Crystal systems
A collection of point groups that in common give
caracteristic symmetry operations
Symmetry
The unit cell is chosen so that the mention symmetry
elements are easily observed.
By describing the symmetry of the unitcell the symmetry
of the condensed material is described fully.
Bravais Lattices
Hexagonal
Cubic
Tetragonal
Trigonal
Orthorombic
Monoclinic
Triclinic
Symmety operations
Identity
Mirrorplane
Rotationaxis
Invertionaxis
Sentrosymmetry
Glidemirrorplane
Screwaxis
i
m
n (2,3,4,6)
n (1,2…)
1
n, d, a, b, c
21, 31, ..63
Identity i
Pointgroup
symmetry
Spechial
symmetry
operations
Rotationaxis n
Mirrorplane m
4-fold rotation axis
α = 360/n
Screw axis, Xy
Inversionaxis n
Translation: y/x
Rotation: 360º/x
Rotation + inversion
α = 360o/n
ordinary 6-fold rotation axis
Screw axis
Glide plane
screw rotation axis 65
21
Glide mirror plane
2
2
a
1
1
0
0
41
r’ = R.r
(-x,-y,z+½)
(x,y,z)
(y,x,z+¾)
r
(x’,y’,z’)
R is a symmetry operator
 x '  a 11 a12
 y' = a
   21 a 22
 z'  a 31 a 32
r’
a 13   x 
a 23   y 
a 33   z 
Symmetry operations:
(-y,x,z+¼)
Identity: I
(x,y,z)
r
aij = 0
aii = 1
1 0 0
0 1 0 


0 0 1
r
{4 | t } = (3 × 3)r + t = Rr + t
{4 | rt } = {1 | rt }
4
Inversion / centrosymmetry: 1 i
Rotation: n, Cn
+ rotation axis [uvw]
n[uvw] or Cn[uvw]
2
3
4
2
{1(i)}(x, y, z ) = (− x,− y,−z )
6
− 1 0
{2[001]}(xyz) =  0 − 1
 0 0
0 − 1
{4[001]}(xyz) = 1 0
0 0
0  x   − x 
0  y  =  − y 
1  z   z 
0  x   − y 
0  y  =  x 
1  z   z 
4
i
Right hand becomes left hand
Rotation axis
Other symmetry elements
Mirrorplane: m
H2O
{m[010]}(x, y, z ) = (x,− y, z )
z
Vertical mirrorplane
C3z
3
Horizontal mirrorplane
Vertical mirrorplane
y
m is defined from the normal of the plane
x
X6
2
m
BF3
Rotation axis
C2z
Rotation axis
Other symmetry elements
Other symmetry elements
y
C6z
6
x
Two-fold axis
Mirror planes
Inversion center
C5H5
y
C5z
5
x
PtCl4-
y
x
C4z
4
C2 axis
Mirror planes
Inversion center
Left hand rule for x,y,z
Highest rotation axis || z
5 C2 axis
Mirror planes
Triclin system
Pointgroups
1 and 1 does not imply any restrictions for a,b,c or α,β,γ
•A chracteristic collection of symmetry elements
•The symmetry elements has origo as common point
1 I
1 I,i
Point group with elements:
•Symmetry elements and point group symbol:
Two schemes:
Schönflies
σ, Cn
Hermann Mauguin
m, n
If a 2-fold axis is added, then the system becomes:
Monoclinic system
2 I, C2
m I, σh
with the symmetry operations 2 (C2) or 2 = m (σh)
•Illustrated in form of stereographic projection
Presence of further 2 or m is a criteria for an orthorombic system
What about m normal to 2?
It does not change the criteria for a,b,c or α,β,γ and is hence possible.
1 0 0   − 1 0 0   − 1 0 0 
{m[001]}{2[001]} = 0 1 0   0 − 1 0 =  0 − 1 0  = {1}
0 0 − 1  0 0 1  0 0 − 1
2/m sentrosymmetric, with the operations: I,C2,σh,i
2
Stereographic projection
•The crystal is surrounded by a sphere
•We are interested in the projected surface in a xy-plane through the sphere
•The projected point is determined by the intersection of a connection line
from the point of interrest to the pole of the opposite side.
•The two halves of the spheres are noted by assigning + and – and to use
filled and open sumbols
H
Cl
Cl
Cl
Cl
Cl
H
H
H
H
Cl
m
m
Pole
H
mm2
xy-plane
+
Generally 4 pos.
Pole
Spechially
2 pos.
2 pos.
1 pos.
Symmety operations
Mirrorplane
Rotationaxis
Invertionaxis
Sentrosymmetry
Glidemirrorplane
Screwaxis
m
n (2,3,4,6)
n (1,2…)
1
n, d, a, b, c
21, 31, ..63
Examples of stereograms of point groups
Points
2
(C2)
Pointgroup
symmetry
2/m
(C2h)
Spechial
symmetry
operations
mmm (D2h)
Symmetry
operations
(fullt symbol 2/m 2/m 2/m)
4mm (C4v)
Point groups:
Point groups:
A crystallographic pointgroup is a selection of symmetry elements that
can operate on a three dimensional lattice.
This is only met by 32 pointgroups.
A crystallographic pointgroup is a selection of symmetry elements that
can operate on a three dimensional lattice.
