Crystallographic direction

A crystallographic direction is defined as a line directed between two points, or a vector. The following steps are used to determine the three directional indices  

A right-handed x-y-z coordinate system is first constructed. As a matter of convenience, its origin may be located at a unit cell corner. 

The coordinates of two points that lie on the direction vector (referenced to the coordinate system) are determined—for example, for the vector tail, point 1 Xj, yi, and Zu whereas for the vector head, point 2 X2, y2, and Z2-WileyPLUS 3. Tail point coordinates are snbtracted from head point components—that is, 

These coordinate differences are then normalized in terms of (i.e., divided by) their respective a, h, and c lattice parameters—that is, 

If necessary, these three numbers are multiplied or divided by a common factor to reduce them to the smallest integer values. 

Since the direction vectors are specified in the smallest set of whole numbers, they specify only direction, not actual lengths. The vector [222] reduces to [111] and both specify the same direction. 

Generalized vector directions. Using vector subtraction, vector A would be designated [136] and vector B would be [232], 

Notice that the lattice directions are written within square brackets with no commas. Negative directions are indicated with a bar over the number as [110]. Families of similar directions are denoted by an angle bracket (100), which consists of the six vectors, [100], [010], [001], [100], [010], and [001]. Note that [100] is antiparallel to [100], etc. 

The above rules apply for any geometry of the unit cell. For the case of cubic unit cells, it is easy to obtain the angles the various direction vectors make with each other by using the definition of the dot product A B = A B cos(AB). For example, the direction from the central atom in a bcc unit cell to one of the comer atoms can be written as [111], and the direction from the central atom to the atom in the opposite comer on the same face is [111]. Taking the dot product, the angle between these two comer atoms may be foimd  

Taking the arc cosine of 1/3 gives the angle between A and B, which is foimd to be 109.47°. 

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Effects of Rate Conditions. It is essential for commercial a-quartz crystals to have usable perfection growth at a high rate and at pressure and temperature conditions that allow economical equipment design. The dependence of rate on the process parameters has been studied (8,14) and may be summarized as follows. Growth rate depends on crystallographic direction the (0001) is one of the fastest directions. Because AS is approximately linear with AT, the growth rate is linear with AT. Growth rate has an Arrhenius equation dependence on the temperature in the crystallization zone ... 

Barium carbonate also reacts with titania to form barium titanate [12047-27-7] BaTiO, a ferroelectric material with a very high dielectric constant (see Ferroelectrics). Barium titanate is best manufactured as a single-phase composition by a soHd-state sintering technique. The asymmetrical perovskite stmcture of the titanate develops a potential difference when compressed in specific crystallographic directions, and vice versa. This material is most widely used for its strong piezoelectric characteristics in transducers for ultrasonic technical appHcations such as the emulsification of Hquids, mixing of powders and paints, and homogenization of milk, or in sonar devices (see Piezoelectrics Ultrasonics). 

A problem arises in using platelet reinforcements if their naturally mechanically weak crystallographic direction is aligned perpendicular to the crack front. The platelets easily fracture in this orientation. Further research is needed to grow platelets with favorable crystallographic orientations. 

Within the elastic range, loading applied along nonspecific crystallographic directions results in propagation of both longitudinal and shear waves which may be of considerable amplitude [80C01],... 

Here 4) = 7 T Cp/L is a capillary length proportional to the isotropic part of the surface energy 7 0 is the angle between the normal n to the interface and some fixed crystallographic direction at which d Q) is minimal e is the strength of the anisotropy. 

Mechanical treatment of samples, ranging from comminution by grinding to imposition of stresses along defined crystallographic directions in... 

This is often referred to as the contracting volume (cube or sphere) equation and is the simplest example of a more general family of expressions [28—31,432,453,458,459], which includes consideration of different rates of interface advance in different crystallographic directions and of variations in crystallite dimensions and shapes. The approach is readily extended, by use of solid geometry, to allow for angles between planar surfaces. Some examples of characteristic behaviour are conveniently discussed with reference to the expression... 

It is usually assumed in the derivation of isothermal rate equations based on geometric reaction models, that interface advance proceeds at constant rate (Chap. 3 Sects. 2 and 3). Much of the early experimental support for this important and widely accepted premise derives from measurements for dehydration reactions in which easily recognizable, large and well-defined nuclei permitted accurate measurement. This simple representation of constant rate of interface advance is, however, not universally applicable and may require modifications for use in the formulation of rate equations for quantitative kinetic analyses. Such modifications include due allowance for the following factors, (i) The rate of initial growth of small nuclei is often less than that ultimately achieved, (ii) Rates of interface advance may vary with crystallographic direction and reactant surface, (iii) The impedance to water vapour escape offered by... 

The structural principles of Prl2 can be derived either from 4" nets (Prl2-I) or 3 nets (all other modifications) of iodine atoms that are stacked along a prominent crystallographic direction, in most cases the [001] direction. Between these layers, half of the respective interstices are filled with praseodymium atoms (but see Prl2-V below). Please note that 4" and 3 nets are closely related to each other, it only needs a shear procedure to transform one net to the other (Fig. 4.3). In the iodine layers I-I distances are even shorter in the 4" net (386 in Prl2-I [4]) than in the 3 net of Prl2-IV (426.5 pm [6, 9]). 

The sum on the right side of (9.14) corresponds to g(E,k) in a single crystal. The PDOS g(E) for randomly oriented systems requires an averaging over all crystallographic directions. 

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