Complementary Error Function Research Articles

Integral partition bounds for fast and effective screening of general one-, two-, and many-electron integrals.

We introduce tight upper bounds for a variety of integrals appearing in electronic structure theories. These include electronic interaction integrals involving any number of electrons and various integral kernels such as the ubiquitous electron repulsion integrals and the three- and four-electron integrals found in explicitly correlated methods. Our bounds are also applicable to the one-electron potential integrals that appear in great number in quantum mechanical (QM), mixed quantum and molecular mechanical (QM/MM), and semi-numerical methods. The bounds are based on a partitioning of the integration space into balls centered around electronic distributions and their complements. Such a partitioning leads directly to equations for rigorous extents, which we solve for shell pair distributions containing shells of Gaussian basis functions of arbitrary angular momentum. The extents are the first general rigorous formulation we are aware of, as previous definitions are based on the inverse distance operator 1/r12 and typically only rigorous for simple spherical Gaussians. We test our bounds for six different integral kernels found throughout quantum chemistry, including exponential, Gaussian, and complementary error function based forms. We compare to previously developed estimates on the basis of significant integral counts and their usage in both explicitly correlated second-order Møller-Plesset theory (MP2-F12) and density functional theory calculations employing screened Hartree-Fock exchange.

Diffusion coefficient calculated by complementary error function for the sublimation diffusion of disperse dye

In order to find a simple and reliable method for the calculation of the diffusion coefficient, the correlation equation of concentration and distance in the form of complementary error function was derived from solving an ordinary differential equation of the diffusion equation. A disperse dye in paste was treated at 170°C–190°C for 3–4 h for the sublimation diffusion into polyethylene terephthalate using a film roll method. Quadratic regression analysis on the profile of dye concentration–distance was used to determine the surface concentration. The diffusion coefficient of each layer was calculated by obtaining the variable value of the complementary error function from the ratio of the mean dye concentration of each layer to the surface concentration. From linear regression analysis on the Arrhenius plot of the logarithm of the diffusion coefficient versus the reciprocal of absolute temperature, the correlation coefficient for the diffusion of 3 h was 0.9978 and that of 4 h was 0.9991. Thus, it was expected that the diffusion coefficients determined by the equation of complementary error function adopted in this experiment were reliable. The activation energy of diffusion for 3 h was 30.5 kcal mol−1 and that for 4 h was 27.4 kcal mol−1.

Effect of Microwave Radiation on Internal Inoculum of Ascochyta Blight in Lentil Seeds at Different Seed Moisture Contents

Abstract. Considering the development of pest and pathogen resistance to chemical treatments and the cost of developing new chemicals, proper physical methods for integrated pest management seem to be gaining interest. Microwave treatment has been considered a potential postharvest pest and disease management strategy in legume and grain seeds. The effect of this process on pathogens related to legumes, which could be expressed as a very fast and volumetric thermal treatment, has not been fully studied. In this study, a domestic microwave oven with full output power of 1100 W (company information) or 485 W (calibrated with 150 g of water) was used to treat red lentil seeds that were naturally infected with the Ascochyta blight (AB) pathogen, a problematic seedborne fungi of lentil. Microwave radiation was applied at three moisture contents of red lentil seeds (9.1%, 16%, and 19% wet basis) and two power levels (50% and 100%). Three exposure times for each moisture content and power level were selected to achieve the same final average temperatures with low and high moisture contents (approx. 65°C to 85°C). The complementary error function was used to model and compare the pathogen and seed death rates based on the normally distributed susceptibility of the AB pathogen and seed germination to microwave heating. At 9% moisture content with water activity of 0.4, the percentage of infected seeds decreased from 17% to 9% without significant loss of seed viability when a power level of 100% (9.7 W g-1) and exposure time of 51 s were applied, and the final average surface temperature reached 84°C. However, seed vigor, measured by the electrical conductivity of the seed leachate, decreased at this power level and exposure time. Despite the feasibility of fungi elimination at higher moisture content at the same final average temperatures, eliminating fungi from the seeds without harming their viability and vigor was challenging. Keywords: Ascochyta blight, Lentils, Microwave energy, Pathology.

