GLG212
Part II, Lecture 1:
Indicatrix and
interference figures
Objectives
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Understand principles of:
• Birefringence (continued)
• Optical indicatrix
• Interference figures
The Optical Indicatrix
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If we construct vectors
whose direction
corresponds to the
vibration direction, and
whose length
corresponds to the
refractive index of the
material for light with
that vibration direction,
their tips will define an
imaginary surface
called the indicatrix.
The Optical Indicatrix
The indicatrix is a shape
called an ellipsoid. Every
cross section of an ellipsoid
is an ellipse.
The longest direction of the
ellipsoid is its major axis.
The shortest direction,
perpendicular to the major
axis, is called the minor
axis.
Perpendicular to the major
and minor axes is the
intermediate axis.
These three axes are called
the principal axes.
The Optical Indicatrix
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A geometric figure that shows the index of
refraction and vibration direction for light
passing in any direction through a material is
called an optical indicatrix.
n = refractive index
The indicatrix is constructed by plotting
indices of refraction as radii parallel to the
vibration direction of the light.
Ray p, propagating along Y, vibrates parallel
to the Z-axis so its index of refraction (np) is
plotted as radii along Z.
Ray q, propagating along X, vibrates parallel
to Y so its index of refraction (nq) is plotted
as radii along Y.
If the indices of refraction for all possible light
rays are plotted in a similar way, the surface
of the indicatrix is defined. The shape of the
indicatrix depends on mineral symmetry.
The Optical Indicatrix
Isotropic Indicatrix
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Optically isotropic minerals all
crystallize in the isometric
crystal system.
One unit cell dimension (a) is
required to describe the unit
cell and one index of refraction
(n) is required to describe the
optical properties because light
velocity is uniform in all
directions for a particular
wavelength of light.
The indicatrix is therefore a
sphere.
All sections through the
indicatrix are circles and the
light is not split into two rays.
Birefringence may be
considered to be zero.
Uniaxial Indicatrix
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Minerals that crystallize in the tetragonal,
trigonal and hexagonal crystal systems
have two different unit cell dimensions (a
and c) and a high degree of symmetry about
the c axis.
Two indices of refraction are required to
define the dimensions of the indicatrix,
which is an ellipsoid of revolution whose axis
is the c crystal axis.
The semiaxis of the indicatrix measured
parallel to the c axis is called ne, and the
radius at right angles is called nw. The
maximum birefringence of uniaxial minerals
is always [ne - nw].
All vertical sections through the indicatrix
that include the c axis are identical ellipses
called principal sections whose axes are nw
and ne. Random sections are ellipses
whose dimensions are nw and ne’ where ne’ is
between nw and ne.
The section at right angles to the c axis
is a circular section whose radius is nw.
Because this section is a circle, light
propagating along the c axis is not doubly
refracted as it is following an optic axis.
Because hexagonal and tetragonal minerals
have a single optic axis, they are called
optically uniaxial.
Uniaxial Indicatrix - Positive
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Z (longest) axis
= optic axis = c
Uniaxial Indicatrix - Negative
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X (shortest) axis
= optic axis = caxis
Use of the Indicatrix
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In this example the mineral is oriented
so that its optic axis is horizontal
We will assume that the mineral is
uniaxial positive
The wave normal is through the centre
of the indicatrix and light is incident
normal to the bottom surface of the
grain.
Because the optic axis is horizontal,
this section is a principal section, which
is an ellipse whose axes are nw and ne.
Therefore
•
•
Ordinary ray = nw
Extraordinary = ne
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Maximum because the mineral is optically positive.
The extraordinary ray vibrates parallel
to the trace of the optic axis (c axis)
and the ordinary ray vibrates at right
angles.
Therefore, birefringence and hence
interference colors are maximum
values.
Use of the Indicatrix
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In this case the mineral sample is
oriented so that the optic axis is
vertical.
The section through the indicatrix
perpendicular to the wave normal is
the circular section whose radius is
nw.
