Analysis of line intensity

As asserted in the previous section, the height of the photolines shown in Fig. 2.4 does not provide the correct measure of the intensity of a photoline. It will now be demonstrated that the appropriate measure for intensities is the area A under the line, recorded within a certain time interval, at a given intensity of the incident light, and corrected for the energy dispersion of the electron spectrometer. This quantity, called the dispersion corrected area AD, then depends in a transparent way on the photoionization cross section er and on other experimental parameters. In order to derive this relation, the photoionization process which occurs in a finite source volume has to be considered, and the convolution procedures described above have to be included. In order to facilitate the formulation, it has to be assumed that certain requirements are met. These concern  

T This prerequisite can be adapted easily to other situations. 

When these requirements are met, the intensity Iexp(Ekin, pass = fUsp) of a photoline with nominal energy °in, obtained by scanning the spectrometer voltage Usp in equal steps and for equal time intervals across the adapted value U°p follows from equ. (2.30) to be 

As demonstrated in Section 10.4.2, the convolution procedure leads to a simple and impressive result if an integration is performed over all spectrometer voltages (for the corresponding treatment of the experimental data see below). One derives (see also [WTW77]) 

Note that it is the fwhm value of the spectrometer function, A sp, which appears in the formula and not the value A exp attached to the observed photoline, as might have been expected naively (for the role of A exp see below). Hence, it is convenient to introduce the constant relative resolution R = (A sp/ °jn) of electrostatic deflection analysers introduced in equ. (1.49) which leads to the dispersion corrected area f 

JGsp(Ekin, Epass) dEkin = 1.06 AEsp AEsp. (2.32b) 

The cross section o is assumed to be constant in the energy range relevant for the selected photoionization process. (This condition is not fulfilled for a resonance which, therefore, needs a slightly different treatment.) Its angle-dependent terms in equ. (1.50) are neglected, i.e., 

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The spatial localization of H atoms in H2 and HD crystals found from analysis of the hyperfine structure of the EPR spectrum, is caused by the interaction of the uncoupled electron with the matrix protons [Miyazaki 1991 Miyazaki et al. 1991]. The mean distance between an H atom and protons of the nearest molecules was inferred from the ratio of line intensities for the allowed (without change in the nuclear spin projections. Am = 0) and forbidden (Am = 1) transitions. It equals 3.6-4.0 A and 2.3 A for the H2 and HD crystals respectively. It follows from comparison of these distances with the parameters of the hep lattice of H2 that the H atoms in the H2 crystal replace the molecules in the lattice nodes, while in the HD crystal they occupy the octahedral positions. 

Qualitative analysis may be made by searching the emission spectrum for characteristic elemental lines. With modem high resolution optics and computer control, the emission spectrum may be readily examined for the characteristic lines of a wide range of elements (Figure 8.13). Quantitative measurements are made on the basis of line intensities which are related to the various factors expressed in equation (8.1). Under constant excitation... 

When the spinning speed is much less than the magnitude of the CSA, spinning sidebands are generated at multiples of the spinning rate, reminiscent of the spinning sidebands for liquid samples discussed in Section 3.3. The envelope of the sidebands approximates the shape of the CSA-broadened line in the absence of spinning, and analysis of the intensity pattern provides the values of cru (X2, and cr3. 

The intensity ratio may be used to determine V3 independently of the analysis of line frequencies v and line splittings Av. The measurement of intensity ratios is a delicate experimental problem. After the determination of I0/Iv there still remain difficulties with some factors contained in the coefficient in Eq. (15). By measurement of the temperature dependence of I0/Iv the coefficient /may be eliminated and a value of Ev — E0 obtained. As this energy difference is a function of V3 and Ff an error in F influences V3 also. 

Figure 3. Statistical analysis of the intensity distribution of the high-resolution spectrum of C2 H2 at about 26,500 cm 1, including nearly 4000 lines. (Adapted from Ref. 55.) The solid line is the maximum entropy distribution (cf. Ref. 56) given by Eq. (3) with v = 3.2.

Changes in the ratios of line intensities were used to analyze the X-ray diffractograms of a series of isomorphously substituted MFI molecular sieves (30). The analysis revealed that, although large cations such as Al or Ga substitute in an ordered manner for Si, substitution by the small B cation leads to statistical replacement of Si. The results were interpreted as leading to a maximum theoretical substitution of about 4 (Al or Ga)/unit cell in the crystalline framework, but a Si/B mole ratio as low as 1 was predicted. 

The diffraction pattern of a liquid resembles a powder photograph except that the very sharp lines of the powder photograph are replaced by a few broad bands of reflected radiation. From an analysis of the intensity distribution in these broad bands, we can construct the radial distribution function for particles around a central particle in the liquid. This distribution function is interpreted in terms of the average number of atoms surrounding a central atom at the distance corresponding to the peak. 

An analysis of the intensities of the lines 220, 311, 222, 400, 331, 420, 422, and 333-511 showed that the best agreement between the calculated and measured values could be obtained... 

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