Post on 28-Jun-2020
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
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
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
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
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
3-D Symmetry Operators
• Inversion (i) – symmetry with respect to a
point, called an inversion center
11
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
3-D Symmetry
New Symmetry
Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
2: Invert
3-D Symmetry
New Symmetry
Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
2: Invert
3-D Symmetry
New Symmetry
Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
4: Invert
3-D Symmetry
New Symmetry
Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
4: Invert
3-D Symmetry
New Symmetry
Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
5: Rotate 360/4
6: Invert
3-D Symmetry
New Symmetry
Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
This is also a unique operation
3-D Symmetry
New Symmetry
Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
A more fundamental
representative of the pattern
3-D Symmetry
New Symmetry
Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
This is unique
1
6
5
2
3
4
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
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
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
3-D SymmetryThe 32 3-D Point Groups
Regrouped by Crystal System
(more later when we consider translations)
Crystal System No Center Center
Triclinic 1 1
Monoclinic 2, 2 (= m) 2/m
Orthorhombic 222, 2mm 2/m 2/m 2/m
Tetragonal 4, 4, 422, 4mm, 42m 4/m, 4/m 2/m 2/m
Hexagonal 3, 32, 3m 3, 3 2/m
6, 6, 622, 6mm, 62m 6/m, 6/m 2/m 2/m
Isometric 23, 432, 43m 2/m 3, 4/m 3 2/m
Table 5.3 of Klein (2002) Manual of Mineral Science, John Wiley and Sons
Crystal Morphology (habit)
Nicholas Steno (1669): Law of Constancy of
Interfacial Angles
Quartz
120o
120o
120o 120o 120o
120o
120o
Crystal
MorphologyGrowth 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!
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