3D Symmetry_1(2 weeks)

Next we would move a step further into 3D symmetry.

Leonhard Euler :

http://en.wikipedia.org/wiki/Leonhard_Euler

Google search: Euler

For convenience, set R = 1Great circle (GC), R=1

Small circleR<1

A B

Spherical trigonometry

Distance: AOB = (GC)o

Pole 90o to arc AB.OP plane defined by OAB

A B

o

P

Well defined

angle

B’POA = /2; POB = /2; POB’ = /2

A B

C

ab

c

arc BC = aarc AC = barc AB = c.

GC

Spherical Angles

A

B C

BAC = B’OC’

B’OC’

oGC

?

Trigonometry: points on a surface of a sphere (directions that intersect the sphere) are connected using arcs of great circles

OA OBOA OCOB OC

angle

Center of the sphere

C’B’

A is the pole for plane defined by B’OC’

ABC and ABC are mutually polar!

Proof: B: pole of arc AC B is 90o away from point A. C: pole of arc AB C is 90o away from point A. A:pole of arc BC. Similarly, B: AC, C: pole of arc AB.

A

B C

A

B C

Polar triangle

A, pole of arc BCB, pole of arc ACC, pole of arc AB

GC

GC

A

B

C

B

C

P

Q

Proof: BAC = , arc BC = a, + a = .

a

B : pole of arc AC arc BQ = /2 C : pole of arc AB arc CP = /2 arc BQ + arc CP = = (arc BP+ arc PQ) + (arc PQ+ arc QC) = (arc BP+ arc PQ+ arc QC) + arc PQ = a + arc PQ

A A C

o

B

QC

Why! See pictures of spherical angle in page 4 (bottom)!

= POQ =

Law of cosines

Plane Trigonometry

AB C

a

bcAbccba cos2222

How about law of cosines in spherical trigonometry?

length

angle

Triangle is defined as

C is spherical angle at point u.

vOu = awOu = bvOw = c

a, b, c

Cbaba costantan2tantan 22

cbaba cossecsec2tantan2 22 cbaba cossecsec2secsec 22

cbaCba cossecsec1costantan

Cbabac cossinsincoscoscos

u

v

w

o

yz

b

a

90o

(1)

(2)

1a

(1)

(2)

(1)= tana

1/(2)= cosa (2) = seca

1b

(3)

(4)

(3)= tanb

1/(4)= cosb (4) = secbUnit circle

(3)

(4)

http://en.wikipedia.org/wiki/Spherical_law_of_cosines

OO

u uz y

2

yz (From uyz)(From oyz)

Stop here about spherical trigonometry!

We obtain all the relations needed forfurther discussion of the 3D point groups!

Combination of two rotation operations in 3 D:

A B

1 R

2 R

3 R

C ? AB

A : (1) (2)B : (2) (3)

(1) and (3) relation?

3-D: translation, reflection, rotation, and inversion.

CAB

must be crystallographic

c

A B

c

A BcA Bc

C

Locate the position of axis C

b a A B

C

c

ab

Euler construction:

A B

M

M’

N’

N C’

C

A: AMAM’. B: BNBN’.

C (the point unmoved). OC: the axis

A symmetry element is the locus of apoint that is left unmoved by an operation.

(1) A: leave A unmoved.(2) B: move A to A’.

A B

MN’ CA’

/2 /2

ABC = A’BC = /2 AB = A’B ABC = A’BC ACB = A’CB /2

A/2B

C

/2

/2

c

ab

The law of cosine (spherical trigonometry)

2cossinsincoscoscos

babac

2cossinsincoscoscos

cbcba

CAB

A/2 B

C

/2/2

c

ab180o-a 180o-b

180o-c

180o-/2180o-/2

180o-/2

Law of cosine to the polar triangle

)180cos()2

180sin()2

180sin(

)2

180cos()2

180cos()2

180cos(

ooo

ooo

c

2sin2sin

2cos2cos2coscos

c

Polartriangle

2sin2sin

2cos2cos2coscos

c

2sin2sin

2cos2cos2coscos

a

2sin2sin

2cos2cos2coscos

b

All the rotation combinations possible in 3D that need to betested: B

A 1

1 111112113114116

2 212213214216313314

3316

2

222223224226323

326324

4 414416

6 616

3 4 6

626 636 646 666

424426

434436

444446

333

336334

Axis atA, B, or C , , or

/2/2/2

1-fold2-fold3-fold4-fold6-fold

360o

180o

120o

90o

60o

180o

90o

60o

45o

30o

-10

1/21/21/2

31/2/2

01

31/2/2 1/21/2

1/2

)2cos(

)2cos(

)2cos(

)2sin(

)2sin(

)2sin(

Case: 11n

A: 1, = 360o, cos( /2) = -1; sin( /2) = 0

A/2B

C

/2

/2

c

ab

B: 1, = 360o, cos( /2) = -1; sin( /2) = 0

C: n, = 360o /n , cos( /2); sin( /2)

