# Why is the commensurate charge density wave for 1T-$\rm TaSe_2$ based on non-integer dimensions?

I have read several papers that have stated that the commensurate charge density wave (CDW) for 1T-$$\rm TaSe_2$$ comes from a $$\sqrt{13}a_0$$ x $$\sqrt{13}a_0$$ periodic lattice distortion where $$a_0$$ is a length parameter of the undistorted lattice, as shown below (Image from: Appl. Phys. Lett. 113, 173103 (2018); https://doi.org/10.1063/1.5052722).

However, my understanding of commensurate CDWs was that it occurs when the period of the density wave modulation was an integer multiple of that of the underlying lattice ($$a_0$$). Does anyone know how this relates to the $$\sqrt{13}a_0$$ x $$\sqrt{13}a_0$$ which is based on a non-integer? Is it that the period of the CDW modulation is not quantitavely defined by the dimensions of the superlattice?

• Hint - count how many "old" unit cells fit in the new unit cell. You should find that this number is an integer for the commensurate case. Commented Jan 12, 2023 at 13:43
• @KFGauss I just want to check my understanding. By taking the "old" unit cell to have dimensions along the hexagonal lattice vectors 1 by 1 (in units of $a_0$) and taking the 'area' of a hexagonal unit cell (ie a parallelogram) as A=bh, then the 'area' of the old unit cell is 1 ($a_0$^2) compared to the distorted unit cell that has 'area' 13 ($a_0$^2). So, is it considered commensurate by fitting 13 "old" unit cells (ie an integer) into the new one, so that there's an integer relationship and the effect of the 13.9 degree rotation is being removed/ignored?
– John
Commented Jan 12, 2023 at 16:27
• The angle isn't being ignored per se, it just tells you which unit cells fit into the new one. Commented Jan 13, 2023 at 3:41