Use Of Analytical Expressions Research Articles

Field emission in ordered arrays of ZnO nanowires prepared by nanosphere lithography and extended Fowler-Nordheim analyses

A multistage chemical method based on nanosphere lithography was used to produce hexagonally patterned arrays of ZnO vertical nanowires, with 1 μm interspacing and aspect ratio ∼20, with a view to study the effects of emitter uniformity on the current emitted upon application of a dc voltage across a 250 μm vacuum gap. A new treatment, based on the use of analytical expressions for the image-potential correction functions, was applied to the linear region below 2000 V of the Fowler-Nordheim (FN) plot and showed the most suitable value of the work function φ in the range 3.3–4.5 eV (conduction band emission) with a Schottky lowering parameter y ∼ 0.72 and a field enhancement factor γ in the 700–1100 range. A modeled γ value of ∼200 was calculated for an emitter shape of a prolate ellipsoid of revolution and also including the effect of nanowire screening, in fair agreement with the experimental value. The Fowler-Nordheim current densities and effective emission areas were derived as 1011 Am−2 and 10−17 m2, respectively, showing that field emission likely takes place in an area of atomic dimensions at the tip of the emitter. Possible causes for the observed departure from linear FN plot behavior above 2000 V were discussed.

Preparation of chemotherapy drugs: Planning policy for reduced waiting times

This study investigates the impact of pharmacy policies on patient waiting time in the Chemotherapy Day Unit of the Netherlands Cancer Institute—Antoni van Leeuwenhoek hospital. The project evaluated whether a reduction in waiting time resulting from medication orders being prepared in advance of patient appointments was justified, given that medications prepared in advance are wasted when patients arrive too sick for treatment. Within this context, we derive analytic expressions to approximate patient waiting times and wastage costs, allowing management to see the tradeoff between these two metrics for different policies. Using a case study and a simulation model, the approximations are evaluated. The use of analytic expressions allows the analysis to be easily repeated when medication costs change or when new medications/protocols are introduced. In the same vein, other hospitals with different patient case mixes can easily complete the analysis in their settings. Finally, the outcome from this study resulted in a new policy at the cancer center which is expected to decrease the waiting time by half, while only increasing pharmacy's costs by 1–2%.

Analytical model of microfluidic transport of non-magnetic particles in ferrofluids under the influence of a permanent magnet

This study describes an analytical model and experimental verifications of transport of non-magnetic spherical microparticles in ferrofluids in a microfluidic system that consists of a microchannel and a permanent magnet. The permanent magnet produces a spatially non-uniform magnetic field that gives rise to a magnetic buoyancy force on particles within ferrofluid-filled microchannel. We obtained trajectories of particles in the microchannel by (1) calculating magnetic buoyancy force through the use of an analytical expression of magnetic field distributions and a nonlinear magnetization model of ferrofluids, (2) deriving governing equations of motion for particles through the use of analytical expressions of dominant magnetic buoyancy and hydrodynamic viscous drag forces, (3) solving equations of motion for particles in laminar flow conditions. We studied effects of particle size and flow rate in the microchannel on the trajectories of particles. The analysis indicated that particles were increasingly deflected in the direction that was perpendicular to the flow when size of particles increased, or when flow rate in the microchannel decreased. We also studied “wall effect” on the trajectories of particles in the microchannel when surfaces of particles were in contact with channel wall. Experimentally obtained trajectories of particles were used to confirm the validity of our analytical results. We believe this study forms the theoretical foundation for size-based particle (both synthetic and biological) separation in ferrofluids in a microfluidic device. The simplicity and versatility of our analytical model make it useful for quick optimizations of future separation devices as the model takes into account important design parameters including particle size, property of ferrofluids, magnetic field distribution, dimension of microchannel, and fluid flow rate.

