Matrix correction factors

The correction factor matrix is particularly simple to calculate since it too is diagonal with all diagonal elements equal... 

Since [A ] and [0] are diagonal we can calculate the correction factor matrix and the... 

The method of successive substitution can be a very effective way of computing the from Eqs. 8.3.24 when the mole fractions at both ends of the diffusion path y-g and y g, are known. In practice, we start from an initial guess of the fluxes and compute the rate factor matrix [< >]. The correction factor matrix [a] may be calculated from an application of Sylvester s expansion formula (Eq. A.5.20)... 

For this special case Eqs. 8.3.46 for the correction factor matrix simplify to... 

Interestingly, even for equimolar transfer N = 0, the correction factor matrix [3] does not reduce to [/] (although it is well approximated by [/]). This means that the composition profiles (as computed from Eq. 8.3.12 and shown in Fig. 8.7) will not be truly linear (although it is hard to discern this fact from Fig. 8.7). Contrast this with a binary system for which Nf = 0 leads to linear composition profiles and 3o equal to unity. 

The eigenvalues of the correction factor matrix follow from Eq. 8.4.28 as... 

It is the eigenvalues (literally characteristic values ) of [0] that characterize the correction factor matrix [S]. Thus, the scalar rate factor 0 and correction factor S when multiplied by identity matrices frequently are quite good models for the behavior of the complete matrices [0] (or [ ]) and [H] in the exact and linearized methods. 

The interaction phenomena discussed earlier for the ideal gas case will also be possible for nonideal fluid mixtures, for which [T] contribute to the matrix [A ] by means of its separate influence on [A ], the zero flux matrix, and [3], the correction factor matrix. 

The first estimate of the diffusion fluxes is calculated from Eq. 9.3.35 with the correction factor matrix taken to be the identity matrix... 

The above computations were carried out with Eqs. 9.3.39 for the eigenvalues of the correction factor matrix. As noted earlier, this involves the computation of the error function that is significantly more time consuming than the exponential function needed for the film model correction factor. With the eigenvalues of [H] given by the film model... 

These values are within 5% of the values calculated with the penetration theory correction factor matrix and support our earlier suggestion that it is sufficient to use the simpler film model correction factor matrix in multicomponent mass transfer calculations at high mass transfer rates. ... 

The eigenvalues of the correction factor matrix are obtained from the film theory expression (Eq. 10.4.35), and the eigenvalues of the high flux mass transfer coefficient matrix follow from Eq. 10.4.32... 

The Compton scatter matrix correction is based on the observation, referred to above, that the intensity of the Compton scatter peak is inversely proportional to the bulk matrix attenuation correction factor. Matrix corrections may then be applied by simply normalizing all fluorescence measurements from a sample to the intensity of the Compton scatter line derived from one of the characteristic fluorescence lines from the X-ray source. This procedure is, however, subject to an important restriction. Corrections are only valid providing no significant absorption edge intervenes between the energy of the Compton scatter peak and the fluorescence line... 

Another principal difficulty is that the precise effect of local dynamics on the NOE intensity cannot be determined from the data. The dynamic correction factor [85] describes the ratio of the effects of distance and angular fluctuations. Theoretical studies based on NOE intensities extracted from molecular dynamics trajectories [86,87] are helpful to understand the detailed relationship between NMR parameters and local dynamics and may lead to structure-dependent corrections. In an implicit way, an estimate of the dynamic correction factor has been used in an ensemble relaxation matrix refinement by including order parameters for proton-proton vectors derived from molecular dynamics calculations [72]. One remaining challenge is to incorporate data describing the local dynamics of the molecule directly into the refinement, in such a way that an order parameter calculated from the calculated ensemble is similar to the measured order parameter. 

Since the composition of the unknown appears in each of the correction factors, it is necessary to make an initial estimate of the composition (taken as the measured lvalue normalized by the sum of all lvalues), predict new lvalues from the composition and the ZAF correction factors, and iterate, testing the measured lvalues and the calculated lvalues for convergence. A closely related procedure to the ZAF method is the so-called ())(pz) method, which uses an analytic description of the X-ray depth distribution function determined from experimental measurements to provide a basis for calculating matrix correction factors. 

Requirements for standards used In macro- and microspectrofluorometry differ, depending on whether they are used for Instrument calibration, standardization, or assessment of method accuracy. Specific examples are given of standards for quantum yield, number of quanta, and decay time, and for calibration of Instrument parameters. Including wavelength, spectral responslvlty (determining correction factors for luminescence spectra), stability, and linearity. Differences In requirements for macro- and micro-standards are considered, and specific materials used for each are compared. Pure compounds and matrix-matched standards are listed for standardization and assessment of method accuracy, and existing Standard Reference Materials are discussed. 

