Extracting the pure profile

In Chapter 3, we saw that the distribution of the measured intensity h(x) can be expressed as the convolution product of the pure profile f(x) and the instrumental function g(x). We can write  

This relation was first suggested by Jones in 1938 [JON 38]. We have already noted, however, that this expression neglects not only the measurement noise, denoted by e(x), but also the contribution from the continuous background, denoted by b(x). In fact, the profile of a diffraction peak looks more like  

1 This relation and the next one have already been explained in Chapter 3 in references [3.1] and [3.2], respectively. They are shown here only to simplify the reading. 

The problem is therefore to determine the nature and the density of the stractural defects from the measurement of the experimental profile h(x) which contains the contribution from the instrament. There are two ways to go about solving this problem. The first method consists of deconvoluting this equation by using, in particular, the properties of Fourier transforms and extracting the pure profile which induced only by the defects. The second approach is described as convolutive . This time, the stractural defects are described without extracting the pure profile, but instead by taking into account the instrument s contribution, which is assumed to be an analytical function, either known or directly calculated from the characteristics of the diffractometer. This instrumental function is then convoluted with the functions expressing the contributions from the various microstructural effects that are assumed to be present. 

Regardless of which approach is used, it is necessaiy, of course, to know the instrumental function. As we have previously recalled, one of the methods for determining the contribution from the instrament is to calculate it from the diffractometer s physical characteristics [CHE 04]. This aspect has been the subject of research for about 20 years and was described in Chapter 3. However, the most commonly used method consists of producing a diffraction pattern with a sample assumed to be comprised of perfect crystals. In practice, several materials are used. Only one of them, lanthanum hexaboride which is sold by the NIST [NIST] is a standard acknowledged by the international community. 

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The essential apphcation of peak profile analysis involves the microstractural studies of the samples, which means that the machine s contribution has to be as small as possible. Digital processing techniques make it possible to properly extract the pure profile, but the quality of the study is higher if the widening of the peaks is essentially due to the sample. In practice, for this type of study, high resolution diffractometers are chosen, since their instmmental function can be very accurately determined. 

The different approaches used to extract the pure profile by deconvolution were described and compared in particular by Louer and Weigel [LOU 69b], and more recently by Armstrong and Kalceft [ARM 99]. Cernansky [CER 99] gave a detailed description of the inherent mathematical aspects of this deconvolution process. Aside from Stokes historical method [STO 48], the various approaches will not be explained here, only the basic ideas will. 

The oldest method used to extract the pure profile was suggested in 1948 by Stokes [STO 48]. It consists of neglecting the experimental noise and the contribution from the continuous background, and inverting the convolution product by a Fourier analysis of the peak profile. 

This method of extracting the pure profile by Fourier analysis has seen major developments and modem computational capabilities have made it rather easy to implement. However, the calculation of the Fourier coefficients imposes that there must be no peak overlaps. This condition considerably limits the application field of this method. Additionally, as we have already mentioned, the experimental noise is assumed to be zero. The presence of a non-zero noise causes oscillations in the resulting signal after deconvolution. This problem can be solved by using the methods described below. 

In 1968, Ergun [ERG 68] suggested extracting the pure profile of X-ray diffraction peaks by using an iterative method initially described by van Citter and... 

In some investigations it may be important to extract the pure component spectra (S) and the associated concentration (C) for the different species present within a mixture. A generalized solution to Eq. (4) has an infinite number of solutions available, while analysis methods such as PCA produce abstract factors that are linear combinations of the pure component spectra and concentrations [Eq. (15)]. An additional transformation step is required to rotate these abstract factors into the pure component spectra and pure concentration profiles. Two specific examples that allow this transformation step are presented below. 

The purpose of the process we explained in the previous sections is to extract from the experimental profile, h(x), the pure profile, f(x), which will then be used to determine the microstmctural characteristics of the sample. 

In order to learn about the effect of substituents close to the ester bond of surface-active esters on the kinetics of the hydrolysis, a series of well-defined PEG esters of fatty acids were synthesized and their hydrolysis rates were investigated both below and above the critical micelle concentration (CMC) [1]. The ester surfactants studied are shown in Fig. 1. They were synthesized in pure form by reacting the acid chloride with a large excess of tetra(ethylene glycol) using pyridine as nonnucleophilic base. The desired product, i.e., the PEG monoester, was removed from the excess tetra(ethylene glycol) by extraction into ethyl acetate from a saturated sodium chloride solution (so-called Weibull extraction). The degradation profile at various pH values was... 

To uniquely associate the unusual behavior of the collision observables with the existence of a reactive resonance, it is necessary to theoretically characterize the quantum state that gives rise to the Lorentzian profile in the partial cross-sections. Using the method of spectral quantization (SQ), it is possible to extract a Seigert state wavefunction from time-dependent quantum wavepackets using the Fourier relation Eq. (21). The state obtained in this way for J = 0 is shown in Fig. 7 this state is localized in the collinear F — H — D arrangement with 3-quanta of excitation in the asymmetric stretch mode, and 0-quanta of excitation in the bend and symmetric stretch modes. If the state pictured in Fig. 7 is used as an initial (prepared) state in a wavepacket calculation, one observes pure... 

