Chapter 9Optical Properties

Objectives

Understand principles of:• Birefringence (continued)• Pleochroism• Optical indicatrix• Dispersion

Birefringence (continued)

Birefringence: the double refraction oflight in a transparent, molecularly orderedmaterial, which is manifested by the existence of orientationdependent differences in refractive index Calculated as:

• Birefringence (B) = |ne - no|, with e – extraordinary ray; O – ordinary ray This is the birefringence that a specific ray of light experience through a specific direction

of a crystal

Minerals has specific values of birefringence (measure how high is the degree of birefringence that can be caused by that mineral)

The maximum possible birefringence (also called the retardation) that can be caused a specific mineral is calculated as follows, (t = the thickness of the mineral, thus the thickness of the thin section):• Isotropic: Δn = t.|na - na| = 0

• Anisotropic uniaxial: Δn = t.|nω - nε|

• Anisotropic biaxial: Δn = t.|nγ - nα|

PLEOCHROISM Caused by double refraction (birefringence) of

light Light is refracted different amounts when it

passes at different directions through the crystal - follows different paths at different speeds

Each path absorb different colors of light When the mineral is observed at one angle, light

gets birefracted (split into two polarized rays with perpenidicular vibrations) and follows a combination of paths, each of which will have had light of different colors absorbed

How is this observed in the microscope?• When mineral grains appear to have different colors

when observed at different angles under a polarizing petrographic microscope (in plane polarized light)

Pleochroism

Absorption of light varies with• Direction of light relative to the optical

axes/plane• Wave length of light

Under microscope• Observed under plane polarized light as:

Change in colors of mineral with rotation

Pleocroism

The Optical Indicatrix If we construct vectors

whose direction corresponds to the vibration direction, and whose length corresponds to the refractive index of the material for light with that vibration direction, their tips will define an imaginary surface called the indicatrix.

The Optical IndicatrixThe indicatrix is a shape called an ellipsoid. Every cross section of an ellipsoid is an ellipse.

The longest direction of the ellipsoid is its major axis.

The shortest direction, perpendicular to the major axis, is called the minor axis.

Perpendicular to the major and minor axes is the intermediate axis.

These three axes are called the principal axes.

The Optical Indicatrix A geometric figure that shows the index of

refraction and vibration direction for light passing in any direction through a material is called an optical indicatrix.

n = refractive index

The indicatrix is constructed by plotting indices of refraction as radii parallel to the vibration direction of the light.

Ray p, propagating along Y, vibrates parallel to the Z-axis so its index of refraction (np) is plotted as radii along Z.

Ray q, propagating along X, vibrates parallel to Y so its index of refraction (nq) is plotted as radii along Y.

If the indices of refraction for all possible light rays are plotted in a similar way, the surface of the indicatrix is defined. The shape of the indicatrix depends on mineral symmetry.

The Optical Indicatrix

Optical Indicatrix with Wave Normal

Shown is an indicatrix showing a propagating direction (PD) An elliptical section through the indicatrix perpendicular to PD is parallel to

the wave front (vibration direction). The long axis of this elliptical section is parallel to the slow ray vibration

direction and the radius parallel to this direction is equal to the slow ray index of refraction (nslow).

The short axis of the ellipticalsection is parallel to the fastray vibration direction andthe radius parallel to thisdirection is equal to the fastray index of refraction (nfast)

Ray Paths for Optical Indicatrix

To find the ray paths, which are paths followed by an image through the mineral such as those seen with the calcite rhomb, tangents to the indicatrix are constructed parallel to the vibration directions of the slow and the fast rays.

In the general case in which the indicatrix is a triaxial ellipsoid, both rays diverge from their associated wave normals.

Isotropic Indicatrix Optically isotropic minerals all

crystallize in the isometric crystal system.

One unit cell dimension (a) is required to describe the unit cell and one index of refraction (n) is required to describe the optical properties because light velocity is uniform in all directions for a particular wavelength of light.

The indicatrix is therefore a sphere.

All sections through the indicatrix are circles and the light is not split into two rays.

Birefringence may be considered to be zero.

Uniaxial Indicatrix Minerals that crystallize in the tetragonal,

trigonal and hexagonal crystal systems have two different unit cell dimensions (a and c) and a high degree of symmetry about the c axis.

Two indices of refraction are required to define the dimensions of the indicatrix, which is an ellipsoid of revolution whose axis is the c crystal axis.

The semiaxis of the indicatrix measured parallel to the c axis is called ne, and the radius at right angles is called nw. The maximum birefringence of uniaxial minerals is always [ne - nw].

