Empirical-coefficient method

The working equations relating line intensity to concentration that emerge from fundamental-parameter calculations are similar in form to the equations used in the empirical-coefficient method. These calculations therefore yield the ay coefficients directly. Jenkins et al. [15.10] discuss the use of a large computer to calculate coefficients for a particular class of samples and the subsequent application of these coefficients to the analysis of particular samples of that class by means of a small on-line computer. They also compare experimental and calculated coefficients. See also various papers in [15.11]. 

Empirical-coefficient method. This method requires standards, and it is very difficult to obtain standards that are chemically homogeneous on a micron scale. 

Alternatively, standardless fundamental parameter (FP) techniques are based on built-in mathematical algorithms that describe the physics of the detector response to pure elements. In this case, the typical composition of a sample must be known, while the calibration model may be verified and optimized by one single standard sample. The techniques include the fundamental parameter method,the influence coefficient method, and the empirical coefficient method. 

The empirical parameters method uses simple mathematical approximation equations, whose coefficients (empirical parameters) are predetermined from the experimental intensities and known compositions and thicknesses of thin-film standards. A large number of standards are needed for the predetermination of the empirical parameters before actual analysis of an unknown is possible. Because of the difficulty in obtaining properly calibrated thin-film standards with either the same composition or thickness as the unknown, the use of the empirical parameters method for the routine XRF analysis of thin films is very limited. 

Quantitative XRF analysis has developed from specific to universal methods. At the time of poor computational facilities, methods were limited to the determination of few elements in well-defined concentration ranges by statistical treatment of experimental data from reference material (linear or second order curves), or by compensation methods (dilution, internal standards, etc.). Later, semi-empirical influence coefficient methods were introduced. Universality came about by the development of fundamental parameter approaches for the correction of total matrix effects... 

Alternatively, fundamental parameter methods (FPM) may be used to simulate analytical calibrations for homogeneous materials. From a theoretical point of view, there is a wide choice of equivalent fundamental algorithms for converting intensities to concentrations in quantitative XRF analysis. The fundamental parameters approach was originally proposed by Criss and Birks [239]. A number of assumptions underlie the application of theoretical methods, namely that the specimens be thick, flat and homogeneous, and that, for calibration purposes, the concentrations of all the elements in the reference material be known (having been determined by alternative methods). The classical formalism proposed by Criss and Birks [239] is equivalent to the fundamental influence coefficient formalisms (see ref. [232]). In contrast to empirical influence coefficient methods, in which the experimental intensities from reference materials are used to compute the values of the coefficients, the fundamental influence coefficient approach calculates... 

Mills and colleagues58 describe the use of these formulations to predict aerobic biodegradation in surface waters and present methods of adjusting for temperature and nutrient limitations. This approach to predicting biodegradation is problematic because it is difficult to obtain empirical coefficients in the deep-well setting. 

The relationship between the herbicidal activity of 1,2,5-oxadiazole iV-oxides and some physicochemical properties potentially related to this bioactivity, such as polarity, molecular volume, proton acceptor ability, lipophilicity, and reduction potential, were studied. The semi-empirical MO method AMI was used to calculate theoretical descriptors such as dipolar moment, molecular volume, Mulliken s charge, and the octanol/water partition coefficients (log Po/w) <2005MOL1197>. 

The empirical approach [7] was by far the most fruitful first attempt. The idea was to fit a few Fourier coefficients or form factors of the potential. This approach assumed that the pseudopotential could be represented accurately with around three Fourier form factors for each element and that the potential contained both the electron-core and electron-electron interactions. The form factors were generally fit to optical properties. This approach, called the Empirical Pseudopotential Method (EPM), gave [7] extremely accurate energy band structures and wave functions, and applications were made to a large number of solids, especially semiconductors. [8] In fact, it is probably fair to say that the electronic band structure problem and optical properties in the visible and UV for the standard semiconductors was solved in the 1960s and 1970s by the EPM. Before the EPM, even the electronic structure of Si, which was and is the prototype semiconductor, was only partially known. 

