Theory of Electron Diffraction by Crystals

1967 ◽  
Vol 22 (4) ◽  
pp. 422-431 ◽  
Author(s):  
Kyozaburo Kambe
Keyword(s):  
Integral Equation ◽  
Electron Diffraction ◽  
General Theory ◽  
Green’S Function ◽  
Green's Function ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Two Dimensions ◽  
The Third ◽  
Third Dimension

A general theory of electron diffraction by crystals is developed. The crystals are assumed to be infinitely extended in two dimensions and finite in the third dimension. For the scattering problem by this structure two-dimensionally expanded forms of GREEN’S function and integral equation are at first derived, and combined in single three-dimensional forms. EWALD’S method is applied to sum up the series for GREEN’S function.

Integral equation approach to the problem of scattering by a number of fixed scatterers

1977 ◽  
Vol 55 (16) ◽  
pp. 1442-1452
Author(s):  
M. Hron ◽  
M. Razavy
Keyword(s):  
Integral Equation ◽  
Wave Function ◽  
Green’S Function ◽  
Integral Equations ◽  
Green's Function ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inhomogeneous Term ◽  
Integral Equation Approach ◽  
An Array Of Cylinders

In the derivation of the Lippmann–Schwinger integral equation for scattering of a wave ψ(r) by the potential ν(r), one constructs the Green's function for the operator [Formula: see text], and treats νψ as the inhomogeneous term. However, in certain cases, it is desirable to formulate the scattering problem in terms of an integral equation by obtaining the Green's function for the operator [Formula: see text], and by considering (−k2ψ) as the inhomogeneous term. An important aspect of this formulation is that the resulting integral equation can be used to generate a low energy expansion of the wave function for some separable and nonseparable systems. For two-dimensional scattering, if the geometry of the scatterers is simple enough, the Laplace equation with the prescribed boundary conditions on the surface of the scatterers is separable in a certain coordinate system, then one can write the solution of the wave equation as an inhomogeneous integral equation. In this way the problems of scattering by two cylinders, an array of cylinders, and a grating can be formulated in terms of integral equations. For three-dimensional scattering, one can consider either the spherically symmetric cases or nonseparable problems. In the former case, for certain types of force laws, a Volterra integral equation in one variable can be found for the wave function. In the latter case, integral equations in two or three variables can be obtained for scattering by two spheres or by a torus.

People Planning by a Fixed Resource Method

1972 ◽  
Vol 34 (3) ◽  
pp. 799-806 ◽  
Author(s):  
John C. Baird ◽  
Virgil Graf ◽  
Richard Degerman
Keyword(s):  
Cluster Analysis ◽  
Outer Space ◽  
Three Dimensional ◽  
Two Dimensions ◽  
The Third ◽  
Complex Stimuli ◽  
The Mean ◽  
Highly Correlated ◽  
Third Dimension ◽  
The Ideal

Results are presented from a new method to determine a person's conception of complex stimuli. In three related experiments Ss expressed their views of ideal organisms by distributing a fixed resource among hypothetical properties of the ideal. The results from the experiments were highly correlated, lending weight to the reliability and generality of the approach. Cluster analysis and multidimensional scaling were used to group the properties in two dimensions, while the mean amount allocated to a property was represented in the third dimension. A three-dimensional plot was constructed for each of four ideals: the only organism on earth, a member of the only species on earth, an organism going into outer space, and an organism coming to earth from outer space.

scholarly journals Concentration of dynamic stresses in an elastic space with twoperiodic array of elliptical cracks

2019 ◽  
pp. 18-25
Author(s):  
Igor Zhbadynskyi
Keyword(s):  
Integral Equation ◽  
Green’S Function ◽  
Boundary Integral Equation ◽  
Green's Function ◽  
Scattering Problem ◽  
Mapping Method ◽  
Boundary Integral ◽  
Elastic Space ◽  
Wide Range ◽  
Elliptical Cracks

