symmetry elements ii
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Symmetry Elements II. Lecture 6. 3-D Symmetry. We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 . Combinations of these elements are also possible - PowerPoint PPT PresentationTRANSCRIPT
Symmetry Elements IISymmetry Elements II
Lecture 6Lecture 6
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
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).
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
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
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
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
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
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
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
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
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
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?
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
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
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
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
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
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.
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
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
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
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
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
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
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
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
Crystal System CharacteristicsCrystal System Characteristics
Isometric/CubicIsometric/Cubic
HexagonalHexagonal
TetragonalTetragonal
OrthorhombicOrthorhombic
MonoclinicMonoclinic
TriclinicTriclinic
ALL AXES EQUAL
AXES UNEQUAL
BirefringenceBirefringence
Isometric/CubicIsometric/Cubic
HexagonalHexagonal
TetragonalTetragonal
OrthorhombicOrthorhombic
MonoclinicMonoclinic
TriclinicTriclinic
ISOTROPIC
ANISOTROPIC
Crystal System CharacteristicsCrystal System Characteristics
Isometric/CubicIsometric/Cubic
HexagonalHexagonal
TetragonalTetragonal
OrthorhombicOrthorhombic
MonoclinicMonoclinic
TriclinicTriclinic
ALL AXES EQUAL
ALL AXES UNEQUAL
TWO AXES EQUAL
Interference FigureInterference Figure
Isometric/CubicIsometric/Cubic
HexagonalHexagonal
TetragonalTetragonal
OrthorhombicOrthorhombic
MonoclinicMonoclinic
TriclinicTriclinic
UNIAXIAL
BIAXIAL
Crystal System CharacteristicsCrystal System Characteristics
Isometric/CubicIsometric/Cubic
HexagonalHexagonal
TetragonalTetragonal
OrthorhombicOrthorhombic
MonoclinicMonoclinic
TriclinicTriclinic
ALL AXES EQUAL
AXES NON-ORTHOGONAL
AXES ORTHOGONAL
ExtinctionExtinction
Isometric/CubicIsometric/Cubic
HexagonalHexagonal
TetragonalTetragonal
OrthorhombicOrthorhombic
MonoclinicMonoclinic
TriclinicTriclinic
PARALLEL
INCLINED