Transcript Slide 1

Physics 590
“International Tables of Crystallography”
“Everything you wanted to know about beautiful flies, but were afraid to ask.”
Gordie Miller (321 Spedding)
Schönflies
Proposed Plan
(1) What basic information is found on the space group pages…
(2) Stoichiometry of the unit cell (Wyckoff sites)
(3) Site (point) symmetry of atoms in solids
(4) Solid-solid phase transitions (group-subgroup relationships)
(5) Diffraction conditions – what to expect in a XRD powder pattern.
References
Space Groups for Solid State Scientists, G. Burns and A. M. Glazer
(Little mathematical formalism; prose style)
International Tables for Crystallography (http://it.iucr.org/)
Bilbao Crystallographic Server (www.cryst.ehu.es)
(Comprehensive resources for all space groups)
1
What can we learn from the International Tables?
BaFe2As2
c
Space Group: I4/mmm
Lattice Constants: a = 3.9630 Å
c = 13.0462 Å
Asymmetric Unit:
Ba (2a):
0
0
0
Fe (4d):
½
0
¼
As (4e):
0
0
0.3544
Intensity
(Arb. Units)
(hkl) Indices
h + k + l = even integer (2n)
(013)
(200)
(112)
a
(116)
(213)
b
(004)
(002)
(015)
(215) (028)
(011)
2θ (Cu Kα)
2
Typical Space Group Pages…
Symbolism
Point Symmetry Features
Stoichiometry
Structure of Unit Cell
Diffraction
Extinction
Conditions
Subgroup/Supergroup
Relationships
3
Symbolism
Point Group of Crystal
the Space Group System
Space Group
Molecules
Solids
Symmetry Operation
Schönflies Notation
International Notation
Proper Rotation (by 2π/n)
Cn (C2, C3, C4, …)
“n” (2, 3, 4, …)
Identity
E = C1
Sn = sh  Cn (S3, S4, S5, …)
Improper Rotation
1
n  1  n (3, 4, 5,
)
Inversion (x,y,z)  (–x,–y,–z)
i = S2
1
Mirror plane  Principal Axis
sh = S1
/m (n/m is the designator: 4/m)
Mirror plane  Principal Axis
sv , sd (= S1)
m= 2
S2  2
NOTE: S n  n
( x, y,z )
( x, y,z )
S2 = s h  C 2
,
In Schönflies notation, what does the symbol S2 mean?
S2 = inversion
2 = reflection
(x,y,z)
C2 rotation followed by sh  C2 axis
In International notation, what does the symbol 2 mean?
x
2  12
( x, y,z )
y
2-fold (C2) rotation followed by inversion ( 1 )
Why are the symbols S2 and 2 not used?
(z)
y
,
x
(x,y,z)
( x, y,z )
(z)
4
Symbolism: Crystal Systems
What rotational symmetries are consistent with a lattice (translational symmetry)?
C1 C2 (2π/2) C3 (2π/3)
C4 (2π/4)
C6 (2π/6)
Crystal System
Minimum Symmetry
Primitive Unit Cell
Triclinic
None
a  b  c;     
Monoclinic
One 2-fold axis (b-axis)
a  b  c;  =  = 90,   90
Orthorhombic
Three orthogonal 2-fold axes
a  b  c;  =  =  = 90
Tetragonal
One 4-fold axis (c-axis)
a = b  c;  =  =  = 90
Cubic
Four 3-fold axes
a = b = c;  =  =  = 90
Trigonal
One 3-fold axis
a = b = c;  =  = 
a = b  c;  =  = 90,  = 120
Hexagonal
One 6-fold axis (c-axis)
a = b  c;  =  = 90,  = 120
c
a
b
Lattice Types
 = angle between b and c
 = angle between a and c
 = angle between a and b
5
Symbolism: Bravais Lattices
7 Crystal Systems = 7 Primitive Lattices (Unit Cells):
P
Crystal System
Minimum Symmetry
Primitive Unit Cell
Lattice Types
Triclinic
None
a  b  c;     
P
Monoclinic
One 2-fold axis (b-axis)
a  b  c;  =  = 90,   90
P C
Orthorhombic
Three orthogonal 2-fold axes
a  b  c;  =  =  = 90
P C (A) I F
Tetragonal
One 4-fold axis (c-axis)
a = b  c;  =  =  = 90
P I
Cubic
Four 3-fold axes
a = b = c;  =  =  = 90
P I F
Trigonal
One 3-fold axis
a = b = c;  =  = 
a = b  c;  =  = 90,  = 120
R (rhombohedral)
P
Hexagonal
One 6-fold axis (c-axis)
a = b  c;  =  = 90,  = 120
P
“Centered Lattices”
c
a
I
F
Body-
(All) Face-
C
B
Base-
?
