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Symmetry Elements II Symmetry Elements II

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Page 1: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Symmetry Elements IISymmetry Elements II

Page 2: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

3-D Symmetry3-D Symmetry

We now have We now have 88 unique unique 3D3D symmetry operations: symmetry operations:

1 2 3 4 6 m1 2 3 4 6 m 3 4 3 4

CombinationsCombinations of these elements are also possible of these elements are also possible

A A completecomplete analysis of analysis of symmetry about a point in spacesymmetry about a point in space requires that we try all possible combinations of these requires that we try all possible combinations of these symmetry elementssymmetry elements

Page 3: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Point GroupPoint Group

The set of symmetry operations that leave The set of symmetry operations that leave the appearance of the crystal structure the appearance of the crystal structure unchanged. unchanged.

There are 32 possible point groupsThere are 32 possible point groups(i.e., unique combinations of symmetry (i.e., unique combinations of symmetry operations). operations).

Page 4: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Try combining a 2-fold rotation axis with a mirrorTry combining a 2-fold rotation axis with a mirror

The result is The result is Point Group 2mmPoint Group 2mm

““2mm” indicates 2mm” indicates 22 mirrors mirrors

The mirrors are differentThe mirrors are different

2-D Symmetry2-D Symmetry

Page 5: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Now try combining a 4-fold rotation axis with a mirrorNow try combining a 4-fold rotation axis with a mirror2-D Symmetry2-D Symmetry

Page 6: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Now try combining a 4-fold rotation axis with a mirrorNow try combining a 4-fold rotation axis with a mirror

Step 1: reflectStep 1: reflect

2-D Symmetry2-D Symmetry

Page 7: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Now try combining a 4-fold rotation axis with a mirrorNow try combining a 4-fold rotation axis with a mirror

Step 1: reflectStep 1: reflect

Step 2: rotate Step 2: rotate 11

2-D Symmetry2-D Symmetry

Page 8: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Now try combining a 4-fold rotation axis with a mirrorNow try combining a 4-fold rotation axis with a mirror

Step 1: reflectStep 1: reflect

Step 2: rotateStep 2: rotate 2 2

2-D Symmetry2-D Symmetry

Page 9: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Now try combining a 4-fold rotation axis with a mirrorNow try combining a 4-fold rotation axis with a mirror

Step 1: reflectStep 1: reflect

Step 2: rotate Step 2: rotate 33

2-D Symmetry2-D Symmetry

Page 10: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Now try combining a 4-fold rotation axis with a mirrorNow try combining a 4-fold rotation axis with a mirror

Any other elements?

2-D Symmetry2-D Symmetry

Page 11: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Now try combining a 4-fold rotation axis with a mirrorNow try combining a 4-fold rotation axis with a mirror

Yes, two more mirrors

Any other elements?

2-D Symmetry2-D Symmetry

Page 12: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Now try combining a 4-fold rotation axis with a mirrorNow try combining a 4-fold rotation axis with a mirror

Point group name??

Yes, two more mirrors

Any other elements?

2-D Symmetry2-D Symmetry

Page 13: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Now try combining a 4-fold rotation axis with a mirrorNow try combining a 4-fold rotation axis with a mirror

4mm

Point group name??

Yes, two more mirrors

Any other elements?

2-D Symmetry2-D Symmetry

Why not 4mmmm?

Page 14: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

3-fold rotation axis with a mirror creates point group 3-fold rotation axis with a mirror creates point group 3m3m

Why not 3mmm?Why not 3mmm?

2-D Symmetry2-D Symmetry

Page 15: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

6-fold rotation axis with a mirror creates point group 6-fold rotation axis with a mirror creates point group 6mm6mm

2-D Symmetry2-D Symmetry

Page 16: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

The original 6 elements plus the 4 combinations The original 6 elements plus the 4 combinations creates creates 1010 possible possible 2-D Point Groups2-D Point Groups::

1 2 3 4 6 m 2mm 3m 4mm 6mm1 2 3 4 6 m 2mm 3m 4mm 6mm

AnyAny 2-D pattern of objects surrounding a point 2-D pattern of objects surrounding a point must conform to one of these groupsmust conform to one of these groups

2-D Symmetry2-D Symmetry

Page 17: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

3-D Symmetry3-D Symmetry

As in 2-D, the number of possible combinations is As in 2-D, the number of possible combinations is limited only by limited only by incompatibilityincompatibility and and redundancyredundancy

There are only There are only 2222 possible unique 3-D possible unique 3-D combinations, when combined with the combinations, when combined with the 1010 original 3-D elements yields the original 3-D elements yields the 32 3-D Point 32 3-D Point GroupsGroups

Page 18: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

3-D Symmetry3-D SymmetryThe 32 3-D Point GroupsThe 32 3-D Point Groups

Every 3-D pattern must conform to Every 3-D pattern must conform to oneone of them. of them.

