I have calculated a conductivity tensor of a rhombohedral crystal. I am now trying to convert this tensor to the hexagonal setting. I have been searching on which matrix transformation to use and the method to do this but am still unsure. Any help would be much appreciated.
Thanks
Alex
I think it should be like this:
1- First define the new basis (hexagonal, H) from the old basis (rhombohedral, R):
aH = bR - cR
bH = -aR + cR
cH = aR +bR +cR
then the transpose of the matrix of the above coefficients is the so-called transition matrix:
P = {{0, -1, 1}, {1, 0, 1}, {-1, 1, 1}}
in such a way that the coordinates of any vector v are transformed from R to H according to the rule:
[v]H = P-1 [v]R (eqn 1)
and from R to H according to the rule:
[v]R = P [v]H (eqn 2)
2- Now to obtain the transformation rule for the tensors, consider two vectors u and v related by the tensor T:
u = T v
In the R system the coordinates obey:
[u]R=[T]R [v]R
and therefore, using eqn 2:
P [u]H = [T]R P [v]H
Then multiplying by P-1 on the left:
[u]H = P-1 [T]RP [v]H
and therefore:
[T]H = P-1 [T]RP
which (I think) is the rule you are looking for... Can anyone please correct this if wrong?
Checked the above rule and it does work.
I also realised of something else that is useful to make the transformation easier.
According of the so-called Neumann's principle, the symmetry elements of any tensorial physical property of a crystal must include all the symmetry elements of the crystal. Therefore, your tensor in the hexagonal setting must be constrained to the form:
[T]H = {{Ta, 0, 0}, {0, Ta, 0}, {0, 0, Tc}}
i.e., with only two independent components, Ta and Tc. On the other hand, in the rhombohedral setting the tensor must be constrained to the form:
[T]R = {{Td, To, To}, {To, Td, To}, {To, To, Td}}
i.e., it also has two independent components: the diagonal value Td and the off-diagonal value To.
If you apply the transformation rule [T]H = P-1 [T]R P now, you can see that:
Ta=Td - To
and
Tc=Td+ 2To
This will allow you to transform the tensor from rhombohedral to hexagonal setting without having to do any matrix multiplication. :-)
Note that 2Ta + Tc = 3 Td , i.e. the trace does not change with a change of basis, as expected.
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