Mineral = “… a highly ordered atomic arrangement.”
- external ordering – crystal faces, only some minerals
- internal ordering – atomic arrangements, all minerals

 
Crystal = “a homogeneous solid with long-range, three dimensional order”
• All minerals are crystals, not all crystals are minerals
• Modifiers:
 - euhedral = well developed faces
 - subhedral = imperfect faces
 - anhedral = without faces

 
Crystalline = any solid with an ordered arrangement of atoms

Amorphous = solid lacking ordered arrangements.
• Examples: Mineraloids, glass
• Processes
- Metamict: destruction of structure through radiation
- Fulgarite: melting from lightning strike
- Clinker: melting from burning coal beds
 
Symmetry:
• result of ordered arrangements
• reflected in crystal faces
• reflected in internal structure
• Constant for any individual mineral!!!
 

What is symmetry?
 
Symmetry:
• correspondence in size, shape and position of parts on opposite sides of a dividing line or median plane or about a center or axis
• a rigid motion of a geometric figure that determines a one-to-one mapping onto itself, i.e. a symmetry operation
 

Two types of symmetry operations:
 - translational (through a volume)
 - repetition (around a point)
 
Translation in a plane: Translation in a plane
• Involves repetition of a single point
 - one and two directions
• Resulting pattern = plane lattice
 - extends to infinity in the plane
• Each spot = lattice node

 
Results of symmetry operations:
• Only five different plane lattices produced by translation in 2 dimensions
• Only four fundamentally different shapes, called- Unit meshes
• Primitive (p) – nothing at center of mesh
• Centered (c)- node at center of mesh
• Axes parallel edges of unit mesh
 - labeled a and b
 - angle between them is g

 
Translation in Three dimensions:
• Completely analogous to 2D
• Arrangement of nodes called space lattice
• Volume outlined is Unit Cell
• Edges of unit cell are crystal axes
 - axes names are a, b, c (front and back)
 - intersect at a point called the origin
 - distances along edges also a, b, and c
 - angles are a, b, g
 - very specific arrangement of axes
 
Bravais Lattices and Crystal systems

• 5 plane lattices repeated in third dimension to make 14 Bravais lattices
 
Crystal Systems:
• Shape of unit cell divides 14 Bravais lattices into 6 crystal systems
 - triclinic
 - monoclinic
 - orthorhombic
 - hexagonal
 - tetragonal
 - isometric
• Each system has identically shaped unit cell
 
Systems may have variations depending on locations of “extra” lattice points:

• Primitive (P) – lattice points only at corners
• Body centered (I) – additional lattice point at center
• C face-centered (C) – additional lattice points at two opposite sides
• Face-centered (F) – additional lattice points at each face

 
Arrangements of crystal axes:

Triclinic
• Translate oblique plane lattice distance c not perpendicular to a or b
 • a¹ b¹ c and a¹ b¹ g
 • No convention to arrangement
 • Generally
- c is parallel to prominent elongation
- b is down and to right
- a is down and to front

 
Monoclinic
• Translating rectangular plane lattices not perpendicular to other two axes
• a ¹ b  ¹ c, any axis may be longest and/or shortest
• a = g ¹ b > 90º

Orthorhombic
• Translating rectangle at right angles to other axes
• a  ¹ b  ¹ c, commonly c < a < b, but other conventions possible
• a = g = b = 90º
 
Hexagonal
• Translating hexagonal space lattice perpendicular to a and b.
• a₁ = a₂ = a₃  ¹ c, c < or > a
• angles between a axes 120º
 

Tetragonal
• Translating square plane lattice
• a1 = a2  ¹ c, c < or > a
• a = g = b = 90º
 
Isometric
 • Translating square plane lattice
• a1 = a2 = a3
• a = g = b = 90º
 
Point Symmetry

• Differs from translation symmetry:
- repetition of “motif” around a point
- motif includes crystal faces or group of atoms
- point repeated around includes center of crystal or origin of unit cell

• Four Point symmetry operations
- Reflection
- Rotation
- Inversion
- Rotoinversion (compound operation)
 
Reflection
• created by “mirror” plane
• generates mirror image on opposite sides of plane
• creates one new motif
• handedness changes
• notation = m
 
Rotation
• repeated motif by rotation around an axis
• motif may be crystal face, lattice, atom etc.
• Five types of rotation axes, depends on number of repeats:
 - one fold = 360º rotation
 - two fold = 180º rotation
 - three fold = 120º rotation
 - four fold = 90º rotation
 - six fold = 60º rotation
• notation = A_n, n = 1,2,3,4,6 and reflects numbers of rotation
 
Inversion
• symmetry through a point: center of symmetry
• a line drawn from a point through the center will hit identical point equal distance on opposite side of center
• notation = i

 
Rotoinversion
• Combines rotation and inversion
• Rotation may be 1, 2, 3, 4, or 6.
• notation = A_n, where n = 1, 2, 3, 4, 6
• Only A₄ is unique operation, other 4 can be duplicated with other operations:
 - A₁ = i
 - A₂ = m
 - A₃ = A₃ + i
 - A₄ is unique
 - A₆ = A₃ + m
 
Symmetry notation

• Consider form with the following symmetry:
 - center of symmetry
 - three two fold rotation axes
 - three mirror planes
- i, 3A₂, 3m

 
• Could also written as 2/m2/m2/m
 - implies that there are three 2 fold rotation axes perpendicular to a mirror
 - this symmetry implies there is a center of symmetry

• Significance is that symmetry elements can be combined
 - limited number of ways
 
32 Point Groups

• First consider combinations of plane symmetry
- In two dimensions, only have mirrors and rotations
- Inversions only in three dimensions
- total of 10 possible combinations

• Add in third dimension and inversion
 - total of 32 unique combinations
 - point groups = crystal classes
 
• The 32 crystal classes can be grouped into the six different crystal systems
- crystal systems grouped according to unit cell dimensions
- 32 point groups fall into systems on basis of common symmetry elements
- e.g. TABLE 2.2

• Note: there is a strict relationship between symmetry elements and crystallographic axes
 
Steno’s Law

• Identifying a mineral’s crystal class is a valuable tool to ID the mineral
• Problem is that crystal are rarely euhedral
• Angles between faces (even poorly formed) are always the same
Steno’s Law: Angle between equivalent faces on crystal of same mineral are always the same

 
Identifying crystal class:

• determine if crystal has center of symmetry
• Identify all mirror planes
• Identify rotation axes, and number of rotations
• Compile all symmetry