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Crystal Systems

Crystal Systems. GLY 4200 Fall, 2012. William Hallowes Miller. 1801 -1880 British Mineralogist and Crystallographer Published Crystallography in 1838 In 1839, wrote a paper, “treatise on Crystallography” in which he introduced the concept now known as the Miller Indices. Notation.

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Crystal Systems

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  1. Crystal Systems GLY 4200 Fall, 2012

  2. William Hallowes Miller • 1801 -1880 • British Mineralogist and Crystallographer • Published Crystallography in 1838 • In 1839, wrote a paper, “treatise on Crystallography” in which he introduced the concept now known as the Miller Indices

  3. Notation • Lattice points are not enclosed – 100 • Lines, such as axes directions, are shown in square brackets [100] is the a axis • Direction from the origin through 102 is [102]

  4. Miller Index • The points of intersection of a plane with the lattice axes are located • The reciprocals of these values are taken to obtain the Miller indices • The planes are then written in the form (h k l) where h = 1/a, k = 1/b and l = 1/c • Miller Indices are always enclosed in ( )

  5. Plane Intercepting One Axis

  6. Reduction of Indices

  7. Planes Parallel to One Axis

  8. Isometric System • All intercepts are at distance a • Therefore (1/1, 1/1, 1/1,) = (1 1 1)

  9. Isometric (111) • This plane represents a layer of close packing spheres in the conventional unit cell

  10. Faces of a Hexahedron • Miller Indices of cube faces

  11. Faces of an Octahedron • Four of the eight faces of the octahedron

  12. Faces of a Dodecahedron • Six of the twelve dodecaheral faces

  13. Octahedron to Cube to Dodecahedron • Animation shows the conversion of one form to another

  14. Negative Intercept • Intercepts may be along a negative axis • Symbol is a bar over the number, and is read “bar 1 0 2”

  15. Miller Index from Intercepts • Let a’, b’, and c’ be the intercepts of a plane in terms of the a, b, and c vector magnitudes • Take the inverse of each intercept, then clear any fractions, and place in (hkl) format

  16. Example • a’ = 3, b’ = 2, c’ = 4 • 1/3, 1/2, 1/4 • Clear fractions by multiplication by twelve • 4, 6, 3 • Convert to (hkl) – (463)

  17. Miller Index from X-ray Data • Given Halite, a = 0.5640 nm • Given axis intercepts from X-ray data • x’ = 0.2819 nm, y’ = 1.128 nm, z’ = 0.8463 nm • Calculate the intercepts in terms of the unit cell magnitude

  18. Unit Cell Magnitudes • a’ = 0.2819/0.5640, b’ = 1.128/0.5640, c’ = 0.8463/0.5640 • a’ = 0.4998, b’ = 2.000, c’ = 1.501 • Invert: 1/0.4998, 1/2.000, 1/1.501 = 2,1/2, 2/3

  19. Clear Fractions • Multiply by 6 to clear fractions • 2 x 6 =12, 0.5 x 6 = 3, 0.6667 x 6 = 4 • (12, 3, 4) • Note that commas are used to separate double digit indices; otherwise, commas are not used

  20. Law of Huay • Crystal faces make simple rational intercepts on crystal axes

  21. Law of Bravais • Common crystal faces are parallel to lattice planes that have high lattice node density

  22. Zone Axis • The intersection edge of any two non-parallel planes may be calculated from their respective Miller Indices • Crystallographic direction through the center of a crystal which is parallel to the intersection edges of the crystal faces defining the crystal zone • This is equivalent to a vector cross-product • Like vector cross-products, the order of the planes in the computation will change the result • However, since we are only interested in the direction of the line, this does not matter

  23. Generalized Zone Axis Calculation • Calculate zone axis of (hkl), (pqr)

  24. Zone Axis Calculation • Given planes (120) , (201) • 1│2 0 1 2│0 • 2│0 1 2 0│1 • (2x1 - 0x0, 0x2-1x1, 1x0-2x2) = 2 -1 -4 • The symbol for a zone axis is given as [uvw] • So,

  25. Common Mistake • Zero x Anything is zero, not “Anything’ • Every year at least one student makes this mistake!

  26. Zone Axis Calculation 2 • Given planes (201) , (120) • 2│0 1 2 0│1 • 1│2 0 1 2│0 • (0x0-2x1, 1x1-0x2,2x2-1x0) = -2 1 4 • Zone axis is • This is simply the same direction, in the opposite sense

  27. Zone Axis Diagram • [001] is the zone axis (100), (110), (010) and related faces

  28. Form • Classes of planes in a crystal which are symmetrically equivalent • Example the form {100} for a hexahedron is equivalent to the faces (100), (010), (001), , ,

  29. Isometric [111] • {111} is equivalent to (111), , , , , , ,

  30. Closed Form – Isometric {100} • Isometric form {100} encloses space, so it is a closed form

  31. Closed Form – Isometric {111} • Isometric form {111} encloses space, so it is a closed form

  32. Open Forms –Tetragonal {100} and {001} • Showing the open forms {100} and {001}

  33. Pedion • Open form consisting of a single face

  34. Pinacoid • Open form consisting of two parallel planes • Platy specimen of wulfenite – the faces of the plates are a pinacoid

  35. Benitoite • The mineral benitoite has a set of two triangular faces which form a basal pinacoid

  36. Dihedron • Pair of intersecting faces related by mirror plane or twofold symmetry axis • Sphenoids - Pair of intersecting faces related by two-fold symmetry axis • Dome - Pair of intersecting faces related by mirror plane

  37. Dome • Open form consisting of two intersecting planes, related by mirror symmetry • Very large gem golden topaz crystal is from Brazil and measures about 45 cm in height • Large face on right is part of a dome

  38. Sphenoid • Open form consisting of two intersecting planes, related by a two-fold rotation axis • (Lower) Dark shaded triangular faces on the model shown here belong  to a sphenoid • Pairs of similar vertical faces that cut the edges of the drawing are pinacoids • Top and bottom faces are two different pedions

  39. Pyramids • A group of faces intersecting at a symmetry axis • All pyramidal forms are open

  40. Apophyllite Pyramid • Pyramid measures 4.45 centimeters tall by 5.1 centimeters wide at its base

  41. Uvite • Three-sided pyramid of the mineral uvite, a type of tourmaline

  42. Prisms • A prism is a set of faces that run parallel to an axes in the crystal • There can be three, four, six, eight or even twelve faces • All prismatic forms are open

  43. Diprismatic Forms • Upper – Trigonal prism • Lower – Ditrigonal prism – note that the vertical axis is an A3, not an A6

  44. Citrine Quartz • The six vertical planes are a prismatic form • This is a rare doubly terminated crystal of citrine, a variety of quartz

  45. Vanadinite • Forms hexagonal prismatic crystals

  46. Galena • Galena is isometric, and often forms cubic to rectangular crystals • Since all faces of the form {100} are equivalent, this is a closed form

  47. Fluorite • Image shows the isometric {111} form combined with isometric {100} • Either of these would be closed forms if uncombined

  48. Dipyramids • Two pyramids joined base to base along a mirror plane • All are closed forms

  49. Hanksite • Tetragonal dipyramid

  50. Disphenoid • A solid with four congruent triangle faces, like a distorted tetrahedron • Midpoints of edges are twofold symmetry axes • In the tetragonal disphenoid, the faces are isosceles triangles and a fourfold inversion axis joins the midpoints of the bases of the isosceles triangles.

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