Chemistry - Why are there only 7 types of unit cells and 14 types of Bravais lattices?

Solution 1:

All quotes will be from Solid State Physics by Ashcroft and Mermin.

Bravais Lattice:

A fundamental concept in the description of any crystalline solid is that of the Bravais lattice, which specifies the periodic array in which the repeated units of the crystal are arranged. The units themselves may be single atoms, groups of atoms, molecules, ions, etc., but the Bravais lattice summarizes only the geometry of the underlying periodic structure, regardless of what the actual units may be."

Primitive Unit Cell:

A volume of space that, when translated through all the vectors in a Bravais lattice, just fills all of space without either overlapping itself or leaving voids is called a primitive cell or primitive unit cell of the lattice.

Unit Cell; Conventional Unit Cell:

One can fill space up with nonprimitive unit cells (known simply as unit cells or conventional unit cells). A unit cell is a region that just fills space without any overlapping when translated through some subset of the vectors of a Bravais lattice. The conventional unit cell is generally chosen to be bigger than the primitive cell and to have the required symmetry.

Crystal Structure:

A physical crystal can be described by giving its underlying Bravais lattice, together with a description of the arrangement of atoms, molecules, ions, etc. within a particular primitive cell.

So, one comes up with 14 Bravais lattices from symmetry considerations, divided into 7 crystal systems (cubic, tetragonal, orthorhombic,monoclinic, triclinic, trigonal, and hexagonal). This comes solely by enumerating the ways in which a periodic array of points can exist in 3 dimensions.

Now, what is on those points is a unit cell, which will itself have some symmetry. Thus, the combination of Bravais lattice and unit cell symmetry can again be enumerated and one comes up with 230 space groups.

Now for some of your related questions:

All cubic-related Bravais lattices will have 90 degree angles because they are based on cubic symmetry. The trigonal Bravais lattice has no 90 degree angles, but isn't talked about much in more basic textbooks because, well, it looks weird.

Why no pentagonal unit cells? Well, because you can't fill space with a 5-fold symmetric Bravais lattice. Quasicrystals, while they have 5-fold symmetry, are a tiling through space that does not obey the rules for a Bravais lattice.

Solution 2:

If you want to follow the rule, that a crystall is formed by endless translational symmetry of a unit cell, then the only possibilites to start from are two structures: parallelepipeds and hexagonal prisms.

You can always, and only, except for hexagonal prisms, fill space by stacking parallelepipeds in all three directions. Rhombohedral, cubic, trigonal etc. are all special cases of the "triclinic" unit cell with higher symmetry, it is obvious that there are not endlessly more options that are more symmetric. Those make up for six of the seven crystal systems, and hexagonal is the special case making up the seventh.

The Bravais lattices come from unit cells which have an internal symmetry. You could go without these by describing them with one of the less symmetric crystal systems, but the rule is to assign the crystal system with highest symmetry. There are again not so many possibilities to have an internal symmetry, so this only makes 14 Bravais lattices out of the 7 crystal systems.

The paradigm is not to think of ways to make the system endlessly more complicated, but to start from the most odd system that is able to fill space, and think of the (limited) possibilities to make it more simple (i.e. symmetric).