Indefinite theta series and generalized error functions

Theta series for lattices with indefinite signature $(n_+,n_-)$ arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case ($n_+=1$), but have remained obscure when $n_+\geq 2$. Using a higher-dimensional generalization of the usual (complementary) error function, discovered in an independent physics project, we construct the modular completion of a class of `conformal' holomorphic theta series ($n_+=2$). As an application, we determine the modular properties of a generalized Appell-Lerch sum attached to the lattice $A_2$, which arose in the study of rank 3 vector bundles on $\mathbb{P}^2$. The extension of our method to $n_+>2$ is outlined.

Average BER and Noisy Reference Loss of Partially Coherent PSK Demodulation Over Shadowed Multipath Fading Channel

In this correspondence paper, we present novel results related to the error performance of coherent systems in the presence of the imperfect carrier phase estimation over Gamma shadowed Nakagami- $m$ fading environment. Taking the Tikhonov-distributed phase error process into account, binary and quaternary phase-shift keying signal detection is analyzed. Based on the Maclaurin series expansion of the complementary error function, approximated closed-form expressions for the bit error rate (BER) are derived. These expressions are simplified to those already reported in literature, which are valid in the case without shadowing. The asymptotic closed-form expressions for noisy reference loss are also presented. We examine the effect of phase-locked loop parameters on both BER and noisy reference loss. Analytical results are validated by independent Monte Carlo simulations.

Water vapor interaction with borosilicate glass

Hydration of borosilicate glass (Corning Pyrex glass) in isotopic water H218O vapor was performed at different temperatures. The depth distributions of 1H and 18O in the borosilicate glass measured by the Time-of-Flight Secondary Ion Mass-Spectrometry (TOF-SIMS) were found to be drastically different. The activation energies for the 18O diffusion and for the glass hydrogenation were measured experimentally. The depth distribution of the 18O isotope was described well by the complementary error function, whereas the depth distribution of hydrogen represented the so-called “S-curve”. A similar hydrogen distribution was observed earlier in hydrated natural aluminosilicate glasses (obsidians). This led us to the conclusion that the hydrogenated layer was formed due to a solid state chemical reaction between protons and the glass network. The protons were the result of another solid state chemical reaction between the absorbed water molecules and glass network occurring in a top surface layer (<30 nm).

Estimation of infiltration rate based on complementary error function peak for Ozat watershed in Gujarat (India)

Infiltration refers to the downward movement of water into the soil from the surface driven by force of gravity and capillary attraction Infiltration of water in unsaturated soils has long been an important impact in soil science hydrology and geotechnical engineering Fieldmeasurement of infiltration is very tedious and time consumingand hence it is often estimated from different conceptual models Thus in the present study a model based on complementary error function peak ERFC is proposed to determine infiltration rate for the Ozat Watershed of Gujarat India The two quantitative standard statistical performance evaluation measures refined Willmott rsquo s index dr and mean absolute error MAE are employed in comparingand evaluating the performance of theproposed model with existing Horton rsquo s Modified Kostiakov and Green Ampt infiltration models The results of the models are validated with the Double ring infiltrometer field data of different soil types The proposed ERFC based model was judged to be more consistent with dr to and MAE to cm hr The results presented in this work are quite encouraging Further the findings indicate that the use of ERFC based proposed model appears to be the most suitable and appropriate for estimation of infiltration rate in Ozat watershed

Case study of MHD blood flow in a porous medium with CNTS and thermal analysis

This articles deals with unsteady MHD free convection flow of blood with carbon nanotubes. The flow is over an oscillating vertical plate embedded in a porous medium. Both single-wall carbon nanotubes (SWCNTs) and multiple-wall carbon nanotubes (MWCNTs) are used with human blood as base fluid. The problem is modelled and then solved for exact solution using the Laplace transform technique. Expressions for velocity and temperature are determined and expressed in terms of complementary error functions. Results are plotted and discussed for embedded parameters. It is observed that velocity decreases with increasing CNTs volume fraction and an increase in CNTs volume fraction increases the blood temperature, which leads to an increase in the heat transfer rates. A validation of the present work is shown by comparing the current results with existing literature.

Accurate and Efficient Parallel Implementation of an Effective Linear-Scaling Direct Random Phase Approximation Method.

