Frequency = # of waves/sec to pass a given point (hz)
Optical Mineralogy Wave Theory = 2 X Amplitude Frequency = # of waves/sec to pass a given point (hz) f = v/l v = velocity Electromagnetic spectrum & visible portion
Violet (400 nm) Red (700 nm) White = ROYGBV (can be separated by dispersion) Refraction Incident ray and reflected ray: Refracted ray:
1) of incidence i = of reflection r' 2) coplanar plane of incidence (in plane ^ interface) Refracted ray: 1) Slower in water(or glass) 2) r I Depends on D v Incident i Reflected r air water r Refracted Index of refraction For a substance x: nx = vair/vx nair =
Index of refraction For a substance x: nx = vair/vx nair = ??light is slower in water, glass, crystals Is nwatergreater or less than 1?? Larger n associated with slower V !!Snells Law: ni sin i = nr sin r for 2 known media (air/water) sin i/sin r = nr / ni = const So can predict angle change! Polarization Non-polarized (usual) light:
Each photon vibrates as a wave form in a single plane Light beam = numerous photons, each vibrating in adifferent plane Vibration in all directions ~ perpendicular to propagationdirection Polarization incoming ray is non-polarized
reflected and refracted rays both become polarized Polarization Microscopes have two polarizers:
polarizer (below stage) is E-W analyzer(above stage) is N-S The Optical Indicatrix
Shows how ni varies with vibration direction. Vectors radiating from center Length of each proportional to ni for light vibrating in the direction of the vector Indicatrix = surface connecting tips of vectors (a shape to represent changes in n with direction) Isotropic media have all ni the same (by definition) What is the shape of an isotropic indicatrix? a spherical indicatrix The Optical Indicatrix
North South A P P For isotropic minerals When analyzer inserted = crossed-nicolsor XPL shorthand (vs PPL) no light passes extinct, even when the stage is rotated Fig. 6-6 A Fig 6-6 Bloss, Optical Crystallography, MSA West P P East Anisotropic crystals Calcite experiment and double refraction Anisotropic crystals Calcite experiment and double refraction
O E Double images: Ray 2 rays withdifferent propagationand vibration directions Each is polarized ( ^ eachother) Fig 6-7 Bloss, Optical Crystallography, MSA Anisotropic crystals Calcite experiment and double refraction
O E O-ray (Ordinary) Obeys Snell's Law and goes straight Vibrates ^ plane containing ray and c-axis (optic axis) E-ray (Extraordinary) deflected Vibrates in plane containing ray and c-axis ..also doesn't vibrate ^ propagation, but we'll ignore this as we said earlier Fig 6-7 Bloss, Optical Crystallography, MSA Called privileged directions Each ray has a different n w = no e = nE
IMPORTANT:A given ray ofincoming light is restricted to only 2(mutually perpendicular) vibrationdirections once it enters ananisotropic crystal Called privileged directions Each ray has a different n w = no e = nE w < e(in the case of calcite) O E Fig 6-7 Bloss, Optical Crystallography, MSA n > 1 for all anisotropic substances
n = f(vibration direction) Indicatrix no longer a sphere Indicatrix = ellipsoid Hexagonal and tetragonal xls have one unique xl axis (caxis) ^ 2 identical ones--UNIAXIAL MINERALS The optical properties reflect this as well: ellipsoid ofrotation about c Fig 6-10 Bloss, Optical Crystallography, MSA
For light travelling parallel c, all vibration directions ^c are the same: circular section of indicatrix ( ^ c) thus behaves as isotropic (no unique plane containing ray and c-axis) only one ray (O-ray) with n = w(doesnt split to two rays) extinct with analyzer in and stays that way as rotate stage For light travelling ^ c get elliptical principal section of indicatrix:
get 2 rays O-ray with n = w E-ray with n = e this e (parallel c) is the maximum possible deviation in nfrom w (true e) Fig. 6-12 For random vibration direction same situation as above Except that E-ray has some n between e and w All intermediate values are called e (a variable value betweene and w) ellipsoid and conventions:
(+) crystal = prolatee > w (-) crystal = oblate e < w (-) crystal: w > e oblate (+) crystal: e > w prolate Fig 6-11 Bloss, Optical Crystallography, MSA Summary: Circular Section Principal Sections
(^ optic axis: all w's) extinct Principal Sections (have w and true e: max & min n's)largest birefringence! Random Sections(e' and w) always have w!! Any cut through center of a uniaxial indicatrix will have w as one semiaxis Fig. 6-12 Color chart Shows the relationship between retardation, crystal thickness, and interference color 550 mm red violet 800 mm green 1100 mm red-violet again (note repeat ) 0-550 mm = 1st order mm = 2nd order mm= 3rd order... Higher orders are more pastel Example: Quartz w = 1.544 e = 1.553 w e 1.544 1.553
Data from Deer et al Rock Forming Minerals John Wiley & Sons Example: Quartz w = 1.544 e = 1.553 Sign?? (+) because e > w
e - w = and is called the birefringence (d) = maximum interference color What color is this?? 1) Follow line in toward origin 2) Where it crosses 30 micron thickness (the standard for thin sections) we get a yellowish tan (see when quartz oriented with OA in plane of stage) For other orientations get e' - w progressively lower color Extinct when priv. direction N-S (every 90o) 360orotation 4 extinction positions exactly 90o apart Conoscopic Viewing A condensing lens below the stage and a Bertrand lens above it Arrangement essentially folds planes of Fig cone Light rays are refracted by condensing lens & pass through crystal in different directions Thus different properties Only light in the center of field of view is vertical & like ortho Interference FiguresVery useful for determining optical properties of xl Fig 7-13 Bloss, Optical Crystallography, MSA Uniaxial Figure Circles of isochromes Note vibration directions:
w tangential e' radial & variable magnitude Blackcross (isogyres) results from locus of extinction directions Center of cross (melatope) represents optic axis Approx 30o inclination of OA will put it at margin of field of view Fig. 7-14 Uniaxial Figure Centered axis figure as 7-14: when rotate stage cross does not rotate Off center:cross still E-W and N-S, but melatope rotates around center Melatope outside field:bars sweep through, but always N-S or E-W at center Flash Figure:OA in plane of stage Diffuse black fills field brief time as rotate Fig. 7-14 Accessory Plates Use a 1st-order red (gypsum) plate
Fig 8-1 Bloss, Optical Crystallography, MSA Use a 1st-order red (gypsum) plate Slow direction is marked N on plate Fast direction (n) || axis of plate The gypsum crystal is oriented and cut so thatD = (N-n) 550nm retardation thus it has the effect of retarding the N ray550 nm behind the n ray If insert with no crystal on the stage 1- order red in whole field of view Accessory Plates N Suppose we view an anisotropic crystal with D = 100 nm (1-order gray) at 45o from extinction n If Ngyp || Nxl Addition Addition since ray in xl || Ngyp already behind by 100nm & it gets further retarded by 550nm in the gypsum plate 650nm On your color chart what will result? Original 1o grey 2o blue N Optic Sign Determination
For all xls remember e' vibrates in plane of ray and OA, w vibr normal to plane of ray and OA O E e ' w 1) Find a uniaxial crystal in which the optic axis (OA) is vertical (normal to the stage) How? 2) Go to high power, insert condensing and Bertrand lenses to optic axis interference figure (+)crystals: e > w so w faster Fig 7-13 Bloss, Optical Crystallography, MSA Optic Sign Determination
