Indicatrix Imaginary figure, but very useful The figures show and/or define: Three type each with characteristic shape:

Location of optic axis Positive and negative minerals Relationship between optical & crystallographic axes Isotropic Uniaxial (anisotropic) Biaxial (anisotropic) Primary use is to understand/visualize vibration directions of slow and fast rays Indicatrix Primary uses: Determine vibration directions within mineral

Vibration direction determines index of refraction of slow and fast rays and thus birefringence and interference colors Determine wave front direction and ray paths if refracted Show relationship between optics and crystallographic axis/crystallographic features Indicatrix

Possible shapes: A sphere or oblate/prolate spheroid Radii of the figures represent vibration directions Length of radii represent the values of n Plots of all possible values of n generates figure Shows vibration directions and

associated n for all ray paths Biaxial Indicatrix Construction Plot primary indices of refraction along three primary axes: X, Y, and Z Always 90 to each other nq & np are two of the

principle vibration directions Fig. 7-22 Biaxial Indicatrix Observe slice of figure perpendicular to wave Wave front normal. plane perpendicular to

Vibration directions perpendicular to wave wave normal Long axis = normal nslow Principle vibration short axis = directions and values ofnfast index of refraction shown by semi-major and semiminor axes of ellipse Fig. 7-22 Biaxial Indicatrix Ray paths constructed by tangents to the

surface of the indicatrix that parallel vibration directions Ray directions Procedure to use Imagine a section through the center of the indicatrix and perpendicular to the wave normal Axes of section are parallel to fast

(short axis) and slow (long axis) rays Ray paths of fast and slow rays are found by constructing tangents parallel to vibration directions Generally used in a qualitative way: Understanding difference between isotropic, uniaxial, and biaxial minerals Understanding the relationship between optical properties, crystallographic axes, and crystallographic properties

Isotropic Indicatrix Isometric minerals only: Unit cell has only one dimension Minerals have only one index of refraction Crystallographic axis = a

Different for each mineral Shape of indicatrix is a sphere All sections are circles Light not split into two rays Birefringence is zero Isotropic indicatrix Ray path and Wave normal coincide Length of radii of sphere represent value for n Circular Section Light does not split

into two rays, polarization direction unchanged Uniaxial Indicatrix Tetragonal and hexagonal minerals only: two dimensions of unit cell (a and c) Two values of ns required to define indicatrix

High symmetry around c axis One is epsilon , the other is omega Remember infinite values of n Range between n and n Uniaxial Indicatrix Ellipsoid of revolution (spheroid) with axis of rotation parallel the c crystallographic axis One semi-axis of ellipsoid parallels c

Other semi-axis of ellipsoid perpendicular to c n n Maximum birefringence is positive difference of n and n Note n < or > n, just as c > or < a

Uniaxial Indicatrix X=Y n>n n

circular section is optic axis (5)Optic axis always c 7-23 crystallographicFig. axis Optic Sign Defined by n and n Optically positive (+) n > n, Z = c = n Optically negative (-) - n < n, Z = a =

n Ordinary and extraordinary rays In uniaxial minerals, one ray always vibrates perpendicular to optic axis Called ordinary or ray Always same index = n Vibration always within the (001) plane

The other ray may be refracted Called extraordinary or ray Index of refraction is between n and n Note that n < or > n Ordinary Ray Ordinary ray vibrates in (001) plane: index = n C

cr ys ta l ax log is rap hi c Fig. 7-24 Extraordinary Ray Refracted extraordinar y ray

vibrates in plane of ray path and c axis Index = n How the mineral is cut is critical for what N the light experiences and its value of Sections of indicatrix Cross section perpendicular to the wave normal usually an ellipse It is important:

Vibration directions of two rays must parallel axes of ellipse Lengths of axes tells you magnitudes of the indices of refraction Indices of refraction tell you the birefringence expected for any direction a grain may be cut Indices of refraction tell you the angle that light is refracted 3 types of sections to indicatrix

Principle sections include c crystallographic axis Circular sections cut perpendicular to c crystallographic axis (and optic axis) Random sections dont include c axis Principle Section Orientation of grain

