Bravais Crystal Lattices and Voids

What is the difference between Hexagonal Close Packing (HCP) structures and the hexagonal crystal Bravais lattice? I've looked up some sites but can't seem to understand them. It would help if you backed up the answer with your specific explanation of Bravais lattices and unit cells.

Edit: I thought the difference was that the lattice consisted of many unit cells and the unit cell packing could be like hexagonal, cubic or simple. But then I looked at the stoichiometric defects which talked about molecules being in interstitial spaces. But in the unit cell, the spaces are already filled. So how is that possible.

1. 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.

1. 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.

Here are some important definitions:

A space lattice provides the framework with reference to which a crystal structure can be described. A lattice is different from crystal. In fact, a lattice gives rise to crystal when lattice points are replaced by atoms, ions, or molecules.

A 2-D or 3-D lattice is a regular arrangements of points. In order to specify it completely, only a small part of the lattice is described. This figure is known as a unit cell.

In 2-D, there are 5 possible lattices namely, square, rectangle, hexagonal, parallelogram and rhombic.

In 3-D, there are 14 possible lattices, and these lattices are called Bravais lattices (after the French mathematician who first described them) like cubic primitive, hexagonal primitve, etc.

For example, In a cubic system there are 3 possible Bravais lattices possible namely, primitive, body centered and face centered.

Similarly in hexagonal crystal system there is only one Bravais lattice viz, Primitive.

Crystallographers have been able to divide 32 point groups and 14 space lattices into seven crystal systems and 14 Bravais lattices. Remember that the primitive cells of any two crystals are not the same. Primitive unit cells mean that the lattice point must be at the corners of the unit cell.

HCP is ABA-ABA arrangement of layers in which tetrahedral void of second layer are covered by the third layer.

• So each lattice site doesn't represent a site that could be replaced by a unit cell but actually by the constituents(atoms/ions) of the unit cell? Feb 18, 2017 at 6:52
• Yes, Unit cell contains many lattice points in/on them and the constituents are placed at those lattice points. Lattice is like a framework, a blueprint but without the constituents. A crystal is the building defined by that blueprint but with the constituents at those lattice points.. Feb 18, 2017 at 7:16
• Also, what do you mean by point groups? Feb 18, 2017 at 9:31