Number of reflections

A number of rotation photographs were made with molybdenum. K-radiation filtered through a zirconium oxide filter to isolate the Ka line. The positions of useful reflections, the indices of the planes producing them, and their visually estimated intensities are given in Table V. The factor placed beside the estimated intensity is a correction for the varying time of reflection, namely Vi — (wl/dsin 20)2, where l is the wave-length, and u and d represent respectively the index of the axis of rotation and the unit translation along it1). (A number of reflections... 

Recognizing the impracticability of determining the positions of the oxygen atoms from X-ray data, we have predicted a set of values for the oxygen parameters with the use of assumed minimum interatomic distances which is found to account satisfactorily for tbe observed intensities of a large number of reflections and which also leads to a structure which is physically reasonable. 

The main disadvantages of the design are the number of reflections, the implied emissivity", and the difficulty to manufacture the quaternary mirror. ... 

The simplex procedure can be used when there is experimental error. Under these circumstances one point may be so much higher than any of the surrounding points that it never becomes eliminated. The result could be that the system rotates forever around this point. To prevent this, the age (number of reflections made after its determination) of each point is determined after each reflection. If the age of any point is greater than a certain number, that point is reevaluated. If it is still higher than any of the surrounding points, the step size should be reduced or the region should be checked to determine if this is a true optimum. 

The fifth class listed above contains a 1-cycle and a 2-cycle in its permutation part, with an odd number of reflections (in this case, one reflection) on sites involved in the two-cycle. The seventh class has a permutation part consisting of a cyclic permutation of the three sites, with an odd number of reflections (either one or three). 

A special O-ring cell design is needed for in situ infrared (IR) vibrational characterization of an electrochemical interface. The absorption of one monolayer (i.e. <1015 cm 2 vibrators) can be measured if the silicon electrode is shaped as an attenuated total reflection (ATR) prism, which allows for working in a multiple-in-ternal-reflection geometry. A set-up as shown in Fig. 1.9 enhances the vibrational signal proportional to the number of reflections and restricts the equivalent path in the electrolyte to a value close to the product of the number of reflections by the penetration depth of the IR radiation in the electrolyte, which is typically a tenth of the wavelength. The best compromise in terms of sensitivity often leads to about ten reflections [Oz2]. 

Oblique texture electron diffraction patterns (Fig.7) are main experimental material for the electron diffraction structure analysis since they have a large number of reflections, so that only one pattern can provide an almost three dimensional set of diffraction reflections. The formation of textures in specimens is achieved by the use of orienting agents. As in the preparation of single crystal films, use can be made of orienting supports, mechanical action or even the application of an electrical field. Electron... 

Figure 11. (a) formation of layer lines in the reciprocal lattice of a texture for a orthogonal unit cell, (b) the doubling of number of circular scattering regions in the reciprocal lattice of a texture and therefore the number of reflections on an ellipse of a pattern for a non-orthogonal unit cell, (c) measurement of a values of 2r and 2 D on a texture pattern. 

Where Jo and S are the current density of the primary beam and the area of the irradiated sample, Z is the wave length, Ohki the structure factor amplitude, Q the volume cell, Z a factor that takes the microstructure of sample into account (Zm - for a mosaic single crystalline film, Zt - for a texture film), t is the sample thickness, dhu the interplanar spacing, a represents the mean angular distribution of the microcrystallites in the film, p is a multiplicity factor (accounts for the number of reflections of coincidence), R is a horizontal coordinate of a particular reflection in DP from textures and (p is the tilt angle of the sample. In the case of polycrystalline films, a local intensity is usually measured and the corresponding relation is ... 

One big problem, which arises mainly in crystals with rather large unit cells, is the overlapping of reflections. This prevents an accurate measurement of the local integrated intensities of a large number of reflections. A solution to this problem can be the 2D pattern decomposition method, which is based on the same principles as in X-ray powder diffraction. This method takes into account the dependence of intensities on the particle orientation function and the size of microcrystals. It is therefore necessary to establish the mathematical formalism that describes the dififiaction pattern taking into account these parameters. 

The addition of a single monochromating crystal gives the device great flexibil ity, since one has independent control over the collimation, by the number of reflections, and the bandpass, controlled by the width of the reflection of the monochromating crystal. Figure 2.16 shows one variant, a three-reflection... 

Survey experiments involving the rapid examination of large numbers of samples, or large numbers of reflections in order to obtain detailed statistical information relating to growth technique or detailed three-dimensional strain analysis. 

Phase Designation Approximate Composition 8 8e Number of Reflections 26 max (°) R-value Bond length (A) Bond Angle ) Ref. 

Figure 11.3 The ratio of the number of reflections to the number of parameters in the XYZB crystallographic refinement of a protein model as a function of the resolution of the data. The data are assumed to be complete and the solvent content to be 50%.

The number of reflection intensities measured in a crystallographic experiment is large, and commonly exceeds the number of parameters to be determined. It was first realized by Hughes (1941) that such an overdetermination is ideally suited for the application of the least-squares methods of Gauss (see, e.g., Whittaker and Robinson 1967), in which an error function S, defined as the sum of the squares of discrepancies between observation and calculation, is minimized by adjustment of the parameters of the observational equations. As least-squares methods are computationally convenient, they have largely replaced Fourier techniques in crystal structure refinement. 

Sodium fluoride, NaF, is a favorable choice for X-ray analysis of the lattice energy of an ionic crystal. Both Na and F are relatively light atoms, and the Na 3s-radial distribution, though diffuse, is not quite as spread out as the Li 2s shell (single-C values are 0.8358 and 0.6396 au-1, respectively see appendix F), and therefore contributes to a larger number of reflections. 

At this point it is important to realize that from single crystals the full crystallographic information is usually obtained by XRD methods (space group, lattice parameters and atomic coordinates) but that for thin films this is hardly the case, because of the physical limitation imposed by the substrate, in particular when the microcrystals are oriented, in which case only a reduced number of reflections are available. Hence only partial information is obtained with conventional diffraction methods. Removing the microcrystals, e.g., by scratching, in order to perform XRD measurements from the powder is not always possible and handling such small and fragile microcrystals with the usual tools is almost impossible. 

The number of reflections is also limited by the size of the image striking the entrance and the size of the mirror M2. As seen in Fig. 11.1b, the images that are refocused from Ml and M3 onto the field mirror M2 are stacked beside each other. The width of M2 therefore determines how many of these images can be accommodated (i.e., how many reflections are possible). 

The accuracy of a structure deduced from such spacings depends on the number of reflections observed. For a semicrystalline polymer showing only three or four reflections, only a rough estimate of the structure can be obtained. However, the structure of a metal or inorganic salt, which may show more than a thousand reflections, can be determined with very high reliability. 

This situation is obviously useless for spectroscopy. Hence, valuable films are very thin, on the order of few nanometers, depending on the material. Figure 4 shows that even for a thin metal film the overall reflectivity of the system can be severely reduced. This characteristic is even more important for multiple reflection elements, for which the total reflectivity Rtot is the reflectivity for a single reflection R to the power of the number of reflections N. Rtot = R - For a reflectivity R of 90% (90% of the incident intensity is reflected), only 35% of the incident intensity will reach the detector after 10 reflections (N — 10). The smaller the reflectivity Rtot> the smaller is the signal at the detector and the larger the noise. On the other hand, the intensity of an absorption... 

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