Absorption spectra have also been used in the reexamination of pH-dependent color and structural transformations in aqueous solutions of some nonacylated anthocyanins and synthetic flavylium salts." ° In a recent study, the UV-Vis spectra of flower extracts of Hibiscus rosasinensis have been measured between 240 and 748 nm at pH values ranging from 1.1 to 13.0." Deconvolution of these spectra using the parallel factor analysis (PARAFAC) model permitted the study of anthocyanin systems without isolation and purification of the individual species (Figure 2.21). The model allowed identification of seven anthocyanin equilibrium forms, namely the flavylium cation, carbinol, quinoidal base, and E- and Z-chalcone and their ionized forms, as well as their relative concentrations as a function of pH. The spectral profiles recovered were in agreement with previous models of equilibrium forms reported in literature, based on studies of pure pigments. 

Characterisation of the aromatic profile in commercial guava essence and fresh fruit puree extracted with solvent yielded a total of 51 components [29]. Commercial essence was shown to be rich in components with low molecular weight, especially alcohols, esters, and aldehydes, whereas in the fresh fruit puree terpenic hydrocarbons and 3-hydroxy-2-butanone were the most abundant components. 

One way to achieve this is to replace the column by a loop of three to six smaller columns, as shown in Figure 12.10c. This is the principle of multi-column continuous chromatography (MCC). Since only pure fractions are collected, leaving mixed fractions to re-circulate through the columns, there is no need to achieve a complete separahon. Inlet (eluent, feed) and outlet (extract-most retained component, raffinate-least retained component) streams are moved periodically by one column according to the direction of the liquid flow and following the concentration profile inside the column. 

The determination of the true principal components was then performed by a sequential target transformation. No pure spectra were added to the data set so that the choice of the single components had to be made in the experimental set of spectra on the base of XRD, RHEED, and the compositions extracted from the sputter profiles. The spectra chosen to be the eigenvectors (i.e. the pure components), were then those with the smallest spoil. 

Example 13.4. The result of a typical X-ray measurement is shown in Fig. 13.10 for a galactocerebroside [605], The plot on the left side shows the normalized reflected X-ray beam intensity versus the incident angle a for two different film pressures. The pressure-area isotherm is shown in the inset, together with the points of measurement a and b. On the right side are the extracted electron density profiles normal to the film surface taken at the same film pressures. At 0 A we find the monolayer surface (top of the alkyl chains), a depth of -40 A corresponds to pure water. In between is the film. The measurement is so sensitive that we even find two different electron densities within the monolayer. This is illustrated by the dashed boxes denoted by film 1 and film 2 (shown for curve b only) which represents the simplified electron density distribution in the so-called two-box model. A box is defined as a part in the film of a certain thickness where the electron density is constant. In the two-box model the film is divided into two layers. Film 1 represents the hydrocarbon tails, film 2 corresponds to the mean electron density of the head groups. 

CFD has also been applied to analyze the flow patterns in a special counter-current solvent extraction column (Angelov et al, 1990). They used a singlephase flow representation and a k- turbulence model to compute the flow patterns in a periodic structure of the column. Validation of the computational results was achieved by applying LDA to obtain experimental data on the velocity profiles. CFD is a very useful tool here because the optimization of the performance of the extraction column from a geometrical point of view can be achieved with relative ease in comparison with a pure empirical strategy. 

Figure 17.3 illustrates the concentration profiles in the column train at the time immediately before the moment when the columns are switched. is positive, so the concentrations of both components in the columns III and IV are lower than the feed concentrations, as shown in Eq. 17.17. This results from the dilution of the feed, at the feed node, by the mobile phase stream entering into column III. From the raffinate node mass balance equation (Eq. 17.7g), we know that the concentration of the first component in the raffinate stream is exactly the same as its concentration at the inlet of column IV. At this stage, a stream of raffinate begins to be collected, at the end of this period. By contrast, the second component does not appear at the extract node during this first cycle and the fraction collected there is pure mobile phase. 

The ethanol column (C-1) has practically only stripping zone. Fig. 9.31-left shows composition profile both for liquid and vapour phase. The examination of the composition profiles highlights the role of the entrainer. In the zone close to the top the benzene extracts the ethanol in the liquid phase, and as a result increases the volatility of water, so that on lower stages the water is completely removed. In the lower part practically only the binary ethanol/benzene remains. The distillation trajectory starts from the ternary azeotrope, goes along the ethanol/ benzene saddle and terminates in the ethanol vertex. Because the boiling point of the azeotrope ethanol-water is close to the pure ethanol, the profile could easily jump to the ethanol/water azeotrope. Consequently, the design and operation of the column (C-1) is very sensitive. 

