MTE 583MTE 583 Advanced Structure of MaterialsAdvanced Structure of MaterialsMTE 583MTE 583 Advanced Structure of MaterialsAdvanced Structure of MaterialsMTE 583 MTE 583 -- Advanced Structure of MaterialsAdvanced Structure of MaterialsMTE 583 MTE 583 -- Advanced Structure of MaterialsAdvanced Structure of Materials

Class 10Crystallography and Crystal Structures

Suggested Reading• Chs. 3 – M. DeGraef and M.E. McHenry, Structure of Materials, Cambridge (2007) 55-69.• Ch. 1 – C. Kittel, Introduction to Solid State Physics, 3rd Edition, Wiley (1956).• Excerpt from ASM Metals Handbook.• Chs. 1 and 3 – S.M. Allen and E.L. Thomas, The Structure of Materials, Wiley (1999).• Chs. 3 and 4 – R. Tilley, Crystals and Crystal Structures, Wiley (2006).

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Questions to PonderQuestions to PonderQuestions to PonderQuestions to Ponder• What is a crystal system?

• What are unit cells?

• What information is needed to specify a crystalWhat information is needed to specify a crystal structure?

H d t l l tti diff f t l• How does a crystal lattice differ from a crystal structure?

• What is a point group?

• What is a space group?

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• What is a space group?

Salt crystals

N ClNa Cl

http://healthfreedoms.org/2009/05/24/table-salt-vs-unrefined-sea-salt-a-primer/

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Shapes of crystals give us a clue about atomic arrangement

CRYSTALSCRYSTALSCRYSTALSCRYSTALS

• Solids where atoms are arranged in periodic(i.e., repeating, symmetric, etc.) patterns.(i.e., repeating, symmetric, etc.) patterns.

SYMMETRYSYMMETRYSYMMETRYSYMMETRY

• Describes the repetitions of a pattern within a crystal.

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Crystal Structure = Lattice + Motif[B ][Basis]

A lattice must be symmetric!Motifs must be arranged symmetrically!

17M. DeGraef and M.E. McHenry, Structure of Materials, Cambridge University Press (2007) p. 56

OneOne--Dimensional LatticeDimensional LatticeOneOne--Dimensional LatticeDimensional LatticeT = 3a

►Each lattice point must have identical surroundings.(i e same “environment”)

aa

(i.e., same environment ).

• In a given direction, all lattice points must be separated by an identical distance a (which we call a l tti t )identical distance, a (which we call a lattice parameter).

• In a 1D lattice, a translation of na from one lattice point to another, where n is an integer, brings you to an “identical”*lattice point.

* l h

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T = na * A lattice point with identical surroundings.

TwoTwo--Dimensional LatticeDimensional LatticeTwoTwo--Dimensional LatticeDimensional Latticey

bx

bT = 1a+2b

• There are two non-collinear lattice vectors (a and b).aa

• A translation of na + pb from one lattice point to another must bring you to an ‘equivalent’ lattice point.

• The interaxial angle defines the relationship between the two lattice vectors.

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T = na + pb

ThreeThree--Dimensional LatticeDimensional LatticeThreeThree--Dimensional LatticeDimensional Lattice• There are 3 non-collinear

basis vectors called lattice parameters (a, b and c).

• 3 interaxial angles.

• All points can be defined by a series of vectors:

– T = na + pb + qc

•c

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a b

ThreeThree--Dimensional LatticeDimensional Latticecont’dcont’dThreeThree--Dimensional LatticeDimensional Latticecont’dcont’d

• There are 3 non-collinear basis vectors called lattice parameters (a, b and c).

• 3 interaxial angles.

• All points can be defined by a series of vectors:

– T = na + pb + qc

• Basis vectors define aunit cell.

c

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a b

An entire lattice can be described by a unit cell

b Unit cell2-D Unit cell3-D

c

c

Lattice points

b a

Unit Cell – A small part of the lattice; retains allcharacteristics (structural and physical) of the lattice

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of the lattice.

Unit Cell + translation = lattice

Lattice Axes, Interaxial Angles, and Unit CellsLattice Axes, Interaxial Angles, and Unit CellsLattice Axes, Interaxial Angles, and Unit CellsLattice Axes, Interaxial Angles, and Unit Cells

The basis vectors are not

necessarily cnecessarily orthogonal.

b

a

Axis a b c a

LATTICE PARAMETERS

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Axis a b c aInter-axial angle

Unit Cell ShapesUnit Cell ShapesUnit Cell ShapesUnit Cell Shapes• Unit cells in crystals have specific shapes.

