Thomas Fermi Dirac Model Research Articles

Relativistic effective charge model of a multi-electron atom

A relativistic version of the effective charge model (ECM) for computation of observable characteristics of multi-electron atoms and ions is developed. A complete and orthogonal Dirac hydrogen basis set, depending on one parameter—effective nuclear charge Z*—identical for all single-electron wave functions of a given atom or ion, is employed for the construction of the secondary-quantized representation. The effective charge is uniquely determined by the charge of the nucleus and a set of occupied single-electron orbitals for a given state. We thoroughly study the accuracy of the leading-order approximation for the total binding energy and demonstrate that it is independent of the number of electrons of a multi-electron atom. In addition, it is shown that the fully analytical leading-order approximation is especially suited for the description of highly charged ions since our wave functions are almost coincident with the Dirac–Hartree–Fock ones for the complete spectrum. Finally, we evaluate various atomic characteristics, such as scattering factors and photoionization cross-sections, and thus envisage that the ECM can replace other models of comparable complexity, such as the Thomas–Fermi–Dirac model for all applications where it is still utilized.

Electronic properties ofδ-doped Si:P and Ge:P layers in the high-density limit using a Thomas-Fermi method

The Thomas-Fermi-Dirac (TFD) approximation and an sp3d5s* tight binding method were used to calculate the electronic properties of a delta-doped phosphorus layer in silicon. This self-consistent model improves on the computational efficiency of "more rigorous" empirical tight binding and ab initio density functional theory models without sacrificing the accuracy of these methods. The computational efficiency of the TFD model provides improved scalability for large multi-atom simulations, such as of nanoelectronic devices that have experimental interest. We also present the first theoretically calculated electronic properties of a delta-doped phosphorus layer in germanium as an application of this TFD model.

Application of the Englert–Schwinger model to the equation of state and the fullerene molecule

The Englert–Schwinger model (ESM) is applied to two problems. One is the calculation of zero-temperature equation of state (EOS) of elements within the spherically symmetric Wigner–Sietz cell approximation. The other is to obtain the equilibrium radius of fullerene molecule using March’s approach [N. H. March, Proc. Camb. Philos. Soc. 48, 665 (1952)]. In each case, the results of the ESM are compared with those of Thomas–Fermi–Dirac (TFD) and Thomas–Fermi–Dirac–Weizsacker (TFDW) models. Zero-temperature equation of state calculations are done for Al and Cu. The results of the ESM show an enormous improvement over those of the TFD model. Also, the ESM is in good agreement with the TFDW model for compressions greater than 2. In the regime of validity of TFDW theory, i.e., compressions greater than 20 and 10 for Al and Cu, respectively, the deviations between the results of the two models are negligible. Hence, the ESM may be used in lieu of the TFDW model for EOS calculations. In the fullerene case, we have obtained the cohesive energy using the models assuming the radius obtained from accurate calculations of the fullerene molecule. We have also obtained the equilibrium radius predicted by each model. The results obtained show that the ESM results are not much of an improvement over the TFD results. This shows that the ESM cannot always improve the results of the TFD model and be a replacement for the TFDW model. However, as in the EOS case, it would give results in good agreement with TFDW results for properties that are dependent on the electron density at the outer reaches of the atom.

The study of relatively low density stellar matter in presence of strong quantizing magnetic field

The effect of strong quantizing magnetic field on the equation of state of matter at the outer crust region of magnetars is studied. The density of such matter is low enough compared to the matter density at the inner crust or outer core region. Based on the relativistic version of semi-classical Thomas–Fermi–Dirac model in presence of strong quantizing magnetic field a formalism is developed to investigate this specific problem. The equation of state of such low density crustal matter is obtained by replacing the compressed atoms/ions by Wigner–Seitz cells with nonuniform electron density. The results are compared with other possible scenarios. The appearance of Thomas–Fermi induced electric charge within each Wigner–Seitz cell is also discussed.