For overlapping peaks the data matrix contains linear combinations of the pure spectra of the overlapping components in its rows, and combinations of the pure elution profiles in its columns. Multivariate analysis of the data matrix may allow extraction of useful information from either the rows or columns of the matrix, or an edited form of the data matrix [107,116-118]. Factor analysis approaches or partial least-squares analysis can provide information on whether a given spectrum (known compound) or several known compounds are present in a peak. Principal component analysis and factor analysis can be used to estimate the maximum number of probable (unknown) components in a peak cluster. Deconvolution or iterative target factor analysis can then be used to estimate the relative concentration of each component with known spectra in a peak cluster. 

As a first step, test other pure chemicals, such as compounds from the same chemical class and structurally related compounds, chemicals that might have been administered to the commodity or its environment during production and potential (likely) environmental contaminants. Next, test for interference from co-extractives from target matrices by testing representative matrices collected from various sources that reflect the expected profile of sample submissions. Testing of material from a single source is not sufficient. Check also for interference from known metabolites and degradation products. 

Conventional approaches based on electrochemical techniques, surface tension, and extraction methods have allowed the estabhshment of thermodynamic and kinetic information concerning partition equilibrium, rate of charge transfer, and adsorption of surfactant and ionic species at the hquid/Uquid interface [4—6]. In particular, electrochemical methods are tremendously sensitive to charge transfer processes at this interface. For instance, conventional instm-mentation allowed the monitoring of ion transfer across a hquid/hquid interface supported on a single micron-sized hole [7, 8]. On the other hand, the concentration profile of species reacting at the interface can be accurately monitored by scanning electrochemical microscopy [9, 10]. However, a detailed picture of the chemical environment at the junction between the two immiscible liquids caimot be directly accessed by purely electrochemical means. The implementation of in-situ spectroscopic techniques has allowed access to key information such as ... 

For the countercurrent extractor, select column then extract in the bottom menu bar. The acetic acid feed should be input at the top of the column and the pure solvent at the bottom of the column. For thermal option select specify tenperature profile and use 293 on the stages. Select acetic acid as the key component for the first liquid phase and triethylamine as the key component for the second liquid phase. 

Preliminary studies have shown an increased level of radioimmu-noassayable VIP in the CSF of schizophrenic patients compared to controls (Morris et ai, 1980) however, following HPLC, this immunoreactivity splits into several fractions only one of which corresponds to VIP. Our suspicions with regard to the specificity of a purely immunoassay approach on crude extracts is again exemplified by the HPLC profile of immuno-reactive TRH and LH-RH in CSF (Fig. 7). In the control patient, multiple peaks of LH-RH activity are observed which are not present in the schizophenic sample. 

Fmin is minimal at given R and V if N= is necessary to achieve the specified separation (xs to Xd) even with zero recovery. We determine F R=c6) because Fmin(F=< )>Fnu (FExtractive profiles at FFm - As is seen in Figure 3, SN just sits on the rectification profile of a pure enough distillate composition if F=F. ... 

An example of multiple ion monitoring using a standard labelled with stable isotopes is the quantitation of morphine (9) and 6-acetylmorphine (11) in the blood of rabbits (16). The compounds were analyzed as the trifluoroacetyl derivatives (10 and 11) using the corresponding N-trideuteriomethyl derivatives (13 and 14) as the internal standards. The base peak in the mass spectra of each pair of compound appeared at m/e 364 and 367 (Fig. 5). As an internal check, two other ions at m/e 363 and 365 were also monitored. An example of the selected ion profiles for blank plasma and the plasma extracts of rabbits administered heroin are shown in Fig. 6. The sensitivity of detection is such that 500 pg can be detected although the pure compound can be detected in the range of 5 to 10 pg. In comparison, immunoassay methods for morphine... 

A wide range of bacteria and fungi, and some protozoan, algal and mammalian cells have the ability to hydrolyse fluorescein diacetate. Recently Schnurer and Rosswall have used this substrate to develop an assay for esterase activity in soil and straw based on the spectrophotometric determination of extracted fluorescein. The amounts formed increased linearly with time (up to 3h). The rates of hydrolysis were directly proportional to the amounts of soil used, to the amounts of fungal mycelium (Fusarium culmorum) or bacterial biomass (Pseudomonas denitrifleans) in buffered (pH 7.6) pure culture suspensions, and, in the case of fungi, to the amounts added to autoclaved soil, which alone was inactive. Esterase activities correlated with respiratory (O2 uptake) activities for samples from different depths in the soil profile. 

Some research was performed for s-triazines in the aqueous environments [58]. Figure 6 shows the GC-SIOMS results obtained by TIC and by SIM (m/z 213). For the TIC profile, a 10-p.l river water extract was injected (concentrated 100-fold, spiked with the model compounds [see caption] at concentrations of 0.8 mg/ml each). The peaks yield easily identified mass spectra that are almost identical to those obtained from the direct introduction of the pure s-triazines. Interestingly, both chromatograms in the SI mode show little interference from other compounds in the river water. 

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