A Method for Coil Design of Superconducting Quadrupoles Based on Sector Coils and Fourier Series

In this paper we present a novel method to optimize quadrupole coil cross-sections based on the cos 2?-type. We first approximate real coil blocks made of cables with concentric sectors powered with a constant current density to make use of analytical expressions of the magnetic field based on the Fourier series development. This allows a rapid evaluation of the field harmonics which permits an almost exhaustive scan on sectors position and dimensions for coil layouts made of up to five sectors. The coil layouts having a larger short-sample gradient can be further explored, approximating them with blocks of Rutherford cables. This second step requires a second fine tuning of the cross-section around the previously found optimum. The method has been used to design a proposed coil cross-section for the new inner triplet quadrupole for the LHC.

A Semianalytic Model for the Productivity Testing of Multiple Wells

SummaryWe present a semianalytic method for modeling the productivity testing of vertical, horizontal, slanted, or multilateral wells. The method is applicable to both oil and gas reservoirs and automatically accounts for well interference. The use of analytic expressions ensures that short-time transient behavior and long-time semisteady-state behavior are handled appropriately, whether close to the well or further into the reservoir. Calculation times are still very limited—on the order of a few minutes to a few seconds when all wells are vertical. This makes the tool suitable for evaluating well testing and determining well productivity.We based the approach on an earlier derived productivity prediction tool, in which the steady-state equations were solved. It has now been extended to solve the time-dependent diffusion equation. In our current method, the equations have first been transformed using the Laplace transformation. The expressions for the producing wells are combined with auxiliary sources outside the reservoir. The crux of the semianalytic method involves an adjustment of the positions and strengths of these sources in order to approximate the boundary conditions at the reservoir boundaries. The solution obtained is transformed back into the time domain by use of a Stehfest algorithm.The new approach has been validated with numeric tools, including both reservoir simulators and well-test interpretation software. Validations were performed with artificial cases and with field production data, using both single-well and multiple-well production tests. The results of these tests were excellent.

Accurate Evaluation of the Cubic Lattice Green Functions Using Binomial Expansion Theorems

A new, simple, and efficient technique for the accurate evaluation of the lattice Green functions is presented. Using binomial expansion theorems, these functions are expressed through the binomial coefficients and basic integrals. The extensive test calculations show that the proposed algorithm in this work is the most efficient method in practical computations. Finally, in order to show the practical use of analytical expressions found some computation examples and comparisons with literature are made.

Kinetics of hydrogen abstraction O( 3P) + alkane → OH + alkyl reaction class: An application of the reaction class transition state theory

This paper presents an application of the reaction class transition state theory (RC-TST) for prediction of thermal rate constants of the O( 3P) + alkane → OH + alkyl reaction class. Parameters of the RC-TST were derived from first principles from a set of 19 reactions representing hydrogen abstractions from primary, secondary, and tertiary carbon atoms so that rate constants of any reaction in this class can be estimated without any further information or with its reaction energy calculated either at the density functional theory BH&HLYP/cc-pVDZ level or the semiempirical AM1 level. Detailed error analyses show that when compared to explicit theoretical calculations, the systematic errors in the calculated rate constants arising from the use of analytical expressions to approximate different reaction class factors in the RC-TST method are less than 40% on the average over the temperature range from 300 to 3000 K. In addition, we found that the rate constants estimated using either approach are in good agreement with available data in the literature.

On the Use of Analytical Expressions for the Voltage Distribution to Analyze Intracellular Recordings

Different analytical expressions for the membrane potential distribution of membranes subject to synaptic noise have been proposed and can be very helpful in analyzing experimental data. However, all of these expressions are either approximations or limit cases, and it is not clear how they compare and which expression should be used in a given situation. In this note, we provide a comparison of the different approximations available, with an aim of delineating which expression is most suitable for analyzing experimental data.

Towards increased expressiveness in service level agreements

AbstractThe aim of this paper is to argue for the benefits of increased expressiveness in service level agreements (SLAs). Such benefits may be obtained from the use of analytical expressions to specify a SLA's agreement terms as functions and not as variable or constant values or ranges. The main idea behind this thinking is that functions may contain variables defined in the SLA or be drawn from the known set of reference variables, such as wall‐clock time, job start time, current bandwidth of the resource, etc. Experiences and conclusions drawn are in the context of SLA‐based job management systems. We demonstrate that the use of analytical expressions in SLAs can potentially reduce the overheads associated with job renegotiation and/or reduce the number of failed agreements. Copyright © 2006 John Wiley & Sons, Ltd.

