Retroreflection by Refraction

Abstract

A sphere of radius a whose refractive index at radius r is n(r) = (2a/r ? 1)^1/2 for r < a (with n = 1 for r > a) is shown to be a perfect ‘retrorefractor’ in geometrical optics; every ray incident on the sphere is bent through 180°. This compares favourably with a conventional retroreflector of three circular mirrors intersecting at right angles (eight corner reflectors) which retroreflects only one-seventh of all incident rays. However, the retrorefracting sphere suffers serious impracticalities.     

On quadrature and singular finite elements

Special quadrature rules are described for elastic finite elements that have r^q behaviour (0 < q < 1) directly induced in natural element co-ordinates. In general, the quadrature points and weights can vary with the exponent q. For two-dimensional problems with a square-root singularity (q = 1/2), the use of special quadrature results in significant improvements over regular Gauss quadrature. The development of special quadrature rules for three-dimensional elements is shown to be a difficult task. Several special case rules are developed and tested for a line-type singular element, and a precise rule is given for a point-type singular element in three dimensions.     

Mapped infinite elements for exterior wave problems

Recently, a new type of infinite elements which uses r^?1 decay was proposed. They were applied to exterior wave problems and good results were obtained. In two-dimensional problems, however, it was necessary to move the origin of the r^?1 decay in order to model the outgoing wave more accurately, because it decays roughly as r^?1/2. In this paper, the mapped infinite elements with r^?1/2 decay and the necessary numerical integration procedure are presented. These elements do not require any artificial movement of the origin. Several example problems are solved. The results show that the infinite elements with r^?1/2 decay here give much more accurate values than the infinite elements with exponential decay and any damper elements.     

Paramagnetic susceptibility and ultrasonic attenuation of itinerant antiferromagnetism above the Neel temperature

Making use of the two-band model of the itinerant antiferromagnetism, we calculate the wave-vector-dependent paramagnetic susceptibility (i.e., the fluctuation propagator of the order parameter) and the ultrasonic attenuation coefficient above the Neel temperatureT _N .We distinguish two regions depending on the degree of the nesting between the electron and the hole bands. The degree of the nesting is characterized byH, which measures the difference of the Fermi radii of the electron and the hole bands. WhenH is less than the critical nestingH*, the fluctuation is of the diffusion type indicating the transition into the commensurate spin density wave (CSDW) state; the fluctuation propagator diverges atT=T _N forq=0 (the paramagnetic susceptibility diverges forq=G _i /2;G _i are the reciprocal lattice vectors) whereq is the wave number vector. WhenH>H*, on the other hand, the fluctuation propagator diverges atT=T _N ,on the sphere in the wave vector space with the radiusq ₀,signalling the transition into the incommensurate spin wave (ISDW) state; the paramagnetic susceptibility diverges now forq=(G _i /2)+q, where |q|=q ₀.The attenuation coefficient is much enhanced due to the spin fluctuation nearT _N . We find that the attenuation coefficient of the longitudinal sound wave diverges atT=T _N as ^–3/2 forHH* and ^–5/2 forH>H*, where =(T/T _N )–1. The latter prediction is in reasonable agreement with the recent measurement of the attenuation coefficient in pure Cr by Imai and Sawada.     

On the Theory of Stimulated Raman Radiation in an Optical Resonator

Stimulated Raman radiation involving an arbitrary number of Stokes components in an optical resonator is considered for the case when a pumping wave is directly incident on a Raman-active condensed substance at an angle θ to the resonator axis. The expressions are derived for the steady-state radiation intensities of all the Stokes components generated, and the number of the components versus the pumping wave intensity is discussed. The conditions of the effective conversion of the pumping wave into radiation of Stokes components are elucidated. It is shown that in a frequency region close to the point ?² n/? ω² = 0 (n is the effective refractive index of the substance inside the resonator) locking Stokes components is feasible due to a ‘tunnel’ traversal of femtosecond pulses through a dispersive medium, provided the latter occupies a characteristic tunnel length along the resonator axis. For the case of the three components generated, the wave band over which locking occurs is derived and a frequency modulation of the first and the third Stokes components beyond this band is predicted. The possibility of an experimental indication of the band discussed, due to a discontinuous disappearance of the frequency modulation in the band, is noted. For locking four Stokes components the possibility is shown of the change of their stable phase relation on variation of the pumping wave intensity.     

