Anisotropic Minerals

1) The velocity of light varies depending on direction through the mineral

2)They show double refraction (the light entering the mineral is split into two rays with different velocities)

3) Each Anisotropic mineral has either 1 (uniaxial) or 2 (biaxial) optic axes along which they act isotropic.

Optically Uniaxial Minerals

1) Minerals that belong to the Hexagonal and Tetragonal systems

2) Have one optic axes

Optically Biaxial Minerals

1) Minerals that belong to the orthorhombic, monoclinic, and triclinic systems.

2) Have two optic axes

Interference Colors

1) The colors  produced because of light split into two rays passing through a mineral under crossed polars.  

Ray Retardation (Δ)

Δ = d(V/V_s-V/V_f)

1) The distance that the slow ray is behind the fast ray after both have excited the crystal.

2) The magnitude of the retardation depends on the thickness of the crystal plate (d) and the differences in the velocity of the slow ray (V_s) and the fast ray (V_f) 

Anisotropic Double Refracted Light Rays

1) Plane polarized and vibrate at right angles to each other.

2) Ray with lower index of refraction (n) is called the fast ray, and the ray with the higher index of refraction is called the slow ray (retarded).

Birefringence (δ)

δ = (n_s - n_f)

1) The difference between the indices of refraction of the slow (n_s) and fast (n_f) rays.

2) Numerical value depends on path of light through the mineral.

3) Paths along optic axes show zero Birefringence. 

4) May vary depending on wavelength of light.

Interference of Two Rays

Two waves travel in the same plane and the same direction.

1) In Phase: two rays are vibrating at right angles, the resolved components are in opposite directions and therefore destructively interfere and cancel each other. No light passes through the polarizer and the mineral grain appears dark

2) Out of Phase: resolved components are in the same direction, light constructively interferes and light passes the upper polarizer.

Orders of Interference Colors

Go through a repeating sequence, with change from red to blue occurring at retardations of approximately 550, 1100, and 1650 nm.

1st order: less than 550 nm

2nd order: between 550 - 1100 nm

3rd order: between 1100-1650 nm

The higher order, the more washed out the color

Parallel Extinction

The mineral is extinct when the cleavage or length is aligned with one or the other of the cross hairs. The extinction angle is zero degrees

Inclined Extinction

The mineral is extinct when the cleavage or length is at some angle to the cross hairs. 

Symmetrical Extinction

The mineral displays either two cleavage directions or two distinct crystal faces to which two extinction angles can be measured, one from each cleavage or crystal face. If the two extinction angles are the same, the mineral displays symmetrical extinction.

No Extinction Angle

Undulatory Extinction

Extinction in a grain follows an irregular or wavy pattern. Due to deformation of rocks, grains may be bent or strained and parts of a single grain are in slightly different orientations and therefore go extinct at different times. 

Pleochroism

Anisotropic minerals that display a change of color as the stage is rotated in plane light (upper polarizer removed). It is produced because the two rays of light are absorbed differently as they pass through the colored mineral and therefore have different colors. 

Extraordinary Ray (ε Ray)

Ray in Uniaxial Mineral that vibrates perpendicular to the ordinary ray (ω ray) and parallel to the c-axis. Velocity varies upon direction. 

Optically positive: index of extraordinary ray (n_ε) is greater than index of ordinary ray (n_ω).

In other words, if the ε ray is slow, the mineral is optically positive and therefore negative if fast.

Ordinary Ray (ω ray)

Ray in Uniaxial Mineral that vibrates perpendicular to the c-axis. Ray has the same velocity regardless of path through mineral.

Optically Negative: index of Ordinary Ray (n_ω) is greater than index of Extraordinary Ray (n_ε)

Indicatrix

[image]

Geometric figure constructed so that the indices of refraction are plotted as radii that are parallel to the vibration direction of the light. Includes a principle, circular, and random section.

Indicatrix: Principal Section

A section through the indicatrix that includes the optic axis, which is an ellipse whose axes are n_ω and n_ε. 

Indicatrix: Circular Section

A section through the indicatrix perpendicular to the optic axis whose radius is n_ω.

Indicatrix: Random Section

Any random cut through the indicatrix produces an ellipse whose axes are n_ω and n_ε' where n_ε' is between n_ω and n_ε^.

Tetragonal Minerals

Typically prismatic and either elongate or stubby parallel to the c-axis. The faces are commonly combinations of prisms parallel to the c-axis , pinacoids perpendicular to c, and pyramids, although other forms are possible. (See Tetragonal: case 1-3)

Tetragonal: Case 1

The crystal is cut perpendicular to the optic axis (circular section) so that light follows the optic axis and travels with index n_ω^.

If polarizers are crossed, the section should be uniformly dark on rotation, so no extinction angle can be measured.

Tetragonal: Case 2

Crystal is cut parallel to the optic axis (Principle Section with ε axis parallel to the length of the crystal and the ω axis perpendicular to the length). Light travels through the crystal perpendicular to the optic axis. Must be parallel extinction b/c vibration directions for ω and ε are parallel to the width and length. 

Tetragonal: Case 3

Crystal is cut along a random angle. Birefringence is intermediate. All three cleavages are visible. n_ε intermediate. Extinction to the prismatic cleavage is parallel, symmetrical, or anything in between. The ω ray vibrates parallel to the trace of the pinacoidal cleavage, so extinction is always parallel to cleavage.

Hexagonal Minerals

Common forms: prisms, pinacoids, pyramids, and rhombohedrons, although a number of other forms are possible. 

Common cleavages: prismatic, pinacoidal, and rhombohedral

See Hexagonal: Case 1-3

Rhombohedral Cleavage

Three cleavage planes that intersect at angles other than 90°

[image]

Prismatic and Pinacoidal Cleavage

Crystal with three prismatic {100} cleavages that intersect at angles of 60° and 120° and a {001} pinacoidal cleavage at right angles to the c-axis.

Interaction of light with minerals

1) Reflection: luster

2) Speed

3) Index of Refraction: n = velocity of light in vacuum/ velocity of light in mineral; measures how effective a mineral is at bending light as it passes from one material to another.

Snell's Law

sinθ₁/sinθ₂ = n₂/n₁ = speed of light in mineral/ seed of light in air

large difference between n of mineral and n of its surroundings means that it will stand out and have high relief. Small difference means it will have low relief. 

Normal Dispersion

Increase in index of refraction (n) with decrease in wavelength (λ)

Abnormal Dispersion

Decrease in index of refraction (n) with a decrease in wavelength (λ). 

Example: Halides; some light absorbed by mineral

Translation Operations

1) Periodic repetitions along vectors

2) Translation of ions - achieved in:

1-D: Row Lattices

2-D: Plane Lattices

3-D: Space Lattices

Lattice

An imaginary pattern of points (or nodes) that has an environment that is identical to any other node in the pattern; repetition of a patter

Space Lattices

Produced by translations along two vectors- repeated with constant distances and angles.

Plane Lattices

1. Oblique Net 
2. Rectangular Net
3. Centered Rectangular Net 
4. Diamond Net
5. Hexagonal Net
6. Square Net

Plane Lattice: Oblique Net

p = primitive, nodes only at corners

Plane Lattice: Rectangular Net

P = primitive, nodes only at the corners

Plane Lattices: Centered Rectangular

and

Diamond Net

c = centered, node in the middle and at the corners (diamond is primitive)

Stacking of Oblique Nets

Stacking of an oblique net (or plane lattice) at an arbitrary angle results in primitive triclinic lattices

7 Lattice Systems

In order from least to most symmetric

1) Triclinic: No symmetry

2) Monoclinic: 1 diad symmetry

3) Orthorhombic: 3 perpendicular diads

4) Rhombohedral: 1 triad

5) Tetragonal: 1 tetrad

6) Hexagonal: 1 hexad

7) Cubic: 4 triads

Triclinic Crystal System

Described by vectors of unequal length, as in the orthorhombic system. All three vectors are not mutually orthogonal (δ,β,α ≠ 90°). Least Symmetric and is the only lattice type that has no mirror planes. Mineral examples include plagioclase, microcline, rhodonite, 

turquoise, wollastonite and amblygonite, all in triclinic normal (bar 1).

Monoclinic Crystal System

Orthorhombic Crystal System

Lattices result from stretching a cubic lattice along two of its orthogonal pairs. (a≠b≠c) and (δ,α,β=90°). Lattices include: simple orthorhombic, base centered, body centered, and face centered. Common minerals include Olivine, Aragonite.

Tetragonal Crystal System

Two Lattices: Simple Tetragonal, Centered Tetragonal. Common minerals: rutile, zircon, pyrolusite, mulfenite, chalcopyrite. (a=b≠c) and (δ,α,β=90°).

Hexagonal Crystal System

has only one lattice type: simple hexagonal. Common Mineral: Graphite.(a=b=c) intersecting at 120°. Has 4 Miller indices. 

Isometric Crystal System

"cubic" crystal system where unit cell is a cube. The most common and simplest shapes found in crystals and minerals. Three varieties of crystals: simple cubic, body-centered cubic, and face-centered cubic. Native minerals are common (lead, aluminum copper, gold, iron, chromium)