This includes every crystal, and every point within a This includes every crystal, and every point within a crystalcrystal

Rotation axis only 1 2 3 4 6

Rotoinversion axis only 1 (= i ) 2 (= m) 3 4 6 (= 3/m)

Combination of rotation axes 222 32 422 622

One rotation axis mirror 2/m 3/m (= 6) 4/m 6/m

One rotation axis || mirror 2mm 3m 4mm 6mm

Rotoinversion with rotation and mirror 3 2/m 4 2/m 6 2/m

Three rotation axes and mirrors 2/m 2/m 2/m 4/m 2/m 2/m 6/m 2/m 2/m

Additional Isometric patterns 23 432 4/m 3 2/m

2/m 3 43m

Increasing Rotational Symmetry

Table 5.1 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

Page 19: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Crystal SystemsCrystal Systems

A grouping point groups that require a A grouping point groups that require a similar arrangement of axes to describe similar arrangement of axes to describe the crystal lattice. |the crystal lattice. |

There are seven unique crystal systems. There are seven unique crystal systems.

Page 20: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

3-D Symmetry3-D SymmetryThe 32 3-D Point GroupsThe 32 3-D Point Groups

Regrouped by Regrouped by Crystal SystemCrystal System

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

Page 21: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

TriclinicTriclinic

Three axes of Three axes of unequal length unequal length

Angles between axes Angles between axes are not equalare not equal

Point group: 1Point group: 1

Page 22: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

MonoclinicMonoclinic

Three axes of Three axes of unequal length unequal length

Angle between two Angle between two axes is 90axes is 90°°

Point groups: Point groups: 2, m, 2/m2, m, 2/m

Page 23: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

OrthorhombicOrthorhombic

Three axes of Three axes of unequal length unequal length

Angle between all Angle between all axes is 90axes is 90°°

Point groups: 222Point groups: 2222/m2/m/2/m, 2mm2/m2/m/2/m, 2mm

Page 24: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

TetragonalTetragonal

Two axes of equal Two axes of equal length length

Angle between all Angle between all axes is 90axes is 90°°

Point groups: 4, 4, Point groups: 4, 4, 4/m, 4mm, 422, 42m, 4/m, 4mm, 422, 42m, 4/m2/m2/m4/m2/m2/m

Page 25: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

HexagonalHexagonal Four axes, three equal Four axes, three equal

axes within one plane axes within one plane

Angle between the 3 Angle between the 3 co-planar axes is 60co-planar axes is 60°°

Angle with remaining axis Angle with remaining axis is 90is 90°°

Point groups: 6, 6, 6/m, Point groups: 6, 6, 6/m, 6mm, 622, 62m, 6mm, 622, 62m, 6/m2/m2/m6/m2/m2/m

Page 26: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Trigonal Trigonal (Subset of Hexagonal)(Subset of Hexagonal)

Four axes, three equal Four axes, three equal axes within one plane axes within one plane

Angle between the 3 Angle between the 3 co-planar axes is 60co-planar axes is 60°°

Angle with remaining axis Angle with remaining axis is 90is 90°°

Point groups: 3, 3, 3/m, Point groups: 3, 3, 3/m, 32, 32/m32, 32/m

Page 27: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Cubic / IsometricCubic / Isometric

All axes of equal All axes of equal length length

Angle between all Angle between all axes is 90axes is 90°°

Point groups: 23, 423, Point groups: 23, 423, 2/m3, 43m, 4/m32/m2/m3, 43m, 4/m32/m

Page 28: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Crystal System CharacteristicsCrystal System Characteristics

Isometric/CubicIsometric/Cubic

HexagonalHexagonal

TetragonalTetragonal

OrthorhombicOrthorhombic

MonoclinicMonoclinic

TriclinicTriclinic

ALL AXES EQUAL

AXES UNEQUAL

Page 29: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

BirefringenceBirefringence

Isometric/CubicIsometric/Cubic

HexagonalHexagonal

TetragonalTetragonal

OrthorhombicOrthorhombic

MonoclinicMonoclinic

TriclinicTriclinic

ISOTROPIC

ANISOTROPIC

Page 30: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Crystal System CharacteristicsCrystal System Characteristics

Isometric/CubicIsometric/Cubic

HexagonalHexagonal

TetragonalTetragonal

OrthorhombicOrthorhombic

MonoclinicMonoclinic

TriclinicTriclinic

ALL AXES EQUAL

ALL AXES UNEQUAL

TWO AXES EQUAL

Page 31: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Interference FigureInterference Figure

Isometric/CubicIsometric/Cubic

HexagonalHexagonal

TetragonalTetragonal

OrthorhombicOrthorhombic

MonoclinicMonoclinic

TriclinicTriclinic

UNIAXIAL

BIAXIAL

Page 32: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

Crystal System CharacteristicsCrystal System Characteristics

Isometric/CubicIsometric/Cubic

HexagonalHexagonal

TetragonalTetragonal

OrthorhombicOrthorhombic

MonoclinicMonoclinic

TriclinicTriclinic

ALL AXES EQUAL

AXES NON-ORTHOGONAL

AXES ORTHOGONAL

Page 33: Symmetry Elements II 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also

ExtinctionExtinction

Isometric/CubicIsometric/Cubic

HexagonalHexagonal

TetragonalTetragonal

OrthorhombicOrthorhombic

MonoclinicMonoclinic

TriclinicTriclinic

PARALLEL

INCLINED


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