• They are based upon:1.Highest symmetry2 Consistency with past convention2.Consistency with past convention3.Minimized unit cell volume4.Satisfaction of minimal symmetry requirements.

• We call the shapes crystal systems[*].

• All crystal structures evolve from crystal systems.

24[*] A set of reference axes used to define the geometry of crystal and crystal structures

Crystal SystemsCrystal SystemsCrystal SystemsCrystal Systems• In 2D there are only four (4).

1. ObliqueThe crystal systems are the only possible

2. Rectangular3. Hexagonal4. Square

shapes for unit cells

With these shapes, 4. Square

• In 3D there are only seven (7).

you can fill all available space and leave no voids!

1. Triclinic (anorthic)2. Monoclinic3. Hexagonal

leave no voids!

g4. Rhombohedral (trigonal)5. Orthorhombic6 Tetragonal

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6. Tetragonal7. Cubic

M DeGraef and M E McHenry Structure of Materials Cambridge University Press (2007) p 66

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M. DeGraef and M.E. McHenry, Structure of Materials, Cambridge University Press (2007) p. 66

R Tilley Crystals and Crystal Structures Wiley New York (2006) p 4

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R. Tilley, Crystals and Crystal Structures, Wiley, New York (2006) p. 4

† The angle β is usually chosen to be larger than 90°

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M. DeGraef and M.E. McHenry, Structure of Materials, Cambridge University Press (2007) p. 68

Keep in mindKeep in mindKeep in mindKeep in mind

• Crystal shape reveals underlying symmetry ofCrystal shape reveals underlying symmetry of crystal.

• Must place lattice points on shape to build up a l ilattice.

All lattice points must be identical

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How do we define lattices and unit cells? How do we define lattices and unit cells? How do we define lattices and unit cells? How do we define lattices and unit cells?

• Must consider symmetry.

• Symmetry limits the number of possibilities.

• Crystal lattices must exhibit a specific minimal amount of symmetryamount of symmetry.

E h t l t h t i t• Each crystal system has a certain symmetry. (Lattice points have certain internal arrangements)

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Symmetry OperatorsSymmetry OperatorsSymmetry OperatorsSymmetry Operators• All motions that allow a pattern to be transformed fromAll motions that allow a pattern to be transformed from

an initial position to a final position such that the initial and final patterns are indistinguishable.

1. Translation*

2 Reflection2. Reflection

3. Rotation

4. Inversion (center of symmetry)

5. Roto-inversion (inversion axis)These are compound6. Roto-reflection

7. Glide (translation + reflection)

8 Screw (rotation + translation)

These are compound symmetry operators (combinations of 1-4)

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8. Screw (rotation + translation)

Axial Interaxial Minimum # of

Symmetry of Crystal SystemsSymmetry of Crystal SystemsSymmetry of Crystal SystemsSymmetry of Crystal Systems

Crystal System Relationships Angles Symmetry Elements

Cubic a = b = c = = = 90° Four 3-fold rotation or roto-inversion axes parallel to body diagonals

Hexagonal a = b ≠ c = = 90°; = 120° One 6-fold rotation or rotoinversionaxis parallel to z-axis

try

Tetragonal a = b ≠ c = = = 90° One 4-fold rotation or roto-inversion axis parallel to z-axis

Rhombohedrala = b = c = = ≠ 90° One 3-fold rotation or roto-inversion

i ll l i ng s

ymm

e

(Trigonal)a b c ≠ 90 axis parallel to z-axis

Orthorhombic a ≠ b ≠ c = = = 90° Three 2-fold rotation or roto-inversion axes parallel to x,y,z-axes In

crea

sin

Monoclinic a ≠ b ≠ c = = 90° ≠ One 2-fold rotation or roto-inversion axis parallel to y-axis

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Triclinic a ≠ b ≠ c ≠ ≠ ≠ 90° None

I’d memorize these if I were you

Crystal System

+

Symmetric Arrayof Lattice Pointso a ce o s

=

Bravais LatticeWe can classify

Bravais lattices in terms of the number

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terms of the number of lattice points in

the unit cell