Suppression of grain growth in dual phase steels

Abstract

The growth behaviour of ferrite (α) and austenite (γ) grains at 1173, 1273, and 1373 K in an α + γ region was observed in a 3Si- 0.3Cr - 0.1C (wt-%) steel with a ferrite volume fraction of ~50%. The results are well summarised by the equation (r)³ - (r ₀)³ = kt, where r, r₀, k, and t are the mean grain radius, the initial grain radius, a constant, and the annealing time, respectively. When a microalloying element Sm, P, B, Al, or Ti was added singly to the dual phase steel, grain growth was clearly retarded. Therefore, it is concluded that ultra refinement of grains can be achieved by using a combination of the dual phase effect and the pinning or dragging effect.     

Growth kinetics of (θ′– and θ-phases in Al–3Cu alloy

Abstract

The precipitation of θ′ and θ in Al–3Cu alloy has been investigated, using a resistivity and a transmission electron microscopy (TEM) method. Both stages of the transformation obey the Avrami equation, ξ = 1? exp (?ktⁿ), where ξ is the precipitated fraction of excess solute. The growth parameters are n = 1·68±0·10 for θ′ precipitation and n = 0·63 ± 0·04 for θ precipitation, and the corresponding activation energies are 85·3 ± 10·9 and 111·1±5·2 kJ, respectively. The n-value for the θ′ precipitation stage corresponds to the model proposed by Ham for diffusion-limited growth of disc-like particles, and is in agreement with the TEM results of this investigation. The activation energy for θ′ precipitation is in the range for θ′ boundary migration reported by Aaronson and Laird. The n-value and activation energy for θ precipitation suggests a growth model in which concurrent dissolution of θ′ supplies solute to θ particles via dislocation paths.

MST/480     

The Use of Incomplete Beta Functions for Prior Distributions in Binomial Sampling

Let θ be the proportion of defectives in a batch selected at random from a fixed population of batches. The paper deals with the determination of the prior distribution of θ, using prior knowledge obtained from batches examined in the past. Assuming a beta type prior distribution, the following three situations are investigated: a. Past records provide the numbers r ₁, r ₂, …, r _N of defectives found in N samples of size n from N previously inspected batches;

b. The expected fraction defective in a batch, E(θ), and the probability, P, of a batch exceeding twice the expected fraction defective are approximately known;

c. (c) The probabilities of θ exceeding a certain value θ₁ and of θ falling below another value θ₂ are approximately known.

Methods of estimating the parameters of the prior distribution are given, and charts are provided to facilitate their determination.

The effects of errors in the parameters on the posterior distribution are discussed in relation to sample size and number of defectives found in the sample. The effects of assuming a beta distribution when in fact some other distribution would have been appropriate are found to be negligible in most practical applications.     

Coarsening of particulate silicon in aluminium based matrices

Abstract

Published measurements of the coarsening parameter K = (r³2r³_o)/t of particulate silicon in Al – Si alloy base matrices are compared with values predicted for volume diffusion control according to the LSW model corrected for finite volume fraction. While some measurements are within a factor of 10 of LSW predictions, others exceed these by factors as high as 10³ to 10⁴. Possible reasons for such large apparent enhancements of coarsening rate are discussed.     

Dispersion corrections to the collective mode spectrum in axial and planar phases of superfluid He3

Using the path integration technique developed by Brusov and Popov earlier, we have calculated for the first time the dispersion corrections to the collective mode spectrum in A- and 2D- phases of superfluidHe ³ for arbitrary directions of the collective excitation momentumk, taking the damping of collective modes into account. In axial-phase clapping (cl) mode remains six fold degenerate fork l, while for other directions a three fold splitting takes place, which reaches a maximum fork l. The pairbreaking modes remain degenerate even taking the dispersion induced corrections into account. In the planar phase the degeneracy of clapping (cl) and quasi-Goldstone (qGd) modes depends on the direction of the collective mode momentumk with respect to the external magnetic fieldH: the mode degeneracy remains the same as in the case of zero momentumk fork H only, and for any other directions two fold splitting of these modes takes place. The dispersion induced splitting of collective modes could be observed (at least in the A-phase) in sound absorption experiments.     

The spin 1/2 Heisenberg-star with frustration: Ground state and thermodynamics

We investigate a spin 1/2 Heisenberg antiferromagnet (AFM) H = J_{1 i} ^N =1 s₀s _i + J₂H_Rs _i describing a central spin which interacts with a surrounding medium h_r. We present general relations for the eigen values, the eigen states and the spin correlation function and study the model in more detail for H_r being either a linear chain or a Lieb-Mattis AFM. Additional to the ground state properties we calculate the full thermodynamics in the case of H_r being a Lieb-Mattis AFM.     

