Statistical Model Of Atom Research Articles

Statistical Model for Quasicontinuum of Heavy Ions in Hot Plasma

A statistical description is proposed for the shape of heavy impurity spectra observed in tokamaks and stellarators, corresponding to the quasicontinuous radiation distribution, and called the quasicontinuum. For the tungsten ions, in particular, it is located in the spectral range of ∼2 to 7 nm. The method is based on the statistical plasma model of atom, which allows expressing the quasicontinuum structure through the electron density distribution of multielectron ions. The Slater distribution is used as a model density distribution. The quasicontinuum is formed by the great number of closely spaced lines of heavy ions with different degrees of ionization, and its calculation is very complicated. A comparison of the results obtained within the statistical approach with the results of the level-by-level quantum mechanical calculations shows that such a model plausibly describes the envelopes of the individual line arrays in the ion spectra. In contrast to the level-by-level calculation codes, the statistical atom model is rather simple and it can be used by a large number of users. The quasicontinuum was simulated for the tungsten, gold, lead, and gadolinium ions, which are observed in plasmas of fusion facilities. The simulation results are in fairly good agreement with the experimental data. The studies of the tungsten quasicontinuum are of interest from the point of view of plasma diagnostics, as well as determination of the tungsten impurity density and its transport in fusion facilities.

Self-consistent assessment of Englert-Schwinger model on atomic properties

Our manuscript investigates a self-consistent solution of the statistical atom model proposed by Berthold-Georg Englert and Julian Schwinger (the ES model) and benchmarks it against atomic Kohn-Sham and two orbital-free models of the Thomas-Fermi-Dirac (TFD)-λvW family. Results show that the ES model generally offers the same accuracy as the well-known TFD-15vW model; however, the ES model corrects the failure in the Pauli potential near-nucleus region. We also point to the inability of describing low-Z atoms as the foremost concern in improving the present model.

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.

Statistical Model of Dielectronic Recombination of Heavy Ions in Plasmas

AbstractCalculation of the total dielectronic recombination (DR) rates was done in the frame of a statistical model of atoms. The model is based on the idea of collective excitations of atomic electrons with the local plasma frequency, which depends on atomic electrons density distribution. The electron density is described in a frame of the Thomas‐Fermi model of atoms. Simple scaling laws for temperature Te and nuclear charge Z dependences follow from the statistical model of DR. Results of the statistical model were compared with other numerical data following detailed level‐by‐level computations for different multielectron ions. The specific attention is paid to Ni‐like ion sequences of different chemical elements in order to check the Z ‐dependence of DR rates. A comparison with numerical data of Flexible Atomic Code (FAC) is presented for tungsten ions. The reasonable correspondence between the statistical model and the detailed numerical data is demonstrated. The application of the statistical model provides very simple and fast calculations of the DR rates useful in modern plasma modelling. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A statistical model to predict the steady-state creep rate

For a long time, empirical formulars have been used to predict the steady-state creep rate due to lack of clear microscopic description of the mechanism, which frequently leads to unreliable predictions. In this work, a statistical model of single atom developed recently is used to predict the steady-state creep rate at an atomic diffusion level. To test the model, we measure the creep rates of three kinds of materials, i.e., 42CrMoA, 2Cr12Ni, and 1Cr12Mo, and collect the experimental data of other materials, such as IN738LC and K435. The results show that our theoretical predicts are in good agreement with the experimental results.

Correlation between diffraction and viscosity data for Bi–Ga molten alloys

Structure of Bi $_{\boldsymbol{100-x}}$ Ga $_{\boldsymbol{x}}$ molten alloys containing 38·5, 50, 70 and 91·5 at. % Ga has been studied by means of X-ray diffraction method and compared with viscosity measurements data. Significant changes in the structure factor profile were observed in vicinity of the concentration 70 at. % Ga. The dynamic viscosity coefficient was calculated by use of a statistical atomic distribution model and a Born–Green kinetic theory. The concentration dependence of viscosity is in agreement with change of structure parameters obtained from diffraction data.

