Why are there only 14 types of Bravais lattices and not 28 when there are 7 types of unit cells and each can have four variations?

Question

As the title suggests, I can't understand why certain kinds of variations (like Face-centred or Body-centred) are restricted to certain types of unit cells. An orthorhombic unit cell has Primitive, Body-centred, Face-centred and End-centred variations, why can't the same be applicable to other crystal systems too?

Has this something to do with symmetry, but in that case a cubic unit cell should have four variations (it only has three). There are some patterns, like all crystal systems which have angles $=90^\circ$ have Body-centred variations, but beyond that it doesn't make sense.

Answer 1

Essentially, certain combinations of the possible point-group symmetries (cubic, tetragonal, hexagonal, trigonal, orthorhombic, monoclinic, triclinic) and possible translational symmetries (simple, base-centered, face-centered, body-centered) end up having identical overall lattice symmetries and thus you don't get $7×4$ unique lattices.

For example, suppose you propose a base-centered cubic lattice. Base-centered means you get the same lattice back if you translate the corners of each unit cell to the centers of a pair of specified opposing faces (the "bases"). But "cubic" means you get the same unit cell back when you rotate each face to match an adjacent one (rotating around a body diagonal of the cube). So to have both the base-centered translational symmetry and the cubic point-group symmetry, you have to allow translation of the unit cell corners to the centers of all the faces, not just one opposing pair, and your intended base-centered cubic lattice is really face-centered cubic.

Let's try a different example. Suppose you try to construct a base-centered tetragonal lattice by allowing translations of the corners onto the centers of the opposing square faces of the prism. Tetragonal point-group symmetry does not include a rotation around a body diagonal or any other operation that would shift the square faces onto another face, so you avoid the trap of turning "base-centered" into "face-centered" like what happened with cubic symmetry. You really do have a specific pair of "bases". But now there is a different trap: you can draw a smaller unit cell, with smaller square faces, that is a simple tetragonal lattice. So again your intended base-centered lattice is not unique; in this case it is just another simple tetragonal lattice.

When we work through all the constraints with each of the seven point-group symmetries, we find that a unique base-centered lattice exists only for the orthorhombic point-group symmetry, a unique body-centered lattice exists only for cubic, tetragonal and orthorhombic point-group symmetries, and so on. Thus only 14 out of the apparent 28 point-group/translational symmetry combinations actually form different Bravais lattices.

Answer 2

Whether to use a centered cell or the smaller primitive cell is a question of convention. The conventions are guided by making life easy. You could take a primitive triclinic cell and redefine the cell axes and lengths to make it body-centered but that would be kind of silly. There is no gain in simplifying symmetry operations, and you just made your cell volume twice as large and added a symmetry operation you didn’t have for the primitive case.

On the other hand, a body centered orthorhombic cell is easier than the corresponding smaller cell with non right angles and symmetry elements going along diagonals.

For cubic space groups, we don’t expect an A-centered cell because the presence of a 3-fold along the body diagonal would result in centering of the other faces as well.