Parallelogram Identity

With homogeneous strain, the deformation is proportionately identical for each volume element of the body and for the body as a whole. Hence, the principal axes, to which the strain may be referred, remain mutually perpendicular during the deformation. Thus, a unit cube (with its edges parallel to the principal strain directions) in the unstrained body becomes a rectangular parallelepiped, or parallelogram, while a circle becomes an ellipse and a unit sphere becomes a triaxial ellipsoid. Homogeneous strain occurs in crystals subjected to small uniform temperature changes and in crystals subjected to hydrostatic pressure. 

FIGURE 7-3 Hexagonal Close Packing, (a) The hexagonal prism with the unit cell outlined in bold, (b) Two layers of an hep unit cell. The parallelogram is the base of the unit cell. The third layer is identical to the first, (c) Location of the atom in the second layer. 

This set of points can be formed as follows. Imagine space to be divided by three sets of planes, the planes in each set being parallel and equally spaced. This division of space will produce a set of cells each identical in size, shape, and orientation to its neighbors. Each cell is a parallelepiped, since its opposite faces are parallel and each face is a parallelogram. The space-dividing planes will intersect each other in a set of lines (Fig. 2-1), and these lines in turn intersect in the... 

The structure of a crystal is characterized by the fact that it is formed by the indefinite repetition in three dimensions of the contents of a parallelo-piped, termed the unit cell if the contents of one unit cell is known, the structure of the whole crystal is given by stacking identical cells in parallel orientation in such a way that each corner is common to eight of these cells. (In an analogous way the pattern of a wallpaper is formed by the indefinite repetition in two dimensions of the design contained within a unit parallelogram.) Unit cells in crystals commonly have linear dimensions of the order of io A, and contain a small whole number of formula units. 

This symmetry element I generates an operation of inversion through a point called the inversion center, Fig. 2.3. Therefore now we deal with inversion symmetry. We can take any point of an object and connect it by a straight line with the center O. Then, along the same line behind the center and at the equal distance from it we must find the point equivalent to the first one. A good example is a parallelogram. Note, that two inversions result in the identical object, I = E. 

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