# Hexagonal unit cells

I know 3 types of unit cells, the simple cubic, the body centered and the face centered.

But what is a hexagonal unit cell, and why does it have a hexagon base while others are all cut into a cubic? And how can I calculate the number of atoms in every primitive hexagonal unit cells?

• Welcome to chemistry.SE! I've taken the liberty to edit your question some, while retaining the original meaning. Commented Feb 21, 2014 at 8:27

The cubic primitive, cubic body-centered and the cubic face-centered are just the three possible Bravais lattices of the cubic crystal system.

In total, we know seven crystal systems, namely (in order of increasing symmetry) triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic.

Among these, the triclinic crystal system has the lowest symmetry ($a \ne b \ne c;\ \alpha, \beta, \gamma \ne 90°$), the cubic crystal system has the highest symmetry ($a = b = c;\ \alpha = \beta = \gamma = 90°$).

A careful look at illustrations of the hexagonal closest packing ($7+3+7$ atoms in $ABA$ layers) in a hexagonal prism and a bit of thinking gives the number of atoms:

• $3$ atoms in the $B$ layer are not shared by any other prism. They fully count as $\mathbf{3}$.

• $2$ atoms in the centre of the top and bottom $A$ layer are each shared by any other prism (above and below). Each of them counts $\frac{1}{2}$. So there's another $2 \times\frac{1}{2}= \mathbf{1}$ atom.

• Top and bottom $A$ layers each have $6$ atoms on corner positions. Each atoms is shared by three neighbors in the the same layer and with another hexagonal prism on top or below. So they count as $2\times6\times\frac{1}{3}\times\frac{1}{2}=\mathbf{2}$ atoms.

The total count thus is $\mathbf{6}$.

These diagrams of the hexagonal closest packing, do not represent the the unit cell as the most simple unit from which the crystal is build by symmetry operations.

Cutting the hexagonal prism (along the axis through the atoms centering the top and bottom faces) into three pieces gives identical rhombohedra. These can be transferred into each other by rotations along the axis described above.

• So how many molecules are in each hexagonal unit cells? And why I saw many people cut hexagonal unit cells into three structures that look like body centered unit cells. Check this out youtube.com/watch?v=vXtO8afu8wg Commented Feb 21, 2014 at 3:17
• @user40003 Did that answer your questions? Commented Feb 22, 2014 at 9:10
• I havent't got time to read it yet. I will do it probably tmr or tonight!Thanks for your answer though Commented Feb 23, 2014 at 16:22
• So when we usually say hexagonal unit cells, we refer to the hexagonal structure one or the one has been cut into rhombohedra Commented Feb 25, 2014 at 2:59
• @user40003 The unit cell of the hexagonal crystal cystem is a rhombohedron. Commented Feb 25, 2014 at 6:27