Normal incidence of the plane time-harmonic longitudinal wave on double-periodic array of coplanar elliptical cracks, which are located in 3D infinite elastic space is considered. Corresponding symmetric wave scattering problem is reduced to a boundary integral equation for the displacement jump across the crack surfaces in a unit cell by means of periodic Green’s function, which is presented in the form of Fourier integrals. A regularization technique for this Green’s function involving special lattice sums in closed forms is adopted, which allows its effective calculation in a wide range of wave numbers. The boundary integral equation is correctly solved by using the mapping method. The frequency dependencies of mode-I stress intensity factor in the vicinity of the crack front points for periodic distances in the system of elliptical cracks are revealed.

Developing a Better Understanding of the Engineer's Creative Role in Design

2002 ◽  
Vol 17 (2-3) ◽  
pp. 129-133
Author(s):  
Bill Addis
Keyword(s):  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Space Structures ◽  
The Third ◽  
Structural Problems ◽  
The World ◽  
Third Dimension ◽  
Dimension Space ◽  
Creative Role

Both architects and engineers are unconsciously drawn towards the two dimensional world – the ubiquity of the plan and elevation, and the ease of analysing 2-D structures. Yet the best architecture always exploits the three dimensional world, and the majority of structural problems and collapses occur when engineers have failed to think in the third dimension. Space structures offer an ideal learning environment for students of both architecture and engineering. They stimulate and challenge both the imagination and the intellect by forcing students out of the cosy, and often dull familiarity of two dimensions. They encourage students to conceive structures in three dimensions and drop down to two when necessary or convenient, rather than the other way round. In a world where form and forces so strongly interact, space structures force architects to step into the world of statics, and engineers into the world of geometry. An important result is a better understanding, for both architects and engineers, of the role engineers can play in helping create imaginative and practical structures.

scholarly journals Solving the Dirichlet acoustic scattering problem for a surface with added bumps using the Green's function for the original surface

2002 ◽  
Vol 31 (11) ◽  
pp. 687-694 ◽  
Author(s):  
Maxim J. Goldberg ◽  
Seonja Kim
Keyword(s):  
Integral Equation ◽  
Dirichlet Problem ◽  
Green’S Function ◽  
Green's Function ◽  
Scattering Problem ◽  
Acoustic Scattering ◽  
Alternative Formulation ◽  
Original Surface ◽  
Local Integral ◽  
Special Case

We solve the Dirichlet problem for acoustic scattering from a surface which has been perturbed by the addition of one or more bumps. We build the solution for the bumpy case using the Green's function for the unperturbed surface, and the solution of a local integral equation in which the integration is carried out only over the added bumps. We conclude by giving an alternative formulation of our method for the special case of a bump on a plane.

An Integral Equation Method for Free Vibration of Circular Plate with Concentrated Mass at Arbitrary Positions

2011 ◽  
Vol 255-260 ◽  
pp. 1830-1835 ◽  
Author(s):  
Gang Cheng ◽  
Quan Cheng ◽  
Wei Dong Wang
Keyword(s):  
Integral Equation ◽  
Eigenvalue Problem ◽  
Free Vibration ◽  
Green’S Function ◽  
Circular Plate ◽  
Green's Function ◽  
Integral Equation Method ◽  
Free Vibrations ◽  
Equation Method ◽  
Concentrated Mass

The paper concerns on the free vibrations of circular plate with arbitrary number of the mounted masses at arbitrary positions by using the integral equation method. A set of complete systems of orthogonal functions, which is constructed by Bessel functions of the first kind, is used to construct the Green's function of circular plates firstly. Then the eigenvalue problem of free vibration of circular plate carrying oscillators and elastic supports at arbitrary positions is transformed into the problem of integral equation by using the superposition theorem and the physical meaning of the Green’s function. And then the eigenvalue problem of integral equation is transformed into a standard eigenvalue problem of a matrix with infinite order. Numerical examples are presented.

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