Crystals isotropic

Whereas in many metals with relatively simple and isotropic crystal structures the parameter / has values between 0.5 and 1, it can have much more extreme values in materials in which the mobile species move through much less isotropic structures with 1-D or two-dimensional (2-D) channels, as is often the case with insertion reaction electrode materials. As a result, radiotracer experiments can provide misleading information about self-diffusion kinetics in such cases. 

Ionic transport in solid electrolytes and electrodes may also be treated by the statistical process of successive jumps between the various accessible sites of the lattice. For random motion in a three-dimensional isotropic crystal, the diffusivity is related to the jump distance r and the jump frequency v by [3] ... 

When the two vectors are parallel, the crystal planes perpendicular to the line form a helix, and the dislocation is said to be of the screw type. In a nearly isotropic crystal structure, the dislocation is no longer associated with a distinct glide plane. It has nearly cylindrical symmetry, so in the case of the figure it can move either vertically or horizontally with equal ease. 

Next, let us compile some quantitative relations which concern the stress field and the energy of dislocations. Using elastic continuum theory and disregarding the dislocation core, the elastic energy, diS, of a screw dislocation per unit length for isotropic crystals is found to be... 

For example, consider a binary alloy in which the stress-free molar volume is a function of concentration, V(cs). The linear expansion due to the composition change can be inferred from diffraction experiments under stress-free conditions (Vegard s effect) and is characterized by Vegard s parameter, ac [e.g., in cubic or isotropic crystals e ° = e°y 0 = = ac(c — c0)]. The assumption of coherency... 

The (3 dependence of the amplification factor in an elastically isotropic crystal (for which R is independent of the direction of 0) is plotted for a temperature inside the coherent spinodal in Fig. 18.9. For (3 < /3crit> the amplification factor R (3) > 0 and the system is unstable—that is, the composition waves in Eq. 18.44 will grow exponentially. The wavenumber /3max, at which dR f3)/df3 = 0, receives maximum amplification and will dominate the decomposed microstructure. Outside the coherent spinodal, where d2 fhom/dc2B+2a2Y(h) > 0, all wavenumbers will have R(/3) < 0 and the system will be stable with respect to the growth of composition waves. 

Figure 18.9 Amplification factor vs. wavenumber plot for an elastically isotropic crystal at a temperature inside the coherent spinodal where d2 fhom /dc% + 2ct2 Y < 0.

Up to now we have been discussing in this Chapter many-particle effects in bimolecular reactions between non-interacting particles. However, it is well known that point defects in solids interact with each other even if they are not charged with respect to the crystalline lattice, as it was discussed in Section 3.1. It should be reminded here that this elastic interaction arises due to overlap of displacement fields of the two close defects and falls off with a distance r between them as U(r) = — Ar 6 for two symmetric (isotropic) defects in an isotropic crystal or as U(r) = -Afaqjr-3, if the crystal is weakly anisotropic [50, 51] ([0 4] is an angular dependent cubic harmonic with l = 4). In the latter case, due to the presence of the cubic harmonic 0 4 an interaction is attractive in some directions but turns out to be repulsive in other directions. Finally, if one or both defects are anisotropic, the angular dependence of U(f) cannot be presented in an analytic form [52]. The role of the elastic interaction within pairs of the complementary radiation the Frenkel defects in metals (vacancy-interstitial atom) was studied in [53-55] it was shown to have considerable impact on the kinetics of their recombination, A + B -> 0. 

In non-polar, isotropic crystals or in glasses, there is no crystallographic direction distinguished and the linear electro-optic effect is absent. Nevertheless a static field may change the index by displacing ions with respect to their valence electrons. In this case the lowest non-vanishing coefficients are of the quadratic form, i.e. the refractive index changes proportionally to the square of the applied field Kerr effect . 

The direction of the principal axes of the index of refraction tensor n can be described by the indicatrix. For isotropic crystals the indicatrix is a sphere. For positive uniaxial crystals it is a prolate spheroid (ns > n0j) for negative uniaxial crystals it is an oblate spheroid (nol > n,). For orientations away from the principal axis orientations, the extraordinary ray will have a refractive index h - intermediate between nm and ne. 

