Indices of reflection

For the spots on layer lines above and below the equator, one index (l) is given by inspection. It should be remembered that the indices of reflections represent phase-differences between waves diffracted by neighbouring units aloiig the three axial directions (see p. 141). The spots on the first layer line above the equator lie oh a cone for which n in the equation nX = c cos (see p. 149) is 1 this means that waves coming from any one diffracting unit are one wavelength behind those from the next diffracting unit above it in fact, n in the cone equation is Z, the third index number. Thus, all spots on the fourth layer line (fourth cone) above the equator are from hk4 planes (those on the fourth layer line below the equator are from hk4 planes), and so on. 

It is not coincidental that I use the variable names h, k, and I for both the indices of planes in the crystal and the indices of reflections in the diffraction pattern (Chapter 2, Section V). I will show later that in fact the set of planes (hkl) produces the reflection hkl of the diffraction pattern. In the terms used in Chapter 2, each set of parallel planes in the crystal produces one reflection, or one term in the Fourier series that describes the electron density within the unit cell. The intensity of that reflection depends upon the electron distribution and density along the planes that produce the reflection. 

Equation (5.18) tells us how to calculate p(jc,y,z) simply construct a Fourier series using the structure factors Fhkl. For each term in the series, h, k, and 1 are the indices of reflection hkl, and Fhkl is the structure factor that describes the reflection. Each structure factor Fhkl is a complete description of a diffracted ray recorded as reflection hkl. Being a wave equation, Fhkl must specify frequency, amplitude, and phase. Its frequency is that of the X-ray source. Its amplitude is proportional to (- j /)1/2, the square root of the measured intensity Ihkl of reflectionhkl. Its phase is unknown and is the only additional information the crystallographer needs in order to compute p(x,y,z) and thus... 

X-ray powder diffraction (XRPD) is the analysis of a powder sample. The typical output is a plot of intensity versus the diffraction angle (20). Such a plot can be considered a fingerprint of the crystal structure, and is useful for determination of crystallographic sameness of samples by pattern comparison. A crystalline material will exhibit peaks indicative of reflections from specific atomic planes. The patterns are representative of the structure, but do not give positional information about the atoms in the molecule. One peak will be exhibited for all repeating planes with the same spacing. An amorphous sample, on the other hand, will exhibit a broad hump in the pattern called an amorphous halo, as shown in Fig. 5. 

After noting down the indices of reflecting planes, the next task is to find whether there are any systematic absences. The absences due to centering of the lattice (f.c.c., b.c.c., etc.) occur throughout the reciprocal space and a general survey will reveal the lattice type which is to be determined. The presence of glide planes and screw axis will affect the absences of certain layers or rows of points in the reciprocal lattice. 

The band structure of semiconducting SmSe shown in Fig. 57 is calculated with the augmented plane-wave method (APW). The approximation is used to obtain the muffin-tin potentials with a = 0.67 for the exchange, assuming an intermediate state for Sm with the configuration 4f d instead of pure 4f . The atomic wave functions are derived in the Hartree-Fock-Slater approximation, Farberovich [1]. The band structure model of [1] is qualitatively confirmed by an analysis of the reflection and electroreflection spectra of SmSe single crystals with the minimum direct gap located at the X point. However, the next direct gap is at r and no indications of reflection structures which are attributable to K point excitations are observed, Kurita et al. [2]. Earlier, the band structure for the T-X (i.e. (100)) direction was calculated by... 

Figure 5 (a) Sample preparation/presentation for Raman spectroscopy. Adapted partly from Everall, N. In Analytical Applications of Raman Spectroscopy, Pelletier, M. J., Ed. Blackwell Publishers Ames, 2000 Chapter 4, p 127. Sample preparation/presentation for (b) NIR spectroscopy and (c) IR ATR spectroscopy (dp, penetration depth of radiation into the sample rti//)2. refractive indices of reflection element/sample A, wavelength , angle of incidence of radiation on the interface of the reflection element/sample). 

Fig. 5. shows six ultrasonic reflection tomograms. Three of these are from the Plexiglas specimen (shown left) and three are from the AlSi-alloy (shown right). The tomograms are reconstructed from reflection data measured across the plane (b), (c) and (e), respectively. The dark regions indicate high reflectivity and represent specimen interfaces and discontinuities. 

