Thomas-Fermi Statistical Model Research Articles

DERIVATION OF THE DIMENSIONAL THOMAS-FERMI EQUATION FOR THE THOMAS-FERMI ATOM MODEL

This article presents a method for calculating the energies of the values of the set of electron-neutral atoms. In this case, the interaction of electrons other than the Coulomb bond of the nucleus makes an important contribution to energy. Quantitative calculation of the interaction of these interactions with the introduction of the theory of approximation in the framework of the Thomas-Fermi statistical model by the method of self-correction field is particularly inefficient for complex atoms. However, for complex atoms, the method of approximation is shown, and its essence lies in its simplicity. Among the various methods for systems consisting of the same number of particles, the statistical method derived from the Thomas-Fermi statistical model of the atom plays an important role. This method (E. Fermi, L. Thomas, 1927) is based on the fact that in complex atoms with a large number of electrons, most electrons have relatively large quantum numbers. In this case, a semi-classical approximation is used. As a result, the concept of "cells in phase space" can be used for the state of the individual electrons of the atom. This model has been developed by researchers over a long period of time and has led to a consistent, complete doctrine without the defects of previous models, for example, its field of application is wider than the original Thomas-Fermi.

Application of the Thomas Fermi quark model to multiquark mesons

The possibility of the existence of mesons with two or more quark-antiquark pairs is investigated with a new application of the Thomas-Fermi (TF) statistical quark model. Quark color couplings are treated in a mean field manner similar to a previous application to baryons, and short and concise expressions for energies are derived. We find that, on average, quarks only interact with antiquarks in such systems. The TF differential equation is constructed and systems with heavy-light quark content are examined. Three types of mesonic systems are defined. In the case of charm quarks, multi-charmonium, multi-Z meson and multi-D meson family types are examined. System analogs for bottom quarks are also constructed. Quantitative trends for system energies of mesonic quark matter are extracted as a function of the number of quark pairs. We find indications from energy plots that multi-Z type mesons (and their bottom quark analogs) are actually stable for a range of quark number pairs. At this initial stage we have not yet included explicit spin interaction couplings between quarks, but we can take one level of degeneracy into account in our two-inequivalent TF function construction.

Charge-state-dependent energy loss of slow ions. II. Statistical atom model

A model for charge-dependent energy loss of slow ions is developed based on the Thomas-Fermi statistical model of atoms. Using a modified electrostatic potential which takes the ionic charge into account, nuclear and electronic energy transfers are calculated, the latter by an extension of the Firsov model. To evaluate the importance of multiple collisions even in nanometer-thick target materials we use the charge-state-dependent potentials in a Monte Carlo simulation in the binary collision approximation and compare the results to experiment. The Monte Carlo results reproduce the incident charge-state dependence of measured data well [see R. A. Wilhelm et al., Phys. Rev. A 93, 052708 (2016)], even though the experimentally observed charge exchange dependence is not included in the model.

Raman scattering and electrical studies of the phase stability in the Hg1−xCdxTe

Raman scattering is performed to access phase stability in the boron-implanted Hg0.7Cd0.3Te with fluences ranging from 1 × 1012 to 1 × 1015 cm−2. Threshold fluence for the formation of an amorphous phase is invoked here using Thomas–Fermi statistical model. Asymmetric broadening and red shift of the Raman active HgTe-like LO phonon mode are observed with varying fluencies. Electrical properties such as sheet carrier concentration and mobility are also changed with the fluence and reach their saturated values beyond threshold fluence of 5 × 1013 cm−2. Threshold fluence for the formation of amorphous phase is also validated by the Raman measurements and electrical transport properties in the implanted layers. The excess free energy of 6.8 kJ/mole is found corresponding to the threshold fluence for phase transition.

Shell correction to the Thomas—Fermi statistical model of plasma with different atomic composition at high and low temperatures

The refined semiclassical method based on the Thomas-Fermi statistical model takes into account the shell effects by means of an additive correction. The method has been verified in the calculations of a plasma equation of state at high temperatures. To expand its application range to low temperatures some assumptions are revised that have been made to obtain the simple formula for a shell correction. The validity of the assumptions is discussed and the results of their refusal are analyzed. As examples, a number of single-particle states in the ideal plasma of a few elements with strongly differing atomic numbers is calculated.

