Matrix dependence

Approach to restoring of stresses SD in the three-dimensional event requires for each pixel determinations of matrix with six independent elements. Type of matrixes depends on chosen coordinate systems. It is arised a question, how to present such result for operator that he shall be able to value stresses and their SD. One of the possible ways is a calculation and a presenting in the form of image of SD of stresses tensor invariants. For three-dimensional SDS relative increase of time of spreading of US waves, polarized in directions of main axises of stresses tensor ... 

The second step determines the LCAO coefficients by standard methods for matrix diagonalization. In an Extended Hiickel calculation, this results in molecular orbital coefficients and orbital energies. Ab initio and NDO calculations repeat these two steps iteratively because, in addition to the integrals over atomic orbitals, the elements of the energy matrix depend upon the coefficients of the occupied orbitals. HyperChem ends the iterations when the coefficients or the computed energy no longer change the solution is then self-consistent. The method is known as Self-Consistent Field (SCF) calculation. 

The two equations couple because the alpha Fock matrix depends on both the alpha and the beta solutions, C and cP (and sim ilarly for the beta Fock matrix). The self-consistent dependence of the Fock matrix on molecular orbital coefficients is best represen ted, as before, via the den sity matrices an d pP, wh ich essen -tially state the probability of describing an electron of alpha spin, and the probability of finding one of beta spin ... 

A quantitative solution to an equilibrium problem may give an answer that does not agree with the value measured experimentally. This result occurs when the equilibrium constant based on concentrations is matrix-dependent. The true, thermodynamic equilibrium constant is based on the activities, a, of the reactants and products. A species activity is related to its molar concentration by an activity coefficient, where a = Yi[ ] Activity coefficients often can be calculated, making possible a more rigorous treatment of equilibria. 

Although EXo /ATcd is standard-state potential for the analyte s half-reaction, a matrix-dependent formal potential is used in its place. After the equivalence point, the potential is easiest to calculate using the Nernst equation for the titrant s half-reaction, since significant quantities of its oxidized and reduced forms are present. 

Another problem is that the Nernst equation is a function of activities, not concentrations. As a result, cell potentials may show significant matrix effects. This problem is compounded when the analyte participates in additional equilibria. For example, the standard-state potential for the Fe "/Fe " redox couple is +0.767 V in 1 M 1TC104, H-0.70 V in 1 M ITCl, and -H0.53 in 10 M ITCl. The shift toward more negative potentials with an increasing concentration of ITCl is due to chloride s ability to form stronger complexes with Fe " than with Fe ". This problem can be minimized by replacing the standard-state potential with a matrix-dependent formal potential. Most tables of standard-state potentials also include a list of selected formal potentials (see Appendix 3D). 

Spike recoveries for samples are used to detect systematic errors due to the sample matrix or the stability of the sample after its collection. Ideally, samples should be spiked in the field at a concentration between 1 and 10 times the expected concentration of the analyte or 5 to 50 times the method s detection limit, whichever is larger. If the recovery for a field spike is unacceptable, then a sample is spiked in the laboratory and analyzed immediately. If the recovery for the laboratory spike is acceptable, then the poor recovery for the field spike may be due to the sample s deterioration during storage. When the recovery for the laboratory spike also is unacceptable, the most probable cause is a matrix-dependent relationship between the analytical signal and the concentration of the analyte. In this case the samples should be analyzed by the method of standard additions. Typical limits for acceptable spike recoveries for the analysis of waters and wastewaters are shown in Table 15.1. ... 

Ema data can be quantitated to provide elemental concentrations, but several corrections are necessary to account for matrix effects adequately. One weU-known method for matrix correction is the 2af method (7,31). This approach is based on calculated corrections for major matrix-dependent effects which alter the intensity of x-rays observed at a particular energy after being emitted from the corresponding atoms. The 2af method corrects for differences between elements in electron stopping power and backscattering (the correction), self-absorption of x-rays by the matrix (the a correction), and the excitation of x-rays from one element by x-rays emitted from a different element, or in other words, secondary fluorescence (the f correction). 

Standards used to constmct a cahbration curve must be prepared such that the matrix of the standard is identical to the sample s matrix because the values of the parameters k and b associated with a linear cahbration curve are matrix dependent. Many areas of chemical analysis are plagued by matrix effects, and it is often difficult to duphcate the sample matrix when preparing external standards. Because it is desirable to eliminate matrix effects, cahbration in the sample matrix itself can be performed. This approach is called the standard addition method (SAM) (14). In this method, the standards are added to the sample matrix and the response of the analyte plus the standard is monitored as a function of the added amount of the standard. The initial response is assumed to be Rq, and the relationship between the response and the concentration of the analyte is... 

