Three-parameter Form Research Articles

γ*NΔform factors from a relativistic dynamical model of pion electroproduction

We obtain the electromagnetic form factors of the $\gamma N\Delta$ transition by analyzing recent pion-electroproduction data using a fully relativistic dynamical model. Special care is taken to satisfy Ward-Takahashi identities for the Born term in the presence of form factors thereby allowing the use of realistic electromagnetic form factors of the nucleon and pion. We parametrize the $Q^2$ dependence of the {\it bare} $\gamma N \Delta$ form factors by a three-parameter form which is consistent with the asymptotic behavior inferred from QCD. The parameters of the bare $\gamma N \Delta$ form factors are the only free parameters of the model and are fitted to the differential cross-section and multipole-analysis data up to $Q^2=4$ (GeV/c)$^2$ in the $\Delta(1232)$-resonance region. This analysis emphasizes the significance of the pion-cloud effects in the extraction of the resonance parameters.

Review of the Thermodynamic Properties of Mg(NO3)2(aq) and Their Representation with the Standard and Extended Ion-Interaction (Pitzer) Models at 298.15 K

Published thermodynamic data yielding the osmotic coefficients, relative apparent molar enthalpies, and apparent molar heat capacities of Mg(NO3)2(aq) have been collected, recalculated consistently, and critically assessed. The more reliable of these data have been used to evaluate the parameters of the standard three-parameter form of Pitzer's ion-interaction model to higher molalities than previously available, along with the parameters of Archer's four-parameter, extended ion-interaction model at 298.15 K. Published experimental thermodynamic data were essentially represented equally well by these two models, provided that the exponential coefficient α1 of the standard Pitzer model is fixed at the optimum value of α1 = 1.55 kg1/2·mol-1/2 rather than the traditional value of α1 = 2.0 kg1/2·mol-1/2. The use of the standard Pitzer model with this modified α1 value is recommended for Mg(NO3)2(aq). In addition, an empirical equation is given for the variation of the water activity of a saturated solution with temperature, from 273.54 to 328.20 K, with Mg(NO3)2·6H2O(s) as the solid phase.

Parameters for the attractive coefficient of the patel-teja-valderrama equation of state

A three-parameter form of the temperature dependence of the attractive term in cubic van der Waals type equations of state (AS; Almeida et al., 1991; Aznar and Silva-Telles, 1992) has been applied to a generalized equation of state known as the Patel-Teja-Valderrama equation (PTV; Valderrama, 1990). The parameters of the AS model in the PTV equation of state were calculated for 457 substances of different polarity, molecular size, and chemical nature. The parameters were calculated using vapor pressure data determined through accurate equations available in the literature (Wagner 1973, Antoine 1888, Frost and Kalkwarf, 1953). The deviations between the equation of state calculations and vapor pressure information are lower than 0.5% for most fluids. The study shows that both the generalized PTV cubic equation and the AS represent good tools for pressure-temperature calculations in engineering applications.

Use of weibull statistics to quantify property variability in TiAl alloys

Tensile experiments have been performed on specimens of four different investment-cast TiAl-based alloys with variations in casting conditions. The average ductilities obtained in these experiments vary between approximately 0.8 to 2.0 pct plastic strain to failure in tension. By using the three-parameter form of the Weibull relation, with the 0.2 pct offset yield strength as the minimum failure strength, the resulting variability in the data can be quantified and is found to be similar for those alloys with similar microstructural scale (grain size). Large variations in lamellar volume fraction, segregation, and phase distribution have a minor influence on property variability, compared to changes in the scale of the grain structure caused by either variations in cooling rate during casting or the addition of grain refiners.

Determination of temperature dependence of limiting activity coefficients for a group of moderately hydrophobic organic solutes in water

