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. nq & np are two of the principle vibration directions
Biaxial Indicatrix
Fig. 7-22
Construction
Plot primary indices of refraction along three primary axes: X, Y, and Z
Always 90º to each other
nq & np are 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 epsilon e, 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. Uniaxial Indicatrix ne>nw 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
Fig. 7-23

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

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

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. What is birefringence of this section?
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

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. What is birefringence of this section?
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. 7-25c

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

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

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

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

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

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

44. Triclinic minerals
Fig. 7-28

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