Yet another strength of SNMS is the ability to measure elemental concentrations accurately at interfaces, as illustrated in Figure 8, which shows the results of the measurement of N and O in a Ti thin film on Si. A substantial oxide film has formed on the exposed Ti surface. The interior of the Ti film is free of N and O, but significant amounts of both are observed at the Ti/Si interface. SNMS is as sensitive to O as to N, and both the O and N contents are quantitatively measured in all regions of the structure, including the interface regions. Quantitation at the interface transition between two matrix types is difficult for SIMS due to the matrix dependence of ion yields. 

Although the RSF contains matrix-dependent quantities, their variations are damped to some extent by virtue of taking ratios, and in practice the RSF is assumed constant for low concentrations of A (e. g. <1 atom%). It can be evaluated from measurements on a well-characterized set of standards containing A in known dilute concentrations. The accuracy of the method, however, is not as high as in laser-SNMS and XPS. 

The ratio Db/Da is a so-called relative sensitivity factor D. This ratio is mostly determined by one element, e. g. the element for insulating samples, silicon, which is one of the main components of glasses. By use of the equation that the sum of the concentrations of all elements is equal to unity, the bulk concentrations can be determined directly from the measured intensities and the known D-factors, if all components of the sample are known. The linearity of the detected intensity and the flux of the sputtered neutrals in IBSCA and SNMS has been demonstrated for silicate glasses [4.253]. For SNMS the lower matrix dependence has been shown for a variety of samples [4.263]. Comparison of normalized SNMS and IBSCA signals for Na and Pb as prominent components of optical glasses shows that a fairly good linear dependence exists (Fig. 4.49). 

Diagonal elements of the HF matrix depend only on the nature of the atom on which x is centred. 

The change in the Fock matrix depends on the change in the density matrix. Combined... 

Without considering photon noise, Le Louam and Tallon (2002) have found that the 4 LGSs 3D mapping system delivers a Strehl ratio of 80% on axis when a single LGS would have been limited to S f 10%. These numbers have to be compared with S f 85% obtained with the same MCAO device fed with a NGS. With a field of view of 100 arcsec, fhey gef S f 30% wifh little anisoplanatism. Performances are weakly dependent on errors in the altitude of the turbulent layers (which could be measured from the 3D mapping system, at the expenses of the linearity of the equations, since the interaction matrix depends on layer altitudes). Unsensed layers do not produce a significant anisoplanatism, as the central obscuration within which no measurement is available. 

The relative extinction-coefficients for Agi,2,s determined by pho-toaggregation procedures were found not to be strongly matrix-dependent (see Table VIII). Moreover, the results for Agj were in good agreement with those obtained by quantitative, metal-atom deposition-techniques. 

For ease of use and wavelength accuracies of 1-2 nm, organic materials or inorganic ions in solution have been recommended as standards (Table II). However, these must be used carefully because (a) the peak maxima are matrix dependent, (b) narrow Instrumental bandpasses are necessary, (c) impurities may affect peak location, and (d) the peak wavelength values have generally not been certified (11). 

The extraction procedure begins the process to separate the analytes from the matrix and present the material in a form that can be more easily analyzed. The type of extraction step that is used for a particular matrix depends on the nature of the matrix and analytes. There are two competing views in the extraction process among chemists. Some prefer to extract the analytes exhaustively from the matrix and rely on extensive cleanup to remove matrix co-extractives. Others prefer the just enough extraction concept, in which the selectivity of the extraction process is honed as much as possible... 

Without isotope dilution, the simultaneous measurement of U and Th is essential in LA-MC-ICPMS, although large (10-100% level) elemental, matrix-dependent fractionation effects can still be observed between U and Th (e g., Stirling et al. 2000). As a result, Th/U ratios can be systematically lower, and apparent U- Th-ages... 

Where is the initial analyte concentration in the liquid phase, C( the concentration of analyte in the gas phase, K the gas-liquid partition coefficient for the analyte at the analysis temperature, V, the volume of liquid phase, and V, the volume of gas phase (318-321,324,325). From equation (8.3) it can be seen that the concentration of the analyte in the headspace above a liquid in equilibrium with a vapor phase will depend on the volume ratio of the geis and liquid phases and the compound-specific partition coefficient which, in turn, is matrix dependent. The sensitivity 1 of the headspace sampling method can be increased in some instances adjusting the pH, salting out or raising the... 

Strong matrix dependency (physical conditions of sample influence SCF penetration)... 

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