5 symm2
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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
a
b
Pick any point
Every point that is exactly n repeats from that point is an equipoint to the original
Translations
Exercise: Escher print
1. What is the motif ?
2. Pick any point and label it with a big dark dot
3. Label all equipoints the same
4. Outline the unit cell based on your equipoints
5. What is the unit cell content (Z) ??
Z = the number of motifs per unit cell
Is Z always an integer ?
TranslationsWhich 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 ?
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
and translation
Step 1: reflect
(a temporary position)
Step 2: translate
repeat
Translations
There are 5 unique 2-D plane lattices.
Name vectors anglesCompatible Point
Group Symmetry*
Oblique a ¹ b 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)
Rectangular a ¹ b g = 90o 2, m, 1, (g)
Primitive (P)
Centered (C)
* any rotation implies the rotoinversion as well
2-D Lattice Types
There are 5 unique 2-D plane lattices.
a
b
g
Oblique Net
a ¹bg ¹90o
p2 p2mm
Rectangular P Net
a ¹bg = 90o
b
a g
Rectangular C Net
a ¹bg = 90o
p2mm
b
a
Diamond Net
a =b
g ¹ 90o, 120o, 60o
a1a2
g
g
Hexagonal Neta1 = a2
g = 60o
p6mm
Square Neta1 = a2
g = 90o
g
p4mm
a
a1
a2
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
Combining translations and point groups
Plane Group Symmetry
p211
Plane Group Symmetry
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)
32 Point Groups fall into 6
categoriesName axes angles
Triclinic a ¹ b ¹ c ¹¹g ¹ 90o
Monoclinic a ¹ b ¹ c g = 90o ¹90
o
Orthorhombic a ¹ b ¹ c g = 90o
Tetragonal a1 = a2 ¹ c g = 90o
Hexagonal
Hexagonal (4 axes) a1 = a2 = a3 ¹ c = 90o g120
o
Rhombohedral a1 = a2 = a3 g ¹90o
Isometric a1 = a2 = a3 g = 90o
3-D Lattice Types
+c
+a
+b
g
Axial convention:
“right-hand rule”
a
b
c
PMonoclinic
g 90o ¹
a ¹b ¹c
a
b
c
I = Ca
b
PTriclinic¹¹g
a ¹b ¹c
c
c
aP
Orthorhombic g 90o a ¹b ¹c
C F I
b
a1
c
PTetragonal
g 90o a1 = a2¹c
I
a2
a1
a3
PIsometric
g 90o a1 = a2 = a3
a2
F I
a1
c
P or C
a2
RHexagonal Rhombohedral
90og0o
a1a2¹cg¹90o
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
+c
+a
+b
g
Axial convention:
“right-hand rule”
3-D SymmetryCrystal Axes
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
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