Indicatrix n n Imaginary figure but very useful

Indicatrix n n Imaginary figure, but very useful The figures show and/or define: n n Three type – each with characteristic shape: n n 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 n Primary uses: n Determine vibration directions within mineral n 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 n Show relationship between optics and crystallographic axis/crystallographic features n

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

Biaxial Indicatrix n Construction Plot primary indices of refraction along three primary axes: X, Y, and Z n Always 90º to each other n n nq & np are two of the principle vibration directions Fig. 7 -22

Biaxial Indicatrix n n n Observe slice of figure perpendicular to wave Wave front – plane normal. perpendicular to wave normal Vibration directions • Long axis = nslow perpendicular to wave • short axis = nfast normal Principle vibration directions and values of index of refraction shown by semi-major and semiminor axes of ellipse Fig. 7 -22

Biaxial Indicatrix n Ray paths constructed by tangents to the surface of the indicatrix that parallel vibration directions Ray directions

Procedure to use n n n 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

n Generally used in a qualitative way: Understanding difference between isotropic, uniaxial, and biaxial minerals n Understanding the relationship between optical properties, crystallographic axes, and crystallographic properties n

Isotropic Indicatrix n Isometric minerals only: Unit cell has only one dimension n n Minerals have only one index of refraction n Crystallographic axis = a Different for each mineral Shape of indicatrix is a sphere All sections are circles Light not split into two rays n Birefringence is zero n

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 n Tetragonal and hexagonal minerals only: two dimensions of unit cell (a and c) n n Two values of n’s required to define indicatrix n n High symmetry around c axis One is epsilon e, the other is omega w Remember – infinite values of n n Range between ne and nw

Uniaxial Indicatrix n n Ellipsoid of revolution (spheroid) with axis of rotation parallel the c crystallographic axis One semi-axis of ellipsoid parallels c n n Other semi-axis of ellipsoid perpendicular to c n n ne nw Maximum birefringence is positive difference of nw and ne n Note nw < or > ne, just as c > or < a

Uniaxial Indicatrix X=Y ne>nw ne<nw Y=Z Note: (1) Axes designated X, Y, Z (2) Z axis always long axis for uniaxial indicatrix (3) May be c axis or a axis (4) Axis perpendicular to circular section is optic axis (5) Optic axis always c crystallographic axis Fig. 7 -23

Optic Sign n Defined by nw and ne Optically positive (+) – ne > nw, Z = c = ne n Optically negative (-) - ne < nw, Z = a = nw n

Ordinary and extraordinary rays n In uniaxial minerals, one ray always vibrates perpendicular to optic axis n n Called ordinary or w ray Always same index = nw Vibration always within the (001) plane The other ray may be refracted n n n Called extraordinary or e ray Index of refraction is between ne and nw Note that ne < or > nw

Ordinary Ray Ordinary ray vibrates in (001) plane: index = nw C cr ys ta ll ax ogr is ap h ic Fig. 7 -24

Extraordinary Ray Refracted extraordinary ray – vibrates in plane of ray path and c axis Index = ne’ How the mineral is cut is critical for what N the light experiences and it’s value of D and d

Sections of indicatrix n n Cross section perpendicular to the wave normal – usually an ellipse It is important: Vibration directions of two rays must parallel axes of ellipse n Lengths of axes tells you magnitudes of the indices of refraction n Indices of refraction tell you the birefringence expected for any direction a grain may be cut n Indices of refraction tell you the angle that light is refracted n

3 types of sections to indicatrix n n n Principle sections include c crystallographic axis Circular sections cut perpendicular to c crystallographic axis (and optic axis) Random sections don’t include c axis

Principle Section n Orientation of grain Optic axis is horizontal (parallel stage) n Ordinary ray = nw ; extraordinary ray = ne n We’ll see that the wave normal and ray paths coincide (no double refraction) n

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 n n Optic axis is perpendicular to microscope stage Circular section, with radius nw 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? Extinction? Fig. 7 -25

Random Section now an ellipse with axes nw and ne’ Find path of extraordinary ray by constructing tangent parallel to vibration direction Most common of all the sections

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

Biaxial Indicatrix n n Crystal systems: Orthorhombic, Monoclinic, Triclinic Three dimensions to unit cell n n a≠b≠c Three indices of refraction for indicatrix na < nb < ng always n Maximum birefringence = ng - na always n

Indicatrix axes n n Plotted on a X-Y-Z system Convention: na = X, nb = Y, ng = Z Z always longest axis (same as uniaxial indicatrix) n X always shortest axis n Requires different definition of positive and negative minerals n n Sometimes axes referred to as X, Y, Z or nx, ny, nz etc.

Biaxial Indicatrix Note – differs from uniaxial because nb ≠ na Fig. 7 -27

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

Circular sections Fig. 7 -27

n Both optic axes occur in the X-Z plane Must be because nb = Y n Called the optic plane n Angle between optic axis is called 2 V n Can be either 2 Vx or 2 Vz depending which axis bisects the 2 V angle n

Optic sign n n Acute angle between optic axes is 2 V angle Axis that bisects the 2 V angle is acute bisectrix or Bxa Axis that bisects the obtuse angle is obtuse bisectrix or Bxo The bisecting axis determines optic sign: n n n If Bxa = X, then optically negative If Bxa = Z, then optically positive If 2 V = 90º, then optically neutral

+ X-Z plane of Biaxial Indicatrix Optically positive Optically negative Fig. 7 -27

n Uniaxial indicatrixes are special cases of biaxial indicatrix: n If nb = na Mineral is uniaxial positive n na = nw and ng = ne, note – there is no nb n n If nb = ng Mineral is uniaxial negative n na = ne and nc = nw 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 n Optic axis vertical n Random section n

Optic normal – Maximum interference colors: contains na and ng Optic axis vertical = Circular section – Extinct: contains nb only Random section – Intermediate interference colors: contains na’ and ng’ Fig. 7 -29

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

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

Orthorhombic Minerals Here optic orientation is: Z=c Y=a X=b Fig. 7 -28

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

n Optic orientation defined by 1. 2. n n 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 b > 90º Symmetry – rotation axis or perpendicular to mirror plane Negative angle because in acute angle Fig. 7 -28

Triclinic minerals n n Indicatrix axes not constrained to follow crystallographic axes One indicatrix axis may (but usually not) parallel crystallographic axis

Triclinic minerals Fig. 7 -28

P. 306 – olivine information Optical orientation All optical properties Optic Axes