What are lattices and unit cells?

The repeating, atomic level structure of a crystal can be represented by a lattice and by the repeating unit of the lattice, the unit cell.

It was apparent very early in the study of crystals that the shapes of crystals stem from an ordered array of smaller structural units. Although we now know a great deal about the nature of these units, it remains very profitable to consider the ways in which points, each with identical surroundings, that are not found characterized can be arranged to give a repeating array.

The limitations on the types of arrangements that can give a repeating pattern in which each point has identical surroundings can best be appreciated from the two dimensional patterns. Only these five essentially different patterns can be constructed. One can verify that any other two dimensional patterns that one attempts to draw is identical, except for the relative magnitudes fo the spacings a and b and the angle∝.

In a similar way there are, as A. bravais showed in 1848, only 14 different types of lattices that can be drawn in three dimensions. Units of these lattices, which when repeated in three dimensions produce the lattice. Any three dimensional array, such as real crystal, must have an internal structure that corresponds on one of the 14 Bravais lattices.

Each crystal, although made up of atoms or simple or complex ions or molecules, must correspond in internal structure to one of the 14 bravais lattices. This does not mean that atoms, ions, or molecules need to be positioned as the lattice points are arranged so that points with identical environment are arranged in the pattern of the Bravais lattices.

One feature of the different Bravis lattices that shows that they are indeed different is the number and arrangement of nearest neighbors of each lattice point. Thus the three cubic lattices give to each lattice point 6, 8 and 12 nearest neighbor lattice points. No other arrangements that produce an extended array with cubic symmetry are possible.

We have already seen that any crystal can be assigned to one of the seven crystal systems on the basis of its symmetry. The repeating units that one constructs to describe the internal patterns of crystals must also have symmetry characteristics that allow them to be associated with the crystal systems. The three lattices at the top, for example, have at least four threefold axes of symmetry and therefore belong to the cubic system. Just as one assigns crystals, such as those which to crystal systems on the basis of symmetry, so can one assign the 14 possible lattice arrangements of these crystals systems?

Unit cells: the three cubes at the top of the figure clearly show the cubic symmetry of these three lattices. Such units of the lattice are known as unit cells. There is some freedom in the choice of the unit cell for a particular lattice, and the selection is made primarily to exhibit the symmetry of the lattice.

The simplest type of unit cell has lattice points, i.e. points which are identical surroundings, only at the corners. Such cells are known as primitive cells. Other unit cells drawn to exhibit the lattice symmetry have additional lattice points either within the cell, to give body centered unit cells, symbol l.

Now with the concept of unit cells, we describe lattices on the basis of:
    
Whether the lattice is primitive, face centered, or body centred.
    
The axes that most conveniently allow points within the unit cell to be located.
    
The symmetry of the unit cell 

The concept of lattices, the existence of only 14 types, and the association of these lattices, with the help of unit cells, to the symmetry based crystal systems provide a suitable connection between internal structure and crystal form.    

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