A new deformable catenary element for the analysis of cable net structures

The formulation of a deformable catenary element permitting the simulation of nets systems composed of multiple cables is presented. The formulation proposed, stemming as a modification of the conventional equations for inextensible cables, ensures exact equilibrium after the deformation of the cable. The use of analytical expressions for both the geometrical description of the element and the corresponding tangent stiffness provides high numerical stability to the calculation procedures. The accuracy and efficiency of the formulation are assessed by comparison with available results obtained by other authors using different analytical or numerical approaches.

Differential geometry of normalized Stokes vector trajectories in anisotropic media

Trajectory of the normalized Stokes vector on the Poincaré sphere corresponding to light propagation in anisotropic tissues with birefringence and biattenuance is derived. Analytic expressions are determined from the Serret-Frenet formulas and derivatives of arc length for five quantities including the tangent, normal, and binormal vectors with curvature and torsion. Depth variation of curvature and torsion of normalized Stokes vector trajectories corresponding to light propagating in rodent tail tendon are given. Use of analytic expressions for depth variation of curvature and torsion of the normalized Stokes vector trajectories on the Poincaré sphere is discussed for analysis of polarization-sensitive optical coherence tomography data recorded from anisotropic biological tissues with birefringence and biattenuance.

Link between enhanced carrier mobilities and DOS profiles in a-Si:H

High carrier mobilities have been reported in a-Si:H samples prepared at high deposition rate and high substrate temperature in an expanding thermal plasma (ETP). Those mobilities were found to be field independent, and subsequent analysis showed that tail-state distributions with a steeper than exponential decrease are required to account for this field independence. Such steeper profile implies an increased ratio of shallow to deep states and, given that the shallow states dominate the carrier mobility, this fact might explain the observed enhanced mobility. Through the use of analytical expressions for the time-of-flight transients, the circumstances whereby the high, field-independent mobility can be achieved are explored. It is found that a prominent feature close to the mobility edge is required to account for the observations.

Displacement and stress Green's functions for a constant slip-rate on a quadrantal fault

SUMMARY I present, in a rigorous form, the Green’s functions that represent the dynamic displacement response of an infinite, homogeneous and isotropic medium to a constant slip rate on a quadrantal fault that continues perpetually after the slip onset. I also present analytical expressions for the corresponding stress Green’s functions that are a major simplification over previously known solutions. The equations derived can be utilized in the formulation of boundary element methods that discretize a fault plane into a set of small rectangular elements and assume the slip rate to be piecewise constant within each discrete element. 1I N T R O DUCTION In the boundary element method (BEM) modelling of the elastodynamic field created by slip on a fault in a three-dimensional (3-D) medium, the fault plane is often discretized into a set of small rectangular elements, on each of which the slip is assumed to follow certain types of spatio-temporal variation profiles. One of the simplest and most popular categories of such assumption is that of the so-called piecewise constant slip rate, which states that the slip rate takes a constant value everywhere and at every moment within a given rectangular element and within a given discrete time window. The elastodynamic field in the medium is calculated by convolving the discrete spatio-temporal profile of the fault slip with Green’s functions corresponding to a constant slip-rate that takes place on each discrete fault element and within each discrete time window. The knowledge of these Green’s functions therefore makes an indispensable part of the BEM modelling of fault dynamics. In mathematical terms, a constant slip-rate profile localized on a rectangular fault element and within a finite time window is equivalent to a superposition, with appropriate offsets in space and time and with appropriate sign reversals, of eight identical profiles of slip rate that takes a constant value everywhere on a quadrantal part of the fault and at every moment after the onset of slip (Fukuyama & Madariaga 1998). The question of calculating Green’s functions for a constant slip rate on a rectangular space‐time element thus reduces to the question of calculating the Green’s functions for a constant slip rate on a quadrantal fault that goes on perpetually after the onset of slip. The stress Green’s functions for such a quadrantal fault model placed in an infinite, homogeneous and isotropic medium were obtained analytically by Aochi (1999) and Aochi et al. (2000a) and were fully utilized in their numerical simulation of fault dynamics. The analytical expressions for the displacement Green’s functions have remained unknown, however, because of the complexity of the algebra involved. In the present study I have derived, in a rigorous form, Green’s functions that represent the displacement response of an infinite, homogeneous and isotropic medium to a constant slip rate on a quadrantal fault that continues perpetually after the slip onset. Although these Green’s functions can be evaluated numerically, the use of analytical expressions is highly beneficial from the viewpoint of computational efficiency. I have double-checked my equations by deriving them with two different methods and confirming their consistency with a corresponding two-dimensional (2-D) theory. I have also been able to substantially simplify the expressions for the stress Green’s functions over those obtained by Aochi (1999) and Aochi et al. (2000a).

