- Elastic deformation (stiffness tensor)
- Reciprocal piezo-electric effect
- Reciprocal piezo-magnetic effect
- Thermal dilatation
8.3 Crystal Symmetries
Intrinsic Symmetries
As seen in chapter 8.2, we can get the stiffness tensor by taking the derivative of the strain in respect
to the stress.
dij
dσkl= sijklThe strain itself is calculated out of the Gibbs free energy:
−
(
∂G
∂σij)
= ijSo the stiffness tensor can be written as the second derivative of the Gibbs free energy in respect to
two different elements of the stress matrix:
−
(
∂G
∂σij∂σkl)
= sijklBecause of Schwarz’ theorem^3 we get the same result if we can change the order of the derivatives:
−
(
∂G
∂σkl∂σij)
= sklij =! sijklThis way we can find those symmetries for a lot of different properties, for example the electric and
the magnetic susceptibility:
−
(
∂G
∂Ej∂Ek)
=(
∂Pk
∂Ej)
= χkj = −(
∂G
∂Ek∂Ej)
=(
∂Pj
∂Ek)
= χjk−
(
∂G
∂Hk∂Hl)
=(
∂Ml
∂Hk)
= ψlk = −(
∂G
∂Hl∂Hk)
=(
∂Mk
∂Hl)
= ψkl(^3) Hermann Schwarz, 1864 till 1951