Optical indicatrix of uniaxial and biaxial minerals, and how it helps determining the optically positive and negative crystals

The Optical Indicatrix: A Foundation

The optical indicatrix is a three-dimensional surface that represents the refractive index (n) for all directions within a crystal. Each point on the surface corresponds to a specific direction, and the distance from the center to that point represents the refractive index for light traveling in that direction. The shape of the indicatrix is governed by the crystal system and its symmetry elements.

Uniaxial Minerals and Their Indicatrix

Uniaxial minerals, belonging to the tetragonal, hexagonal, and trigonal crystal systems, exhibit a single optic axis. Their indicatrix is shaped like a sphere that has been either stretched or compressed along the optic axis. This results in an ellipsoid of revolution.

• ω (omega) and ε (epsilon): Uniaxial minerals have two principal refractive indices: ω, the refractive index for light vibrating perpendicular to the optic axis, and ε, the refractive index for light vibrating parallel to the optic axis.
• Optical Sign: The optical sign of a uniaxial mineral is determined by whether ω > ε (optically negative) or ω < ε (optically positive).
• Example: Calcite (CaCO₃) is a uniaxial mineral that is optically positive (ω < ε). Titanite (CaTiSiO₅) is uniaxial negative (ω > ε).

In a uniaxial mineral, the optic axis is the direction of minimum or maximum refractive index, depending on whether it is positive or negative. Light traveling parallel to the optic axis experiences a single refractive index (ε or ω), while light traveling in other directions experiences different refractive indices depending on their orientation relative to the optic axis.

Biaxial Minerals and Their Indicatrix

Biaxial minerals, belonging to the orthorhombic, monoclinic, and triclinic crystal systems, possess two optic axes. Their indicatrix is an ellipsoid with three unequal principal refractive indices: α, β, and γ.

• Principal Refractive Indices: α, β, and γ represent the maximum, intermediate, and minimum refractive indices, respectively. These indices are associated with three mutually perpendicular directions called the principal sections.
• Optical Plane: The plane containing the two optic axes and the direction of intermediate refractive index (β) is called the optical plane.
• Optical Sign: The optical sign of a biaxial mineral is determined by the relative positions of the refractive indices with respect to the optical plane.

Determining Optical Sign in Biaxial Minerals

The optical sign of a biaxial mineral is determined by the acute angle (2V) between the two optic axes. This angle is measured in the optical plane.

• Optically Positive (2V > 90°): In optically positive minerals, the acute bisectrix (the direction of the fastest vibration) lies within the acute angle between the two optic axes. The β ray is faster than the α and γ rays.
• Optically Negative (2V < 90°): In optically negative minerals, the acute bisectrix lies outside the acute angle between the two optic axes. The α ray is faster than the β and γ rays.
• Example: Augite ((Ca,Mg,Fe)₂Si₂O₆) is a common biaxial mineral that is typically optically negative. Orthoclase (KAlSi₃O₈) is biaxial positive.

Table Summarizing Key Differences

Feature	Uniaxial Minerals	Biaxial Minerals
Optic Axes	One	Two
Principal Refractive Indices	ω, ε	α, β, γ
Optical Plane	Not applicable	Present
Determining Optical Sign	ω > ε (negative), ω < ε (positive)	2V angle (positive if >90°, negative if <90°)