3D Symmetry _2

1
3D Symmetry _2 (Two weeks)
2
3D lattice Reading crystal7.pdf
Building the 3D lattices by adding another
translation vector to existing 2D lattices
Oblique (symmetry 1)
triclinic
General
Triclinic Primitive
3
Oblique (symmetry 2)
projection
Is this position OK?
X
4 choices in order to maintain 2 fold rotation
symmetry
4
Double cell side centered
Double cell side centered
5
Double cell body centered
Some people use based centered, some use body
centered.
monoclinic
6
Rectangular (symmetry m)
90o
90o
already exist!
the same.
Rectangular (symmetry g)
cm ?
7
Rectangular (symmetry 2mm)
P2mm P2mg p2gg
Orthorhombic primitive
8
Double cell side centered
Orthorhombic base-centered
Orthorhombic base-centered
Double cell side centered
9
Orthorhombic body-centered
rectangular
10
Centered Rectangular (symmetry 2mm)
C2mm
the same
11
Face centered
orthorhombic
12
Square (symmetry 4, 4mm)
P4 P4mm p4gm
Tetragonal primitive
13
Tetragonal Body centered
Tetragonal
14
Hexagonal (symmetry 3, 3m)
p3
p31m
p3m1
not in this category Why?
Hexagonal primitive
Rhombohedral
15
Hexagonal primitive
Rhombohedral
triple cell
16
2/3
2/3
1/3
?
1/3
a b c ? ? ? ? 90o
17
Hexagonal (symmetry 3m, 6, 6mm)
p6
p6mm
can only located at positions
Hexagonal primitive
p31m
Hexagonal 6 related can only fit 3P!
18
11 lattice types already
cubic (isometric)
Special case of orthorhombic (222) with a b
c Check the 3 fold rotation symmetry in 111
direction
Primitive (P) Body centered (I) Face centered
(F) Base center (C)
Tetragonal (I)?
Cubic
a b ? c
100/010/001
111
Tetragonal (P)
19
Another way to look as cubic Consider an
orthorhombic and requesting the
diagonal direction to be 3 fold rotation symmetry
Primitive
222P ? 23P
Body centered
222I ? 23I
222F ? 23F
Face centered
222C ? 23C 3 fold rotation symmetry does not
exist in base centered cell
20
Bingo! 14 Bravais lattices!
21
Lattice type - compatibility with - point
group reading crystal9.pdf.
Crystal Class Bravais Lattices Point Groups
Triclinic P (1P)
Monoclinic P (2P), C(2I) 2, m, 2/m
Orthorhombic P(222P), C(222C) F(222F), I(222I) 222, mm2, 2/m 2/m 2/m
Rhombohedral P (3P), 3R
Hexagonal P (3P)
Tetragonal P (4P), I (4I)
Isometric (Cubic) P (23P), F(23F), I (23I)
22
?
http//www.theory.nipne.ro/dragos/Solid/Bravais_t
able.jpg
P
I
T P
23
http//users.aber.ac.uk/ruw/teach/334/bravais.php
24
Next, we can put the point groups to the
compatible lattices, just like the cases in 2D
space group.
3D Lattices (14) 3D point groups ?? 3D Space
group
There are also new type of symmetry shows up in
3D space group, like glide appears in 2D space
(plane) group!
25

• The naming (Herman-Mauguin space group symbol) is
  the same as previously mentioned in 2D plane
  group!
• The first letter identifies the type of lattice
• P Primitive I Body centered F Face centered
• C C-centered B B-centered, A A-centered
• The next three symbols denote symmetry elements
  in certain directions depending on the crystal
  system. (See next page)

