# 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. – tschoppi Feb 21 '14 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 – user40003 Feb 21 '14 at 3:17
• @user40003 Did that answer your questions? – Klaus-Dieter Warzecha Feb 22 '14 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 – user40003 Feb 23 '14 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 – user40003 Feb 25 '14 at 2:59
• @user40003 The unit cell of the hexagonal crystal cystem is a rhombohedron. – Klaus-Dieter Warzecha Feb 25 '14 at 6:27