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

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

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 |

Mineral

Polarisedlight (E_W)

Fast wavewith 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 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

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)….

Quartzn > n

uniaxial positive

Calciten < n

uniaxial negative

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

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

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 nnb = intermediate nng = 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)

As n < n < n, there must be a point between n und n with n = n

• 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)

OA OA

Cut ^ nb

ng

na

= nb

= nb

The Bisectrix & 2V

Angle between the optic axes 2V angle 2VX and 2VZ Bisector of this angle Bisectrix BX or BZ

If the angle is acute acute bisectrix (2V < 90°)If the angle is obtuse obtuse bisectrix (2V > 90°)

2VX

2VZ

OAOA

BX

BZ

Optical Sign (+ or -)

Biaxial positive (+) defined as 2VZ < 90°

…or… n closer to n than to n

‘RUGBY BALL’ like

Biaxial negative (+) defined as 2VZ > 90°

…or… n closer to n than to n

‘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 c-axis is NOT at 90° to all the other crystallographic axes….

INCLINED EXTINCTION

Extinction Angle

Extinction anglee = 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?

Mineral

Polarisedlight (E_W)

Fast wavewith 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 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

Retardation, = d ∙ Δn (in nm)

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) ∙ /2k = 0, 1, 2, 3, …

Explanation of interference colours

Example: a mineral with retardation of 550 nm in the diagonal position

Retardation, 550 550 550 550 550 550Wavelength, 400 440 489 550 629 733

13/8 l 11/4 l 11/8 l 1 l 7/8 l 3/4 l

550 nm is lost, other wavelengths will be partly or fully transmitted.

Retardation, 550 550 550 550 550 550Wavelength, 400 440 489 550 629 733

13/8 l 11/4 l 11/8 l 1 l 7/8 l 3/4 l

No green (absorbed) red + violet purple interference colour

Fig 7-7 Bloss, Optical Crystallography, MSA

Retardation, 800 800 800 800 800 800 800Wavelength, 400 426 457 550 581 711 800

2 l 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

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 ()