Self-consistent method for determining the boundary shape between a plasma and a magnetic field.
Reiterative method for determining boundary shape between plasma and magnetic field
BOUNDARY LAYER↗
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Study of the extra-ionic electron distributions in semi-metallic structures by nuclear quadrupole resonance techniques
A straightforward self-consistent method was developed to estimate solid state electrostatic potentials, fields and field gradients in ionic solids. The method is a direct practical application of basic electrostatics to solid state and also helps in the understanding of the principles of crystal structure. The necessary mathematical equations, derived from first principles, were presented and the systematic computational procedure developed to arrive at the solid state electrostatic field gradients values was given.
Murty, A. N.↗
Method of the self-consistent field in general relativity and its application to the gravitational geon.
Free-space einstein equations for concentrations of radiation held together by gravitational attraction, investigating gravity wave perturbations on background metric
GRAVITATIONAL FIELD↗
Numerical results in the statistical theory of thermal turbulence
Several statistical theories of turbulence are applied to thermal convection problem, and are compared in a series of numerical calculations. Results are presented for both the case of very large and very small Prandtl numbers, and for moderate Rayleigh numbers. The theories compared are the direct-interaction approximation, the self-consistent field approximation, the quasi-linear approximation, the quasi-normal approximation, and a 'Markovianized' quasi-normal approximation. Results for these theories are then compared with some simple exact solutions to the statistical problem. The results indicate that the direct-interaction and the self-consistent field method give satisfactory results at all Rayleigh numbers investigated, whereas the procedures based on the quasi-normal approximation give sensible results only at small Rayleigh numbers.
Herring, J. R.↗
The closure approximation in the hierarchy equations.
The expectation of the solution process in a stochastic operator equation can be obtained from averaged equations only under very special circumstances. Conditions for validity are given and the significance and validity of the approximation in widely used hierarchy methods and the ?self-consistent field' approximation in nonequilibrium statistical mechanics are clarified. The error at any level of the hierarchy can be given and can be avoided by the use of the iterative method.
Adomian, G.↗
Application of the method local potential to the analysis of turbulent shear flows
It has been found that, in general, the local potential cannot be employed to obtain approximate solutions for the various correlations of turbulent properties which appear in the time averaged form of the conservation equations. Although the method of local potential is equivalent to the Galerkin method when the self-consistent condition is applied, the local potential can also be applied as an iterative algorithm in place of using the selfconsistent condition. This procedure offers an alternative to the Galerkin method and may be useful in obtaining approximate solutions for the total turbulent velocity. In addition, for certain simple turbulent shear flows the iterative algorithm may permit approximate, but non-empirical, solutions by modeling only the mean velocity and the Reynolds stress.
Reed, T. D.↗
The use of projection in SCF spin density calculations.
Spin-projected unrestricted self-consistent field /SCF/ methods for spin density calculations used to determine hyperfine coupling constants
SPIN-ORBIT INTERACTION↗
Calculations of molecular ionization energies using a self-consistent-charge Hartree-Fock-Slater method
A numerical-variational method for performing self-consistent molecular calculations in the Hartree-Fock-Slater (HFS) model is presented. Molecular wavefunctions are expanded in terms of basis sets constructed from numerical HFS solutions of selected one-center atomlike problems. Binding energies and wavefunctions for the molecules are generated using a discrete variational method for a given molecular potential. In the self-consistent-charge (SCC) approximation to the complete self-consistent-field (SCF) method, results of a Mulliken population analysis of the molecular eigenfunctions are used in each iteration to produce 'atomic' occupation numbers. The simplest SCC potential is then obtained from overlapping spherical atomlike charge distributions. Molecular ionization energies are calculated using the transition-state procedure; results are given for CO, H2O, H2S, AlCl, InCl, and the Ni5O surface complex. Agreement between experimental and theoretical ionization energies for the free-molecule valence levels is generally within 1 eV. The simple SCC procedure gives a reasonably good approximation to the molecular potential, as shown by comparison with experiment, and with complete SCF calculations for CO, H2O, and H2S.
Rosen, A.↗
Electronic states of molecules. i - self- consistent field calculations of the ground, ionized, and excited states of n sub 2 and o sub 2.
Ionization, excitation and total energies of oxygen and nitrogen, using self-consistent field molecular orbital method
SELF-CONSISTENT FIELD /SCF/↗
Can Malin's gravitational-field equations be modified to obtain a viable theory of gravity
Malin's (1975) gravitational theory, which was recently shown by Lindblom and Nester (1975) to be incorrect, is modified by means of a recently proposed method for obtaining viable gravitational theories. The resulting self-consistent theory, which is in effect a Rastall-type modification of the Einstein theory, exhibits nonconservation of momentum, yet agrees with all experimental limits known to date within the post-Newtonian approximation framework.
Smalley, L. L.↗
Self-consistent Pauli corrections in Brueckner-Hartree-Fock calculations
A scheme is introduced which makes it feasible to make completely self-consistent Brueckner-Hartree-Fock (BHF) and renormalized BHF calculations for spherical, closed-shell and axially-symmetric deformed nuclei. The usual requirement or orbital self-consistency has been imposed, as well as self-consistency in the starting energies and occupation probabilities. Previously, only approximate forms were used for the Pauli operator. This approximation is removed and a method for making the necessary Pauli corrections to the reaction matrix during the approach to self-consistency is presented. A discussion of the symmetries which reduce the problem to one of manageable proportions is included.
Braley, R. C.↗
Dynamics of J equal 302 super plus K xi and KN systems in the 27-fold representation, some remarks on a typical feature of the Balazs- type procedure.
Dynamics of baryon-meson systems in 27-fold representation, obtaining self-consistent bound state and low lying resonant solutions with Balazs method
YUKAWA POTENTIAL↗
Spin-projected and extended SCF calculations.
Spin restricted, unrestricted, projected unrestricted and extended SCF wave functions energies compared, discussing calculation method for spin extended SCF functions
WAVE FUNCTION↗
Computation on collisionless steady-state plasma flow past a charged disk
A computer method is presented using the 'inside-out' approach, for predicting the structure of the disturbed zone near a moving body in space. The approach uses fewer simplifying assumptions than other available methods, and is applicable to large ranges of the values of body and plasma parameters. Two major advances concerning 3-dimensional bodies are that thermal motions of ions as well as of electrons are treated realistically by following their trajectories in the electric field, and the technique for achieving self-consistency is promising for very large bodies. Three sample solutions were obtained for a disk-shaped body, charged negatively to a potential 4kT/e. With ion Mach number 4, and equal ion and electron temperatures, the wakes of a relatively small body (radius 5 Debye lengths) and a relatively large body (radius 100 Debye lengths) both begin to fill up between 2 and 3 body radii downstream. For the large body there is in addition a potential well (about 6kT/e deep) behind the body. Increasing the ion Mach number to 8 for the large body causes the potential well to become wider and longer but not deeper. For the large body, the quasineutrality assumption is validated outside of a cone-shaped region in the very near wake. For the large as well as the small body, the disturbed zone behind the body extends transversely no more than 2 or 3 body radii, a result of significance for the design of spacecraft boom instrumentation.
Parker, L. W.↗
A model study of diurnal behavior of the ionosphere and the protonosphere coupling
A new method to study dynamic behavior of the ionosphere-protonosphere coupling within the plasmasphere is developed and used to calculate ion distributions above 500 km using observed electron densities at 500 km. The method is based on the relation between the total magnetic flux tube content of H(+) above some reference height (e.g., 3000 km) and the H(+) flux at that height, which is uniquely determined by the coupled momentum and continuity equations for the quasi-steady state. The O(+) profile is perturbed from diffusive equilibrium only by ion drag with H(+). The time dependency is taken into account by applying self-consistent boundary conditions which are obtained from the solution of the time-dependent equation for the tube content of H(+). Calculations are carried out for two models. The first model deals with the ion dynamics in a tube of force which rotates with the earth. In the second model, effects of the cross-L plasma drifts are considered. Both models produce diurnal variations of H(+) and O(+) densities near 1000 km similar to observational results.
Marubashi, K.↗
Departures of the electron energy distribution from a Maxwellian in hydrogen. I - Formulation and solution of the electron kinetic equation. II - Consequences
The problem of calculating the steady-state free-electron energy distribution in a hydrogen gas is considered in order to study departures of that distribution from a Maxwellian at sufficiently low degrees of ionization. A model kinetic equation is formulated and solved analytically for the one-particle electron distribution function in a steady-state partially ionized hydrogen gas, and it is shown that the formal solution can be accurately approximated by using the WKB method. The solutions obtained indicate that the high-energy tail of the distribution is susceptible to distortion by imbalanced inelastic collisions for ionization fractions not exceeding about 0.1 and that such departures from a Maxwellian can lead to significant changes in the collisional excitation and ionization rates of ground-state hydrogen atoms. Expressions for the electron-hydrogen collision rates are derived which explicitly display their dependence on the hydrogen departure coefficients. The results are applied in order to compare self-consistent predictions with those based on the a priori assumption of a Maxwellian distribution for models of the thermal ionization equilibrium of hydrogen in the optically thin limit, spectral-line formation by a gas consisting of two-level atoms, and radiative transfer in finite slabs by a gas of four-level hydrogen atoms.
Shoub, E. C.↗
N-body simulations of disks
The methods used in large-scale n-body simulations are discussed. However, the present review concentrates on results already obtained in n-body simulations using systems containing up to 200,000 simulation stars. Results are presented which show that the stability criterion developed for flattened systems applies only to truly axisymmetric instabilities. Purely stellar disks acquire rather large velocity dispersions, generally two or more times the velocity dispersion required by Toomre (1964) for axisymmetric stability. In computer simulations, the bar-forming instability can be prevented only by comparatively large velocity dispersions. However, simulations including the effects of the galactic halo and core as a fixed background field show that bar formation can be prevented for fixed halo components as large or larger than the self-consistent disk component. Experiments performed to determine the collisional relaxation time for large-scale gravitational n-body calculations show that these models are indeed 'collisionless'.
Hohl, F.↗
A modified Monte Carlo model for the ionospheric heating rates
A Monte Carlo method is adopted as a basis for the derivation of the photoelectron heat input into the ionospheric plasma. This approach is modified in an attempt to minimize the computation time. The heat input distributions are computed for arbitrarily small source elements that are spaced at distances apart corresponding to the photoelectron dissipation range. By means of a nonlinear interpolation procedure their individual heating rate distributions are utilized to produce synthetic ones that fill the gaps between the Monte Carlo generated distributions. By varying these gaps and the corresponding number of Monte Carlo runs the accuracy of the results is tested to verify the validity of this procedure. It is concluded that this model can reduce the computation time by more than a factor of three, thus improving the feasibility of including Monte Carlo calculations in self-consistent ionosphere models.
Mayr, H. G.↗