26
Monoclinica ? b 90o c ? b 90o. b axis is
chosen to correspond to a 2-fold axis of
rotational symmetry axis or to be perpendicular
to a mirror symmetry plane. Convention for
assigning the other axes is c lt a. a ? c is
obtuse (between 90º and 180º). OrthorhombicThe
standard convention is that c lt a lt b. 
Once you define the cell following the convention
? A, B, C centered
27
Crystal System Symmetry Direction Symmetry Direction Symmetry Direction
Crystal System Primary Secondary Tertiary
Triclinic None    
Monoclinic 010    
Orthorhombic 100 010 001
Tetragonal 001 100/010 110
Hexagonal/Rhombohedral 001 100/010 120/1 0
Cubic 100/010/001 111 110
28
Monoclinic 2
Consider 2P Monoclinic 2
2D
p2
P2
?/2
P2
?/2
29
How about 2I Monoclinic 2
c
b
a
There is a lattice point in the cell centered!
30
z
(1)
(2)
(3)
z
z 1/2
(3)
(1)
(2)
New type of operation
In general
31
A
B
(3)
(1)
(2)
(3)
(2)
(1)
32
Crystal diffraction 2.ppt page 31
x
x
x
33
Specifying
34
For a 3-fold screw axis
3
4-fold screw axis
35
(No Transcript)
36
31
32
3
21
2
41
42
43
4
37
61
62
63
64
65
6
38
62
39
Example to combine lattice with screw symmetry
D
A
A 2-fold translation (to arise at B, C, or D)
B
C
Rotation symmetry of B, C, and D is the same as A.
A 2
P 2 P2
40
A 21
21
P 21 P21
21
21
I 2 I2 or I 21 I21
A
A 2 ? E 21
Same, only shifted
E
A 21 ? E 2
I2 I21
41
Hexagonal lattice (P and R) with 3, 31, 32. Case
P first!
All translations in P have component on c of 0 or
unity!
A
B??
C?
B?
C??
B? and C?? same point B?? and C? equivalent
point
Having
42
P3
P31
P32
43
Case R!
A
E?
2/3
D??
All translations of R has component on c of 1/3
or 2/3!
E??
D?
1/3
Screw at
Designation of Space group
A
?
D E
3 31 32
0 c/3 2c/3
2?/3 2?/3 2?/3
c/3 2c/3 c
2c/3 c 4c/3
2?/3 2?/3 2?/3
31 32 3
32 3 31
R3 R31 R32
R3 R3


Hexagonal lattice (P, R) 3, 31, 32 ? P3, P31,
P32, R3.
44
(No Transcript)
45
Square lattice P with 4, 41, 42, 43.
The translation of P have component on c of 0 or
unity!
C?
A
B?
B???
B??
C??
C???
A 4 41 42 43
? 0 c/4 c/2 3c/4
B? ?/2 0 ?/2 c/4 ?/2 c/2 ?/2 3c/4
B?? ? 0 ? c/2 ? c ? 3c/2
B? 4 41 42 43
B?? 2 21 2 21
P4 P41 P42 P43
46
P41
P4
P43
P42
47
How to obtain Herman-Mauguin space group symbol
by reading the diagram of symmetry elements?
First, know the Graphical symbols used for
symmetry elements in one, two and three
dimensions!
International Tables for Crystallography (2006).
Vol. A, Chapter 1.4, pp. 711. http//www.kristal
l.uni-frankfurt.de/media/exercises/Symbols-for-sym
metryelements-ITC-Vol.A2.pdf
48
Symmetry planes normal to the plane of projection
Symmetry plane Graphical symbol Translation Symbol
Reflection plane None   m
Glide plane 1/2 along line   a, b, or c
Glide plane 1/2 normal to plane   a, b, or c
Double glide plane 1/2 along line 1/2 normal to plane (2 glide vectors)   e
Diagonal glide plane 1/2 along line, 1/2 normal to plane (1 glide vector)   n
Diamond glide plane 1/4 along line 1/4 normal to plane   d
49
1/2 along line
a, b, c
1/2 normal to plane
a, b, c
50
n-glide
d-glide
51
e-glide
52
Symmetry planes parallel to plane of projection
Symmetry plane Graphical symbol Translation Symbol
Reflection plane None   m
Glide plane 1/2 along arrow   a, b, or c
Double glide plane 1/2 along either arrow   e
Diagonal glide plane 1/2 along the arrow   n
Diamond glide plane 1/8 or 3/8 along the arrows   d
3/8
1/8
The presence of a d-glide plane automatically
implies a centered lattice!
53
Symmetry Element Graphical Symbol Translation Symbol
Identity None None   1
2-fold ? page None   2
2-fold in page None   2
2 sub 1 ? page 1/2   21
2 sub 1 in page 1/2   21
3-fold None   3
3 sub 1 1/3   31
3 sub 2 2/3   32
4-fold None   4
4 sub 1 1/4   41
4 sub 2 1/2   42
4 sub 3 3/4   43
6-fold None   6
6 sub 1 1/6   61
6 sub 2 1/3   62
6 sub 3 1/2   63
54
Symmetry Element Graphical Symbol Translation Symbol
6 sub 4 2/3   64
6 sub 5 5/6   65
Inversion None   1
3 bar None   3
4 bar None   4
6 bar None   6 3/m
2-fold and inversion None   2/m
2 sub 1 and inversion None   21/m
4-fold and inversion None   4/m
4 sub 2 and inversion None   42/m
6-fold and inversion None   6/m
6 sub 3 and inversion None   63/m
55
c-glide
? b
n-glide
c
21
? c
a
n
2
21
2
m
b-glide
m
b
? c
? a
56
From the point group mmm ? orthorhombic
For orthorhombic primary direction is (100),
secondary direction is (010), and tertiary is
(001).
lattice
for orthorhombic
C
Short symbol
No. 17 orthorhombic that can be derived
57
Principles for judging crystal system by space
group
58

