advertisement

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 nen 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)