[Pw_forum] Transformation of a tensor from the crystal basis to the cartesian basis and viceversa

[Iurii TIMROV]

Dear All,

I would appreciate if someone can help me to understand what crystal basis
we deal with (composed from direct or reciprocal vectors) in the routines
PH/trntnsc.f90 and PW/trntns.f90.

In the routine PH/trntnsc.f90 there is a comment that it does a
transformation of a complex tensor (like the dynamical matrix) from
crystal to cartesian axes or viceversa. I have a question: Which crystal
axes we are dealing with, the crystal basis of direct vectors
(at(i),i=1,2,3) or the crystal basis of reciprocal vectors
(bg(i),i=1,2,3)?

Crystal -> Cartesian : D_cart  = B * D_cryst * B^T ,
Crystal <- Cartesian : D_cryst = A^T * D_cart * A ,

where D_cryst, D_cart are the tensors of the dynamical matrix in the
crystal and cartesian basis respectively, A is the matrix composed from
the direct lattice vectors (at(i),i=1,2,3), B is the matrix composed from
the reciprocal lattice vectors (bg(i),i=1,2,3), the symbol * stands for
the matrix multiplication, and "T" stands for the transposition of the
matrix.

It seems to me that in these routines (PH/trntnsc.f90 and PW/trntns.f90)
we are dealing with the crystal basis of reciprocal vectors, not sure.

And I think that if one wants to do a transformation from the cartesian
basis to the crystal basis of direct vectors then the equations would
read:

Crystal(direct vectors) -> Cartesian : W_cart  = A * W_cryst * A^T ,
Crystal(direct vectors) <- Cartesian : W_cryst = B^T * W_cart * B ,

here W is some tensor.

Is it true?

Yours faithfully,
Iurii Timrov


Iurii TIMROV
Doctorant (PhD student)
Laboratoire des Solides Irradies
Ecole Polytechnique
F-91128 Palaiseau
+33 1 69 33 45 08
timrov at theory.polytechnique.fr