1. Lecture 12 Crystallography
Internal Order and 2-D Symmetry
Plane Lattices
Planar Point Groups
Plane Groups

2. Internal Order and Symmetry
Repeated and symmetrical arrangement (ordering) of atoms and ionic complexes in minerals creates a 3-dimensional lattice array
Arrays are generated by translation of a unit cell – smallest unit of lattice points that define the basic ordering
Spacing of lattice points (atoms) are typically measured in Angstroms (= m) About the scale of atomic and ionic radii

3. Two-Dimensional Plane Lattice
Generating an 2D Lattice Array (Plane Lattice) involves translation of a motif in two directions; possible directions not unique
Translation in two directions: x and y axes
Angle between x and y axes is called g gamma
Translation distance: a along x and b along y
Replacing motifs with points (or nodes) creates a plane lattice
Unit Cell defined by a choice
of lengths and directions.

4. 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

5. 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

6. Translations (Lattices) 2-D translations = a net
Symmetry
Translations (Lattices)
2-D translations = a net

7. Translations (Lattices) 2-D translations = a net
Symmetry
Translations (Lattices)
2-D translations = a net
A 2-D Unit Cell
Unit Cell: the basic repeat unit that, by translation only, generates the entire pattern

8. Translations (Lattices) 2-D translations = a net
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

9. There are 5 Types of Plane Lattices
Memorize these names and rules
Preferred

10. Symmetry Elements of Planar Motifs: Planar Point Groups
Point groups have labels that are similar to Hermann Mauguin symbols.
For example: 2mm shown has the a axis with a two fold rotational axis, and b and c have mirrors
A point group is a group of geometric symmetries that keep at least one point fixed.
10 Possible symmetry combinations; called Planar Point Groups
Limitations of rotational symmetries: (1,2,3,4, & 6)
dark lines added “found mirrors”

11. Translations
The lattice and point group symmetries interrelate, because both are properties of the overall symmetry pattern

12. Translations
The lattice and point group symmetry interrelate, because both are properties of the overall symmetry pattern
Choose:
Smallest
Most orthogonal
Most in line with symmetry
2 Nodes per Lattice Vector
Most Primitive (non-centered)
Good unit cell choice.

13. Defining a 2-D Unit Cell Rules that help us Choose the: Smallest
Most orthogonal
Most in line with symmetry
Use 2 Nodes per Lattice Vector
Pick the Most Primitive (non-centered)

14. total 17 point groups

15. Translations
There is a new 2-D symmetry operation when we consider translations
The Glide Line, g:
A combined reflection
and translation
repeat
Step 2: translate
Step 1: reflect
(a temporary position)

16. There are 5 unique 2-D plane lattices.
There are also 17 2-D Plane Groups that combine translations with compatible symmetry operations. The bottom row are examples of Plane Groups that correspond to each lattice type
Note: p refers to a primitive cell, as opposed to c, a 2-end (opposite ends) centered cell. More on this in 3-D

17. 17 Plane Groups
10 H-M Point Groups and 5 Lattices combine to form 17 Plane Groups.

18. Lecture 13 3-D Crystallography
3-D Internal Order & Symmetry Space (Bravais) Lattices Space Groups
So far we examined the five 2-D plane lattices and combined them with the 10 planar point groups to generate the 17 2-D plane (space) groups. Next we study the 14 Bravais 3-D lattices and combine them with the 32 3-D point groups to generate D space groups.

19. 3-D Translations and Lattices
Different ways to combine 3 axes
Translations compatible with 32 3-D point groups
(~ crystal classes)
32 Point Groups fall into 6 systems

20. 3-D Translations and Lattices+b g b a
Axial convention:
“right-hand rule”
Different ways to combine 3 axes
Translations compatible with 32 3-D point groups (~ crystal classes)
32 Point Groups fall into 6 Crystal Systems

21. Unit Cell Types in 14 Bravais Lattices
P – Primitive; nodes at corners only
C – Side-centered; nodes at corners and in center of one set of faces
F – Face-centered; nodes at corners and in center of all faces
I – Body-centered; nodes at corners and in center of cell

22. On combining 7 Crystal Classes with 4 possible unit cell types we get 14 Bravais Lattices

23. P P I = C P C F I a b Triclinic ¹ b ¹ g ¹ c a b c Monoclinic a = g
= 90 o
¹ b
I = C c a P
Orthorhombic
a = b = g
= 90 o C F I
There are also several non-primitive lattice choices
P = Primitive
C = C-face Centered
F = all-Face centered
I = body-centered
One might expect that each choice is possible for every lattice type, but not so. Why not ?

24. P I P F I (body-centered) a c Tetragonal a = b = g = 90 = a ¹ a1 c P
Tetragonal
a = b = g
= 90 o
= a 2 I a 1 3 P
Isometric
a = b = g
= 90 o
= a 2 F
I (body-centered)
There are also several non-primitive lattice choices
P = Primitive
C = C-face Centered
F = all-Face centered
I = body-centered
One might expect that each choice is possible for every lattice type, but not so. Why not ?

25. Crystal Axes Conventions
Triclinic:
No symmetry constraints.
No reason to choose C (white) when can choose simpler P (blue)
Do so by convention, so that all mineralogists do the same

26. Crystal Axes Conventions+b g b a
Axial convention:
“right-hand rule”

27. System Conventions

28. System Conventions

29. System Conventions

30. System Conventions

31. 3-D Space Groups
As in the 17 2-D Plane Groups, the 3-D point group symmetries can be combined with translations to create the D Space Groups
Also as in 2-D there are some new symmetry elements that combine translation with other operations
Glides: Reflection + translation
Screw Axes: Rotation + translation

32. A point group is a group of geometric symmetries that keep at least one point fixed.
A space group is some combination of the translational symmetry of a unit cell including lattice centering, the point group symmetry operations of reflection, rotation and rotoinversion, and the screw axis and glide plane symmetry operations. The combination of all these symmetry operations results in a total of 230 unique space groups describing all possible crystal symmetries.

33. 230 Space Groups
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Hexagonal
Isometric
Notation indicates lattice type (P,I,F,C) and Hermann-Mauguin notation for basic symmetry operations (rotation and mirrors)
Screw Axis notation as previously noted
Glide Plane notation indicates the direction of glide – a, b, c, n (diagonal) or d (diamond)