A somewhat more obviously empirical variation on the multilevel approach is the multi-coefficient method of Truhlar and co-workers. Although many different variations of this approach have now been described, it is simplest to illustrate the concept for the so-called multi-coefficient G3 (MCG3) model (Fast, Sanchez, and Truhlar 1999). In essence, the model assumes a G3-like energy expression, but each term has associated with it a coefficient that is not restricted to be unity, as is the case for G3. Specifically... 

In all these figures, we used the competitive Langmuir isotherm model to calculate the band profiles. However, the coefficients of the isotherms used for Figures 11.21 are the coefficients of the single-component isotherms determined by frontal analysis, while the coefficients of the isotherms used to calculate the profiles in Figure 11.22 are measured by the simple wave method (Chapter 4, Section 4.2.4). These latter coefficients are certainly empirical coefficients, and their use would not permit an accurate prediction of single-component bands. However, they permit the calculation of band profiles that are in much better agreement... 

Three methods are used for quantitative analysis calibration curves, empirical coefficients, and fundamental parameters. 

A graphical method, as represented by a calibration curve, is no longer adequate because a whole family of such curves would be needed. Instead, an analytical approach is adopted. A set of simultaneous equations is written, involving measured line intensities, the desired concentrations, and empirical coefficients determined from previous measurements on standard specimens. 

Note 2 Numerical fire test results obtained using different ASTM procedure generally cannot be comparable and/or cannot be translated to expected results of, say, method ASTM E 84, using some empirical coefficients. There are too many noncontrollable factors involved in small and large-scale burning. 

Ihe empirical isotherm method considers equilibrium distribution of only one component and requires the experimental determination of its sorption isotherms in specific conditions. However, ground water contains multiple ions, which compete for the location in the exchange capacity. For this reason, ion exchange equilibrium directly correlates with the ground water composition and adsorption capacity of its components. This effect of ions on one another may be identified only computational methods based on exchange coefficients. 

In principle, an empirical correction procedure can be described as the correction of an analyte element intensity for the influence of interfering elements, using the product of the intensity from the interfering element line and a constant factor, as the correction term [14], This constant factor is today generally referred to as an influence coefficient, since it is assumed to represent the influence of the interfering element on the analyte. Commonly employed influence coefficient methods may use either the intensity or the concentration of the interfering element as the correction term. These methods are referred to as intensity correc-... 

Cp = empirical coefficient depending on soil type and method of construction (see Table 9.11) D = pile diameter... 

This example and the use of these two equations indicates that the empirical coefficients of 1100 in the Newitt method for fine coal and sand, or the empirical coefficient of 100 for sand from the Hayden and Stelson equation, do not converge for similar results. Testing would be recommended to confirm the magnitude of these coefficients. 

The WUson-Thomas method for full flow in closed channels does not rely on empirical coefficients such as the Darby method, but is based on the assumption that a thick sublayer lubricates the wall surface of the pipe. It has not been modified yet for open channel flows. This is a topic well worth further research. 

Quantitative prediction methods for volumetric mass transfer coefficienfs that rely on empirical coefficients for each particular packing and packing size have been developed and can be found in the relevant literature. We do not often resort to complete predictions of this type, and it is more common to use relations that will extend known coefficients, such as those listed in Table 5.6, to a different set of conditions. This can be done in an approximafe fashion using fhe following proportionalities ... 

This relation, based on values of0.423 and 0.674 for the empirical coefficients a and b respectively, suggests a high flux immediately adjacent to the surface where the difference methods do not provide an estimate, and well approximates the log-law values for F above z = 0.15 m (Table 16.5). With regard to Equation 16.8, the empirical model fit would suggest that the settling velocity (wt) for the sampled dust from the nickel slag is around 0.1 ms . ... 

Molecular orbitals were one of the first molecular features that could be visualized with simple graphical hardware. The reason for this early representation is found in the complex theory of quantum chemistry. Basically, a structure is more attractive and easier to understand when orbitals are displayed, rather than numerical orbital coefficients. The molecular orbitals, calculated by semi-empirical or ab initio quantum mechanical methods, are represented by isosurfaces, corresponding to the electron density surfeces Figure 2-125a). 

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