A THEOREM ON NEUTRON MULTIPLICATION

1947 ◽  
Vol 25a (4) ◽  
pp. 276-292 ◽  
Author(s):  
G. Placzek ◽  
G. Volkoff
Keyword(s):  
Point Source ◽  
Asymptotic Form ◽  
Homogeneous Medium ◽  
Integral Form ◽  
Exact Expression ◽  
Neutron Distribution ◽  
Chain Reaction ◽  
Neutron Multiplication ◽  
Fission Neutrons ◽  
The Fourier Transform

The asymptotic behaviour of the neutron distribution due to a point source in an infinite homogeneous medium in which a convergent chain reaction (multiplication constant k<1) takes place is investigated without special assumptions about the properties of the medium and the mechanism of neutron diffusion. It is shown under very general assumptions that at large distances r from the point source the neutron distribution always has the form[Formula: see text]General expressions for the constants μ and A of this asymptotic form of the distribution are given for any k<1 in terms of the Fourier transform of the spatial distribution of primary fission neutrons. These expressions reduce to particularly simple form for (1 − k) << 1. The exact expression for the neutron distribution throughout the medium is given in integral form. Four special frequently occurring cases are discussed as illustrations of the general result.

The isotropic scattering of a concentrated ray pencil from a point source

1964 ◽  
Vol 60 (1) ◽  
pp. 105-114 ◽  
Author(s):  
M. G. Smith
Keyword(s):  
Integral Equation ◽  
Laplace Transform ◽  
Point Source ◽  
Transport Equation ◽  
Asymptotic Form ◽  
Homogeneous Medium ◽  
Isotropic Scattering ◽  
Spherically Symmetric

AbstractThe two-sided Laplace transform is used to obtain a solution of a certain integral equation. The equation is that associated with the transport equation for isotropic scattering, when there is a point source of radiation giving a concentrated ray pencil, in a spherically symmetric homogeneous medium extending to infinity. The solution is then used to find the asymptotic form of the intensity in the forward and backward directions.

A STATISTICAL THEORY OF APERTURE SYNTHESIS

1967 ◽  
Vol 45 (6) ◽  
pp. 2041-2052
Author(s):  
Ralph J. Gagnon
Keyword(s):  
Point Source ◽  
Random Noise ◽  
Circular Disk ◽  
Brightness Distribution ◽  
Source Distribution ◽  
Base Line ◽  
Fourier Inversion ◽  
Visibility Function ◽  
The Fourier Transform ◽  
Infinite Set

The usual methods of interferometry make use of the Fourier transform relationship which holds between a radio-noise brightness distribution and the complex visibility function which is measured with a pair of antennas. The visibility function is a function of the distance or base line between the antennas. If it were known for all base lines, then the brightness distribution could be found by Fourier inversion. Unfortunately, the visibility function is not known for all base lines and the Fourier inversion is not unique. If the observer wishes to interpret his data by displaying a single possible brightness distribution, then he must choose from the infinite set of brightness distributions which could have produced his data. Previously, the author suggested that this be accomplished by representing the set of possible distributions as a statistical ensemble, and making the choice on a statistical basis so as to minimize the expected mean-square error.In the present communication, the results of the previous paper are presented for the two-dimensional case. The inversion formulas are worked out in detail for the cases of uniform point-source distributions in a square (or rectangle) and in a circular disk, and also for a point-source distribution with a Gaussian envelope taper. It is shown how to extend the point-source results to a distribution of nonpoint sources, and as an example the inversion equations are computed for the case of a distribution of Gaussian-shaped sources distributed with a Gaussian amplitude or density envelope. Finally, the appropriate inversion equations are derived for an observed visibility function which is contaminated with additive zero-mean Gaussian random noise, uncorrelated with the true visibility function.

