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 simplest case - cubic minerals (e.g. garnet)

The Optical Indicatrix

• 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?

Spheren 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 indicatrix

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 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

Quartz Calcite

c-axis

Anisotropic Minerals – The Uniaxial Indicatrix

c-axis

What does the indicatrix for each mineral look like?

Uniaxial indicatrix – ellipsoid of rotation

optic axis ≡ c-axis

ne

no b=X

c=Z

a=X

ne

b=Z

c=X

no

a=Z

n > n

uniaxial positive (+)

PROLATE or ‘RUGBY BALL‘

n < n

uniaxial negative (-)

OBLATE or ‘SMARTIE‘

NOTE:no = n

nen

Quartzn > n

uniaxial positive

Calciten < n

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)….

Basal sectionCut 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!

Different slices through the indicatrix

Isotropic section(remains black in XPL)

Cut PERPENDICULAR to the c-axis,

Contains only no (n)

Basal Section

The principal section shows MAXIMUM birefringence and the HIGHEST polarisation colour

DIAGNOSTIC PROPERTY OF MINERAL

n > n

Principal Section

Cut PARALLEL to the c-axis,contains no (n) und ne (n)

A random section shows an intermediate polarisation colour

no use for identification purposes

Random Section

Cut at an angle to the c-axis,contains no (n) and ne‘ (n‘)

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….

Polariser parallel to ne:

only the extraordinary ray is transmitted inserting the analyser BLACK

= EXTINCTION POSITION

Polariser parallel to no:

only the ordinary ray is transmitted inserting the analyser BLACK

= EXTINCTION POSITION

Polariser

ne

no

Parallel position

no

ne

As both rays are forcedto vibrate in the N-S direction,

they INTERFERE

Split into perpendicular two rays (vectors) :

1) ordinary ray where n = no

2) extraordinary ray where n = ne

® Each ray has a N-S component, which are able to pass through the analyser.

® Maximum brightness is in the diagonal position.ne

no

Polariser

Diagonal position

Mineral

Polarisedlight (E–W)

Fast wave with vf

(lower nf)Slow wave with vs

(higher ns)

Polariser(E-W)

= retardation

d

Retardation (Gangunterschied)

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: 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

Retardation, = d ∙ Δn (in nm)

Michel-Lévy colour chart

thic

knes

s of

sec

tion

birefringence (d)

30 mm (0.03 mm)

d = 0.009 d = 0.025

first order second order third order

lines of constant d

Michel-Lévy colour chart

retardation ()

….orders separated by red colour bands….

birefringence (d)

30 mm (0.03 mm)

d = 0.009 d = 0.025

lines of constant d

Which order? - Fringe counting….

retardation ()

Uniaxial indicatrix - summary

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 n;

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 last week;

Distinguishing isotropic from anisotropic minerals;

Calculating retardation;

Calculating and reporting birefringence - fringe counting.

Thinking about vibration directions….