Insofar as small crystals of nonreducible oxides dispersed on the internal interfaces of the basic structural units (platelets) will stabilize the active catalyst surface Fe(lll), the paracrystallinity hypothesis will probably hold true. But the assumption that this will happen on a molecular level on each basic structural unit is not true. The unique texture and anisotropy of the ammonia catalyst is a thermodynamically metastable state. Impurity stabilization (structural promotion) kinetically prevents the transformation of platelet iron into isotropic crystals by Ostwald ripening [154]. Thus the primary function of alumina is to prevent sintering by acting as a spacer, and in part it may also contribute to stabilizing the Fe(lll) faces [155], [156], [298],... 

Screw dislocation. The simplest case to start with is that of a straight screw dislocation of Burgers vector b parallel to the surface of a thin parallel-sided crystal foil, as shown in Figure 5.10. Using the coordinate system defined there, the dislocation AB is parallel to y and at a depth z below the top surface. The dislocation causes a column CD of unit cells parallel to z in the perfect crystal to be deformed. If we assume that the atomic displacements around the dislocation are the same in the thin specimen as in an infinitely large, elastically isotropic crystal, then the components u, v, w of the deformation of the column along thex, y, and z directions will be... 

Figure 5.22. Imaging a spherical inclusion by its strain field in the surrounding isotropic crystal matrix, (a) Diagram illustrating the compressive strain around the inclusion, (b) Schematic diagram illustrating the nature of the image and, in particular, the line of no contrast CC normal to the diffraction vector g.

Figure 5.23. Variation of the 20-percent image width with e, g, Tq, and t for a spherical inclusion in an elastically isotropic crystal matrix at 5 = 0.

Figure 5.24. Variation of the 20-percent image width with g, b, and 0 for a prismatic dislocation loop (or a platelike inclusion) in an isotropic crystal matrix for several values of / ,. Thickness of foil = s = 0 r" = 10/g,...

If a crystal belongs to the cubic system, the velocity of light through it (and therefore its refractive index) is isotropic (the same in all directions). The refractive index of such an isotropic crystal is measured by observing it when it is immersed in a colorless liquid of matching refractive index (obtained by mixing appropriate liquids of known refractive indices in which the crystal is insoluble). When the refractive index of the surrounding mixture of liquids exactly matches that of the crystal, the latter becomes invisible. The refractive index of the liquid mixture can be measured and is equal to the refractive index of the crystal. 

FIGURE 5.8. (a) Waves pa.ssing through an isotropic crystal are spherical, whereas those passing through a birefringent crystal (b) behave as ellipsoidal wavelets. 

When an isotropic crystal is placed between crossed Nicol prisms, there is extinction, which means that nothing is seen, and the field is perfectly dark. On the other hand, if an optically anisotropic, birefrin-gent crystal is similarly viewed, it will appear colored except at certain positions of rotation, usually 90° apart. At these positions, extinction occurs because the vibration directions of the light transmitted through the... 

FIGURE 5.13. Crossed Nicol prisms, (a) No crystal, (b) an isotropic crystal between the polarizer and the analyzer and (c) an anisotropic crystal which rotates the plane of plane polarized light to some extent as shown. 

Crystalline index of refraction As different polymorphs have different internal structures, they belong to different crystal systems therefore, polymorphs can be distinguished using polarized light and a microscope. The crystals can be either isotropic or anisotropic. In isotropic crystals, the velocity of light is the same in all directions, whereas anisotropic crystals have two or three different light velocities or refractive indices. In terms of crystal systems, only the cubic system is isotropic and the other six are anisotropic. 

Equilibrium for a solid would also stipulate the shape, since the surface free energy should be a minimum. For an isotropic crystal this shape would be that of a sphere, but this is hardly ever a factor. The loss in energy for a surface atom is about one-half of the cohesive energy per atom. But the number of atoms on a typical surface is only about 10, or 10 mol, so the surface energy is very small. There is one observable effect, however a collection of small crystals will cohere to form larger crystals, if a mechanism, such as digestion of a precipitate, is provided. 

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