Stereographic Projection Want to represent 3-D crystal on 2-D paper Use a Projection A cubic xl like our model Note poles (normals to xl face planes) Fig 6.3 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

Spherical Projection Click to run animation Case Klein animation for Mineral Science, © John Wiley & Sons

Stereographic Projection The outer sphere is a spherical projection Plot points where poles intersect sphere Planes now = points But still 3-D Fig 6.3

Stereographic Projection Gray plane = Equatorial Plane Want to use it as our 2-D representation and project our spherical poles back to it This is a 2-D stereographic projection Fig 6.5 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

Stereographic Projection D and E are spherical D' and E' are stereographic Distance GD' = f(r) as r  90 D’  G as r  0 D’  O Fig 6.6 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

Stereographic Projection We can thus use the angles and calculate the 2-D distances from the center to find the stereographic poles directly Or we can use special graph paper and avoid the calculation Fig 6.5 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

Inclined Planes and Great Circles Great Circle as stereographic projection calculated from angle r Great circles on stereographic projection = locus of all points projected from the intercept of an inclined plane to the equatorial plane (bowl analogy)- structural geology Use your hand for dip and a pencil for the pole of (011) at 45o from vertical

This is the graph paper for avoiding calculating the distance from the center as a function of r each time It is graduated in increments of 20o

Back to Fig. 2.42 (111) (100) (111) (011) (100) all coplanar (= zone) Thus all poles in a zone are on the same great circle!! How do we find the zone axis?? Fig 6.3 of Klein (2002) Manual of Mineral Science, John Wiley & Sons

Gives angles between any two points on a great circle Small circles Gives angles between any two points on a great circle = the angle between 2 coplanar lines!! 20o

The Wulff Net Combines great circles and small circles in 2o increments

Stereographic Projection How to make a stereographic projection of our crystal Use a contact goniometer to measure the interfacial angles (also measures normals: poles) Fig 6.2 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

Plot Cardboard Model Isometric System (p. 93) Crystallographic Axes “The crystal forms of classes of the isometric system are referred to three axes of equal length that make right angles with each other. Because the axes are identical, they are interchangeable, and all are designated by the letter a. When properly oriented, one axis, a1, is horizontal and oriented front to back, a2 is horizontal and right to left, and a3 is vertical.” +a3 +a1 +a2 90

Plot (100) (001) (010) (110) (101) (011):  = top half o = bottom half How plot (111) ? a) Plot (110) & then plot (111) between (110) and (001) (110)  (111) = 36.5o - go in from primitive b) No measure technique: (111) must lie between (110) & (001) (zone add rule) also between (100) & (011) thus intersection of great circles  (111)

The finished product symmetry elements face poles and principal zones Fig 6.8 of Klein (2002) Manual of Mineral Science, John Wiley and Sons symmetry elements face poles and principal zones

Once finished can determine the angles between any 2 faces w/o measuring. What is (100)  (111) ? (54.5o) (111)  (111) ? (70o)

Model #75- How can you use the position of the (111) face on a stereonet to determine: a/b? b/c? a/c?

Twinning Rational symmetrically-related intergrowth Lattices of each orientation have definite crystallographic relation to each other

Twinning Aragonite twin Note zone at twin plane which is common to each part Although aragonite is orthorhombic, the twin looks hexagonal due to the 120o O-C-O angle in the CO3 group Redrawn from Fig 2-69 of Berry, Mason and Dietrich, Mineralogy, Freeman & Co.

Twinning 1) Reflection (twin plane) Twin Operation is the symmetry operation which relates the two (or more) parts (twin mirror, rot. axis) 1) Reflection (twin plane) Examples: gypsum “fish-tail”, models 102, 108 2) Rotation (usually 180o) about an axis common to both (twin axis): normal and parallel twins. Examples: carlsbad twin, model 103 3) Inversion (twin center) The twin element cannot be a symmetry element of the individuals. Twin plane can't be a mirror plane of the crystal Twin Law is a more exact description for a given type (including operation, plane/axis, mineral…)

Contact & Penetration twins Both are simple twins only two parts

Multiple twins (> 2 segments repeated by same law) Cyclic twins - successive planes not parallel Polysynthetic twins Albite Law in plagioclase

Twinning Mechanisms: 1) Growth Growth increment cluster adds w/ twin orientation Epitaxial more stable than random Not all epitaxis  twins Usually simple & penetration synneusis a special case

Twinning Mechanisms: 1) Growth Feldspars: Plagioclase: Triclinic Albite-law-striations a-c a-c b b

Twinning Mechanisms: 1) Growth Feldspars: Plagioclase: Triclinic Albite-law-striations

cyclic twinning in inverted low quartz Mechanisms: 2) Transformation (secondary) SiO2: High T is higher symmetry High Quartz P6222 Low Quartz P3221

Twinning Mechanisms: 2) Transformation (secondary twins) Feldspars: Orthoclase (monoclinic)  microcline (triclinic) a-c a-c Monoclinic (high-T) Triclinic (low-T) b b

Twinning Mechanisms: 2) Transformation (secondary) Feldspars: K-feldspar: large K  lower T of transformation “tartan twins” Interpretation wrt petrology!

Twinning Mechanisms: 3) Deformation (secondary) Results from shear stress greater stress  gliding, and finally rupture Also in feldspars. Looks like transformation, but the difference in interpretation is tremendous

Mechanisms: 3) Deformation (secondary) Results from shear stress. Plagioclase

Mechanisms: 3) Deformation (secondary) Results from shear stress. Calcite

X-ray Crystallography X-ray wavelengths are on the same order of magnitude as atomic spacings. Crystals thus makes excellent diffraction gratings Can use the geometry of the x-ray spots to determine geometry of grating (ie the crystal)

X-ray Crystallography X-ray generation W C a t h o d e C u A n o d e ( - ) ( + ) electrons X - r a y s

X-ray Crystallography X-ray generation Continuous & characteristic spectrum (Fig. 7.2) Continuous from E loss of collisions Characteristic is quantized I l

X-ray Crystallography Destructive and constructive interference of waves Bragg Equation: in phase in phase Y x q q q d

X-ray Crystallography nl=2dsinq n is the “order” As soon as the crystal is rotated, the beam ceases (This is diffraction, not reflection) Only get diffraction at certain angles! Relation between l and d and q Y x q q d

X-ray Crystallography Methods: 1) Single-Crystal: Laue Method Several directions simultaneously fulfill Bragg equations Good for symmetry, but poor for analysis because distorted Fig 7.39 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

X-ray Crystallography Methods: 1) Single-Crystal: Precession Use motors to move crystal & film to satisfy Bragg equations for different planes without distortions Fig 7.40 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

X-ray Crystallography Methods: 2) Powder- Easiest Infinite orientations at once, so only need to vary q Cameras and diffractometers