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 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 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 no b=X a=X a=Z NOTE: no = n nen n > n n < n uniaxial positive (+) uniaxial negative (-) PROLATE or ‘RUGBY BALL‘ OBLATE or ‘SMARTIE‘ b=Z Quartz n > n uniaxial positive Calcite n < 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)…. 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! Basal Section Cut PERPENDICULAR to the caxis, Contains only no (n) c=Z ne nw a=X b=Y Isotropic section (remains black in XPL) Principal Section Cut PARALLEL to the c-axis, contains no (n) und ne (n) n > 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 no (n) and ne‘ (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…. 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 POSITION ne 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: = retardation • The slow ray has travelled distance d….. • The fast ray has travelled the distance d+….. 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 + /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) 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) 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 dn 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….