# How to derive the k-path for monoclinic lattice structure?

Good afternoon all,

My post is inquiring about the mathematical derivation behind the Monoclinic crystal structure (c axis unique). The Brillouin zone (BZ) I really would like to derive is shown below:

The plan for at least in the 2D system a and b is the following:

1. Get reciprocal lattice vectors a and b from you lattice vectors.
2. Draw vectors a, b, -a, -b, a+b, and -a-b in paper.
3. Bisect them. (It should be the equation of a straight line passing mid-distance from the vector, and forming a 90 degree angle with the vector, which is what a bisector is.)
4. Determine the points where two consecutive bisectors join, by writing down the equations of their straight lines and making them equal.
5. The points where bisectors cross are your high symmetry points. And this process, repeated for all pairs of vectors in point 2, give you the first Brillouin zone.

I got the following:

When i started calculating the bisector equations it got really messy and I hope to get someone to help continue the work or find a suitable reference to know how we got the BZ.

• +1. Would you consider pasting your TeX code into the question though, rather than only giving a screenshot of it? It would look better I think! Aug 8, 2020 at 15:40
• You are absolutely right @NikeDattani I found it but it was too late :( next time for sure. Aug 9, 2020 at 0:47
• Why not edit your question? Aug 9, 2020 at 2:38

Not a direct answer to your question, but I hope you will still find this relevant. A very useful online tool to construct Brillouin zones and conventional $$\mathbf{k}$$-paths for structures of any symmetry is SeeK-path.

Inputs. It accepts input structures in the formats from major DFT codes (Quantum Espresso, Castep, Vasp), as well as other formats like cif, xyz, or pbd. Additionally, you can simply specify a space group without providing a structure.

Outputs. The online tool then generates the Brillouin zone, labelled with the corresponding high symmetry points. Additionally, it generates a proposed $$\mathbf{k}$$-point path connecting high symmetry points.

Link. You can access the website here.

Reference. All is explained in this paper (or arXiv version).

• Thank you @ProfM for the comment. I did use it before but I had my doubts on the K path chosen which is why I wanted to find out why. Aug 9, 2020 at 0:42
• I normally use this reference and worked fine ( arXiv version: 1004.297).
– Camps
Aug 10, 2020 at 12:17