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

2mm, 3m, 4mm, 6mm

• Mathematical maximum of 10 combinations

4mm

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. 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 )

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 )

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

Hexagonal class

Rhombohedral

form Hexagonal

form

Crystal Morphology (habit)

Nicholas Steno (1669): Law of Constancy of

Interfacial Angles

Quartz

120o

120o

120o 120o 120o

120o

120o

Crystal Morphology

Diff planes have diff atomic environments

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

Miller Indices