Symmetry

http://www.popularphilosophy.com/ideas/2005/3/15/symmetry-of-symmetries.html

http://www.wittandwisdom.com/photos/witt_and_wisdom_photos/symmetry.html

http://www.sculpturegallery.com/sculpture/symmetry.html

Reading Assignment:1. W. B-Ott, Crystallography-chapter 5, 7, 82. A. D. Krawitz- chapter 3. D. Sherwood- chapter 3

Contents

Rotation Axis, Reflection, Inversion

Rotoinversion/Rotoreflection

Combination

3

1

2

4

Symmetry, Symmetry Operation

5 32 Point Group

6 Crystal, Molecular Symmetry

Symmetry- Repetition

1. Lattice translation- three non-coplanar lattice translation

space lattice

2. Rotation (회전)

3. Reflection (반사)

4. Inversion (반전)

Symmetry Aspects of M. C. Escher’s Periodic Drawings

Fig. 1.1. Pattern based on a fourteenth-century Persian tiling design.

http://www.iucr.ac.uk/

Fig. 1.2. A teacup, showing its mirror plane of symmetry. (After L. S. Dent Glasser, Crystallography & its applications: Van Nostrand

Reinhold, 1977.)

http://www.iucr.ac.uk/

Fig. 1.3. Some symmetry elements, represented by human figures. (a) Mirror plane, shown as dashed line,in elevation and plan. (b) Twofold axis, lying along broken line in elevation, passing perpendicularlythrough clasped hands in plan. (c) Combination of twofold axis with mirror planes; the position of thesymmetry elements given only in plan. (d) Threefold axis, shown in plan only. (e) Centre of symmetry (incentre of clasped hands). (f) Fourfold inversion axis, in elevation and plan, running along the dashed lineand through the centre of the clasped hands. (After L. S. Dent Glasser, Chapter 19, The Chemistry of Cements: Academic Press, 1964.)

* All repetition operations are called symmetry operations.

Symmetry consists of the repetition of a pattern by the

application of specific rules.

* When a symmetry operation has a locus, that is a point, or a line,

or a plane that is left unchanged by the operation, this locus is

referred to as the symmetry element.

* Symmetry operation symmetry element

reflection mirror plane

rotation rotation axis

inversion inversion center

(center of symmetry)

Symmetry

Rotation Axis

- general plane lattice

180o rotation about the central lattice point A- coincidence- 2 fold rotation axissymbol: 2,

normal or parallel to plane of paper

A

360 2o

n πφ φ

= =

360 2o

n πφ φ

= =-n-fold axis φ: minimum angle required to reach a position indistinguishable

-n > 2 produce at least two other points lying in a plane normal to it- three non-colinear points define a lattice plane- fulfill the conditions for being a lattice plane

(translational periodicity)

- 3 fold axis: φ= 120o, n=3, - 4 fold axis: φ=90o, n=4,

Rotation Axis

-5 fold axis: φ= 72o, n=5, - 6 fold axis: φ=60o, n=6,

II-V and III-IV parallel butnot equal or integral ratio

* In space lattices and consequently in crystals, only 1-, 2-, 3-,

4-, and 6-fold rotation axes can occur.

Rotation Axis

http://jcrystal.com/steffenweber/JAVA/jtiling/jtiling.html

http://jcrystal.com/steffenweber/JAVA/jtiling/jtiling.html

Rotation Axis

- limitation of φ set by translation periodicity

aφ φ

b ma=

A A’

B B’b

where m is an integer

2 cos1 2cos

1cos2

ma a am

m

φφ

φ

= −= −

−=

m cosφ φ n

-1 1 2π 10 ½ π/3 61 0 π/2 42 -½ 2π/3 3 3 -1 π 2

1. 2, 3, 4, 6

Rotation Axis

- Matrix representation of rotation in Cartesian coordinate

직교축계에서 z축을 회전축으로 하고φ각만큼 회전시킨 회전조작

1

cos sin 0( ) sin cos 0

0 0 1zR n

φ φφ φ

− =

1

1 0 0(2 ) 0 1 0

0 0 1zR

− = −

1

1 3 02 23 1(3 ) 0

2 20 0 1

zR

− −

= −

2

1 3 02 23 1(3 ) 0

2 20 0 1

zR

−

= − −

Rotation Axis

1

0 1 0(4 ) 1 0 0

0 0 1zR

− =

2 1

1 0 0(4 ) (2 ) 0 1 0

0 0 1z zR R

− = = −

3

0 1 0(4 ) 1 0 0

0 0 1zR

= −

1 2 3

0 1 0 1 0 0 0 1 0(4 ) (4 ) 1 0 0 0 1 0 1 0 0 (4 )

0 0 1 0 0 1 0 0 1z z zR R R

− − • = − = − =

1

1 3 02 23 1(6 ) 0

2 20 0 1

zR

−

=

5

1 3 02 2

3 1(6 ) 02 2

0 0 1

zR

= −

2 1(6 ) (3 )z zR R=

3 1(6 ) (2 )z zR R=

4 2(6 ) (3 )z zR R=

Rotation Axis

Five Plane Lattices

Rotation Axis

1 2 3

4 6

1

- hexagonal coordinatea1, a2, a3, cMiller-Bravais indices ( h k i l ) i=-(h+k)

Rotation Axis

Standard (0001) projection (hexagonal, c/a=1.86)

Rotation Axis

(x1,y1,z1)

•

•

•

(x2,y2,z2)

(x3,y3,z3)

2 1x y= −2 1 1y x y= −2 1z z=

3 1 1x x y= − +3 1y x= −3 1x z=

1

0 1 0(3 ) 1 1 0

0 0 1zR

− = −

2

1 1 0(3 ) 1 0 0

0 0 1zR

− = −

1

1 3 02 23 1(3 ) 0

2 20 0 1

zR

− −

= −

* Cartesian coordinate

Rotation Axis

- reflection, a plane of symmetry or a mirror plane, m, |,

Tartaric acid

Reflection

rectangular centered rectangular

2 1x x=2 1y y=2 1z z= −

•

mxy (mz)

1 0 0( ) 0 1 0

0 0 1zR m

= −

myz (mx)

2 1x x= −2 1y y=2 1z z=

1 0 0( ) 0 1 0

0 0 1xR m

− =

( ) 1zR m = −

enantiomorph

대장상

Reflection

- inversion, center of symmetry or inversion center, 1

All lattices are centrosymmteric.

2 1x x= −2 1y y= −2 1z z= −

•

1 0 0(1) 0 1 0

0 0 1R

− = − − (1) 1R = −

Inversion

http://www.gh.wits.ac.za/craig/diagrams/

Inversion

Compound Symmetry Operation

- link of translation, rotation, reflection, and inversion operation

-compound symmetry operationtwo symmetry operation in sequence as a single event

- combination of symmetry operationstwo or more individual symmetry operations are combined which are themselves symmetry operations

compound combination

Compound Symmetry Operation

Rotoinversion

-compound symmetry operation of rotation and inversion

-rotoinversion axis

- 1, 2, 3, 4, 6 -

-

n

1, 2, 3, 4, 6

1

••

up, right

down, left

-2( )m≡

•

mxy (mz)

- 3 3 1≡ +

•

•

•

1

2

3 4

5

6

12

Rotoinversion

- 4

- 6

•

•

1

2

3

4

•

•

•

4

2

3

1

5

6

Rotoinversion

1 ≡ 2 ,m≡ 3 3 1,≡ + 4 6 3 ,m≡ ⊥- inversion center, implies

- only rotoinversion axes of odd order imply the presence ofan inversion center

2,

Rotoreflection

1S m= 2 1S = 3 6S = 4 4S = 6 3S =

1S m=

•12

- The axes and , including and , are called point-symmetryelement, since their operations always leave at least one pointunmoved

n n 1 m

Rotoinversion

Enantiomorphous objects

1-fold rotation axis Center of inversion

2-fold rotation axis 2bar = m

3-fold rotation axis 3barPecharsky page 15, 16, 17

6-fold rotation axis 6bar

4-fold rotation axis 4bar

Pecharsky page 18, 19

When two symmetry elements interact, they result in

additional symmetry element(s)

Interaction of symmetry elements

Start with 2 and 1bar (on 2) m

New symmetry element m emerged as

the result of the sequential

application of two symmetry elements

(2 then 1bar) to the original object

2 X 1bar (on 2) = 1bar (on 2) X 2 = m (⊥ 2 thru 1bar)

2 X m (⊥ 2) = m (⊥ 2) X 2 = 1bar (@ m⊥ 2)

m X 1bar (on m) = 1bar (on m) X m = 2 (⊥ m thru 1bar)

Pecharsky page 20

proper vs improper

• Proper symmetry elements

– Rotation axes, screw axes, translation vectors

• Improper symmetry elements

– Inverts an object in a way that may be imaged by comparing right and left hands

– Inverted object is called an enantiomorph of the direct object (right vs left hand)

– Center of inversion, rotoinversion axes, mirror plane, glide plane

Combination

- a mirror plane is added normal to the rotation axis,Xm

φ

m

1 ( )mm

− ≡

•1 2

•

1

22m

−

•

3

43 ( 6)m

− ≡

•

•

•

4

6

2

61

3

5

- a mirror plane is added normal to the rotation axis,Xm

4m

− 6m

−

•

•

4

7

2

5

••

3

1

68

•

•

•

11

2

105

9•

•

•

12

4

6

78

12

Combination

- a mirror plane is added parallel to the rotation axis, Xm

φ

m

1 ( )m m− ≡

12

1

2

2 ( 2 , 2)m mm mm− ≡

3

4

3m−

4

6

2

61

3

5generated

Combination

- a mirror plane is added parallel to the rotation axis, Xm

1

3

4 ( 4 )m mm− ≡

5

7

6 ( 6 )m mm− ≡

7

11

3

21

5

9

generated

2

4 68

4

6

8

10

12

generated

Combination

Combination of rotation axes- space lattice

regarded as a stack of plane latticesonly have symmetry axes with n=1,2,3,4,or 6, normal to a netmany possible planeseach such plane conform to the symmetry of a particularn-fold axis

