Inverse Cube Root Research Articles

Cr stable isotope fractionation by evaporation from silicate melts

We present new experimental results of Cr isotope fractionation during degassing of silicate melts. Our experiments at oxidizing conditions (in air) at 1 bar total pressure show that evaporative loss of Cr is substantial and depends on run duration and on temperature (between 1350 and 1500 °C). The stable Cr (δ53Cr) isotope compositions of the evaporation residues were analyzed by thermal ionization mass spectrometry using the double spike method. Our results indicate that during degassing the lighter isotope 52Cr is enriched in the gas phase and consequently the residual melt becomes enriched in the heavier 53Cr isotope in a mass-dependent fashion. The kinetic isotopic fractionation factor, α, derived from our experimental data is 0.9995 ± 0.0001, which is significantly different from the fractionation factor predicted for an ideal gas composed of CrO3, αkin = 0.995. Our experimentally-determined value of α is consistent with diffusive transport of CrO3 (g) away from the melt surface, proportional to the inverse cube-root of the reduced masses of the two isotopologues. We hypothesize that Cr loss by evaporation may be recorded by Cr isotopes in terrestrial magmas with sufficiently high oxygen fugacities erupting near the surface, which are plausibly found in lava lakes or erupting arc magmas.

An Effective Solution of the Cube-Root Truly Nonlinear Oscillator: Extended Iteration Procedure

The cube-root truly nonlinear oscillator and the inverse cube-root truly nonlinear oscillator are the most meaningful and classical nonlinear ordinary differential equations on behalf of its various applications in science and engineering. Especially, the oscillators are used widely in the study of elastic force, structural dynamics, and elliptic curve cryptography. In this paper, we have applied modified Mickens extended iteration method to solve the cube-root truly nonlinear oscillator, the inverse cube-root truly nonlinear oscillator, and the equation of pendulum. Comparison is made among iteration method, harmonic balance method, He’s amplitude-frequency formulation, He’s homotopy perturbation method, improved harmonic balance method, and homotopy perturbation method. After comparison, we analyze that modified Mickens extended iteration method is more accurate, effective, easy, and straightforward. Also, the comparison of the obtained analytical solutions with the numerical results represented an extraordinary accuracy. The percentage error for the fourth approximate frequency of cube-root truly nonlinear oscillator is 0.006 and the percentage error for the fourth approximate frequency of inverse cube-root truly nonlinear oscillator is 0.12.

Fast Calculation of Cube and Inverse Cube Roots Using a Magic Constant and Its Implementation on Microcontrollers

We develop a bit manipulation technique for single precision floating point numbers which leads to new algorithms for fast computation of the cube root and inverse cube root. It uses the modified iterative Newton–Raphson method (the first order of convergence) and Householder method (the second order of convergence) to increase the accuracy of the results. The proposed algorithms demonstrate high efficiency and reduce error several times in the first iteration in comparison with known algorithms. After two iterations 22.84 correct bits were obtained for single precision. Experimental tests showed that our novel algorithm is faster and more accurate than library functions for microcontrollers.

Gravitational backreaction near cosmic string kinks and cusps

We find the leading-order effect of gravitational backreaction on cosmic strings for points near kinks and cusps. Near a kink, the effect diverges as the inverse cube root of the distance to the kink and acts in a direction transverse to the world sheet. Over time, the kink is rounded off, but only regions fairly close to the kink are significantly affected. Near cusps, the effect diverges inverse linearly with the distance to the cusp and acts against the direction of the cusp motion. This results in a fractional loss of string energy that diverges logarithmically with the distance of closest approach to the cusp.

Coherent terahertz radiation from beam-driven upper hybrid wave in magnetostatic plasma

ABSTRACTA scheme for generating tunable terahertz radiation using relativistic electron beam-induced upper hybrid wave immersed in a magnetized plasma is investigated. The ponderomotive force applied to the beam electrons because of electromagnetic radiation wave and pump wave generates the non-linear current density in the high-frequency regime. For an appropriate pump wave frequency, under the condition of phase matching the THz radiation wave is obtained. The enhancement in the radiation power occurs along the axial direction in the vicinity of Cerenkov resonance. Our analytical expression shows that the growth rate in the unstable modes of the THz radiation wave increases with beam density (beam current) and the modulation index. In addition, the growth rate of the instability decreases with an increase in radiation wave frequency and scales as the inverse cube root of radiation frequency. An interesting result obtained from our model is that an increase in cyclotron frequency causes the growth decrement in THz radiation emission, while the efficiency of THz radiation emission increases with the growth rate and the modulation index. Also, the emission of the THz radiation wave becomes larger as the modulation index approaches unity.

