WS 2008/2009

So far ….
• Light - properties and behaviour;
• Refraction - the refractive index ( n ) of minerals leads to many of their optical properties  Snell’s law - learn it!;
• Double refraction;
• Polarization and the polarizing microscope;
• Plane polarised light observations (PPL):
• crystal shape/habit
• colour/pleochroism
• cleavage/fracture
• relief, Becke test  refractive index estimation

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 n at that point.

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 (= cubic)
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

Isotropic minerals and polarized light
West (links)
Ost (rechts)
Polarisator z.B. Granat
Süden
(vorne)
Analysator
Norden
(hinten)
Schwarz!!
Beobachtungen unter gekreuzten Polarisatoren
(gekreuzte Nicols)
XPL = crossed nicols (crossed polars)

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
One of the rays (the fast ray for calcite) obeys Snell’s Law ordinary ray ( n o
)
The other ray does not obey Snell’s law extraordinary ray ( n e
)
 Birefringence = Δ n = n e
− n o

Anisotropic Minerals – The Uniaxial Indicatrix c-axis c-axis
Quartz
Calcite
What does the indicatrix for each mineral look like?

Uniaxial indicatrix – ellipsoid of rotation c=Z optic axis ≡ c-axis c=X n e n o b=Y=X n e n o a=X n e
> n o uniaxial positive (+)
PROLATE or ‘RUGBY BALL‘
NOTE: n o n e
= n

 n
 a=X n e
< n o uniaxial negative (-)
OBLATE or ‘SMARTIE‘ b=Y

Quartz n e
> n o uniaxial positive
Calcite n e
< n o uniaxial negative

Uniaxial Indicatrix
All minerals belonging to the TRIGO NAL , TETRAGO NAL and HEXAGO NAL crystal systems have a uniaxial indicatrix….
This reflects the dominance of the axis of symmetry (= c-axis) in each system ( 3-, 4and 6 fold respectively)….

Different slices through the indicatrix (+)

Basal section
Cut perpendicular to the optic axis: only n o
 No birefringence (isotropic section)

Principal section
Parallel to the optic axis: n o
 Maximum birefringence
& n e

Random section
 n e' and n o
 n e' is between n e and n
 Intermediate birefringence o

All sections contain n o
!

c=Z b=Y n  n  a=X
Basal Section
Cut PERPENDICULAR to the caxis,
Contains only n o
(n

)
Isotropic section
(remains black in XN)

n

> n

Principal Section
Cut PARALLEL to the c-axis, contains n o
(n

) und n e
(n

)
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 n o
(n

) und n e‘
(n
 ‘
)
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 ….

n e n o n o
Polariser
Parallel position
Polariser parallel to n e
:
 only the extraordinary ray is transmitted inserting the analyser  BLACK
= EXTINCTION POSITION
Polariser parallel to n o
:
 only the ordinary ray is transmitted inserting the analyser  BLACK
= EXTINCTION POSITION n e

Diagonal position n e
Polariser n o
Split into perpendicular two rays (vectors) :
1) ordinary ray where n = n o
2) extraordinary ray where n = n e

Each ray has a N-S component, which are able to pass through the analyser.

Maximum brightness is in the diagonal position.
As both rays are forced to vibrate in the N-S direction, they INTERFERE

So why do we see polarisation colours?

Fast wave with v f
(lower n f
) d
Mineral
Retardation ( Gangunterschied )
Polariser
(E-W)

= retardation
Slow wave with v s
(higher n s
)
Polarised light (E_W)
After time, t, when the slow ray is about to emerge from the mineral:
• The slow ray has travelled distance d…..
• The fast ray has travelled the distance d +
 …..
Slow wave :
Fast wave:
…and so t = d/v s t = d/v f
+

/v air d/v s
= d/v f
+

/v air

= d(v air
/v s
- v air
/v f
)

= d(n s
- n f
)
 = d ∙ Δn
Retardation,
 = d ∙ Δn (in nm)
….and we know d (= 30  m)….

Interference



Analyser forces rays to vibrate in the N-S 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
550 550
440 489
550
550
550 550
629 733
1 3 /
8 l 1 1 /
4 l 1 1 /
8 l 1 l 7 /
8 l
 550 nm is lost, other wavelengths will be partly or fully transmitted.
3 /
4 l

Retardation,

Wavelength,

550 550 550 550
400 440 489 550
1 3 /
8 l 1 1 /
4 l 1 1 /
8 l 1 l
550 550
629 733
7 /
8 l 3 /
4 l
No green
(absorbed)
 red + violet
 purple interference colour
Fig 7-7 Bloss, Optical
Crystallography, MSA

Retardation,

Wavelength,

800 800 800 800
400 426 457 550
800
581
2 l 1 7 /
8 l 1 3 /
4 l 1 1 /
2 l 1 3 /
8 l
800 800
711 800
1 1 /
8 l 1 l
No red or violet
(absorbed)
 green interference colour
Fig 7-7 Bloss, Optical
Crystallography, MSA

MichelLévy colour chart

MichelLévy colour chart birefringence ( d
) d
= 0.009
d
= 0.025
30
 m
(0.03 mm) retardation (

) first order second order third order
….orders separated by red colour bands….

Uniaxial indicatrix - summary



Can be positive or negative;
Mierals of the tetragonal, 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 n w
;
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






Isotropic (cubic) minerals show no birefringence and remain black in XN;
Anisotropic minerals have variable n and therefore show polarisation colours;
The larger
 n 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…..






Making the PPL observations you made in the previous 2 weeks;
Putting a scale on your sketches to estimate grain sizes;
Distinguishing isotropic from anisotropic minerals;
Calculating retardation;
Calculating and reporting birefringence - fringe counting.
Thinking about vibration directions….