restriction to the angular relationships between intersectingn-fold axes

n

n

- Euler construction - select for attention line AM

- select for attention line BN

- intersection of AM’ and BN is C’intersection of BN’ and AM is C

- 1: Aα brings C to C’2: Bβ brings C’ to C

' / 2MAB M AM α∠ = ∠ =

' / 2NAB N AM β∠ = ∠ =

Combination of rotation axes

- euler construction - Aα and Bβ leaves C unchanged- if there is a motion of points on the sphere due to Aα and Bβ,

it must be a rotation about an axis OC- 1: Aα leaves A unmoved

2: Bβ moves A to A’- consider the spherical triangle BA’C

- rotation about C carries Ato A’ through twiceor through angle γ

-

' / 2ABC A BC φ∠ = ∠ = 'AB A B='ABC A BC∆ = ∆ ' / 2ACB A CB γ∠ = ∠ =

ACB∠

/ 2, / 2, / 2U V Wα β γ= = =, , :u v w arcs

cos cos coscossin sin

W U VwU V

+=

Combination of rotation axes

Combination of rotation axes

Combination of rotation axes

Combination of rotation axes, n2

- 12(≡2) -222 -32

•

1

2

4

3

generated

•

•4

5

2

61

3

6

•

•

•

- 422 -622

2

3•

•1

4

5

6

7

8••

generated7

11

3

21

5

9

4

6

8

10

12

generated

•

•

• •

•

•

Combination of rotation axes

- 23 -432

1

2

•

•

3

4

5

6

••

••

1

2

•

•3

4

56

••

••

•7

8

9• 10

•11

•

••

http://neon.materials.cmu.edu/degraef/pg/

Combination of rotation axes,

•

13

•

•

•5

6

7

7

11

9

8

10

12

•

•

•

2n212( )m

− ≡ 22( 2 )mm− ≡232( 3 )m

− ≡

42( 42 )m− ≡ 62( 62 6 2)m m− ≡ ≡1

2

•

4

3

•24

•

•

•

1

2

3 4

5

6•

•

7

9

11

8

10

••

1

24

3

8

•

•

•

•

4

2

3

1

5

6

7

11

9

8

10

12

•

•

•

•

•

•

4

2

3

1

5

6

•

•

•

•

•

•

62m 6 2m

<c><a><210> <c><a><210>

1 ( 2 )m mmm

− ≡ 2 2 2 2( )mm m m m

− ≡ 3 ( 6 2)m mm

− ≡

4 4 2 2( )mm m m m

− ≡6 6 2 2( )mm m m m

− ≡

•

1 2

•34

•

1

2

•

3

4•

•

5

6

7

8•

•

•

4

6

2

1

3

5

7

129

8

10

•

•

•

11

•

•

4

7

2

5

••

3

1

68

•

•

••

913

1410

1612

1511

•

•

•11

3

105

9•

•

•

1

2

4

6

7

8

12

• •

•

•

•

•

1319

2 3( 3)mm

− ≡

1

2

•

•

9

10

• •••

•

•

3

4

•

5 6

•7 8

•

•

43m−

2

•

•

6

1

••

• •• 5

•

•

7

3

4

8

9

•

•

•

•

9 1

•

•10

•

•6

3

5

7

8 •

•

•

4 23 ( 3 )m mm m

− ≡

152

4 16

1211

1314

•

•

••

•

• ••

••

••

••

••

http://neon.materials.cmu.edu/degraef/pg/

32 Point Group

- Schonflies symbol vs. International symbol

•

1

22m

−

•

3

4

•

12

m−2−

1

2

Monoclinic

Crystal Symmetry

-galena (PbS)4 23m m

Cube(정육면체)의 대칭요소• 정6면체는 대칭심(center of symmetry), 9개의 거울면(mirror plane),

6개의 2회전축(diad axis), 4개의 3회전축(triad axis), 그리고 3개의 4

회전축 (tetrad axis)을 가지고 있다.

i

m면 대각선 방향 :6개

직각 방향 :3개

X=4X=3

X=2

http://www.gh.wits.ac.za/craig/diagrams/benze.gif

- benzene(C6H6)

Molecular Symmetry6 2 2m m m

6

•

•

•

•

•

•

• •

•

•

•

•

http://www.gh.wits.ac.za/craig/diagrams/thiou.gif

- thiourea

Molecular Symmetry

2mm

Molecular Symmetry

Homework 1

Ott chapter 5 --- 1, 3, 4, 5, 6, 9

chapter 8---1, 2, 3, 4, 5, 8, 9, 10,

11(1,2,3,4,5,6,7,8), 17