1. 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 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

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

12. 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

13. 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

14. 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)

15. 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

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

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

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

19. 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 of Berry, Mason and Dietrich, Mineralogy, Freeman & Co.

20. 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…)

21. Contact & Penetration twins
Both are simple twins only two parts

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

23. 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

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

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

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

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

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

29. 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

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

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

32. 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)

33. 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

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

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

36. 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

37. 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

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

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