(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
Geometrical crystallography: the study of patterns and their symmetry
Example
Motif
Are any of these patterns the same or are there all different?
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 ,,
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
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: