1. Miller indices/crystal forms/space groups

2. Crystal Morphology How do we keep track of the faces of a crystal?
Sylvite a= Å
Fluorite a = Å
Pyrite a = Å
Galena a = 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 Miller Index is the accepted indexing method
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. Given the following crystal:
Crystal Morphology
Given the following crystal:
2-D view
looking down c b a b a c

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

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

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

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

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

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

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

13. Miller index of face XYZ using ABC as the reference face
3-D Miller Indices (an unusually complex example)
a b c c
unknown face (XYZ) 2 2 2
reference face (ABC) 1 4 3 C
invert 1 2 4 3 Z
clear of fractions (1 3) 4
Miller index of face XYZ using ABC as the reference face O A X Y 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. What are the Miller Indices of face Z?b a w
(1 1 0)
(2 1 0) z

16. 1 The Miller Indices of face z using x as the reference a b c 1 ¥ ¥ 1
unknown face (z) 1
reference face (x) 1 1 1
invert 1 b w
(1 1 0)
clear of fractions 1
(2 1 0)
Miller index of
face z using x (or any face) as the reference face
(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. Isometric forms include
Cube Octahedron
Dodecahedron
111 _ __
110
101
011 _

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

29. General form Special form Rectangle block
{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
4/m, 2/m, 2/m

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) 1-D translations = a row
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