Coefficient of thermal expansion and biaxial Young's modulus in Si-rich silicon nitride thin films

The coefficient of thermal expansion (α) and biaxial Young's modulus is determined by comparing the differential thermal stress induced in Si-rich silicon nitride thin films deposited on single-crystal Si and sapphire substrates. The amorphous films are deposited in mixtures of dichlorosilane and ammonia, by low pressure chemical vapor deposition, in a temperature range of 1050–1115 K. Temperature-dependent wafer curvature measurements are performed to determine the differential thermal stress, across a temperature range spanning 300–800 K. Observations indicate that both α and the biaxial modulus decrease as the silicon content in the films increases. The trend of reduction in α is consistent with the relative α values for the limiting-case compositions of cubic-Si3N4 and amorphous Si. The decrease in α is attributed to a reduction in anharmonicity associated with Si–Si bonds relative to Si–N bonds. The biaxial modulus is observed to be proportional to the inverse cube root of the amorphous Si volume fraction. This observation is shown to be consistent with volumetric dilation of basal Si-SiyN4-y tetrahedra (y = 0–4), when substoichiometric films are formed through the substitution of Si for N. Additionally, the intrinsic film stress is observed to vary in proportion to the square of the inverse cube root of the amorphous Si volume fraction, which is shown to be consistent with compositionally induced intrinsic strain variation in Si-rich silicon nitride thin films.

On the Size‐Determining Role of the Comonomer in the Nucleation and Growth of Cationic Polystyrene Latex via Emulsion Polymerization

AbstractThe formation of positively charged polystyrene latex particles via the surfactant‐free emulsion polymerization of styrene with cationic initiator is investigated. Using the comonomer (vinylbenzyl)trimethyl‐ammonium chloride (VBTMAC) it is shown that the particle size can be tuned rather precisely between above 350 nm and below 70 nm. The simple functional dependence of the particle size on the inverse cube root of the comonomer concentration leads to propose, with reasonable assumptions, that VBTMAC comonomer addition allows the separation of nucleation and growth for this reaction. Thereby the comonomer concentration determines the nucleation rate and number of particles produced, while the styrene amount determines by how much the nucleated particles grow. Support for this model is obtained through zeta potential measurements at different pH values. The latter confirmed that the surface density of cationic groups resulting from the comonomer relative to that of groups resulting from the initiator decreases with decreasing comonomer concentration.

Crater Formation on Electrodes during Charge Transfer with Aqueous Droplets or Solid Particles.

We report that metallic electrodes are physically pitted during charge transfer events with water droplets or other conductive objects moving in strong electric fields (>1 kV/cm). Post situ microscopic inspection of the electrode shows that an individual charge transfer event yields a crater approximately 1-3 μm wide, often with features similar to a splash corona. We interpret the crater formation in terms of localized melting of the electrode via resistive heating concurrent with dielectric breakdown through the surrounding insulating fluid. A scaling analysis indicates that the crater diameter scales as the inverse cube root of the melting point temperature T_{m} of the metal, in accord with measurements on several metals (660 °C≤T_{m}≤3414 °C). The process of crater formation provides a possible explanation for the longstanding difficulty in quantitatively corroborating Maxwell's prediction for the amount of charge acquired by spheres contacting a planar electrode.

Absolute ion hydration enthalpies and the role of volume within hydration thermodynamics

This paper reports that various thermodynamic properties in aqueous media for certain individual ions and for compounds are linear functions of the inverse cube root of the solid respective ionic and compound solid state volumes, Vm−1/3.

On Properties of the Inverse Cube Transformation of Error Component of the Multiplicative Time Series Model

This paper examines the inverse cube transformation of error component of multiplicative time series model. The probability density function (pdf) of the inverse cube root transformation of the multiplicative time series model was established, Further the was mathematically proved as a proper pdf since The Statistical properties (mean and variance) of the inverse cube transformation were equally shown.

Equi-flux streamline seeding for three-dimensional vector fields

Appropriate streamline placement is important in the visualisation of 3D vector fields and many streamline seeding algorithms exist that produce evenly spaced or feature-based streamlines. However, in many areas of applied science, such as 3D coil design for magnetic resonance imaging hardware or in meteorology, interest lies with obtaining streamlines that represent equal flux across the entire 3D vector field. In this article, an equi-flux streamline seeding method is presented in the context of a 3D volumetric current density. Streamline density is controlled by a density map that evolves with the placement of each streamline. A novel Gaussian filter is described as part of the process, which has amplitude equal to the updated density map maximum and variance related to the inverse cube root of this amplitude such that each streamline represents approximately fixed flux or integrated current. The method is verified by applying it to simple 1D, 2D and 3D examples for which exact results can be obtained for comparison. In addition, the method allows streamline count to be varied in a straightforward manner, and can be applied to complicated 3D vector fields for which no exact result can be obtained.

