Particle diffusion in degenerate, ionized gases
Particle diffusion distribution function of two component ionized degenerate gas from Boltzmann transport equation, giving tables for various spin and mass ratios
Rosen, L. C.↗
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Ion velocity distributions in the high-latitude ionosphere
The theory and observations relating to ion velocity distributions in the high-latitude F region are reviewed. The review covers three basic aspects, including methods for calculating ion velocity distributions, experimental evidence for the existence of non-Maxwellian ion velocity distributions, and the ionospheric consequences of non-Maxwellian distributions. Specific topics covered include closed form solutions to Boltzmann's equation for a simple collision model, series solutions to the Boltzmann equation for more general collision models, retarding potential analyzer measurements, excitation of VLF electrostatic emissions, ion-neutral reaction rates, and transport effects.
St-Maurice, J.-P.↗
Electrical characterization of Hg(1-x)Cd(x)Te alloys
Theoretical models are described for calculations of charge-carrier concentrations, Fermi energy, and conduction electron mobility as functions of x, temperature, and ionized and neutral defect concentrations of Hg(1-x)Cd(x)Te alloys. Measurements are reported of electron concentration and electron mobility from 5-300 K for alloys with x values of between 0.17 and 0.30. The electrical data are in reasonable agreement with theory, and were analyzed to obtain estimates of donor and acceptor state concentrations. The electron mobilities are calculated in terms of a microscopic theory of electrical conduction derived from the solution of the Boltzmann equation for the perturbed steady-state electron distribution function, and they show that longitudinal optical-phonon and charged and neutral defect scattering are the dominant mobility-limiting mechanisms.
Lehoczky, S. L.↗
Surface-slip equations for multicomponent nonequilibrium air flow
Equations are presented for the surface-slip (or jump) values of species concentration, pressure, velocity, and temperature in the low-Reynolds number, high-altitude flight regime of a space vehicle. The equations are obtained from closed form solutions of the mass, momentum, and energy flux equations using the Chapman-Enskog velocity distribution function. This function represents a solution of the Boltzmann equation in the Navier-Stokes approximation. The analysis, obtained for nonequilibrium multicomponent air flow, includes the finite-rate surface catalytic recombination and changes in the internal energy during reflection from the surface. Expressions for the various slip quantities were obtained in a form which can be employed in flowfield computations. A consistent set of equations is provided for multicomponent, binary, and single species mixtures. Expression is also provided for the finite-rate, species-concentration boundary condition for a multicomponent mixture in absence of slip.
Gupta, R. N.↗
Slip-boundary equations for multicomponent nonequilibrium airflow
Equations are presented for the surface-slip (or jump) values of species concentration, pressure, velocity, and temperature in the low Reynolds number, high-altitude flight regime of a space vehicle. These are obtained from closed-form solutions of the mass, momentum, and energy flux equations by using the Chapman-Enskog velocity distribution function. This function represents a solution of the Boltzmann equation in the Navier-Stokes approximation. The analysis, obtained for nonequilibrium multicomponent airflow, includes the finite-rare surface catalytic recombination and changes in the internal energy during reflection from the surface. Expressions for the various slip quantities have been obtained in a form which can readily be employed in flow-field computations. A consistent set of equations is provided for multicomponent and binary mixtures and single-species gas. An expression is also provided for the finite-rate species-concentration boundary condition for a multicomponent mixture in the absence of slip.
Gupta, R. N.↗
Surface-slip equations for multicomponent, nonequilibrium air flow
Equations are presented for the surface slip (or jump) values of species concentration, pressure, velocity, and temperature in the low-Reynolds-number, high-altitude flight regime of a space vehicle. These are obtained from closed-form solutions of the mass, momentum, and energy flux equations using the Chapman-Enskog velocity distribution function. This function represents a solution of the Boltzmann equation in the Navier-Stokes approximation. The analysis, obtained for nonequilibrium multicomponent air flow, includes the finite-rate surface catalytic recombination and changes in the internal energy during reflection from the surface. Expressions for the various slip quantities have been obtained in a form which can readily be employed in flow-field computations. A consistent set of equations is provided for multicomponent, binary, and single species mixtures. Expression is also provided for the finite-rate species-concentration boundary condition for a multicomponent mixture in absence of slip.
Gupta, Roop N.↗
A method for direct numerical integration of the Boltzmann equation
The principal difficulties in numerical solution of the Boltzmann equation are considered. The study is aimed at formulating a numerical solution in such a manner that it contains a minimum amount of excess information at the distribution function level. It is pointed out that the accurate calculation of the distribution function at each point in phase space requires a tremendous number of operations, due to the necessity of solving five-fold quadratures in the collision integral. This results in the operational memory of the digital computer being insufficient to store all the data on the distribution functions at the necessary points in phase space. An algorithm is constructed involving successive iterations of the Boltzmann equation which does not require storage of each step of the new distribution function.