Three-dimensional surface singularity of an interface crack

The three-dimensional singularity field near the terminal point of an interface crack at the free surface of an elastic bimaterial is investigated. The Finite Element Iterative Method (FEIM) is used for evaluating the asymptotic field. A spherical coordinate system r, , is used and the singular displacement field is assumed to be of a product form r g(, ), where and g(.) are in general complex. To validate the model, the method is first applied to the three dimensional surface crack in a homogeneous elastic material. The results for this case show excellent agreement with previously published analytical and numerical results. For an extreme effect of bimaterial property mismatch, on the surface crack singularity, an elastic material bonded to a rigid substrate is investigated (E1/E2=). The results show that the complex power singularity depends strongly on Poisson's ratio . The real part of the stress singularity is greater than 0.5 of the plane strain case and the imaginary part becomes almost zero at 0·25 instead of at =0.5. The second term in the expansion of the asymptotic field was shown to have a singularity of 0.5.     

Localized fringes from identical twin image sources

Abstract

Twin coherent monochromatic point sources of light produce Young's optical interference fringes throughout space. Often in interferometry, there are effectively identical twin sources (partially reflected images of a single source by plane surfaces). They are small, but are far from being point sources. The consequence is long known: the fringes are ‘localized’, being visible in a certain region only. Here the isometry between the image sources (their identical shape and size) is exploited to deduce the three-dimensional shape of this region and the form of the fringes within it by means of wave optics. In the short wavelength limit the fringes are localized in a definite tube about a certain curve in space, given in cylindrical polar coordinates (r, φ, z) by z = Z tan φ/ tan φ, r = R, where (R, ± φ, ±, Z) locate the sources (and describe their mutual orientation). Also the fringe sheets within this tube differ from the hyperboloids of Young's fringes, being instead nested pieces of concentric cylindrical surfaces.     

Mechanical properties and drug release of venlafaxine HCl solid mini matrices prepared by hot-melt extrusion and hot or ambient compression

Objective/significance: To elucidate the role of plasticizers in different mini matrices and correlate mechanical properties with drug release.

Methods: Cylindrical pellets were prepared by hot-melt extrusion (HME) and mini tablets by hot (HC) and ambient compression (AC). Venlafaxine HCl was the model drug, Eudragit^® RSPO the matrix former and citric acid or Lutrol^® F127 the plasticizers. The matrices were characterized for morphology, crystallinity, and mechanical properties. The influence of plasticizer’s type and content on the extrusion pressure (P_e) during HME and ejection during tableting was examined and the mechanical properties were correlated with drug release parameters.

Results: Resistance to extrusion and tablet ejection force were reduced by Lutrol^® F127 which also produced softer and weaker pellets with faster release, but harder and stronger HC tablets with slower release. HME pellets showed greater tensile strength (T) and 100 times slower release than tablets. P_e correlated with T and resistance to deformation of the corresponding pellets (r²?=?0.963 and 0.945). For both HME and HC matrices the decrease of drug release with T followed a single straight line (r²?=?0.990) and for HME the diffusion coefficient (D_e) and retreat rate constant (k_b) decreased linearly with T (r²?=?0.934 and 0.972).

Conclusions: Lutrol^® F127 and citric acid are efficient plasticizers and Lutrol^® F127 is a thermal binder/lubricant in HC compression. The different bonding mechanisms of the matrices were reflected in the mechanical strength and drug release. Relationships established between T and drug release parameters for HME and HC matrices may be useful during formulation work.     

A reappraisal of transition elements in linear elastic fracture mechanics

This article offers a detailed comparison of the transition elements described by P.P. Lynn and A.R. Ingraffea [International Journal for Numerical Methods in Engineering 12,1031–1036] and C. Manu[Engineering Fracture Mechanics 24,509–512]. The source of a numerical phenomenon in using Manu's transitionelement (TE) is explained. The effect of eight-noded TEs with differentquarter-point elements (QPE) on the calculated stress intensity factors (SIFs) isinvestigated. Strain at the crack tip is shown to be singular for any ray emanating from the crack tip within an eight-noded TE, but strain has bothr ^–1/2andr ^–1singularities, withr ^–1/2dominating for large TEs. Semi-transition elements (STEs) are defined and shown to have a marginal effect on the calculated SIFs. Nine-nodedtransition elements are formulated whose strain singularity is shown to be the same as that of eight-noded TEs. Then the effect of eight-noded and nine-noded TEs with collapsed triangular QPEs, and rectangular and nonrectangular quadrilateral eight-noded and nine-noded QPEs, is studied, and nine-noded TEs are shown to behave exactly like eight-noded TEs with rectangular eight-noded and nine-noded QPEs and to behave almost the same with other QPEs. The layered transition elements proposed by V. Murti and S.Valliapan [Engineering Fracture Mechanics 25, 237–258] areformulated correctly. The effect of layered transition elements is shown by two numerical examples.     

