# Is the difference between cubic and hexagonal diamond structure in 2 dimensions or 3 dimensions?

I was reading the book Solid State Physics by Charles Kittel. It was explained that the difference between Cubic F or FCC and the Hexagonal Closed Packed structure or the HCP was as follows -

(Please note that I know that atoms are 3 dimensional spherical entities but here when I say a dimension I am interested in how many directions the atoms are stacked)

1. Both have the same type of packing of atoms in 2 dimensions
2. The different placement of one 2D-layer above the other causes the structure to become FCC or HCP.
3. FCC follows an ABCABCABC type stacking in 3 dimensions
4. HCP follows ABABAB type stacking in 3 dimensions

Now for the Diamond structures,

It was explained that in cubic diamond the layers numbered 0,3,6... lie exactly above each other and in hexagonal diamond the layers 0,2,4,6.... lie exactly above each other. Hence Cubic Diamond was called 012012 type and Hexagonal Diamond was called 010101 type. It was also mentioned in the caption of the figure depicting both these structures that these 012012 and 010101 structures are similar to FCC and HCP respectively.

But the figure hinted that the stacking difference between these 012012 and 010101 types was in the 2 dimensional stacking and not in the 3 dimensional stacking.

(Please note that I know that in diamond there are tetrahedrons which are 3 dimensional entities but I am assuming a tetrahedron to be a single entity and here when I say a dimension I am interested in how many directions the tetrahedrons are stacked)

So are the 012012 and 010101 stackings in 2 dimensions or are they in 3 dimensions?

• Tetrahedrons exist in your head. Diamond is made of atoms. Ditto for the 2D layers. That being said, cubic diamond and lonsdaleite can be thought of as made of the same layers, but stacked differently in 3D. Commented Apr 4 at 9:21
• Ok so just like FCC and HCP are made of same 2D layers but their stacking in 3D causes the difference, 012012 and 010101 are made of same 2D layers and their 3D stacking causes the differences. Thank you. Commented Apr 4 at 11:28
• Also posted on Physics. Please pick one site. Commented Apr 4 at 11:31
• Yes, just like that. Remember, though, that there are no layers in nature. They are in our heads. (This applies to diamond; some other things may contain actual, physically connected layers.) Commented Apr 4 at 11:33
• Ok thank you. And sorry if I caused any disturbance by posting on both sites, I simply wished to increase the viewer base. Commented Apr 4 at 14:22

### The layers

In FCC or HCP packing, the layers could be considered 2-dimensional because the atom centers fall into a single plane. If you take the diamond structure or the Lonsdaleite structure and extract layers common to both, you get a system of fused cylcohexanes.

The cyclohexanes are all in a chair conformation, so not flat. Looking perpendicular onto the layers, half of the atoms are a bit lower (with the dots), and half a bit higher (without the dots).

### Combining two layers

To add the next layer, the atoms that are a bit higher (no dots) on the lower layer have to line up with the atoms that are a bit lower (dots) on the upper layer. If the two layers are related by a pure translation, the only way to do this is with a diamond structure. The diamond structure has adamantane (four fused cyclohexanes in the chair conformation) at its core.

So looking from the top, the upper atoms of the upper layer are in the center of the cyclohexanes of the lower layer. Here is an animation of the two layers (blue on the bottom, tan on the top moving in) combining to form the diamond structure.

To get the hexagonal structure, you need cyclohexane rings to align from layer to layer. Lining cyclohexanes from neighboring layers results in a unit of iceane, five fused cyclohexane rings (with the bottom and top in the chair conformation and the three on the sides in the boat conformation):

A pure translation of a layer is insufficient because dots would line up with dots, not resulting in a tetrahedral surrounding of the carbon atoms. Instead, you have to rotate and translate (you can rotate 60 degrees or 180 degrees in the plane, or turn the layer upside down). In the animation, we are rotating by 180 degrees before aligning the dots on the upper layer with the carbons lacking dots on the lower one.

### More layers

For the diamond structure, you have to arrange the layers in a ABCA... fashion, i.e. looking from the top, every third layer has a matching position (in x and y). For the Lonsdaleite structure, the layering is A* A A* A, with the star referring to the rotation of the layer. Looking from above, all hexagons align.

[OP] Is the difference between cubic and hexagonal diamond structure in 2 dimensions or 3 dimensions?

It is in 3 dimensions. You can find layers one atom thick and even two atoms thick that are the same in the two crystal structures. The difference is in how you combine them.

[OP] It was explained that in cubic diamond the layers numbered 0,3,6... lie exactly above each other and in hexagonal diamond the layers 0,2,4,6.... lie exactly above each other.

Yes, if you take the chair-conformation cyclohexane fused in one plane as the layer, as I did above (even though technically, it is not a 2-D layer because there are two distinct z-coordinates).

[OP] Hence Cubic Diamond was called 012012 type and Hexagonal Diamond was called 010101 type.

That is a confusing nomenclature. The "1" in diamond and the "1" in Lonsdaleite are not in the same position (and one is rotated and the other is not). This is why I used A B C A B C A for diamond and A A* A A* A for Lonsdaleite.

[OP] It was also mentioned in the caption of the figure depicting both these structures that these 012012 and 010101 structures are similar to FCC and HCP respectively.

If you want to see the relationship, you would have to remove every second carbon (i.e. those without the dot in my figures). Then, every layer would indeed be purely 2D, and atoms would have 6 nearest neighbors within a layer, 3 above and 3 below for a total of 12, the known coordination number of closest packed spheres. The diamond structure would turn into an FCC packing (with gaps because the distances are too large), and the Lonsdaleite structure would turn into an HCP packing.

You could also go the other direction by starting with FCC or HCP, picking 4 atoms that are mutual neighbors, and placing another atom in the center (adding an atom to a "tetrahedral hole"). Making sure that every new and every old atom is tetrahedrally surrounded while you add more atoms, this would determine which other "tetrahedral holes" to fill and which to skip. In the end, you would double the number of atoms and arrive at either the diamond or the Lonsdaleite structure.

[OP] So are the 012012 and 010101 stackings in 2 dimensions or are they in 3 dimensions?

Maybe the layers have 2.$$\epsilon$$ dimensions, and then they stack to give 3 dimensions?

• Thank you so much. This explanation has been really helpful. Commented Apr 6 at 3:39