In anisotropic minerals, the refractive index varies with the crystal direction. This relationship can be illustrated with the help of a geometrical figure called the ”optical indicatrix”. The optical indicatrix of an anisotropic crystal is an ellipsoid whose semiaxes are proportional to the refractive indices in the respective directions. In the crystals of cubic system as the refractive index is the same in all directions, the indicatrix will have the form of a sphere.

### (1). Uniaxial Indicatrix:

The crystals of tetragonal and hexagonal systems have two main values of refractive index: (i) for vibrations parallel to the c-axis, and (ii) for vibrations in the direction normal to the c-axis. The optical indicatrix of such crystals is an ellipsoid of rotation which has only one circular section is a semi-axis, the length of which may be greater or smaller than the radius of this section.

- If the length of the semi-axis is greater than the radius of the circular section, the ellipsoid will be prolate (i.e extended along the optic axis). and the crystal is said to be optically ”positive”.
- If the length of the semi-axis is less than the radius of the circular section, the ellipsoid is oblate (i.e. flattened along the optic axis) and the crystal is said to be optically ”negative”.

The direction normal to the circular section is called the ”optic axis”. The light moving parallel to this axis is not doubly refracted. In all other directions, the light is doubly refracted and gives rise to ordinary and extraordinary rays. We can fully understand the concept of Uni-axial indicatrix by means of the following video;

### (2). Bi-axial Indicatrix:

In crystals belonging to orthorhombic, monoclinic and triclinic systems, the optical indicatrix is a triaxial ellipsoid. The lengths of its three semi-axis are proportional to the refractive indices **α, β, **and **γ****: **(i) **α **along X direction, (ii) **β **along Y direction, and (iii) **γ **along Z direction.

This triaxial elliptical has two circular sections having diameter equal to its intermediate axis. The directions normal to these circular sections are ”optic axes”. As there are two optic axes, the crystals of this group are called ”biaxial”.

Three principal sections through the indicatrix along planes XY, YZ, and XZ are shown in the following figure.

They are all ellipses. The lengths of semi-major and semi-minor axes of each represent the refractive indices in the corresponding direction. Of these three sections, the XZ section is most important and its main characters are as follows:

- The lengths of semi-major and semi-minor axes of the XZ section are proportional to
**γ**and**α**respectively. Between these two extreme values, it is possible to locate points on the ellipse where the radius is proportional to the intermediate refractive index**β.**This radius is shown by*S*in the above figure. - There are two circular sections of radius
*S.*The directions perpendicular to these sections are the ”optic axes”. The XZ plane that contains them is the ”optic axial plane”. - The intermediate axis Y which is perpendicular to the XZ plane is called the ”optic normal”.
- The acute angle between the two optic axes is called the ”optic axial angle”. It is designated by 2V.
- As the Z direction is the acute bisectrix, the crystal shown in the above figure.

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