# Supercell Elastic Constant

Suppose, that one wants to study a bulk structure which is doped by several impurities. More precisely, I want to calculate the elastic constants using a DFT code (VASP) by fitting stress to strain (Voigt Notation):

$$\sigma_i = \sum_{i=1}^6 C_{ij} \varepsilon_j$$

From the literature I know, that in a cubic crystal structure the number of independent elastic constant reduces to three. For my purpose, I construct a tetragonal supercell (large enough to cover a low concentration of impurities), e.g. 3x3x4 unit cells.
Following questions arise:
1)In the case without doping, there are still three elastic constants (due to the periodic boundary condition), because the symmetry of the unit cell transfers to the supercell ?
2) Considering the doped case, the relaxed supercell is probably not tetragonal anymore. The unit cell is also not cubic anymore, at least if impurities are present. Does this mean I have to regard all elastic constants ?

I ask this question, because with the loss of symmetry I have to regard more strained systems, resulting in more computations. My system is large (ca. 150 atoms) and the calculation IBRION=6 in VASP takes too long.

Best,
Luca

• +1 Welcome to our forum!
– Camps
Jan 23 at 15:50

In the undoped case, when considering a cubic crystal structure and using a tetragonal supercell not changes point symmetry, Hence number of independent elastic constants remains three. If you rotate the crystal to other orientation you may get different $$C_{ij}$$ components. However, Eigen values of $$C_{ij}$$ remains three.