Slab Of Finite Thickness Research Articles

Nucleation of Superconductivity aboveHc3

The recent calculations of van Gelder concerning an upper critical field ${H}_{c4}>{H}_{c3}$ for a wedge-shaped specimen are discussed in the light of the surface-nucleation field of a slab of finite thickness. It is shown that ${H}_{c4}$ is not a new critical field and that it is related to the size-dependent surface-nucleation field.

A Quantal Approach to the Problem of Neutron Multiple Scattering in a Slab

The solution of a neutron multiple scattering problem is obtained by using a quantal approach. The elastic isotropic scattering is considered in the approximation where the scattering length is much less than both the neutron mean free path and the average distance between nuclei. In the case of a finite thickness slab, the probability density of the neutron momenta as a function of the traversed depth and of the scattering cosine is written directly in quadratures without using the transport equation.

A Simplified Orders-of-Scattering Technique for Calculating the Absorption of X Rays

A simple, efficient technique is described for calculating the energy absorption in a planar slab of finite thickness from a planar source of x rays. Compton scattering and photoelectric absorption are considered as the attenuation processes. The computational model allows two scattering angles per Compton event, and in this form is amenable to closed-form solution. Results are given for several absorbers in the energy range from 25 to 400 keV, and comparisons are made with Monte Carlo calculations. Good agreement is obtained at depths of at least three mean free paths. In all cases, the Monte Carlo calculation required at least 100 times the computation time of the simplified orders-of-scattering technique.

Three Geometrical Modifications of the Surface-Impedance Experiment in Low Magnetic Fields

Bound states of electrons trapped by a field along a smooth metal surface are known as magnetic surface levels. Transitions between these levels have been observed in conventional surface-impedance measurements. We here consider three geometrical modifications of the conventional semi-infinite half-space bounded by a plane surface. The first geometry employs a slab of finite thickness. The second considers a surface which is circularly cylindrical, rather than plane. The third considers a surface ruled periodically, like a grating. The experimental consequences are explored, and it is found that parameters inaccessible by other methods may be extracted from such experiments. There are also definite new effects predicted which will provide crucial tests of the magnetic-surface-level hypothesis.

A general starting solution for melting and solidifying slabs

General starting and short-time solutions for melting or solidifying slabs are developed. The following effects are included: arbitrary heating conditions, arbitrary rates of liquid removal (including the limiting cases of stationary and instantaneously removed liquid), temperature-dependent properties, and finite slab thickness. The solution is obtained by means of the ‘embedding technique’, in which a body with time-dependent boundaries is considered mathematically as part of an embedding body of fixed geometry.

Temperature stresses in a slab of finite thickness in the presence of oscillations of the ambient temperature on one or both sides

On the basis of a solution of the problem of thermoelasticity for a free slab of finite thickness the author presents formulas and graphs useful for practical calculations of the temperature stresses in the presence of ambient-temperature fluctuations on one or both sides and convective heat transfer at the surfaces.

Transmission of Electromagnetic Waves through a Conducting Slab. II. The Peak at Cyclotron Resonance

This is the second in a series of three papers concerned with the propagation of electromagnetic waves through a metallic slab of finite thickness. In the situation studied here, the relation between the current in the slab and the electric field which drives it is highly nonlocal because of the finite electron mean free path. We calculate the transmission amplitude under these conditions, assuming that there is a steady magnetic field normal to the faces, and assuming also that electrons within the slab scatter diffusely at the surfaces. A transmission peak is predicted at cyclotron resonance which, in the limit of slab width much larger than an electron mean free path, has a line shape which approaches the square root of a Lorentzian and a phase which shifts by $\ensuremath{\pi}$ across the line. All of this is superposed on the Gantmakher-Kaner oscillations which are also present. This resonant behavior, which is absent when the electrons are assumed to be scattered specularly from the surface of the slab, can be understood in terms of electron transport from Fresnel zones on the Fermi sphere which expand at resonance so that the first zone covers the entire hemisphere. The absence of this resonance when the electrons are scattered specularly is tentatively ascribed to a fortuitous cancellation which arises because the equatorial electrons, whose response gives rise to currents which shield the interior of the metal, themselves undergo a resonance which excludes the field from the metal. In carrying out these calculations using the two-sided Wiener-Hopf technique for the nonlocal wave equation, we found that the technique itself can be simplified considerably if the quantity of interest is the transmission amplitude (the field at the emergent face of the slab) rather than the full-field amplitude in the slab. We show that this simplification arises when certain short-range parts of the field are eliminated.