The role of glass-viscosity on the growth of semiconductor quantum dots in glass matrices

The dependence of glass viscosity on the growth of semiconductor (CdS) quantum dots in borosilicate glass matrices is studied systematically. The glass viscosity was varied (3000–260 000 Pa s) by varying the glass compositions of constituent oxides and growth temperatures. Thus, grown dot sizes at various growth times (2–48 h) were in the range 2–30 nm. The linear variation of dot size with cube root of growth time confirms the diffusion controlled growth. The growth of dot size was found to vary linearly with the inverse cube root of glass viscosity. The excitonic peak energy of the dots was found to vary linearly with inverse square of the dot size consistent with the quantum confinement effect. This analysis yields the bulk value of energy gap Eg (bulk) = 2.47 eV and the reduced effective masses as μ = 0.138 for CdS consistent with the values available in the literature. The emission (photoluminescence) spectra further support the absorption results.

Flow Effects on Mush Coarsening

Gradient annealing experiments of a near-eutectic AlCu30 alloy with artificial stirring induced by a rotating magnetic field (RMF) of 6 mT were performed. The specific surface area of the primary phase was measured on metallographic sections perpendicular to the sample axis with a fixed amount of fraction solid. The variation of the specific surface area with fluid flow is compared to flow free experiments: the specific surface area varies as the inverse cube root of annealing time if no RMF is applied, but varies as an inverse forth root at 6 mT. The experimental procedure and results are presented in detail and discussed with models of convective coarsening of dendrites.

Mushy zone coarsening with fluid flow

Cylindrical samples of a near-eutectic AlCu30 alloy are annealed applying constant axial temperature gradients to directionally molten samples in an aerogel furnace. During annealing with various times and gradients also a rotating magnetic field (RMF) of 6 mT was applied leading to azimuthal and meridional flows of well known magnitude. The specific surface area of the primary phase was measured on metallographic in section perpendicular to the sample axis with a fixed amount of fraction solid. The specific surface area varies as the inverse cube root of annealing time if no RMF is applied, but varies as an inverse forth root at 6 mT. The experimental procedure and results are presented in detail and compared to isothermal coarsening measurements of Voorhees and co-workers.

EXTENDED-SOURCE EFFECT AND CHROMATICITY IN TWO-POINT-MASS MICROLENSING

We explore the sensitivity of two-point-mass gravitational microlensing to the extended nature of the source star, as well as the related sensitivity to its limb darkening. We demonstrate that the sensitive region, usually considered to be limited to a source-diameter-wide band along the caustic, is strongly expanded near cusps, most prominently along their outer axis. In the case of multicomponent caustics, facing cusps may form a region with a non-negligible extended-source effect spanning the gap between them. We demonstrate that for smaller sources the size of the sensitive region extending from a cusp measured in units of source radii increases, scaling as the inverse cube root of the radius. We study the extent of different sensitivity contours and show that for a microlensed Galactic bulge giant the probability of encountering at least a 1% extended-source effect is higher than the probability of caustic crossing by 40-60% when averaged over a typical range of lens-component separations, with the actual value depending on the mass ratio of the components. We derive analytical expressions for the extended-source effect and chromaticity for a source positioned off the caustic. These formulae are more generally applicable to any gravitational lens with a sufficiently small source. Using exactly computed amplifications we test the often used linear-fold caustic approximation and show that it may lead to errors on the level of a few percent even in near-ideal caustic-crossing events. Finally, we discuss several interesting cases of observed binary and planetary microlensing events and point out the importance of our results for the measurement of stellar limb darkening from microlensing light curves.

