Symmetry
Translations (Lattices)
A property at the atomic level, not of crystal shapes
Symmetric translations involve repeat distances
The origin is arbitrary
1-D translations = a row
Symmetry
Translations (Lattices)
A property at the atomic level, not of crystal shapes
Symmetric translations involve repeat distances
The origin is arbitrary
1-D translations = a row
a

a is the repeat vector
Symmetry
Translations (Lattices)
2-D translations = a net
a
b
Symmetry
Translations (Lattices)
2-D translations = a net
a
b
Unit cell
Unit Cell: the basic repeat unit that, by translation only, generates the entire pattern
How differ from motif ??
Symmetry
Translations (Lattices)
2-D translations = a net
b
a
Pick any point
Every point that is exactly n repeats from that point is an equipoint to the original
Translations
Exercise: Escher print
1.
2.
3.
4.
5.
What is the motif ?
Pick any point and label it with a big dark dot
Label all equipoints the same
Outline the unit cell based on your equipoints
What is the unit cell content (Z) ??
Z = the number of motifs per unit cell
Is Z always an integer ?
Translations
Which unit cell is
correct ??
Conventions:
1. Cell edges should,
whenever possible,
coincide with
symmetry axes or
reflection planes
2. If possible, edges
should relate to each
other by lattice’s
symmetry.
3. The smallest possible
cell (the reduced cell)
which fulfills 1 and 2
should be chosen
Translations
The lattice and point group symmetry interrelate, because
both are properties of the overall symmetry pattern
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Translations
The lattice and point group symmetry interrelate, because
both are properties of the overall symmetry pattern
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Good unit cell choice. Why? What is Z?
Are there other symmetry elements ?
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Translations
The lattice and point group symmetry interrelate, because
both are properties of the overall symmetry pattern
This is why 5-fold and > 6-fold rotational symmetry
won’t work in crystals
Translations
There is a new 2-D symmetry operation when we
consider translations
The Glide Plane:
A combined reflection
repeat
and translation
Step 2: translate
Step 1: reflect
(a temporary position)
Translations
There are 5 unique 2-D plane lattices.
2-D Lattice Types
vectors
angles
Compatible Point
Group Symmetry*
Oblique
a¹b
g ¹ 90o
1, 2
Square
a=b
g = 90o
4, 2, m, 1, (g)
Hexagonal
a=b
g = 120o
3, 6, 2, m, 1, (g)
Name
Rectangular
a¹b
g = 90o
Primitive (P)
Centered (C)
* any rotation implies the rotoinversion as well
2, m, 1, (g)
There are 5 unique 2-D plane lattices.
Oblique Net
Diamond Net
Rectangular P Net
a ¹ b
g ¹90o
a = b
g ¹ 90o, 120o, 60o
a ¹b
g = 90o
Rectangular
C Net
a
a
Square Net
a1 = a2
g = 60o
a1 = a2
g = 90o
a ¹b
g = 90o
b
b
Hexagonal Net
a2
b
g
g
a
g
a
g
g
a2
p2
p2mm
a1
a1
p2mm
p6mm
There are also 17 2-D Plane Groups that combine translations
with compatible symmetry operations. The bottom row are
examples of plane Groups that correspond to each lattice type
p4mm
Plane Group Symmetry
Combining translations and point groups
Plane Group Symmetry
p211
Tridymite: Orthorhombic C cell
3-D Translations and Lattices

Different ways to combine 3 non-parallel, non-coplanar axes

Really deals with translations compatible with 32 3-D point
groups (or crystal classes)

32 Point Groups fall into 6 categories
3-D Translations and
Lattices

Different ways to combine 3
non-parallel, non-coplanar axes

Really deals with translations
compatible with 32 3-D point
groups (or crystal classes)
+c
3-D Lattice Types
32 Point Groups fall
axesinto 6
Tricliniccategories
a¹b¹c

Name
angles
¹¹g ¹ 90
o
Monoclinic
a¹b¹c
g = 90o ¹90o
Orthorhombic
a¹b¹c
g = 90o
a1 = a 2 ¹ c
g = 90
a1 = a2 = a3 ¹ c
 = 90 g120
a1 = a 2 = a 3
g ¹90o
a1 = a 2 = a 3
g = 90
Tetragonal
Hexagonal
Hexagonal (4 axes)
Rhombohedral
Isometric
o
+a

g

o
o
o
+b
Axial convention:
“right-hand rule”
c
c
c
b
a
b
b
P
P
a I
Monoclinic
g 90o ¹
a ¹b ¹c
a
Triclinic
¹¹g
a ¹b ¹c
c
a
b
P
C
F
Orthorhombic
 g 90o a ¹b ¹c
I
=C
c
c
a2
a1
a2
P
a1
I
Tetragonal
 g 90o a1 = a2 ¹c
P or C
R
Hexagonal
Rhombohedral
90og 0o g¹90o
a1 = a2 = a3
a1a2¹c
a3
a2
a1
P
F
I
Isometric
 g 90o a1 = a2 = a3
3-D Translations and Lattices
Triclinic:
No symmetry constraints.
No reason to choose C when can choose simpler P
Do so by convention, so that all mineralogists do the same
Orthorhombic:
Why C and not A or B?
If have A or B, simply rename the axes until  C
3-D Symmetry
Crystal Axes
+c
+a

g

+b
Axial convention:
“right-hand rule”
3-D Symmetry
3-D Symmetry
3-D Symmetry
3-D Symmetry
3-D Space Groups
As in the 17 2-D Plane Groups, the 3-D point group
symmetries can be combined with translations to create
the 230 3-D Space Groups
Also as in 2-D there are some new symmetry elements
that combine translation with other operations
Glides: Reflection + translation
4 types. Fig. 6.52 in Klein
Screw Axes: Rotation + translation
Fig. 5.67 in Klein