Accuracy (systematic error or bias) expresses the closeness of the measured value to the true or actual value. Accuracy is usually expressed as the percentage recovery of added analyte. Acceptable average analyte recovery for determinative procedures is 80-110% for a tolerance of > 100 p-g kg and 60-110% is acceptable for a tolerance of < 100 p-g kg Correction factors are not allowed. Methods utilizing internal standards may have lower analyte absolute recovery values. Internal standard suitability needs to be verified by showing that the extraction efficiencies and response factors of the internal standard are similar to those of the analyte over the entire concentration range. The analyst should be aware that in residue analysis the recovery of the fortified marker residue from the control matrix might not be similar to the recovery from an incurred marker residue. 

Iteration. Matrix correction factors are dependent on the composition of the specimen, which is not known initially. Estimated concentrations are initially used in the correction factor calculations and, having applied the corrections thus obtained the calculations are repeated until convergence is obtained, i.e. when the concentrations do not change significantly between successive calculations. 

For a balanced historical record I should add that the late W. E. Blumberg has been cited to state (W. R. Dunham, personal communication) that One does not need the Aasa factor if one does not make the Aasa mistake, by which Bill meant to say that if one simulates powder spectra with proper energy matrix diagonalization (as he apparently did in the late 1960s in the Bell Telephone Laboratories in Murray Hill, New Jersey), instead of with an analytical expression from perturbation theory, then the correction factor does not apply. What this all means I hope to make clear later in the course of this book. 

Fundamental Parameters (FP) are universal standardless, factory built-in calibration programs that describe the physics of the detector s response to pure elements, correction factors for overlapping peaks, and a number of other parameters to estimate element concentration while theoretically correcting for matrix discrepancies (e.g., Figure 1987). FP should be used for accurately measuring samples of unknown chemical composition in which concentrations of light and heavy elements may vary from ppm to high percent levels. 

Internal standards are also used in trace metal analysis by inductively coupled plasma atomic emission spectrometry (ICP-AES) and inductively coupled plasma mass spectrometry (ICP-MS) techniques. An internal standard solution is added to ICP-MS and ICP-AES samples to correct for matrix effects, and the response to the internal standard serves as a correction factor for all other analytes (see also chapter 2). 

Griffiths et al. [92] quantified Pt, Pd and Rh in autocatalyst digests by ICP with a CCD detector array. They compared univariate techniques (pure standards, pure standards with interelement correction factors and matrix-matched standards) and PLS, the latter being superior in general, although less effective... 

In this equation a correction factor Gxv l — m) appears. We can order the sites appearing in this equation with respect to their importance the site l is the main site (first order), n is the second site and m is a site which is far away. The matrix Gx now becomes non-symmetric. The equation must fulfill all the conditions which are fulfilled by the ansatz (9.1.20). To this end Gxv l — rn) must be unity for large distances 11 — m. We use equation (9.1.30) and define a lattice average ... 

Measured and computed values of the matrix coefficient are shown in Table 7.13. The values agree within a few percent except for Fe and Mn in radishes, where the difference is 6%. A matrix correction factor of 2 means that the combined attenuation of the exciting and fluorescent X-rays is 50%. In radishes, about half of this figure is from the cellulose and the other half from the presence of 6% potassium. 

To this effect, in analytical chemistry it has been good practice that the analyst obtains a reference material with a matrix Z, similar - but of course not identical - to the one of the material X which needs to be measured. By performing the same operations on a sample of matrix reference material Z as performed on the sample with matrix X, an estimate can be obtained of the overall correction factor X(aE,Z). The value of the amount content 6(aE,Z)RM cert of the reference material as supplied by the reference material producer is known. The value b(aE,Z)sampie Qbs is observed by the analyst Hence, a correction factor can be calculated X(aE,Z) for losses during digestion and recovery etc. (the chemical operations in Fig. 2) as determined with the help of the reference material. It can be applied to the measurement on the unknown sample. In short, this process can be described by Eq 2 ... 

As shown in the central chain, upper part in Fig. 1, ( HELP from matrix RM ),the analyst can now use the correction factor from Eq 2 and substitute it for the unknown correction factor X(aE,X) in Eq 1. This enables a correction to be made of the value observed in the unknown sample. In summary the analyst simply substitutes X(aE,X) for K(aE,Z). 

Thus the matrix reference material has fulfilled another function than the AS function for the analyst, a function which is not located in the traceability chain, but outside the chain it enables the analyst to make an independent assessment of the possible magnitude of the conversion factor K, thus assessing - possibly reducing - the uncertainty of the measurement by carrying out a correction . But even this correction carries an uncertainty which must be evaluated. The problem of this correction factor has been treated elsewhere in more technical detail under the name recovery factor ... 

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