On the Molière theory of multiple scattering of charged particles (1947–1948) and its critique in subsequent years

Several “misconceptions” regarding the theory of multiple scattering of fast charged particles in matter developed by Moliere in 1947–1948 and its application in the analysis of experimental results are discussed. It is shown that the critics of this theory misinterpreted the Moliere method for determining the cross section of particle scattering by atoms with the screening of their nuclear fields by electron shells described by the Thomas-Fermi statistical model. If the original Moliere method is applied consistently, the obtained scattering cross section generally agrees with the results of later classical calculations carried out by Lindhard and his collaborators and other authors.

Inclusion of the discreteness of the electronic spectrum in the statistical model of free ions

The characteristics of free positive ions have been studied with a semiclassical method based on the Thomas-Fermi statistical model. An efficient method has been proposed for the inclusion of the discreteness of the electronic spectrum and the calculation of a shell correction to the number of states by the statistical model in the case of the strong irregularity of the correction. The calculation of the free-ion ionization potentials within this approach reproduces the “sawtooth” dependence of these potentials on the ion charge and strongly improves the description of the experimental data by the statistical model.

Equation of state of Al for compressed and expanded states from first-principles calculations

Results of theoretical calculations of equation of state and critical temperature of Al are presented using a three-term EOS model. In this model cold (0K) term is calculated from first-principles method near normal conditions. Cold curve is extrapolated to ultrahigh pressures using Thomas– Fermi–Dirac model and to expanded states using a soft sphere function. Electron thermal term is calculated using Thomas–Fermi statistical model. Ion-thermal term is calculated using the modified Cowan model. In expanded state, the adjustable parameters of the modified Cowan model are tuned using quantum molecular dynamics (QMD) results. In compressed state, P−ρ and Us−Up Hugoniots derived using our results show good agreement with the reported experimental results. In expanded state, the estimated critical temperature shows good agreement with the reported results and pressure versus internal energy along isochores show reasonably good agreement with the reported experimental results.

Thomas–Fermi statistical models of finite quark matter

I introduce and discuss models of finite quark matter using the formalism of the Thomas–Fermi statistical model. Similar to bag models, a vacuum energy density term is introduced to model long distance confinement, but the model produces bound states from the residual color Coulomb attraction even in the absence of such a term. I discuss three baryonic applications: an equal mass nonrelativistic model with and without volume pressure, the ultra-relativistic limit confined by volume pressure, and a color-flavor locking massless model. These models may be extended to multi-meson and other mixed hadronic states. Hopefully, it can help lead to a better understanding of the phenomenology of high multi-quark states in preparation for more detailed lattice QCD calculations.

Ettore Majorana’s Forgotten Publication on the Thomas-Fermi Model

We analyze the forgotten communication of Ettore Majorana (1906–1938?) on the Thomas-Fermi statistical model of the atom, which he presented on December 29, 1928, during the XXII General Meeting of the Italian Physical Society in Rome, and which was published in Il Nuovo Cimento, the Society’s journal, in 1929. His communication was not mentioned subsequently in any of the numerous publications of Enrico Fermi (1901–1954) and his group in Rome, nor in any of the later accounts of Majorana’s life and work. We place Majorana’s contribution within the context of contemporary research on the subject, point out its influence on the final formulation of the Thomas-Fermi statistical model by Fermi and Edoardo Amaldi (1908–1989) in 1934, and discuss Majorana’s other scientific contributions before his mysterious disappearance in 1938.

Ionization potentials and partition functions of ions in a semiclassical model

The characteristics of an isolated ion (its radius, energy, ionization potential, bound-state energy, and excitation spectrum), which appear in the chemical plasma model, are calculated via the Thomas-Fermi statistical model. The expression for the partition function of the ion excitation at a given temperature and plasma density has been derived. A comparison with the empirical results is performed.

