1. Crystal Morphology Remember: Space groups for atom symmetry
Point groups for crystal face symmetry
Crystal Faces = limiting surfaces of growth
Depends in part on shape of building units & physical cond. (T, P, matrix, nature & flow direction of solutions, etc.)

2. Crystal Morphology Observation:
The frequency with which a given face in a crystal is observed is proportional to the density of lattice nodes along that plane

3. Crystal Morphology Observation:
The frequency with which a given face in a crystal is observed is proportional to the density of lattice nodes along that plane

4. Crystal Morphology
Because faces have direct relationship to the internal structure, they must have a direct and consistent angular relationship to each other

5. Nicholas Steno (1669): Law of Constancy of Interfacial Angles
Crystal Morphology
Nicholas Steno (1669): Law of Constancy of Interfacial Angles
Quartz

6. Diff planes have diff atomic environments
Crystal Morphology
Diff planes have diff atomic environments
A note on directional properties: H, n, thermal exp...

7. Crystal Morphology
Crystal symmetry conforms to 32 point groups  32 crystal classes in 6 crystal systems
Crystal faces act just as our homework: symmetry about the center of the crystal so the point groups and the crystal classes are the same

8. Crystal Morphology
Crystal Axes: generally taken as parallel to the edges (intersections) of prominent crystal faces a b c

9. Crystal Morphology
Crystal Axes: generally taken as parallel to the edges (intersections) of prominent crystal faces
The more faces the better  prism faces & quartz c-axis, halite cube, etc.
We must also keep symmetry in mind: c = 6-fold in hexagonal
With x-ray crystallography we can determine the internal structure and the unit cell directly and accurately
The crystallographic axes determined by XRD and by the face method nearly always coincide
This is not coincidence!!

10. How do we keep track of the faces of a crystal?
Crystal Morphology
How do we keep track of the faces of a crystal?

11. 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

12. 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

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

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

15. 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

16. 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

17. 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

18. 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)

19. 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)

20. 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)

21. Which choice is correct?
1) x = (1 1 0)
y = (2 1 0)
2) x = (1 2 0)
y = (1 1 0) b a x y
The choice is arbitrary
What is the difference?

22. What is the difference? axial ratio = a/b = 0.80y
unit cell shape if
y = (1 1 0)
unit cell shape if
x = (1 1 0) b b a a x b y a
axial ratio = a/b = 0.80
axial ratio = a/b = 1.60

23. The technique above requires that we graph each face
A simpler (?) way is to use trigonometry
Measure the interfacial angles b a x y b a w x y z
148o ? ?
interfacial angles
141o

24. The technique above requires that we graph each face
A simpler (?) way is to use trigonometry b a x y b a w x y z
tan 39 = a/b = 0.801
tan 58 = a/b = 1.600
58o
148o
39o
141o

25. What are the Miller Indices of all the faces if we choose x as the reference?
Face Z? b a w
(1 1 0)
(2 1 0) z

26. 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

27. b
Can you index the rest?
(1 1 0)
(2 1 0)
(1 0 0) a

28. 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)

29. 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

30. Demonstrate MI on cardboard cube model

31. We can get the a:b:c axial ratios from the chosen (111) face
We can also determine the true unit cell by XRD and of course determine the a:b:c axial ratios from it
If the unit face is correctly selected, the ratios should be the same
If not, will be off by some multiple - i.e. picked (211) and called it (111)
Best to change it
Mineralogy texts listed axial ratios long before XRD
We had to change some after XRD developed

32. 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)

33. Form = a set of symmetrically equivalent faces
braces indicate a form {210}
Multiplicity of a form depends on symmetry
{100} in monoclinic, orthorhombic, tetragonal, isometric

34. Form = a set of symmetrically equivalent faces
braces indicate a form {210}
F in your text (p )
pinacoid prism pyramid dipryamid
related by a mirror or a 2-fold axis
related by n-fold axis or mirrors

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

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

37. Octahedron to Cube to Dodecahedron
Click on image to run animation

38. All three combined:

39. Zone Any group of faces || a common axis
Use of h k l as variables for a, b, c intercepts
(h k 0) = [001]
If the MI’s of 2 non-parallel faces are added, the result = MI of a face between them & in the same zone

40. BUT doesn't say which face
(100)
Which??
(010)
(110)?

41. BUT doesn't say which face
(100)
(010)
(210)
(110)
(100)
Which??
(010)
(110)?
(100)
(010)
(110)
(120)
Either is OK