This document last updated on 25-Oct-2010 EENS 2110 MineralogyTulane University Prof. Stephen A. NelsonUniaxial Minerals, Uniaxial Indicatrix, Optic Sign, & Ray Path Introduction to Uniaxial Minerals Uniaxial minerals are a class of anisotropic minerals that include all minerals that crystallize in the tetragonal and hexagonal crystal systems. They are called uniaxial because they have a single optic axis. Light traveling along the direction of this single optic axis exhibits the same properties as isotropic materials in the sense that the polarization direction of the light is not changed by passage through the crystal. Similarly, if the optic axis is oriented perpendicular to the microscope stage with the analyzer inserted, the grain will remain extinct throughout a 360o rotation of the stage. The single optic axis is coincident with the c-crystallographic axis in tetragonal and hexagonal minerals. Thus, light traveling parallel to the c-axis will behave as if it were traveling in an isotropic substance because, looking down the c-axis of tetragonal or hexagonal minerals one sees only equal length a-axes, just like in isometric minerals. z Like all anisotropic substances, the refractive indices of uniaxial crystals varies between two extreme values. For uniaxial minerals these two extreme values of refractive index are defined as ω (or No) and ε (or Ne). Values between ω and ε are referred to as ε'. z Uniaxial minerals can be further divided into two classes. If ω > ε the mineral is said have a negative optic sign or is uniaxial negative. In the opposite case, where ε > ω the mineral is said to have a positive optic sign or is uniaxial positive. z The absolute birefringence of a uniaxial minerals is defined as | ω − ε | (the absolute value of the difference between the extreme refractive indices). Double Refraction All anisotropic minerals exhibit the phenomenon of double refraction. Only when the birefringence is very high, however, is it apparent to the human eye. Such a case exists for the hexagonal (and therefore uniaxial) mineral calcite. Calcite has rhombohedral cleavage which means it breaks into blocks with parallelogram - shaped faces. If a clear rhombic cleavage block is placed over a point and observed from the top, two images of the point are seen through the calcite crystal. This is known as double refraction. What happens is that when unpolarized light enters the crystal from below, it is broken into two polarized rays that vibrate perpendicular to each other within the crystal. Uniaxial Minerals10/25/2010Page 1 of 8One ray, labeled o in the figure shown here, follows Snell's Law, and is called the ordinary ray, or o-ray. It has a vibration direction that is perpendicular to the plane containing the c-axis and the path of the ray. The other ray, labeled e in the figure shown here, does not follow Snell's Law, and is therefore referred to as the extraordinary ray, or e-ray. The e ray is polarized with light vibrating within the plane containing the c-axis and the propagation path of the ray. Since the angle of incidence of the light is 0o, both rays should not be refracted when entering the crystal according to Snell's Law, but the e-ray violates this law because it's angle of refraction is not 0o, but is r, as shown in the figure. Note that the vibration directions of the e-ray and the o-ray are perpendicular to each other. These directions are referred to as the privileged directions in the crystal. If one separates out the e-ray and the o-ray as shown here, it can be seen that the o-ray has a vibration direction that is perpendicular to the propagation direction. On the other hand, the vibration direction of the e-ray is not perpendicular to the propagation direction. A line drawn that is perpendicular to the vibration direction of the e-ray is called the wave normal direction. It turns out the wave normal direction does obey Snell's Law, as can be seen by examining the diagram of the calcite crystal shown above. In the case shown, the wave normal direction would be parallel to the o-ray propagation direction, which is following Snell's Law. Uniaxial Indicatrix Just like in isotropic minerals, we can construct an indicatrix for uniaxial minerals. The uniaxial indicatrix is constructed by first orienting a crystal with its c-axis vertical. Since the c-axis is also the optic axis in uniaxial crystals, light traveling along the c-axis will vibrate perpendicular to the c-axis and parallel to the ω refractive index direction. Light vibrating perpendicular to the c-axis is associated with the o-ray as discussed above. Thus, if vectors are drawn with lengths proportional to the refractive index for light vibrating in that direction, such vectors would define a circle with radius ω. This circle is referred to as the circular section of the uniaxial indicatrix. Uniaxial Minerals10/25/2010Page 2 of 8Light propagating along directions perpendicular to the c-axis or optic axis is broken into two rays that vibrate perpendicular to each other. One of these rays, the e-ray vibrates parallel to the c-axis or optic axis and thus vibrates parallel to the ε refractive index. Thus, a vector with length proportional to the ε refractive index will be larger than or smaller than the vectors drawn perpendicular to the optic axis, and will define one axis of an ellipse. Such an ellipse with the ε direction as one of its axes and the ω direction as its other axis is called the Principal Section of the uniaxial indicatrix. Light vibrating parallel to any direction associated with a refractive index intermediate between ε and ω will have vector lengths intermediate between those of ε and ω and are referred to as ε' directions. Thus, the uniaxial indicatrix is seen to be an ellipsoid of revolution. Such an ellipsoid of revolution would be swept out by rotating the ellipse of the principal section by 180o. Note that there are an infinite number of principal sections that would cut the indicatrix vertically. Light propagating along one of the ε' directions is broken into two rays, one vibrates parallel to an ε' direction and the other vibrates parallel to the ω direction. An ellipse that has an ε' direction and a ω direction as its axes is referred to as a random section of the indicatrix. Optic Sign Recall that uniaxial minerals can be divided into 2 classes based on the optic sign of the