Chemistry - Bravais Crystal Lattices and Voids
2 brief observations about the answer by Mitchell.
- a hexagonal lattice does not exist in 2-dimensions.
By definition,A lattice is an infinite array of points in space, in which each point has identical surroundings to all others. For example consider a 2d cubic lattice; now, for example, choose one of the lattice point (we call it point A) and connect it with an arrow with the point just above and with another arrow with the point on its right. Now move these two arrows as a solid object to another point of the lattice (we call it point B); you will see that the arrow will again connect this other lattice point with other two lattice points, one just above, the other at its right. You find the same thing for both point A and point B, whatever choice you make.
You can do the same job with a hexagonal arrangment of points. Choose again an arbitrary point A in the hexagonal arrangement and plot the two arrows connecting point A to two first neighbouring point. Then again move the two arrows as a solid object from point A to one of the 3 first neighbouring three points. You will see that now arrows will end where no point is present in the space. This is because the 2D hexagonal arrangment of points is actually constituted by two different sublattices, whose symmetries are trigonal.
- the different crystal systems are defined by the symmetry operations that are present, not by their metric.
The group of the symmetry operations define the symmetry along the different main crystallographic directions. For example, in the cubic system we have a=b=c and alfa=beta=gamma=90 metric because the symmetry operations that are present impose this.
The tetragonal symmetry simply dictates a=b and alfa=beta=gamma=90; there's no METRIC restriction on the c-axis. The SYMMETRY along the c-axis is different from that along a and b, but in principle it is possible to find a crystal structure that has a tetragonal structure (= symmetry), but a cubic metric, that is a=b=c (and it is still tetragonal!). there are several compounds where this is observed. In conclusion, inequalities strictly hold on symmetry, not on the metric of the crystal structure.