Direction in a crystal

In the same way that close-packed directions in a crystal have larger refractive indices, so too can the application of a tensile stress to an isotropic glass increase the index of refraction normal to the direction of the applied stress. Uniaxial compression has the reverse effect. The resulting variation in refractive index with direction is called birefringence, which can be used as a method of measuring stress. 

It is important to nolice that this method not only reveals polar directions in a crystal structure it does much more—it determines in... 

Crystals, except those belonging to the cubic system, are anisotropic in this respect the force of repulsion varies with the orientation of the crystal with respect to the direction of the field. The graph representing vectorialiy the diamagnetic susceptibility in all directions in a crystal is an ellipsoid, whose orientation with respect to the unit cell is restricted by symmetry in exactly the same way as that of the optical indicatrix. Thus, for uniaxial crystals the magnetic ellipsoid an ellipsoid of revolution whose unique axis coincides with the threefold, fourfold, or sixfold axis of the crystal for orthorhombic crystals the ellipsoid has three unequal axes which necessarily coincide with the three axes of the crystal for monoclinie crystals the only restriction is that one of the principal axes of the magnetic ellipsoid must coincide with the b axis of the crystal while for triclinic crystals the orientation of the ellipsoid is not restricted in any way. 

Stereographic projection provides a convenient way of displaying the angular relations between planes and directions in a crystal in two dimensions. The system involves first projecting planes and directions of interest onto a spherical surface and then mapping the spherical surface. Figure 4.1 illustrates how planes and directions are projected onto a sphere. If an infinitesimal crystal were placed at the center of a sphere and its planes extended, they would intersect the sphere as great circles and their directions would intersect the sphere as points. 

Directions in a crystal are specified in the same way. In these instances the integers u, v, w are applied to the unit cell vectors, a-, b- and c-. As with Miller indices, it is sometimes convenient to group together all directions that are identical by virtue of the symmetry of the structure. These are represented by the notation (uvw). In a cubic crystal the symbol (100) represents the six directions [100], [100], [010], [0T0],... 

A polar direction in a crystal is a direction [ yir] that is not related by symmetry to the opposed direction [nvw]. The two senses of the direction are physically different. No polar direc-... 

Diffraction of X-rays is the basic technique for obtaining information on the atomic structure of crystalline soHds and is one of the key standard laboratory techniques. XRD is based on the interference of X-ray waves elastically scattered by a series of atoms orientated along a particular direction in a crystal characterized by a vector A. The waves scattered by two atoms a and b interfere constructively with each other when the path difference PQR is equal to an integer number of wavelengths PQR = W. This condition is valid for orientations K of the scattered waves which satisfy the Laue condition ... 

A direction in a crystal is given as a set of three integers in square brackets [uvw] u, v, and w correspond to the above definition of the translation vector r. A direction in a cubic crystal can be described also by Miller indices, as a plane can be defined by its normal. The indices of a direction are expressed as the smallest integers which have the same ratio as the components of a vector (expressed in terms of the axis vectors a, b, c) in that direction. Thus the sets of integers 1, 1, 1 and 3, 3, 3 represent the same direction in a crystal, but the indices of the direction are [111] and not [333]. To give another example, the x axis of an orthogonal x, y, z coordinate system has Miller indices [ 100] the plane perpendicular to this direction has indices (100). 

Besides the planes, also the directions in a crystal are very important and should be rationalized to the fundamental referential system at the unit cell level. 

Further, we have to take into account that the constraints of the translational lattice, namely the translation T with n units along t-direction in a crystal should equal (or to express themselves through) m integer units of the vector t-size in the elementary cell ... 

A lattice node is defined by its coordinates (in the units of the vector length), which are placed in double square parentheses (for instance, [[001]]). All parallel directions in a crystal are equivalent. So a straight crystal line (or simply line) is conducted through the origin and its indexes are defined by the coordinate of the first node, lying on this line a line s indexes are enclosed in brackets (so, direction [001] complies with the direction to axis z, since the first node on it has the coordinates [[001]]). 

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