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Indicatrix

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# Indicatrix - PowerPoint PPT Presentation

Indicatrix. Imaginary figure, but very useful The figures show and/or define: Location of optic axis Positive and negative minerals Relationship between optical &amp; crystallographic axes Three type – each with characteristic shape: Isotropic Uniaxial (anisotropic) Biaxial (anisotropic)

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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)
• Primary use is to understand/visualize vibration directions of slow and fast rays
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
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

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

Ray 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 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 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 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 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

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

Ordinary Ray

Fig. 7-24

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

C crystallographic

axis

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
• 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
• 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
• 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)

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
• 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

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
• 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

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 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
• 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 Indicatrix

Note – differs from uniaxial because nb ≠ na

Fig. 7-27

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

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 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

X-Z plane of Biaxial Indicatrix

Optically positive

Optically negative

• 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
• Optic normal section – Y axis vertical so X and Z in plane of thin section
• Optic axis vertical
• Random section
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
• Optic orientation
• Angular relationship between crystallographic and indicatrix axes
• Three systems (biaxial) orthorhombic, monoclinic, & triclinic
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

Orthorhombic Minerals

Here optic orientation is:

Z = c

Y = a

X = b

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

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. 7-28

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

Triclinic minerals

P. 306 – olivine information

Optical orientation

All optical properties

Optic Axes