Thomas-Fermi Approximation Research Articles

Light and heavy clusters in warm stellar matter

Light and heavy clusters are calculated for warm stellar matter in the framework of relativistic mean-field models, in the single-nucleus approximation. The cluster abundances are determined from the minimization of the free energy. In-medium effects of light cluster properties are included by introducing an explicit binding energy shift analytically calculated in the Thomas–Fermi approximation, and the coupling constants are fixed by imposing that the virial limit at low density is recovered. The resulting light cluster abundances come out to be in reasonable agreement with constraints at higher density coming from heavy-ion collision data. Some comparisons with microscopic calculations are also shown.

Inconsistency between linearized Thomas–Fermi approximation and electron-ionized impurity scattering rate in the first Born approximation

We show that by computing the electron-impurity scattering rate at the first order via Fermi’s golden rule, and assuming that the localized impurity potential is of Yukawa form, one obtains a wave vector transfer distribution which is inconsistent with the finite temperature linearized Thomas–Fermi approximation for n-type semiconductors. Our previous findings show that this is not the case for the carrier nondegenerate dynamics, because the average wave vector transferred being in general negligible in this regime. Moreover, we examine the behavior of the electron-impurity differential cross-sections in the first Born approximation for relevant values of the wave vector transfer. We find that in the majority of collisions, the scattering probabilities differ at the most by 1% from the estimates computed by means of the impurity potential at random phase approximation level.

Thomas–Fermi approximation for β-stable nuclear matter in the Landau Fermi-liquid theory

An isentropic description of the equation of state (EOS) for β-stable nuclear matter (BSNM) is presented in a semi-classical mean-field model. Within the Thomas–Fermi (TF) approximation, the Landau Fermi-liquid theory (LFT) is used in a self-consistent approach. Accordingly, isentropic thermodynamics of BSNM is investigated for the neutrino-free and neutrino-trapped cases. We find that the EOS for neutrino-trapped matter is stiffer than the one for neutrino-free matter. Our findings can set the stage for profound studies about the structure and composition of proto-neutron stars (PNSs).

The ground state properties of two dimensional Fermi gas system confined in a potential composed of harmonic and a Gaussian terms

In this work, the ground state properties of two dimensional Fermi gas system interacting in a potential consisting of harmonic and Gaussian terms are investigated in the frame of Thomas-Fermi approximation. The depth and the curvature of the potential are changed by varying confinement parameters and the influence of the constraining conditions on the system properties like the density profile, the kinetic and the potential energy of the fermionic system is analyzed comprehensively. The deviations of the results due to the Gaussian potential are also determined by comparing the results with those obtained for pure harmonic potential. Calculations are also performed analytically for non-interacting case for comparative purposes. The results show that the confinement parameters play crucial role on the ground state properties of confined system.

Common Features in Multi-species Bose–Einstein Condensates and the Origin of the Inherent Critical Phenomena

A general approach is proposed to solve the coupled Gross–Pitaevskii equations (CGP) for K-species Bose–Einstein condensates under the Thomas–Fermi approximation. We aim at finding out the common features of these condensates different in K. In particular, two types of phase transitions, full-state transition and partial-state transition, are found. In the former, all species are involved in the transition, while in the latter only a few specified species are significantly involved. This leads to the criticality and the hidden criticality (namely, the criticality found in a condensate with fewer species recovers in a condensate with more species). The former originates from the singularity of the whole matrix of the CGP, while the latter originates from the singularity of a specified sub-matrix (which is related to a few specified species). In general, for any many-body system, the singularity inherent in the equations that govern the system is crucial to its critical behavior.

Effect of oxygen vacancies at the Fe/SrTiO3(001) interface: Schottky barrier and surface electron accumulation layer

