Lecture 12 CrystallographyLecture 12 Crystallography

Internal Order and 2-D SymmetryInternal Order and 2-D Symmetry

Plane LatticesPlane Lattices

Planar Point GroupsPlanar Point Groups

Plane GroupsPlane Groups

Internal Order and SymmetryInternal Order and Symmetry

Repeated and symmetrical arrangement Repeated and symmetrical arrangement (ordering) of atoms and ionic complexes in (ordering) of atoms and ionic complexes in minerals creates a 3-dimensional lattice arrayminerals creates a 3-dimensional lattice array

Arrays are generated by Arrays are generated by translationtranslation of a of a unit cellunit cell – smallest unit of lattice points that define the – smallest unit of lattice points that define the basic orderingbasic ordering

Spacing of lattice points (atoms) are typically Spacing of lattice points (atoms) are typically measured in Angstroms (= 10measured in Angstroms (= 10-10 -10 m) About the m) About the scale of atomic and ionic radiiscale of atomic and ionic radii

Two-Dimensional Plane LatticeTwo-Dimensional Plane Lattice

Translation in two directions: Translation in two directions: xx and and y y axesaxes

Angle between x and y axes is called Angle between x and y axes is called gamma gamma

Translation distance: Translation distance: aa along along xx and and bb along along yy

Replacing motifs with points (or nodes) creates Replacing motifs with points (or nodes) creates a plane latticea plane lattice

Generating an 2D Lattice Array (Plane Lattice) involves translation of a motif in two directions; possible directions not unique

Unit Cell defined by a choice Unit Cell defined by a choice of lengths and directions.of lengths and directions.

SymmetrySymmetry

Translations (Lattices)Translations (Lattices)A property at the atomic level, not of crystal A property at the atomic level, not of crystal

shapesshapes

Symmetric translations involve Symmetric translations involve repeatrepeat distancesdistances

The The originorigin is is arbitraryarbitrary

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

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

aa is the is the repeat vectorrepeat vector

SymmetrySymmetryTranslations (Lattices)Translations (Lattices)

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

a

b

SymmetrySymmetryTranslations (Lattices)Translations (Lattices)

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

a

b

A 2-D Unit CellA 2-D Unit 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

SymmetrySymmetryTranslations (Lattices)Translations (Lattices)

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

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

There are 5 Types of Plane LatticesThere are 5 Types of Plane Lattices

Preferred

Memorize these names and rules

Symmetry Elements of Planar Motifs: Planar Symmetry Elements of Planar Motifs: Planar Point GroupsPoint Groups

10 Possible symmetry combinations; called Planar Point Groups

Limitations of rotational symmetries: (1,2,3,4, & 6)

dark lines added “found mirrors”

Point groups have labels that are similar to Hermann Mauguin symbols.For example: 2mm shown has the a axis with a two fold rotational axis, and b and c have mirrors

A point group is a group of geometric symmetries that keep at least one point fixed.

TranslationsTranslationsThe lattice and point group symmetries The lattice and point group symmetries interrelateinterrelate, because , because

both are properties of the overall symmetry patternboth are properties of the overall symmetry pattern

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TranslationsTranslations

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. Good unit cell choice.

Choose: SmallestMost orthogonalMost in line with symmetry2 Nodes per Lattice VectorMost Primitive (non-centered)

Defining a 2-D Unit CellDefining a 2-D Unit Cell

Choose the: SmallestMost orthogonalMost in line with symmetryUse 2 Nodes per Lattice VectorPick the Most Primitive (non-centered)

Rules that help us

total 17 point groups

TranslationsTranslations

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

The The Glide Line, g:Glide Line, g:

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

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 examplesexamples of Plane Groups that correspond to each lattice type of Plane Groups that correspond to each lattice type

Note: p refers to a primitive cell, as opposed to c, a 2-end (opposite ends) centered cell. More on this in 3-D

17 Plane Groups17 Plane Groups

10 H-M Point Groups and 5 Lattices combine to form 17 Plane Groups.

Lecture 13Lecture 13 3-D 3-D CrystallographyCrystallography

3-D Internal Order & Symmetry3-D Internal Order & SymmetrySpace (Bravais) LatticesSpace (Bravais) LatticesSpace GroupsSpace Groups

So far we examined the five 2-D plane lattices and So far we examined the five 2-D plane lattices and combined them with the 10 planar point groups to combined them with the 10 planar point groups to generate the 17 2-D plane (space) groups. Next we generate the 17 2-D plane (space) groups. Next we study the 14 Bravais 3-D lattices and combine them study the 14 Bravais 3-D lattices and combine them with the 32 3-D point groups to generate 230 3-D with the 32 3-D point groups to generate 230 3-D space groups.space groups.

3-D Translations and Lattices3-D Translations and Lattices Different ways to combine 3 axesDifferent ways to combine 3 axes

Translations compatible with 32 3-D point groups Translations compatible with 32 3-D point groups

(~ crystal classes)(~ crystal classes)

32 Point Groups fall into 6 systems32 Point Groups fall into 6 systems

3-D Translations and 3-D Translations and LatticesLattices

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”

Different ways to combine 3 axesDifferent ways to combine 3 axes Translations compatible with 32 3-D point Translations compatible with 32 3-D point

groups (~ crystal classes)groups (~ crystal classes) 32 Point Groups fall into 6 Crystal Systems32 Point Groups fall into 6 Crystal Systems

Unit Cell Types Unit Cell Types in 14 Bravais in 14 Bravais

LatticesLattices

P – Primitive; nodes at corners only

C – Side-centered; nodes at corners and in center of one set of faces

F – Face-centered; nodes at corners and in center of all faces

I – Body-centered; nodes at corners and in center of cell

On combining 7 Crystal Classes with 4 possible unit cell types we get 14 Bravais Lattices

a

b

c

PMonoclinic

abc

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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 (body-centered)

a1

c

P or C

a2

RHexagonal Rhombohedral

a1a2

c

a1 = a2 = a3

Crystal Axes ConventionsCrystal Axes Conventions

Triclinic:Triclinic:

No symmetry constraints.No symmetry constraints.No reason to choose C (white) when can choose simpler P (blue)No reason to choose C (white) when can choose simpler P (blue)Do so by Do so by conventionconvention, so that all mineralogists do the same, so that all mineralogists do the same

+c

+a

+b

Axial convention:“right-hand rule”

Crystal Axes ConventionsCrystal Axes Conventions

System ConventionsSystem Conventions

System ConventionsSystem Conventions

System ConventionsSystem Conventions

System ConventionsSystem Conventions

3-D Space Groups3-D Space GroupsAs 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 the symmetries can be combined with translations to create the 230 3-D Space Groups230 3-D Space Groups

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

Glides:Glides: Reflection + translation Reflection + translation Screw Axes:Screw Axes: Rotation + translation Rotation + translation

A point group is a group of geometric symmetries that keep at least one point fixed.

A space group is some combination of the translational symmetry of a unit cell including lattice centering, the point group symmetry operations of reflection, rotation and rotoinversion, and the screw axis and glide plane symmetry operations. The combination of all these symmetry operations results in a total of 230 unique space groups describing all possible crystal symmetries.

230 Space 230 Space GroupsGroups

Notation indicates lattice type (P,I,F,C) and Hermann-Mauguin notation for basic symmetry operations (rotation and mirrors)

Screw Axis notation as previously noted

Glide Plane notation indicates the direction of glide – a, b, c, n (diagonal) or d (diamond)

Triclinic

Monoclinic

Orthorhombic

Tetragonal

Hexagonal

Isometric