Crystal Chem Crystallography Chemistry behind minerals and how

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

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 • Mirror Planes (m) – reflection along a plane A line denotes mirror planes

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 2 mm, 3 m, 4 mm, 6 mm • Mathematical maximum of 10 combinations 4 mm

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

3 -D Symmetry Operators • Rotation Axes (1, 2, 3, 4, or 6 a. k. a. A 1, A 2, A 3, A 4, A 6) – 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 1 1

3 -D Symmetry Operators • Rotoinversion (1, 2, 3, 4, 6 a. k. a. A 1, A 2, A 3, A 4, A 6) – 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 )

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

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

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

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 ) 5 3 1 This is unique 4 6 2

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 = 2 mm 3 || m = 3 m, also 4 mm, 6 mm 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, 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 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

Hexagonal class Rhombohedral form Hexagonal form

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

Crystal Morphology Diff planes have diff atomic environments

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!

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

Miller Indices