Transcript Indicatrix - University of Florida

Indicatrix
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Imaginary figure, but very useful
The figures show and/or define:
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Three type – each with characteristic shape:
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Location of optic axis
Positive and negative minerals
Relationship between optical & crystallographic axes
Isotropic
Uniaxial (anisotropic)
Biaxial (anisotropic)
Primary use is to understand/visualize vibration
directions of slow and fast rays
Indicatrix
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Primary uses:
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Determine vibration directions within mineral
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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
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Possible shapes:
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A sphere or oblate/prolate spheroid
Radii of the figures represent vibration
directions
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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
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Construction
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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
Fig. 7-22
Biaxial Indicatrix
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Observe slice of figure
perpendicular to wave
Wave front – plane
normal.
perpendicular to
wave normal
Vibration directions
• Long axis = nslow
perpendicular to wave • short axis = nfast
normal
Principle vibration
directions and values of
index of refraction shown
by semi-major and semiminor axes of ellipse
Fig. 7-22
Biaxial Indicatrix
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Ray paths
constructed by
tangents to the
surface of the
indicatrix that
parallel vibration
directions
Ray directions
Procedure to use
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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
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Generally used in a qualitative way:
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Understanding difference between isotropic,
uniaxial, and biaxial minerals
Understanding the relationship between
optical properties, crystallographic axes, and
crystallographic properties
Isotropic Indicatrix
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Isometric minerals only: Unit cell has only
one dimension
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Minerals have only one index of refraction
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Crystallographic axis = a
Different for each mineral
Shape of indicatrix is a sphere
All sections are circles
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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
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Tetragonal and hexagonal minerals only:
two dimensions of unit cell (a and c)
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Two values of n’s required to define
indicatrix
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High symmetry around c axis
One is epsilon e, the other is omega w
Remember – infinite values of n
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Range between ne and nw
Uniaxial Indicatrix
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Ellipsoid of revolution (spheroid) with axis of
rotation parallel the c crystallographic axis
One semi-axis of ellipsoid parallels c
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Other semi-axis of ellipsoid perpendicular to c
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ne
nw
Maximum birefringence is positive difference of
nw and ne
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Note nw < or > ne, just as c > or < a
Uniaxial Indicatrix
X=Y
ne>nw
ne<nw
Y=Z
Note:
(1) Axes designated X, Y, Z
(2) Z axis always long axis
for uniaxial indicatrix
(3) May be c axis or a axis
(4) Axis perpendicular to
circular section is optic
axis
(5) Optic axis always c
crystallographic axis
Fig. 7-23
Optic Sign
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Defined by nw and ne
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Optically positive (+) – ne > nw, Z = c = ne
Optically negative (-) - ne < nw, Z = a = nw
Ordinary and extraordinary rays
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In uniaxial minerals, one ray always vibrates
perpendicular to optic axis
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Called ordinary or w ray
Always same index = nw
Vibration always within the (001) plane
The other ray may be refracted
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Called extraordinary or e ray
Index of refraction is between ne and nw
Note that ne < or > nw
Ordinary Ray
Ordinary ray
vibrates in (001)
plane: index = nw
Fig. 7-24
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
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Cross section perpendicular to the wave
normal – usually an ellipse
It is important:
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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
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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
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Orientation of grain
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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)
Principle Section
Emergent point – at
tangents
Indicates wave normal
and ray path are the
same, no double
refractions
Semi major axis
Semi-minor axis
What is birefringence of this section?
Fig. 7-25
How many times does it go extinct with 360 rotation?
Circular Section
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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
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
Random Section
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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
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
Biaxial Indicatrix
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Crystal systems: Orthorhombic,
Monoclinic, Triclinic
Three dimensions to unit cell
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a≠b≠c
Three indices of refraction for indicatrix
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na < nb < ng always
Maximum birefringence = ng - na always
Indicatrix axes
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Plotted on a X-Y-Z system
Convention: na = X, nb = Y, ng = Z
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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
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Biaxial indicatrix has two circular sections
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Radius is nb
The circular section ALWAYS contains the Y
axis
Optic axis:
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perpendicular to the circular sections
Two circular sections = two optic axes
Neither optic axis is parallel to X, Y, or Z
Circular sections
Fig. 7-27
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Both optic axes occur in the X-Z plane
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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
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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:
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If Bxa = X, then optically negative
If Bxa = Z, then optically positive
If 2V = 90º, then optically neutral
+
X-Z plane
of Biaxial
Indicatrix
Optically positive
Optically negative
Fig. 7-27
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Uniaxial indicatrixes are special cases of
biaxial indicatrix:
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If nb = na
Mineral is uniaxial positive
 na = nw and ng = ne, note – there is no nb
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If nb = ng
Mineral is uniaxial negative
 na = ne and nc = nw
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Like the uniaxial indicatrix – there are
three primary sections:
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Optic normal section – Y axis vertical so X and
Z in plane of thin section
Optic axis vertical
Random section
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
Crystallographic orientation of
indicatrix
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Optic orientation
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Angular relationship between crystallographic
and indicatrix axes
Three systems (biaxial) orthorhombic,
monoclinic, & triclinic
Orthorhombic minerals
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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.
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Aragonite: X = c, Y = a, Z = b
Anthophyllite: X = a, Y = b, Z = c
Orthorhombic
Minerals
Here optic
orientation is:
Z=c
Y=a
X=b
Fig. 7-28
Monoclinic
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One indicatrix axis always parallels b axis
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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
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Optic orientation defined by
1.
2.
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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
Monoclinic minerals
Positive angle
because in obtuse
angle
b > 90º
Symmetry – rotation
axis or perpendicular to
mirror plane
Negative angle
because in acute
angle
Fig. 7-28
Triclinic minerals
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Indicatrix axes not constrained to follow
crystallographic axes
One indicatrix axis may (but usually not)
parallel crystallographic axis
Triclinic
minerals
Fig. 7-28
P. 306 – olivine information
Optical
orientation
All optical
properties
Optic Axes