Limiting activity coefficients γ 1 ∞ in water were measured for 17 moderately hydrophobic mono- and bifunctional compounds containing hydroxy, keto, ether, ester and nitrile groups at several temperatures from 298 to 353 K. Three experimental techniques, viz. Rayleigh distillation, inert gas stripping (IGS), and headspace analysis (HSA), were alternatively employed. With the objective to establish the reliable γ 1 ∞( T) dependences, the γ 1 ∞ values were combined with calorimetric data on limiting partial molar excess enthalpies H ̄ 1 E,∞ and heat capacities C ̄ P,1 E,∞ determined previously in our laboratories. This database was further supplemented with critically evaluated literature data. The equation, ln γ 1 ∞=A+B/τ+C ln τ+Dτ , where τ=T/(298.15 K) , was employed for the simultaneous thermodynamic treatment of γ 1 ∞, H ̄ 1 E,∞ and C ̄ P,1 E,∞ data. With the exception of 1-butanol, for all solutes studied the three-parameter form of this equation ( D=0) assuming temperature independent C ̄ P,1 E,∞ was found to represent available information adequately. For all the solutes, γ 1 ∞( T) exhibits a maximum between 273 and 373 K. As shown, by processing simultaneously the equilibrium and thermal data, thermodynamically consistent, accurate γ 1 ∞( T) dependences can be obtained for the range 273–373 K even in cases of minimum underlying information (room temperature values of H ̄ 1 E,∞ and C ̄ P,1 E,∞ , a single temperature value of γ 1 ∞). In general, the described methodology should be the route of choice when determining γ 1 ∞( T) of aqueous semi-hydrophobic or non-hydrophobic solutes. Furthermore, prediction of γ 1 ∞ for the systems studied was attempted by two group contribution methods: the modified UNIFAC (Dortmund) and the method of Cabani et al. Both were found to perform unsatisfactorily for the bifunctional derivatives.

Osmotic and Activity Coefficients Using a Modified Pitzer Equation for Strong Electrolytes 1:1 and 1:2 at 298.15 K

We have correlated simultaneously experimental data for osmotic and activity coefficients of strong electrolyte solutions using the Pitzer equation and a modification of it. The optimal value for the Pitzer b parameter is different from the traditional value of 1.2 when using higher concentrations in its determination. An analysis of the equation shows that it is possible to reduce the model to a three-parameter form that represents the coefficients at molalities as high as 25 with better accuracy than the model as proposed by Pitzer.

A maximum likelihood method for estimating P S N curves

A maximum likelihood method for estimating P S N curves is proposed in order to minimize the time and the number of specimens required for laboratory testing. The S N curves are assumed in a three-parameter form. The proposed maximum likelihood method can conveniently be used to estimate the three parameters of the S N curves. The required number of test specimens for the maximum likelihood method is less than that for the conventional method. Based on the test results of notched steel specimens subject to axial loads, it is shown here that the P S N curves obtained by the maximum likelihood method agree well with those obtained by the conventional method.

A DATA BANK OF PARAMETERS FOR THE ATTRACTIVE COEFFICIENT OF THE PENG-ROBINSON EQUATION OF STATE

A data bank of parameters for two expressions for the temperature-dependence of the attractive coefficient, <FONT FACE="Symbol">a</font> (T), of the Peng-Robinson equation of state was compiled for 468 pure substances of varied polarity. The performances of the two expressions, a two-parameter form by Melhem et al. (1989) and a three-parameter form by Almeida et al. (1991) were compared through the root mean square (rms) and maximum (max) deviations in the prediction of the vapor pressure for these substances. The results show the superiority of the three-parameter form

Solubility of Anthracene in Binary Alcohol + Dibutyl Ether Solvent Mixtures

Solid-liquid equilibrium data of organic nonelectrolyte systems are becoming increasingly important in the petroleum industry, particularly in light of present trends toward heavier feedstocks and known carcinogenicity/mutagenicity of many of the larger polycyclic aromatic compounds. Experimental solubilities are reported for anthracene dissolved in seven binary mixtures containing dibutyl ether with 1-propanol, 2-propanol, 1-butanol, 2-butanol, 1-octanol, 2-methyl-1-propanol, and 3-methyl-1-butanol at 25 C. Results of these measurements are used to test two mathematical representations based upon the combined nearly ideal binary solvent (NIBS)/Redlich-Kister equation and modified Wilson model. For the systems studied, the combined NIBS/Redlich-Kister provided the better mathematical representations, with deviations between experimental and back-calculated values being on the order of {+-} 1.5% or less in the case of the three-parameter form. Slightly larger deviations were noted for the modified Wilson mathematical representation.

The Monomolecular and Rectangular Hyperbola as Empirical Models of the Response of Photosynthetic Rate to Photon Flux Density, with Applications to Three Veronica Species

This paper discusses and formalises the use of the monomolecular function as an empirical model of the photosynthetic rate/PFD relationship. We shall treat this model, and the rectangular hyperbola in a three-parameter form which describes net photosynthesis and allows dark respiration and the light compensation point to be estimated

Channeling, exponential tails, and analytical modeling of Si implants into GaAs

The depth distribution N(x) of Si/sup 29/ ion beam implanted into GaAs for a wide range of implant doses, energies, and SiN randomizing-layer thicknesses is discussed. It is found that the variation of implant angle with position on the wafer (typically encountered in scanned-ion-beam implantation systems) results in significant channeling-induced nonuniformity of the doping profiles, even in some cases where a randomizing layer was used. Because of the existence of exponential tails in virtually all implants, an extension of the standard LSS theoretical profile is necessary for accurate characterization of the implants and prediction of the device parameters. The measured profiles were fitted to two different modified-Gaussian analytical forms: the Pearson four-parameter form, which offers excellent flexibility in fitting arbitrary distributions but is relatively devoid of physical content, and a new three-parameter form that is an extension of a Gaussian to profiles hyperbolic in log N. In almost all cases, the hyperbolic-Gaussian form fits the data as well as the Pearson form, and the hyperbolic-Gaussian form is simpler. Parameters characterizing the implants for both types of fit are presented. >

A consistent cluster model treatment of exotic decays and alpha decays from heavy nuclei

A cluster model employing a local, effective cluster-core potential based on a folding procedure is used to investigate exotic and alpha decays from heavy nuclei as a quantum tunnelling phenomenon within a semiclassical approximation. For the exotic decays, excellent agreement with all reported experimental measurements of the partial lifetimes for 14C and 24Ne emission from heavy nuclei is obtained. Further consideration of the structure of these nuclei indicates preformation or spectroscopic factors generally in the range 0.1-0.5. For alpha decay, a similar cluster-core potential is represented by a simple three-parameter form having a fixed diffuseness, a radius proportional to A1/3 and a depth which is adjusted to fit the Q value of the particular decay. The calculations yield excellent agreement with the vast majority of the available data for ground state to ground state alpha decays. Some examples are presented.

Vapor—liquid equilibrium measurements for the n-pentane—methanol—acetone ternary at 372.7 K and their prediction from the constituent binaries

Vapor—liquid equilibrium measurements for the n-pentane—methanol—acetone ternary system have been obtained along the 372.7 K isotherm. For the measurements of this polar-nonpolar system, the pressure varied from 348.5 to 850.7 kPa. Of all the equations tested, these measurements are correlated best with the four-suffix Margules equation using three adjustable ternary parameters. Nevertheless, the two- and three-parameter forms of the Wilson equation and the NRTL equation are capable of predicting these ternary measurements when the associated parameters are determined from binary information only. The UNIQUAC equation does not perform as well for this sytem primarily because it cannot model the n-pentane—methanol binary behavior satisfactorily. The predictive capabilities of the four-suffix Margules equation and the Peng-Robinson equation of state are poor. However, if an equation of state such as the Peng-Robinson equation of state is to be used, it has been clearly shown that utilizing binary interaction coefficients is a requirement, not a refinement. Finally, this ternary system does not possess a ternary azeotrope at 372.7 K. The maximum pressure is that corresponding to the azeotrope belonging to the n-pentane—methanol binary.

An extended Kramers equation for photoisomerization

The experimental data from seven different studies of liquid-state cis-trans isomerization over reasonably high barriers are fit to a simple three-parameter form of the Kramers equation. These fits suggest that in all the systems discussed here the memory kernal remains effectively constant over the time scale of the barrier crossing, as in a frozen solvent model. The various parameters derived from these fits seem reasonable.

A new statistical distribution function for fracture toughness

A statistical view of fracture at cracks is presented that is also appropriate for failures in singularity-dominated, self-similar fields other than those at crack tips. Consideration of the behaviour of the distributions of stress and strain near crack tips results in the development of a new two-parameter distribution function for the probability of failure. The two fundamental premises on which the function is based are, firstly, that the failure of any part of the material near to the crack tip leads to total failure along the whole crack front or at least represents total failure; and secondly, that the variability of strength in material is due to micro-structural inhomogeneity. The new function is tested by means of several large sets of toughness data from other workers, and is found to give with only two parameters better fits than can the three-parameter distribution function of Weibull. The Weibull function is capable of giving reasonable fits in its extremely flexible three-parameter form, but that very flexibility means that these fits may be no more than descriptions without theoretical foundation. It is found also that the new function is applicable equally to ceramics and steels. The very good fits afforded by the new function are further support for previous findings in two basic areas in the science of fracture. Firstly, previous work concerned with the distributions of stress and strain at crack tips and with crack-opening displacement upon which the new function is based, is supported, as is the idea that the crack-opening displacement is fundamental in determining the possibility and prob­ability of failure. Secondly, the present work is in agreement with widely accepted ideas concerned with stress-controlled mechanisms of failure in materials.

