Invariant Embedding Method Research Articles

Reflections of nonstationary signals in media with intense fluctuations of irregularities

With the help of the invariant embedding method, we obtain an equation for the kernel of the backscattering operator and find numerically its solutions corresponding to the response of the medium to an incident delta-impulse. The statistical characteristics of reflections of such impulses are determined in the case of intense fluctuations of irregularities given the logarithmic derivative of the medium impedance is a continuous stationary centered Gaussian process with the unit variance and a finite correlation radius. We study the temporal evolution of the statistical distribution of reflected signals. Examples of reflections for various shapes of incident impulses are discussed.

Improvement of Air Transport Data and Wall Transmission/Reflection Data in the SKYSHINE Code

Transmission and reflection data of concrete and steel for 6.2 MeV gamma-ray in the SKYSHINE code have been generated using up-to-date data and method with a view to improving an accuracy of results. The transmission and reflection data depend on energy and angle. The invariant embedding method, which has merits of producing no negative angular flux and of taking small computer time, is suitable and adopted to the present purpose. Transmission data were calculated for concrete of 12 ~ 160 cm thick and steel of 4~39 cm thick based on the PHOTX library. Reflection data were calculated for semi-infinite slabs of concrete and steel. Consequently, smooth and consistent differential data over whole angle and energy were obtained compared with the original data calculated by discrete ordinates Sn code and Monte Carlo code. In order to use these data in the SKYSHINE code, further verification is needed using various calculation method or experimental data.

Calculation of Gamma Ray Buildup Factors for a Very Thick Slab with Angular Eigenvalue Method

The buildup factors of gamma rays are still widely used in the practical shielding calculations. Although the early compilation of buildup factors was published back to 50’s, it is still a tedious, though straightforward, task even for state-of-the-art computers. The most recent database of buildup factors covers the distance up to 40 mean free paths. The angular eigenvalue method, which was proposed by one of the author (A. S.) to calculate the transmission of gamma rays through a homogeneous slab, is based fundamentally on the invariant embedding method. This method has been applied to obtain the gamma ray buildup factors in homogeneous medium. It was found that the buildup factors were easily obtained for the distance over 40 mean free paths.

States ladder model for electrons backscattered in X-ray microanalysis. Application of the invariant embedding method

The advantage of the Invariant Embedding method for the study of transport process in electron probe microanalysis (EPMA), is illustrated considering a model where the energy of the impinging electrons slows down on a ladder of states. Differential equations systems for the probability of the electrons being backscattered and for the contribution of these electrons to the detected X-ray intensity are obtained. Theoretical predictions of the model are compared with experimental results. It is shown that the proposed model describes the general trend of experimental data and leads to a good agreement for a wide range of atomic numbers.

Self-consistent electronic structure of quantum wells by invariant embedding method

A new approach based on the invariant embedding method for the self-consistent calculation of electronic structure of quantum wells is presented and is applied to both neutral quantum well and parabolic quantum well. Numerical results obtained for these structures agree very well with those of previous theoretical and experiment studies. The present approach is expected to lead to a more efficient and stable scheme for the calculation of electronic band structure of quantum structures. Realistic boundary conditions are naturally taken into account in the present calculation which provides a convenient way for studying boundary effects.

Analytical Synthesis of the Probabilistic Characteristics of One Class of Non-Markovian Processes

Nowadays analytical methods for synthesis of probabilistic characteristics of random processes exist only for Markovian processes and are based on solution of the Fokker-Plank-Kolmogorov equation. The article presents an approach that allows us to solve such a problem for non-Markovian processes, which are described by approximated solutions of stochastic nonlinear integrated equations in general form. The basis for the method proposed is a representation of the integral equation by two-point boundary-value problem with its subsequent solution by the invariant embedding method.

Application of the invariant embedding method to x-ray microanalysis

The invariant embedding principle was applied to obtain closed analytical expressions of several magnitudes of interest in electron probe microanalysis [ionization distribution function, ϕ(ρz) backscattering probability, etc.]. This approach permits one easily to obtain solutions for a model which explicitly considers the mean free path between effective electron collisions. © 1998 John Wiley & Sons, Ltd.