Light coming from below is not
doubly refracted, birefringence is
zero and the light preserves
whatever vibration direction it
initially had.
Between crossed polars this mineral
should behave like an isotropic
mineral and remain dark as the
stage is rotated.
However, because the light from the
substage condenser is moderately
converging, some light may pass
through the mineral.
The mineral may display
interference colors but they will be
the lowest order found in that
mineral.
Use of the Indicatrix
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In this sample the mineral
sample is oriented in a random
orientation so that the light path
is at an angle q to the optic axis.
The section through the
indicatrix parallel to the bottom
surface of the mineral is an
ellipse whose axes are nw and
ne
The extraordinary ray vibrates
parallel to the trace of the optic
axis as seen from above, while
the ordinary ray vibrates at right
angles
Both birefringence and
interference colors are
intermediate because ne’ is
intermediate between nw and ne.
Biaxial Indicatrix
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Minerals that crystallize in the orthorhombic, monoclinic and
triclinic crystal systems require three dimensions (a, b and c) to
describe their unit cells and three indices of refraction to define
the shape of their indicatrix
The three principal indices of refraction are na, nb and ng where na
< nb < ng.
• The maximum birefringence of a biaxial mineral is always ng - na
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Construction of a biaxial indicatrix requires that three indices of
refraction are plotted
• However, while three indices of refraction are required to describe
biaxial optics, light that enters biaxial minerals is still split into two
rays.
• As we shall see, both of these rays behave as extraordinary rays for
most propagation paths through the mineral
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The index of refraction of the fast ray is identified as na’ where na
< na’ < nb and the index of refraction of the slow ray is ng’ where
nb < ng’ < ng
Biaxial Indicatrix
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The biaxial indicatrix contains three principal sections, the YZ, XY and XZ
planes.
The XY section is an ellipse with axes na and nb,
The XZ section is an ellipse with axes na and ng
YZ section is an ellipse with axes nb and ng.
Random sections through the indicatrix are ellipses whose axes are na’ and ng’.
The indicatrix has two circular sections with radius nb that intersect the Y axis.
The XZ plane is an ellipse whose radii vary between na and ng. Therefore radii
of nb must be present.
Radii shorter than nb are na’ and those that are longer are ng’.
The radius of the indicatrix along the Y axis is also nb
Biaxial Indicatrix
An ellipse with three unequal
principal axes has two circular
cross sections. The plane
containing the major and minor
axis cuts the ellipsoid in an
ellipse with the maximum and
minimum possible radii.
Somewhere in between is a
radius equal to the
intermediate axis. The two
circular sections have radius
equal to the intermediate axis
and intersect along the
intermediate axis. They are
shown in blue and purple at
left.
Biaxial Indicatrix
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The directions perpendicular to each one
of the circular sections are the optic axes.
For a generic ellipsoid with three unequal
axes, there are two optic axes. Such
materials are called biaxial.
Biaxial Indicatrix – XZ plane
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Therefore the Y axis and the nb radii in the XZ plane define the two
circular sections.
Like uniaxial minerals, the circular sections in biaxial minerals are
perpendicular to the optic axes, hence the term biaxial.
Because both optic axes lie in the XZ plane of the indicatrix, that
plane is called the optic plane.
The angle between the optic axes bisected by the X axis is also called
the 2Vx angle, while the angle between the optic axes bisected by the
Z axis is called the 2Vz angle
where 2Vx + 2Vz = 180°.
The Y axis, which is
perpendicular to the optic plane
is called the optic normal.
Optic Sign – Biaxial Minerals
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The acute angle between the optic axes is called the optic
angle or 2V angle.
The axis (either X and Z)that bisects the acute angle
between the optic axes is the acute bisectrix or Bxa.
The axis (either Z or X) that bisects the obtuse angle
between the optic axes is the obtuse bisectrix or Bxo.
The optic sign of biaxial minerals depends on whether the X
or Z indicatrix axis bisects the acute angle between the
optic axes.