b
 = angle between b and c
 = angle between a and c
 = angle between a and b
A
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Symbolism: Point Groups
Schönflies
Notation
Type
Symbol
Features
Uniaxial
n
Single rotation axis Cn
Low
Symmetry
Dihedral
Polyhedral
nh
+ mirror plane  Cn axis
nv
+ n mirror planes || Cn axis
1
Asymmetric (NO symmetry)
s
Mirror plane only
i
Inversion center only
n
Rotation axis Cn + n C2 axes  Cn axis
nd
+ n mirror planes || Cn axis
nh
+ mirror plane  Cn axis
T, Th , Td
Tetrahedral; 4 C3 axes (cube body-diagonals)
O, Oh
Octahedral; 4 C3 axes + 3 C4 axes (cube faces)
I, Ih
Icosahedral; 6 C5 axes
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Symbolism: Crystallographic Point Groups
Allowed Rotations = C1 C2 C3
Crystal
System
Schönflies
Symbol
Triclinic
(Holohedral)
Monoclinic
(Holohedral)
Orthorhombic
(Holohedral)
Tetragonal
(Holohedral)
C4
32 Point Groups
C6
International Symbol
Directions
Order /
Inversion?
Full
Abbrev.
1
1
1
1 / No
i
1
1
2 / Yes
s
1m1 or 11m
m (2)
2
121 or 112
2
2h
12/m1 or 112/m
2/m
2v
2mm
2mm
2
222
222
2h
2/m 2/m 2/m
mmm
4
4
4
4
4
4
4h
4/m
4/m
2d
42m
42m
8 / No
4v
4mm
4mm
8 / No
4
422
422
8 / No
4h
4/m 2/m 2/m
4/mmm
[010] or [001]
b
c
2 / No
Yes:
Laue Groups
2 / No
4 / Yes
[100][010][001]
a
b
c
4 / No
m2m or mm2
4 / No
8 / Yes
[001]{100}{110}
c
a
b
a+b
a–b
4 / No
4 / No
8 / Yes
4 m2
16 / Yes
8
Questions for Friday…
For the following space group symbol
Cmm2
(a) What is the crystal class?
Orthorhombic
(b)What is the lattice type?
Base (C)-centered
Which face(s) are centered?
ab-faces
(c) What is the point group of the space group
using Schönflies notation?
C
2v
(d)Does the point group contain the inversion
operation?
No
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Symbolism: Crystallographic Point Groups (cont.)
International Symbol
Crystal
System
Schönflies
Symbol
Full
Abbrev.
Directions
Trigonal
3
3
3
[001]{100}{210}
6
3
3
3v
3m or 3m1
3m or 3m1
3
32 or 321
32 or 321
3d
32 / m or 32 / m1
3 m or 3 m1
6
6
6
3h
6
6
6h
6/m
6/m
3h
6 m2
6 m2
12 / No
6v
6mm
6mm
12 / No
6
622
622
12 / No
6h
6/m 2/m 2/m
6/mmm
24 / Yes
T
23
23
Th
2 / m3
m3
Td
43m
43m
O
432
432
24 / No
Oh
4 / m32 / m
m3m
48 / Yes
(Holohedral)
Hexagonal
(Holohedral)
Cubic
(Holohedral)
c
a
b
[001]{100}{210}
c
a
b
{100}{111}{110}
a
b
c
Order /
Inversion?
