Optical Mineralogy
WS 2012/2013
Crystal systems and symmetry
The crystal systems are sub-divided by their degree of symmetry….
CUBIC > TETRAGONAL, HEXAGONAL, TRIGONAL >
ORTHORHOMBIC, MONOCLINIC, TRICLINIC
The Optical Indicatrix
•  The
optical indicatrix is a 3-dimensional graphical
representation of the changing refractive index of a
mineral;
•  The shape of the indicatrix reflects the crystal system
to which the mineral belongs;
•  The distance from the centre to a point on the surface
of the indicatrix is a direct measure of the refractive
index (n) at that point;
•  Smallest n = X, intermediate n = Y, largest n = Z
The Optical Indicatrix
The simplest case - cubic minerals (e.g. garnet)
•  Cubic minerals have highest
symmetry (a=a=a);
•  If this symmetry is reflected in the
changing refractive index of the
mineral, what 3-d shape will the
indicatrix be?
Isotropic indicatrix
Sphere
n is constant is every direction isotropic minerals do not change the
vibration direction of the light - no
polarisation
Indicatrix = 3-d representation of refractive index
Isotropic indicatrix
Anisotropic minerals – Double refraction
Example: Calcite
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The incident ray is split into 2 rays that
vibrate perpendicular to each other.
These rays have variable v (and therefore
variable n)  fast and slow rays
As n ∞ 1/v, fast = small n, slow = big n
One of the rays (the slow ray for calcite)
obeys Snell’s Law - ordinary ray (no)
The other ray does not obey Snell’s law extraordinary ray (ne)
 Birefringence = Δn = ne − no
Double Refraction
O
E
Anisotropic Minerals – The Uniaxial Indicatrix
c-axis
Quartz
c-axis
Calcite
What does the indicatrix for each mineral look like?
Uniaxial indicatrix – ellipsoid of rotation
c=Z
optic axis ≡ c-axis
c=X
ne
no
ne
b=X
a=X
nε > nω
no
a=Z
NOTE:
no = nω
ne = nε
nε < nω
uniaxial positive (+)
uniaxial negative (-)
PROLATE or ‘RUGBY BALL‘
OBLATE or ‘SMARTIE‘
b=Z
Quartz
nε > nω
uniaxial positive
Calcite
nε < nω
uniaxial negative
Uniaxial Indicatrix
All minerals belonging to the TRIGONAL, TETRAGONAL and
HEXAGONAL crystal systems have a uniaxial indicatrix….
This reflects the dominance of the axis of symmetry (= c-axis)
in each system (3-, 4- and 6-fold respectively)….
Different slices through the indicatrix

Basal section
Cut perpendicular to the optic axis: only nω
 No birefringence (isotropic section)
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Principal section
Parallel to the optic axis: nω & nε
 Maximum birefringence
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Random section
 nε' and nω
 nε' is between nε and nω
 Intermediate birefringence
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All sections contain nω!
Basal Section
Cut PERPENDICULAR to the caxis,
Contains only no (nω)
c=Z
n!
n"
a=X
b=Y
Isotropic section
(remains black in XPL)
Principal Section
Cut PARALLEL to the c-axis,
contains no (nω) und ne (nε)
nε > nω
The principal section shows MAXIMUM birefringence
and the HIGHEST polarisation colour
 DIAGNOSTIC PROPERTY OF MINERAL
Random Section
Cut at an angle to the c-axis,
contains no (nω) and ne‘ (nε‘)
A random section shows an intermediate
polarisation colour
 no use for identification purposes
Double Refraction
Privileged Vibration directions
In any random cut through an anistropic
indicatrix, the privileged vibration directions are
the long and short axis of the ellipse. We know
where these are from the extinction positions….
Parallel position
Polariser parallel to ne:
ne
 only the extraordinary ray is transmitted
inserting the analyser  BLACK
no
Polariser
= EXTINCTION
POSITION
Polariser parallel to no:
 only the ordinary ray is transmitted
inserting the analyser  BLACK
no
= EXTINCTION
ne
POSITION
Diagonal position
Split into perpendicular two rays (vectors) :
1) ordinary ray where n = no
2) extraordinary ray where n = ne
no
ne
→
Each ray has a N-S component, which are able to
pass through the analyser.
→
Maximum brightness is in the diagonal position.
Polariser
As both rays are forced
to vibrate in the N-S direction,
they INTERFERE
Retardation (Gangunterschied)
After time, t, when the slow ray is about to
emerge from the mineral:
Γ = retardation
•  The slow ray has travelled distance d…..
•  The fast ray has travelled the distance d+Γ…..
Fast wave with
vf
(lower nf)
Slow wave
with vs
(higher ns)
d
Mineral
Polarised
light (E–W)
Slow wave:
t = d/vs
Fast wave:
t = d/vf + Γ/vair
…and so
d/vs = d/vf + Γ/vair
Γ = d(vair/vs - vair/vf)
Γ = d(ns - nf)
Γ = d · Δn
Polariser
(E-W)
Retardation, Γ = d · Δn (in nm)
Michel-Lévy colour chart
Michel-Lévy colour chart
thickness of section
birefringence (δ)
δ = 0.009
δ = 0.025
lines
tant
s
n
o
c
of
30 µm
(0.03 mm)
δ
retardation (Γ)
first order
second order
third order
….orders separated by red colour bands….
Which order? - Fringe counting….
birefringence (δ)
δ = 0.009
δ = 0.025
lines
retardation (Γ)
tant
s
n
o
c
of
30 µm
(0.03 mm)
δ
Uniaxial indicatrix - summary
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Can be positive or negative;
Mierals of the tertragonal, trigonal and hexagonal crystal
systems have a uniaxial indicatrix;
All sections apart from the basal section show a
polarisation colour;
All sections through the indicatrix contain nω;
The basal section is isotropic and means you are looking
down the c-axis of the crystal;
The principal section shows the maximum polarisation
colour characteristic for that mineral.
Polarisation colours
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Isotropic (cubic) minerals show no birefringence and
remain black in XN;
Anisotropic minerals have variable n and therefore show
polarisation colours;
The larger δn is, the higher the polarisation colour;
The polarisation colour is due to interference of rays of
different velocities;
THE MAXIMUM POLARISATION COLOUR IS THE
CHARACTERISTIC FEATURE OF A MINERAL (i.e.,
look at lots of grains);
Polarisation colours should be reported with both
ORDER and COLOUR (e.g., second order blue, etc.).
Todays practical…..
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Making the PPL observations you made last week;
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Distinguishing isotropic from anisotropic minerals;
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Calculating retardation;
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Calculating and reporting birefringence - fringe
counting.
Thinking about vibration directions….