Simple Analytical Equations Research Articles

Relaxation of the magnetization in uniaxial single-domain ferromagnetic particles driven by a strong ac magnetic field

The magnetization spectrum and reversal time of uniaxial single-domain ferromagnetic particles driven by an ac magnetic field of arbitrary strength applied along the easy axis are evaluated by solving the infinite hierarchy of recurrence equations for the Fourier components of the relaxation functions governing the frequency dependence of the magnetization. A simple analytic equation for the reversal time is also obtained via the mean first passage time method exhibiting pronounced nonlinear effects such as a rapid decrease in the reversal time as the strength of the ac field increases. For weak ac fields, the results agree with perturbation theory.

An objective analysis of the dynamic nature of field capacity

Field capacity is one of the most commonly used, and yet poorly defined, soil hydraulic properties. Traditionally, field capacity has been defined as the amount of soil moisture after excess water has drained away and the rate of downward movement has materially decreased. Unfortunately, this qualitative definition does not lend itself to an unambiguous quantitative approach for estimation. Because of the vagueness in defining what constitutes “drainage of excess water” from a soil, the estimation of field capacity has often been based upon empirical guidelines. These empirical guidelines are either time, pressure, or flux based. In this paper, we developed a numerical approach to estimate field capacity using a flux‐based definition. The resulting approach was implemented on the soil parameter data set used by Schaap et al. (2001), and the estimated field capacity was compared to traditional definitions of field capacity. The developed modeling approach was implemented using the HYDRUS‐1D software with the capability of simultaneously estimating field capacity for multiple soils with soil hydraulic parameter data. The Richards equation was used in conjunction with the van Genuchten‐Mualem model to simulate variably saturated flow in a soil. Using the modeling approach to estimate field capacity also resulted in additional information such as (1) the pressure head, at which field capacity is attained, and (2) the drainage time needed to reach field capacity from saturated conditions under nonevaporative conditions. We analyzed the applicability of the modeling‐based approach to estimate field capacity on real‐world soils data. We also used the developed method to create contour diagrams showing the variation of field capacity with texture. It was found that using benchmark pressure heads to estimate field capacity from the retention curve leads to inaccurate results. Finally, a simple analytical equation was developed to predict field capacity from soil hydraulic parameter information. The analytical equation was found to be effective in its ability to predict field capacities.

Modelling and characterization of thin film planar capacitors: inherent errors and limits of applicability of partial capacitance methods

The analytical partial capacitance methods (PCM) widely accepted for calculation of properties of capacitors with planar electrodes, coplanar strip waveguides (CPS) and coplanar waveguides (CPW) are reviewed based on the challenges met during the development and tailoring of (Ba, Sr)TiO3 thin films fabricated on different types of substrates. An alternative view on the conformal mapping method is given, a correction for electrodes of finite thickness is introduced and the applicability of easy-to-use simplified analytical equations is reviewed and extended. Calculation results obtained with different models are compared against the results of FEM numerical simulation in the parameter range relevant for development of tuneable ferroelectric films. The models were found to be less accurate than previously reported. Accordingly, guidelines for extending the applicability domain and for minimizing the inherent errors of the analytical models are presented. The discussion is focused on the possible improvements of the measurement techniques.

Radiative properties of optically thick fluorescent turbid media

In this paper simple analytical equations for the reflection and transmission coefficients of fluorescent turbid media are given. The case of weakly absorbing optically thick media is considered (e.g., paper, textiles, tissues). The calculations are performed in the framework of the two-flux approximation for finite layers under monochromatic illumination conditions. The relationships of Kubelka-Munk parameters to the true absorption and transport extinction coefficients of fluorescent turbid media are derived. The results can be used for the development of various optimization procedures in the paper and textile industries and also in the area of fluorescence spectroscopy of turbid media.

Radiative properties of optically thick fluorescent turbid media

In this paper simple analytical equations for the reflection and transmission coefficients of fluorescent turbid media are given. The case of weakly absorbing optically thick media is considered (e.g., paper, textiles, tissues). The calculations are performed in the framework of the two-flux approximation for finite layers under monochromatic illumination conditions. The relationships of Kubelka-Munk parameters to the true absorption and transport extinction coefficients of fluorescent turbid media are derived. The results can be used for the development of various optimization procedures in the paper and textile industries and also in the area of fluorescence spectroscopy of turbid media.

Interfacial Metal Flux in Ligand Mixtures. 1. The Revisited Reaction Layer Approximation: Theory and Examples of Applications

Understanding the physical chemical behaviors of each metal species in a solution containing a mixture of ligands is a prerequisite, e.g., for studying metal bioavailability or making predictions on dynamic risk assessment in ecotoxicology. For many years, the reaction layer concept has been used fruitfully due to its simplicity for understanding and making predictions on diffusion/reaction processes. Until now, it has been applied mainly to solutions containing one ligand. Here, we reconsider the fundamentals of this approach and extend it to multiligand systems. It is shown that each metal complex has its own reaction layer (so-called composite reaction layer), which results from the interplay of this particular complex with all the other complexes. Moreover, it is shown that the overall metal flux can be computed by assuming the existence of one single fictitious equivalent reaction layer thickness for the whole of the complexes. This equivalent reaction layer is a mathematical combination of all the composite reaction layers. Simple analytical equations are obtained, which make it possible to readily interpret the role of the various types of metal species in a mixture. The revisited reaction layer approach, denoted as the reaction layer approximation (RLA), is validated by comparing the total metal flux, the individual fluxes of each metal species, and their concentration profiles computed by the RLA with those obtained by a rigorous mathematical approach. The examples of Pb(II) in a modified Aquil medium and of Cu(II) in solutions of nitrilotriacetic acid and N-(2-carboxyphenyl)glycine are treated in detail. In particular, an original result is obtained with the Cu/NTA/N-(2-carboxyphenyl)glycine system, namely an unexpected flux enhancement is observed, which is specific to solutions with ligand mixtures. The corresponding physicochemical mechanism is not readily understood by the rigorous mathematical (either numerical or analytical) solutions due to their involved combination of parameters. On the other hand, we show that, due to the simplicity of the RLA concept, the RLA facilitates elucidation of the physicochemical mechanism underlying complicated processes.

Analytical characterization of RF phase‐cycled balanced steady‐state free precession

AbstractBalanced steady‐state free precession (bSSFP) techniques, commonly called True‐FISP, FIESTA, or balanced FFE, are rapid, efficient, high signal‐to‐noise ratio sequences used in angiographic, cardiac and abdominal MR imaging. One of their major drawbacks is their sensitivity to off‐resonance effects arising from Bo inhomogeneities, which can produce the familiar dark bands in the images. Radio frequency (RF) phase cycling is known to minimize this artifact. Although RF phase‐cycled bSSFP is an accepted and firmly established technique, a simple and intuitive analytical signal equation has not yet been provided. Here, a complete theoretical framework is developed from which insightful characterizations such as signal uniformity, optimal flip angle, SNR efficiency, and banding effect minimization are provided. The analytical predictions and guidelines are substantiated via direct numerical simulations for clinically relevant tissues (e.g., water‐like organs and lipids) using various acquisition conditions, and also verified experimentally with phantoms. We demonstrate that the banding artifact can be virtually eliminated, even for phantoms immersed in significantly nonuniform magnetic fields. © 2009 Wiley Periodicals, Inc. Concepts Magn Reson Part A 34A: 133–143, 2009.

Water sorption on composite “silica modified by calcium nitrate”

Composite SWS-8L was synthesized by impregnating a mesoporous silica KSK with a saturated aqueous solution of Ca(NO 3) 2. Its equilibrium with water vapor was studied by measuring sorption isobars and isosters at T = 30–150 °C. The water sorption increased up to 0.2–0.3 g/g due to the presence of the salt inside the silica pores. The water uptake was not a simple addition of those by the bulk salt and the silica, because the confined salt changed its properties. The isosteric sorption heat decreased from 52 to 47 kJ/mol when the water uptake increased from 0.07 to 0.23 g/g. The X-ray diffraction study revealed no formation of crystalline dihydrate in the confined state during the hydration process that can be caused by the dihydrate dissolution inside the pores. For approximation of the sorption equilibrium, a simple analytical equation based on the Polanyi principle of temperature invariance was suggested. Sorption kinetics was studied by the Isothermal Differential Step method at T = 50 and 70 °C and the pressure range 10–35 mbar. Confinement of calcium nitrate to the silica pores significantly hastened the sorption with respect to the bulk salt. The hydration of the confined salt was limited by the intraparticle water diffusion instead of intrinsic chemical transformation of the bulk salt. Comparison of the new composite adsorbent with those based on CaCl 2, MgSO 4 and LiBr confined to the same host matrix showed that SWS-8L can be envisaging further development for adsorptive air conditioning driven by low temperature heat.

Influence of optical interference and carrier lifetime on the short circuit current density of organic bulk heterojunction solar cells

Based on simple analytical equations, short circuit current density (JSC) of the organic bulk heterojunction solar cells has been calculated. It is found that the optical interference effect plays a very important role in the determination of JSC; and obvious oscillatory behaviour of JSC was observed as a function of thickness. At the same time, the influence of the carrier lifetime on JSC also cannot be neglected. When the carrier lifetime is relatively short, JSC only increases at the initial stage and then decreases rapidly with the increase of active layer thickness. However, for a relatively long carrier lifetime, the exciton dissociation probability must be considered, and JSC behaves wave-like with the increase of active layer thickness. The validity of this model is confirmed by the experimental results.

Destructive Testing and Finite Element Analysis to Determine Ultimate Capacity of Skewed Steel I-Girder Bridges

Current bridge design and rating techniques are based at the component level and thus cannot account for the increase in ultimate capacity that bridges experience because of system-level interactions. Compared with that of a normal bridge, the ultimate capacity of a bridge increases to an even greater extent as bridge supports are skewed because of changes in load paths. Although advances in computer technology have made it possible to conduct accurate system-level analyses, which would allow for more efficient bridge design and rating, the knowledge base surrounding system-level bridge behavior is still too small to make it a highly accurate or intuitive tool. To advance system-level design and rating, two studies were undertaken. First, to evaluate the ultimate capacity of a skewed simple-span steel bridge, a ⅕-scale, slab-on-steel girder bridge was tested to ultimate capacity and then modeled by using finite element analysis. This test provided both insight into the system behavior and validation of an ABAQUS Explicit analysis algorithm and associated modeling techniques. Second, to investigate the effects of skew on the ultimate capacity of simple-span bridges, a parametric study was conducted with finite element analysis. A four-girder, simple-span bridge was modeled with skews varying from 0 to 75 degrees. The results showed that the ultimate capacity of the bridge increased with skew; these results were compared with simple analytical equations to provide insight into the fundamental behavior and load distribution characteristics of skewed bridges.

WATER, O2, AND ICE IN MOLECULAR CLOUDS

We model the temperature and chemical structure of molecular clouds as a function of depth into the cloud, assuming a cloud of constant density n illuminated by an external FUV (6 eV < E < 13.6 eV) flux G_0 (scaling factor in multiples of the local interstellar field). Extending previous photodissociation region models, we include the freezing of species, simple grain surface chemistry, and desorption (including FUV photodesorption) of ices. We also treat the opaque cloud interior with time-dependent chemistry. Here, under certain conditions, gas phase elemental oxygen freezes out as water ice and the elemental C/O abundance ratio can exceed unity, leading to complex carbon chemistry. Gas phase H2O and O2 peak in abundance at intermediate depth into the cloud, roughly A_V~3-8 from the surface, the depth proportional to ln(G_0/n). Closer to the surface, molecules are photodissociated. Deeper into the cloud, molecules freeze to grain surfaces. At intermediate depths photodissociation rates are attenuated by dust extinction, but photodesorption prevents total freezeout. For G_0 < 500, abundances of H2O and O2 peak at values ~10^(-7), producing columns ~10^(15) per cm^2, independent of G_0 and n. The peak abundances depend primarily on the product of the photodesorption yield of water ice and the grain surface area per H nucleus. At higher values of G_0, thermal desorption of O atoms from grains enhances the gas phase H2O peak abundance and column slightly, whereas the gas phase O2 peak abundance rises to ~10^(-5) and the column to ~2x10^(16) per cm^2. We present simple analytic equations for the abundances as a function of depth which clarify the dependence on parameters. The models are applied to observations of H2O, O2, and water ice in a number of sources, including B68, NGC 2024, and Rho Oph.

A Simple and Accurate Model for RFID Rectifier

In this paper, a simple model for the UHF low power rectifier circuit is proposed. Using a novel approach to model the rectifier current waveform, simple analytical equations are derived. The output DC voltage and the efficiency of the rectifier are derived analytically. Simulation results of the rectifier using actual models are very close to those predicted by the proposed model. The derived formulas for the output DC voltage and the efficiency are simple and physically meaningful and can be used to optimize the performance of the rectifier.

IGSHIELD: A new interactive point kernel gamma ray shielding code

IGSHIELD is an interactive gamma ray shielding code (stand alone) developed in Visual Basic Version 6.0 for windows operating system. The computational methodology is based on the point kernel technique employed in QADCGPIC (and hence QADCGGP) code. In IGSHIELD, all the features of QADCGPIC code written in FORTRAN and its user interface GUI2QAD-3D written in visual basic have been incorporated and several additional improvements have been made. The new improvements made in the code are (i) handling of multiple source shapes with accompanying energy distributions, (ii) incorporation of hexagonal cylinder as a shield body in the combinatorial geometry routine, (iii) provision to declare hexagonal cylinder and truncated right circular cone as source bodies, (iv) provision to declare point, line and plane as source bodies and (v) arbitrary orientation of sources such as cylinder, hexagonal cylinder and truncated cone. The newly introduced source bodies will be of immense use for calculations involving hexagonal shaped cylindrical sub-assemblies of fast reactors and cone shaped radioactive plume release from stack. The source array option will be of use for calculations involving stacked radioactive drums in vast storage areas. The code is validated by comparing the results with (i) simple source-shield configurations that can be solved using analytical equations and MCNP code, (ii) a point source in an infinite shield medium configuration that is solved using simple analytical equation and (iii) a standard problem in which an irradiated fuel is surrounded by various shielding materials and the problem is solved using MCNP. Several sample problems encountered in nuclear industry and other useful information required for shield designers and analysts are the other highlights of the code. The input for any kind of situations can be prepared easily using the forms that flash with appropriate prompts. Since entry of data is the important part of the input, its validity is checked immediately and the user is cautioned against incorrect data. The input module accepts only the correct data. If the data happens to be an object, the code instantaneously display the 3D view of all the objects built so far. Thus, the Interactive Gamma Ray Shielding Code will cater the needs of a shield designer as well as serve as an excellent educational tool.

Adsorption properties of synthetic piceatannol

The adsorption isotherms of synthetic piceatannol (trans-3,5,3′,4′-tetrahydroxystilbene) from n-hexane-ethyl acetate at nanodispersed silicon, titanium, and aluminum oxides were measured under static conditions. The adsorption properties of piceatannol and other hydroxystilbenes were compared with those of polyphenols. It was found that piceatannol adsorbs at titanium oxide better than at alumina and silica and almost as well as phloroglucinol adsorbs at these oxides. The adsorption isotherms can be satisfactorily approximated by the Freundlich equation for the model of localized adsorption at energetically heterogeneous sites. The influence of the nature and concentration of the polar components of a binary organic solvent on the adsorption of piceatannol at silica gel was studied by normal-phase chromatography and UV spectroscopy. The dependences of the retention factors of piceatannol and other hydroxystilbenes on the solvent composition were described by a simple analytical equation. The optimum composition of the mobile phase for separation of piceatannol and other stilbenoids was found.

Extension to Multiple Coupled Circuit Modeling of Induction Machines to Include Variable Degrees of Saturation Effects

Multiple coupled circuit models (MCCM) of cage induction machines (CIMs), also called the winding function approach, is the most detailed and complete model used to analyze the performance of CIM especially under various faults. However, the inclusion of saturation effects to this model is still in the beginning stages. This paper extends the model to include variable degrees of saturation effects using an air gap function which incorporates saturation factor as a parameter. Simple exact analytic equations have been obtained to update time varying inductances used in the model and their derivatives versus rotor position. This eliminates the need for huge lookup tables, inexact analytic equations, and numerical differentiations which are common in MCCM. Also, this facilitates the introduction of the saturation factor to the model as a continuous variable. A novel technique has been introduced to locate the angular position of the air gap flux density, which is required with the air gap function. Results obtained from the simulation of a general purpose CIM is verified by the results obtained from the experiments and this demonstrates the efficiency of the extended model.

Modelling of time delay of electrical breakdown for nitrogen-filled tubes at pressures of 6.6 and 13.3 mbar in the increase region of the memory curve

Experimentally measured electrical breakdown time delay data versus the afterglow period (representing ‘memory curves’) for nitrogen-filled tubes at pressures of 6.6 and 13.3 mbar have been shown. The influence of N(4S) nitrogen atoms on secondary electron emission from the cathode (the SEE process) in late afterglow has been discussed. N(4S) atom concentration decay over relaxation time τ, N(4S)(τ), has been analysed by a numerical model and two analytical models. N(4S) decay analytical models are combined with different yield models that describe the SEE process by N(4S) and these combinations are employed to fit the experimental data. It has been shown that in late afterglow solving of very simple analytical equations instead of numerical solving of partial differential equations for N(4S)(τ) fitting can be used and that the combination of the first and the second order of the SEE process by N(4S) in yield modelling should be used in the case of 13.3 mbar pressure.

The influence of product decomposition on the probability of nonthermal transitions in charge transfer reactions

The paper considers charge transfer under substantially nonequilibrium conditions. A model taking into account the excitation of the high-frequency intramolecular vibrational mode of the products along with medium reorganization is suggested. A simple analytic equation for the probability of a nonthermal transition taking into account product vibrational relaxation was obtained within the framework of the stochastic approach. The fast relaxation of high-frequency vibrational modes was shown to be capable of substantially accelerating the reaction if the slope of the term of the products was smaller than or comparable with the slope of the term of the reagents in the vicinity of the intersection point.

Optimized nonadiabatic nanofocusing of plasmons by tapered metal rods

Using rigorous numerical methods of analysis, this paper investigates nonadiabatic nanofocusing in tapered nanorods with the major emphasis on structural optimization for achieving maximal possible local field enhancement. Simple analytical equations for the determination of the optimal length of the tapered rod are presented and discussed. It is also shown that for the considered structures, optimal taper angle and optimal length of the rod only very weakly depend on the radius of curvature of the rounded tip of the rod. Contrary to this, enhancement of the local electric field at the rounded tip strongly increases with decreasing radius of the tip. Comparison of the numerical results with the adiabatic theory of nanofocusing results in accurate verification of the applicability conditions for adiabatic approximation in tapered nanorods.

Deep-Trap Stress Induced Leakage Current Model for Nominal and Weak Oxides

We have developed a model of the stress-induced leakage current (SILC) based on the inelastic trap-assisted tunneling (ITAT) by introducing a trap with a deep energy level of 3.6 eV from the bottom of the conduction band. This model can explain both of two field dependencies, i.e., a field dependence of the direct tunneling (DT) for A-mode SILC and that of the Fowler–Nordheim (FN) tunneling for B-mode SILC by analytical equations of a common form. For simple analytical equations, we introduce the most favorable trap position (MFTP), which gives the largest contribution to the leakage current. The trap area density for A-mode SILC of around 1×1010 cm-2 and the area density of the leakage paths for B-mode SILC of 1×102 cm-2 were obtained by comparisons between the experimental results and the present model.

Peak intensity analysis as a method for estimation of fluorescent probe binding to artificial and natural nanoparticles: Tetramethylrhodamine uptake by isolated mitochondria

A modified version of fluorescence correlation spectroscopy (FCS) closely related to the photon counting histogram (PCH) method, which is used in the case of a mixture of molecules with similar diffusion coefficients, was applied here for analyzing the binding of the potential-sensitive dye tetramethylrhodamine ethyl ester, TMRE, to isolated mitochondria both in energized and deenergized states. Fluorescence time traces of suspensions of TMRE-doped mitochondria representing sequences of peaks of different intensity appeared to be similar to those of fluorescent beads and TMRE-doped latex particles. The experimental data were obtained under stirring conditions which increased the number of events by about three orders of magnitude thus substantially enhancing the resolution of the method. The statistics of the brightness of identical fluorescent particles reflecting their random walk through the confocal volume was described by a simple analytical equation which enabled us to perform the peak intensity analysis (PIA) of TMRE-doped mitochondria. The validity of PIA was tested with fluorescent beads of different sizes and TMRE-doped latex particles. Mitochondrial energization in the presence of TMRE led to the increase in the number and the intensity of the peaks in fluorescence time traces, the PIA of which allowed us to determine mitochondrial membrane potential and additionally a number of mitochondrial particles per ml of the suspension. The value of the membrane potential on a single mitochondrion was estimated to be about 180 mV in agreement with the data related to mitochondrial suspensions. Importantly, the PIA method required less than 1 microgram of mitochondrial protein per measurement.