Analysis of ionization and recombination processes under the coronal equilibrium conditions by using a statistical atomic model

Ionization and recombination processes accompanying collisions of free electrons with plasma ions are considered using a statistical atomic model in which ionization and recombination are regarded as the processes of pair electron collisions in the electron gas of an atom. An expression for the ionization rate as a function of the ionization energy I and temperature T is derived. According to this expression, the ionization rate at I ≫ T is proportional to exp(−I/T). The statistical atomic model provides an estimate of the recombination rate for an ion with an arbitrary nuclear charge number Z, whereas more exact calculations of the recombination rate can be performed only for large Z. The model explains relatively low values of I/T (as compared to those given by the Saha equation) under the coronal equilibrium conditions and predicts a reduction in I/T with increasing Z. The values of I/T and the average ion charge number obtained from the balance equation for multielectron ions with the use of one fitting coefficient agree with the tabulated data calculated in the multilevel coronal model.

Statistical models in the polarization radiation theory

The survey is devoted to general results for polarization free–free and free–bound radiative transitions in collisions of charged particles with heavy atoms and ions following from statistical atomic models. The atomic plasma model results for dynamic atomic polarizabilities are presented. Polarization Bremsstrahlung in collisions of fast and moderate energy electrons with Thomas–Fermi atoms is analyzed in details. A new polarization electron-heavy ions recombination channel is considered and compared with known radiative recombination channel. It is shown that recombination channels can be compared or even dominates over standards static radiation channels. Polarization radiation of fast multicharged ions in condensed media is also under consideration.

Statistical atomic models with complete neglect of differential overlap for the study of free and confined systems

AbstractTotal ground‐state energies of free and confined many‐electron atoms are studied through use of the Thomas–Fermi–Dirac–Weizsäcker energy density functional using known properties of the orbital electron densities. The concept of complete neglect of differential overlap put forward by Pople and Segal is combined with the original ideas of Wang and Parr on the development of statistical atomic models. For free atoms, it is found that total energies within 0.1% difference relative to Hartree–Fock values can be obtained using a prefactor λ = 1/8 in the Weizsäcker inhomogeneity correction. The corresponding description of total radial densities and shell structure is found to be overall moderate when compared with Hartree–Fock calculations. It is also shown, for the first time, that this approach properly accounts for the ground‐state energy evolution of many‐electron atoms confined by hard spherical walls. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Fermi, Majorana and the Statistical Model of Atoms

We give an account of the appearance and first developments of the statistical model of atoms proposed by Thomas and Fermi, focusing on the main results achieved by Fermi and his group in Rome. Particular attention is addressed to the unknown contribution to this subject by Majorana, anticipating some important results reached later by leading physicists.

Simple stopping power formula for low and intermediate energy electrons

A simple stopping power formula, modified from that of Rohrlich and Carlson is presented. In this study analytical expressions for effective charge and effective mean excitation energies of target atoms in the modified Rohrlich and Carlson stopping power formula are used, while for effective charge of incoming electrons, Sugiyama's semiempirical formula is used from Peterson and Green. The Lenz–Jensen statistical atomic density model has been used for calculations of effective charge and effective mean excitation energies. The calculated results of stopping power for electrons in the energy range from a few tens of electron volts to 10 MeV are found to be in good agreement to within 10% with the experimental data and a number of other calculations.

A rigorous test of the statistical model for atom–diatom insertion reactions

The statistical model of atom–diatom insertion reactions is combined with coupled-channel capture theory and used to calculate differential cross sections for the reactions of C(1D), N(2D), O(1D) and S(1D) with H2. In the case of C(1D) and S(1D), the resulting statistical differential cross sections are found to be in excellent agreement with the recent quantum reactive scattering calculations of Honvault and Launay. They are therefore also in good agreement with molecular beam experiments for the S(1D)+H2 reaction, in contrast to the results of earlier calculations based on a less rigorous statistical theory. However, because the exact quantum mechanical differential cross sections for N(2D) and O(1D) exhibit a slight forward–backward asymmetry, the agreement with the statistical model for these reactions is not quite so good. The difference between the two cases can be rationalized in terms of the greater exoergicities of the N(2D) and O(1D) reactions, which lead to broader resonances and hence to shorter lifetimes of the H2O and NH2 collision complexes than those of CH2 and H2S.