We have investigated the interface formation at room temperature between Fe and ${\mathrm{TiO}}_{2}$-terminated ${\mathrm{SrTiO}}_{3}(001)$ surface using x-ray photoelectron spectroscopy. Oxygen vacancies within the ${\mathrm{SrTiO}}_{3}$ lattice in the first planes beneath the $\mathrm{Fe}\text{/}{\mathrm{SrTiO}}_{3}$ interface are induced by the Fe deposition. Through a detailed analysis of the Fe $2p$, Sr $3d$, and Ti $2p$ core-level line shapes we propose a quantitative description of the impact of the vacancies on the electronic properties of the $\mathrm{Fe}\text{/}{\mathrm{SrTiO}}_{3}$ system. While for an abrupt $\mathrm{Fe}\text{/}{\mathrm{SrTiO}}_{3}$ junction the Schottky barrier height for electrons is expected to be about 1 eV, we find that the presence of oxygen vacancies leads to a much lower barrier height value of 0.05 eV. The deposition of a fraction of Fe monolayer also pushes the surface conduction band edge of the ${\mathrm{SrTiO}}_{3}$ below the Fermi level in favor of the formation of a surface electron accumulation layer. This change in the band bending stems from the incorporation of oxygen vacancies in the near-surface region of ${\mathrm{SrTiO}}_{3}(001)$. We deduce the conduction band profile as well as the carrier density in the accumulation layer as a function of the surface potential by solving the one-dimensional Poisson equation within the modified Thomas-Fermi approximation. Owing to the electric-field dependence of the dielectric permittivity, the ${\mathrm{SrTiO}}_{3}$ with oxygen vacancies at the surface shows original electronic properties. In particular, our simulations reveal that variations of a few percent of the vacancies concentration at the surface can cause changes of several tenths of an eV in the band bending that can lead to important lateral surface inhomogeneities for the potential. We also find through our modeling that the defect states density related to oxygen vacancies at the ${\mathrm{SrTiO}}_{3}$ surface cannot exceed, at room temperature, a critical value of $\ensuremath{\sim}8\ifmmode\times\else\texttimes\fi{}{10}^{13}/\mathrm{c}{\mathrm{m}}^{2}$.

Stability analysis of a Bose–Einstein condensate trapped in a generic potential

We investigate the dynamical behavior of the Gross-Pitaevskii equation for a Bose-Einstein condensate trapped in a spherical power law potential restricted to the repulsive case, from the dynamical system formalism point of view. A five-dimensional dynamical system is found (due the symmetry of the Gross-Pitaevskii equation interacting with a potential), where the Thomas-Fermi approximation constrains the parameter space of solutions. We show that for values of the power law exponent equal or smaller than 2 the system seems to be stable. However, when the corresponding exponent is bigger than 2, the instability of the system grows when the power law exponent grows, indicating that large values of the aforementioned parameter can be related to a loss in the number of particles from the condensed state. This fact can be used also to show that the stability conditions of the condensate are highly sensitive to the exponent associated with the external potential.

Systematic corrections to the Thomas–Fermi approximation without a gradient expansion

We improve on the Thomas–Fermi approximation for the single-particle density of fermions by introducing inhomogeneity corrections. Rather than invoking a gradient expansion, we relate the density to the unitary evolution operator for the given effective potential energy and approximate this operator by a Suzuki–Trotter factorization. This yields a hierarchy of approximations, one for each approximate factorization. For the purpose of a first benchmarking, we examine the approximate densities for a few cases with known exact densities and observe a very satisfactory, and encouraging, performance. As a bonus, we also obtain a simple fourth-order leapfrog algorithm for the symplectic integration of classical equations of motion.

Nuclear skin and the curvature of the symmetry energy

The effect of correlations between the slope and the curvature of the symmetry energy on ground state nuclear observables is studied within the extended Thomas-Fermi approximation. We consider different isovector probes of the symmetry energy, with a special focus on the skin of $^{208}{\rm Pb}$. We use a recently proposed meta-modelling technique to generate a large number of equation of state models, where the empirical parameters are independently varied. The results are compared to a set of calculations using 17 different Skyrme interactions. We show that the curvature parameter plays a non-negligible role on the neutron skin, while the effect is reduced in Skyrme functionals because of the correlation with the slope parameter.

Properties of magnetized Coulomb crystals of ions with polarizable electron background

We have studied phonon and thermodynamic properties of a body-centered cubic (bcc) Coulomb crystal of ions with weakly polarized electron background in a uniform magnetic field B. At B = 0, the difference between phonon moments calculated using the Thomas-Fermi (TF) and random phase approximations is always less than 1% and for description of phonon properties of a crystal, TF formalism was used. This formalism was successfully applied to investigate thermodynamic properties of magnetized Coulomb crystals. It was shown that the influence of the polarization of the electron background is significant only at κTFa>0.1 and T≪Tp(1+h2)−1/2, where κTF is the Thomas-Fermi wavenumber, a is the ion sphere radius, Tp≡ℏωp is the ion plasma temperature, h≡ωB/ωp, ωB is the ion cyclotron frequency, and ωp is the ion plasma frequency.

