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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)

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indicatrix
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
indicatrix2
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
indicatrix3
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

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
biaxial indicatrix5
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

biaxial indicatrix6
Biaxial Indicatrix
  • Ray paths constructed by tangents to the surface of the indicatrix that parallel vibration directions

Ray directions

procedure to use
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
slide8

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
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
isotropic indicatrix10
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
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
uniaxial indicatrix12
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
fig 7 23
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

optic sign
Optic Sign
  • Defined by nw and ne
    • Optically positive (+) – ne > nw, Z= c = ne
    • Optically negative (-) - ne < nw, Z = a = nw
ordinary and extraordinary rays
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
fig 7 24

Ordinary Ray

Fig. 7-24

Ordinary ray vibrates in (001) plane: index = nw

C crystallographic

axis

extraordinary ray
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
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
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
principle section
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)
fig 7 25

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?

circular section
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
fig 7 2523

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?

random section
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
fig 7 25c
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?

biaxial indicatrix26
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
indicatrix axes
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.
biaxial indicatrix28
Biaxial Indicatrix

Note – differs from uniaxial because nb ≠ na

Fig. 7-27

slide29

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
slide31

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
optic sign32
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
fig 7 27

+

-

Fig. 7-27

X-Z plane of Biaxial Indicatrix

Optically positive

Optically negative

slide34

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
slide35

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
fig 7 29

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’

crystallographic orientation of indicatrix
Crystallographic orientation of indicatrix
  • Optic orientation
    • Angular relationship between crystallographic and indicatrix axes
    • Three systems (biaxial) orthorhombic, monoclinic, & triclinic
orthorhombic minerals
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
fig 7 28
Fig. 7-28

Orthorhombic Minerals

Here optic orientation is:

Z = c

Y = a

X = b

monoclinic
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
slide41

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
fig 7 2842
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

triclinic minerals
Triclinic minerals
  • Indicatrix axes not constrained to follow crystallographic axes
  • One indicatrix axis may (but usually not) parallel crystallographic axis
fig 7 2844
Fig. 7-28

Triclinic minerals

p 306 olivine information
P. 306 – olivine information

Optical orientation

All optical properties

Optic Axes