Indentation size effect at the initiation of plasticity for ceramics and metals

In nanoindentation, the plasticity size effect has been observed for several years, where a higher hardness is measured as contact size decreases. For spherical indenters, Lim and Chaudhri (1999 Phil. Mag. 79 2979) first showed that the entire flow curve appears at higher contact pressures for smaller radius indenters in copper. However, few papers have reported the initial yield size effect due to the difficulty in defining the yield point. Recently, Spary et al (2006 Phil. Mag. 86 5581) demonstrated that the initial yield strength of metals increases linearly with inverse cube root of indenter radius, by nanoindentation together with finite elemental modelling. Here, we use a clear method to determine the onset of plasticity in spherical nanoindentation without the uncertainties of modelling. This enables us to measure the yield pressure of tungsten metal and a series of ceramics with a high degree of accuracy and over a large range of indenter radii (hundreds of nanometres to several tens of micrometres). Our data of all ceramics and metals show clearly that there is a significant yield strength enhancement, which is inversely proportional to the cube root of the indenter radius. Normalizing the data for each material by its yield pressure for an infinite radius indenter, we find that the data for metals and ceramics fall on lines of different slope, indicating that material parameters influence the indentation yield strength size effect as well as the geometrical size effect.

Study of the interaction between the indentation size effect and Hall–Petch effect with spherical indenters on annealed polycrystalline copper

Methods to obtain tensile stress–strain properties of materials from a practically non-destructive indentation test are of great industrial interest. Nanoindentation is a good candidate. However, to do this successfully, indentation size effects must be accounted for. An indentation size effect with spherical indenters has been shown for a range of fcc metals with relatively large grain size (Spary et al 2006 Phil. Mag. 86 5581–93); the increase in yield stress being proportional to the inverse cube root of indenter radius. Here, we investigate these differences further and present results for the indentation size effect with spherical indenters on Cu samples with a range of different grain sizes from 1 µm to single crystal. The important experimental control parameter, of the relative size of the indentation compared with the grain size, is also explored by using indenters of different radii on the different grain sized samples. When the grain size, d, is less than 6 times the radius of the projected contact area, a, a Hall–Petch-like behaviour is observed superimposed on the indentation size effect. For d > 6a the indentation size effect dominates. The two effects may be combined by addition in quadrature. This new parametric function is able to predict the indentation pressure in annealed copper given input values of indenter radius and grain size.

Size effect in the initiation of plasticity for ceramics in nanoindentation

The indentation size effect has been observed for many years and is usually associated with increasing hardness as the depth of indentation is reduced for pyramidal indenters. The indentation size effect for spherical indenters has recently been associated with an increase in the yield stress of metals proportional to the inverse cube root of indenter radius [Spary, I.J., Bushby, A.J., Jennett, N.M., 2006. On the indentation size effect in spherical indentation. Philos. Mag. 86 (33), 5581–5593]. Here we investigate ceramic materials where the yield point is high enough to be easily distinguished in nanoindentation tests. A robust method for determining the yield point from a nanoindentation test with spherical indenters is presented. The results for a range of ceramics confirm that the increase in yield pressure is directly proportional to the inverse cube root of indenter radius. Furthermore, the yield pressure is also shown to be proportional to the inverse square root of the contact radius. Revisiting data in the literature shows that this inverse square root relationship is also true for pyramidal indenters. This implies that the indentation size effect is driven by the contact area rather than by the depth of indentation or by the indenter radius.

Optimal third root asymptotic bounds in the statistical estimation of thresholds

This paper is concerned with estimating the intersection point of two densities, given a sample of both of the densities. This problem arises in classification theory. The main results provide lower bounds for the probability of the estimation errors to be large on a scale determined by the inverse cube root of the sample size. As corollaries, we obtain probabilistic bounds for the prediction error in a classification problem. The key to the proof is an entropy estimate. The lower bounds are based on bounds for general estimators, which are applicable in other contexts as well. Furthermore, we introduce a class of optimal estimators whose errors asymptotically meet the border permitted by the lower bounds.

Volume-Based Thermodynamics: Estimations for 2:2 Salts

The lattice energy of an ionic crystal, U(POT), can be expressed as a linear function of the inverse cube root of its formula unit volume (i.e., Vm(-1/3)); thus, U(POT) approximately 2I(alpha/Vm(1/3) + beta), where alpha and beta are fitted constants and I is the readily calculated ionic strength factor of the lattice. The standard entropy, S, is a linear function of Vm itself: S approximately kVm + c, with fitted constants k and c. The constants alpha and beta have previously been evaluated for salts with charge ratios of 1:1, 1:2, and 2:1 and for the general case q:p, while values of k and c applicable to ionic solids generally have earlier been reported. In this paper, we obtain alpha and beta, k and c, specifically for 2:2 salts (by studying the ionic oxides, sulfates, and carbonates), finding that U(POT)[MX 2:2]/(kJ mol(-1)) approximately 8(119/Vm(1/3) + 60) and S degree [MX 2:2]/(J K(-1) mol(-1)) approximately 1382V(m) + 16.