Gamma Ray Albedo Data Generated by the Invariant Embedding Method

The energy-and-angle dependent doubly differential γ-ray albedos for homogeneous semi-infinite medium have been calculated for water, ordinary concrete, soil, heavy concrete, iron, tin and lead. The procedure of calculation employed in the present paper is the invariant embedding method which is being developed to solve neutral particle transport problems in homogeneous one dimensional medium. The calculated γ-ray albedos are stored in the data base. One can easily obtain from this data base the various kinds of albedos (number, dose, energy, etc.) with a simple interface program. The accuracy of the calculated γ-ray albedos is ascertained by comparing with the Monte Carlo calculations (MCNP4A and EGS4).

Gamma Ray Albedo Data Generated by the Invariant Embedding Method.

The energy-and-angle dependent doubly differential γ-ray albedos for homogeneous semi-infinite medium have been calculated for water, ordinary concrete, soil, heavy concrete, iron, tin and lead. The procedure of calculation employed in the present paper is the invariant embedding method which is being developed to solve neutral particle transport problems in homogeneous one dimensional medium. The calculated γ-ray albedos are stored in the data base. One can easily obtain from this data base the various kinds of albedos (number, dose, energy, etc.) with a simple interface program. The accuracy of the calculated γ-ray albedos is ascertained by comparing with the Monte Carlo calculations (MCNP4A and EGS4).

Initial Value Problem for Exit Function for Multiple Light Scattering

To alleviate the nonuniqueness problem of the solution of the integral equation derived by Hovenier for the exit function, a generating function for reflection and transmission functions for homogeneous, plane-parallel media, we have derived an integrodifferential equation, so that the function can be obtained in the form of the initial value problem. The numerical stability of our equation was then tested against the problem of multiple scattering in layers of a wide range of optical thickness, characterized by the Henyey-Greenstein phase function of varying degrees of anisotropy. The main features of the present technique are (1) an application of the fast invariant embedding method of Sato et al. and (2) an effective interpolation scheme to estimate numerically the limiting values of the singular terms involved in the equation. The results were compared with those of the doubling method. It was found that the maximum relative errors of the reflection function, ranging from 0.005% to 0.03%, generated from the exit function was 1 to 2 orders of magnitude smaller than those of the transmission function. Nevertheless, even the transmission functions rarely show errors larger than 2%, unless the optical thickness is substantially less than 0.1. Our results indicate that the equation presented in this work has a high degree of numerical stability and that it can be potentially useful for multiple scattering computations.

Allergy of latex, avocado pear, and banana: Evidence for a 30-kilodalton antigen in immunoblotting

In this chapter, we want to show, without completeness, that the success of the invariant embedding method for the factorization of elliptic boundary value problems is not bound to cylindrical domains. The cylindrical case is especially simple as the subdomains are just sub-cylinders and an infinitesimal variation of the domain is achieved by moving the limiting boundary by an infinitesimal amount in the orthogonal direction. In the non-cylindrical case, we need to define, on the one hand, a family of surfaces continuously sweeping over the initial domain and which limit the subdomains, and, on the other hand, a displacement field between these surfaces. In the first step, we study the situation where there is a transformation preserving orthogonality that allows us to reduce the problem to the cylindrical case. Then, the case of subdomains limited by a plane surface but with a non-orthogonal displacement is studied. Further, we turn to the general case where the surfaces of the family have no edge. A particular situation is studied where the domain is star shaped and the subdomains are obtained from the initial domain by a homothety.

Reflectivity/transmissivity for one-dimensional inhomogeneous random elastic media: dynamic-equivalent-medium approach

SUMMARY We study the influence of 1-D randomly stratified media on both the transmissivity and reflectivity of obliquely incident elastic plane waves. For this, a combination of the invariant embedding method with the small-perturbation expansion is used. The resulting analytic expressions for the plane time-harmonic and transient P- and S I/-wave transmissivities and reflectivities can be used for both deterministically and statistically specified stratifications. They do not have any essential restrictions concerning the frequency range of validity. However, it is assumed that the fluctuations of the density and compressional and shear velocities are small in comparison with the average values of these parameters. The analytic expressions for the attenuation coefficients and phase velocities of the P and W transmissivities predict their smoothed behaviour well in one single realization of a 1-D random medium. Both expressions can be considered as fundamental wave attributes for a ‘dynamic-equivalent medium’. They can also be looked upon as a generalization of the well-known O’DohertyAnstey formula, which can now account for oblique incidence and mode-conversion effects in an inhomogeneous 1-D medium with depth-dependent statistics. Likewise, the resulting frequency-dependent phase velocities can be considered as a broadfrequency-range generalization of their static approximations, obtained by what is known as Backus averaging. They postulate a frequency-dependent elastic transverse isotropy of the dynamic-equivalent medium, which replaces the random 1-D medium. We use the dynamic-equivalent-medium description to formulate some simple approximations of the P- and SV-wave reflectivities.