Singular finite element efficiency in estimating stress intensity factors

Singular finite element displacement functions for linear elastic fracture mechanics applications are investigated. The displacement fields are expressed in terms of polynomials. These polynomials are expanded in varying degrees of r and θ in order to examine the effect of the number and type of terms on the solution to various problems with known solutions. Methods of estimating the stress intensity factors for mode 1 and mode 2 are presented. Numerical results evaluating the effect of various r and θ terms on the solutions are presented.     

The Folded Normal Distribution: Two Methods of Estimating Parameters from Moments

The general formula for the rth moment of the folded normal distribution is obtained, and formulae for the first four non-central and central moments are calculated explicitly.

To illustrate the mode of convergence of the folded normal to the normal distribution, as μ/σ = θ increases, the shape factors β _f1 and β _f2 were calculated and the relationship between them represented graphically.

Two methods, one using first and second moments (Method I) and the other using second and fourth moments (Method II) of estimating the parameters μ and σ of the parent normal distribution are presented and their standard errors calculated. The accuracy of both methods, for various values of θ, are discussed.     

Left atrial volume index as a predictor of ventricle repolarization abnormalities in adult dialyzed patients

This study was performed to investigate the relationship between left atrium (LA) volume index (LAVI) and left ventricle electrical activity presumably repolarization in end‐stage renal disease patients. Study group was consisted of 120 dialyzed patients divided into two subgroups: 57 (age 50.7 ± 7.1) were on continuous ambulatory peritoneal dialysis (CAPD) and 73 (age 51.6 ± 7.6) were hemodialyzed (HD). All patients were undergoing three‐dimensional vectorcardiographic (VCG) monitoring to assess parameters concerning T vector: QRS‐T angle, Tel, and Taz. Standard echocardiography was performed to assess: LA_max, LA_short, LA_long. LAVI was calculated due to formula: LAVI = (π/6X [LA_max × LA_short × LA_long])/m². LAVI in HD as well as in CAPD patients was significantly higher compared with controls (respectively: 36.29 ± 10.92; 36.41 ± 11.06; 20.64 ± 6.77 mL/m²). The calculated cutoff value of LAVI was 36.32 mL/m². In HD patients, the strong correlations between LAVI and QRS‐T angle and Tel were determined (respectively: r = 0.407, P < 0.001 and r = 0.359, P = 0.006). Similarly in CAPD group were significant associations between LAVI and QRS‐T angle and Tel (respectively: r = 0.423, P < 0.001 and r = 0.374, P = 0.004). The QRS‐T angle, Tel and Taz are independently and markedly associated with LAVI in both HD and CAPD patients. LAVI and VCG indices are higher in both HD and CAPD patients. Correlation between QRS‐T angle and LAVI may reflect unfavorable influence on the electrical activity of the heart in dialyzed patients with left ventricle diastolic dysfunction. LAVI cutoff value is useful biomarker for stratification of ventricle repolarization disturbances in those patients.     

Binding of Two Helium Atoms in Confined Geometries

We carry out a comprehensive study of the binding of two helium atoms in unrestricted and, in particular, in restricted geometries in both two and three dimensions. Besides the well known binding of the ⁴ He dimer in unrestricted geometry in two and three dimensions, we also find weakly bound states of the ³ He- ⁴ He molecule and the ³ He dimer in 2 dimensions. Furthermore, any combination of two ⁴ He or ³ He atoms can form a molecule if their motion is sufficiently confined. Our calculations are carried out by numerically solving the Schrödinger equation as well as by constructing a suitable variational wave function.     

Typical flow between enclosed corotating disks and its dependence on Reynolds number

Abstract

The flow in an enclosed co‐rotating disk pair is investigated by Laser Doppler Velocimetry (LDV) measurements and flow visualizations. First, the typical flow structure at Re = 5.25 × 10⁵ and S = 0.09 is clarified. The flow fields in the r – θ and the r – z planes are both investigated and then divided into several flow regions based on the distinct flow types observed. The flow regions found in the two different planes are also compared and integrated. Second, with S fixed, the dependence of the flow field structure upon the Reynolds number is discussed. Three regimes of the r – θ plane flow with different Reynolds numbers are identified based on the measured mean velocity and spectral intensity. When Re < 1.6 × 10⁵, no solid body region is found and the flow is in a laminar regime. In the range 1.6 × 10⁵ ≤ Re ≤ 2.0 × 10⁶, the solid body region and the outer region vortices coexist, and an empirical equation is developed to estimate the number of vortices. When Re > 2.0 × 10⁶, the flow becomes turbulent. As Re increases from 9.3 × 104 to 5.25 × 10⁵, the spectral intensity initially increases and then decreases before increasing again to an even higher level, resulting in an increasing sawtooth pattern.