1. Crystal Chem  Crystallography
Chemistry behind minerals and how they are assembled
Bonding properties and ideas governing how atoms go together
Mineral assembly – precipitation/ crystallization and defects from that
Now we will start to look at how to look at, and work with, the repeatable structures which define minerals.
This describes how the mineral is assembled on a larger scale

2. Symmetry

3. Symmetry Introduction
Symmetry defines the order resulting from how atoms are arranged and oriented in a crystal
Study the 2-D and 3-D order of minerals
Do this by defining symmetry operators (there are 13 total)  actions which result in no change to the order of atoms in the crystal structure
Combining different operators gives point groups – which are geometrically unique units.
Every crystal falls into some point group, which are segregated into 6 major crystal systems

4. 2-D Symmetry Operators Mirror Planes (m) – reflection along a plane
A line denotes mirror planes

5. 2-D Symmetry Operators
Rotation Axes (1, 2, 3, 4, or 6) – rotation of 360, 180, 120, 90, or 60º around a rotation axis yields no change in orientation/arrangement
2-fold
3-fold
4-fold
6-fold

6. 2-D Point groups
All possible combinations of the 5 symmetry operators: m, 2, 3, 4, 6, then combinations of the rotational operators and a mirror yield 2mm, 3m, 4mm, 6mm
Mathematical maximum of 10 combinations
4mm

7. 3-D Symmetry Operators
Mirror Planes (m) – reflection along any plane in 3-D space

8. 3-D Symmetry Operators
Rotation Axes (1, 2, 3, 4, or 6 a.k.a. A1, A2, A3, A4, A6) – rotation of 360, 180, 120, 90, or 60º around a rotation axis through any angle yields no change in orientation/arrangement

9. 3-D Symmetry Operators
Inversion (i) – symmetry with respect to a point, called an inversion center 1 1

10. 3-D Symmetry Operators
Rotoinversion (1, 2, 3, 4, 6 a.k.a. A1, A2, A3, A4, A6) – combination of rotation and inversion. Called bar-1, bar-2, etc.
1,2,6 equivalent to other functions

11. New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )

12. 3-D Symmetry New Symmetry Elements 4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4

13. 3-D Symmetry New Symmetry Elements 4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
2: Invert

14. 3-D Symmetry New Symmetry Elements 4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
2: Invert

15. 3-D Symmetry New Symmetry Elements 4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4

16. 3-D Symmetry New Symmetry Elements 4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
4: Invert

17. 3-D Symmetry New Symmetry Elements 4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
4: Invert

18. 3-D Symmetry New Symmetry Elements 4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
5: Rotate 360/4

19. 3-D Symmetry New Symmetry Elements 4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
5: Rotate 360/4
6: Invert

20. 3-D Symmetry New Symmetry Elements 4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
This is also a unique operation

21. 3-D Symmetry New Symmetry Elements 4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
A more fundamental representative of the pattern

22. 3-D Symmetry New Symmetry Elements 4. Rotoinversion This is unique 3 5
c. 3-fold rotoinversion ( 3 )
This is unique 3 5 1 4 2 6

23. 3-D Symmetry Operators
Mirror planes ┴ rotation axes (x/m) – The combination of mirror planes and rotation axes that result in unique transformations is represented as 2/m, 4/m, and 6/m

24. 3-D Symmetry 3-D symmetry element combinations
a. Rotation axis parallel to a mirror
Same as 2-D
2 || m = 2mm
3 || m = 3m, also 4mm, 6mm
b. Rotation axis  mirror
2  m = 2/m
3  m = 3/m, also 4/m, 6/m
c. Most other rotations + m are impossible

25. Point Groups
Combinations of operators are often identical to other operators or combinations – there are 13 standard, unique operators
I, m, 1, 2, 3, 4, 6, 3, 4, 6, 2/m, 4/m, 6/m
These combine to form 32 unique combinations, called point groups
Point groups are subdivided into 6 crystal systems

26. 3-D Symmetry The 32 3-D Point Groups Regrouped by Crystal System
(more later when we consider translations)
Table 5.3 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

27. Hexagonal class
Rhombohedral
form
Hexagonal
form

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

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

31. Crystal Morphology
Growth of crystal is affected by the conditions and matrix from which they grow. That one face grows quicker than another is generally determined by differences in atomic density along a crystal face
Note that the internal order of the atoms can be the same but the crystal habit can be different!

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

33. Miller Indices
Scan in figure – miller indices