1.7 Indicatrix - Indicatrix Imaginary figure, but very...

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Unformatted text preview: Indicatrix Imaginary figure, but very useful; shows/defines: Three type each with characteristic shape: Optic axis Positive and negative minerals Relationship to crystallographic axes Isotropic Uniaxial (anisotropic) Biaxial (anisotropic) Primary use is to determine n and vibration directions of slow and fast rays Indicatrix Possible shapes: Radii of the figure parallel vibration directions A sphere or oblate/prolate spheroid Length of radii represent the value of n Plots of all possible ray paths generate figure Shows vibration directions and associated n for all ray paths Biaxial Indicatrix Construction: plot indices of refraction along three primary axes: X, Y, and Z Always 90 to each other Vibration directions perpendicular to ray path/wave normal. Biaxial Indicatrix Random wave determine vibration directions and index of refraction by semi major and semiminor axes of ellipse Ray paths constructed by tangents to the surface of the indicatrix that parallel vibration 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: Unit cell has only one dimension Minerals have only one index of refraction Crystallographic axis = a Shape of indicatrix is a sphere All sections are circles Different for each mineral 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: two dimensions of unit cell (a and c) Two values of n's required to define indicatrix High symmetry around c axis Remember infinite values of n One is epsilon , the other is omega Uniaxial Indicatrix Ellipsoid of revolution with axis of rotation parallel the c crystallographic axis One semiaxis of ellipsoid parallels c Other semiaxis of ellipsoid perpendicular to c Called n Maximum birefringence is positive difference of n and n Called n Note n < or > n , just as c > or < a Uniaxial Indicatrix Note: (1)Axes designated X, Y, Z (2)Z axis always long axis for uniaxial inidicatrix (3)May be c axis (O.A.) or a axis Fig. 723 Optic Sign Defined by n and n Optically positive n > n , Z = n Optically negative n < n , Z = n Sections of indicatrix The section of indicatrix cross section perpendicular to the wave normal It is important: 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 Sections to indicatrix Principal sections include c axis Circular sections cut perpendicular to c Random sections don't include c axis Principal section orientation of grain Optic axis is horizontal (parallel stage) Indicatrix section defines the n and vibration directions Vibration directions of two rays must parallel axes of ellipse Ordinary ray = n ; extraordinary ray = n Note that the wave normal and ray paths coincide (no double refraction) Principal Section Emergent point at tangents Indicates wave normal and ray path are the same, no double refractions Semi major axis Semiminor axis What is birefringence of this section? Fig. 725 How many times does it go extinct with 360 rotation? Circular section Optic axis is perpendicular 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? Extinction? Fig. 725 Random section Section now an ellipse with axes n and n ' Find path of extraordinary ray by constructing tangent parallel to vibration direction Random Section Point of emergence for ray vibrating parallel to index ' What is birefringence of this section? Extinction? Fig. 725c Ordinary and extraordinary rays In uniaxial minerals, one ray always vibrates perpendicular to optic axis Called ordinary or ray Always same index = n The other ray may be refracted Vibration always within the (001) plane Called extraordinary or ray Index of refraction is between n and n Note that n < or > n Refracted extraordinary ray vibrates in plane of ray path and c axis Ordinary ray Fig. 724 Biaxial Indicatrix Crystal systems: Orthorhombic, Monoclinic, Triclinic Three dimensions to unit cell Three indices of refraction for indicatrix a b c n < n < n Maximum birefringence = n n Indicatrix axes Plotted on a XYZ system Convention: n = X, n = Y, n = Z Sometimes axes referred to as X, Y, Z or nx, ny, nz etc. Z always longest axis X always shortest axis Requires different definition of positive and negative minerals Biaxial Indicatrix Note differs from uniaxial because n n Biaxial indicatrix has three axes Sometimes wave normal and ray path coincide Light split into two rays Sometimes both rays act like extraordinary ray Ray path and wave normal diverge One always faster than other Called fast and slow rays Values of n intermediate between n and n called n' and n ' Biaxial indicatrix has two circular sections Radius is n Optic axis: The circular section contains the Y axis perpendicular to the circular sections Two circular sections = two optic axes Neither optic axis is parallel to X, Y, or Z Circular sections Both optic axes occur in the XZ plane 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 axis 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 + Biaxial Indicatrix Optically positive Optically negative Fig. 727 Uniaxial indicatrixes are special cases of biaxial indicatrix: If n = n, the optic axis converges on Z Mineral is uniaxial positive n = n and n = n If n = n , the optic axis converges on X Mineral is uniaxial negative n = n and n = n Like the uniaxial there are three primary sections: Optic normal section optic axis is in plane of the 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' and n ' Fig. 729 Crystallographic orientation of indicatrix Optic orientation Angular relationship between crystallographic and indicatrix axes Orthorhombic minerals Three crystallographic axes (a, b, c) coincide with X,Y,Z indicatrix axes 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 Orthorhombic Minerals Fig. 728 Monoclinic One indicatrix axis always parallels b 2fold rotation or perpendicular to mirror plane Could be X, Y, or Z Other two axes lie in [010] plane One additional indicatrix axis may (but usually not) parallel to crystallographic axis Optic orientation defined by 1. 2. 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 Which indicatrix axis parallels b Angles between other indicatrix axes and a and c crystallographic axes Monoclinic minerals > 90 Positive angle because in obtuse angle Symmetry rotation axis or perpendicular to mirror plane Negative angle because in acute angle Fig. 728 Triclinic minerals Indicatrix axes not constrained to follow crystallographic axes One indicatrix axis may (but usually not) parallel crystallographic axis Triclinic minerals Fig. 728 P. 306 olivine information Optical orientation All optical properties Optic Axes ...
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