A Model for Computing Equation of State of Plasma of the FST Injector of Electromagnetic Railgun

Fast shock tube (FST) is a launcher, which can be used as the injector of electromagnetic railgun, and its working fluid often chooses the inert gas, which is ionized to high-temperature and high-pressure plasma by strong shock wave in the process of launching. This paper presents a simplified model of computing equation of state (EOS) of plasma of the FST injector; we gain the analytic function of the ionization potential versus the average degree of ionization by fitting the numerical results of the ionization potential calculated with Thomas-Fermi (TF) statistical model. Combined the function of the ionization potential with the Saha equation of multiple ionizations, we establish a simplified model for calculating the average degree of ionization, which is the analytic function of the average degree of ionization of argon versus temperature and density in local thermal equilibrium (LTE) case. The curve of pressure p versus specific volume V in argon plasma is calculated according this simplified model, and the results calculated are basically in agreement with the several sets of experimental date and the results calculated with the Saha-equation theory in the high-density region. This simplified model can also be used to calculate the equation of state of plasmas mixture and is expected to have a wider use in the field of electromagnetic launch technology involving the strongly ionized plasma

A simplified model for computing state equation of argon plasma

In this paper, We first fitted the numerical results of the ionization potential calculated by Thomas-Fermi statistical model and gained the analytical function of the potential versus the degree of ionization, then calculated the ionization potential and the average degree of ionization for argon versus temperature and density in local thermal equilibrium (LTE) case. The curve of pressure versus specific volume V of argon plasmas is calculated according this simplified model. The calculated results by this simplified model are in agreement with experimental data and the calculated results of the Saha-equation theory. This simplified model can be used to the calculation of the equation of state of plasmas mixture and is expected to be widely used in the field of EML technology involving the strongly ionized plasmas.

Equation of state of compressed matter: a simple statistical model

We propose a simple approach for studying systems of compressed matter based on the Thomas–Fermi statistical model of single atom. The central point of our work is the development of the concept of “statistical ionization” by compression; in simple terms, we calculate the fraction of electrons within the atom whose positive energy, due to the compression, exceeds the negative binding energy electron–nucleus. Next we extend this concept from a single atom to macroscopic systems and write the corresponding equation of state. Positive aspects as well as limitations of the model are illustrated and discussed throughout the paper.

Polarization bremsstrahlung of a fast charged particle on a Thomas-Fermi atom

An universal description of the polarization bremsstrahlung of a fast charged particle on a multi-electron atom (Z ≫ 1, Z is the nuclear charge) is obtained using the local electron density method and the Thomas-Fermi statistical model. It is shown that the cross section of the process can be represented in the form \(d\sigma ^{PB} (\omega ) = Z^2 d\tilde \sigma ^{PB} (\nu )\), where the function \(d\tilde \sigma ^{PB} (\nu )\) exhibits approximate scaling with respect to the parameter ω/Z = v, and the corresponding R factor (ratio of the cross sections in the polarization and ordinary channels) is greater than 1 in a wide range of frequencies and reaches its maximum value at frequencies ω ≈ ZRy. It is demonstrated that in the frequency range pac < ħω < γ2pac (γ is the relativistic factor, pa is the characteristic momentum of the atomic electrons, and c is the speed of light) the angular distribution of the polarization bremsstrahlung of a relativistic charged particle undergoes narrowing due to the compensation, by the momentum of the emitted photon, of the momentum transfer from the incident particle to the atomic core.

Thomas-Fermi Theory of an Inhomogeneous Electron Liquid Generalized to Incorporate Density Gradients

Motivated by exact results for many closed shells in a bare Coulomb field, a generalization of the Thomas-Fermi statistical model is proposed. This generalization includes density gradients in the density-potential relation, and offers the possibility of avoiding the singularity (of the original method) in the density at an atomic nucleus and of embodying Kato's theorem.