Cheremisin, F. G.↗
Charge state distributions in dense plasmas
Charge state distributions in hot, dense plasmas are a key ingredient in the calculation of spectral quantities like the opacity. However, they are challenging to calculate, as models like Saha–Boltzmann become unreliable for dense, quantum plasmas. Here, we present a new variational model for the charge state distribution, along with a simple model for the energy of the configurations that includes the orbital relaxation effect. Comparison with other methods reveals generally good agreement with average atom-based calculations, the breakdown of the Saha–Boltzmann method, and mixed agreement with a chemical model. We conclude that the new model gives a relatively inexpensive, but reasonably high fidelity method of calculating the charge state distribution in hot dense plasmas, in local thermodynamic equilibrium.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Modeling rarefied gas chemistry with QuiPS, a novel quasi-particle method
The goal of this work is to build up the capability of quasi-particle simulation (QuiPS), a novel flow solver, such that it can adequately model the rarefied portion of an atmospheric reentry trajectory. Direct simulation Monte Carlo (DSMC) is the conventional solver for such conditions, but struggles to resolve transient flows, trace species, and high-level internal energy states due to stochastic noise. Quasi-particle simulation (QuiPS) is a novel Boltzmann solver that describes a system with a discretized, truncated velocity distribution function. The resulting fixed-velocity, variable weight quasi-particles enable smooth variation of macroscopic properties. The distribution function description enables the use of a variance-reduced collision model, greatly minimizing expense near equilibrium. This work presents the addition of a neutral air chemistry model to QuiPS and some demonstrative 0D simulations. The explicit representation of internal distributions in QuiPS reveals some of the flaws in existing physics models. Overall, variance reduction, a key feature of QuiPS, can greatly reduce expense of multi-dimensional calculations, but is only cheaper when the gas composition is near chemical equilibrium.
42 ENGINEERING↗
Observing differential spin currents by resonant inelastic X-ray scattering
Controlling spin currents, that is, the flow of spin angular momentum, in small magnetic devices, is the principal objective of spin electronics, a main contender for future energy-efficient information technologies. A pure spin current has never been measured directly because the associated electric stray fields and/or shifts in the non-equilibrium spin-dependent distribution functions are too small for conventional experimental detection methods optimized for charge transport. Here we report that resonant inelastic X-ray scattering (RIXS) can bridge this gap by measuring the spin current carried by magnons—the quanta of the spin wave excitations of the magnetic order—in the presence of temperature gradients across a magnetic insulator. This is possible due to the sensitivity of the momentum- and energy-resolved RIXS intensity to minute changes in the magnon distribution under non-equilibrium conditions. Furthermore, we use the Boltzmann equation in the relaxation time approximation to extract transport parameters, such as the magnon lifetime at finite momentum, essential for the realization of magnon spintronics.
36 MATERIALS SCIENCE↗
Thermal fluctuations in the Landau-Lifshitz-Bloch model
A formulation for thermal noise in the stochastic form of the Landau-Lifshitz-Bloch equation used for modeling the magnetization dynamics at elevated temperatures is presented here. The diffusion coefficients for thermal fluctuations are obtained via the Fokker-Plank equation using the mean-field approximation of the field in defining the free energy. The presented model leads to a mean magnetization consistent with the equilibrium magnetization for small and large particles. The distribution of the magnetization magnitude is of the Maxwell-Boltzmann type. The presented model was tested by studying the equilibrium magnetization in macrospin particles at high temperatures. The model is appealing for multiscale modeling, such as modeling heat assisted magnetic recording systems and all-optical magnetization reversal.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Exact results on the temperature dependence of the specific equilibrium recombination rate coefficient
Two theorems based on Laplace transforms are used to relate threshold and high energy behavior to a temperature dependent variable. A recombination reaction involving electrically neutral species is presented to illustrate the dynamics of the activation energy and the associated rate coefficient. The distribution functions are shown to correspond with those predicted by Boltzmann's equations.
Mickens, R. E.↗
Role of thermal conduction in the acceleration of the solar wind
The thermal and suprathermal processes involving solar wind electrons are discussed from a theoretical point of view. A model for the electron distribution function, f(e), based on the solutions of the Boltzmann equation in Krook's approximation is outlined: the angular and energy dependences of for various distances from the Sun between the coronal base and the Earth are presented with the express purpose of obtaining the radial profile of the heat flux generated by the suprathermals. The basic fluid-dynamical equations for the solar wind plasma as a whole along any given magnetic field line are solved. Results are in good agreement with observations. The predicted density and temperature profiles for positive ions exhibit the characteristics of recent measurements, both in the corona (above the coronal holes) and between 0.3 and 1 A.U. (the Helios region). It is concluded that the suprathermal electrons play an important role in the formation and the dynamics of the solar wind.
