1.4 Crystal Lattices and Unit Cells

You must have noticed that when tiles are placed to cover a floor, a repeated pattern is generated. If after setting tiles on floor we mark a point at same location in all the tiles (e.g. Centre of the tile) and see the marked positions only ignoring the tiles, we obtain a set of points.

This set of points is the scaffolding on which pattern has been developed by placing tiles. This scaffolding is a space lattice on which two-dimensional pattern has been developed by placing structural units on its set of points (i.e. tile in this case). The structural unit is called basis or motif. When motifs are placed on points in space lattice, a pattern is generated. In crystal structure, motif is a molecule, atom or ion.

A space lattice, also called a crystal lattice, is the pattern of points representing the locations of these motifs. In other words, space lattice is an abstract scaffolding for crystal structure. When we place motifs in an identical manner on points of space lattice, we get crystal structure. Figure shows a motif, a two-dimensional lattice and a hypothetical two-dimensional crystal structure obtained by placing motifs in the two-dimensional lattice. Spacial arrangement of lattice points gives rise to different types of lattices. Figure shows arrangement of points in two different lattices.

In the case of crystalline solids, space lattice is a three-dimensional array of points. The crystal structure is obtained by associating structural motifs with lattice points.

Each repeated basis or motif has same structure and same spacial orientation as other one in a crystal. The environment of each motif is same throughout the crystal except for on surface. Following are the characteristics of a crystal lattice:

(a) Each point in a lattice is called lattice point or lattice site.

(b) Each point in a crystal lattice represents one constituent particle which may be an atom, a molecule (group of atoms) or an ion.

(c) Lattice points are joined by straight lines to bring out the geometry of the lattice.

We need only a small part of the space lattice of a crystal to spacify crystal completely. This small part is called unit cell. One can choose unit cell in many ways.

Normally that cell is chosen which has perpendicular sides of shortest length and one can construct entire crystal by translational displacement of the unit cell in three dimensions. Figure. shows movement of unit cell of a two-dimensional lattice to construct the entire crystal structure.

Also, unit cells have shapes such that these fill the whole lattice without leaving space between cells. In two dimensions a parallelogram with side of length ‘a’ and ‘b’ and an angle r between these sides is chosen as unit cell. Possible unit cells in two dimensions are shown in figure.

A portion of three-dimensional crystal lattice and its unit cell is shown in Figure. In the three-dimensional crystal structure, unit cell is characterised by:

(i) its dimensions along the three edges a, b and c. These edges may or may not be mutually perpendicular.

(ii) angles between the edges, a (between b and c), b (between a and c) and g (between a and b).

Thus, a unit cell is characterised by six parameters a, b, c, a , b and g. These parameters of a typical unit cell are shown in figure.

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