $\ensuremath{\sim}200\text{ }\text{ }\mathrm{m}$, which reaches a few kilometers if we neglect the interface effects and adopt Gibbs construction. The masses and radii of $ud\mathrm{QM}$ dwarfs are smaller than typical white dwarfs, which would increase if the interface effects are neglected.

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.