Thomas-Fermi theory of the breathing mode and nuclear incompressibility

A Thomas-Fermi theory with a linear scaling assumption is proposed for the breathing mode of nuclear collective motion. It leads to a general result ${K}_{A}=〈K(\ensuremath{\rho},\ensuremath{\delta})〉{+K}_{\mathrm{GD}}\ensuremath{-}{2E}_{C}/A$ which states that the incompressibility ${K}_{A}$ of a finite nucleus $A$ mainly equals the nuclear matter incompressibility $K(\ensuremath{\rho},\ensuremath{\delta})$ averaged over the nucleon density distribution $\ensuremath{\rho}(\mathbf{r})$ of nucleus $A$, added to a term ${K}_{\mathrm{GD}}$ contributed from the gradients of nucleon densities, with twice the Coulomb energy per nucleon ${E}_{C}/A$ subtracted. The nuclear matter equation of state given by the Thomas-Fermi statistical model with a Seyler-Blanchard-type interaction is employed to calculate the nuclear matter incompressibility $K(\ensuremath{\rho},\ensuremath{\delta})$ and a localized approximation of the Seyler-Blanchard-type interaction, which is shown to be similar to the Skyrme-type interaction, is developed to calculate the value of ${K}_{\mathrm{GD}}.$ ${K}_{\mathrm{GD}}$ and $\ensuremath{-}{2E}_{C}/A$ contribute about 20--10 % and 1--5 %, respectively, to the nuclear incompressibility ${K}_{A},$ from the light to the heavy nuclei. The shell and the even-odd effects are discussed by a scaling model which shows that these effects can be neglected for medium and heavy nuclei. The anharmonic effect is shown to be significant only for light nuclei. The leptodermous expansion of ${K}_{A}$ is obtained and the contribution from the curvature term proportional to ${A}^{\ensuremath{-}2/3}$ is discussed. The calculated isoscalar giant monopole resonance energy ${E}_{M}$ for a variety of nuclei are shown to be in agreement with experimental measurements.

Impact-parameter dependence of inelastic energy losses in slow atom-atom collisions

A quasiclassical expression has been derived for the inelastic energy losses in atom-atom collisions below the Bohr velocity as a function of the impact parameter. A collision of slow atoms is treated on the basis of the Thomas-Fermi statistical model. The gas of atomic electrons is characterized by the space dependent value of the Fermi energy. The inelastic energy losses are believed to be due to the electron momentum transfer between colliding partners. The excitation of the Fermi gas at each point of the phase space is assumed to be small compared with the Fermi energy. The projectile trajectory shape is accounted for classically by means of the Thomas-Fermi-Firsov potential. It has been found that the Coulomb repulsion between the nuclei of the projectile and the target atom decreases greatly the contribution of small impact-parameter collisions in the limiting case of low reduced energies.

Large‐Z and ‐N dependence of atomic energies from renormalization of the large‐dimension limit

AbstractBy combining Hartree–Fock results for nonrelativistic ground‐state energies of N‐electron atoms with analytic expressions for the large‐dimension limit, we have obtained a simple renormalization procedure. For neutral atoms, this yields energies typically threefold more accurate than the Hartree–Fock approximation. Here, we examine the dependence on Z and N of the renormalized energies E(N, Z) for atoms and cations over the range Z, N = 2 → 290. We find that this gives for large Z = N an expansion of the same form as the Thomas–Fermi statistical model, E → Z7/2(C0 + C1Z−1/3 + C2Z−2/3 + C3Z−3/3 + ⃛), with similar values of the coefficients for the three leading terms. Use of the renormalized large‐D limit enables us to derive three further terms. This provides an analogous expansion for the correlation energy of the form δE δZ4/3(δC3 + δC5Z−2/3 + δC6Z−3/3 + ⃛); comparison with accurate values of δE available for the range Z ⩽ 36 indicates the mean error is only about 10%. Oscillatory terms in E and δE are also evaluated. © 1994 John Wiley &amp; Sons, Inc.

Study of the planar surface potential in grazing proton-surface scattering

The energy loss of 28.5–72.2 keV H + specularly reflected off a graphite surface has been measured as a function of the surface temperature. The experimental data enable one to derive the interatomic potential between projectile and graphite surface atom for the intermediate range of interatomic distances. The interatomic potential thus found is in fair agreement with a first order perturbation theory calculation where the Hartree-Fock electron densities have been approximated by a continuous piecewise exponentially decreasing function. The potential is much weaker than the potential derived from the Thomas-Fermi statistical model of the atom.