Download Crystallography 1

References for the first four parts:
(1) Course of MIT 3.60
Symmetry, Structure and Tensor Properties of Materials
(abbreviation: SST)
http://www.youtube.com/watch?v=vT_6DlaHcWQ&feature=PlayList&p=7
E7E396BF006E209&playnext_from=PL&index=1
Fall 2005, lectures given by Professor Bernhardt Wuensch
(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
Crystal
Mapping or geometry
X-ray crystallography
crystallography
Optical crystallography (polarized light)
Geometrical crystallography (symmetry
theory)
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
A
location of rotation axis
Rotation: A
angle of rotation
A 2 fold rotation

T
How about this one?
m? Yes!
New type of transformation! Reflection! Symbol used for
reflection is m (mirror).
m? No!
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?

x, y  x  a, y  b  x  2a, y  2b
Translation: T
Reflection: m
Rotation: A in the above case
x, y
Reflection:
y
x, y
m
x
x, y  x , y (m  x & pass through the origin )
y
Rotation:
A
x, y
x, y  x , y
Translation:
Reflection:
Rotation:
x, y
x, y  x  a , y  b
x, y  x , y
x, y  x , y
That is all we can do in 2D!
A
x
In 3-D, one more operation
z
Inversion
R
y
x, y, z  x , y, z
x
L
1D:

T Translation
m Rotation
x xa
xx
Analytical
symbol
Individual
Operation
m

Rotation axis
Reflection
Rotation
2

n
Geometrical
symbol
n = integer
Analytical
symbol
Individual
Operation
m
n

A
Geometrical
symbol
n - gon
1 (no symmetry)

T1

Add another translation vector T2
exist

T1

T1

T2

T2
X
colinear.
 O
Already covered by T1


 T2  pT1 :(p integer) not a new translation vector


T1 and T2 are non-colinear.

T2

T1
 
T1 ,T2  2D space lattice.


nT1  mT2 ;    n, m  
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Lattice: frame work of a periodic crystalline structure
(same environment for every point)
In 2D lattices:
Define the area uniquely associated with a lattice point.
Unit cell

T2

T1
 
T1 ,T2  Array of lattice points  cell
There are many ways to choose a cell with the same area.
 
T1 ,T2
  ' conjugate translations
T1 ,T2

T2
'
T2

T1
Different cells with
the same area.
Which one to use?
Rules: (1) pick theshortest translations;
(2) pick T1 ,T2 that display the symmetry of the lattice.
Handedness
chiral-molecules
chirality
'
T2

T2

T

T1



 T  2T1  1T2 Rational direction
integer

'


'
T1  T2
T  2.43T1  1.03T2
Cartesian coordinate
Use lattice net to describe is much easier!



In general
2D
T  uT1  vT2
u, v, w: integer




Extended to 3D  T  uT1  vT2  wT3
y
Notation for rational planes:
 2D case – line: line equation
Bt2
x y
 1
A B
x
At1
 3D case – plane: plane equation
x y z
  1
A B C
y
Bt2
x
convert to integers
Ct
At1
3
ABCx ABCy ABCz


 ABC
z
A
B
C
BCx  ACy  ABz  ABC Equation of intercept plane
1 1 1
h

BC
;
k

AC
;
l

AB
h:k :l  : :
hx  ky  lz  ABC
A B C
Rational intercept plane (h k l)
 How many planes are there?
 2D: AB lines
x y
  1  Bx  Ay  AB
A B
A = 2, B = 3
A = 2, B = 2
Bt2
Bt2
At1
At1
3x  2 y  0
3x  2 y  6
x y 0
x y 2
 3D: ABC planes
x y z
  1
A B C
Bt2
hx  ky  lz  ABC
1 1 1
h:k :l  : :
A B C
y
At1
Ct3
1/k
1st plane
2nd plane
3rd plane
nth plane
hx  ky  lz  1
hx  ky  lz  2
hx  ky  lz  3
hx  ky  lz  n n = ABC
r
A BC
p q
Common factor
1/h
x
1/l
z
x
y
z


1
(1 / h) (1 / k ) (1 / l )
ABC
number of planes =
pqr
Crystallographic equivalent?
(hkl) Individual plane
{hkl} Symmetry related set
Example:
{100}
z
(100) ( 1 00)
(010) (0 1 0)
y (001) (00 1 )
x
z
{100}
(100) ( 1 00)
(001) (00 1 )
x
y
Different
Symmetry
related set
Coordination of an atom in a cell:

T3

T1

T2



xT1  yT2  zT3



1T1  1T2  0T3
110
xyz
coordinate of an atom

x: fraction of unit length of T1

y: fraction of unit length of T2

z: fraction of unit length of T3
  
Where T1 , T2 , T3 are basic translation vectors of the cell