Light clusters in warm stellar matter: Explicit mass shifts and universal cluster-meson couplings

In-medium modifications of light cluster properties in warm stellar matter are studied within the relativistic mean-field approximation. In-medium effects are included by introducing an explicit binding energy shift analytically calculated in the Thomas-Fermi approximation, supplemented with a phenomenological modification of the cluster couplings to the $\sigma$ meson. A linear dependence on the $\sigma$ meson is assumed for the cluster mass, and the associated coupling constant is fixed imposing that the virial limit at low density is recovered. The resulting cluster abundances come out to be in reasonable agreement with constraints at higher density coming from heavy ion collision data. Some comparisons with microscopic calculations are also shown.

Comparison between two scalar field models using rotation curves of spiral galaxies

Scalar fields have been used as candidates for dark matter in the universe, from axions with masses ∼ 10-5eV until ultra-light scalar fields with masses ∼ Axions behave as cold dark matter while the ultra-light scalar fields galaxies are Bose-Einstein condensate drops. The ultra-light scalar fields are also called scalar field dark matter model. In this work we study rotation curves for low surface brightness spiral galaxies using two scalar field models: the Gross-Pitaevskii Bose-Einstein condensate in the Thomas-Fermi approximation and a scalar field solution of the Klein-Gordon equation. We also used the zero disk approximation galaxy model where photometric data is not considered, only the scalar field dark matter model contribution to rotation curve is taken into account. From the best-fitting analysis of the galaxy catalog we use, we found the range of values of the fitting parameters: the length scale and the central density. The worst fitting results (values of much greater than 1, on the average) were for the Thomas-Fermi models, i.e., the scalar field dark matter is better than the Thomas- Fermi approximation model to fit the rotation curves of the analysed galaxies. To complete our analysis we compute from the fitting parameters the mass of the scalar field models and two astrophysical quantities of interest, the dynamical dark matter mass within 300 pc and the characteristic central surface density of the dark matter models. We found that the value of the central mass within 300 pc is in agreement with previous reported results, that this mass is ≈ 107 M⊙/pc2, independent of the dark matter model. And, on the contrary, the value of the characteristic central surface density do depend on the dark matter model.

Anomalous Statistics of Bose-Einstein Condensate in an Interacting Gas: An Effect of the Trap's Form and Boundary Conditions in the Thermodynamic Limit.

We analytically calculate the statistics of Bose-Einstein condensate (BEC) fluctuations in an interacting gas trapped in a three-dimensional cubic or rectangular box with the Dirichlet, fused or periodic boundary conditions within the mean-field Bogoliubov and Thomas-Fermi approximations. We study a mesoscopic system of a finite number of trapped particles and its thermodynamic limit. We find that the BEC fluctuations, first, are anomalously large and non-Gaussian and, second, depend on the trap’s form and boundary conditions. Remarkably, these effects persist with increasing interparticle interaction and even in the thermodynamic limit—only the mean BEC occupation, not BEC fluctuations, becomes independent on the trap’s form and boundary conditions.

Intermediate band formation in a δ-doped like QW superlattices of GaAs/AlxGa1−xAs for solar cell design

It is reported a numerical computation of the local density of states for a δ-doped like QW superlattices of AlxGa1−xAs, as a possible heterostructure that, being integrated into a solar cell device design, can provide an intermediate band of allowed states to assist the absorption of photons with lower energies than that of the energy gap of the solar-cell constituent materials. This work was performed using the nearest neighbors sp3s* tight-binding model including spin. The confining potential caused by the ionized donor impurities in δ-doped impurities seeding that was obtained analytically within the lines of the Thomas-Fermi approximation was reproduced here by the Al concentration x variation. This potential is considered as an external perturbation in the tight-binding methodology and it is included in the diagonal terms of the tight-binding Hamiltonian. Special attention is paid to the width of the intermediate band caused by the change in the considered aluminium concentration x, the inter-well distance between δ-doped like QW wells and the number of them in the superlattice. In general we can conclude that this kind of superlattices can be suitable for intermediate band formation for possible intermediate-band solar cell design.