Application of the invariant embedding method for computing nonlinear periodic oscillations

The invariant embedding method is applied to compute periodic solutions of nonlinear systems that depend continuously on the perturbation frequency. Analysis of such systems in the region of multivalued solutions is demonstrated. An example of an essentially nonlinear system is examined.

Electromagnetic scattering from a stratified bi-isotropic (nonreciprocal chiral) slab: numerical computations

A transversely polarized time-harmonic plane wave normally incident on a stratified bi-isotropic (nonreciprocal chiral) slab is considered. The invariant embedding method is used to derive the differential equations for the scattering coefficient matrices, which are solved using appropriate boundary conditions. With the calculated reflected field as the boundary value, the differential equations for the Green functions are solved numerically. Numerical results are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Calculation of Azimuthally Dependent Thermal-Neutron Albedos for Slab by Method of Invariant Embedding

Calculation of Azimuthally Dependent Thermal-Neutron Albedos for Slab by Method of Invariant Embedding

Calculation of Azimuthally Dependent Thermal-Neutron Albedos for Slab by Method of Invariant Embedding.

The method of invariant embedding has been applied to the calculation of differential thermal-neutron albedos for a semi-infinite ordinary concrete slab. The calculations have been performed in both cases of isotropic and anisotropic scattering in the laboratory system. The calculated albedo data are compared with those obtained by the experiments and the semi-empirical formula fitted the detailed data obtained by Monte Carlo method. The calculated results assuming isotropic scattering are in good overall agreement with the values obtained by Monte Carlo and SN methods, but there are some errors for azimuthally anisotropic scattering when azimuthal angle becomes large. In this method, much less computing times within given accuracy are required for azimuthally isotropic scattering, but it is pronounced that the necessary computing times are heavily dependent on N in DP (N/2)-1 (ξ)TN(μ) quadrature sets when the azimuthally anisotropic scattering is considered. It is found that, except for large N for the c...

Method of invariant embedding for the radiation transfer equation and description of the scattering of finite beams

Method of invariant embedding for the radiation transfer equation and description of the scattering of finite beams

Invariant embedding treatment of phase randomisation and electrical noise at disordered surfaces

The author derives the Fokker-Planck equation for the distribution of the phase of the reflection amplitude for a disordered conductor of length L, using the invariant embedding method. The limiting (L= infinity ) stationary distribution in the strong-reflection regime is uniform provided that the localisation length is large compared with a Fermi wavelength. Next the author studies the joint distribution of the phase shift and of the time delay experienced by an incident electron in the conductor before being back-scattered. He obtains the explicit form of the marginal distribution for time delays for large L and strong reflections, using the uniformity of the phase distribution. The author applies it to study the spectral density of the surface charge fluctuation noise. An approximate analytical calculation yields an f-12/(3- alpha 2)/ spectrum where alpha >or=1.

Exact and Glauber amplitudes in multi-channel scattering

A model for one-dimensional many-channel scattering of a particle from a composite system is considered. It is assumed that the target cannot break up. Using Bellman's method of invariant embedding, a system of non-linear first-order differential equations with one-point boundary conditions is obtained for the transmission and reflection coefficients. These equations have stable numerical solutions. An alternative invariant embedding with the approximation of no reflection at any stage of the scattering process yields the Glauber approximation for the transmission coefficients. These formulations are used to compare the exact and the Glauber amplitudes for a projectile scattering from a bound two-particle system. It is found that unlike the inelastic Glauber amplitudes, the elastic Glauber amplitudes are very good approximations to the exact results.

Invariant embedding method for the solution of coupled Burgers' equations

A combined technique based on linear approximation and invariant embedding is proposed for solving coupled Burgers' equations over irregular regions. The given region is embedded in a conveniently chosen regular region. A difference analog of Crank-Nicolson type is derived for the linearized equation and is analyzed for its local stability. An invariant embedding algorithm is proposed for solving the resulting algebraic problem. The existence of inverses of some matrices appearing in the recurrence relations and the stability of the algorithm are also established. The method is illustrated by solving a test example. The numerical results indicate the existence of steady-state solutions. The proposed method may be used for solving nonlinear parabolic equations over irregular regions.