8 Crystal Physics
The main subject of Crystal Physics is how the symmetries of the crystal affect the physical properties.
Knowledge of the symmetries can ease the measurements of the properties. If we take a simple cubic
crystal for example, repetition of a measurement after a rotation of 90°wouldn’t make much sense.
Tensors, with which one can describe such properties must take that symmetry into account. Knowing
the structure of the tensors, we would know how many independent properties we would have to
measure to be able to explain the whole behavior of the crystal.
8.1 Stress and Strain
To give an example for a physical property that is related to a certain tensor we will look at the stress
and strain relation in solids. If we apply a force to a crystal, it will most likely be deformed. The
former right-angled basis vectorsx,y, andzwill move to not-right-angled onesx’,y’andz’, as shown
in fig. 55. This distortion of the material is described by the strain matrixij.
Figure 55: Shift of the basis vectors because of strain, former right angled vectors (a) get displaced
(b)
x′ = (1 +xx)x + xyy+ xzz
y′ = yxx+ (1 + yy)y + yzz
z′ = zxx + zyy+ (1 + zz)zThe strain matrix enables us to describe shear, the distortion in a different direction than the force
acts.
Correspondingly, stress describes the forces per unit area applied to the material. We can find
nine independent forces Xx, Xy, Xz, Yx, Yy, Yz, Zx, Zy and Zz as shown in fig. 56. The capital
letter describes the direction of the force, the small ones the, to the plane on which the force acts on,