This is only met by 32 pointgroups.
Crystal system
Triklinic
Monoklininc
Orthorombic
Tetragonal
Trigonal
Hexagonal
Cubic
Crystallographic point group
1,-1
2, m, 2/m
222, mm2, mmm
4, -4, 4/m, 422, 4mm, -42m, 4/mmm
3, -3, 32, 3m, -3m
6, -6, 6/m, 622, 6mm, -6m2, 6/mmm
23, m-3, 432, -43m, m-3m
Of these are:
11 centrosymmetric
21 non-centrosymmetric
10 polar
11 enantiomorphic (chiral)
Crystal system
Triklinic
Monoklininc
Orthorombic
Tetragonal
Trigonal
Hexagonal
Cubic
Crystallographic point group, full symbols
1,-1
2, m, 2/m
222, mm2, 2/m2/m2/m
4, -4, 4/m, 422, 4mm, -42m, 4/m2/m2/m
3, -3, 32, 3m, -32/m
6, -6, 6/m, 622, 6mm, -6m2, 6/m2/m2/m
23, 2/m-3, 432, -43m, 4/m-32/m
Of these are:
11 centrosymmetric
21 non-centrosymmetric
10 polar
11 enantiomorphic (chiral)
There are 230 spacespace-groups!
groups!
Space group symbol:
Bravais lattice:
P (R)
F, I
A,B,C
Xefg
Symmetry for characteristic directions
(dependent on crystal system)
Symmetry operations without translation:
Inversion
-1,1
Rotation
n
Mirror
m
Rotation-inversion n
Symmorfe space groups (73 groups)
Symmetry operations with translation:
21,63, etc.
Screw-axis
nm,
Glideplane
a,b,c,n,d
Non-symmorfe space groups (157 groups)
Symmetry planes
Symbol Translation
Mirror
m
none
Axial glide
a
b
c
a/2
b/2
c/2
Diagonal glide
n
(a+b)/2, (b+c)/2, (a+c)/2
(a+b+c)/2 for cubic and tetragonal only
Diamond glide
d
(a±b)/4, (b±c)/4, (a±c)/4
(a±b±c)/4 for cubic and tetragonal only
7 Crystal systems
CuO
a = 465 pm, b = 341 pm, c = 511 pm, β = 99,5°
Spacegroup C2/c
Cu in 4c
O in 4e y = 0.416
14 Bravais lattices
32 Point groups
230 Space groups
In order to identify the pointgroup of a space group one must:
Change symbols for symmetry elements with translation with
corresponding symbol for symmetry elements without translation
Crystal system: a = b = c, α = γ = 90°, β = 90°
Bravais-lattice: C monoclinic, side centered
Corresponding crystallographic point group: 2/m
Viz.
for a screwaxis
nm -> n
a,b,c,d or n -> m for glide plane
Example:
P63/mmc
Pnma
63/mmc
nma
6/mmm
mmm
O
Cu
a = 465 pm, b = 341 pm, c = 511 pm, β = 99,5°
Spacegroup C2/c
Cu in 4c
O in 4e y = 0.416
CuO
Crystal system: a = b = c, α = γ = 90°, β = 90°
Bravais-lattice: C monoclinic, side centered
Corresponding crystallographic point group: 2/m
Atom coordinates:
Cu
O
¼, ¼, 0
0, 0.416, ¼
Symmetry operations on a general position:
(x,y,z)
(-x,y,½ -z)
(-x,-y,-z)
(½+x, ½+y,z)
(½-x, ½+y,½ -z) (½-x, ½-y,-z)
(x,-y, ½+z)
(½+x, ½-y, ½+z)
CuO
Density
•Wetting
•Experimental (pyknometric)
•Pores in the material
•Calculated; X-ray density
based on the assumption that the unit cell is known or
that a model exists
ρx-ray > ρexp.
Formula weight ⋅ number of units / cell
m
ρ X − ray =  
=
Unitcell volume ⋅ N A
 V  unitcell
V = a ⋅ (b × c )
Density and defects
ρobs
•V unitcell; is determined experimentally
•Formula weight
ρcalc
NaCl
CaC2
Model assumptions:
A/B < 1
Cubic
z=4
AB1+y
A1-xB
ρ(interstitial B) > ρ(perfect) > ρ(vacant space A)
Non-cubic
NaCl
Cubic
z=4
CaC2
CaC2 (lt)
CaC2 (ht)
Non-cubic
Tetragonal
z=2
fcc (F)
Z=4
bcc (I)
ABO3
Z=2
c
Perovskite
a=b=c, α=β=γ=90°
cubic
b
Disordered Cu0.75Au0.25
High temp
Disordered Cu0.50Au0.50
High temp
Low temp
Ordered Cu3Au
Low temp
Ordered CuAu
a
b
Projection on the
ab-plane:
Cell dimensions are
determined by:
A-O-A
a
ac-plane
bc-plane
Identical
c
A3B3O6
YBa2Cu3O6
Perovskite
a=b=c, α=β=γ=90°
Tetragonal
b
a
Assume that the ab-plane is unchanged
viz. a=b. Assume changed c-axis
b
b
Projection on the
ab-plane:
a
a=b tetragonal
a
a=b orthorombic