All vertical sections through the indicatrix that include the c axis are identical ellipses called principal sections whose axes are nw and ne. Random sections are ellipses whose dimensions are nw and ne’ where ne’ is between nw and ne.

The section at right angles to the c axis is a circular section whose radius is nw. Because this section is a circle, light propagating along the c axis is not doubly refracted as it is following an optic axis.

Because hexagonal and tetragonal minerals have a single optic axis, they are called optically uniaxial.

Ordinary and Extraordinary Rays Recall that if a cleavage rhomb of calcite is placed on a dot or other

image on a piece of paper, two images appear, each composed of plane-polarised light vibrating at right angles to the other.

The light passing up through the calcite can be considered to be incident at right angles. • Based on Snell’s Law, the wave normal for this light is not bent, it remains

perpendicular to the bottom surface of the rhomb.

When light moves in uniaxial crystals in any direction other than parallel to the c axis, it is broken into two rays travelling with different velocities, what we have previously called the fast ray and the slow ray• One ray vibrates in the basal plane – ordinary ( )w ray• One ray vibrates in principal section - extraordinary (e) ray (at right angles

to basal plane and thus in a plane that includes the c axis

Uniaxial negative - extraordinary faster than ordinary Uniaxial positive - extraordinary slower than ordinary

Optic Sign – Uniaxial Minerals

The dimensions of the indicatrix along the c axis may be either greater or less than the dimensions at right angles.

We can use this to define the optic sign in uniaxial minerals

1. In optically positive minerals, ne > nw and thus the extraordinary rays are slow rays.

2. In optically negative minerals, ne i> nw and thus extraordinary rays are fast rays.

Uniaxial Indicatrix - Positive

Z (longest) axis = optic axis = c

Uniaxial Indicatrix - Negative

X (shortest) axis = optic axis = c-axis

Use of the Indicatrix

We can now examine the behaviour of light passing through grains of a uniaxial minerals in different orientations either in a thin section or a grain mount.

Light strikes the bottom surface of the sample more or less normal to the surface.

This means that the wave normal of the light entering the mineral is not bent and the wave front is parallel to the bottom surface of the mineral.

Use of the Indicatrix In this example the mineral is oriented

so that its optic axis is horizontal We will assume that the mineral is

uniaxial positive The wave normal is through the centre

of the indicatrix and light is incident normal to the bottom surface of the grain.

Because the optic axis is horizontal, this section is a principal section, which is an ellipse whose axes are nw and ne.

Therefore• Ordinary ray = nw• Extraordinary = ne

Maximum because the mineral is optically positive.

The extraordinary ray vibrates parallel to the trace of the optic axis (c axis) and the ordinary ray vibrates at right angles.

Therefore, birefringence and hence interference colors are maximum values.

Use of the Indicatrix In this case the mineral sample is

oriented so that the optic axis is vertical.

The section through the indicatrix perpendicular to the wave normal is the circular section whose radius is nw.

Light coming from below is not doubly refracted, birefringence is zero and the light preserves whatever vibration direction it initially had.

Between crossed polars this mineral should behave like an isotropic mineral and remain dark as the stage is rotated.

However, because the light from the substage condenser is moderately converging, some light may pass through the mineral.

The mineral may display interference colors but they will be the lowest order found in that mineral.

Use of the Indicatrix

In this sample the mineral sample is oriented in a random orientation so that the light path is at an angle q to the optic axis.

The section through the indicatrix parallel to the bottom surface of the mineral is an ellipse whose axes are nw and ne

The extraordinary ray vibrates parallel to the trace of the optic axis as seen from above, while the ordinary ray vibrates at right angles

Both birefringence and interference colors are intermediate because ne’ is intermediate between nw and ne.

Biaxial Indicatrix Minerals that crystallize in the orthorhombic, monoclinic and

triclinic crystal systems require three dimensions (a, b and c) to describe their unit cells and three indices of refraction to define the shape of their indicatrix

The three principal indices of refraction are na, nb and ng where na < nb < n .g

• The maximum birefringence of a biaxial mineral is always n - g na

Construction of a biaxial indicatrix requires that three indices of refraction are plotted• However, while three indices of refraction are required to describe

biaxial optics, light that enters biaxial minerals is still split into two rays.