An efficient algorithm for calculating the random phase approximation (RPA) correlation energy is presented that is as accurate as the canonical molecular orbital resolution-of-the-identity RPA (RI-RPA) with the important advantage of an effective linear-scaling behavior (instead of quartic) for large systems due to a formulation in the local atomic orbital space. The high accuracy is achieved by utilizing optimized minimax integration schemes and the local Coulomb metric attenuated by the complementary error function for the RI approximation. The memory bottleneck of former atomic orbital (AO)-RI-RPA implementations ( Schurkus, H. F.; Ochsenfeld, C. J. Chem. Phys. 2016 , 144 , 031101 and Luenser, A.; Schurkus, H. F.; Ochsenfeld, C. J. Chem. Theory Comput. 2017 , 13 , 1647 - 1655 ) is addressed by precontraction of the large 3-center integral matrix with the Cholesky factors of the ground state density reducing the memory requirements of that matrix by a factor of [Formula: see text]. Furthermore, we present a parallel implementation of our method, which not only leads to faster RPA correlation energy calculations but also to a scalable decrease in memory requirements, opening the door for investigations of large molecules even on small- to medium-sized computing clusters. Although it is known that AO methods are highly efficient for extended systems, where sparsity allows for reaching the linear-scaling regime, we show that our work also extends the applicability when considering highly delocalized systems for which no linear scaling can be achieved. As an example, the interlayer distance of two covalent organic framework pore fragments (comprising 384 atoms in total) is analyzed.

Effects of Dufour and fractional derivative on unsteady natural convection flow over an infinite vertical plate with constant heat and mass fluxes

In this paper, we analyze the effects of Dufour number and fractional-order derivative on unsteady natural convection flow of a viscous and incompressible fluid over an infinite vertical plate with constant heat and mass fluxes. The fractional constitutive model is obtained using fractional calculus approach. The Caputo fractional derivative operator is used in this problem. The dimensionless system of equations has been solved by employing Laplace transformation technique. Closed form solutions for concentration, temperature and velocity are presented in the form of Wright function and complementary error function. Effects of fractional and physical parameters on temperature and velocity profiles are illustrated graphically.

Exact analytical solution to the electron density for monolayer and bilayer graphene

This article presents a mathematical method for analytically evaluating the electron density for both monolayer and bilayer graphene. Once an appropriate probability function is chosen for finding the electronic density of states, a complete analytical solution is found for the electronic density without recourse to any analytical approximations or asymptotic formulations. The form of the solution is in terms of a convergent series involving the complementary error function. The analytical solution developed in this paper is valid for all temperatures and for all Fermi energies greater than or equal to zero. The method is easily extended to include Fermi energies less than zero.

Model for threading dislocations in metamorphic tandem solar cells on GaAs (001) substrates

We present an approximate model for the threading dislocations in III–V heterostructures and have applied this model to study the defect behavior in metamorphic triple-junction solar cells. This model represents a new approach in which the coefficient for second-order threading dislocation annihilation and coalescence reactions is considered to be determined by the length of misfit dislocations, LMD, in the structure, and we therefore refer to it as the LMD model. On the basis of this model we have compared the average threading dislocation densities in the active layers of triple junction solar cells using linearly-graded buffers of varying thicknesses as well as S-graded (complementary error function) buffers with varying thicknesses and standard deviation parameters. We have shown that the threading dislocation densities in the active regions of metamorphic tandem solar cells depend not only on the thicknesses of the buffer layers but on their compositional grading profiles. The use of S-graded buffer layers instead of linear buffers resulted in lower threading dislocation densities. Moreover, the threading dislocation densities depended strongly on the standard deviation parameters used in the S-graded buffers, with smaller values providing lower threading dislocation densities.

Analysis of Back Surface Field (BSF) Performance in P-Type And N-Type Monocrystalline Silicon Wafer

