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The unit cell
Since the crystal lattice is made up of a regular arrangement which repeats in three dimensions, we can save ourselves a great deal of work by considering the simple repeating unit rather than the entire crystal lattice. The basic repeating unit is known as the unit cell. Crystalline solids often have flat, well-defined faces that make definite angles with their neighbors and break cleanly when struck. These faces lie along well-defined directions in the unit cell.
The unit cell is the smallest, most symmetrical repeating unit that, when translated in three dimensions, will generate the entire crystal lattice.
It is possible to have a number of different choices for the unit cell. By convention, the unit cell that reflects the highest symmetry of the lattice is the one that is chosen. A unit cell may be thought of as being like a brick which is used to build a building (a crystal). Many bricks are stacked together to create the entire structure. Because the unit cell must translate in three dimensions, there are certain geometrical constraints placed upon its shape. The main criterion is that the opposite faces of the unit cell must be parallel. Because of this restriction there are only six parameters that we need to define in order to define the shape of the unit cell. These include three edge lengths a, b, and c and three angles $\alpha $ , $\beta $ and $\gamma $ . Once these are defined all other distances and angles in the unit cell are set. As a result of symmetry, some of these angles and edge lengths may be the same. There are only seven different shapes for unit cells possible. These are given in the chart below.
Unit Cell Type | Restrictions on Unit Cell Parameters | Highest Type of Symmetry Element Required |
Triclinic | a is not equal to b is not equal to c; $\alpha $ is not equal to $\beta $ is not equal to $\gamma $ . | no symmetry is required, an inversioncenter may be present |
Monoclinic | a is not equal to b is not equal to c $\alpha $ = $\gamma $ $\text{90}\xb0$ $\beta $ is not equal to $\text{90}\xb0$ . | highest symmetry element allowed is aC2 axis or a mirror plane |
Orthorhombic | a is not equal to b is not equal to c $\alpha $ = $\beta $ = $\gamma $ $\text{90}\xb0$ | has three mutually perpendicularmirror planes and/or C2 axes |
Tetragonal | a =b is not equal to c $\alpha $ = $\beta $ = $\gamma $ $\text{90}\xb0$ | has one C4 axis |
Cubic | a =b =c $\alpha $ = $\beta $ = $\gamma $ $\text{90}\xb0$ | has C3 and C4 axes |
Hexagonal, Trigonal | a =b is not equal to c $\alpha $ = $\beta $ = $\text{90}\xb0$ $\gamma $ $\text{120}\xb0$ | C6 axis (hexagonal); C3 axis (trigonal) |
Rhombohedral* | a =b =c $\alpha $ = $\beta $ = $\gamma $ is not equal to $\text{90}\xb0$ | C3 axis (trigonal) |
*There is some discussion about whether the rhombohedral unit cell is a different group or is really a subset of the trigonal/hexagonal types of unit cell.
You will be asked to count the number of atoms in each unit cell in order to determine the stoichiometry (atom-to-atom ratio) or empirical formula of the compound. However, it is important to remember that solid state structures are extended, that is, they extend out in all directions such that the atoms that lie on the corners, faces, or edges of a unit cell will be shared with other unit cells, and therefore will only contribute a fraction of that boundary atom. As you build crystal lattices in these exercises you will note that eight unit cells come together at a corner. Thus, an atom which lies exactly at the corner of a unit cell will be shared by eight unit cells which means that only⅛of the atom contributes to the stoichiometry of any particular unit cell. Likewise, if an atom is on an edge, only¼of the atom will be in a unit cell because four unit cells share an edge. An atom on a face will only contribute½to each unit cell since the face is shared between two unit cells.
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