introduction to geometric and structural...
TRANSCRIPT
Introduction to geometric and structural
Crystallography
Lecture No. 10:
Crystallographic space groups
Crystallographic space groups
• Space groups needed for the description of symmetry properties in the 3-dimensional space
• Space group: Totality of all symmetry operations (isometries) of 3-dimensional, infinite, and ideal crystal structure
• Notation of SG according to Hermann-Mauguin
• Number of different Types of space groups in 3-d space: 230
• But note: There is an infinite number of space groups!
• 73 types of space groups with identical point symmetry elements as compared to the crystallographic point group but additional translation: symmorphic space groups
H. Kirmse, HU Berlin, Physik, AG SEM VL zur Einführung in die geometrisch-Strukturelle Kristallographie
VL 10 Raumgruppen 2
Crystallographic space groups
• Description of space groups by coset decomposition:
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
3
• Decomposition of space groups of infinite order into cosets leads to:
230 types of space groups (3-d)
17 types of layer groups (2-d)
E = W1 W2 W3 … … Wi
T1 T1W2 T1W3 … … T1Wi
T2 T2W2 T2W3 … … T2Wi
T3 T3W2 T3W3 … … T3Wi
… … … … … …
… … … … … …
E = W1: unity
W: symmetry operation
T = {Ti}: normal subgroup of space group
H. Kirmse, HU Berlin, Physik, AG SEM
5 VL zur Einführung in die geometrisch-Strukturelle Kristallographie
VL 10 Raumgruppen H. Kirmse, HU Berlin, Physik, AG SEM
The 14 Bravais lattice types in 3-d space (A. Bravais, 1850)
Crystal system Centering Symbol
Triclinic Primitive aP
Monoclinic Primitive mP
Face-centered mA
Orthorhombic Primitive oP
Body-centered oI
Single face- cenntered
oC
All face-centered
oF
Crystal system Centering Symbol
Tetragonal Primitive tP
Body-centered tI
Trigonal Rhombohedral hR
Trig. + Hexagonal Primitive hP
Cubic Primitive cP
Body-centered cI
All face- centered
cF
Crystallographic space groups
• Asymmetric unit and unit cell
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
6 H. Kirmse, HU Berlin, Physik, AG SEM
Crystallographic space groups
• Asymmetric unit and unit cell
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
7 H. Kirmse, HU Berlin, Physik, AG SEM
Symmetry operation: translation
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
8
• Shift of a motive (asymmetric unit) by translation vector t
t
• Consequences: No longer point symmetry only
Space filling
New symmetry operations
• Application to: 1 dimension line groups
2 dimensions layer groups
3 dimensions space groups
H. Kirmse, HU Berlin, Physik, AG SEM
Combination of translation and reflection
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
9
• Translation • Reflection
H. Kirmse, HU Berlin, Physik, AG SEM
Symmetry elements: glide reflection
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
10
• Glide reflection
Combination of translation and reflection :
Step 1: translation by t = ½ a0
Step 2: reflection
H. Kirmse, HU Berlin, Physik, AG SEM
Symmetry elements: glide reflection
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
11
m a, b c n e = a;c
H. Kirmse, HU Berlin, Physik, AG SEM
(a+b)/4, (a+c)/4, or (b+c)/4
• Symbols
Symmetry elements: glide reflection
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
12
• Glide components
a, b
c
n
d
½ a or ½ b ½ a simultaneous to ½ b; or ½ a simult. to ½ c, or ½ b simult. to ½ c
(a+b)/2, (a+c)/2, or (b+c)/2
e
new!
H. Kirmse, HU Berlin, Physik, AG SEM
Symmetry elements: glide reflection
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
13
n d
(a+b)/4, (a+c)/4, or (b+c)/4
(a+b)/2, (a+c)/2, or (b+c)/2
H. Kirmse, HU Berlin, Physik, AG SEM
Symmetry elements: screw axis
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
14
• 2-fold rotation axis • 2-fold screw axis
= ½ ao
• Translation period of a screw axis np:
2 21
H. Kirmse, HU Berlin, Physik, AG SEM
= p/n
Symmetry elements: screw axis
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
15
• 3-fold screw axis:
• 31 and 32 are enantiomorphic screw axes
32
31 32
= 2/3
Rotation in math. positive sense!