Hydration of the bentonite buffer in a KBS-3 repository

The present study addresses the problem of bentonite hydration in a KBS-3 type repository. A finite element program, Code_Bright, that has been specifically developed to analyze coupled Thermo-Hydro-Mechanical (THM) processes in unsaturated porous media, is used to estimate the time required to reach full saturation of the bentonite buffer. The finite element model analyzed here concerns a horizontal slice of the annular bentonite-filled space between canister and rock wall at canister midheight and considers T-H aspects of the bentonite material behavior. Processes included in the model are transport of heat and moisture. Moisture movements caused by suction gradients as well as by thermally induced vapor mass fraction gradients are considered. All these T-H transport processes are coupled and highly nonlinear. In particular, the thermally driven vapor transport is sensitive to temperature gradients and to temperature levels. Therefore, an important issue is to determine relevant thermal boundary conditions for the FEM model. This is done here by use of analytical expressions which take the initial canister power, the fuel power decay characteristics, the repository layout and the rock thermal properties into account. Results are presented that show the buffer saturation as function of distance to the canister surface at different times. Effects of incomplete saturation on the temperature development are analyzed for a number of assumptions regarding bentonite properties. In particular are conditions that may lead to delayed saturation, such as low permeability, high vapor diffusivity, etc., analyzed. For the reference KBS-3 repository, a general result is that the buffer filling out the annular space between canister and rock wall will be fully water saturated after about 3 or 4 years, provided that there is access to water at the buffer/rock interface of the individual deposition holes. This general result seems to apply for a wide range of assumptions regarding values of transport parameters, gas confinement, porosity, canister power etc. There are, however, a number of uncertainties in the material model. Some of these are either unimportant or can be handled conservatively, while others, such as the validity of the saturation/suction relations are concluded to be important and will need further attention in the future. A general conclusion is that the saturation time scale may turn out to depend more on hydrological conditions around individual deposition holes, on the hydrological buffer/rock interaction and on the repository-scale restoration of the ground water pressure after repository closure, than on details of the bentonite material model.

Radiative transfer codes applied to hyperspectral data for the retrieval of surface reflectance

The present investigation evaluates surface reflectance retrieved from Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) and Compact Airborne Spectrographic Imager ( casi™) data using the atmospheric radiative transfer (RT) codes: atmospheric removal program (ATREM), Canadian advanced modified 5S (CAM5S) and moderate atmospheric radiance and transmittance model (MODTRAN4). The retrieved surface reflectances were compared with ground-based reflectances acquired with a GER3700™ spectroradiometer for a playa and canola target. The results showed that the best overall performance was achieved with MODTRAN4 (average relative error of 2.3%), followed by ATREM (3.6%) and CAM5S (4.2%). Major differences occur in the stronger gas absorption regions. At wavelengths unaffected by strong gaseous absorption, the performance was similar for the three RT codes even though ATREM and CAM5S make extensive use of analytical expressions and, therefore, have faster execution times.

