Mineralogy and Petrology:Mineralogy and Petrology:Crystallography IICrystallography II

Dr Geoff BromileySchool of GeoSciences

Room 318Email: geoffrey.bromiley@ed.ac.uk

Symmetry elements (a recap)Symmetry elements (a recap)

• mirror planes

• inversion (centres of symmetry)

• rotation

• rotoinversion: combination of rotation and inversion

Symmetry elements 4: Symmetry elements 4: rotoinversionrotoinversion

• Combination of axes of rotation with inversion

• To differentiate between normal rotation and rotoinversion we add a bar ¯above the rotation symbol

Sketch stereogram: 3̄

Combinations of symmetry elements: tetrad with a Combinations of symmetry elements: tetrad with a diaddiad

Symmetry elements also operate on each other, and may combine to automatically generate further elements!

Result: a set of two new

diads

D

B

C

A

Symbol: 422

A B

C D

Combinations of symmetry elements: 3 mirror planesCombinations of symmetry elements: 3 mirror planesVertical mirror planes (perpendicular)

Horizontal Mirror plane

Result: 3 new diads

Symbol: 2/m 2/m 2/m

No new symmetry generatedSymbol: 4/m

Combinations of symmetry elements: rotation (tetrad) with Combinations of symmetry elements: rotation (tetrad) with a mirror planea mirror plane

Combinations of symmetry elementsCombinations of symmetry elements

• combination of mirror, rotation, inversion and rotoinversion symmetry elements leads to 32 possible, non-identical symmetry combinations

• These are the 32 crystal classes (or point groups)

• Symbols named after Hermann-Mauguin (sometimes called the ‘International Symbols’ (e.g. 4/m spoken as ‘four over m’)

Crystal system Symmetry of Crystal classesTriclinic 1 and ⎯1

Monoclinic 2, m, and 2/m

orthorhombic 222, 2mm, 2/m 2/m 2/m

tetragonal 4,⎯4, 4/m, 422, 4mm, ⎯42m, and 4/m 2/m 2/m

Hexagonal (incl. trigonal)

3, ⎯3, 32, 3m, ⎯32/m, 6, ⎯6, 6/m, 622, 6mm, ⎯6m2, and 6/m 2/m 2/m

cubic 23, 2/m ⎯3, 432, ⎯43/m, 4/m ⎯32/m

The 32 crystal classesThe 32 crystal classes

Crystal system Hermann-Mauguin Conventions and choice of axes

No constraints (three unequal axes intersecting at oblique angles)Two-fold axis is y-axis, mirror plane (if present) is vertical to y (a-c plane)Symmetry elements in x, y, z order, two-fold axes coincide with crystallographic axesTetrad is z-axis, second symbol (if present) refers to x, third symbol refers to a direction 45º to x

First symbol in z-axis direction, second symbol refers to symmetry elements parallel to x, the third symbol refers to sym. elements perpendicular to x

First symbol: all 3 axes, second symbol: the 4 diagonal directions of 3-fold symmetry (i.e. the [111] directions pointing to corners of cube), third symbol: the 6 directions between edges of cube [110]

triclinic 1 and ⎯1

monoclinic 2, m, and 2/m

orthorhombic 222, 2mm, 2/m 2/m 2/m

tetragonal 4, ⎯4, 4/m, 422, 4mm, ⎯42m, and 4/m 2/m 2/m

hexagonal (incl. trigonal)

3, ⎯3, 32, 3m,⎯32/m, 6, ⎯6, 6/m, 622, 6mm, ⎯6m2, and 6/m 2/m 2/m

cubic 23, 2/m ⎯3, 432,⎯43/m, 4/m,⎯32/m

The 32 crystal classesThe 32 crystal classes

Crystal system Characteristic symmetry

Triclinic One-fold symmetry only

Monoclinic Only one diad or mirror

orthorhombic Three diads or m's at 90º

tetragonal One tetrad

Hexagonal (incltrigonal)

One triad or one hexad

cubic Four triads each inclined at 54º44' to crystallographic axes (in [111] direction)

The 32 crystal classesThe 32 crystal classes

Exercise: Draw a sketch stereogram of 422Exercise: Draw a sketch stereogram of 422

Guide lines only

FIRST STEP: Crystal system?FIRST STEP:

Crystal system?tetragonal as characteristic

symmetry element is only one tetrad

FIRST STEP: Crystal system?

