Approximation Of Green's Function Research Articles

Correlated random walk on a bcc lattice with next-nearest-neighbor hops: Self-consistent decoupling approximation.

The self-consistent decoupling scheme recently introduced to improve the theory of Tahir-Kheli and Elliott for tracer diffusion on a lattice, in the limit of a static background, is extended here to include next-nearest-neighbor hops on a bcc lattice. Results for ${f}^{\mathrm{tr}}$, previously conjectured by Tahir-Kheli, are obtained through the new, approximate Green's-function equations. It is shown that the method applies generally to nth-neighbor hopping on any hypercubic lattice.

Various functionals for the kinetic energy density of an atom or molecule.

Various approximate density functionals for the kinetic energy density of atoms and molecules are analyzed. These include the results of a gradient expansion to first and second orders and a form recently derived from a new Green's function approximation [W. Yang, preceding paper, Phys. Rev. A 34, 4575 (1986)]. All the approximate functionals studied diverge to minus infinity at a nucleus, due to the ${\ensuremath{\nabla}}^{2}$\ensuremath{\rho} term that is in them, while the exact functional is positive and finite everywhere. Away from nuclei, however, the Hartree-Fock results are well reproduced, including the atomic shell structure. New functionals are proposed to correct the divergent behavior, and accurate total kinetic energy values are obtained from a new formula for kinetic energy density ${t}_{\mathrm{MP}(\mathrm{r})={C}_{k}\mathrm{\ensuremath{\rho}}{(\mathrm{r})}^{5/3}}$ +(1/72)\ensuremath{\Vert}\ensuremath{\nabla}\ensuremath{\rho}(r)${\ensuremath{\Vert}}^{2}$/\ensuremath{\rho}(r)+( 1/12)${\ensuremath{\nabla}}^{2}$\ensuremath{\rho}(r), with a divergence correction.

Approximate Green functions as a tool to prove correctness of a formal approximation in a model of competing and diffusing species

The purpose of this paper is two-fold; firstly, we shall derive some new results concerning a singular perturbation problem describing the stationary distribution of two competing and diffusing species; secondly, we want to demonstrate the power of a technique using approximate Green functions to prove the validity of a constructed formal approximation in a singular perturbation problem.

Inversion of long period regional body waves for crustal structure

Long‐period regional distance (2° < Δ < 12°) body waves are strongly influenced by the waveguide nature of the crust. If the seismic source is known, it is possible to parameterize the waveforms in terms of gross structural properties of crustal thickness and Pn velocity. Since the regional distance waveforms are a product of interference of multiple reflected, and mode converted phases, changes in crustal thickness or Pn velocity changes the relative timing of the various phases. Approximate Green's function for a suite of crustal structures can be constructed by stretching or squeezing the Green's function of an average crustal model, which is the basis of an interative, least squares inversion procedure for crustal structure. The crustal structure of the western U.S. is inverted as a test of the technique.

Atoms in a laser field

The method of quasienergy states (QES) for describing a quantum system in the field of a monochromatic light wave is presented. The perturbation theory for QES and quasistationary (decay) QES is developed. Various methods of obtaining approximate Green functions of valence electrons in atoms used for numerical calculations of the multiphoton-transition cross sections are discussed. On the basis of the developed formalism the variation of the atomic spectrum in an intense field, the multiphoton ionization of atoms and the influence of the light field on electron-atom and atom-atom collision processes are examined.

A two-dimensional imaging theory of surface discontinuities with the scanning acoustic microscope

This paper discusses theoretical aspects of the V ( z ) response of the scanning acoustic microscope (s. a. m.) when used to examine specimens with lateral discontinuities. The problem is introduced in terms of a simple ray model to establish a physical picture of the processes involved. An approximate Green function is then developed which enables use of a modified form of the Fourier optical formulations to calculate V ( z ) for a cylindrical lens. These calculations explain two different types of contrast observed when imaging specimens in the reflection s. a. m., such as (i) the ability to image fine discontinuities and display them with enhanced contrast and an apparent width determined by the acoustic wavelength; and (ii) to give a quantitative account of the amplitude of periodic ripple often observed running parallel to cracks on acoustic micrographs. Both these types of contrast may be predicted by using the same model and arise naturally from variation of the reflection and transmission properties of the discontinuity, the relative value of these parameters determines which type of contrast predominates. At an interface between media with different elastic properties, the contrast is affected not only by the scattering properties of the boundary but also by the very fact that surface waves must propagate in media with different elastic properties. This effect alone can provide a powerful contrast mechanism which enables one to understand the light to dark contrast reversals often observed at grain boundaries in polycrystalline specimens at different values of defocus.

