1. Crystal Systems
GLY 4200
Fall, 2012

2. William Hallowes Miller
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
Source:

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 = nm
Given axis intercepts from X-ray data
x’ = nm, y’ = nm, z’ = nm
Calculate the intercepts in terms of the unit cell magnitude

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

19. Clear Fractions Multiply by 6 to clear fractions
2 x 6 =12, 0.5 x 6 = 3, 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│ │1
(2x1 - 0x0, 0x2-1x1, 1x0-2x2) =
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│ │0
(0x0-2x1, 1x1-0x2,2x2-1x0) =
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
Source:

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
Source:

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
Source:

45. Vanadinite Forms hexagonal prismatic crystals
Images:
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
Photo:

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

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.

51. Dodecahedrons A closed 12-faced form
Dodecahedrons can be formed by cutting off the edges of a cube
Form symbol for a dodecahedron is isometric{110} 
Garnets often display this form

52. Tetrahedron
The tetrahedron occurs in the class bar4 3m and has the form symbol {111}(the form shown in the drawing) or {1 bar11}
It is a four faced form that results form three bar4 axes and four 3-fold axes
Photo:
Tetrahedrite, a copper sulfide mineral

53. Forms Related to the Octahedron
Trapezohderon - An isometric trapezohedron is a 12-faced closed form with the general form symbol {hhl}
The diploid is the general form {hkl} for the diploidal class (2/m bar3)

54. Forms Related to the Octahedron
Hexoctahedron
Trigonal trisoctahedron

55. Pyritohedron
The pyritohedron is a 12-faced form that occurs in the crystal class 2/m bar3
The possible forms are {h0l} or {0kl} and each of the faces that make up the form have 5 sides
Photo:

56. Tetrahexahedron
A 24-faced closed form with a general form symbol of {0hl}
It is clearly related to the cube

57. Scalenohedron A scalenohedron is a closed form with 8 or 12 faces
In ideally developed faces each of the faces is a scalene triangle
In the model, note the presence of the 3-fold rotoinversion axis perpendicular to the 3 2-fold axes

58. Trapezohedron
Trapezohedron are closed 6, 8, or 12 faced forms, with 3, 4, or 6 upper faces offset from 3, 4, or 6 lower faces
The trapezohedron results from 3-, 4-, or 6-fold axes combined with a perpendicular 2-fold axis
Bottom - Grossular garnet from the Kola Peninsula (size is 17 mm)

59. Rhombohedron
A rhombohedron is 6-faced closed form wherein 3 faces on top are offset by 3 identical upside down faces on the bottom, as a result of a 3-fold rotoinversion axis
Rhombohedrons can also result from a 3-fold axis with perpendicular 2-fold axes
Rhombohedrons only occur in the crystal classes bar3 2/m , 32, and bar3 .

60. Application to the Core
Fig. 1. (left) An illustrated cutaway of the Earth’s interior reveals seismic anisotropy of the inner core. (middle) Representative elastic anisotropies of single-crystal hexagonal close packed (hcp; red dashed line) and body-centered cubic (bcc; red dash-dotted line) iron from theory [Belonoshko et al., 2008], prefer-oriented hcp iron from static experiments at 112 gigapascals (green diamond) [Antonangeli et al., 2004], and seismic observations in the inner core (blue lines) are shown for comparison. Elastic anisotropy could be explained if the bcc iron-based alloy in the inner core had preferable orientation along Earth’s rotation axis (similar to that of Atomium, right, a 102-meter-tall monument built for the 1958 World’s Fair in Brussels, Belgium, which forms the shape of an iron crystal magnified 165 billion times) or a dominant amount of prefer-oriented hcp iron.
From EOS, v.90, #3, 1/20/09