(1) Course of MIT 3.60Symmetry, Structure and Tensor Properties of Materials(abbreviation: SST)

http://www.youtube.com/watch?v=vT_6DlaHcWQ&feature=PlayList&p=7E7E396BF006E209&playnext_from=PL&index=1Fall 2005, lectures given by Professor Bernhardt Wuensch

References for the first four parts:

(2) Ref. “Elementary crystallography”, Martin J. Buerger, 1963 (out of print, available in Physics Library)

(3) International Tables for Crystallography(International Unions for Crystallography) V. A, B, C, …http://it.iucr.org/Ab/contents/

crystallography

X-ray crystallography

Optical crystallography (polarized light)

Geometrical crystallography (symmetry theory)

crystallography

Crystal Mapping or geometry

Basic Symmetry(Two hours)

Geometrical crystallography: the study of patterns and their symmetry

Example

Motif

Are any of these patterns the same or are there all different?

T

T

: operation of translation magnitude, direction, no unique origin, like a plain vector

Other symmetry?

A

Rotation: A

location of rotation axis

angle of rotation

A

A 2 fold rotation

How about this one?T

New type of transformation! Reflection! Symbol used forreflection is m (mirror).

m? No! m? Yes!

m? Yes!

Definition of Symmetry element:Symmetry element is the locus of points left unmoved (invariant) by the operation.

What we have found for 2-dimensional symmetry operations?

T

Translation:mReflection:

ARotation: in the above case

byaxbyaxyx 2,2,,

mReflection:

x

yyx,

yx,

) ( ,, xmyxyx & pass through the origin

ARotation:x

y yx,

yx, A

yxyx ,,

Translation:Reflection:Rotation:

That is all we can do in 2D!

byaxyx ,,yxyx ,, yxyx ,,

In 3-D, one more operation

x

y

z

zyxzyx ,,,,

R

L

Inversion

Rotation

1D:

TranslationT

maxx

xx

Analyticalsymbol

m

IndividualOperation

Geometricalsymbol

Rotation axisn

2 n = integer

Analyticalsymbol

m

IndividualOperation

Geometricalsymbol

n A n - gonReflectionRotation

1 (no symmetry)

Add another translation vector

1T

1T

2T

2T X

1T

2T

Already covered by 1T

1T

2T

1T

2T

and are non-colinear.

21,TT

2D space lattice.

mnTmTn , ;21

exist

Ocolinear.

12 TpT

:(p integer) not a new translation vector

Lattice: frame work of a periodic crystalline structure (same environment for every point)

There are many ways to choose a cell with the same area.

In 2D lattices:Define the area uniquely associated with a lattice point.

1T

2T

Unit cell

21,TT

Array of lattice points cell

21,TT

1T

2T '

2T

conjugate translations'21,TT

Different cells withthe same area.

Which one to use? Rules: (1) pick the shortest translations; (2) pick that display the symmetry of the lattice.

21,TT

Handednesschiral-moleculeschirality

T

1T

2T

21 12 TTT

'21 03.143.2 TTT

'

2T

'21 TT

Cartesian coordinate

Rational direction

integer

Use lattice net to describe is much easier!

Extended to 3D

In general 2D

321 TwTvTuT

21 TvTuT

u, v, w: integer

Notation for rational planes: 2D case – line: line equation

At1

Bt2

1B

y

A

x

3D case – plane: plane equation

At1

Bt2

Ct3

1C

z

B

y

A

x

convert to integers

ABCC

ABCz

B

ABCy

A

ABCx

ABCABzACyBCx

ABClzkyhx ABlACkBCh ;;

Rational intercept plane (h k l)

Equation of intercept plane

x

y

x

y

z

CBAlkh

1:

1:

1::

How many planes are there? 2D: AB lines

At1

Bt2

At1

Bt2

A = 2, B = 3 A = 2, B = 2

ABAyBxB

y

A

x 1

023 yx 623 yx 0 yx

2 yx

1)/1()/1()/1(

l

z

k

y

h

x

1 lzkyhx

2 lzkyhx3 lzkyhx

1st plane2nd plane

3rd plane

1/l

1/k1/h

x

y

znlzkyhx nth plane n = ABC

At1

Bt2

Ct3

3D: ABC planes

CBAlkh

1:

1:

1::

1C

z

B

y

A

x ABClzkyhx

A B C

p q

r

Common factornumber of planes =

pqr

ABC

(hkl) Individual plane

Symmetry related set{hkl}

DifferentSymmetry related set

)100(

)010(

)001(

)001(

)010(

)100(

)100( )001(

)001( )100(

xy

z

x y

z

{100}

{100}

Crystallographic equivalent?

Example:

Coordination of an atom in a cell:

1T

321 011 TTT

110

321 TzTyTx

xyz

coordinate of an atom

x: fraction of unit length of

y: fraction of unit length of

z: fraction of unit length of

Where are basic translation vectors of the cell

3T

2T

1T

3T2T

321 , , TTT