Hi,
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 (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?
thanks!
michael
Monoclinic medium has its "crystal frame" which is unique. In the crystal frame, the monoclinic frame plane of symmetry is horizontal, the local z axis is normal to this plane (i.e., the local z is vertical), while local x and local y are two mutually orthogonal polarizations of the fast shear and the slow shear propagating in the local vertical direction z. In the local frame, a monoclinic medium is characterized by 12 parameters only. In the global frame, the 6x6 Voigt symmetric matrix is fully populated. It can be converted to the local frame, where the stiffness tensors of two monoclinic media can be compared. In order to rotate the stiffness tensor from the global frame to the local, we rotate the matrix. It is more suitable to use the Kelvin presentation than the Voigt. Such rotation was first described by Bond for the matrix in the Voigt form (Bond, W., Jan. 1943, The mathematics of the physical properties of crystals: The Bell System Technical Journal, 1-72.), and by Slawinski (Slawinski, M., 2015, Waves and Rays in Elastic Continua, Third Edition, World Scientific Publishing Co, ISBN 978-981-4641-75-3) for the matrix in the (more suitable) Kelvin form. To create he rotation matrix that you asked, an eigenvalue-eigenvector/eigentensor analysis of the stiffness tensor is needed
You need the transformation matrix (and you may even need to know how your tensor transforms, that may depend on whether it is a real-space or a reciprocal-space tensor, I would have to look that up). I am not 100% sure that knowing that the space groups are P21/n and P21/a is sufficient to determine the transformation matrix: names like "P21/n" and "P21/a" are just names, and it is possible that multiple sets of symmetry operators formally have the same name. So you really need to know the actual unit-cell parameters and the actual matrix representations of the symmetry operators. There are some other subtleties (e.g. you have implicitly assumed that both coordinate frames have the same handedness).
I have code that can do this, feel free to contact me by e-mail. It should be trivial to find my e-mail address using Google.
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