Orthogonal Surfaces Research Articles

X-ray focusing using square channel-capillary arrays

A class of imaging, condensing, and collimating devices for x rays is investigated which is based on the use of an array of small channels of square cross section. The focusing and collimating effect arises from external reflection of near-grazing-incidence rays at the interior channel surfaces. Rays are redirected by being singly reflected from two orthogonal channel surfaces and are imaged from a source point to a square region with a side length MT+1 times that of the channel side length, where MT is the transverse magnification. The image and source locations are related by a thin-lens formula. The point spread function and the efficiency of these focusing devices are calculated. Two energy regimes with different channel reflectivity characteristics are examined in detail: the hard x-ray regime (E>8 keV) and the soft x-ray regime (E<200 eV). For these cases the efficiency of focusing x rays depends only on the channel aspect ratio and reflectivity parameters. A discussion is made of channel plates of other configurations, such as arrays of rectangular channels. The results are supported by Monte Carlo simulations.

Surface polaron in the vicinity of two orthogonal surfaces.

When a slow electron is in the vicinity of a corner made of two orthogonal ionic surfaces such as LiF, it interacts with the surfaces in two forms, one representing the ionic polarization and the other representing electronic polarization. The comparison between the binding energy of the system with the related surface polarization quanta indicates that the latter takes the static image-potential form and the former creates a surface polaron in the vicinity of the corner. The coupling constant between the electron and surface phonon in LiF is about 7 and, therefore, in the limit of small total momentum, we can adopt the simple Tomonaga approximation for the wave function to calculate the binding energy and effective mass of the system. The result shows the presence of the second surface increasing both the binding energy and effective mass of the surface polaron.

A MULTI-DIMENSIONAL TUNNELING THEORY WITH APPLICATION TO SCANNING TUNNELING MICROSCOPE

We have developed a formal general approach for evaluating the WKB wave function in a forbidden region of a non-separable potential in a three-dimensional space. The wave function is described by two sets of orthogonal wave-fronts, the equi-phase and equiamplitude surfaces, or equivalently by two sets of paths defined to be normal to these surfaces respectively. We have extended a Huygens type construction to the forbidden region to obtain the multi-dimensional wave function. The construction determines the wave-fronts and the paths. It is found that the equations for the paths obtained from the construction are coupled to each other and do not reduce to a set of ordinary differential equations. However, if the incident wave is normal to the turning surface, the equations satisfied by the paths can be de-coupled. These equations are found to be equivalent to Newton's equations of motion for the inverted potential and energy. This characteristics has been used to calculate the current density distribution for an STM. In this paper, we use the same approach to derive the expressions for the lateral resolutions and corrugations for a model STM.

Electron bound states in the vicinity of two orthogonal surfaces.

In a number of physical systems, it has been shown that there are electron bound states on a surface due to a potential well created by the interaction of the electron with its image and a potential barrier or a band gap keeping the electron out of the interior of the solid or liquid. We consider a configuration of two orthogonal solid surfaces and determine the binding energies and the wave functions of an electron that is in the vicinity of the corner. Our calculations indicate that the presence of the second surface gives a corner state, extended along the intersection of two surfaces, whose energy is below that of the plane-surface case, for both dielectric and metal surfaces.

Effectiveness of Geologic Characterization Techniques, Climax Granitic Stock, Nevada Test Site

Several different techniques were used to characterize the structural geology of the quartz monzonite unit of the Climax Stock. This study examined major geologic structural features and yielded detailed data on over 1,500 fractures. The data were used to identify joint sets, faults and shear zones as well as to statistically analyze joint frequencies and spacings. Eight joint sets, three major shear sets, and a fault zone were studied. The results of previous geologic characterization work performed by many other investigators were compared with the results of our studies. These data provided the basis for evaluating how effective various techniques were in describing the structural geology of the Climax Stock. The specific techniques that were evaluated included the use of unoriented diiamond drilled core, mappoing of only major fractures within the drifts, detailed mapping using modified scan-line techniques, data reduction techniques to account for orientation biases, and surface trench mapping. The results of this study identified several issues for consideration during characterization studies. Firstly, geologic structural features that are often assumed to be uniform and continuous were shown to change characteristics within very short distances. Secondly, we demonstrated that when applied properly to an adequate data set, orientation bias correctionmore » techniques were effective in identifying all fracture sets. Thirdly, we demonstrate that unless three nearly equal length orthogonal surfaces are mapped, interpretations of spacing or of joint prevalence can be misleading. Finally, we demonstrate the usefulness of mapping only isolated and limited sized areas in detail with more general or reconnaissance mapping of the entire facility to tie the detailed map sections together and to identify major features. 10 refs., 23 figs., 2 tabs.« less

Orthogonal Central Composite Designs of the Third Order in the Evaluation of Sensitivity and Plant Growth Simulation Models

Abstract The U.S. Department of Agriculture has used response-surface techniques as applied by Baker and Bargmann (1981) to plant process simulation models as an aid in the identification of interrelationships among yield and various single-valued and functional parameters. Orthogonal cubic surfaces have provided insight into higher order relationships as well as a measure of the relative sensitivity of yield to experimentally determined parameter values. Several examples investigate the effectiveness of those higher order surfaces and illustrate how less precise (and less costly) measurements may be possible in building and using these simulation models.