3 / No
6 / Yes
6 / No
31m
6 / No
312
12 / Yes
31m
6 / No
6 / No
12 / Yes
62m
12 / No
24 / Yes
24 / No
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Symbolism: Symmetry Operations (4h = 4/mmm = 4/m 2/m 2/m)
Schönflies
International
Coordinates
Schönflies
International
Coordinates
(1)
E
1
x, y, z (𝑥, 𝑦, 𝑧)
(9)
i
1
–x, –y, –z (𝑥, 𝑦, 𝑧)
(2)
C2 = C42
2 0,0,z
–x, –y, z (𝑥, 𝑦, 𝑧)
(10)
σh
m x,y,0
x, y, –z (𝑥, 𝑦, 𝑧)
(3)
C4
4+ 0,0,z
–y, x, z (𝑦, 𝑥, 𝑧)
(11)
S43
4+ 0,0,z; 0,0,0
y, –x, –z (𝑦, 𝑥, 𝑧)
(4)
C43
4– 0,0,z
y, –x, z (𝑦, 𝑥, 𝑧)
(12)
S4
4− 0,0,z; 0,0,0
–y, x, –z (𝑦, 𝑥, 𝑧)
(5)
C2
2 0,y,0
–x, y, –z (𝑥, 𝑦, 𝑧)
(13)
σv
m x,0,z
x, –y, z (𝑥, 𝑦, 𝑧)
(6)
C2
2 x,0,0
x, –y, –z (𝑥, 𝑦, 𝑧)
(14)
σv
m 0,y,z
–x, y, z (𝑥, 𝑦, 𝑧)
(7)
C2
2 x,x,0
y, x, –z (𝑦, 𝑥, 𝑧)
(15)
σd
m x,𝑥,z
–y, –x, z (𝑦, 𝑥, 𝑧)
(8)
C2
2 x,𝑥,0
–y, –x, –z (𝑦, 𝑥, 𝑧)
(16)
σd
m x,x,z
y, x, z (𝑦, 𝑥, 𝑧)
Proper Rotations
4/mmm
33 Matrices:
 0 1 0 
C4   1 0 0 
0 0 1


(Determinant = +1)
+
, –
–
, +
Improper Rotations
–
+
,
,
+
–
–
+
+
–
,
,
y
+ ,
–
– ,
+
x
 0 1 0 
S 4   1 0 0 
 0 0 1


(Determinant = 1)
11
Symmorphic Space Groups
General
Position
Special
Positions
Point Group = {Symmetry operations intersecting in one point}
(32)
Space Group = {Essential Symmetry Operations}  {Bravais Lattice}
(230)
12
Symmorphic Space Groups
Ba (2a):
4/mmm (D4h)
Fe (4d):
𝟒m2 (D2d)
As (4e):
4mm (C4v)
General
Position
Special
Positions
“Ba2Fe4As4”
Z=2
Space Group: I4/mmm
Lattice Constants: a = 3.9630 Å c = 13.0462 Å
Asymmetric Unit:
BaFe2As2
Ba (2a):
Fe (4d):
As (4e):
0
½
0
0
0
0
0
¼
0.3544
13
Space Groups (230)
Symmorphic Space Groups (73): {Essential Symmetry Operations} is a group.
Point Group of the Space Group
Nonsymmorphic Space Groups (157): {Essential Symmetry Operations} is a not a group.
14
Space Group Operations: Screw Rotations and Glide Reflections
I41/amd
4/mmm
Screw Rotations:
Rotation by 2/n (Cn) then
Displacement j/n lattice vector || Cn axis
(allowed integers j = 1,…, n–1)
Symbol = nj
21, 31, 32, 41, 42, 43, 61, 62, 63, 64, 65
Point Group of the
Space Group
P42/ncm
Glide Reflections:
Reflection then
Displacement 1/2 lattice vector || reflection plane
Axial: a, b, c (lattice vectors = a, b, c)
Diagonal: n (vectors = a+b, a+c, b+c)
Diamond: d (vectors = (a+b+c)/2, (a+b)/2, (b+c)/2,
(b+c)/2)
Nonsymmorphic Space Groups (157)
15
The Origin!
Si: 0, 0, 0
16