Approximate atomic calculations and stability of negative ions with a modified Thomas–Fermi–Dirac model

AbstractBy means of an improved modified Thomas–Fermi–Dirac approach we obtain fair shell‐averaged total energies of atoms and ions. In particular, stable single‐charged negative ions can be described with this method. In addition, the possibility of developing hybrid schemes with this method coupled to quantum mechanical calculations for the evaluation of quantities such as ionization potentials is also investigated. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Effective temperature rise during propagation of shock wave and high-speed deformation in metals

Two theoretical models are developed for the calculations of temperature rise during high-speed deformation and shock wave propagation. In the first model the calculations of the temperature distribution in metals during high-speed deformation are based on a model where the stationary high-speed deformation is considered as a propagation of shock wave with some fixed velocity in these metals. In this model the self-consistent system of equations describing the equation of state of metals and the conservation laws for momentum, energy and flow of energy is used for the determination of the temperature profile in the front of shock wave. The numerical calculations of the temperature distribution profile in shock wave front have been performed using the microscopic Thomas–Fermi–Dirac model for such metals as Al, Cu and Fe. In the second theoretical model the process of high-speed deformation is considered as an adiabatic process where a fraction of plastic deformation is converted into heat. The results of the numerical calculations of temperature rise during high-speed deformation in the dependence of strain to fracture are presented for metals: Al, Cu, Ni and Fe. It was shown that using these models the temperature during high-speed deformation can increase in different metals up to 1000 K.

A comparative study of interatomic potentials for copper and aluminum gas phase sputter atom transport simulations

A comparative study of interatomic potential models for use in gas phase sputter atom transport simulations is presented. Quantum chemical interatomic potentials for argon–copper and argon–aluminum are calculated using Kohn–Sham density functional theory utilizing the PW91 functional. These potentials (PW91) are compared to the commonly used Born–Mayer potentials calculated by Abrahamson [Phys. Rev. 178 (1969) 76] using the Thomas–Fermi–Dirac model (TFD) and the screened Coulomb potentials derived from the “universal” form calculated by Ziegler, Biersack and Littmark (ZBL). Monte Carlo simulations of gas phase sputter atom transport were performed to determine the average energy of atoms arriving at the substrate versus pressure for the three potential models. Overall, the ZBL potential gave results in much better agreement with the PW91 potential than the TFD potential. A characteristic thermalization pressure–distance product of ∼0.11 mTorr cm was found for both copper and aluminum using the PW91 potential.

Reduction of the tensile stress in CoSi 2 films by pre-deposition carbon ion implantation

The tensile stress in CoSi 2 films formed by deposition of Co films on Si(1 0 0) substrates and subsequent ex situ rapid thermal annealing was reduced by implantation of carbon ions into Si substrates before the deposition of Co films. It was found that the stress in the CoSi 2 films decreases linearly with the increase of the C + implantation doses. The reason for the reduction in the tensile stress is explained in terms of a new mechanism on the origin of the intrinsic stress in thin solid films based on a refined Thomas–Fermi–Dirac model. The theoretical calculations according to this model agree quite well with the experimental results.

Homogeneities in density of various LDA energy functionals

Expanding on the work of March (Phys. Rev. A 56 (1997) 1025), partial differential equations are obtained for self-consistent atomic electron densities in the local density approximation. Local models treated include the Thomas–Fermi model, the Thomas–Fermi–Dirac model, and the Thomas–Fermi–Dirac model plus correlation. It is shown that within these models, the Thomas–Fermi kinetic energy functional can be expressed as a functional homogeneous of degree one with respect to the self-consistent-density scaling. In addition, the Dirac exchange functional and the Liu–Parr correlation functional each also can be rewritten as a functional homogeneous of degree one in the density scaling.

