Point Groups Roya Majidi 1393

Point Groups Space Groups 7 Crystal Systems 32 point groups in 3D How symmetry operators without translation combine Point Groups 7 Crystal Systems 32 point groups in 3D Lattices have only 7 distinct point groups How symmetry operators with translation combine Space Groups 14 Bravais Lattices Lattices have only 14 distinct space groups 230 space groups in 3D

Classification of Symmetry Operators Dimension of the Operator Takes an object to its mirror form or not Based on If the operator acts at a point or moves a point (i.e. outside a unit cell) If it plays a role in the shape of a crystal or not (Macroscopic/Microscopic)

Rotation Reflection Inversion Translation Screw Rotation Basic Symmetry Operations Inversion Translation Screw Rotation Compound Symmetry Operations Glide Reflection Rotoinversion

Operator Symmetry Inversion Rotation Mirror Central (I, O, C) Classification based on the dimension invariant entity of the symmetry operator Operator Symmetry Inversion Central (I, O, C) Rotation Axial (A) Mirror Planar (m, P)

Mirror m

Vertical Mirror y x z (x y z) (-x y z)

Vertical Mirror y x z (x y z) (x -y z)

Horizontals Mirror y x z (x y z) (x y -z)

Inversion 1

6 6

(x y z) (-x –y –z)

n Rotation Axis If an object come into self-coincidence through smallest non-zero rotation angle of  then it is said to have an n-fold rotation axis where: Crystals can only have 1, 2, 3, 4 or 6 fold symmetry

1, 2, 3, 4, 6 A1, A2, A3, A4, or A6 C1, C2, C3, C4, or C6

Two-fold rotation = 360o/2 rotation

 = 360 n = 1 1-fold rotation axis 1

Symbol for 2-fold axis  = 180 n = 2 2-fold rotation axis 2

Symbol for 3-fold axis  = 120 n = 3 3-fold rotation axis 3

 = 90 n = 4 4-fold rotation axis 4

 = 60 n = 6 6-fold rotation axis 6

Roto-inversion operations A roto-inversion operator rotates a point/object and then inverts it (inversion operation) in one go. Roto-inversion operations

Try combining a 2-fold rotation axis with a mirror 2-D Symmetry Try combining a 2-fold rotation axis with a mirror

2-D Symmetry Try combining a 2-fold rotation axis with a mirror Step 1: reflect (could do either step first)

2-D Symmetry Try combining a 2-fold rotation axis with a mirror Step 1: reflect Step 2: rotate (everything)

2-D Symmetry Try combining a 2-fold rotation axis with a mirror Step 1: reflect Step 2: rotate (everything) Is that all??

2-D Symmetry Try combining a 2-fold rotation axis with a mirror Step 1: reflect Step 2: rotate (everything) No! A second mirror is required

2-D Symmetry Try combining a 2-fold rotation axis with a mirror The result is Point Group 2mm “2mm” indicates 2 mirrors The mirrors are different (not equivalent by symmetry)

Now try combining a 4-fold rotation axis with a mirror 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror

Now try combining a 4-fold rotation axis with a mirror Step 1: reflect 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect

2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect Step 2: rotate 1

2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect Step 2: rotate 2

2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect Step 2: rotate 3

Now try combining a 4-fold rotation axis with a mirror 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements?

Now try combining a 4-fold rotation axis with a mirror 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements? Yes, two more mirrors

Now try combining a 4-fold rotation axis with a mirror 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements? Yes, two more mirrors Point group name??

Now try combining a 4-fold rotation axis with a mirror 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements? Yes, two more mirrors Point group name?? 4mm

Now try combining a 4-fold rotation axis with a mirror 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements? Yes, two more mirrors Point group name?? 4mm

3-fold rotation axis with a mirror creates point group 3m 2-D Symmetry 3-fold rotation axis with a mirror creates point group 3m

6-fold rotation axis with a mirror creates point group 6mm 2-D Symmetry 6-fold rotation axis with a mirror creates point group 6mm

Example 4mm 622

Example mm m

Example 4mm m

Example No symmetry 4 This is Amorphous!!

We now have 10 unique 3-D symmetry operations: 1 2 3 4 6 i m 3 4 6 Combinations of these elements are also possible A complete analysis of symmetry about a point in space requires that we try all possible combinations of these symmetry elements

Rotoinversion a. 1-fold rotoinversion ( 1 ) 3-D Symmetry Rotoinversion a. 1-fold rotoinversion ( 1 )

3-D Symmetry Rotoinversion a. 1-fold rotoinversion ( 1 ) Step 1: rotate 360/1 (identity)

3-D Symmetry Rotoinversion a. 1-fold rotoinversion ( 1 ) Step 1: rotate 360/1 (identity) Step 2: invert This is the same as i, so not a new operation

3-D Symmetry Rotoinversion b. 2-fold rotoinversion ( 2 ) Step 1: rotate 360/2 Note: this is a temporary step, the intermediate motif element does not exist in the final pattern

3-D Symmetry Rotoinversion b. 2-fold rotoinversion ( 2 ) Step 1: rotate 360/2 Step 2: invert

Rotoinversion b. 2-fold rotoinversion ( 2 ) The result: 3-D Symmetry Rotoinversion b. 2-fold rotoinversion ( 2 ) The result:

3-D Symmetry Rotoinversion b. 2-fold rotoinversion ( 2 ) This is the same as m, so not a new operation

Rotoinversion c. 3-fold rotoinversion ( 3 ) 3-D Symmetry Rotoinversion c. 3-fold rotoinversion ( 3 )