Core excitation of Li by electron impact.

Cross sections for the excitation of a core electron, which leads to autoionization, in lithium (Li) atomic system by electron impact have been calculated with use of the single-configuration Hartree-Fock wave function within the asymptotic Green's-function approximation (AGFA) in the low-bombarding-energy region. Comparison is made with available results. Our investigation demonstrates that the AGFA supports the R-matrix as well as the distorted-wave Born-approximation behavior.

STRESS INTENSITY FACTORS FOR CRACKS IN NOTCHED FINITE PLATES SUBJECTED TO BIAXIAL LOADING

AbstractStress intensity factors were calculated, based on Bueckner's principle for cracks in both infinite and finite plates with notches subjected to biaxial loading. Approximate Green's functions have been obtained by modifying two existing Green's functions, originally for unnotched plates. Values of stress intensity factors calculated using Bueckner's principle with the approximate Green's functions are in good agreement with published stress intensity factors for cracks in both infinite and finite plates containing a circular notch or an elliptical notch, previously found by the method of boundary collocation.

Scattering of electrons by hydrogen atoms

We have applied our asymptotic Green function approximation (AGFA) to calculate the integrated cross sections for the elastic as well as the inelastic transitions (1s−1s, 1s−2s, 1s−2p m : m = 0, ±1) in hydrogen atoms by electron impact. The calculated results are compared with experimental observations and other theoretical predictions.

Approximate Green function for a semi-infinite solid with varying properties

An expression is derived for the spatially dependent Green function or response function in a solid occupying a half-space and having generally varying properties. The expression is valid in the same range as the WKB approximation, namely for slowly varying properties. Some generalisation is indicated and the time-dependent Green function or the resolvent is obtained as a Fourier transform.

Application of the asymptotic Green's-function approximation toe−-H scattering

We have performed calculations for the excitation of the $2{p}_{m}$ states of the hydrogen atom by electron impact for $\ensuremath{\lambda}$, $\ensuremath{\chi}$, and $\frac{d\ensuremath{\sigma}}{d\ensuremath{\Omega}}$ within our recently proposed asymptotic Green's function approximation, and the results are compared with the experimental observations and other theoretical predictions. Agreement between our calculated results and experimental data is reasonably good.

Improved-Brooks Mobility Formula for Disorder Scattering in Semiconductor Alloys

The Green function method is used to study disorder scattering in compound semiconductor alloys, taking into account many-body scattering. The present calculation and the Brooks disorder scattering mobility are compared. For typical semiconductor alloys, the discrepancy is as much as ∼20%, but the error is much smaller for alloys with small band edge difference and effective mass. By using an approximate Green function, an improved Brooks formula is derived in the form of the original Brooks formula multiplied by a correction factor.

Wave equation Green's function approximations for nearly-planar half-planes and half-spaces

Studies of subsurface wave-scattering often rely on wave equation Green's functions for half-planes or half-spaces with plane surfaces. Small deviations from the planar are assumed to have negligible effect away from the surface. Based on extensively-developed perturbation schemes, this article gives analytical first-order approximations for 2D and 3D wave equation Green's functions in the interiors of respectively, half-planes and half-spaces by two approaches. The first approach involves a boundary perturbation, while the second approach invokes a coordinate transformation leading to a perturbation of the governing equations themselves. The cases of, respectively, the functions and their normal derivatives vanishing on the surface are treated. Some numerical results for specific examples are presented and discussed.

Effect of distance-dependent exchange interactions in MnO on elastic constants and magnetic susceptibility

Measurements of velocity by an ultrasonic technique have been performed on antiferromagnetic MnO for longitudinal and transverse waves propagating along the [100], [110] and [111] directions, from 77 up to 350 K. Near the Néel temperature, we found a remarkable softening for the elastic constant C 44 and rather small anomalies for C 11 and C 11- C 12. These results are clarified by comparing them with calculations in the Green function approximation, using the distance dependences of the nn and nnn exchange constants, γ 1 = 23 and γ 2 = 12, respectively. Reciprocals of the exchange part C ex 44 determined from the observed C 44 exhibits a linearity with temperature and leads to λ 1 = 24 in comparison with the result in the high temperature approximation. We also explain the magnetic susceptibility of MnO modified by exchange striction, computing the temperature variations with and without the striction in the Green function approximation.

