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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
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)
L tti P t & At i
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Lattice Parameter & Atomic
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
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)
Isotropy vs Anisotropy
Lattice Parameter & Atomic
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Lattice Parameter & Atomic
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
Isotropy vs Anisotropy
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
Isotropy vs Anisotropy
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’
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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
Structural Aspects of Phase
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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
Structural Aspects of Phase
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Transformations of Bond TypeC (diamond) → C (graphite)
Transformations - 3
covalent metallic
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Compound Structures - 2
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Compound Structures 2
Coordinates of atomsNumber of ions/unitcellCoordination number
Nearestneighbor(s)
Interstitial site(s)
Charge neutrality
Compound Structures - 3
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Compound Structures 3
Coordinates of atomsNumber of ions/unit
cellCoordination number
Nearestneighbor(s)
Interstitial site(s)Charge neutrality
Compound Structures - 4
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Compound Structures 4
Coordinates of atoms
Number of ions/unitcellCoordination number
Nearest
neighbor(s)Interstitial site(s)Charge neutrality
Perovskite Structure