5 Symm2

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