Materials Science
Processing – Structure – Properties Relationships
Processing
Materials
Optimization loop
Structure
observational
Composition
Architecture
Properties
(performance)
MSE 421/521 Structural Characterization
Levels of Structure
Macrostructure – visible with unaided eye or detectable
by common non-destructive methods.
(large pores, inclusions, cracks...)
Mesostructure – features on the borderline of the visible.
Microstructure –structures made by assemblies of crystals,
around 1 µm to 1 mm. Characterised by grain sizes, grain
size distribution, phase assemblage, and porosity. Optical
microscopes are often sufficient, but very fine structures may
require the use of an electron microscope.
Nanostructure – sub-micron features (grain boundaries, precipitates
in early stages of nucleation, ordered regions in amorphous solids...)
Crystal structure – assemblies of atoms, geometries determined by
bonding
MSE 421/521 Structural Characterization
10µ
µm
Crystals
Crystal: A periodic arrangement of atoms in space
Lattice: An infinite array of points in space
Each point has identical surroundings to all other points.
Basis:
A group of atoms associated with each lattice point
(aka motif) – NOT the same as a unit cell
Basis
Sr2+ 000
Ti4+ ½½½
O2- ½½0
O2- ½0½
O2- 0½½
Lattice
+
Crystal
=
MSE 421/521 Structural Characterization
Bravais Lattices
The 14 Bravais Lattices
Combination of 7 crystal systems
and 5 centering operations
(e.g., bcc, fcc - not hcp)
Hexagonal-R
Rhombohedral-P
MSE 421/521 Structural Characterization
Space Groups
Point Group + Bravais Lattice
Crystal System
Axial lengths and angles
Primitive
Body-Centered
Face-Centered
Primitive
Bravais
Lattice
cP
cI
cF
tP
Body-Centred
tI
Primitive
Body-Centered
Base-Centered
Face-Centered
oP
oI
oC, oA, oB
oF
Primitive
hP
Primitive
rP (hR)
Primitive
hP
α = γ = 90°
(second setting)
Primitive
Base-Centered
mP
mA, mB
P6, P6 , P6/m, P622, P6mm,
P6 m2 , P6 2m , P6/mmm
P2, Pm, P2/m
C2, Cm, C2/m (second setting)
no conditions
Primitive
aP
P1, P 1
Cubic
Four 3 or 3
a = b = c, α = β = γ = 90°
Tetragonal
One 4 or 4
a=b
Orthorhombic
Three 2 or 2
α = β= γ = 90°
α = β = γ = 90°
Trigonal
One 3 or 3
(Rhombohedral)
One 3 or 3
Hexagonal
One 6 or 6
Monoclinic
One 2 or 2
Triclinic
1 or 1
α = β = 90°,
γ = 120°
a=b
a = b = c, α = β = γ
α = β = 90°,
γ = 120°
a=b
Centring
Symmorphic Space Group
P23, Pm3, P432, P4 3m , Pm3m
I23, Im3, I432, I4 3m , Im3m
F23, Fm3, F432, F4 3m , Fm3m
P4, P4 , P4/m, P422, P4mm
P42m, P4 m2 , P4/mmm
I4, I4 , I4/m, I422, I4mm
I42m, I4m2, I4/mmm
P222, Pmm2, Pmmm
I222, Imm2, Immm
C222, Cmm2, Amm2, Cmmm
F222, Fmm2, Fmmm
P3, P3 , P312, P321, P3m1, P31m,
P 3 1m , P 3 m1
R3, R 3 , R32, R3m, R 3 m
MSE 421/521 Structural Characterization
Crystal Planes
(010)
dhkl =
(222)
(111)
(120)
a
2
h + k2 + l 2
Note: These are NOT families of planes.
Find length of direction perpendicular to plane and divide by dhkl
MSE 421/521 Structural Characterization
Hexagonal Axes
planes
{1010} prism plane
{1101} pyramidal plane
{100}
{111}
c
c
a3
a1
a3
(110) = (1120)
a2
(120) = (1210)
a1
{0001} basal plane
{001}
MSE 421/521 Structural Characterization
Hexagonal Axes
directions
[h ’k ’l ’] = [hkil]
where i = -h -k
h’=h–i
k’=k–i
l’=l
b
a
h = (2h’ – k’)/3
k = (2k’ – h’)/3
i=-h-k
l = l’
a = [100] or [2110], b = [010] or [1210], c = [001] or [0001].
Red direction = [110] or [1100]
MSE 421/521 Structural Characterization