Low-energy electron impact excitation of the lowest lying autoionising level in Cs

Calculates the electron impact differential as well as the integral cross sections for the excitation of the lowest lying autoionising level in caesium (Cs) atomic system within a multi-channel T-matrix equation with an asymptotic Green's function approximation in the low bombarding energy region. The autoionising level has been assumed to arise from the inner-shell excitation of an electron to the valence shell and no account has been taken of the electron-electron couplings in the target atoms. The authors also calculated the alignment parameter ( lambda ) within the first Born and the asymptotic Green's function approximations for the Cs5p56s2 2P3/2 autoionising state at incident energy 30 eV.

Magnetic Interactions in α‐MnSeO4, α‐FeSO4, and β‐FeSO4

AbstractThe energy distributions of neutrons scattered by α‐MnSeO4, α‐FeSO4, and β‐FeSO4 in their paramagnetic state are measured by means of a time‐of‐flight technique. The values of two exchange parameters can be estimated from a simultaneous analysis of the second moment of these energy distributions and of the Néel temperature computed in the Green's function approximation. Various two‐parameter analyses are performed and checked by comparing calculated and experimental values for θ and χ(TN). For α‐FeSO4 the dominant interactions occur between the first and second neighbours. For β‐FeSO4 two‐parameter analyses do not lead to a satisfactory description of all data; however, one can conclude that the interactions between third neighbours are important, in agreement with the stability conditions for the observed spin configuration.

Self-consistent cluster theory for systems with off-diagonal disorder

A self-consistent cluster theory for elementary excitations in systems with diagonal, off-diagonal, and environmental disorder is presented. The theory is developed in augmented space where the configurational average over the disorder is replaced by a ground-state matrix element in a translationally invariant system. The analyticity of the resulting approximate Green's function is proved. Numerical results for the self-consistent single-site and pair approximations are presented for the vibrational and electronic properties of disordered linear chains with diagonal, off-diagonal, and environmental disorder.

The Role of Algebraic Formulations of Approximate Green's Functions for Systems With a Finite Number of Electrons

Some of the difficulties encountered in the approximation of Green's functions are reviewed with particular emphasis on the self-consistent polarization propagator. Some new basic theorems are presented, involving a general procedure, which lead to both a derivation and alternate prescriptions for the computations. As demonstrated by Linderberg and Öhrn the ground state is an AGP (antisymmetrized geminal power). An energy optimized AGP (EAGP) in a Brueckner orbital representation can be used to generate a manifold of excited states|K⟩ = OK+|AGP⟩, OK|AGP⟩ = 0 with the excitation operators, O+K, and killers, OK, being combinations of particle-hole and hole-particle operators. The SCPP can be expressed then as ⟨⟨X, Y⟩⟩E = - ⟨AGP|XT0(-E + EAGP)Y + YT0(E + EAGP)X|AGP⟩in terms of the resolvent T0(ϵ) = P(ϵ -PHP)-1 with P spanned by the states|K⟩⟨K|, in the sense of Löwdin. Our results establish a natural relationship between hermiticity, completeness, non-redundance of operator manifolds with stationarity and killer conditions.

Magnetic Interactions in α‐MnSO4 and β‐MnSeO4

AbstractThe energy distributions of neutrons scattered by α‐MnSO4 and β‐MnSeO4 in their paramagnetic state are measured by means of a time‐of‐flight technique. From a simultaneous analysis of the second moment of these energy distributions and of the Néel temperature computed in the Green's function approximation it is concluded that the exchange interactions between fourth neighbours play an important role in these manganese compounds. Values for J1 and J4 are obtained from which reasonable values for Θ and χ(TN) are predicted in the case of α‐MnSO4.

Estimates of Green's functions and Schauder estimates for solutions of elliptic boundary problems in a dihedral angle

Estimates of Green's functions and Schauder estimates for solutions of elliptic boundary problems in a dihedral angle