Advanced Solid State Physics

Figure 56: Forces applyable in the x-direction

orthogonal direction. We can define a stress tensor:

σkl =







Xx

Ax

Xy

Ay

Xz

Az

Y x

Ax

Y y

Ay

Y z

Az

Zx

Ax

Zy

Ay

Zz

Az







The relationship between those two can be described with a rank four stiffness tensor

ij = sijklσkl

Of course, this relationship exists in two directions, the inverse of the stiffness tensor is called the
compliance tensor

σij = cijklkl

Most physical properties can be connected to each other via tensors, as we will see in chapter 8.2.
One further example is the electric susceptibilityχ, which connects the electric field to the electric
polarisation. Since both properties are vectors, the susceptibility is a rank two tensor, a matrix. So if
we want to relate two properties, like the strain and the electric field, we know that since the electric
field is a vector and the strain is a matrix, we will need a tensor of rank three.

8.2 Statistical Physics

As usual in statistical physics we start out with a microcannonical ensemble. There, the internal
energy is described by the extrinsic variables (Just to remind you: Extrinsic variables scale with the
size of the system). We can take the total derivative:

dU =

∂U

∂S

dS +

∂U

∂ij

dij +

∂U

∂Pk

dPk +

∂U

∂Ml

dMl (61)

Or, using the intrinsic variables:

dU = TdS + σijdij + EkdPk + HldMl