New particle transport diagnostics with tracer-encapsulated solid pellet

A new diagnostic method for local particle transport, which is based oninjection of the tracer-encapsulated solid pellet (TESPEL), was appliedfor the first time in experiments on the CHS device. Such aconfiguration of the pellet allows the lithium hydride tracer to reachthe core plasma region and be deposited within a few centimetres in theradial direction, which was confirmed by measurements withphotomultipliers and CCD imaging. The radial diffusion of the fullyionized tracer is observed by means of charge exchange recombinationspectroscopy with the heating neutral beam as a source. The local tracerdeposition and complete ionization of the tracer greatly simplify thetransport analysis and allow the use of analytic expressions forderiving the diffusion coefficient, D. With this procedure, thediffusion coefficient was determined for various plasma conditions andwas found to be larger for discharges with higher electron temperature.An impurity transport code was also applied to the experimental data,which allowed more precise calculation of the transport coefficientsincluding the pinch velocity, V. It is expected that a higher accuracywill be achieved for the case of TESPEL injection into a larger-scaleplasma.

Power series expansion of the roots of a secular equation containing symbolic elements: Computer algebra and Moseley’s law

We use computer algebra to expand the Pekeris secular determinant for two-electron atoms symbolically, to produce an explicit polynomial in the energy parameter ε, with coefficients that are polynomials in the nuclear charge Z. Repeated differentiation of the polynomial, followed by a simple transformation, gives a series for ε in decreasing powers of Z. The leading term is linear, consistent with well-known behavior that corresponds to the approximate quadratic dependence of ionization potential on atomic number (Moseley’s law). Evaluating the 12-term series for individual Z gives the roots to a precision of 10 or more digits for Z⩾2. This suggests the use of similar tactics to construct formulas for roots vs atomic, molecular, and variational parameters in other eigenvalue problems, in accordance with the general objectives of gradient theory. Matrix elements can be represented by symbols in the secular determinants, enabling the use of analytical expressions for the molecular integrals in the differentiation of the explicit polynomials. The mathematical and computational techniques include modular arithmetic to handle matrix and polynomial operations, and unrestricted precision arithmetic to overcome severe digital erosion. These are likely to find many further applications in computational chemistry.

Mean stress effects in fatigue of welded steel joints

Mean stress effects in steel weldments were examined under both constant and random narrowband amplitude fatigue loadings. The purpose of these tests was to provide experimental data with which to substantiate the use of analytical expressions to account for mean stress effects. Fatigue tests were performed under both tensile and compressive mean stress levels. Test results indicate agreement with the modified Goodman equation to be favorable in accounting for the effect of tensile mean stresses on fatigue life. However, test results from high fatigue loadings (maximum stresses nominally above half ultimate) were found to possess better agreement with the Gerber formulation than with the modified Goodman one. Behavior under compressive mean stresses indicated a linear correction relationship was required, which was less conservative than any of the relationships considered. Test results obtained under random amplitude fatigue loadings exhibited trends similar to those observed under constant amplitude loadings. This finding, along with supporting analysis, indicates that the same correction relationship can be used in the same manner for both constant amplitude and random (narrowband) amplitude loadings.

Analysis of aperture-coupled microstrip antenna using the segmentation method

The authors present an original analysis of an aperture-coupled microstrip antenna. The theory is based on the segmentation method, which considers the patch as a multiport network whose impedance matrix is deduced from a hybrid matrix, and the use of analytical expressions of the cavity admittance at the slot centre. The theory is presented for the firdt time, and the theoretical results are in good agreement with previous published measurements.

Trace element standards for the electron microanalyser using layered and ion-implanted materials

The merits and limitations of thin surface film deposition and ion implantation for preparing reference materials for quantitative analysis of trace elements using wavelength-dispersive X-ray spectrometry with the electron probe microanalyser are illustrated and discussed. The objective is to generate X-ray intensities typical of elements present at trace levels and to relate the measured intensities to the mass concentration of these elements when assumed to be homogeneously distributed within bulk materials. For this purpose, X-ray intensity calculations based on the use of analytical expressions for both the depth–ionization and the depth–concentration distributions, respectively, are presented. It will be demonstrated that gold-layered and materials ion implanted with gold can be advantageously used to validate the experimental approaches used for X-ray intensity measurements in the presence of distortions in the shape of an X-ray peak and the underlying continuous emission distribution, and also to test the algorithms and the physical parameters used in the analytical descriptions of the ionization and concentration profiles as a function of depth. The contribution of layered and ion-implanted compounds to a better understanding of surface and bulk charging phenomena in strongly insulating specimens is also presented.