Orthogonal Central Composite Designs of the Third Order in the Evaluation of Sensitivity and Plant Growth Simulation Models

Abstract The U.S. Department of Agriculture has used response-surface techniques as applied by Baker and Bargmann (1981) to plant process simulation models as an aid in the identification of interrelationships among yield and various single-valued and functional parameters. Orthogonal cubic surfaces have provided insight into higher order relationships as well as a measure of the relative sensitivity of yield to experimentally determined parameter values. Several examples investigate the effectiveness of those higher order surfaces and illustrate how less precise (and less costly) measurements may be possible in building and using these simulation models.

Optical anisotropy of coals as an indicator of tectonic deformation, Broad Top Coal Field, Pennsylvania

The anisotropic “reflectance fabric” of coal is used as a finite strain marker in the structural analysis of deformed strata in the Broad Top Coal Field of central Pennsylvania. Oriented blocks of coal were collected at various field locations. The reflectances of three orthogonal surfaces were used to estimate the magnitudes and orientations of the three principal reflectance axes. Broad Top coals are shown to have three principal reflectance values, in contrast with most bituminous coals, which have only two. Reflectance values can be represented by a biaxial indicatrix that may be either prolate or oblate. The oblate indicatrices, showing flattening in the bedding plane, are associated with gently deformed strata. The prolate indicatrices, elongated parallel to the fold axes (N32°E), are associated with more intensely folded strata and show the influence of strong lateral compression superimposed upon gravitational loading. The reflectance indicatrices maintain a constant spatial orientation, regardless of the inclination of the enveloping bedding. The minimum principal reflectance axis plunges steeply toward the west, indicating that during deformation the direction of maximum compression was plunging toward the northwest. These results suggest that in the Broad Top field maximum coalification and, therefore, maximum burial were penecontemporaneous with the later stages of deformation, postdating most of the folding.

The Elliptic Solution of the Secondary Flow Problem

This paper presents the elliptic solution of the inviscid incompressible secondary flow in curved passages. The three-dimensional flow field is synthesized between 3 sets of orthogonal nonstream surfaces. The two-dimensional flow field on each set of surfaces is considered to be resulting from a source/sink distribution. The distribution and strength of these sources are dependent on the variation in the flow properties normal to the surfaces. The dependent variables in this formulation are the velocity components, the total pressure, and the main flow vorticity component. The governing equations in terms of these dependent variables are solved on each family of surfaces using the streamlike function formulation. A new mechanism is implemented to exchange information between the solutions on the three family surfaces, resulting into a unique solution. In addition, the boundary conditions for the resulting systems of equations are carefully chosen to insure the existence and uniqueness of the solution. The numerical results obtained for the rotational inviscid flow in a curved duct are discussed and compared with the available experimental data.

Pressure bar for a capo tasto

The invention is a pressure bar for a capo tasto which attaches itself to a stem of a stringed instrument, having a fret board and strings adjacent to the fret board, and which depresses the strings against the fret board in order to change the key of the stringed instrument. The capo tasto includes a pressure bar having a length adapted to extend across the fret board of the stringed instrument, a mounting member for mounting the pressure bar on the stem of the stringed instrument so that it is disposed perpendicular to the longitudinal axis of the stem adjacent to the fret board and a pressure applicating device for applying pressure to the surface of the stem oppositely disposed to the surface on which the fret board is placed so that pressure is transmitted through the mounting member to the pressure bar. The pressure bar is a multi-surface member which is formed from a pliable material so that it will not damage the strings of the stringed instrument and which is mechanically coupled to the mounting member so that it may rotate about its longitudinal axis in order to allow its various surfaces to contact the strings and the fret board of the stringed instrument. One type of pressure bar has three flat orthogonal surfaces and one concave surface and the second type of pressure bar has four flat surfaces and two concave surfaces, which differ in degrees of concavity.