Radiation loss of electrons under scattering by a Thomas-Fermi atom

A universal theory and calculation results for the bremsstrahlung of electrons on complex atoms are presented. The theory accounts for the dynamic polarization of the core in the energy range from 0.5 to 10 keV, which is characteristic of radiation energy losses in a hot plasma with heavy ions. The treatment is based on the statistical atom model and the quasi-classical approximation of the incident electron. The model accounts for the penetration of the incident electron into the atomic core, which affects the relationship between the polarization and static radiation channels. The contribution of the polarization channel in both the spectral and the total radiation loss of electrons at various frequencies, nucleus charges, and energies of the incident particle is analyzed. It is shown that the contribution of the polarization channel is comparable with that of the static channel (which was calculated elsewhere) in a wide range of parameters. The results obtained are in a reasonable quantitative agreement with the detailed quantum-mechanical calculation carried out for individual atoms.

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.

Statistical Models in Theory of Polarization Bremsstrahlung

Universal relations have been derived to describe the spectral-angular dependence of the polarization bremsstrahlung of a fast charged particle, based on a statistical model for the electron skeleton of the target. Three statistical atomic models are considered in terms of which the target characteristics that determine the cross section for polarization bremsstrahlung are calculated and analyzed.

NLTE ionization and energy balance in high- Z laser-plasmas including two-electron transitions

We describe a new non-LTE statistical atomic kinetics model of plasmas in which the two-electron transitions of auto-ionization and its inverse, resonant-capture, play a dominant role in establishing ionization and energy balance. We show that, compared with a familiar collisional–radiative-equilibrium model which includes only the one-electron bound–bound and bound–free transitions: (1) the two-electron transitions force recombination of the plasma with decreasing density, (2) the two-electron transitions nevertheless further act to greatly increase the radiative emissivity of the plasma, and (3) the relaxation of the two-electron transition driven sysems proceeds much faster.

Atomization Energy of the Cu, Ag, Au Crystals in Terms of the Statistical Model

AbstractThe statistical atom model is used for ab initio calculations of the atom binding energy in crystals. The original calculations by the above method of atomization energy at T = 0 K of the A1 modification crystals of IB elements group of the Periodic System are consistent with the experimental data. The atomization energy estimates of the A2, A3 modifications are made.

Thomas - Fermi-type dielectric screening of statistical atomic models of donor-specific impurities in a semiconductor-like valence electron gas at zero temperature

Thomas - Fermi- (TF-) type theories are applied to the problem of static dielectric screening in a semiconductor at absolute zero when the impurity-charge-density distribution is given by a variational statistical approximation. This treatment of the external perturbation is consistent with the valence electron-gas formulation of the host crystal and in contrast with the point-charge probe and other donor-specific or pseudocharge distributions that have been previously studied in this context. One-parameter exponential bound-electron charge densities are employed in this report. For purposes of illustration, atomic parameters specifying the chemical identity of an impurity are characteristic of phosphorus (group V) and sulphur (group VI) substitutional single and double donors, respectively, and a reference silicon ion. Linearized TF and Thomas - Fermi - Dirac (TFD) screening equations for the exponential case and its point-charge limit are solved in closed analytical form with silicon as the host semiconductor. The corresponding nonlinear TF and TFD equations are solved numerically and nonlinear screened impurity potentials are compared with linear results. Dielectric screening of statistical impurities follows the same general trend as noted for a point charge, that is, the nonlinear theory is more effective than its linear approximation. Furthermore, for a given statistical impurity, with a given degree of ionization, the ion potential is more effectively screened in the following order: linear TF, linear TFD, nonlinear TF, and nonlinear TFD. Corresponding screening radii decrease in this order. Within the set of statistical ions under consideration, linear and corresponding nonlinear screening radii and their differences decrease as the degree of ionization decreases. Further development is planned to test the usefulness of this approach to donor and host potentials in conventional and generalized effective-mass calculations of isocoric and nonisocoric binding energies.

Two-Center Problem for a Dense Plasma

The Thomas-Fermi-Dirac-Weizsacker statistical model of atoms is applied to a compressed diatomic molecule in a dense plasma. As a two-dimensional problem, we analyze numerically a spatial profile of the screening function and the electron density inside a closed boundary surface. We also calculate the Helmholtz's excess free energy as a function of the internuclear distance, to examine a possibility of the binding state.

Multiple ionization of atoms by powerful circularly polarized laser field.

We present a theoretical description of the multiple ionization of atoms by a poweful circularly polarized laser field. Our calculations are based on the statistical model of the atom. The modified Thomas-Fermi equation in the rotating frame has been derived. Appearance intensities versus laser frequency have been plotted and discussed. The instability of the solution for circularly polarized laser light of sufficiently high frequency is expected.