1 / 60

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.

caraf
Download Presentation

Crystal Systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

More Related