Offset‐dependent geometrical spreading in a layered medium

Geophysics ◽  
1990 ◽  
Vol 55 (4) ◽  
pp. 492-496 ◽  
Author(s):  
Bjørn Ursin
Keyword(s):  
Point Source ◽  
Taylor Series ◽  
Layered Medium ◽  
Exact Expression ◽  
Second Order ◽  
Geometrical Spreading ◽  
Approximate Expressions

The geometrical spreading for a point source in a horizontally layered medium has been computed by Ursin (1978) and Hubral (1978) as a Taylor series in the offset coordinate. The coefficients in the Taylor series depend on the thicknesses and the velocities of the layers. Here, I start with the exact expression for geometrical spreading and show that it can be expressed as a function of the velocity in the first layer, the offset, and the first‐ and second‐order traveltime derivatives. A shifted hyperbolic traveltime approximation (Castle, 1988) and the usual hyperbolic traveltime approximation are used to derive approximate expressions for geometrical spreading. These expressions can also be derived from a truncated Taylor series by making additional approximations, but this procedure is not so obvious.

Tissue dosage from a point source of fission neutrons in air

1969 ◽  
Vol 26 (6) ◽  
pp. 625-626
Author(s):  
I. V. Goryachev ◽  
�. I. Zeinalov ◽  
V. I. Kukhtebich ◽  
G. M. Obaturov ◽  
L. A. Trykov ◽  
...  
Keyword(s):  
Point Source ◽  
Fission Neutrons

Transient Thermo-Hydraulic Analysis of the Windowless Target System for the Lead Bismuth Eutectic Cooled Accelerator Driven System

2006 ◽  
Author(s):  
Fosco Bianchi ◽  
Roberta Ferri ◽  
Vincent Moreau
Keyword(s):  
Operating Conditions ◽  
Target System ◽  
Material Damage ◽  
Hydraulic Analysis ◽  
Chain Reaction ◽  
Critical Part ◽  
Hydraulic Behaviour ◽  
Driven System ◽  
Accelerator Driven System ◽  
Fission Neutrons

The target system, whose function is to supply an external neutron source to the ADS sub-critical core to sustain the neutron chain reaction, is the most critical part of an ADS being subject to severe thermo-mechanical loading and material damage due to accelerator protons and fission neutrons. A windowless option was chosen as reference configuration for the target system of the LBE-cooled ADS within the European PDS-XADS project in order to reduce the material damage and to increase its life. This document deals with the thermo-hydraulic results of the calculations performed with STAR-CD and RELAP5 codes for studying the behaviour of the windowless target system during off-normal operating conditions. It also reports a description of modifications properly implemented in the codes needed for this analysis. The windowless target system shows a satisfactory thermo-hydraulic behaviour for the analysed accidents, except for the loss of both pumps without proton beam shut-off and the beam trips lasting more than one second.

Distortion in resistivity logging at shallow depth

Geophysics ◽  
1995 ◽  
Vol 60 (4) ◽  
pp. 1058-1069 ◽  
Author(s):  
Tien‐Chang Lee ◽  
Brian N. Damiata
Keyword(s):  
Point Source ◽  
Line Source ◽  
Apparent Resistivity ◽  
Homogeneous Medium ◽  
Radial Distance ◽  
Ground Surface ◽  
Current Electrode ◽  
Potential Electrode ◽  
Resistivity Logging ◽  
Source Array

Owing to the proximity of an insulating ground surface, normal resistivity logging at shallow depths (less than 30 m) can yield an apparent resistivity that exceeds 200% of the formation resistivity for a homogeneous medium. The distortion is more acute for long‐normal than for short‐normal logging. Three examples from a landfill site in southern California are presented to show such distortion. The patterns of distortion are similar for logging devices consisting of either two point‐source electrodes or one point‐source and one finite length, line‐source electrode. The former electrode array is a generally accepted approximation of the latter. However, the simulated apparent resistivity for the line‐source array is greater than that for the point‐source array at any given depth. A resistivity contrast between the formation and the borehole fluid can shift the magnitude of the background apparent resistivity but does not significantly alter the pattern of distortion. The magnitude of the distortion can be reduced by placing the reference‐ground potential electrode at a radial distance that is about equal to the spacing between the downhole upper potential electrode and the upper current electrode. It can also be removed by including the radial distance in an array‐dependent geometric factor that accounts for the resistivity of the borehole fluid and the proximity of the logging device to the ground surface.

