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Miller indices/crystal forms/space groups

Miller indices/crystal forms/space groups. Crystal Morphology. How do we keep track of the faces of a crystal? Sylvite a= 6.293 Å Fluorite a = 5.463 Å Pyrite a = 5.418 Å Galena a = 5.936 Å. Crystal Morphology. How do we keep track of the faces of a crystal?

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Miller indices/crystal forms/space groups

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  1. Miller indices/crystal forms/space groups

  2. Crystal Morphology • How do we keep track of the faces of a crystal? • Sylvite a= 6.293 Å • Fluorite a = 5.463 Å • Pyrite a = 5.418 Å • Galena a = 5.936 Å

  3. Crystal Morphology How do we keep track of the faces of a crystal? Remember, face sizes may vary, but angles can't Note: “interfacial angle” = the angle between the faces measured like this

  4. Crystal Morphology How do we keep track of the faces of a crystal? Remember, face sizes may vary, but angles can't Thus it's the orientation & angles that are the best source of our indexing Miller Index is the accepted indexing method It uses the relative intercepts of the face in question with the crystal axes

  5. Crystal Morphology 2-D view looking down c b a Given the following crystal: b a c

  6. b a Crystal Morphology • How reference faces? • a face? • b face? • -a and -b faces? Given the following crystal:

  7. Crystal Morphology b w x y Suppose we get another crystal of the same mineral with 2 other sets of faces: How do we reference them? b a a z

  8. Pick a reference face that intersects both axes Which one? b b Miller Index uses the relative intercepts of the faces with the axes w x x y y a a z

  9. b w x y a z Either x or y. The choice is arbitrary. Just pick one. Suppose we pick x b Which one? x y a

  10. 2 1 1 b invert  1 1 2 1 0 clear of fractions x Miller index of face y using x as the a-b reference face y (2 1 0) a a b c 1 1 unknown face (y)  1 reference face (x) 2 1 MI process is very structured (“cook book”)

  11. 1 1 1 b invert  1 1 1 1 0 clear of fractions (1 1 0) x Miller index of the reference face is always 1 - 1 y a a b c 1 1 unknown face (x)  1 reference face (x) 1 1 What is the Miller Index of the reference face? (2 1 0)

  12. 1 1 1 b invert  1 2 1 2 0 clear of fractions (1 2 0) x Miller index of the reference face is always 1 - 1 y a a b c 2 1 unknown face (x)  1 reference face (y) 1 1 What if we pick y as the reference. What is the MI of x? (1 1 0)

  13. 1 4 3 invert 2 2 2 (1 4 3) clear of fractions Miller index of face XYZ using ABC as the reference face 3-D Miller Indices (an unusually complex example) a b c c 2 2 2 unknown face (XYZ) 1 4 3 reference face (ABC) C Z O A Y X B a b

  14. Miller indices • Always given with 3 numbers • A, b, c axes • Larger the Miller index #, closer to the origin • Plane parallel to an axis, intercept is 0

  15. b w (1 1 0) (2 1 0) a z What are the Miller Indices of face Z?

  16. 1 1 1 invert ¥ ¥ 1 1 0 0 clear of fractions Miller index of face z using x (or any face) as the reference face The Miller Indices of face z using x as the reference a b c ¥ ¥ 1 unknown face (z) 1 1 reference face (x) 1 b w (1 1 0) (2 1 0) (1 0 0) a z

  17. What do you do with similar faces on opposite sides of crystal? b (1 1 0) (2 1 0) (1 0 0) a

  18. b (0 1 0) (1 1 0) (1 1 0) (2 1 0) (2 1 0) (1 0 0) a (1 0 0) (2 1 0) (2 1 0) (1 1 0) (1 1 0) (0 1 0)

  19. Demonstrate MI on cardboard cube model

  20. If you don’t know exact intercept: • h, k, l are generic notation for undefined intercepts

  21. You can index any crystal face

  22. Crystal habit • The external shape of a crystal • Not unique to the mineral • See Fig. 5.2, 5.3, and 5.4

  23. Crystal Form = a set of symmetrically equivalent faces braces indicate a form {210} b (0 1) (1 1) (1 1) (2 1) (2 1) (1 0) a (1 0) (2 1) (2 1) (1 1) (1 1) (0 1)

  24. Form = a set of symmetrically equivalent faces braces indicate a form {210} Multiplicity of a form depends on symmetry

  25. Form = a set of symmetrically equivalent faces braces indicate a form {210} What is multiplicity? pinacoid prism pyramid dipryamid related by a mirror or a 2-fold axis related by n-fold axis or mirrors

  26. Form = a set of symmetrically equivalent faces braces indicate a form {210} • Quartz = 2 forms: • Hexagonal prism (m = 6) • Hexagonal dipyramid (m = 12)

  27. _ 111 111 __ _ 111 111 011 101 _ 110 110 _ _ 011 101 Isometric forms include Cube Octahedron Dodecahedron

  28. Crystal forms • Forms can be open or closed • Cube block demo • Forms on stereonets • Cube faces on stereonet

  29. General form • {hkl} not on, parallel, or perpendicular to any symmetry element • Special form • On, parallel, or perpendicular to any symmetry element • Rectangle block • Find symmetry, plot symmetry, plot special face, general face, determine multiplicity

  30. Space groups • Point symmetry: symmetry about a point • 32 point groups, 6 crystal systems • Combine point symmetry with translation, you have space groups • 230 possible combinations

  31. Symmetry • Translations (Lattices) • A property at the atomic level, not of crystal shapes • Symmetric translations involve repeat distances • The origin is arbitrary • 1-D translations = a row a  a is the repeat vector

  32. Symmetry Translations (Lattices) 2-D translations = a net b a Pick any point Every point that is exactly n repeats from that point is an equipoint to the original

  33. Translations There is a new 2-D symmetry operation when we consider translations The Glide Plane: A combined reflection and translation repeat Step 2: translate Step 1: reflect (a temporary position)

  34. 32 point groups with point symmetry • 230 space groups adding translation to the point groups

  35. 3-D translation • Screw axes: rotation and translation combined

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