A B

C

180o

180o/n

c

ab 180o

2sin2sin

2cos2cos2coscos

c

0

12coscos

c None exist! Except, = 360 o

111

Case: 22n

A: 2, = 180o, cos( /2) = 0; sin( /2) = 1A

/2B

C

/2

/2

c

ab

B: 2, = 180o, cos( /2) = 0; sin( /2) = 1

C: n, = 360o /n , cos( /2); sin( /2)

A B

C

90o

180o/n

c

ab 90o

2sin2sin

2cos2cos2coscos

c

2cos1

02coscos

c

oo ccn 90 ; 0cos ; 02/cos ; 180 ; 2 oo ccn 60 ; 2/1cos ; 2/12/cos ; 120 ; 3

oo ccn 45 ; 2/1cos ; 2/12/cos ; 90 ; 4 oo ccn 30 ; /23cos ; /232/cos ; 60 ; 6

2

c

222

223

224

226

o90c

o60c

o45c

o30c

Angle between A and B axis

AB

A Bo60

o90

A Bo45

A Bo30

C

a b

What are a and b?

0)2/sin(1

)2/cos(00

2sin2sin

2cos2cos2coscos

b

A: 2, = 180o, cos( /2) = 0; sin( /2) = 1

B: 2, = 180o, cos( /2) = 0; sin( /2) = 1

C: n, = 360o /n , cos( /2); sin( /2)

0)2/sin(1

)2/cos(00

2sin2sin

2cos2cos2coscos

a

a = 90o.

b = 90o.

30o

222

90o

223

60o

224

45o

226

AB

C

A

B

C

A

B

C

A

B

C

Case: 23n

A: 2, = 180o, cos( /2) = 0; sin( /2) = 1

B: 3, = 120o, cos( /2) = 0.5; sin( /2) = 30.5/2

C: n, = 360o /n , cos( /2); sin( /2)

A B

C

90o

360o/n

c

ab 60o

)2/3(1

)5.0(02cos

2sin2sin

2cos2cos2coscos

c

'4454 ; 3/1cos ; 2/12/cos ; 120 ; 3 oo ccn

'1635 ; 6/2cos ; 2/12/cos ; 90 ; 4 oo ccn

oo ccn 0 ; 1cos ; /232/cos ; 60 ; 6 None exist

The rest of combination does not exist!

233

234

236

3

1

2/32/3

5.05.00

2sin2sin

2cos2cos2coscos

a

Case: 233

A: 2, = 180o, cos( /2) = 0; sin( /2) = 1

B: 3, = 120o, cos( /2) = 0.5; sin( /2) = 30.5/2

C: 3, = 120o, cos( /2) = 0.5; sin( /2) =30.5/2

a = 70o32.

3

1

2/31

5.005.0

2sin2sin

2cos2cos2coscos

b

b = 54o44.

70o32’

54o44’

54o44’ 233x

y

z

000

]111[]111[

]100[A

B

C

Angle between A and B is

'44543

1coscos311]111[]100[ occc

Angle between A and C is'4454

3

1coscos311]111[]100[ obbb

Angle between B and C is'3270

3

1coscos331]111[]111[ obbb

3

1

)2/1(2/3

)2/1(5.00

2sin2sin

2cos2cos2coscos

a

Case: 234

A: 2, = 180o, cos( /2) = 0; sin( /2) = 1

B: 3, = 120o, cos( /2) = 0.5; sin( /2) = 30.5/2

C: 4, = 90o, cos( /2) = 1/20.5; sin( /2) = 1/20.5

a = 54o44.

2

2

2/11

)2/1(05.0

2sin2sin

2cos2cos2coscos

b

b = 45o.

54o44’

234

45o

35o16’

x

y

z

000

]111[

]110[

]100[

AB

C

Angle between A and B is'1635

6

2coscos322]111[]110[ occc

Angle between A and C isobbb 45

2

1coscos121]100[]110[

Angle between B and C is'4454

3

1coscos131]100[]111[ obbb

Geometry of the permissible nontrivial combination of rotations:

Combination 2A = 2B = 2C = c a b

222223224226233234

180o

180o

180o

180o

180o

180o

90o

60o

45o

30o

54o4435o16

180o

180o

180o

180o

120o

120o

180o

120o

90o

60o

120o

90o

90o

90o

90o

90o

54o4445o

90o

90o

90o

90o

70o3254o44

222

International symbol

322

32(2)

Just like 3m(m)Only one independent

2 fold rotation axis

422

(1)(2)

(3)

(1) (2)

(3)

622

22 operation is basicallyon the plane!

Schonllies notation: T

n22 222 32(2) 422 622 Dn D2 D3 D4 D6

dihedral

233

Tetrahedral

Schonllies notation

different dihedral angleA

2/3C 2/3B

A

2/3B

23 is enough to specify the symmetry!