THOMAS–FERMI MODEL ELECTRON DENSITY WITH CORRECT BOUNDARY CONDITIONS: APPLICATIONS TO ATOMS AND IONS

We propose an electron density in atoms and ions, which has the Thomas–Fermi–Dirac form in the intermediate region ofr, satisfies the Kato condition for smallr, and has the correct asymptotic behavior at large values ofr, whereris the distance from the nucleus. We also analyze the perturbation in the density produced by multipolar fields. We use these densities in the Poisson equation to deduce average values ofrm, multipolar polarizabilities, and dispersion coefficients of atoms and ions. The predictions are in good agreement with experimental and other theoretical values, generally within about 20%. We tabulate here the coefficientAin the asymptotic density; radial expectation values 〈rm〉 form= 2, 4, 6; multipolar polarizabilities α1, α2, α3; expectation values 〈r0〉 and 〈r2〉 of the asymptotic electron density; and the van der Waals coefficientC6for atoms and ions with 2 ≤Z≤ 92. Many of our results, particularly the multipolar polarizabilities and the higher order dispersion coefficients, are the only ones available in the literature. The variation of these properties also provides interesting insight into the shell structure of atoms and ions. Overall, the Thomas–Fermi–Dirac model with the correct boundary conditions provides a good global description of atoms and ions.

Superfluidity near solid surfaces in a Thomas-Fermi-Dirac jellinm model: Potassium and nickel

Generally an almost ideal Fermi system with attraction between fermions possesses nonNewtonian behavior. Electrons described according to the Thomas-Fermi-Dirac model (TFD) form such a system. We investigated these electrons in solids with surfaces. In this kind of non-Newtonian motions, superfluidity appeared under certain condition. This conclusion is shown theoretically when we apply this model to potassium and nickel. For K, there is a layer of thickness only 0.33 bohr near the surface in which electrons move to higher potential locations similar to the climbing effect under gravity although non-Newtonian motions occur throughout the system. For nonmagnetic 4s2 electrons of Ni, there may also be a layer of this kind of thickness 0.25 bohr near the surface after we choose the surface density properly in theory. In all these systems, electrons are reflected from surface to interior layers.

Thomas-Fermi-Dirac models of atoms constrained by nuclear cusp and long-range conditions

The traditional Thomas–Fermi–Dirac model of the electronic structure for a neutral atom is deficient in that it predicts an infinite electron density at the nucleus and a sharp cutoff of the electron density at a finite radius. This study was carried out to remedy these faults in the model. Extending an idea used earlier in Thomas–Fermi (TF) theory [Proc. Natl. Acad. Sci. U.S.A. 83, 3577 (1985)], the Thomas–Fermi–Dirac (TFD) energy functional is minimized under constraints ∫ρ(r) dr = N, ∫e−2kr▽2ρ(r)dr < ∞ and ∫(1 − e−k′r)ρ4/3(r)dr < ∞, with k and k′ determined by the nuclear cusp condition and the correct asymptotic behavior. Optimum coordinate scaling also is considered. It is found that the TFD model is substantially improved by constraining the minimization search domain of the energy functional in this way. Energies are given for five noble gas atoms, and Compton profiles for these atoms are calculated. The behavior of electrons in momentum space is improved in both this modified TFD model and in the corresponding modified TF model.

Scaling laws for pressure, temperature, and ionization with two-temperature equation-of-state effects in laser-produced plasmas

