Crystal expressions

Mossbauer Effect in Similar Glasses and Crystals. Expression of the Mg-Fe content of tektites in terms of enstatite-ferrosilite percentages suggests a comparison with silicate minerals which contain Fe and... 

The term <9>Sphericai-atom crystal (r)> in ET (8-33) is the average potential in the unit cell of the promolecule crystal, equal to 0(0) for the promolecule crystal. Expression (8.33) thus gives the deviation from the average promolecule potential in the crystal. Modification of Eq. (8.33) for the direct space evaluation of the K-modified non-neutral spherical atom densities is straightforward. 

Yosio Sakamoto, Madelung constants of simple crystals expressed in terms of Bom s basic potentials of 15 figures, /. Chem. Phys. 28 164-165 (1958). 

Figure 5.14. (A) Influence of US intensity on calcium carbonate crystallization expressed as free [Ca ]. US intensity (m) 250 W/cm, (o) 105 W/cm (horn tip diameter and immersion depth 3 mm and 3 cm, respectiveiy). (B) Variation of the crystaiiization rate of caicium carbonate with the US intensity at variabie horn tip diameters (m) 3 mm, fAj 14 mm, ( ) 22 mm (horn immersion depth 3 cm). (C) Variation of the crystaiiization rate of caicium carbonate with the product of the US intensity and square root of the horn tip area. Horn tip diameter and immersion depth as in B. (D) Variation of the particie size of hydroxyapatite as a function of the US power (Reproduced with permission of Eisevier, Refs. [142], [144].)...

FIGURE 7.1. The relative orientations of the reciprocal lattice of a crystal (expressed as a and b ), and its indexed X-ray diffraction pattern (expressed as h and k). In the diffraction pattern the intensities of the diffracted beams (/) (the blackness of spots on X-ray film, for example) and the directions of travel (sin 6) (positions of spots on the X-ray film) are measured. Note the relationship of a to h, and b to k. From the positions of spots on the photographic film it is possible to deduce the dimensions of the reciprocal lattice, hence of the crystal lattice, hence the indices hkl of each Bragg reflection. 

Because we are dealing with molecules, two types of lattice vibrations can be distinguished translational and rotational. In order to describe these motions we have to know the potential energy of the crystal, expressed as a function of the center of mass positions and the orientations of all molecules. In Section II, we give a fairly detailed description of the different ways in which the potential can be expressed, each way having its own merits, depending on the subsequent calculations in which it has to be used. 

In this subsection we derive formulas for the operators of the electric and the magnetic field in crystals, expressing them in terms of the Bose amplitudes and p. 

SAXS and XRD measurements elucidated significant differences between hepatic liquid crystals obtained from different species of avian. In Taihe fowl, the SAXS scattering of the hepatic liquid crystal expresses a strong peak at 26. But the peak is weak or absent in its crystal XRD diffraction. In pigeon, the SAXS hepatic liquid crystal peak does have a significant XRD diffraction pattern. This difference indicates that although liquid crystal can be found in the same tissues of the two avians, they likely contain different chemical components (Table 2). 

Figure 7.1 shows the patterns of RRDE measurements recorded on different facets of platinum low-index single crystal (expressed as Pt(hlk)) in 02-saturated 0.1 M KOH aqueous solution. The insets... 

Kapustinskii equation For an ionic crystal composed of cations and anions, of respective charge and z, which behave as hard spheres, the lattice energy (U) may be obtained from the expression... 

In general, the phonon density of states g(cn), doi is a complicated fimction which can be directly measured from experiments, or can be computed from the results from computer simulations of a crystal. The explicit analytic expression of g(oi) for the Debye model is a consequence of the two assumptions that were made above for the frequency and velocity of the elastic waves. An even simpler assumption about g(oi) leads to the Einstein model, which first showed how quantum effects lead to deviations from the classical equipartition result as seen experimentally. In the Einstein model, one assumes that only one level at frequency oig is appreciably populated by phonons so that g(oi) = 5(oi-cog) and, for each of the Einstein modes. is... 

The amplitude and therefore the intensity, of the scattered radiation is detennined by extending the Fourier transfomi of equation (B 1.8.11 over the entire crystal and Bragg s law expresses die fact that this transfomi has values significantly different from zero only at the nodes of the reciprocal lattice. The amplitude varies, however, from node to node, depending on the transfomi of the contents of the unit cell. This leads to an expression for the structure amplitude, denoted by F(hld), of the fomi... 

The integrand in this expression will have a large value at a point r if p(r) and p(r+s) are both large, and P s) will be large if this condition is satisfied systematically over all space. It is therefore a self- or autocorrelation fiinction of p(r). If p(r) is periodic, as m a crystal, F(s) will also be periodic, with a large peak when s is a vector of the lattice and also will have a peak when s is a vector between any two atomic positions. The fiinction F(s) is known as the Patterson function, after A L Patterson [14], who introduced its application to the problem of crystal structure detemiination. 

Temary and quaternary semiconductors are theoretically described by the virtual crystal approximation (VGA) [7], Within the VGA, ternary alloys with the composition AB are considered to contain two sublattices. One of them is occupied only by atoms A, the other is occupied by atoms B or G. The second sublattice consists of virtual atoms, represented by a weighted average of atoms B and G. Many physical properties of ternary alloys are then expressed as weighted linear combinations of the corresponding properties of the two binary compounds. For example, the lattice constant d dependence on composition is written as ... 

The proof that these expressions are equivalent to Eq. (1.35) under suitable conditions is found in statistics textbooks. We shall have occasion to use the Poisson approximation to the binomial in discussing crystallization of polymers in Chap. 4, and the distribution of molecular weights of certain polymers in Chap. 6. The normal distribution is the familiar bell-shaped distribution that is known in academic circles as the curve. We shall use it in discussing diffusion in Chap. 9. 

The direction of the alignment of magnetic moments within a magnetic domain is related to the axes of the crystal lattice by crystalline electric fields and spin-orbit interaction of transition-metal t5 -ions (24). The dependency is given by the magnetocrystalline anisotropy energy expression for a cubic lattice (33) ... 

Under Httle or no illumination,/ must be minimized for optimum performance. The factor B is 1.0 for pure diffusion current and approaches 2.0 as depletion and surface-mode currents become important. Generally, high crystal quality for long minority carrier lifetime and low surface-state density reduce the dark current density which is the sum of the diffusion, depletion, tunneling, and surface currents. The ZM product is typically measured at zero bias and is expressed as RM. The ideal photodiode noise current can be expressed as follows ... 

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