Symmetry Translations (Lattices) A property at the atomic level, not of crystal shapes Symmetric translations involve repeat distances The origin is arbitrary

SymmetrySymmetryTranslations (Lattices)Translations (Lattices)

A property at the atomic level, not of crystal shapesA property at the atomic level, not of crystal shapes

Symmetric translations involve Symmetric translations involve repeatrepeat distances distances

The The originorigin is is arbitraryarbitrary

1-D translations = a 1-D translations = a rowrow


A property at the atomic level, not of crystal shapesA property at the atomic level, not of crystal shapes

Symmetric translations involve Symmetric translations involve repeatrepeat distances distances

The The originorigin is is arbitraryarbitrary

1-D translations = a 1-D translations = a rowrowa

aa is the is the repeat vectorrepeat vector


2-D translations = a 2-D translations = a netnet

a

b



a

b

Unit cellUnit cell

Unit Cell: the basic repeat unit that, Unit Cell: the basic repeat unit that, by translation onlyby translation only, generates the entire pattern, generates the entire pattern

How differ from motif ??How differ from motif ??



a

b

Pick Pick anyany point point

Every point that is exactly n repeats from that point is an Every point that is exactly n repeats from that point is an equipointequipoint to the original to the original

TranslationsTranslations

Exercise: Escher printExercise: Escher print1. What is the motif ?1. What is the motif ?

2. Pick any point and label it with a big dark dot2. Pick any point and label it with a big dark dot

3. Label all equipoints the same3. Label all equipoints the same

4. Outline the 4. Outline the unit cellunit cell based on your equipoints based on your equipoints

5. What is the 5. What is the unit cell content (Z)unit cell content (Z) ?? ??

Z = the number of motifs per unit cellZ = the number of motifs per unit cell

Is Z always an integer ?Is Z always an integer ?

TranslationsTranslationsWhich unit cell is Which unit cell is correct ??correct ??

Conventions:Conventions:1. Cell edges should, 1. Cell edges should,

whenever possible, whenever possible, coincide with coincide with symmetry axes or symmetry axes or reflection planesreflection planes

2. If possible, edges 2. If possible, edges should relate to each should relate to each other by lattice’s other by lattice’s symmetry.symmetry.

3. The smallest possible 3. The smallest possible cell (the reduced cell) cell (the reduced cell) which fulfills 1 and 2 which fulfills 1 and 2 should be chosenshould be chosen


The lattice and point group symmetry The lattice and point group symmetry interrelateinterrelate, because , because both are properties of the overall symmetry patternboth are properties of the overall symmetry pattern

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Good unit cell choice. Why? What is Z?Good unit cell choice. Why? What is Z?Are there other symmetry elements ?Are there other symmetry elements ?



This is why 5-fold and > 6-fold rotational symmetry This is why 5-fold and > 6-fold rotational symmetry won’t work in crystalswon’t work in crystals


There is a new 2-D symmetry operation when we There is a new 2-D symmetry operation when we consider translationsconsider translations

The The Glide Plane:Glide Plane:

A combined reflectionA combined reflection

and translationand translation

Step 1: reflectStep 1: reflect(a temporary position)(a temporary position)

Step 2: translateStep 2: translate

repeatrepeat

TranslationsTranslationsThere are 5 unique 2-D plane lattices.There are 5 unique 2-D plane lattices.

Name vectors anglesCompatible Point Group Symmetry*

Oblique a b 90o 1, 2

Square a = b = 90o 4, 2, m, 1, (g)

Hexagonal a = b = 120o 3, 6, 2, m, 1, (g)

Rectangular a b = 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.There are 5 unique 2-D plane lattices.

a

b

Oblique Net

a b 90o

p2 p2mm

Rectangular P Net

a b = 90o

b

a

Rectangular C Net

a b = 90o

p2mm

b

a

Diamond Net

a =b 90o, 120o, 60o

a1a2

Hexagonal Neta1 = a2 = 60o

p6mm

Square Neta1 = a2 = 90o

p4mm

a

a1

a2

There are also 17 2-D There are also 17 2-D Plane GroupsPlane Groups that combine translations that combine translations with compatible symmetry operations. The bottom row are with compatible symmetry operations. The bottom row are examples of plane Groups that correspond to each lattice typeexamples of plane Groups that correspond to each lattice type

Combining translations and point groupsCombining translations and point groupsPlane Group SymmetryPlane Group Symmetry

p211p211

Plane Group SymmetryPlane Group Symmetry

Tridymite: Orthorhombic C cellTridymite: Orthorhombic C cell

3-D Translations and Lattices3-D Translations and Lattices Different ways to combine 3 non-parallel, non-coplanar axesDifferent ways to combine 3 non-parallel, non-coplanar axes

Really deals with translations compatible with 32 3-D point Really deals with translations compatible with 32 3-D point groups (or crystal classes)groups (or crystal classes)

32 Point Groups fall into 6 categories32 Point Groups fall into 6 categories

3-D Translations and 3-D Translations and LatticesLattices

Different ways to combine 3 Different ways to combine 3 non-parallel, non-coplanar axesnon-parallel, non-coplanar axes

Really deals with translations Really deals with translations compatible with 32 3-D point compatible with 32 3-D point groups (or crystal classes)groups (or crystal classes)

32 Point Groups fall into 6 32 Point Groups fall into 6 categoriescategories

Name axes angles

Triclinic a b c 90o

Monoclinic a b c = 90o 90o

Orthorhombic a b c = 90o

Tetragonal a1 = a2 c = 90o

Hexagonal

Hexagonal (4 axes) a1 = a2 = a3 c = 90o 120o

Rhombohedral a1 = a2 = a3 90o

Isometric a1 = a2 = a3 = 90o

3-D Lattice Types

++cc

++aa

++bb

Axial convention:Axial convention:““right-hand rule”right-hand rule”

a

b

c

PMonoclinic

abc

a

b

c

I = Ca

b

PTriclinicabc

c

c

aP

Orthorhombicabc

C F Ib

a1

c

PTetragonal

a1 = a2c

Ia2

a1

a3

PIsometric

a1 = a2= a3

a2

F I

a1

c

P or C

a2

RHexagonal Rhombohedral

a1a2

c

a1 = a2 = a3

3-D Translations and Lattices3-D Translations and Lattices

Triclinic:Triclinic:

No symmetry constraints.No symmetry constraints.No reason to choose C when can choose simpler PNo reason to choose C when can choose simpler PDo so by Do so by conventionconvention, so that all mineralogists do the same, so that all mineralogists do the same

Orthorhombic:Orthorhombic:

Why C and not A or B? Why C and not A or B?

If have A or B, simply rename the axes until If have A or B, simply rename the axes until C C

+c

+a

+b

Axial convention:“right-hand rule”

3-D Symmetry3-D SymmetryCrystal AxesCrystal Axes

3-D Symmetry3-D Symmetry




3-D Space Groups3-D Space Groups

As in the As in the 17 2-D Plane Groups17 2-D Plane Groups, the 3-D point group , the 3-D point group symmetries can be combined with translations to create symmetries can be combined with translations to create the the 230 3-D Space Groups230 3-D Space Groups

Also as in 2-D there are some new symmetry elements Also as in 2-D there are some new symmetry elements that combine translation with other operationsthat combine translation with other operations

Glides:Glides: Reflection + translation Reflection + translation 4 types. Fig. 6.52 in Klein4 types. Fig. 6.52 in Klein

Screw Axes:Screw Axes: Rotation + translation Rotation + translation Fig. 5.67 in KleinFig. 5.67 in Klein

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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