A two-temperature equation of state (EOS) for a plasma medium was developed. The cold-electron temperature is taken from semiempirical calculations, while the thermal contribution of the electrons is calculated from the Thomas–Fermi–Dirac model. The ion EOS is obtained by the Gruneisen–Debye solid–gas interpolation method. These EOS are well behaved and are smoothed over the whole temperature and density regions, so that the thermodynamical derivatives are also well behaved. Moreover, these EOS were used to calculate the average ionization 〈Z〉 and 〈Z2〉 of the plasma medium. Furthermore, the thermal conductivities have been calculated with the use of an extrapolation between the conductivities in the solid and plasma (Spitzer) states. The two-temperature EOS, the average ionization 〈Z〉 and 〈Z2〉, and the thermal conductivities for electrons and ions were introduced into a two-fluid hydrodynamic code to calculate the laser-plasma interaction in carbon, aluminum, copper, and gold slab targets. It was found that the two EOS are important mainly from the ablation surface outward (toward the laser). In particular, the creation of cavitons in the distribution of the electrons is predicted here, especially for light materials such as aluminum. These studies enable us also to establish that the commonly used exponential scaling laws of the type Pa = A[I/(1014 W/cm2)]α for the ablation pressure and similar laws for the temperature are valid only for absorbed laser intensities in the range 3 × 1012-3 × 1014 W/cm2, while the degree of ionization (at the corona) follows a quite different scaling law. We also found that the parameters A and α. in the above expression are dependent on material, laser wavelength, and pulse shape. Thus we determined for the ablation pressure, using a trapezoidal Nd laser pulse, that α varies between 0.78 and 0.84 and that A varies between 14 and 8 Mbar for 6 ≤ Z ≤ 79. Beyond the range of validity the scaling laws may give values at least twice as large as those obtained by the simulation.

Electron-spin polarization in the Thomas-Fermi and Thomas-Fermi-Dirac atoms.

The Thomas-Fermi (TF) and Thomas-Fermi-Dirac (TFD) statistical models are extended by the electron-spin interaction with an external uniform magnetic field B. The statistical atom, which is spherical, is then described with the aid of two distributions: the electron density n(r) and the relative magnetization \ensuremath{\zeta}(r). In the TF atom the magnetic field polarizes the electron spins totally (\ensuremath{\zeta}=1) in a region near the atomic boundary, while the rest of the TF atom is only partially polarized; however, \ensuremath{\zeta}(r) and n(r) are continuous. A convenient description of this atom is done with the aid of a suitably defined reduced potential f(x) to which n(r) and \ensuremath{\zeta}(r) are related. At B&gt;0 the neutral TF atom remains infinite. Its magnetic moment shows unphysical ${\mathit{B}}^{3/4}$ proportionality for small B. The inclusion of the exchange in the TFD model for B&gt;0 results in two possible types of atom: the atom of the type I---which exists only for B\ensuremath{\le}1.3\ifmmode\times\else\texttimes\fi{}${10}^{7}$ G---has continuous n(r) and \ensuremath{\zeta}(r); in the type-II atom there is a discontinuity in both n(r) and \ensuremath{\zeta}(r) that makes this atom unusable in physical applications. The type-I atoms have finite radii and are assumed to represent approximately the crystal atomic cells. The appropriate differential equation and boundary conditions for \ensuremath{\zeta}(r), together with the relationship between \ensuremath{\zeta}(r) and n(r), have been derived for this atom. The solutions of the equations and the atomic radii have been obtained for a wide range of atomic numbers and magnetic fields. On the basis of these solutions, the coefficients for the volume magnetostriction, the spin susceptibilities, and the ionization energies have been calculated for elements of the whole periodic table. Qualitatively, the experimental tendencies are found to be represented well by the type-I TFD atoms. A modified virial theorem for the spin-polarized TF and TFD atoms has been proved.

Equation of state in metals and cold stars - Evaluation of statistical models

The Thomas-Fermi-Dirac (TFD) statistical model for electronic structures if refined by Weizsacker gradient and correlation energy corrections. The resulting FTD-λWc model is then used to calculate the nonrelativistic, cold matter equation of state (EOS) in the Wigner-Seitz spherical cell approximation. The correction terms are important mainly at low densities near matter equilibrium (i.e., zero pressure). Inclusion of the gradient term removes many of the unphysical features of the TFD model. Results are summarized for several elements of astrophysical and theoretical importance. The mass-radius relation of a low-mass white dwarf model is calculated using the TDF-λWc EOS.

