Experimental intensity

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

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. 

For ail samples, both a.p. and s.o., irrespective of the preparation method, the experimental intensity ratios, V2p/Zr3d, increased proportionally to the V-content up to 3 atoms nm 2 (pjg 2). The ratio approaches those calculated with the spherical model proposed recently by Cimino et al. [27] (full line in Fig. 2). For ZV samples with V-content < 3 atoms nm 2, this finding shows that vanadium species are uniformly spread on the Zr02 surface. On ZV catalysts with a larger V content (not shown in Fig. 2), the intensity ratios were markedly larger than the corresponding values yielded by the spherical model. The results obtained on samples with V-content > 3 atoms nm 2 point therefore to a V surface enrichment. 

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... 

The mass attenuation coefficient values of the elements are available in the literature [46]. Therefore, the mass attenuation coefficient of a compound can be calculated. Thus and (in Eq. 15) can be calculated provided the molecular formulas of components 1 and 2 are known. It is then possible to calculate the intensity ratio, /u/(/ii)o> as a function of xx. This ratio can also be experimentally obtained. The intensity of peak i of a sample consisting of only 1 is determined [(/ii )o] This is followed by the determination of the intensity of the same peak in mixtures containing different weight fractions of 1 and 2. This enables the experimental intensity ratio, /n/(/n)o, to be obtained as a function of xx. The principles discussed above formed the basis for the successful analyses of quartz-beryllium oxide and quartz-potassium chloride binary mixtures [45]. 

Here I is the estimated experimental intensity, var denotes the variance, m is the mixing factor defined by the point spread function and g is... 

Here, is the experimental intensity (in unit of counts) measured from an energy-filtered CBED pattern and i and j are the pixel coordinate of the detector and n is the total number of points. is the theoretical intensity calculated with parameters ai, to and c is the normalization coefficient. The other commonly used GOF is the R-factor... 

Accurate measurements of low order structure factors are based on the refinement technique described in section 4. Using the small electron probe, a region of perfect crystal is selected for study. The measurements are made by comparing experimental intensity profiles across CBED disks (rocking curves) with calculations, as illustrated in fig. 5. The intensity was calculated using the Bloch wave method, with structure factors, absorption coefficients, the beam direction and thickness treated as refinement parameters. 

Figure 3. Total experimental intensities and background lines for tetramethylsilane. The two curves refer to two different scattering point to registration plane distances. ...

Structure determination from X-ray and neutron diffraction data is a standard procedure. Starting with a rough model, the accurate structure is determined using a least-squares structure refinement, which is based on kinematic diffraction and in which the differences between calculated and experimental intensities are minimized. X-ray and neutron diffraction are not applicable to all crystals. To determine crystal structures of thin layers on a substrate or small precipitates in a matrix (see figure 1) only electron diffraction (ED) can lead you to the crystal structure. 

The measurement of vibrational optical activity requires the optimization of signal quality, since the experimental intensities are between three and six orders of magnitude smaller than the parent IR absorption or Raman scattering intensities. To date all successful measurements have employed the principles of modulation spectroscopy so as to overcome short-term instabilities and noise and thereby to measure VOA intensities accurately. In this approach, the polarization of the incident radiation is modulated between left and tight circular states and the difference intensity, averaged over many modulation cycles, is retained. In spite of this common basis, there are major differences in measurement technique and instrumentation between VCD and ROA consequently, the basic experimental methodology of these two techniques will be described separately. 

We shall Illustrate our analysis by using one normalized characteristic linewidth distribution G(T, ) from the CONTIN method based on the experimental intensity-intensity time correlation functlon.measured at scattering angle e = 33, concentration C = 3.44x10 g/ml in MEK at 25°C. With fR - 2.56xll "m and... 

For comparision, one should inspect Fig. 14 with the PE spectrum of 2.4.6-tri-tert-butyl-X -phosphorin 24 (p. 37). In going from 24 to l.l-dimethoxy-2.4.6-tri-tert-butyl-X -phosphorin the first band is shifted by 1.3 eV to lower ionization potential, while the second band remains at the same ionization potential. Due to the experimental intensity ratio of band 1 band 2 = 1 2 in 24, the second band was attributed to the 112 and n MOs. In l.l-dimethoxy-2.4.6-tri-tert-butyl-phosphorin the second band does not include the n MO and has thus the same intensity as the first band. These observations experimentally support the orbital configuration of X -phosphorins and X -phosphorins predicted by Schweig and coworkers... 

Fig. 4.3. Experimental intensity vs. voltage (energy) curves for electron diffraction from at Pt(l 11) surface. Beams are identified by different labels (h,k) representing reciprocal lattice vectors parallel to the surface. An incidence angle of 4° from the surface normal is used...

Data was taken in the electron energy range of 10-200 eV, but little sensitivity to the organic adsorbate is found above 100 eV. The observed diffraction pattern arises from three equivalent 120° — rotated domains of (2 X 2) unit cells. The optimum agree "ent between calculated and experimental intensity data for the metastable acetylene structure is achieved for an atop site coordination. The molecule is located at a z-distance of 2.5 A from the underlying surface platinum atom. However, the best agreement is obtained if the molecule is moved toward a triangular site, where there is a platinum atom in the second layer, by 0.25 A, as shown in Fig. 7.2. 

Warren (1937, 1940) calculates iS from the experimental intensity curve and plots it as a function of 8 he then carries out the integration... 

At this point photoionization cross sections have been computed mostly for diatomic molecules, rr-electron systems, and other relatively small molecules [see Rabalais (242) for a summary of this work up to 1976]. Very few photoionization cross section calculations have been performed (108) on transition metal systems and the agreement with experimental intensities is rather poor. For the most part, therefore, one must rely on empirical trends when dealing with the photoionization of metal-containing molecules. A number of such trends have now emerged and are useful for spectral assignment. 

The standard method for normalisation of diffracted intensity data into electron units, is to compute both the mean square atomic scattering factor and the mean incoherent scatter for the particular molecular repeat over a range of high two theta values (say 40°-60°) where their total value can be considered to be equivalent to the actual diffraction from the molecular system concerned. An appropriate normalisation factor is then applied to the experimental intensity data after geometrical correction and, finally, incoherent scatter is subtracted ( 1 ). 

The bands in an IR spectrum have not just positions ( frequencies , denoted by various wavenumbers), but also intensities. IR intensities present considerably more difficulties in their measurement and theoretical calculation than do frequencies, and in fact experimental intensities are not routinely quantified, but are commonly merely described as weak, medium, or strong. To calculate an IR spectrum for visual comparison with experiment it is desirable to compute both... 

Fig. 2. Experimental intensities (crosses) measured with AgKa-radiation for aconcen-trated erbium(III) chloride solution. The values have been normalized with the use of calculated values for the independent coherent scattering (solid line). The lower part shows the reduced intensities, s-i(s), where its) is obtained by subtracting the independent coherent scattering from the observed intensity values.

Figure 5. Low energy ion scattering on Rh/TiC>2 model catalysts (A) structure models (B) expected intensity profiles (C) experimental intensity profiles (adopted from Ref. 62).

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