# Crystallographic terminology associated with the honeycomb structure

I'm trying to quickly learn basic crystallographic principles in 2D under pressure (it's me whose under pressure, not the lattice) so I'm checking many sources to find those that "speak to me" the best (including this excellent answer). Many sources (books, papers, researcher's websites) cite the "honeycomb lattice" as an example of a lattice which is non-Bravais in order to reinforce their explanation of what a Bravais lattice is.

• some online examples where honeycomb is a "lattice" but not Bravais: slide 3 and here and slide 7

• some online examples where the "lattice" designator is avoided: here and slide 20

• ambiguous or noncommittal as to being a "lattice" slide 6

Briefly (I don't have the various texts with me right now to transcribe the paragraphs) it seems to me that a 2D Bravais lattice any set of points $$m \vec a + n \vec b$$ for all integers $$m, n$$.

The vertices of a honeycomb don't satisfy this because if you try to use that expression, 1/3 of the points are missing. If you want to use a Bravais lattice, you have to define a unit cell with two "atoms" in it.

Question: Can the vertices of a honeycomb or "honeycomb lattice" really be called a proper lattice (but just not a Bravais lattice), or should we really call it either a net or a regular complex apeirogons or something else, but not refer to it as a lattice?

From http://www-personal.umich.edu/~sunkai/teaching/Winter_2018/01302018.pdf (click for full size)

## †The Five nets

From Elizabeth A. Wood's classic Vocabulary of Surface Crystallography Journal of Applied Physics 35, 1306 (1964); https://doi.org/10.1063/1.1713610

In triperiodic structures the equivalent points form a three-dimensional lattice in which the space units are unit cells. In diperiodic structures the equivalent points form a two-dimensional net in which the area units are unit meshes. The fourteen Bravais space lattices are replaced, in the diperiodic case, by five nets, described in Table I and diagrammed in Fig. 2.

• After some thought, I think this is mostly a question of what you need to convey by using a particular term. If 'net' is commonly used in the community you are trying to engage, go for it. But I'm not sure there is a definitive answer to be had - use what somebody else will understand. Jul 23 '19 at 19:19
• @JonCuster sage advice for sure! Yes, the more sources I read, the more I begin to wonder if the term "lattice" doesn't have a universally accepted definition that includes or excludes the honeycomb vertices.
– uhoh
Jul 23 '19 at 22:19
• Well, Bravais lattice is well defined. And the honeycomb is a lattice, just a Bravais lattice with a basis. Like diamond cubic... Jul 23 '19 at 22:34

the concept of lattice is very (too!) often misinterpreted outside crystallography. By definition, the lattice points are all identical, that is each lattice point has identical surroundings to all others. In a honeycomb pattern you can distinguish 2 different sets of points, the red and the blue ones in the figure.

So honeycomb pattern is actually a 2-dimensional structure (not a lattice), resulting from the convolution of a basis (in this casis constituted of one blu + one red point) with a 2D lattice (the hexagonal one in the figure representing the five nets)

• Thank you for your excellent answer, it's great when a new user picks up an old question and answers it nicely (i.e. clearly, completely and concisely). Yes this makes complete sense.
– uhoh
Jan 1 at 12:09
• To the longer form of the question statement in the body of the question post "Question: Can the vertices of a honeycomb or "honeycomb lattice" really be called a proper lattice (but just not a Bravais lattice), or should we really call it either a net† or a regular complex apeirogons or something else, but not refer to it as a lattice?" (bumpy wording because I was grappling with concepts back then) I think the answer is "NO if 'vertices' = atoms or honeycomb points" and "YES if 'vertices' = ring centers or either the red or blue atoms as indicated by the unit cell vectors in the diagram"
– uhoh
Jan 1 at 12:11
• I still don't fully understand the usage of the term "net" here or in general. In the block quoted passage it looks like the author uses "lattice" for 3D lattices and "net" for what I would call a 2D lattice. Perhaps that's a historical artifact. To me I would call the honeycomb pattern a net, and the hexagonal lattice that underpins it a lattice, but I really don't know if net has a specific meaning anymore or not.
– uhoh
Jan 1 at 12:16
• I agree with you. you can find reliable definitions of net here: [Journal of Solid State Chemistry 152, 3 (2000)]; [Journal of Solid State Chemistry 178, 2480 (2005)] Jan 1 at 16:18