Nuclear structure calculations for neutron star crusts

The goal of this paper is to investigate properties of clusterized nuclear matter which is believed to be present in crusts of neutron stars at subnuclear densities. It is assumed that the whole system can be represented by the set of Wigner-Seitz cells, each containing a nucleus and an electron background under the condition of electroneutrality. The nuclear structure calculations are performed within the relativistic mean-field model with the NL3 parametrization. The first set of calculations is performed assuming the constant electron background. The evolution of neutron and proton density distributions was systematically studied along isotopic chains until very neutron-rich system beyond the neutron dripline. Then we have replaced the uniform electron background with the realistic electron distributions, obtained within the Thomas-Fermi approximation in a self-consistent way with the proton distributions. Finally, we have investigated the evolution of the $\beta$-stability valley as well as neutron and proton driplines with the electron density.

The perturbation of the condensed medium in the Thomas–Fermi model

The research presented in this work is the continuation of the previous studies, which gave a qualitative estimating solution of the problem. The aim of this work is the calculation of static perturbation of condensed medium in the Thomas–Fermi approximation.

Spontaneous Separation in Trapped Fermi Gas with p-Wave Interactions: Due to the Mass-Imbalance

Based on density functional theory, the spontaneous separation in the mass-imbalanced Fermi–Fermi mixture is studied. The ground-state energy density functional is constructed with the effective contact interaction, with which the ground-state density profiles of the mixture are calculated under different conditions of mass-imbalance and coupling strength. The influence of mass-imbalance on the separation and the cloud size is analyzed. In addition, the system with both mass- and population-imbalance is also calculated and studied. Despite our rough treatment in the Thomas–Fermi approximation, it is hoped that the results may provide new clues to understand the nature of phase separation of a trapped ultracold gas in both theoretical and experimental researches in the future.

Bose–Einstein condensation under the cubic–quintic Gross–Pitaevskii equation in radial domains

We study stationary and quasi-stationary solutions for the cubic–quintic Gross–Pitaevskii equation modeling Bose–Einstein condensates (BECs) in one, two, and three spatial dimensions under the assumption of radial symmetry with the BEC dynamics influenced by a confining potential. We consider both repulsive and attractive cubic interactions – corresponding respectively to repulsive and attractive two-body interactions – under similar frameworks in order to deduce the effects of the potentials in each case. We also carefully consider the role played by the quintic nonlinearity (modeling the strength of inter-atomic coupling) in modifying the solutions arising due to a purely cubic interaction term. In one spatial dimension, we obtain a variety of exact solutions in the zero-potential limit (including new periodic solutions which generalize known soliton solutions) as well as perturbation solutions for small amplitude confining potentials. For more general forms of the confining potential, we rely on numerical simulations, but these agree with the analytical results when the latter are valid. We also consider the limit where the quintic term dominates the cubic term (with such a limit relevant in the study of a Tonks–Girardeau gas). Under the assumption of radial symmetry, we also consider cylindrical (or, cigar-shaped) and spherical BECs. We consider the nonperturbative regime where either the potential or the amplitude of the solutions is large, obtaining various qualitative analytical results. When the kinetic energy term is small (relative to the nonlinearity and the confining potential), we recover the expected Thomas–Fermi approximation for the stationary solutions. Numerical simulations, under a variety of external confining potentials, are then used to understand the role these potentials play on the BEC solution structure for both the attractive and repulsive regimes. This assortment of analytical and numerical results allows us to better understand the structure of BECs modeled via cubic–quintic Gross–Pitaevskii equation under confining potentials with radial symmetry. Some of the results are natural extensions of those for the cubic Gross–Pitaevskii equation, while others appear new.

A Model for the Characterization of the Polarizability of Thin Films Independently of the Thickness of the Film.

The dielectric properties of thin films can be modified relative to the bulk material because the interaction between film and substrate influences the mobility of the atoms or molecules in the first layers. Here we show that a strong scale effect occurs in nanometer size octadecylammine thin films. This effect is attributed to the different distribution of molecules depending on the size of the film. To accurately describe this effect, we have developed a model which is a reinterpretation of the linearized Thomas-Fermi approximation. Within this model, we have been able to characterize the polarizability of thin films independently of the thickness of the film.

Criterion for strong localization on a semiconductor surface in the Thomas–Fermi approximation

The localization behavior of a two-dimensional electron gas confined at the surface of a heavily doped semiconductor under conditions of the “natural” size effect in the space-charge region is investigated. The scattering of low-energy electrons by the chaotic potential formed at the surface by point charges of ionized impurities is analyzed and the electron mean free path is determined. A criterion for strong localization in this two-dimensional electron system is obtained on the basis of the Ioffe–Regel criterion.