Back Surface Field (BSF) has been used as one of means to enhance solar cell performance by reducing surface recombination velocity (SRV). One of methods to produce BSF is by introducing highly doped layer on rear surface of the wafer. Depending on the type of the dopant in wafer, the BSF layer could be either p+ or n+. This research aims to compare the performance of BSF layer both in p-type and n-type wafer in order to understand the effect of BSF on both wafer types. Monociystalline silicon wafer with thickness of 300 μm. area of 1 cm2, bulk doping level NB = 1.5×1016/cm3 both for p-type wafer and n-type wafer are used. Both wafer then converted into solar cell by adding emitter layer with concentration NE =7.5×1018/cm3 both for p-type wafer and n-type wafer. Doping profile that is used for emitter layer is following complementary error function (erfc) distribution profile. BSF concentration is varied from 1×1017/cm3 to 1×1020/cm3 for each of the cell. Solar cell performance is tested under standard condition, with AM1.5G spectrum at 1000 W/m2. Its output is measured based on its open circuit voltage (Voc). short circuit current density (JSC), efficiency (η). and fill factor (FF). The result shows that the value of VOC is relatively constant along the range of BSF concentration, which is 0.694 V – 0.702 V. The same pattern is also observed in FF value which is between 0.828 – 0.831. On the other hand, value of JSC and efficiency will drop against the increase of BSF concentration. Highest JSC which is 0.033 A/cm2 and highest efficiency which is 18.6% is achieved when BSF concentration is slightly higher than bulk doping level. The best efficiency can be produced when BSF concentration is around 1×1017cm-3.. This result confirms that surface recombination velocity has been reduced due to the increase in cell’s short circuit current density and its efficiency. In general both p-type and n-type wafer will produce higher efficiency when BSF is applied. However, the increase is larger in p-type wafer than in n-type wafer. Better performance for solar cell is achieved when BSF concentration is slightly higher that bulk doping level because at very high BSF concentration the cell’s efficiency will be decreased.

General Solutions for Hydromagnetic Free Convection Flow over an Infinite Plate with Newtonian Heating, Mass Diffusion and Chemical Reaction

The problem of hydromagnetic free convection flow over a moving infinite vertical plate with Newtonian heating, mass diffusion and chemical reaction in the presence of a heat source is completely solved. Radiative and porous effects are not taken into consideration but they can be immediately included by a simple rescaling of Prandtl number and magnetic parameter. Exact general solutions for the dimensionless velocity and concentration fields and the corresponding Sherwood number and skin friction coefficient are determined under integral form in terms of error function or complementary error function of Gauss. They satisfy all imposed initial and boundary conditions and can generate exact solutions for any problem with technical relevance of this type. As an interesting completion, uncommon in the literature, the differential equations which describe the thermal, concentration and momentum boundary layer, as well as the exact expressions for the thicknesses of thermal, concentration or velocity boundary layers were determined. Numerical results have shown that the thermal boundary layer thickness decreases for increasing values of Prandtl number and the concentration boundary layer thickness is decreasing with Schmidt number. Finally, for illustration, three special cases are considered and the influence of physical parameters on some fundamental motions is graphically underlined and discussed. The required time to reach the flow according with post-transient solution (the steady-state), for cosine/sine oscillating concentrations on the boundary is graphically determined. It is found that, the presence of destructive chemical reaction improves this time for increasing values of chemical reaction parameter.

Bi-Parabolic Graded Buffer Layers for Metamorphic Devices

Metamorphic semiconductor devices such as high electron mobility transistors (HEMTs), light-emitting diodes (LEDs), laser diodes, and solar cells are grown on mismatched substrates and typically exhibit a high degree of lattice relaxation. Linearly-graded buffer layers are commonly used to accommodate the lattice mismatch in these structures, but theoretical and experimental studies suggest that non-linear compositional grading might allow lower threading dislocation densities and improved device performance. In this work we have compared the threading dislocation densities in heterostructures employing bi-parabolic, linear, and S-graded (complementary error function) buffer layers. The threading dislocation densities were calculated using the approximate Matthews, Mader, and Light model (1) for plastic flow and by assuming that the coefficient for second-order annihilation and coalescence reactions between threading dislocations is equal to the length of misfit dislocations.

A real distinct poles rational approximation of generalized Mittag-Leffler functions and their inverses: Applications to fractional calculus

The generalized Mittag-Leffler function and its inverse is very useful in solving fractional differential equations and structural derivatives, respectively. However, their computational complexities have made them difficult to deal with numerically. We propose a real distinct pole rational approximation of the generalized Mittag-Leffler function. Under some mild conditions, this approximation is proven and empirically shown to be L-Acceptable. Due to the complete monotonicity property of the Mittag-Leffler function, we derive a rational approximation for the inverse generalized Mittag-Leffler function. These approximations are especially useful in developing efficient and accurate numerical schemes for partial differential equations of fractional order. Several applications are presented such as complementary error function, solution of fractional differential equations, and the ultraslow diffusion model using the structural derivative.

