Indicatrix

1. Indicatrix • Imaginary figure, but very useful • The figures show and/or define: • Location of optic axis • Positive and negative minerals • Relationship between optical & crystallographic axes • Three type – each with characteristic shape: • Isotropic • Uniaxial (anisotropic) • Biaxial (anisotropic) • Primary use is to understand/visualize vibration directions of slow and fast rays

2. 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

3. 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

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

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

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

7. 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

8. 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

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

10. 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

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

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

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

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

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

16. Ordinary Ray Fig. 7-24 Ordinary ray vibrates in (001) plane: index = nw C crystallographic axis

17. 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

18. 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

19. 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 don’t include c axis

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

21. Emergent point – at tangents Indicates wave normal and ray path are the same, no double refractions Principle Section Fig. 7-25 Semi major axis Semi-minor axis What is birefringence of this section? How many times does it go extinct with 360 rotation?

22. Circular Section • Optic axis is perpendicular to microscope stage • Circular section, with radius nw • Light retains its polarized direction • Blocked by analyzer and remains extinct

23. Circular Section Fig. 7-25 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?

24. Random Section • 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

25. Fig. 7-25c 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?

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

27. Indicatrix axes • Plotted on a X-Y-Z system • Convention: na = X, nb = Y, ng = 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.

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

29. Biaxial indicatrix has two circular sections • Radius is nb • 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

30. Circular sections Fig. 7-27

31. Both optic axes occur in the X-Z plane • Must be because nb = 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

32. 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 = 90º, then optically neutral

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

34. Uniaxialindicatrixes are special cases of biaxial indicatrix: • If nb = na • Mineral is uniaxial positive • na = nw and ng = ne, note – there is no nb • If nb = ng • Mineral is uniaxial negative • na = ne and nc = nw

35. 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

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

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

38. Orthorhombic minerals • 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. • Aragonite: X = c, Y = a, Z = b • Anthophyllite: X = a, Y = b, Z = c

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

40. 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

41. Optic orientation defined by • 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

42. Fig. 7-28 Monoclinic minerals Positive angle because in obtuse angle Symmetry – rotation axis or perpendicular to mirror plane b > 90º Negative angle because in acute angle

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

44. Fig. 7-28 Triclinic minerals

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