(1) course of mit 3.60 symmetry, structure and tensor properties of materials (abbreviation: sst)
TRANSCRIPT
(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