A new model integrating short- and long-term aging of copper added to soils.

Aging refers to the processes by which the bioavailability/toxicity, isotopic exchangeability, and extractability of metals added to soils decline overtime. We studied the characteristics of the aging process in copper (Cu) added to soils and the factors that affect this process. Then we developed a semi-mechanistic model to predict the lability of Cu during the aging process with descriptions of the diffusion process using complementary error function. In the previous studies, two semi-mechanistic models to separately predict short-term and long-term aging of Cu added to soils were developed with individual descriptions of the diffusion process. In the short-term model, the diffusion process was linearly related to the square root of incubation time (t1/2), and in the long-term model, the diffusion process was linearly related to the natural logarithm of incubation time (lnt). Both models could predict short-term or long-term aging processes separately, but could not predict the short- and long-term aging processes by one model. By analyzing and combining the two models, we found that the short- and long-term behaviors of the diffusion process could be described adequately using the complementary error function. The effect of temperature on the diffusion process was obtained in this model as well. The model can predict the aging process continuously based on four factors—soil pH, incubation time, soil organic matter content and temperature.

Fano fluctuations in superconducting-nanowire single-photon detectors

Because of their universal nature, Fano fluctuations are expected to influence the response of superconducting nanowire single-photon detectors (SNSPDs). We predict that photon counting rate ($PCR$) as a function of bias current ($I_B$) in SNSPDs is described by an integral over a transverse coordinate-dependent complementary error function. The latter describes smearing of local responses due to Fano fluctuations of the amount of energy deposited into electronic system. The finite width, $\sigma$, of the $PCR$ vs $I_B$ arises from fluctuations in the energy partition between quasiparticles and phonons during the energy down-conversion cascade. In narrow-nanowire SNSPDs the local responses are uniform, and the effect of Fano-fluctuations on $\sigma$ is dominant. In wide-nanowire SNSPDs with strong coordinate dependence of local responses due to vortex-antivortex unbinding and vortex entry from edges, Fano-fluctuations smear singularities imprinted by vorticity on the transition part of $PCR$ curve. We demonstrate good agreement between theory and experiments for a series of bath temperatures and photon energies in narrow-wire WSi SNSPDs. The time-resolved hotspot relaxation curves predicted by Fano fluctuations match the Lorentzian shapes observed in experiments over the whole range of bias currents investigated except for their tails.

Characterisation of case depth in induction-hardened medium carbon steels based on magnetic minor hysteresis loop measurement technique

The magnetic hysteresis behaviours of ferromagnetic materials vary with the heat treatment-induced micro-structural changes. In the study, the minor hysteresis loop measurement technique was used to quantitatively characterise the case depth in two types of medium carbon steels. Firstly, high-frequency induction quenching was applied in rod samples to increase the volume fraction of hard martensite to the soft ferrite/pearlite (or sorbite) in the sample surface. In order to determine the effective and total case depth, a complementary error function was employed to fit the measured hardness-depth profiles of induction-hardened samples. The cluster of minor hysteresis loops together with the tangential magnetic field (TMF) were recorded from all the samples and the comparative study was conducted among three kinds of magnetic parameters, which were sensitive to the variation of case depth. Compared to the parameters extracted from an individual minor loop and the distortion factor of the TMF, the magnitude of three-order harmonic of TMF was more suitable to indicate the variation in case depth. Two new minor-loop coefficients were introduced by combining two magnetic parameters with cumulative statistics of the cluster of minor-loops. The experimental results showed that the two coefficients monotonically linearly varied with the case depth within the carefully selected magnetisation region.

Algorithm and Implementation Details for Complementary Error Function

The algorithm proposed in this article provides high accuracy and requires only double-precision arithmetic. The calculated value obtained by the algorithm is proved to be accurate to <inline-formula><tex-math notation="LaTeX"> $0.785 \cdot \text{ulp}$</tex-math> </inline-formula> as long as the result does not underflow. In the case of subnormal results the error does not exceed <inline-formula><tex-math notation="LaTeX">$0.875 \cdot \text{ulp}$</tex-math></inline-formula> . The testing results have also shown that the implementation of the proposed algorithm has reasonable computational time compared to other implementations.