tetragonal as characteristic

symmetry element is only one tetrad

Now insert symmetry elements

Exercise: Draw a sketch stereogram of 422Exercise: Draw a sketch stereogram of 422

c

Tetrad in c-directionFirst diad in a-

directionSecond diad 45º to

a-axisLet symmetry do the

rest

a

A general face

1. Search for characteristic symmetry elements

Four triads → cubic2. Check Hermann-Mauguin

conventiona, [111], [110]

3. Crystal class

Exercise 2: Identifying crystal classes from sketch Exercise 2: Identifying crystal classes from sketch stereogramsstereograms

4̄ 3 m

Flächendeckung periodischer Muster mit Motiven n-zähliger Drehsymmetrie: (a) n = 2, 3, 4, 6; (b) n = 5, 7, 8

Here are some other crystal classesHere are some other crystal classes

cubic

tetragonal

Hex.

orthorhombic

trigonal

monoclinic

• Holosymmetric crystal class: The class with the highest symmetry (e.g., 2/m in monoclinic, or 4/m 2/m 2/m in tetragonal symmetry)

• Note that 4/m 2/m 2/m is sometimes written as 4/m m m , as the diads are a product of the combination of all m's

• Crystal classes (best described by Hermann-Mauguin symbols) are sometimes described by names of forms, e.g. 4/m 2/m 2/m as ditetragonal -dipyramidal class (see Klein(22nd ed.), p. 203-208 for more information)

Some definitions & extrasSome definitions & extras

• 7 crystal systems (Triclinic, monoclinic, tetragonal, etc)

• Symmetry elements (m, 1, 2, 3, 4, 6,⎯1, ⎯2, ⎯3, ⎯4, ⎯6)

• Combination of symmetry operations: 32 crystal classes (Hermann-Mauguin)

Next, crystal lattices and crystal structures…

Summary: crystal morphologySummary: crystal morphology

• Crystal lattice: imaginary arrangement (an array) of infinitely repeating identical points (or nodes) in space in which every point has anenvironment that is identical to any other point of the lattice

• Crystal Structure: real total arrangement of a motif (atoms etc) on a lattice (or periodical array of points)

• Motif: The simplest repeating group of atoms (ions, groups of atoms or ions, e.g. Ca2+, or CO3

2-). Each motif may be either simply repeated at each lattice point (or lattice node), or be related to other motifs by symmetry operations

• Unit cell: smallest unit of the lattice

• Crystal: homogeneous solid with three-dimensional long-range order (order is results of the repeat of motifs by regular translation in three dimensions)

Some definitionsSome definitions

Part of a crystal structureThe motif

Structure, motif and latticesStructure, motif and lattices

Lattice points

Unit cell (smallest part of the lattice) Script p. 26-28

Structure, motif and latticesStructure, motif and lattices

Abbé René-Just Haüy (1743-1826)

Lattice point

a

b

• Concept of a lattice is similar to early crystallographic models

• Consider translation of unit cell (and lattice points) in 3d space to construct a lattice

…from lecture 1

There are 5 possible spaceThere are 5 possible space--filling 2d lattices, or filling 2d lattices, or BravaisBravais LatticesLattices……....

Four types of 3D Four types of 3D BravaisBravais latticeslattices• Primitive (P) space lattices (with lattice points only at its corners).

• Face-centered (F) space lattices (from the German "flächen zentriert") Centering occurs on all faces of the unit cell

• Body-centered (I) space lattices (from the German "innen zentriert" meaning something like 'inside-centered'. Centering occurs in the center of the unit cell)

• C-centered (C) or base-centered space lattices (centering occurs in a pair of opposite faces in the z-direction (Note that this is a convention))

14 Bravais Lattices in 3d and the 7 crystal systems

Primitive Body-Centered Face-Centered

Primitive Body-Centered

Simple Body-Centered Base-Centered Face-Centered

Primitive Base-Centered

Cubic

Hexagonal

Rhombohedral(trigonal)

Tetragonal

Orthorhombic

Monoclinic

Triclinic

a

aa

a

c

a

a

a

aa

c

ab

c

a bc

α

β

γ

αβ

γ

αβ

γ

ab

c

a=b=c

a≠c

a=b=c

a=b≠c

a≠b≠c

a≠b≠c

a≠b≠c

α≠90°β≠90°γ≠90°

α≠90°β≠90°γ≠90°

α≠90°β=90°γ=90°

• Cells A, B and C are primitive cells

• Cell D, E, and F are non-primitive or multiple unit cells (i.e. more than 1 node per cell)

• E is a centered cell, F and D consist of two smaller units

Best choice would be A, B, C (small) or E (orthogonal

centered)

Choosing unit cells: 2d problemChoosing unit cells: 2d problem

UnitUnit cellscells: : Choice seems somewhat Choice seems somewhat arbitrary but here are some rules for arbitrary but here are some rules for threethree--dimensional latticesdimensional lattices

• Edges of unit cells should coincide, if possible, with symmetry axes of the lattice

• Edges of unit cells should be related to each other by symmetry of the lattice

• The smallest possible cell should be chosen

x

yz

Lattice planes: indexed using Miller system

OriginOur lattice planes intersects x-axis at ½ a,and y-axis at 1b, and is parallel to z→ (210) for this lattice plane

Lattice planesLattice planes

d Lattice spacing (d)

Crystal faces develop along planes defined by the points in the lattice.In other words, all crystal faces must intersect atoms or molecules that make up the points. A face is more commonly developed in a crystal if it intersects a larger number of lattice points. This is known as the Bravais Law

This is related to atomic density of certain planes

Faces will be more common if they develop along the lattice planes labeled 1, somewhat common if they develop along those labeled 2, and less and less common if they develop along planes labeled 3, 4, and 5

Bravais' Bravais' lawlaw

• 230 space groups (developed by Schönflies, Barlov, Federov)

• New symmetry elements: screw axes (combination of rotation with translation) and glide planes(combination of mirror reflection and translation)(see Klein, 22nd ed., p.229-239)

Combination Combination of 14 Bravais of 14 Bravais lattices with lattices with symmetry symmetry of of the the 32 32 crystal classescrystal classes? ?

How many unique patterns are there How many unique patterns are there in in threethree--dimensional spacedimensional space? ?

screw axes (combination of rotation (2, 3, 4, and 6) with translation)

Screw axes Screw axes and and glide glide planes planes

screw axes (combination of rotation with translation (here 1/4, 1/2 and 3/4, symbols 41, 42, and 43)

Screw axes Screw axes and and glide glide planes planes

Main symbol (e.g. 4) is the rotation elementSubscript is the fraction of translation inherent in the operation

42 implies ½ is the translation involved43 implies ¾ is the translation involved (rotation in opposite direction)

glide planes (combination of mirror reflection and translation)

Screw axes Screw axes and and glide glide planes planes

Glide in a-direction: Symbol aGlide in b-or c-direction: Symbol b or cGlide diagonal: Symbol n (when translation 1/2)Glide diamond: Symbol d (when translation 1/4)

d

n

• Space group symbols are composed of– Designation of lattice type

– The crystal class symbol in which the notation for rotation axes and mirror planes is replaced by the notation of the respective screw axes and glide planes (if applicable)

• Example: I 2/b 2/a 2/m, space group with identical symmetry with the point group 2/m2/m2/m and with a body-centered lattice

• Example: P 21/b 21/c 21/a, another space group with symmetry of the point group 2/m2/m2/m, in this case the lattice is primitive.

230 230 space groups space groups

XX--ray diffractionray diffractionX-rays are a type of electromagnetic radiation with wavelengths between roughly 0.1 and 100Å

This is similar to the interatomic distances in crystals, so it allows X-rays to interact with crystal lattices

X-rays

nλ=2d sin θ

Where λ is the wavelength of the X-rays, d the spacing of the lattice, and θ is the glancing angle

Bragg equation

Peak positions give 2θ values, and can be used to determine dhkl of lattice planes

X-ray data (peak position and intensity): full structural refinement (lattice type, unit cell parameters atomic positions, mixing of atoms on different crystallographic sites, presence of vacancies….even composition, crystallite size and deformation)

In Earth Sciences, XRD is most commonly used for phase identification (comparing d values for most prominent peaks) using databases such as JCPDS, and for determining proportions of different minerals in a sample

• Symmetry (Symmetry Elements, 7 crystal systems, 32 crystal classes)

• Interfacial angles (Steno's law)• Stereographic projection (Zones, zone axes, Miller

indices, Wulff net)• Sketch stereograms (combination of sym. elements,…)

• One- and two-dimensional lattices• 14 Bravais lattices• 230 space groups (screw axes, glide planes)• X-ray diffraction

SummarySummary