(left = left-hand rotation) (right)
Periodicity
H. Kirmse, HU Berlin, Physik, AG SEM
Symmetry elements: screw axis
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
16
= 1/4 = 2/4 = 3/4
41 43
42
• 41 and 43 are enantiomorphic screw axes
• 41 right-hand , 43 left-hand, 42 no sense of rotation (like 2)
• 4-fold screw axis:
H. Kirmse, HU Berlin, Physik, AG SEM
Symmetry elements: screw axis
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
17
• 6-fold screw axis:
= 1/6 = 2/6 = 3/6
61 62 63
H. Kirmse, HU Berlin, Physik, AG SEM
Symmetry elements: screw axis
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
18
• 6-fold screw axis:
= 4/6 = 5/6
64 65
enantiomorphic: 61 and 65 as well as 62 and 64
61: right-hand rotation 62: right-hand 31 and 2
63: 21 and 3
64: left-hand 32 and 2
65: left-hand rotation
H. Kirmse, HU Berlin, Physik, AG SEM
Rotation and screw axes running along the viewing direction
Symmetry elements: rotation and screw axes
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
19
Rotation and screw axes running normal or inclined to viewing direction
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
20
½ +
½ +
½ -
½ -
½
+
-
+
-
Projection onto basal plane
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
21
½ +
½ +
½ -
½ -
½
+
-
+
-
Symmetry elements
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
22
½ +
½ +
½ -
½ -
½
+
-
+
-
1. Assignment to crystal system
2. Finding rotation axes
3. Finding mirror planes
4. Finding inversion centers
5. Conclusion of space group
Symmetry elements
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
23
½ +
½ +
½ -
½ -
½
+
-
+
-
1. Assignment to crystal system
2. Finding rotation axes
3. Finding mirror planes
4. Finding inversion centers
5. Conclusion of space group
Symmetry elements
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
24
½ +
½ +
½ -
½ -
½
+
-
+
-
Symmetry elements
Orthorhombic primitive
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
25
½ +
½ +
½ -
½ -
½
+
-
+
-
Symmetry elements
Orthorhombic primitive
1. Assignment to crystal system
2. Finding rotation axes
3. Finding mirror planes
4. Finding inversion centers
5. Conclusion of Space group
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
26
½ +
½ +
½ -
½ -
½
+
-
+
-
Symmetry elements
Orthorhombic primitive
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
27
½ +
½ +
½ -
½ -
½
+
-
+
-
Symmetry elements
¼, ¾ ¼, ¾
¼, ¾ ¼, ¾
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
28
½ +
½ +
½ -
½ -
½
+
-
+
-
Symmetry elements
¼, ¾ ¼, ¾
¼, ¾ ¼, ¾
1. Assignment to crystal system
2. Finding rotation axes
3. Finding mirror planes
4. Finding inversion centers
5. Conclusion of space group
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
29
½ +
½ +
½ -
½ -
½
+
-
+
-
Symmetry elements
¼, ¾ ¼, ¾
¼, ¾ ¼, ¾
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
30
½ +
½ +
½ -
½ -
½
+
-
+
-
Symmetry elements
¼, ¾ ¼, ¾
¼, ¾ ¼, ¾ ¼, ¾
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
31
½ +
½ +
½ -
½ -
½
+
-
+
-
Symmetry elements
¼, ¾ ¼, ¾
¼, ¾ ¼, ¾ ¼, ¾
1. Assignment to crystal system
2. Finding rotation axes
3. Finding mirror planes
4. Finding inversion centers
5. Conclusion of space group
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
32
½ +
½ +
½ -
½ -
½
+
-
+
-
Symmetry elements
¼, ¾ ¼, ¾
¼, ¾ ¼, ¾ ¼, ¾
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
33
½ +
½ +
½ -
½ -
½
+
-
+
-
Symmetry elements
¼, ¾ ¼, ¾
¼, ¾ ¼, ¾ ¼, ¾
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
34
½ +
½ +
½ -
½ -
½
+
-
+
-
Symmetry elements
¼, ¾ ¼, ¾
¼, ¾ ¼, ¾ ¼, ¾
1. Assignment to crystal system
2. Finding rotation axes
3. Finding mirror planes
4. Finding inversion centers
5. Conclusion of space group
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
35
½ +
½ +
½ -
½ -
½
+
-
+
-
¼, ¾ ¼, ¾
¼, ¾ ¼, ¾ ¼, ¾
Space group:
H. Kirmse, HU Berlin, Physik, AG SEM
Example of space group determination
• Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
36
½ +
½ +
½ -
½ -
½
+
-
+
-
Space group:
¼, ¾ ¼, ¾
¼, ¾ ¼, ¾ ¼, ¾
n
2
m
2
n
2P 11
H. Kirmse, HU Berlin, Physik, AG SEM
Example: Marcasite (FeS2)
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
37
n
2
m
2
n
2P 11
H. Kirmse, HU Berlin, Physik, AG SEM
Representation of space groups
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
38 H. Kirmse, HU Berlin, Physik, AG SEM
Symmetry elements: glide reflection
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
39
m a, b c n e = a;c
H. Kirmse, HU Berlin, Physik, AG SEM
P nc2
Symmetry elements: glide reflection
VL zur Einführung in die geometrisch-Strukturelle Kristallographie VL 10 Raumgruppen
40
A em2
H. Kirmse, HU Berlin, Physik, AG SEM