3-D Symmetry Rotoinversion 1 c. 3-fold rotoinversion ( 3 ) Step 1: rotate 360o/3 Again, this is a temporary step, the intermediate motif element does not exist in the final pattern 1

3-D Symmetry Rotoinversion c. 3-fold rotoinversion ( 3 ) Step 1: rotate 360o/3 Step 2: invert through center

3-D Symmetry Rotoinversion 1 2 c. 3-fold rotoinversion ( 3 ) Completion of the first sequence 1 2

Rotoinversion c. 3-fold rotoinversion ( 3 ) Rotate another 360/3 3-D Symmetry Rotoinversion c. 3-fold rotoinversion ( 3 ) Rotate another 360/3

Rotoinversion c. 3-fold rotoinversion ( 3 ) Invert through center 3-D Symmetry Rotoinversion c. 3-fold rotoinversion ( 3 ) Invert through center

3-D Symmetry Rotoinversion 3 1 2 c. 3-fold rotoinversion ( 3 ) Complete second step to create face 3 3 1 2

3-D Symmetry Rotoinversion 3 1 4 2 c. 3-fold rotoinversion ( 3 ) Third step creates face 4 (3  (1)  4) 3 1 4 2

3-D Symmetry Rotoinversion 1 5 2 c. 3-fold rotoinversion ( 3 ) Fourth step creates face 5 (4  (2)  5) 5 1 2

3-D Symmetry Rotoinversion 5 1 6 c. 3-fold rotoinversion ( 3 ) Fifth step creates face 6 (5  (3)  6) Sixth step returns to face 1 5 1 6

Rotoinversion c. 3-fold rotoinversion ( 3 ) This is unique 3-D Symmetry Rotoinversion c. 3-fold rotoinversion ( 3 ) This is unique 3 5 1 4 2 6

Rotoinversion d. 4-fold rotoinversion ( 4 ) 3-D Symmetry Rotoinversion d. 4-fold rotoinversion ( 4 )

Rotoinversion d. 4-fold rotoinversion ( 4 ) 3-D Symmetry Rotoinversion d. 4-fold rotoinversion ( 4 )

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

Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 2: Invert 3-D Symmetry Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 2: Invert

Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 2: Invert 3-D Symmetry Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 2: Invert

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

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

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

Rotoinversion d. 4-fold rotoinversion ( 4 ) 5: Rotate 360/4 3-D Symmetry Rotoinversion d. 4-fold rotoinversion ( 4 ) 5: Rotate 360/4

Rotoinversion d. 4-fold rotoinversion ( 4 ) 5: Rotate 360/4 6: Invert 3-D Symmetry Rotoinversion d. 4-fold rotoinversion ( 4 ) 5: Rotate 360/4 6: Invert

3-D Symmetry Rotoinversion d. 4-fold rotoinversion ( 4 ) This is also a unique operation

3-D Symmetry Rotoinversion d. 4-fold rotoinversion ( 4 ) A more fundamental representative of the pattern

Rotoinversion e. 6-fold rotoinversion ( 6 ) Begin with this framework: 3-D Symmetry Rotoinversion e. 6-fold rotoinversion ( 6 ) Begin with this framework:

Rotoinversion e. 6-fold rotoinversion ( 6 ) 3-D Symmetry Rotoinversion e. 6-fold rotoinversion ( 6 ) 1

Rotoinversion e. 6-fold rotoinversion ( 6 ) 3-D Symmetry Rotoinversion e. 6-fold rotoinversion ( 6 ) 1

Rotoinversion e. 6-fold rotoinversion ( 6 ) 3-D Symmetry Rotoinversion e. 6-fold rotoinversion ( 6 ) 1 2

Rotoinversion e. 6-fold rotoinversion ( 6 ) 3-D Symmetry Rotoinversion e. 6-fold rotoinversion ( 6 ) 1 2

Rotoinversion e. 6-fold rotoinversion ( 6 ) 3-D Symmetry Rotoinversion e. 6-fold rotoinversion ( 6 ) 1 3 2

Rotoinversion e. 6-fold rotoinversion ( 6 ) 3-D Symmetry Rotoinversion e. 6-fold rotoinversion ( 6 ) 1 3 2

Rotoinversion e. 6-fold rotoinversion ( 6 ) 3-D Symmetry Rotoinversion e. 6-fold rotoinversion ( 6 ) 1 3 2 4

Rotoinversion e. 6-fold rotoinversion ( 6 ) 3-D Symmetry Rotoinversion e. 6-fold rotoinversion ( 6 ) 1 3 2 4

Rotoinversion e. 6-fold rotoinversion ( 6 ) 3-D Symmetry Rotoinversion e. 6-fold rotoinversion ( 6 ) 1 3 5 2 4

Rotoinversion e. 6-fold rotoinversion ( 6 ) 3-D Symmetry Rotoinversion e. 6-fold rotoinversion ( 6 ) 1 3 5 2 4

Rotoinversion e. 6-fold rotoinversion ( 6 ) 3-D Symmetry Rotoinversion e. 6-fold rotoinversion ( 6 ) 1 3 5 2 6 4

3-D Symmetry Rotoinversion e. 6-fold rotoinversion ( 6 ) Note: this is the same as a 3-fold rotation axis perpendicular to a mirror plane Top View

Rotoinversion e. 6-fold rotoinversion ( 6 ) A simpler pattern 3-D Symmetry Rotoinversion e. 6-fold rotoinversion ( 6 ) A simpler pattern Top View

3-D Symmetry The 32 3-D Point Groups (sometime crystal classes) Every 3-D pattern must conform to one of them. This includes every crystal, and every point within a crystal Table 5.1 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

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

Crystal has 4mm symmetry Example Crystal has 4mm symmetry

Example Crystal has 4/m 2/m 2/m symmetry