A problem obviously exists in trying to characterise anomalies in concrete due to the limitations of the individual techniques. Even a simple problem such as measurement of concrete thickness can result in misleading data if complementary measurements are not made In Fig. 7 and 8 the results of Impact Echo and SASW on concrete slabs are shown. The lE-result indicates a reflecting boundary at a depth corresponding to a frequency of transient stress wave reflection of 5.2 KHz. This is equivalent to a depth of 530 mm for a compression wave speed (Cp) of 3000 m/s, or 706 mm if Cp = 4000 m/s. Does the reflection come from a crack, void or back-side of a wall, and what is the true Cp ... 

All tenus in the sum vanish if / is odd, so (00/) reflections will be observed only if / is even. Similar restrictions apply to classes of reflections with two indices equal to zero for other types of screw axis and to classes with one index equal to zero for glide planes. These systematic absences, which are tabulated m the International Tables for Crystallography vol A, may be used to identify the space group, or at least limit die... 

Once the atoms arc defined, the bonds between them arc specified in a bond block. Each line of this block specifies which two atoms are bonded, the multiplicity of the bond (the bond type entry) and the stereo configuration of the bond (there arc also three additional fields that arc unused in Molfiles and usually set to 0). The indices of the atoms reflect the order of their appearance in the atom block. In the example analyzed, V relates to the first carbon atom (see also Figure 2-24). "2" to the second one, 3" to oxygen atom, etc. Then the two first lines of the bond block of the analyzed file (Figure 2-29) describe the single bond between the two carbon atoms C1-C2 and the double bond C2=0-5, respectively. 

Polydisperse polymers do not yield sharp peaks in the detector output as indicated in Fig. 9.14. Instead, broad bands are produced which reflect the polydispersity of synthetic polymers. Assuming that suitable calibration data are available, we can construct molecular weight distributions from this kind of experimental data. An indication of how this is done is provided in the following example. 

Several properties of the filler are important to the compounder (279). Properties that are frequentiy reported by fumed sihca manufacturers include the acidity of the filler, nitrogen adsorption, oil absorption, and particle size distribution (280,281). The adsorption techniques provide a measure of the surface area of the filler, whereas oil absorption is an indication of the stmcture of the filler (282). Measurement of the sdanol concentration is critical, and some techniques that are commonly used in the industry to estimate this parameter are the methyl red absorption and methanol wettabihty (273,274,277) tests. Other techniques include various spectroscopies, such as diffuse reflectance infrared spectroscopy (drift), inverse gas chromatography (igc), photoacoustic ir, nmr, Raman, and surface forces apparatus (277,283—290). 

Color. The visual color, from white to dark brown, of sugar and sugar products is used as a general indication of quaUty and degree of refinement. Standard methods are described for the spectrophotometric deterrnination of sugar color that specify solution concentration, pH, filtration procedure, and wavelength of deterrnination. Color or visual appearance may also be assessed by reflectance measurements. 

Because of Bragg s explanation of diffraction of x-rays from a crystal as being like reflections from famihes of planes, the diffraction spots ate usually called "reflections." Each reflection is identified with three integer indices, h, k, and / For the set of planes shown in Figure 7, the indices of the corresponding reflection are /i = 1, = 0, and I = 2. 

Indexings and Lattice Parameter Determination. From a powder pattern of a single component it is possible to determine the indices of many reflections. From this information and the 20-values for the reflections, it is possible to determine the unit cell parameters. As with single crystals this information can then be used to identify the material by searching the NIST Crystal Data File (see "SmaU Molecule Single Stmcture Determination" above). 

Breakings Point (FRAAS, IP 80/53). This test of the Institute of Petroleum is an approximate indication of the temperature at which an asphalt possesses no ductiUty and would reflect brittle fracture conditions. 

Vitrinite Reflectance. The amount of light reflected from a poHshed plane surface of a coal particle under specified illumination conditions increases with the aromaticity of the sample and the rank of the coal or maceral. Precise measurements of reflectance, usually expressed as a percentage, ate used as an indication of coal rank. 

Another indication of the growth of the powder coating market is reflected in the membership statistics of the Powder Coating Institute (PCI). Prom 1987 to 1991 the number of members associated with powder coating manufacturers increased from 5 to 22 equipment suppHers from 3 to 9 custom apphcators from 1 to 29 and suppHers to the powder coating industry from 5 to 38 (78). 

There are alternatives to the addition-elimination mechanism for nucleophilic substitution of acyl chlorides. Certain acyl chlorides are known to react with alcohols by a dissociative mechanism in which acylium ions are intermediates. This mechanism is observed with aroyl halides having electron-releasing substituents. Other acyl halides show reactivity indicative of mixed or borderline mechanisms. The existence of the SnI-like dissociative mechanism reflects the relative stability of acylium ions. 

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