### Chapter 8: Major Elements

```Optical Mineralogy
WS 2012/2013
Next week….
There is NO lecture
REVISE!
Last week….
Indicatrix - 3-d representation of changing n in minerals
Uniaxial indicatrix - ellipsoid of rotation  tetragonal, hexagonal and
trigonal crystal systems
Uniaxial indicatrix can be positive (prolate or ‘rugby ball’) or negative
(oblate or ‘smartie’)
Basal section  circular  o-ray (n) only  isotropic
Random section  ellipse  o-ray and e’-ray (n n' ) 
intermediate polarisation colour
Principal section  ellipse  o-ray and e-ray (n n)  maximum
birefringence (n)  highest polarisation colour
Last week….
Polarisation colours - result of retardation (v) between o- and
e-rays
 = retardation = d ∙ n
Michel-Levy colour chart  find max. polarisation colour  30
m sections  measure birefringence (n) 
CHARACTERISTIC OF MINERAL
Colours reported by ORDER and COLOUR
….fringe counting….
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
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
Anisotropic minerals – Double refraction
Example: Calcite
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 fast 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 |
Retardation (Gangunterschied)
After time, t, when the slow ray is about
to emerge from the mineral:
 = retardation
Fast wave
with vf
(lower nf)
Slow wave
with vs
(higher ns)
d
Mineral
Polarised
light (E_W)
• The slow ray has traveled distance
d…..
• The fast ray has travelled the distance
d + …..
Slow wave:
t = d/vs
Fast wave:
t = d/vf + /vair
…and so
d/vs = d/vf + /vair
 = d(vair/vs - vair/vf)
 = d(ns - nf)
 = d ∙ Δn
Polariser
(E-W)
Retardation,  = d ∙ Δn (in nm)
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)….
Quartz
n > n
uniaxial positive
Calcite
n < n
uniaxial negative
Uniaxial indicatrix – ellipsoid of rotation
c=Z
optic axis ≡ c-axis
c=X
ne
no
ne
no
b=X
a=X
a=Z
NOTE:
no = n 
nen
n > n
n < n
uniaxial positive (+)
uniaxial negative (-)
PROLATE or ‘RUGBY BALL‘
OBLATE or ‘SMARTIE‘
b=Z
Different slices through the indicatrix
Basal section
Cut perpendicular to the optic axis: only n
 No birefringence (isotropic section)
Principal section
Parallel to the optic axis: n & n
 Maximum birefringence
Random section
 n' and n
 n' is between n and n
 Intermediate birefringence
All sections contain n!
Crystal systems and symmetry
The crystal systems are sub-divided by their degree of symmetry….
CUBIC > TETRAGONAL, HEXAGONAL, TRIGONAL >
ORTHORHOMBIC, MONOCLINIC, TRICLINIC
The Biaxial Indicatrix (….the ‘potato’….)
For orthorhombic, monoclinic and triclinic crystal systems:
The indicatrix is a triaxial ellipsoid with the axes X, Y, Z
The indicatrix has 3 principal refractive indices - n < n < n
The XZ plane (maximum n) is the OPTIC AXIAL PLANE
na = smallest n
nb = intermediate n
ng = largest n
Possible vibration directions =
X, Y and Z
 X || na , Y || nb , Z || ng
na < na' < nb < ng' < ng
Biaxial indicatrix - principal section (XZ)
ng
OA
OA
As n < n < n, there must be a point
between n und n with n = n
= nb
na
= nb
• This gives a circular section (= isotropic)
• The OPTIC AXIS is perpendicular to the
circular section
• There must be 2 circular sections
 optically BIAXIAL
The optic axes lie in the XZ plane and are
perpendicular to n
 OPTIC AXIAL PLANE (max n)
Cut ^ nb
The Bisectrix & 2V
BZ
2VZ
OA
OA
2VX
BX
Angle between the optic axes
Bisector of this angle

If the angle is acute
If the angle is obtuse



2V angle
Bisectrix
 2VX and 2VZ
 BX or BZ
acute bisectrix (2V < 90°)
obtuse bisectrix (2V > 90°)
Optical Sign (+ or -)
Biaxial positive (+) defined as 2VZ <
90°
Biaxial negative (+) defined as 2VZ >
90°
…or… n closer to n than to n
…or… n closer to n than to n
‘RUGBY BALL’ like
‘SMARTIE’ like
Biaxial indicatrix - summary
How do we know?