Gradient expansion correction to the Dirac exchange term in statistical models for the Na atom with shell structure

AbstractIn the study of the ground‐state binding energy of atoms, relatively more attention has been paid to the gradient expansion of the kinetic energy term of the Thomas–Fermi–Dirac model than to the gradient expansion of the exchange term of this model. Recently Shih and Shih, Murphy, and Wang have focused on this problem and calculated the first gradient expansion correction to the Dirac exchange term by making use of electron densities constructed from Hartree–Fock wave functions. In previous work, aimed to introduce the shell structure of atoms via a variational procedure using the energy density functional formalism, electron densities have been obtained for the Na atom within the Thomas–Fermi–Dirac model with and without the Weizsäcker and Hodges gradient expansion corrections to the kinetic energy term. In this paper we make use of the respective electron densities and calculate the first gradient expansion corrections to the Dirac exchange term. The results show that, for the Na atom, the magnitude of this correction is about 1% of the magnitude of the total binding energy.

Zero-temperature equation of state of metals in the statistical model with density gradient correction

Abstract The density gradient correction to kinetic energy (nine times smaller than the original von Weizsacker correction) has been used within local density formalism to calculate the cold compression curve of metals. Numerical results are reported for Li, Be, Al and Cu. The modifications to the Thomas-Fermi-Dirac (TFD) results strongly depend, in the low compression range ( ϱ / ϱ 0 ⩽ 5), on the electron density of the material. The relative difference between the pressures in the present model and the TFD model regularly decreases with increasing compression, suggesting that TFD is reliable for ϱ / ϱ 0 ⩾ 50 in Li, ϱϱ 0 ⩾ 20 in Be and Al and ϱ / ϱ 0 ⩾ 10 in Cu.

Note on the Thomas–Fermi and Thomas–Fermi–Dirac model calculation of atomic polarizabilities

This paper discusses a recent calculation by Bruch and Lehnen of the static electric dipole polarizabilities of neutral inert gas atoms. These authors obtained the polarizabilities within the framework of statistical models by making use of an energy functional method. The method, as applied by Bruch and Lehnen, makes use of the exact solutions of either the Thomas–Fermi (TF) or of the Thomas–Fermi–Dirac (TFD) equations. This paper points out that a better agreement than the one obtained by Bruch and Lehnen between calculated and experimental values of the atomic polarizabilities can be obtained upon resorting to approximate (analytical) variational solutions of the TF and TFD equations. The improved agreement, in the case of the TF model, is attributed to the fact that the variational solution of the TF equation permits one to construct a radial electron density for a neutral atom that decreases with the distance from the nucleus exponentially. That this is an improvement is seen from the fact that in the exact TF model the radial electron density drops off with the distance from the nucleus as the inverse fourth power of this quantity. Essentially the same situation prevails in the case of the approximate (analytical) solution of the TFD equation, which makes use of the variational solution for a TF atom. In this case, the improved radial electron density manifests itself in a contraction of the atomic radius, relative to that which is obtained within the exact TFD model.

Statistical model calculations of atomic polarizabilities

The static electric dipole polarizabilities of neutral inert gas atoms are obtained with the Thomas–Fermi (TF) and the Thomas–Fermi–Dirac (TFD) statistical models. The TF model yields an infinite polarizability; the TFD model yields polarizabilities which are much larger than experiment, though the percent error decreases with increasing atomic number.

An Extended Thomas-Fermi-Dirac Theory for Diatomic Molecule

In order to remedy the well-known defect of the Thomas-Fermi-Dirac (TFD) theory which in its usual treatment fails to give a stable binding state of a molecule, the Weizsäcker correction multiplied by a constant weighting factor λ (=1/5) is introduced as a quantum correction. Numerical calculations to solve this extended TFD equation are carried out for the nitrogen molecule, and it is shown that the TFD model with the modified Weizsäcker correction surely yields an energetically stable molecule. The calculated values of the dissociation energy D e and of the equilibrium internuclear distance R e are 0.342 a.u. and 2.425 a.u., respectively. These values are in good agreement with experiment.