Approximate Analysis of a Laminar Elliptical Jet

Theme TPHE past several years have witnessed an upsurge JL of interest in thrust augmenting ejectors for V/STOL aircraft and ejector pumps for lasers. In each of these devices, as well as many others, energy is transferred from a high velocity primary flow to a lower velocity secondary flow in such a compact dimension that the near field of the jet is critical. A parameter of basic importance in determining the success or failure of this transfer of energy is the momentum distribution due to the shape and location of the primary nozzles. The momentum distribution may be characterized by a (defined as the lateral position of the mean velocity), since a majority of the jet's momentum is contained within the half widths. In the case of turbulent nonaxisymmetric jet shapes, Trentacoste and Sforza1 have shown experimentally that the major axis half width initially decreases while on the minor axis, the half width grows monotonically. Such a difference in growth rate has not been predicted by either a laminar or turbulent treatment of the problem. Pai and Hsieh2 have treated the linearized case of the laminar rectangular jet by applying the boundary-layer assumptions in both the major and minor axis directions. The result is a rnonotonic half width growth on both axes. In the present analysis, the Navier-Stokes equations are derived in elliptical coordinates. It is then assumed that at any position within the jet, one of the orthogonal coordinate surfaces coincides with the elliptical isovel passing through that position. At this point it is possible to make some rational assumptions concerning the functional form and order of magnitude of each of the variables employed. These assumptions reduce the equations to boundary-layer (i.e., parabolic) type. They also reduce the 4 equations and 4 unknowns to 2 equations and 2 unknowns. These 2 equations are of twodimensional form with several additional terms, and may be described as quasi-two-dim ensional. This form does not require that the equations be solved throughout the threedimensional field but rather that they be solved on the major and minor axes of the jet cross section at each streamwise position. In this way the required computations are greatly reduced. Although the present analysis is laminar, references are made to turbulent experiments to point out qualitative agreement between the two.

Orthogonal polynomial trend surfaces for irregularly spaced data

The advantages of using orthogonal rather than nonorthogonal polynomials for trend-surface analysis are discussed briefly. A method for calculating orthogonal polynomial trend surfaces of any degree on the basis of irregularly spaced data is described. The method is illustrated with subsurface data for the elevation on top of the Devonian Dundee Limestone, Michigan.

Right-Angle Reflector Technique for the Ultrasonic Examination of Material Properties

An ultrasonic goniometer is described that utilizes a single transducer and a reflector composed of two plane orthogonal surfaces. With the instrument immersed in a suitable coupling liquid, the transducer rotates about an axis colinear with the intersection of the orthogonal surfaces. The amplitude of the reflected signal is automatically recorded versus the angle of incidence measured relative to one of the reflector walls (the removable specimen). Amplitude variations can be related to the excitation of surface waves, Lamb waves, critical refraction conditions for body waves, and other physical properties of the reflecting surfaces. The instrument has been extremely useful in measuring surface wave velocities for both single crystal and polycrystalline specimens. On copper and quartz single crystals, good agreement has been achieved with theoretical predictions of surface wave velocity as a function of propagation direction. [Phys. Rev. 119, 533 (1960); J. Acoust. Soc. Am. 41, 921 (1967)]. The sensitivity of the device to other physical parameters will be discussed. [Work supported in part by the Office of Naval Research].

Isometry Groups with Surface-Orthogonal Trajectories

It is shown that the trajectories of an isometry group admit orthogonal surfaces if the sub-group of stability leaves no vector in the tangent space of the trajectories fixed. A necessary and sufficient condition is given that the trajectories of an Abelian group admit orthogonal surfaces. In spacetimes which admit an Abelian G2 of isometries, the trajectories admit orthogonal 2-surfaces if a timelike congruence exists with the following properties: the curves lie in the trajectories and are invariant under G2; ωα and üα are linearly independent and orthogonal to the trajectories.

10. There are no mutually orthogonal surfaces

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A Class of Orthogonal Surfaces with the Property that from any one of them an Infinite Number of Others can be Deduced by Differentiation Alone

A Class of Orthogonal Surfaces with the Property that from any one of them an Infinite Number of Others can be Deduced by Differentiation Alone

Prizes Proposed by the Paris Academy of Sciences for 1911

GEOMETRY.—The Francœur prize (1000 francs), for discoveries or works useful to the progress of pure or applied mathematics; the Bordin prize (3000 francs), for improving at an important point the theory of triple systems of orthogonal surfaces; the Poncelet prize (2000 francs), for work in applied mathematics.

O soustavách orthogonálných ploch. [II.

O soustavách orthogonálných ploch. [II.

O soustavách orthogonálných ploch. [I.

O soustavách orthogonálných ploch. [I.

Systems of Plane Curves whose Orthogonals form a Similar System

While tracing the lines of motion and the meridian sections of their orthogonal surfaces for an infinite mass of perfect fluid disturbed by a moving sphere :—the question occurred to me “When are such systems similar?” In the problem alluded to, the equations of the curves are, respectively,It was at once obvious that any sets of curves such asare orthogonals. But they form similar systems only whenHence the only sets of similar orthogonal curves, having equations of the above form, are (a) groups of parallel lines and (b) their electric images (circles touching each other at one point).