23

=

International symbol

http://en.wikipedia.org/wiki/Tetrahedron

54o44’

234 or 432

45o

35o16’

2

A3

2B

C

Schonllies notation: O Octahedron

International symbol

http://en.wikipedia.org/wiki/Octahedron

11 axial combinations

1 2 3 4 6222 322 422 622233 432

11 axial combinations + Extender

vertical m

n

horizontal m

n 422 horizontal

vertical diagonal for Dn, T, O

Ways to add m:

Not for Cn

Extender: v, h, d, ! (+ extender create new rotation axis!)1

1 2 3 4 6 222 32 422 622 23 432

v

h

d

1

See readingcrystal4.pdf

Cnv, Dnv

Tv, Ov

Cnh, Dnh

Th, Oh

Cni, Dni

Ti, Oi

Dnd

Td, Od

http://ocw.mit.edu/courses/materials-science-and-engineering/3-60-symmetry-structure-and-tensor-properties-of-materials-fall-2005/readings/crystal4.pdf

4

We will explain it later

m

m

1

-

m

2

2mm

-

m

2

3m

63

m

3

4mm

1 2 3 4 6 222 32

v

h

d

1

Cnv, Dnv

Tv, Ov

Cnh, Dnh

Th, Oh

Cni, Dni

Ti, Oi

Dnd

Td, Od

-

m

4

-

m

4

4

6mm

m

6

-

m

6

mmm

222

mmm

222

mmmmmm

222

m24m

23

26m

26m

m

23

Let’s look at some cases

Two fold rotation (2) + horizontal mirror (h)

1 Ah

A

hR

R

L

?: (1) (3)

at the point of intersection

1

h

A

1

hA

1 A1

A

h

h

1A

h

A h11

1

1

1

11

mA h

2

m2

}1 1{ hA hC2

A

h

(1) R(2)R

(3)L

xyz

zyx

updown

RL

RL

L

R: right-handednessL: left-handedness

2/A

h

1

R

L

mA h

42 hC4

Four fold rotation (4) + horizontal mirror (h)

updown

RL

Four fold rotation (4) + vertical mirror (v)

updownR

R

RR

L

LL

L mmA v 42

The mirror that you put in

Mirror 45o with respect to the firstmirror set

vC4

3/A

h

1hC6

mA h

63

vC6mmA v 63

Six fold rotation (6) + horizontal mirror (h)

R R

R

R R

R

Down Up

L

L

L

L

L

L

Six fold rotation (6) + vertical mirror (v)

Group symmetry elements: 12

63

32 m

A h hC3

} 1{ 3/3/3/23/2 AAAA h

Three fold rotation (3) + horizontal mirror (h)

R R

R

updown

L

L

L

(1) (1) (1) (2) (1) (3) (1) (1) (1) (2) (1) (3)

13/2A

3/23/4 AAh

13/ A

13/ ANew two step operations

Roto-inversion

3/3/ 1 AA

3/3/ 1 AA

(1)

(2)(3)

(1)R

(2)R

(3)L

A~

n~

1~

2~

3~

4~

5~

6~

7~

8~

Roto-reflection

2~

1 1~

2 6~

3 4~

4

5 3~

6 7 8~

8

Roto-inversion

A

(1)R

(2)R

(3)L

nThe one used

313/2 A

(1)

(2) (3)(4)

(5)(6)

(1) (1)(1) (2)(1) (3)(1) (4)(1) (5)(1) (6)

13/2A

3/23/4 AA

3/23/2 1 AA

3/23/4 1 AA

1

}1 1{ 3/23/23/23/2 AAAA

ihv CCC 333 3 m

3 m3

Three fold rotation (3) + vertical mirror (v)

mA v 33/2 vC3

Three fold rotation (3) + inversion ( )1

iC3

R

R RL

LL Down Up

11

down up

12

R

L

111

m

212

?2

m2 SC

You can except 414

4

(1)(2) (1) To (1) 1

(3)

(1) To (2) 2/A

(4)(1) To (3) A

(1) To (4) 2/A

4S Sphenoid (Greek word for axe)

Not tetrahedron

Equal length The rest four: equal length.

nnn 1 odd is If

R L

RL

} 1{ 2/2/ AAA

nnn 1even is If

How about ? 6 Down Up

R

L R

L

R L 6

m

3

4 is a special one that you have to add to the11 axial combination

4

Add h m

4

L

L

R

R R

R R

R R

R

R

422

L R

R L L

R R L

R L

L R

R L L R

mmm

224

+ h

4

Add m

4

L

L

R

1

Homework!

R R

R L

R L

Look at the ppt file that I send you regarding to222 + Extender (v, h, d, ) as an example!1

T 23updown

all down.

all up.

T 23Add a horizontal

mirror plane Th 32

m

Create aninversion center

inversion

RR

R L

L R

R L

L R

Crystal System

Symmetry Direction

Primary Secondary Tertiary

Triclinic None    

Monoclinic [010]    

Orthorhombic [100] [010] [001]

Tetragonal [001] [100]/[010] [110]

Hexagonal/Trigonal [001] [100]/[010] [120]/[1 0]

Cubic[100]/[010]/

[001] [111] [110]

1

Buerger’s book

3D crystallographic point group

2D lattices: chapter 7 (pg. 69-83)

Euler’s construction: pg. 35-43

Some combination theorems: chapter 6

Points group: pg: 59-68