We use CONOSCOPIC light to see
whether a crystal is uniaxial or biaxial,
positive or negative….
….next two lectures….
Vibration directions & EXTINCTION
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….
Extinction Angle
The EXTINCTION ANGLE is the angle between a linear
feature in the crystal (a crystal edge, a cleavage plane, a twin
plane) and the extinction position.
The EXTINCTION ANGLE is (surprise, surprise) directly
related to the CRYSTAL SYSTEM….
…more specifically, the angular relationship with the
c-axis and the other crystallographic axes….
Symmetry and extinction angles
In cubic minerals and those in the tetragonal,
hexagonal and trigonal systems (= uniaxial
minerals), the c-axis is at 90° to the other
crystallographic axes….
 STRAIGHT EXTINCTION
Symmetry and extinction angles
This is also true of orthorhombic minerals  STRAIGHT EXTINCTION
For minerals in the monoclinic and triclinic systems (= biaxial), the caxis is NOT at 90° to all the other crystallographic axes….
 INCLINED EXTINCTION
Extinction Angle
Extinction angle
e = I – II = 29,5°
I = 153,0°
II = 182,5°
Only the MAXIMUM extinction angle is diagnostic of a
mineral  measure lots of grains
Extinction Angle
Only the MAXIMUM extinction angle is diagnostic of a
mineral  measure lots of grains
Tröger….
Look and work it out….
So why do we see polarisation colours?
Retardation (Gangunterschied)
After time, t, when the slow ray is about
to emerge from the mineral:
 = retardation
Fast wave
with vf
(lower nf)
Slow wave
with vs
(higher ns)
d
Mineral
Polarised
light (E_W)
• The slow ray has traveled distance
d…..
• The fast ray has travelled the distance
d + …..
Slow wave:
t = d/vs
Fast wave:
t = d/vf + /vair
…and so
d/vs = d/vf + /vair
 = d(vair/vs - vair/vf)
 = d(ns - nf)
 = d ∙ Δn
Polariser
(E-W)
Retardation,  = d ∙ Δn (in nm)
Interference
Analyser forces rays to vibrate in the NS plane and interfere.
Destructive interference (extinction):
 = k∙
k = 0, 1, 2, 3, …
Constructive interference (maximum
intensity):
 = (2k+1) ∙ /2
k = 0, 1, 2, 3, …
Explanation of interference colours
Example: a mineral with retardation of 550 nm in the diagonal position
Retardation, 
Wavelength, 
550
400
550
440
550
489
550
550
13/8 l
11/4 l
11/8 l
1l
550
629
550
733
7/
8
3/
4
l
 550 nm is lost, other wavelengths will be partly or fully transmitted.
l
Retardation, 
Wavelength, 
550
400
550
440
13/8 l
11/4 l 11/8 l
No green
(absorbed)
 red + violet
 purple
interference
colour
Fig 7-7 Bloss,
Optical
550
489
550
550
1l
550
629
550
733
7/
3/
8
l
4
l
Retardation, 
Wavelength, 
800
400
800
426
800
457
800
550
800
581
800
711
2l
17/8 l
13/4 l
11/2 l
13/8 l
1 1/8 l 1 l
No red or
violet
(absorbed)
 green
interference
colour
Fig 7-7 Bloss,
Optical
Crystallography,
MSA
800
800
Michel-Lévy colour chart
Michel-Lévy colour chart
thickness of section
birefringence (d)
d = 0.009
d = 0.025
30 mm
(0.03 mm)
retardation ()
first order
second order
third order
….orders separated by red colour bands….
Which order? - Fringe counting….
birefringence (d)
d = 0.009
retardation ()
d = 0.025
30 mm
(0.03 mm)
```