Surface tension of pure and mixed simple substances at low temperatures

A new design of cell is described for measuring the gas-liquid surface tension of both pure and mixed liquefied gases by the differential capillary-rise method in the temperature range 90–250 K at pressure up to 1000 Torr. Results for the pure methane, krypton, tetrafluoromethane, ethene, ethane, and dimethyl ether at temperatures below their normal boiling point are presented, compared with literature values, and analysed in terms of simple two- and three-parameter forms of the principle of corresponding states. Results are illustrated for the mixture methane + tetrafluoromethane in the temperature range 98 to 102 K.

Complex spin-orbit potential forPb208(n, n)Pb208at 10 MeV

The analyzing power ${A}_{y}(\ensuremath{\theta})$ and differential cross section $\ensuremath{\sigma}(\ensuremath{\theta})$ have been measured for elastic scattering of 10-MeV neutrons from $^{208}\mathrm{Pb}$. These data were analyzed with a spherical optical model which has a real spin-orbit potential with either a three-parameter conventional form or a one-parameter semimicroscopic form. The predictions with both spin-orbit potentials reproduce the data equally well, suggesting that fewer free parameters can be used in spherical optical model calculations. Including an imaginary spin-orbit term ${W}_{\mathrm{SO}}(r)$ in the optical potential greatly improves the fits to the ${A}_{y}(\ensuremath{\theta})$ measurements and establishes the need for ${W}_{\mathrm{SO}}(r)\ensuremath{\ne}0$ in the n + $^{208}\mathrm{Pb}$ potential.NUCLEAR REACTIONS $^{208}\mathrm{Pb}$(n, n); $E=10$ MeV; measured $\ensuremath{\sigma}(\ensuremath{\theta})$, ${A}_{y}(\ensuremath{\theta})$; deduced complex spin-orbit potential; spherical optical model calculations.

Redshift modifications to HEAO A-1 cluster X-ray luminosities

New redshift measurements for the 24 Abell galactic clusters are presented. The optical observations are described and applied to the cluster X-ray luminosity function, and the results are used to delimit the cluster contribution to the diffuse X-ray background (DXRB). The evolution of the X-ray luminosity function is then investigated. It is found that while simple two-parameter analytic forms do not fit this function very well, three-parameter forms such as the exponential times power law do fit it very well. Future redshift observations will probably not change the volume emissivity significantly, but they may in a statistically significant way affect the parameters of analytic fits which define the shape of the luminosity function. The contribution of rich clusters to the DXRB in the 2-6 keV range is 5.2 percent. Future redshift observations will modify this result by no more than a factor of about 1.1.

Equations of state of polymer melts and temperature dependence of the number of external degrees of freedom in a macromolecule

The theory of the liquid state developed by Flory is reviewed and its assumptions compared with those in the recent model of thermal expansion of polymer melts developed by the author. Methods of evaluation of the characteristic parameters p ∗, V ∗ and T ∗ are discussed. The number c of external degrees of freedom of a polymeric segment does depend on the temperature; consequences of disregarding this dependence are noted. It is shown that a simplification of the Flory equation of state leads to the van der Waals equation. The four-parameter equation of state proposed by the author, simplified to a three-parameter form, is consistent with the principle of corresponding states.

Sampling error in the bending strength distribution of dimension lumber

Information is presented on the magnitude of errors associated with various sampling simulation schemes of the distribution of three different populations, representing actual bending strength of dimension lumber. Errors were determined between the simulated and actual distributions. Graphical evaluations indicated good fits with the three-parameter form of the weibull distribution for both original and simulated bending strength data, as well as with the resulting error terms. Error terms, based on the simulated versus actual distributions, were generated for the lower 5% exclusion limit, for the 50% exclusion limit and for the entire distribution curve. Simulations were carried out with the aid of Monte Carlo techniques using distribution functions fitted to actual test data for dimension lumber. The errors are expressed as functions of confidence levels. The comparison of the erro obtained through the various sampling schemes could provide some initial directions to choose an economical sampling plan for the presently ongoing in-grade lumber testing program.