Transmission of Electromagnetic Waves through a Conducting Slab. I. The Two-Sided Wiener-Hopf Solution

This is the first of a series of papers dealing with propagation of electromagnetic waves through a metallic slab of finite thickness. In this first paper, we present a method for solving the integrodifferential equation governing the electric field in the interior of the metal when the electrons in the interior of the metal suffer diffuse reflection at each surface. The method is potentially of use in a wide class of problems, namely, the finite-slab generalization of all those semi-infinite-medium problems which are conventionally studied by the Wiener-Hopf technique. The solution given here is an iterative one with successive terms converging as e−L/l, where L is the thickness of the slab and l is the range of the kernel of the integral term in the equation.

Neutron diffusion across a temperature discontinuity in crystalline moderators

Calculations on the diffusion of neutrons across a temperature discontinuity in crystalline moderators—graphite and beryllium, are reported. Using plane geometry, with the first slab of finite thickness and the second slab extending to infinity, we have calculated the steady-state neutron spectra at different space points. Consequently, the space variation of total neutron density, as also the space variation of average neutron energy have been studied. Defining the effective temperature T eff, as T eff = ( 2 3k ) × (average energy) (where k is the Boltzmann constant) we have compared our results for graphite with the experimental data of Coates and Gayther (1961), Day and Partridge (1964) and Day (1967). The agreement is good except near the interface and it is shown, that this discrepancy is most probably due to the fact that the steady-state neutron spectra are not Maxwellian and therefore the calculated value of the effective temperature depends upon the way it is defined.

The initial-value transport problem for monoenergetic neutrons in an infinite slab with delayed neutron production

The time-dependent behavior of the neutron density in a material body is described by a nonnegative function which satisfies a linear Boltzmann equation. In multiplying media, in which the only neutron source is fission, neutrons can be produced either instantaneously or with a finite delay time after the absorption of a neutron by a fissile nucleus. The latter are due to fission fragments which act as delayed neutron emitters (precursors), and the Boltzmann equation must be supplemented by an equation for the time rate of change of each group of precursors. It is the purpose of this paper to solve the associated initial-value problem. We restrict our attention to problems involving mono-energetic neutrons, isotropic scattering, and one group of precursors and seek the distribution of neutrons and precursors everywhere in an infinite slab of finite thickness surrounded by a vacuum or by a pure absorber. The appropriate equations for the time rate of change of the neutron density N and the precursor density C inside the slab are

Numerical results for the auxiliary equation of radiative transfer

A basic problem is astrophysics, meteorology and neutron transport theory us the determination of the internal intensity produced when uniform parallel rays illuminate a slab of finite thickness which both absorbs and scatters radiation isotropically. One approach to this problem is through the intermediate step of determining the source function which represents the rate of production of scattered radiation per unit volume per unit solid angle as a function of the depth below the top surface. Ordinarily this function is characterized as the solution of a singular integral equation, which presents difficulties from the numerical viewpoint. In earlier memoranda the authors derived other differential integral equations for the source function viewed as a function of the thickness of the slab. In this memorandum they present the results of extensive calculations of source functions in both graphical and tabular form.

On the theory of the passage of ?-quanta through a slab of finite thickness

On the theory of the passage of ?-quanta through a slab of finite thickness

Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness. By R. E. Bellman, R. E. Kalaba and M. C. Prestrud. Pp. viii, 346. 55s. American Elsevier Publishing Co., New York.

Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness. By R. E. Bellman, R. E. Kalaba and M. C. Prestrud. Pp. viii, 346. 55s. American Elsevier Publishing Co., New York. - Volume 50 Issue 371

Application of Invariant Imbedding to the Reflection and Transmission Problem of Gamma Rays, (I)

Equations were derived for the reflection and transmission functions of γ-rays through slabs of finite thickness, making use of the invariant imbedding principle. Numerical solutions were obtained for the reflection function (albedo) of semi-infinite medium. The results agreed well with the solutions by Monte Carlo method. Comparisons were also made with experimental results. Fair agreement was obtained except for the angular distribution of the reflected photons from light elements.

Transfer-Matrix Method for Gamma-Ray and Neutron Penetration

The problem of radiation transport is formulated in terms of a transfer matrix H which is a 2 × 2 matrix of operators. H is simply related to the more intuitive transmission and reflection operators T and R. An explicit expression for H is derived in slab geometry for radiation distributions that depend on the angle with the slab normal and on energy. H for a multilayer slab is the matrix product of the transfer matrices for the individual layers. A formal expression for H for a homogeneous slab of finite thickness is found in terms of the T and R appropriate to an infinitestimally thin slab. These in turn are related to the single-scattering distribution and therefore can be computed from the microscopic cross section. For purposes of computation, finite matrix representations for the operators must be introduced corresponding to the finite vectors which approximate the distributions. Expansions of the distributions in the cosine of the angle and group representations in energy were chosen in the present work. Some numerical results are presented for gamma rays on aluminum. The extension to problems with internal sources and to nonplanar geometries is outlined.

Effective cadmium cut-off energies in cycindrical geometry for fissile and [formula omitted] detectors

The importance of geometry in detector-filter arrangements used for integral neutron spectrum measurements is discussed, and cadmium transmission functions are presented for a number of detectors ranging from cylinders of 0·040 in. o.d. to slabs of finite thickness and infinite extent, with close-fitting filters of thickness 0·030, 0·040 and 0·050 in. The transmission function as defined in this paper is dependent on the physical size of the detector and the importance of this consideration is also demonstrated. In addition the transmission functions have been used to determine the effective cadmium cut-off energies for 239Pu, 235U, 233U and 1 V detectors.

Application of invariance principles to a radiative transfer problem in a homogeneous spherical medium

The problem of steady-state, radiative transport through a finite, spherically symmetric and uniformly generating medium is considered. The governing equations are applicable to the study of thermal radiation in a heat generating grey gas as well as to the idealization proposed by Cuperman, Engelmann, and Oxenius to study radiation loss in a homogeneous, isothermal spherical plasma due to an optically thick spectral line associated with the de-excitation energy of two-level impurity ions. The analysis is related directly to the study of a one-dimensional problem in a slab of finite optical thickness. This equivalence then permits the direct application of results based on the invariance principles of Ambartsumian and Chandrasekhar. In particular, exact expressions are derived for the source function at the spherical boundary and the radiative flux loss. These results are expressed in terms of tabulated moments of Chandrasekhar's X and Y functions.

Invariant Imbedding and Radiant Transfer in Slabs of Finite Thickness (R. E. Bellman, R. E. Kalaba and M. C. Prestrud)

Previous article Next article Invariant Imbedding and Radiant Transfer in Slabs of Finite Thickness (R. E. Bellman, R. E. Kalaba and M. C. Prestrud)E. H. BareissE. H. Bareisshttps://doi.org/10.1137/1006047PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"Invariant Imbedding and Radiant Transfer in Slabs of Finite Thickness (R. E. Bellman, R. E. Kalaba and M. C. Prestrud)." SIAM Review, 6(2), p. 185 Previous article Next article FiguresRelatedReferencesCited byDetails Volume 6, Issue 2| 1964SIAM Review History Published online:18 July 2006 InformationCopyright © 1964 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1006047Article page range:pp. 185-185ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

The thermal response of a metal slab to a class of radially dependent heat inputs

The temperature fields are deduced for an infinite metal slab of finite thickness produced by a heat flux input which depends on the radial coordinate. The formal solution is obtained by means of transform techniques.

Disturbances in an infinite slab due to torsional forces applied on a cylindrical cavity

The paper deals with the disturbances in an infinite slab of finite thickness, the medium being homogeneous, isotropic and perfectly elastic.