Electromagnetic fields in the presence of cylindrical objects of arbitrary physical properties and cross sections

1970 ◽  
Vol 48 (15) ◽  
pp. 1789-1798 ◽  
Author(s):  
L. Shafai
Keyword(s):  
Physical Properties ◽  
Electromagnetic Fields ◽  
Cross Sections ◽  
Approximate Solutions ◽  
Scattering Matrix ◽  
Two Dimensional ◽  
Matrix Elements ◽  
Approximate Evaluation ◽  
Permittivity And Permeability ◽  
The Matrix

Approximate solutions for two-dimensional problems of electromagnetic fields in the presence of cylindrical objects have been found by approximate evaluation of a scattering matrix. The equations are derived for cylindrical objects of arbitrary physical properties and cross sections and a procedure for evaluation of the matrix elements is discussed. The elements of permittivity and permeability tensors are assumed to be analytic, but otherwise arbitrary functions of the transverse coordinates.

scholarly journals Worldline master formulas for the dressed electron propagator. Part 2. On-shell amplitudes

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
N. Ahmadiniaz ◽  
V. M. Banda Guzmán ◽  
F. Bastianelli ◽  
O. Corradini ◽  
J. P. Edwards ◽  
...  
Keyword(s):  
Cross Sections ◽  
Drop Out ◽  
Matrix Elements ◽  
Fermion Propagator ◽  
Particle Path ◽  
Path Integral Representation ◽  
Fermion Line ◽  
Electron Propagator ◽  
The Matrix ◽  
Standard Linear

Abstract In the first part of this series, we employed the second-order formalism and the “symbol” map to construct a particle path-integral representation of the electron propagator in a background electromagnetic field, suitable for open fermion-line calculations. Its main advantages are the avoidance of long products of Dirac matrices, and its ability to unify whole sets of Feynman diagrams related by permutation of photon legs along the fermion lines. We obtained a Bern-Kosower type master formula for the fermion propagator, dressed with N photons, in terms of the “N-photon kernel,” where this kernel appears also in “subleading” terms involving only N − 1 of the N photons.In this sequel, we focus on the application of the formalism to the calculation of on-shell amplitudes and cross sections. Universal formulas are obtained for the fully polarised matrix elements of the fermion propagator dressed with an arbitrary number of photons, as well as for the corresponding spin-averaged cross sections. A major simplification of the on-shell case is that the subleading terms drop out, but we also pinpoint other, less obvious simplifications.We use integration by parts to achieve manifest transversality of these amplitudes at the integrand level and exploit this property using the spinor helicity technique. We give a simple proof of the vanishing of the matrix element for “all +” photon helicities in the massless case, and find a novel relation between the scalar and spinor spin-averaged cross sections in the massive case. Testing the formalism on the standard linear Compton scattering process, we find that it reproduces the known results with remarkable efficiency. Further applications and generalisations are pointed out.

EARLY TIME PROPERTIES OF TIME EVOLUTION IN TWO-DIMENSIONAL SUBSPACE AND CP VIOLATION PROBLEM

1993 ◽  
Vol 08 (21) ◽  
pp. 3721-3745 ◽  
Author(s):  
K. URBANOWSKI
Keyword(s):  
Time Evolution ◽  
Early Time ◽  
Dimensional Subspace ◽  
Effective Hamiltonian ◽  
Two Dimensional ◽  
Matrix Elements ◽  
The Matrix ◽  
Long Time ◽  
Time Region ◽  
Short Time

Approximate formulae are given for the effective Hamiltonian H||(t) governing the time evolution in a subspace ℋ|| of the state space ℋ. The properties of matrix elements of H||(t) and the eigenvalue problem for H||(t) are discussed in the case of two-dimensional ℋ||. The eigenvectors of H||(t) for the short time region are found to be different from those for the long time region. The decay law of particles described by the eigenvectors of H||(t) is shown to have the form of the exponential function multiplied by some time-independent factor, equal to 1 only in the case of the [Formula: see text]-invariant theory. Some general properties of the matrix elements of H||(t) are tested in the Lee model.

scholarly journals Matrix Elements of the Boltzmann Collision Operator in a Basis Determined by an Anisotropic Maxwellian Weight Function including Drift

1980 ◽  
Vol 33 (2) ◽  
pp. 449 ◽  
Author(s):  
Kailash Kumar
Keyword(s):  
Weight Function ◽  
Cross Sections ◽  
Collision Operator ◽  
Weight Functions ◽  
Anisotropic Pressure ◽  
Matrix Elements ◽  
Methods Of Calculation ◽  
Special Cases ◽  
The Matrix ◽  
Boltzmann Collision Operator

The matrix elements of the linear Boltzmann collision operator are calculated in a Burnett-function basis determined by a weight function which itself describes a velocity distribution with a net drift and an anisotropic pressure (or temperature) tensor. Three different methods of calculation are described, leading to three different types of formulae. Two of these involve infinite summations, while the third involves only finite sums, but at the cost of greater complications in the summands and the integrals over cross sections. Both elastic and inelastic collisions are treated. Special cases arising from particular choices of the parameters in the weight functions are pointed out. The structure of the formulae is illustrated by means of diagrams. The work is a contribution towards establishing efficient methods of calculation based upon a better understanding of the matrix elements in such bases.

