II.3 Calcite

 

As an example of the behaviour of light in an anisotropic crystal, let’s have a look at calcite (CaCO₃). Calcite is a hexagonal mineral that is strongly anisotropic because of the parallel arrangement of the carbonate groups. The main crystallographic direction is the c-axis, along which a threefold rotation axis can be observed.

 

 

Fig. 2.4 Crystal structure of calcite

 

In the following discussion we assume that the light ray will hit the crystal perpendicular to its crystal face. In this case we will take one of the rhombohedral faces (Fig. 2.5, compare also with Fig. 2.2).

 

 

Fig. 2.5 Double refraction in a calcite crystal (the angle between the two rays is exaggerated).

 

 

When a light ray L enters the crystal it will be split in two separate rays. One of these, the o- (ordinary) ray is in the same direction as the incoming ray. It is called the ordinary ray because it obeys Snell’s law. The other, the e- (extraordinary) ray is refracted relative to the incoming ray L and lies in a plane through L parallel to the c-axis.  The extraordinary ray is deflected and does not obey Snell’s law. The splitting of ray L into two separate rays is called double refraction. Both rays are strongly polarised; the e-ray in the plane through L and parallel to the c-axis and the o-ray perpendicular to the e-ray. These two directions are called privileged directions.  The propagation velocity of the e-ray is larger than that of the o-ray and thus the refractive index corresponding with the e-ray is lower. So, ω = n_o (1.658) and ε = n_e (1.486).

 

After leaving the calcite crystal the two rays will be parallel again in their propagation direction, but they remain polarised. When we make a thin section of a calcite crystal in a different orientation we will see similar effects. However, the degree of double refraction depends on the orientation of the crystal, i.e. on the angle between the c-axis and the incoming light ray L. In the special situation that the light ray L comes in perpendicular to the c-axis, the e- and o-ray will coincide but the polarisation will still be perpendicular to one another. The e-ray is now polarised parallel to the c-axis.

 

At this angle of incidence the difference between the velocities of the o- and e-ray is maximal and thus the difference in refractive indices is maximal. When we decrease the angle of incidence this difference will become smaller and will reach zero when the light ray hits the crystal parallel to the c-axis. In this situation the crystal behaves optically isotropic and the refractive index is identical to that of the o-ray. So, the orientation of the section is determining not only the refractive indices but also the polarisation directions of both the o- and e-rays.

 

In general, one can say that for anisotropic crystals the following is true:

-         incoming light will be split at the crystal surface in two perpendicularly polarised rays

-         the degree of double refraction is partly determined by the orientation of the crystal

-         the propagation velocity, and thus the refractive index (because v = c/n), depends on the polarisation direction