1. If the acute bisectrix is the X axis, the mineral is optically
negative and 2Vx is less than 90°
2. If the acute bisectrix is the Z axis, the mineral is optically
positive and 2Vz is less than 90°
3. If 2V is exactly 90° so neither X nor Z is the acute bisectrix,
the mineral is optically neutral.
Optic Sign and Biaxial Indicatrix
Indicatrix Axes and Orthorhombic Crystals
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Orthorhombic crystals have
three mutually perpendicular
crystallographic aces of unequal
length. These crystal axes must
coincide with the three
indicatrix axes and the
symmetry planes in the mineral
must coincide with principal
sections in the indicatrix. Any
crystal axis may coincide with
any indicatrix axis however.
The optic orientation is defined
by indicating which indicatrix
axis is parallel to which mineral
axis.
1. Aragonite X = c, Y = a, Z =
b
2. Anthophyllite X = a, Y = b,
Use of the Biaxial Indicatrix
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The biaxial indicatrix is used in the same way as
the uniaxial indicatrix.
It provides information about the indices of
refraction and vibration direction given the wave
normal direction that light is following through a
mineral.
Birefringence depends on how the sample is cut
or mounted.
Birefringence is:
• A maximum if the optic plane is horizontal
• A minimum if an optic axis is vertical
• Intermediate for all other random orientations
Birefringence
(continued)
Birefringence: the double refraction of
light in a transparent, molecularly ordered
material, which is manifested by the existence of orientation
dependent differences in refractive index
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Calculated as:
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• Birefringence (B) = |ne - no|, with e – extraordinary ray; O – ordinary
ray
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This is the birefringence that a specific ray of light experience through a specific direction
of a crystal
Minerals has specific values of birefringence (measure how high is
the degree of birefringence that can be caused by that mineral)
The maximum possible birefringence (also called the retardation)
that can be caused a specific mineral is calculated as follows, (t =
the thickness of the mineral, thus the thickness of the thin
section):
• Isotropic:
• Anisotropic uniaxial:
• Anisotropic biaxial:
Δn = t.|na - na| = 0
Δn = t.|nω - nε|
Δn = t.|nγ - nα|
Interference colours
First order colors
Second order colors
Third order colors
Birefringence
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A characteristic that all anisotropic minerals
have, intensity differs
• High birefringent minerals – third/fourth order
interference colours
• Med birefringent minerals – second order
interference colours
• Low birefringent minerals - first order interference
colours
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For specific mineral birefringence depends on
orientation:
• Maximum birefringence - orientation of grain
shows highest possible interference colour for the
specific mineral
Observation of interference figures
using convergent light – conoscopic
view
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Insert condenser
lens
Gives convergent
light
Enters sample at
50º - 90º angles
See image of light
source
Interference effects at
different angles
Conoscopic observation of
interference figures
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Isotropic
• No image
Conoscopic observation of
interference figures
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Uniaxial
• Perpendicular to
optical axis
Conoscopic observation of
interference figures
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Uniaxial
• At an angle to the
optical axis
Conoscopic observation of
interference figures
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Uniaxial
• Parallel to the
optical axis
Conoscopic observation of
interference figures
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Biaxial
• Perpendicular to acute bisectrix
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2V angle < 60°
Conoscopic observation of
interference figures
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Biaxial
• Perpendicular to acute bisectrix
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2V angle > 60° (but still < 90 °)
Conoscopic observation of
interference figures
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Biaxial
• Perpendicular to optical axis
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2V < 30°
Conoscopic observation of
interference figures
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Biaxial
• Perpendicular to optical axis
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2V > 30° (but still < 90 °)
Conoscopic observation of
interference figures
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Biaxial
Conoscopic observation of
interference figures
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Biaxial
• Perpendicular to obtuse bisectrix
Conoscopic observation of
interference figures
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Biaxial
• Parallel to the axial plane
Conoscopic observation of
interference figures
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Biaxial
• Off-centre interference figures
Conoscopic observation of
interference figures
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Biaxial
• Off-centre interference figures