Review and Summary
The key points to know from what we have learnt thus far are the following:
- A periodic crystal (one with translational symmetry) can be represented by lattice points with a motif placed at each lattice points.
- The lattice can be represented by a unit cell of lattice points. Each lattice point has an identical environment around it. This is why when lattice points are placed at the vertices, only certain other places in the unit cell can contain lattice points.
- By convention, the first lattice point of a unit cell is placed at the vertices of the unit cell.
- All lattices can be represented by a primitive unit cell, though this may not be the one conventionally used.
- Some centring is not possible for certain crystal systems due to the symmetry e.g. base-centred and cubic.
- Some centring is possible for a given crystal system, but is not typically used as it is a redefinition of an already specified lattice type e.g. The monoclinic body-centred lattice is identical to the monoclinic base-centred lattice.
I would recommend moving onto the section studying space group diagrams. This will solidify an intuitive understanding of symmetry elements before attempting to use coordinate representations with these.