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The three a-axes, a 1, a 2, and a 3, are parallel to edges of a nonprimitive hexagonal unit cell. Although the third a-axis is redundant for describing symmetry or points in 3D space, it has been included in the past to emphasize that there are three identical a-axes perpendicular to the c-axis.
There are 7 types of unit cells (figure 12.1.a), defined by edge lengths (a,b,c) respectively along the x,y,z axis and angles \(\large\alpha\), \(\large\beta\), and \(\large\gamma\).
This describes a body-centered unit cell (I), a 4 1 screw axis perpendicular to an a glide plane, a 3-fold rotoinversion axis, and a proper 2-fold axis perpendicular to a d glide plane. There are, however, nine other space groups that can produce crystals with the same symmetry as garnet.
To define the resulting 3D lattice, it is convenient to specify reference directions, x, y and z, which are chosen to be parallel to three edges of the unit cell (Figure 30b). These are known as crystallographic axes , and it is important to remember that they are not always at 90° to each other.
Two-dimensional Symmetry Elements. Lattice type: p for primitive, c for centred. Symmetry elements: m for mirror lines, g for glide lines, 4 for 4-fold axis etc. Design by M.C. Escher. Bravais Lattices and Crystal Systems.
If a crystal has symmetry 222, mm2, or mmm, it must have a unit cell that is an orthorhombic prism, and so forth. The table below summarizes these relationships. By examining a crystal’s morphology, we can often determine the point group, system, and unit cell shape.
Unit Cell • In 3D space the unit cells are replicated by three noncoplanar translation vectors a 1, a 2, a 3 and the latter are typically used as the axes of coordinate system • In this case the unit cell is a parallelepiped that is defined by length of vectors a 1, a 2, a 3 and angles between them. The volume of the parallelepiped is
