I just calculated the stiffness tensor for a given material (a monoclinic molecular crystal) using molecular simulation. When trying to compare my results to published experimental results I found that the crystal structure I used for simulation, though essentially the same as the one used as reference system in experiment, has different base vectors/reference frame (P21/n vs P21/a) - therefore I can compare invariants and averages but not the individual components c_ij of the stiffness (as matrix in Voigt notation).

So my question is: given the, altogether six, lattice vectors of the two crystal structures, how can I use this information to generate a rotation matrix to transform the Voigt matrix from experiment (based on a crystal in the P21/a space group) so that I can compare it to my calculated numbers from simulation (based on a crystal with P21/n symmetry)?

What comes closest to an answer I found here: http://solidmechanics.org/text/Chapter3_2/Chapter3_2.htm but I am not sure what the two bases (e and m, in section 3.2.11 Basis change formulas for anisotropic elastic constants) are - are these the normalized lattice vectors of the two structures in Cartesian coordinates? if not what else?




1 Answer 1


In order to define the matrix transforming the P21/n structure into P21/a it is imperative to know the unique axis of both space groups.

For example the contracted P21/n symbol can be used for the full P21/n11 symbol (unique monoclinic axis along a), for the full P121/n1 (unique axis along b) and for the full P1121/n (unique axis along c).

The same apply for P21/a (i.e. it could stand for P21/n11,...).

So you have to understand the exact settings of space groups, then you can find the transformation matrix relating them. For example, if you have the initial and final settings P121/n1 -> P121/a1 (same unique b axis in bth space groups), the corresponding transformation matrix is: [-1 0 1] [ 0 1 0] [-1 0 0] that is -a-c,b,a

If you have P121/n1 -> P1121/a, than the transformation matrix is -a-c,c,b

  • $\begingroup$ thanks for your answer! ... I am not sure though whether it answers the question ... I am looking for a rotation matrix to convert the stiffness tensor (either the 4th rank tensor, or preferably the Voigt matrix), and not just to rotate the structure/coordinates - or are these the same? $\endgroup$ Commented Jan 6, 2021 at 15:53
  • $\begingroup$ sorry and one more follow up ... I want to convert C21/a to C21/n (and not the other direction as discussed by you) - would I just use the inverse of the matrix you provided in this case? $\endgroup$ Commented Jan 6, 2021 at 15:56
  • $\begingroup$ what do you mean with C21/a and C21/n? It seems to me that these space groups do not exist. $\endgroup$
    – gryphys
    Commented Jan 6, 2021 at 16:02
  • $\begingroup$ In any case to answer to your first comment, I never made a transformation with a 4th rank tensor, but it shuold be the same as for a 2nd one. $\endgroup$
    – gryphys
    Commented Jan 6, 2021 at 16:05
  • $\begingroup$ sorry ... i meant P21/n or a .... typo ... and I was talknig about a voigt matrix which is not a tensor at all, but it can still be transformed with a nother matrix to account for a differnt base/reference ... so what i meant can i use the rotation matrix you provided for this purpose? $\endgroup$ Commented Jan 6, 2021 at 16:10

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