• Orthorhombic All three symbols following the
  lattice descriptor will be either mirror planes,
  glide planes, 2-fold rotation or screw axes (i.e.
  Pnma, Cmc21, Pnc2)
• Monoclinic The lattice descriptor will be
  followed by either a single mirror plane, glide
  plane, 2-fold rotation or screw axis or an
  axis/plane symbol (i.e. Cc, P2, P21/n)
• Triclinic The lattice descriptor will be
  followed by either a 1 or a (-1).

http//chemistry.osu.edu/woodward/ch754/sym_itc.h
tm
59
What can we do with the space group
information contained in the International
Tables?
1. Generating a Crystal Structure from its
Crystallographic Description
2. Determining a Crystal Structure from Symmetry
Composition
60
Example Generating a Crystal Structure
http//chemistry.osu.edu/woodward/ch754/sym_itc.h
tm
Description of crystal structure of Sr2AlTaO6
 
Atom x y z
Sr 0.25 0.25 0.25
Al 0.0 0.0 0.0
Ta 0.5 0.5 0.5
O 0.25 0.0 0.0
61
From the space group tables
http//www.cryst.ehu.es/cgi-bin/cryst/programs/nph
-wp-list?gnum225
32 f 3m xxx, -x-xx, -xx-x, x-x-x, xx-x, -x-x-x, x-xx, -xxx
24 e 4mm x00, -x00, 0x0, 0-x0,00x, 00-x
24 d mmm 0 ¼ ¼, 0 ¾ ¼, ¼ 0 ¼, ¼ 0 ¾, ¼ ¼ 0, ¾ ¼ 0
8 c ¼ ¼ ¼ , ¼ ¼ ¾
4 b ½ ½ ½
4 a 000
62
Sr 8c Al 4a Ta 4b O 24e
40 atoms in the unit cell stoichiometry
Sr8Al4Ta4O24 ? Sr2AlTaO6
F face centered ? (000) (½ ½ 0) (½ 0 ½) (0
½ ½)
Sr
(000) (½½0) (½0½) (0½½)
8c ¼ ¼ ¼ ? (¼¼¼) (¾¾¼) (¾¼¾) (¼¾¾) ¼ ¼ ¾
? (¼¼¾) (¾¾¾) (¾¼¼) (¼¾¼)
Al
¾ ½ 5/4 ¼
4a 0 0 0 ? (000) (½ ½ 0) (½ 0 ½) (0 ½ ½)
63
(000) (½½0) (½0½) (0½½)
Ta
4b ½ ½ ½ ? (½½½) (00½) (0½0) (½00)
(000) (½½0) (½0½) (0½½)
O
x00
24e ¼ 0 0 ? (¼00) (¾½0) (¾0½) (¼½½)
¾ 0 0 ? (¾00) (¼½0) (¼0½) (¾½½)
-x00
0 ¼ 0 ? (0¼0) (½¾0) (½¼½) (½¾½)
0x0
0-x0
0 ¾ 0 ? (0¾0) (½¼0) (½¾½) (0¼½)
00x
0 0 ¼ ? (00¼) (½½¼) (½0¾) (0½¾)
00-x
0 0 ¾ ? (00¾) (½½¾) (½0¼) (0½0¼)
64
 
 
Sr ion is surrounded by 12 O Sr-O distance d
2.76 Å
65
Determining a Crystal Structure from Symmetry
Composition
 
66
First step calculate the number of formula units
per unit cell Formula Weight SrTiO3 87.62
47.87 3? (16.00) 183.49 g/mol (M)
Unit Cell Volume (3.90?10-8 cm)3 5.93 ? 10-23
cm3 (V)
(5.1 g/cm3)?(5.93 ? 10-23 cm3) weight in a unit
cell
(183.49 g/mole) / (6.022 ?1023/mol) weight of
one molecule of SrTiO3
67
? (5.1 g/cm3)?(5.93 ? 10-23 cm3)/ (183.49
g/mole/6.022 ?1023/mol) 0.99
? number of molecules per unit cell 1 SrTiO3.
From the space group tables (only part of it)
6 e 4mm x00, -x00, 0x0, 0-x0,00x, 00-x
3 d 4/mmm ½ 0 0, 0 ½ 0, 0 0 ½
3 c 4/mmm 0 ½ ½ , ½ 0 ½ , ½ ½ 0
1 b ½ ½ ½
1 a 000
http//www.cryst.ehu.es/cgi-bin/cryst/programs/nph
-wp-list?gnum221
68
Sr 1a or 1b Ti 1a or 1b ? Sr 1a Ti 1b or vice
verse O 3c or 3d
Evaluation of 3c or 3d Calculate the Ti-O bond
distances d (O _at_ 3c) 2.76 Å (0 ½ ½) d (O _at_
3d) 1.95 Å (½ 0 0, Better)
Atom x y z
Sr 0.5 0.5 0.5
Ti 0 0 0
O 0.5 0 0