Optic axis is horizontal (parallel stage) Ordinary ray = n ; extraordinary ray = n Well see that the wave normal and ray paths coincide (no double refraction) Principle Section Emergent point at tangents Indicates wave normal and ray path

are the same, no double refractions Semi major axis Semi-minor axis What is birefringence of this section? Fig. 7-25 How many times does it go extinct with 360 rotation? Circular Section Optic axis is perpendicular to microscope

stage Circular section, with radius n Light retains its polarized direction Blocked by analyzer and remains extinct Circular Section Optic Axis Light not constrained to vibrate in any one direction Ray path and wave normal coincide no double refraction

What is birefringence of this section? Fig. 7-25 Extinction? Random Section Section now an ellipse with axes n and n Find path of extraordinary ray by constructing tangent parallel to vibration direction Most common of all the sections Random Section

Line tangent to surface of indicatrix = point of emergence Point of emergence for ray vibrating parallel to index What is birefringence of this section? Extinction? Fig. 7-25c

Biaxial Indicatrix Crystal systems: Orthorhombic, Monoclinic, Triclinic Three dimensions to unit cell abc Three indices of refraction for indicatrix

n < n < n always Maximum birefringence = n - n always Indicatrix axes Plotted on a X-Y-Z system Convention: n = X, n = Y, n = Z Z always longest axis (same as uniaxial

indicatrix) X always shortest axis Requires different definition of positive and negative minerals Sometimes axes referred to as X, Y, Z or nx, ny, nz etc. Biaxial Indicatrix Note differs from uniaxial because n n Fig. 7-27

Biaxial indicatrix has two circular sections Radius is n The circular section ALWAYS contains the Y axis Optic axis: perpendicular to the circular sections Two circular sections = two optic axes Neither optic axis is parallel to X, Y, or Z

Circular sections Fig. 7-27 Both optic axes occur in the X-Z plane Must be because n = Y Called the optic plane Angle between optic axis is called 2V Can be either 2Vx or 2Vz depending

which axis bisects the 2V angle Optic sign Acute angle between optic axes is 2V angle Axis that bisects the 2V angle is acute bisectrix or Bxa Axis that bisects the obtuse angle is obtuse bisectrix or Bxo The bisecting axis determines optic sign:

If Bxa = X, then optically negative If Bxa = Z, then optically positive If 2V = 90, then optically neutral + X-Z plane of Biaxial Indicatri x Optically positive Optically negative Fig. 7-27

Uniaxial indicatrixes are special cases of biaxial indicatrix: If n = n Mineral is uniaxial positive n = n and n = n , note there is no n

If n = n Mineral is uniaxial negative n = n and n = n Like the uniaxial indicatrix there are three primary sections:

Optic normal section Y axis vertical so X and Z in plane of thin section Optic axis vertical Random section Optic normal Maximum interference colors: contains n and n Optic axis vertical = Circular section Extinct: contains n only Random section Intermediate interference colors: contains n andFig. n7-29

Crystallographic orientation of indicatrix Optic orientation Angular relationship between crystallographic and indicatrix axes Three systems (biaxial) orthorhombic, monoclinic, & triclinic Orthorhombic minerals

Three crystallographic axes (a, b, c) coincide with X,Y, Z indicatrix axes all 90 Symmetry planes coincide with principal sections No consistency between which axis coincides with which one Optic orientation determined by which axes coincide, e.g. Aragonite: X = c, Y = a, Z = b Anthophyllite: X = a, Y = b, Z = c

Orthorhomb ic Minerals Here optic orientation is: Z=c Y=a X=b Fig. 7-28 Monoclinic One indicatrix axis always parallels b axis

2-fold rotation or perpendicular to mirror plane Could be X, Y, or Z indicatrix axis Other two axes lie in [010] plane (i.e. a-c crystallographic plane) One additional indicatrix axis may (but usually not) parallel crystallographic axis Optic orientation defined by 1. 2.

Which indicatrix axis parallels b Angles between other indicatrix axes and a and c crystallographic axes Angle is positive for the indicatrix axis within obtuse angle of crystallographic axes Angle is negative for indicatrix axis within acute angle of crystallographic axes Monoclinic minerals Positive angle because in obtuse

angle > 90 Symmetry rotation axis or perpendicular to mirror plane Negative angle because in acute angle Fig. 7-28 Triclinic minerals

Indicatrix axes not constrained to follow crystallographic axes One indicatrix axis may (but usually not) parallel crystallographic axis Triclinic mineral s Fig. 7-28 P. 306 olivine information Optical orientatio n All optical properties

Optic Axes