• As we shall see, both of these rays behave as extraordinary rays for most propagation paths through the mineral

The index of refraction of the fast ray is identified as na’ where na < na’ < nb and the index of refraction of the slow ray is ng’ where nb < ng’ < ng

Biaxial Indicatrix The biaxial indicatrix contains three principal sections, the YZ, XY and XZ

planes. The XY section is an ellipse with axes na and nb, The XZ section is an ellipse with axes na and ng

YZ section is an ellipse with axes nb and ng. Random sections through the indicatrix are ellipses whose axes are na’ and ng’. The indicatrix has two circular sections with radius nb that intersect the Y axis. The XZ plane is an ellipse whose radii vary between na and ng. Therefore radii

of nb must be present. Radii shorter than nb are na’ and those that are longer are ng’. The radius of the indicatrix along the Y axis is also nb

Biaxial Indicatrix

An ellipse with three unequal principal axes has two circular cross sections. The plane containing the major and minor axis cuts the ellipsoid in an ellipse with the maximum and minimum possible radii. Somewhere in between is a radius equal to the intermediate axis. The two circular sections have radius equal to the intermediate axis and intersect along the intermediate axis. They are shown in blue and purple at left.

Biaxial Indicatrix

The directions perpendicular to each one of the circular sections are the optic axes.

For a generic ellipsoid with three unequal axes, there are two optic axes. Such materials are called biaxial.

Biaxial Indicatrix – XZ plane Therefore the Y axis and the nb radii in the XZ plane define the two

circular sections. Like uniaxial minerals, the circular sections in biaxial minerals are

perpendicular to the optic axes, hence the term biaxial. Because both optic axes lie in the XZ plane of the indicatrix, that

plane is called the optic plane. The angle between the optic axes bisected by the X axis is also called

the 2Vx angle, while the angle between the optic axes bisected by the Z axis is called the 2Vz anglewhere 2Vx + 2Vz = 180°.

The Y axis, which isperpendicular to the optic planeis called the optic normal.

Optic Sign – Biaxial Minerals

The acute angle between the optic axes is called the optic angle or 2V angle.

The axis (either X and Z)that bisects the acute angle between the optic axes is the acute bisectrix or Bxa.

The axis (either Z or X) that bisects the obtuse angle between the optic axes is the obtuse bisectrix or Bxo.

The optic sign of biaxial minerals depends on whether the X or Z indicatrix axis bisects the acute angle between the optic axes.

1. If the acute bisectrix is the X axis, the mineral is optically negative and 2Vx is less than 90°

2. If the acute bisectrix is the Z axis, the mineral is optically positive and 2Vz is less than 90°

3. If 2V is exactly 90° so neither X nor Z is the acute bisectrix, the mineral is optically neutral.

Optic Sign and Biaxial Indicatrix

Crystallographic Orientation of Indicatrix Axes

The term optic orientation refers to the relationship between indicatrix axes and crystal axes.

Because the optical properties of minerals are directly controlled by the symmetry of the crystal structure, optic orientation must be consistent with mineral symmetry.

Indicatrix Axes and Orthorhombic Crystals

Orthorhombic crystals have three mutually perpendicular crystallographic aces of unequal length. These crystal axes must coincide with the three indicatrix axes and the symmetry planes in the mineral must coincide with principal sections in the indicatrix. Any crystal axis may coincide with any indicatrix axis however.

The optic orientation is defined by indicating which indicatrix axis is parallel to which mineral axis.1. Aragonite X = c, Y = a, Z =

b2. Anthophyllite X = a, Y = b, Z

= c

Use of the Biaxial Indicatrix

The biaxial indicatrix is used in the same way as the uniaxial indicatrix.

It provides information about the indices of refraction and vibration direction given the wave normal direction that light is following through a mineral.

Birefringence depends on how the sample is cut or mounted.

Birefringence is:• A maximum if the optic plane is horizontal• A minimum if an optic axis is vertical• Intermediate for all other random orientations

Dispersion Chromatic dispersion - the change of index of refraction with

wavelength In a prism, material dispersion (a wavelength-dependent

refractive index) causes different colors to refract at different angles, splitting white light into a rainbow

Prevent this in optical analysis – use monochromoatic light• If not the RI varies with different wavelengths dispersed within mineral

• Thus RI is dependent of wavelength of light

• Monochromatic light from Na-vapor lamp: 589 nm – used in most reference books and microscopes

NB Terminology Optical indicatrix Optical plane Optical axis Ellipsoid Ellipse Major, minor, intermediate

axes Principal axes Basal plane Principal section General section Circular section Acute bisectrix Obtuse bisectrix

2V angle Optic normal Dispersion