Biaxial Crystals Orthorhombic, Monoclinic, and Triclinic crystals don't have 2 or more identical crystallographic axes The indicatrix is a triaxial 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 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 a circular section

Biaxial Crystals Perpendicular is by definition the optic axis; Looking down b (  to g- a plane) Biaxial Crystals g If a < b < g then there must be some point between a & g with n = b OA OA Perpendicular is by definition the optic axis; and there must be two!  Biaxial Hexagonal and tetragonal are Uniaxial = b a = b

Biaxial Crystals Looking down true b g Has all of the properties of a circular section! all rays = b no preferred vibration direction polarized incoming light will remain so unpolarized “ “ “ “ thus appear isotropic as rotate stage = b a

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 Acute angle between OA's = 2V The axis that bisects the acute angle = acute bisectrix = Bxa The axis that bisects the obtuse angle = obtuse bisectrix = Bxo

Biaxial Crystals B(+) defined as Z (g) = Bxa Looking down true b Biaxial Crystals g B(+) defined as Z (g) = Bxa described by the 2Vz angle Thus b is closer to a than to g OA OA = b a = b

Biaxial Crystals B(-) defined as X (a) = Bxa Looking down true b Biaxial Crystals g B(-) defined as X (a) = Bxa described by the 2Vx angle Thus b is closer to g than to a = b OA a OA = b

Biaxial Interference Figures Bxa figure (Bxa is vertical on stage) As in uniaxial, condensing lens causes rays to emanate out from O OX ® OS ® OA results in decreasing retardation (color) as g ® b OA ® OT ® OU Increase again, but now because b ® a OX ® OQ ® OP  increase retardation (interference colors) OR is random with a' and g' black Fig 10-14 Bloss, Optical Crystallography, MSA

Biaxial Interference Figures Fig 10-15 Bloss, Optical Crystallography, MSA Bxa figure Result is this pattern of isochromes for biaxial crystals

Biaxial Interference Figures Application of B-F rule to conoscopic view Bxa figure + = bisectrices of optic axis planes Isogyres are locus of all N-S (& E-W) vibration directions Since incoming light vibrates E-W, there will be no N-S component ® extinct Fig 10-16 Bloss, Optical Crystallography, MSA

black

Biaxial Interference Figures Centered Bxa Figure Fig 10-16 Bloss, Optical Crystallography, MSA

Biaxial Interference Figures Same figure rotated 45o Optic axes are now E-W 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 Makes use of Bxa tricky

Biaxial Optic Sign B(-) a = Bxa thus b closer to g add subtract add 100 gray + 550 ® 650 blue Fig. 11-1A add subtract add 100 gray - 550 ® 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

Always use Optic Axis Figure & curvature of isogyre to determine optic sign How find a crystal for this? Blue in NW is (-) still works

Estimating 2V Fig 11-5A Bloss, Optical Crystallography, MSA OAP

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

Tourmaline

Sign of Elongation If g || elongation will always add ® length slow If a || elongation will always subtract ® length fast g a N U(-) will also ® length fast U(+) will also ® length slow

Sign of Elongation If b || elongation Sometimes will add ® length slow Sometimes will subtract ® length fast g b b a N

Sign of Elongation If b || elongation Sometimes will add ® length slow Sometimes will subtract ® length fast g b b a N Platy minerals may appear elongated too Can still use sign of elongation on edges

Sign of Elongation

Cleavage Tendency of minerals to break parallel to atomic planes Octahedral Cubic Dodecahedral (12) Prismatic Basal Rhombohedral

Extinction (symmetrical)

Extinction (symmetrical and parallel)

Extinction

Extinction (inclined)

Extinction Angles Extinction angle is taken as the acute angle between a crystallographic direction (as seen by a crystal edge or cleavage direction) and a vibration direction. The extinctin angle is generally indicated e.g. Z^c where Z is the vibration direction and c is the crystallographic direction. Extinction angles are taken as positive or negative with respect to a slow or fast vibration direction One source considers that a counter-clockwise rotation from a crystallographic direction to the slow vibration direction is a positive extinction angle, and a clockwise rotation is negative.

Extinction Angle Trace of a {110} cleavage Fast vibration direction Slow vibration direction -60º +20º Trace of a {110} cleavage