Symmetry elements within the unit cell

We have already covered the symmetry elements that apply to individual molecules (in the Schoenflies notation) in Section 2.3.1, and you should read this section now if you have not already done so. We now need to consider the symmetry operations that map the relationships between entire arrays of atoms or molecules in three dimensions. These are generally expressed using the Herman-Mauguin symbols, in which the inversion operation, i, is replaced by 1, the reflection, t, by m, and rotations, C , simply by n. Examples are shown in 

Alternative descriptions of a unit cell by three lattice vectors a, b and c, and by edge lengths a, b, c and angles a, and y. The axes are conventionally defined to be right-handed. 

Examples of simple crystallographic symmetry elements denoted in Hermann-Mauguin notation. 

Some symmetry operations, known as proper operations, retain the chirality of an object These arc the translations and the rotations. Others, the improper operations, such as the inversion operation, reverse it. 

Because there is translation symmetry in a crystal it is also possible to combine the symmetry operators within the cell with translation elements, but only some combinations meet the demands of both symmetries. The two types of coupled symmetry elements that involve translational elements are screw axes and glide planes. 

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The location of the symmetry elements within the unit cells of the plane groups is illustrated in Figure 3.10. Heavy lines represent mirror lines and heavy dashed lines represent glide lines. The unit cell has a light outline. 

Figure 3.10 The location of the symmetry elements within the unit cells of the 17 plane groups...

The crystal system relates the shape and syrmnetry of the irrrit cell. Table 3.5 sitrmnarizes the seven crystal systems. The simplest and lowest syrmnetry is triclinic, while the highest is cnbic with four threefold axes. It is important to note that the symbol it used in this context means not necessarily unequal, as in some cases coincidental pseudosymmetry may arise with cell dimensions resembling those of a higher syrmnetry crystal system, but without the reqitired symmetry elements within the unit cell. 

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