Olbert, S.↗
A second-order accurate kinetic-theory-based method for inviscid compressible flows
An upwind method for the numerical solution of the Euler equations is presented. This method, called the kinetic numerical method (KNM), is based on the fact that the Euler equations are moments of the Boltzmann equation of the kinetic theory of gases when the distribution function is Maxwellian. The KNM consists of two phases, the convection phase and the collision phase. The method is unconditionally stable and explicit. It is highly vectorizable and can be easily made total variation diminishing for the distribution function by a suitable choice of the interpolation strategy. The method is applied to a one-dimensional shock-propagation problem and to a two-dimensional shock-reflection problem.
Deshpande, Suresh M.↗
Numerical investigations of low-density nozzle flow by solving the Boltzmann equation
A two-dimensional finite-difference code to solve the BGK-Boltzmann equation has been developed. The solution procedure consists of three steps: (1) transforming the BGK-Boltzmann equation into two simultaneous partial differential equations by taking moments of the distribution function with respect to the molecular velocity u(sub z), with weighting factors 1 and u(sub z)(sup 2); (2) solving the transformed equations in the physical space based on the time-marching technique and the four-stage Runge-Kutta time integration, for a given discrete-ordinate. The Roe's second-order upwind difference scheme is used to discretize the convective terms and the collision terms are treated as source terms; and (3) using the newly calculated distribution functions at each point in the physical space to calculate the macroscopic flow parameters by the modified Gaussian quadrature formula. Repeating steps 2 and 3, the time-marching procedure stops when the convergent criteria is reached. A low-density nozzle flow field has been calculated by this newly developed code. The BGK Boltzmann solution and experimental data show excellent agreement. It demonstrated that numerical solutions of the BGK-Boltzmann equation are ready to be experimentally validated.
Deng, Zheng-Tao↗
Validation of the SCEPTRE Boltzmann-CSD Solver
A new Boltzmann-CSD solver has been developed within the SCEPTRE radiation-transport code, based on the 1st-order form of the transport equation, using discontinuous finite elements in space and energy and discrete ordinates in angle. The Boltzmann-CSD solver has been validated against experimental data for electron energy deposition distributions and for electron emission spectra. Comparison of the calculated results with experimental data shows excellent agreement for many of the test configurations and reasonable agreement for other test configurations. The tests have also been modeled with the ITS Monte Carlo code, which also shows excellent to reasonable agreement with the SCEPTRE results and experimental data. The SCEPTRE Boltzmann-CSD solver relies on electron cross sections generated by the legacy CEPXS code, which currently is limited to electron-only Boltzmann-CSD cross sections. Performing full electron-photon radiation transport with the Boltzmann-CSD solver will require further development in the cross section generating code. For the energy- deposition calculations, neglecting photon transport results in at most about 5% overprediction of the energy deposition for high-energy electrons on high-Z targets, and relatively insignificant difference for the other test configurations.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Behaviour of ion velocity distributions for a simple collision model
Calculation of the ion velocity distributions for a weakly ionized plasma subjected to crossed electric and magnetic fields. An exact solution to Boltzmann's equation has been obtained by replacing the Boltzmann collision integral with a simple relaxation model. At altitudes above about 150 km, where the ion collision frequency is much less than the ion cyclotron frequency, the ion distribution takes the shape of a torus in velocity space for electric fields greater than 40 mV/m. This shape persists for one to two hours after application of the electric field. At altitudes where the ion collision and cyclotron frequencies are approximately equal (about 120 km), the ion velocity distribution is shaped like a bean for large electric field strengths. This bean-shaped distribution persists throughout the lifetime of ionospheric electric fields. These highly non-Maxwellian ion velocity distributions may have an appreciable affect on the interpretation of ion temperature measurements.
St-Maurice, J.-P.↗
Information theory lateral density distribution for Earth inferred from global gravity field
Information Theory Inference, better known as the Maximum Entropy Method, was used to infer the lateral density distribution inside the Earth. The approach assumed that the Earth consists of indistinguishable Maxwell-Boltzmann particles populating infinitesimal volume elements, and followed the standard methods of statistical mechanics (maximizing the entropy function). The GEM 10B spherical harmonic gravity field coefficients, complete to degree and order 36, were used as constraints on the lateral density distribution. The spherically symmetric part of the density distribution was assumed to be known. The lateral density variation was assumed to be small compared to the spherically symmetric part. The resulting information theory density distribution for the cases of no crust removed, 30 km of compensated crust removed, and 30 km of uncompensated crust removed all gave broad density anomalies extending deep into the mantle, but with the density contrasts being the greatest towards the surface (typically + or 0.004 g cm 3 in the first two cases and + or - 0.04 g cm 3 in the third). None of the density distributions resemble classical organized convection cells. The information theory approach may have use in choosing Standard Earth Models, but, the inclusion of seismic data into the approach appears difficult.
Rubincam, D. P.↗