Inserting platefor a (+) crystal: subtraction in NW & SE where n||N addition in NE & SW where N||N Whole NE (& SW) quads add 550nm isochromes shift up 1 order Isogyre adds red In NW & SE where subtract Each isochrome loses an order Near isogyre (~100nm) get yellow in NW & SE and blue in NE & SW e ' w sub add add sub (+)crystals: e > w so w faster N (+) OA Figure with plate
(+) OA Figure without plate (+) OA Figure with plate Yellow in NW is (+) (-) OA Figure without plate (same as (+) figure)
(-) OA Figure with plate Blue in NW is (-) Estimating birefringence
1) Find the crystal of interest showing the highest colors (D depends on orientation) 2) Go to color chart thickness = 30 microns (but slides can be thick!) use 30 micron line + color, follow radial line through intersection to margin & read birefringence Suppose you have a mineral with second-order green What about third order yellow? Pleochroism Changes in absorption color in PPL as rotate stage (common in biotite, amphibole) Pleochroic formula: Tourmaline: e = dark green to bluish w = colorless to tan Can determine this as just described by isolating first w and then e E-W and observing the color Biaxial Crystals Orthorhombic, Monoclinic, and Triclinic xls don't have 2 or more identical crystallographic axes The indicatrix is a general ellipsoid with three unequal, mutually perpendicular axes One is the smallest possible n and one the largest Fig 10-1 Bloss, Optical Crystallography, MSA a = smallest n(fastest) b = intermediate n g = largest n(slowest) The principal vibration directions are x, y, and z( x || a, y || b,z || g) By definition a < a' < b < g '< g Biaxial Crystals g If a < b < g then there must be some point between a & g with n = b Because =b in plane, and true b is normal to plane, then the section containing both is acircular section Has all of the properties of a circular section! If look down it: all rays = b no preferred vibration direction polarized incoming light will remain so thus appear isotropic as rotate stage = b a Looking down true b Biaxial Crystals ^ optic axis by definition g
If a < b < g then there must be some point between a & g with n = b OA ^ optic axis by definition = b a Looking down true b Biaxial Crystals ^ optic axis by definition
g If a < b < g then there must be some point between a & g with n = b OA OA ^ optic axis by definition And there must be two! Biaxial Hexagonal and tetragonal are Uniaxial = b a = b Looking down true b Biaxial Crystals Nomenclature:
2 circular sections 2 optic axes Must be in a-g plane = Optic Axial Plane (OAP) Y || b direction ^ OAP= optic normal Fig 10-2 Bloss, Optical Crystallography, MSA Acute angle between OA's = 2V The axisthat bisects acute angle = acute bisectrix = Bxa The axisthat bisects obtuse angle = obtuse bisectrix = Bxo Biaxial Crystals B(+) defined as Z (g) = Bxa
Thus b closer to a than to g OA OA = b a = b Looking down true b Biaxial Crystals B(-) defined as X (a) = Bxa
g B(-) defined as X (a) = Bxa Thus b closer to g than to a = b OA a OA = b Looking down true b Biaxial Interference Figures
Fig Bloss, Optical Crystallography, MSA Bxa figure Result is this pattern of isochromes for biaxial crystals Biaxial Interference Figures
Centered Bxa Figure Fig Bloss, Optical Crystallography, MSA Biaxial Interference Figures
Same figure rotated 45o Optic axes are now E-W Clearly isogyres must swing Fig 10-16B Bloss, Optical Crystallography, MSA Centered Optic Axis Figure Large 2V:
As rotate Centered Optic Axis Figure Large 2V: Bxa Figure with Small 2V: Not much curvature Biaxial Optic Sign B(-) a = Bxa thus b closer to g add subtract add
100 gray 650 blue Fig. 11-1A add subtract add 100 gray 450 yellow Biaxial Optic Sign B(-) a = Bxa thus b closer to g (in stage) add
subtract Centered Bxa 2V = 35o Centered Bxa 2V = 35o With accessory plate Biaxial Optic Sign B(+) g = Bxa thus b closer to a (in stage) sub add
Fig. 11-1A sub add sub Estimating 2V Fig 11-5A Bloss, Optical Crystallography, MSA OAP