Crystallography and Crystal Structures

8/8/2019 Crystallography and Crystal Structures

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Crystallography andCrystal Structures

Guna Selvaduray

MatE 115 – Fall 2006



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Concepts of Crystallinity

Crystalline solids

Features:

Amorphous solids

Features:



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Symmetry Operations

Mirror planes Rotational symmetry,

e.g., 90o, 180o

Translationalsymmetry e.g., chess board with

identical chess pieces 1-D or 2-D

symmetry? Translation vector

Length, direction

Straight-line pathab



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Arrays

1-D arrays line lattice

2-D arrays plane lattice 3-D arrays space lattice Repetition of identical points Lattice: Set of points in space such that

the surroundings of one point are identicalwith those of all others



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Lattice Points

Points arranged periodically in 3-D space

Points with identical surroundings



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Unit Cell

Smallest possible “structural” unit that is

repeated, 3 dimensionally Contains a full description of the structure

as a whole

Complete structure can be generated bythe repeated stacking of adjacent unitcells, face to face, throughout three-dimensional space

Crystallographic analog of “atom”



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Crystal Systems

Seven unique unit cell shapes that can bestacked together to fill three-dimensionalspace



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Bravais Lattices

Crystal systems, combined with the permissibleways in which atoms can be stacked together in aunit cell results in the 14 Bravais Lattices

Not to be confused with “crystal structure”

“Crystal structure” is derived by combining“Bravais Lattice” and “motif” or “basis”

Motif or Basis: A group of one or more atoms orions, located in a particular manner with respect toeach other and associated with one lattice point aka: Number of atoms/ions per lattice point



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Cubic system

Translational symmetry?

Number of lattice points in unit cell?



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Coordinates, Directions & Planes

Intercepts – not actual distances

Coordinates of atoms/ions → (x,y,z) Directions (x2, y2, z2) – (x1, y1, z1) → [xyz]

Planes – Miller Indices (xyz) Determine intercepts

Take reciprocals

Clear fractions



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Coordinates



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Directions





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Planes



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Elemental Cubic System - SC

Coordinates of atoms

Number of atoms/unit cell

Relationship between latticeparameter & atomic radius

Coordination number

Distance to nearest neighbor Number of nearest neighbors

Linear density

Planar density

Volume density (packing fraction)

Interstitial site(s)



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Lattice Parameter & Atomic

Radius



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Coordination Number

aka Number of Nearest Neighbors



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Linear Density

Number of lattice points per unit length

(lattice parameter) in the direction ofinterest

Linear Density Fraction actually covered by atoms in thedirection of interest

[100] vs [110] vs [111]



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Planar Density

Planar Packing Fraction = area of atoms per facearea of face= area

area

Planar Density = atoms per face = numberarea of face area



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Volume Density

aka Packing Factor

(Atomic) Packing Factor =

(# of atoms/cell)(Vol of each atom)

Vol of unit cell



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Interstitial Site



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Elemental Cubic System - BCC




Coordination number


Linear density

Planar density



L tti P t & At i



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Radius



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Coordination Number



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Linear Density

Linear Packing Fraction Planar Density

Planar Packing Fraction

Volume Density (Atomic) Packing Fraction



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Interstitial Sites



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Elemental Cubic System - FCC




Coordination number


Linear density

Planar density



Isotropy vs Anisotropy




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Radius



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Coordination Number



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Linear Density

Linear Packing Fraction Planar Density

Planar Packing Fraction

Volume Density (Atomic) Packing Fraction



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Interstitial Sites



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Diamond Cubic

Si, Ge, α-Sn, diamond

Bravais Lattice = ?

Crystal Structure = ?

Number of Lattice Points = ?

Translational Symmetry = ? Motif/Basis = ?

Number of atoms/lattice point



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Crystallographic Directions

(x2, y2, z2) – (x1, y1, z1)



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Equivalent Directions

Criterion:Indistinguishable

E l l



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Equivalent Planes

El l T l S



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Elemental Tetragonal System

a = b ≠ c

α = β = γ = 90Examples: In, α-Sn,

β-U


H l S



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Hexagonal System

a1 = a2 = a3 ≠ c

α = β = 90; γ = 120

Examples: C(graphite), Be, Cd, Mg,α-Ti, etc


H l Cl P k d



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Hexagonal Close-Packed

Coordination Number = ?

Di ti & Pl



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Directions & Planes

Cubic

3 axes x, y, z

h’, k’, l’

Hexagonal

4 axes a1, a2, a3, z

BUT a3 is redundant

a1 + a2 = - a3

h, k, i, l

x, y z

Pl



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Planes

Intercepts

Reciprocals

Clear Fractions

Miller Indices

Di ti



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Directions



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3 axes: h’, k’, l’ 4 axes: h, k, i, lh = 1(2h’ – k’)

3

k = 1(2k’ – h’)3

i = - 1(h’ + k’) [ h + k = - i]3

l = l’



Examples



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Examples

α-Fe ↔ γ-Fe ↔ δ-Fe ↔ Liquid Fe

α-Ti ↔ β-Ti

Crystal structures for each allotrope?

Volume expansion/contraction? Solubility of gases

910oC 1400oC 1535oC

882oC

Types of Polymorphic



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yp y p

Transformations Enantiotropic: mutually transformable at the transition

(equilibrium) temperature

Examples: H2O (s)↔

H2O (l)α-Fe ↔ γ-Fe

Monotropic: Proceeds only in one direction, from metastable

to stable Examples: SiO2 (glass) → SiO2 (quartz)

Fe (martensite) → Fe (ferrite)C (diamond) → C (graphite)

Stability criteria/regions Metastability



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49Source: R.E. Dickerson, Molecular Thermodynamics, p 224

Phase Transformations



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Phase Transformations

At any given T, P, the phase with Gmin is the moststable phase

Transition from one phase to another involves adiscontinuous change in (dG/dX) dG = dH –TdS → (dG/dT)P = dS

→(dG/dP)T = dV

1st Order Phase Transformations: The freeenergy function is continuous, but all of its 1stderivatives are discontinuous, e.g., V, S, E, etc.

2nd Order Phase Transformations: The freeenergy function and its 1st derivatives arecontinuous, but the 2nd derivative is discontinuous,

e.g., T g, cp, etc.



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51Source: Wert & Thomson, p 445

Structural Aspects of Phase



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Transformations - 1 Transformation of 1st Coordination

Dilatational: aka shear transformation No breakage of bonds

Austenite → Martensite

BaTiO3 (cubic) → BaTiO3 (tetragonal)

Low ∆G* (activation energy barrier)

Reconstructive: Bonds broken and reformed Austenite → Ferrite

High ∆G* (activation energy barrier)

Possible to “bypass” phase transformation by veryrapid cooling

Dilatational Transformation



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Dilatational Transformation

Source: Verma & Krishna, p 47




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Transformation of 2nd Coordination Reconstructive:

1st coordination bonds broken, 2nd coordination bonds broken and reformed, Then 1st coordination bonds reformed in

original format SiO2 (quartz) ↔ SiO2 (tridymite) ↔ SiO2

(cristobalite)

Displacive Change in 2nd coordination without breaking 1st

coordination bonds

Transformations - 2




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Transformations of Bond TypeC (diamond) → C (graphite)

Transformations - 3

covalent metallic



Compound Structures - 2



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Compound Structures 2

Coordinates of atomsNumber of ions/unitcellCoordination number

Nearestneighbor(s)


Charge neutrality




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Coordinates of atomsNumber of ions/unit

cellCoordination number

Nearestneighbor(s)

Interstitial site(s)Charge neutrality




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Number of ions/unitcellCoordination number

Nearest

neighbor(s)Interstitial site(s)Charge neutrality

Perovskite Structure



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Perovskite Structure

Source: Kingery, Bowen & Uhllman, p 68

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Crystallography and Crystal Structures