scholarly journals Transfer scattering matrix of non-uniform surface acoustic wave transducers

1987 ◽  
Vol 10 (3) ◽  
pp. 563-581
Author(s):  
N. C. Debnath ◽  
T. Roy
Keyword(s):  
Acoustic Wave ◽  
Equivalent Circuit ◽  
Surface Acoustic Wave ◽  
Scattering Matrix ◽  
Circuit Model ◽  
Complex Frequency ◽  
Matrix Elements ◽  
Special Cases ◽  
The Matrix ◽  
Scattering Matrix Elements

This paper is concerned with a general mathematical theory for finding the admittance matrix of a three-port non-uniform surface acoustic wave (SAW) network characterized bynunequal hybrid sections. The SAW interdigital transducer and its various circuit model representations are presented in some detail. The Transfer scattering matrix of a transducer consisting ofNnon-uniform sections modeled through the hybrid equivalent circuit is discussed. General expression of the scattering matrix elements for aN-section SAW network is included. Based upon hybrid equivalent circuit model of one electrode section, explicit formulas for the scattering and transfer scattering matrices of a SAW transducer are obtained. Expressions of the transfer scattering matrix elements for theN-section crossed-field and in-line model of SAW transducers are also derived as special cases. The matrix elements are computed in terms of complex frequency and thus allow for transient response determinations. It is shown that the general forms presented here for the matrix elements are suitable for the computer aided design of SAW transducers.

Collisional depolarization of light alkali atoms excited to their lowest 2P levels

1970 ◽  
Vol 48 (24) ◽  
pp. 3047-3058 ◽  
Author(s):  
M. Elbel
Keyword(s):  
Cross Sections ◽  
Transition Matrix ◽  
Inner Product ◽  
Matrix Elements ◽  
Angular Momenta ◽  
Rank Tensor ◽  
Alkali Atoms ◽  
The Matrix ◽  
Transition Matrix Elements ◽  
Depolarization Of Light

Transition matrix elements connecting the Zeeman sublevels of the lowest p doublets in light alkali atoms have been derived using methods of steady-state collision theory. The matrix elements generally consist of two parts which, under rotations of the quantization axis with respect to the scattering plane, behave like components of a first rank and a second rank tensor, respectively. Only the second rank tensor components lead to the selection rule j, mJ↔j, −mj, whereas the first rank tensor components do not. The latter can be ascribed to a formal interaction term which is proportional to the inner product of the orbital angular momenta of the valence electron and the colliding atoms, respectively, thus accounting for molecular coupling phenomena during the collision. Finally, the transition matrix elements are used to calculate the depolarizing cross sections from the van der Waals potential.

28Si + 28Si elastic scattering at energies above the Coulomb barrier

1988 ◽  
Vol 66 (9) ◽  
pp. 813-817 ◽  
Author(s):  
Ashok Kumar ◽  
B. B. Srivastava
Keyword(s):  
Elastic Scattering ◽  
Cross Sections ◽  
Simple Procedure ◽  
Matrix Elements ◽  
Nucleon Potential ◽  
Generator Coordinate ◽  
Group Methods ◽  
The Matrix ◽  
Sharp Cutoff ◽  
Good Agreement

A relatively simple procedure has been used to calculate the differential cross sections for 28Si + 28Si elastic scattering at Elab = 67, 74, 77, and 120 MeV. The nuclear interactions have been calculated microscopically using the equivalence relation between the matrix elements of the generator coordinate and the resonating group methods from the two-nucleon potential, which explains the two-nucleon data fairly well. The absorption effects due to the opening of the nonelastic channels are taken into account approximately by the classical sharp cutoff of partial waves. The calculated results are in quite good agreement with the available experimental data at most of these energies.

Approximate solutions to the one-particle Dirac equation: numerical results

1986 ◽  
Vol 64 (3) ◽  
pp. 297-302 ◽  
Author(s):  
R. A. Moore ◽  
T. C. Scott
Keyword(s):  
Dirac Equation ◽  
Numerical Solutions ◽  
Structure Constant ◽  
Approximate Solutions ◽  
Fine Structure Constant ◽  
Matrix Elements ◽  
Atomic Systems ◽  
The Matrix ◽  
The One ◽  
Significant Figures

The zero-, first-, and second-order differential equations in a previously defined hierarchy of equations giving approximate solutions to the one-particle Dirac equation and the corresponding eigenvalue contributions are each written as power series in α, the fine structure constant, for an arbitrary, spherically symmetric potential. These equations are solved numerically for the hydrogen-atom potential to obtain wave functions to order α2 and eigenvalues to order α4 for all states with n = 1–4, inclusive. The numerical solutions are then used to evaluate a number of matrix elements to order α2. A comparison with the exact expressions shows that the numerical values for the coefficients of the different powers of α have at least six significant figures in the eigenfunctions and eigenvalues and five in the matrix elements. Thus, the procedure is validated and can be applied with confidence to other atomic systems.

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