POTENTIAL AND APPARENT RESISTIVITY OVER DIPPING BEDS

Geophysics ◽  
1956 ◽  
Vol 21 (3) ◽  
pp. 780-793 ◽  
Author(s):  
Jerome Chastenet De Gery ◽  
Geza Kunetz
Keyword(s):  
Point Source ◽  
Current Density ◽  
Potential Field ◽  
Apparent Resistivity ◽  
Exact Expression ◽  
Electrode Configuration ◽  
Vertical Profiles ◽  
The Earth ◽  
Electrode Separation

The potential field due to a point source of current, located on the surface of the earth near a dipping bed, is given in an exact expression and modified expressions are developed for computations. These expressions lead to graphs of the potential field and to apparent resistivity vertical profiles which are presented. The Schlumberger electrode configuration is used. This configuration consists of two current electrodes and two potential electrodes, the latter placed close enough together that the current density between them can be considered to be uniform. With this configuration oriented perpendicular to the strike of the dipping bed, the apparent resistivity is paradoxical in that it approaches either zero or infinity as the electrode separation increases without limit.

scholarly journals The Evolution of Linearized Perturbations in a Magnetohydrodynamic Boundary Layer

2014 ◽  
Vol 19 (2) ◽  
pp. 397-406
Author(s):  
A.R. Vijayalakshmi ◽  
P.M. Balagondar
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Fourier Transform ◽  
Shear Flow ◽  
Point Source ◽  
Piecewise Linear ◽  
Transverse Velocity ◽  
Piecewise Linear Approximation ◽  
Initial Value ◽  
The Fourier Transform

Abstract The evolution of linearized perturbations in a magnetohydrodynamic shear flow is studied using the initial value problem approach. Here the resulting equation in time posed by using the Fourier transform is solved for the Fourier amplitudes for modeled boundary layer for different initial disturbances. The shear flow prototype here is a piecewise linear approximation of a magnetohydrodynamic boundary layer. The initial disturbances that are considered are a point source of the field of transverse velocity and magnetic field. Solutions are obtained for small values of Alfve’n velocity. The velocity plots are drawn for different values of Alfve’n velocity.

Calculation of electrical potentials along a longitudinal section of a 2-D terrain

Geophysics ◽  
2002 ◽  
Vol 67 (2) ◽  
pp. 511-516 ◽  
Author(s):  
Shi-zhe Xu ◽  
Dahai Zhang ◽  
Baiyao Ruan ◽  
Shikun Dai ◽  
Yuguo Li
Keyword(s):  
Fourier Transform ◽  
Point Source ◽  
Apparent Resistivity ◽  
Inverse Fourier Transform ◽  
Vertical Plane ◽  
Longitudinal Section ◽  
Electric Resistivity ◽  
Electrical Potentials ◽  
Terrain Corrections ◽  
The Fourier Transform

The problem of 2-D terrain corrections for point-source electric resistivity data is considered. The total electric potential is divided into normal and anomalous terms. An integral equation is derived for the Fourier transform of the anomalous potential and is solved using a boundary element method. An inverse Fourier transform is applied to recover the anomalous potential along a “longitudinal” profile passing through the point source and oriented perpendicular to the vertical plane containing the 2-D terrain variations. The sum of the normal and anomalous potentials are then used to calculate an apparent resistivity. A sample calculation demonstrates that the longitudinal apparent resistivity calculated in this manner is less sensitive to terrain variations than the traditional “transverse” apparent resistivity that is computed from potential measurements made parallel to the vertical plane containing the 2-D terrain variations.

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