1. “International Tables of Crystallography”
Physics 590
“International Tables of Crystallography”
“Everything you wanted to know about beautiful flies, but were afraid to ask.”
Schönflies
Gordie Miller (321 Spedding)
Proposed Plan
What basic information is found on the space group pages…
Stoichiometry of the unit cell (Wyckoff sites)
Site (point) symmetry of atoms in solids
Solid-solid phase transitions (group-subgroup relationships)
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 (
Bilbao Crystallographic Server (
(Comprehensive resources for all space groups)

2. BaFe2As2 What can we learn from the International Tables? c b a
Space Group: I4/mmm
Lattice Constants: a = Å
c = Å
Asymmetric Unit:
Ba (2a): 0 0 0
Fe (4d): ½ 0 ¼
As (4e): a b c
Intensity
(Arb. Units)
(hkl) Indices
h + k + l = even integer (2n)
(013)
(112)
(200)
(116)
(213)
(015)
(004)
(215)
(028)
(002)
(011)
2θ (Cu Kα)

3. Typical Space Group Pages…
Symbolism
Diffraction
Extinction
Conditions
Point Symmetry Features
Stoichiometry
Structure of Unit Cell
Subgroup/Supergroup
Relationships

4. NOTE: Symbolism In Schönflies notation, what does the symbol S2 mean?
Point Group of
the Space Group
Crystal
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 1
Improper Rotation
Sn = h  Cn (S3, S4, S5, …)
Inversion (x,y,z)  (–x,–y,–z)
i = S2
Mirror plane  Principal Axis
h = S1
/m (n/m is the designator: 4/m)
Mirror plane  Principal Axis
v , d (= S1)
m =
NOTE:
S2 = h  C2 x y ,
(z)
In Schönflies notation, what does the symbol S2 mean?
(x,y,z)
S2 = inversion
C2 rotation followed by sh  C2 axis
In International notation, what does the symbol mean? x y
(z)
2-fold (C2) rotation followed by inversion ( )
(x,y,z) ,
Why are the symbols S2 and not used?

5. 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
Lattice Types
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)
 = angle between b and c
 = angle between a and c
 = angle between a and b c a b

6. “Centered Lattices” 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
Hexagonal
One 6-fold axis (c-axis)
(rhombohedral)
“Centered Lattices”
 = angle between b and c
 = angle between a and c
 = angle between a and b ? c a b I F C B A
Body-
(All) Face-
Base-

7. Symbolism: Point Groups
Schönflies
Notation
Type
Symbol
Features
Uniaxial n
Single rotation axis Cn
nh
+ mirror plane  Cn axis
nv
+ n mirror planes || Cn axis
Low
Symmetry 1
Asymmetric (NO symmetry) s
Mirror plane only i
Inversion center only
Dihedral n
Rotation axis Cn + n C2 axes  Cn axis
nd
nh
Polyhedral
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

8. Symbolism: Crystallographic Point Groups
Allowed Rotations = C1 C2 C3 C4 C6
32 Point Groups
Crystal
System
Schönflies
Symbol
International Symbol
Order /
Inversion?
Full
Abbrev.
Directions
Triclinic 1 1
1 / No
(Holohedral) i
2 / Yes
Monoclinic s
1m1 or 11m m
[010] or [001]
2 / No 2
121 or 112 2
2h
12/m1 or 112/m
2/m
4 / Yes
Orthorhombic
2v
2mm
[100][010][001]
4 / No 2
222
2h
2/m 2/m 2/m
mmm
8 / Yes
Tetragonal 4 4
[001]{100}{110} 4
4h
4/m
2d
8 / No
4v
4mm 4
422
4h
4/m 2/m 2/m
4/mmm
16 / Yes
Yes:
Laue Groups b c
m2m or mm2 a b c c a b
a+b
a–b

9. Cmm2 For the following space group symbol What is the crystal class?
Questions for Friday…
For the following space group symbol
What is the crystal class?
What is the lattice type?
Which face(s) are centered?
(c) What is the point group of the space group using Schönflies notation?
(d) Does the point group contain the inversion operation?
Cmm2
Orthorhombic
Base (C)-centered
ab-faces
C2v No

10. Symbolism: Crystallographic Point Groups (cont.)
System
Schönflies
Symbol
International Symbol
Order / Inversion?
Full
Abbrev.
Directions
Trigonal 3 3
[001]{100}{210}
3 / No 6
6 / Yes
3v
3m or 3m1
6 / No 3
32 or 321
(Holohedral)
3d
12 / Yes
Hexagonal 6 6
3h
6h
6/m
3h
12 / No
6v
6mm 6
622
6h
6/m 2/m 2/m
6/mmm
24 / Yes
Cubic T 23
{100}{111}{110} Th Td
24 / No O
432 Oh
48 / Yes c a b
31m
312 c a b a b c

11. 4 / m m m Symbolism: Symmetry Operations (4h = 4/mmm = 4/m 2/m 2/m)
Schönflies
International
Coordinates
(1) E 1
x, y, z (𝑥,𝑦,𝑧)
(9) i
–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
,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)
2 x,0,0
x, –y, –z (𝑥, 𝑦 , 𝑧 )
(14)
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)
2 x, 𝑥 ,0
–y, –x, –z ( 𝑦 , 𝑥 , 𝑧 )
(16)
m x,x,z
y, x, z (𝑦,𝑥,𝑧)
Proper Rotations
Improper Rotations – +
4 / m m m , , x y + – , , + – + –
33 Matrices: , , – + – + – + , , + –
(Determinant = +1)
(Determinant = -1)

12. Symmorphic Space Groups
General
Position
Special
Positions
Point Group = {Symmetry operations intersecting in one point} (32)
Space Group = {Essential Symmetry Operations}  {Bravais Lattice} (230)

13. “Ba2Fe4As4” BaFe2As2 Symmorphic Space Groups Z = 2 General Position
Ba (2a):
4/mmm (D4h)
Fe (4d):
𝟒 m2 (D2d)
As (4e):
4mm (C4v)
General
Position
“Ba2Fe4As4”
Special
Positions
Z = 2
Space Group: I4/mmm
Lattice Constants: a = Å c = Å
Asymmetric Unit:
Ba (2a): 0 0 0
Fe (4d): ½ 0 ¼
As (4e):
BaFe2As2

14. Point Group of the Space Group
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.

15. Point Group of the Space Group
Space Group Operations: Screw Rotations and Glide Reflections
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
I41/amd
4/mmm
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)

16. The Origin!
Si: 0, 0, 0