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J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
• The field can be summarized in two sentences: – Properties <= composition + microstructure – Microstructure <= composition + processing
• Composition = kind and fraction of atoms present
• Microstructure = How those atoms are arranged – Microstructure is not only essential to understanding properties – It is often much more important than composition – Example: endovascular stents from 316L stainless (Fe-Cr-Ni-Mo)
• Self-expanding (hard in as-drawn condition) • Balloon-expanded (soft in annealed condition)
The Essence of Materials Science
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Microstructure
• Microstructure: – Type and location of all atoms in solid
• All atom positions known in only two ideal cases – Perfect order (“crystals” or “quasi-crystals”) – Perfect disorder (“amorphous solids” or “glasses”)
• Almost all solids prefer the crystalline state – But perfect crystals are not possible in nature – Describe by basic crystal structure + “crystal defects”
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Atomic Resolution TEM Image: Two Crystals (Grains) of Al
grain boundary
-Eric Stach NCEM/LBNL
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Atomic Resolution TEM Image: Two Crystals Meet at an Interface
NiSi
Silicon
-Eric Stach NCEM/LBNL
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Importance of Crystal Structure: Diamond vs. Graphite
• Carbon as diamond – Each atom has four neighbors – Covalent bonding in 3-D – Electrical insulator – Ultrahard material
• Carbon as graphite – 3 neighbors in hexagonal plane – Strong bond in-plane, weak
between planes – Electrical conductor (in-plane) – Mechanically weak (lubricant) – Strong if loaded in-plane (fiber)
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Crystal Structure
• Crystal lattice – Periodic grid of positions in space (“lattice points”)
• Crystal structure – Identical group of atoms placed at each lattice point
• Mathematical description: atom positions (Ri) – Ri = hia1 + kia2 + t (hi, ki = integers, t = constant)
• Physical description: – Infinitely repeatable unit cell that fills space
a1
2a Lattice points
Identical atom groups
Unit cell
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Example: 2d Crystal of Diatomic Molecules
• Filled circles are lattice points – a1, a2 = lattice vectors
• Open circles are atoms – t1, t2 = basis vectors
• Shaded area is unit cell
• Same atom arrangement – at every lattice site – in every unit cell
2a
a1
2a
a1
†1†2
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Primitive and Non-primitive Cells
• One lattice point/cell => “primitive lattice”
• Multiple lattice points/cell => “non-primitive lattice”
• One atom/cell => “primitive cell” – Need primitive lattice plus one
atom/lattice point
a
b
©
a ≠ b© ≠ 90º
b
a
©
a = b© = 90º
b
a
©
a ≠ b© = 90º
n = 4x(1/4) = 1
n = 4x(1/4) + 1 = 2
a1
a2b
a
©
a = b© ≠ 60º, 90º, 120º
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
The Five Two-Dimensional “Bravais Lattices”
• Unit cell – Chosen to reveal symmetry – Even if non-primitive cell
• The five lattice cells: – Parallelogram (primitive) – Rectangle (primitive) – Square (primitive) – Hexagonal – Face-centered rectangle
a
b
©
a ≠ b© ≠ 90º
b
a
©
a = b© = 90º
b
a
©
a ≠ b© = 90º
a1
a2 b
a©
a = b© = 120º
a1
a2b
a
©
a = b© ≠ 60º, 90º, 120º
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Three Dimensional Crystal Structures
• Three non-collinear basis vectors (a1, a2, a3) define lattice
• Identical atom group at each lattice site: – Ri = hia1 + kia2 + lia3 (h, k, l = integers)
• Primitive cell is a parallelepiped
• Distinguish 14 Bravais lattices to show symmetry – Most engineering materials are cubic => simple properties
a12a
3a
a12a
3a
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Locating Points in a Cubic Crystal
• Let the edges of the cell define Cartesian coordinates
• Choose the unit of length to be the cell edge
• (h,k,l) is the point with coordinates h, k and l in this system
• {h,k,l} denotes all points of the same geometric type – Ex.: {1,0,0} is the set of cell corners
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Locating Directions in a Cubic Crystal
• The direction [hkl] is found as follows: – Locate a point that lies along that
direction from the origin – Find the coordinates of that point
(a,b,c) – Clear all fractions to find the simplest
set of whole numbers [hkl] – If a coordinate is negative, place
minus sign above it:
• <hkl> denotes the family of similar directions (±h,±k,±l) – Ex.: <111> is the set of directions
pointing toward cell corners
[123]
[110]
[111]
Directions in a cubic crystal
The family of <111> directions
€
h kl[ ]
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Locating Planes in a Cubic Crystal
• The “Miller Indices” of a plane are found as follows:
– Draw the plane and find its intercepts (a,b,c) with the coordinate axes
– Take the reciprocals: (1/a,1/b,1/c) – Clear all fractions to find the simplest set of
whole numbers (hkl) – If a coordinate is negative, place minus
sign above it
• {hkl} denotes a family of geometrically similar planes
– Ex.: {100} is the set of all cube faces
Crystal showing (100) and (342) planes
The (110) and (111) planes
^^
^
e
e
1
e23
(100)(342)
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Locating Planes in a Cubic Crystal
• Ex: (100) – Intercepts = 1, ∞, ∞ – Reciprocals = 1, 0, 0 – Miller indices are (100)
• Ex: (342) – Intercepts = 2/3, 1/2, 1 – Reciprocals = 3/2, 2, 1 – With common denominator
= 3/2, 4/2, 2/2 – Miller indices are (342)
Crystal showing (100) and (342) planes
^^
^
e
e
1
e23
(100)(342)
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Three Advantages of Miller Indices
• (hkl) are the indices [hkl] of a perpendicular to the plane ⇒ Can easily find the angles between planes
• All parallel planes have the same Miller indices
• The distance between planes with Miller indices (hkl) is
^^
^
e
e
1
e23
(100)(342)
dhkl =
a
h2 + k2 + l2(a = cell edge length)
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Crystal Structures of Interest
• Elemental solids: – Face-centered cubic (fcc) – Hexagonal close-packed (hcp) – Body-centered cubic (bcc) – Diamond cubic (dc)
• Binary compounds – Fcc-based (Cu3Au,NaCl, ß-ZnS) – Hcp-based (α-ZnS) – Bcc-based (CsCl, Nb3Sn)
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
The Common Crystal Structures: Body-Centered Cubic (BCC)
• Atoms at the corners of a cube plus one atom in the center – Is a Bravais lattice, but drawn with 2 atoms/cell to show
symmetry – Bcc is not ideally close-packed – Closest-packed direction: <111> – Closest-packed plane: {110}
• Common in – Alkali metals (K, Na, Cs) – Transition metals (Fe, Cr, V, Mo, Nb, Ta)
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
The Face-Centered Cubic (fcc) and Hexagonal Close-Packed (hcp) Structures
• Fcc: atoms at the corners of the cube and in the center of each face – Is a Bravais lattice, but drawn with 4 atoms/cell to show symmetry – Found in natural and noble metals: Al, Cu, Ag, Au, Pt, Pb – Transition metals: Ni, Co, Pd, Ir
• Hcp: close-packed hexagonal planes stacked to fit one another – Does not have a primitive cell (two atoms in primitive lattice of hexagon) – Divalent solids: Be, Mg, Zn, Cd – Transition metals and rare earths: Ti, Zr, Co, Gd, Hf, Rh, Os
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
fcc and hcp from Stacking Close-Packed Planes
BC
A
A A
AA
A A
B
B
C C
B C
A A A
A A A A
B B
C C →
The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may
A
A A
A A
A A B
B
B C C
C
→
A AB ABA = hcp
ABC = fcc
• There are two ways to stack spheres
• The sequence ABA creates hcp
• The sequence ABC creates fcc
B C
A A A
A A A A
C C B
B B
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Hexagonal Close-Packed
• HCP does not have a primitive cell – 2 atoms in primitive cell of hexagonal lattice – 6 atoms in cell drawn to show hexagonal symmetry
• Common in – Divalent elements: Be, Mg, Zn, Cd – Transition metals and rare earths: Ti, Zr, Co, Gd, Hf, Rh, Os
• Anisotropy limits engineering use of these elements
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Face-Centered Cubic Structure
• FCC is cubic stacking of close-packed planes ({111}) – 1 atom in primitive cell; 4 in cell with cubic symmetry – <110> is close-packed direction
• Common in – Natural and noble metals: Cu, Ag, Au, Pt, Al, Pb – Transition metals: Ni, Co, Pd, Ir
ABC stacking Fcc cell View along diagonal (<111>)
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Interstitial Sites: Octahedral Voids in fcc
• Octahedral interstitial site is equidistant from six atoms – “Octahedral void” – Located at {1/2,1/2,1/2} and {1/2,0,0}
• There are 4 octahedral voids per fcc cell – One per atom
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Interstitial Sites: Tetrahedral Voids in FCC
• Tetrahedral site is equidistant from four atoms – “tetrahedral void” – Located at {1/4,1/4,1/4} - center of cell octet
• There are 8 tetrahedral voids per fcc cell – Two per atom
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Interstitial Sites: Voids between Close-packed Planes
• In both FCC and HCP packing: – Tetrahedral void above and below each atom (2 per atom) – Octahedral void in third site between planes
• Stacking including voids: – Fcc: ...(aAa)c(bBb)a(cCc)b(aAa)… – Hcp: ...(aAa)c(bBb)c(aAa)… (octahedral voids all on c-sites) ⇒ Size and shape of voids are the same in fcc and hcp
C A
A A
A A
A A
C C B
B
B A
A A
A A
A A B
B
B C C
C
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
The Diamond Cubic Structure
• Fcc plus atoms in 1/2 of tetrahedral voids – Close-packed plane stacking is ...AaBbCc… or ... aAbBcC... - Each atom has four neighbors in tetrahedral coordination - Natural configuration for covalent bonding
• DC is the structure of the Group IV elements – C, Si, Ge, Sn (grey) – Are all semiconductors or insulators
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Solid Solutions and Compounds
• Solid solution – Solute distributed through solid - Substitutional: solutes on atom sites - Interstitial: solutes in interstitial sites - Ordinarily small solutes (C, N, O, …)
• Ordered solution (compound) – Two or more atoms in regular pattern
(AxBy) – Atoms may be substitutional or interstitial
on parent lattice – “Compound” does not imply
distinguishable molecules
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Binary Compounds: Examples
• Substitutional: – Bcc: CsCl – Fcc: Cu3Au
• Interstitial: – Fcc octahedral: NaCl – Fcc tetrahedral: ß-ZnS – Hcp tetrahedral: α-ZnS – Bcc tetrahedral: Nb3Sn (A15)
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
BCC Substitutional: CsCl
• BCC parent – Stoichiometric formula AB – A-atoms on edges – B-atoms in centers – Either specie may be “A”
• Found in: – Ionic solids (CsCl)
• Small size difference • RB/RA > 0.732 to avoid like-ion
impingement – Intermetallic compounds
• CuZn (ß-brass)
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
FCC Substitutional: Cu3Au
• FCC parent – Stoichiometric formula A3B – B-atoms on edges – A-atoms on faces
• Found in: – Intermetallic compounds (Cu3Au) – As “sublattice” in complex ionics
• E.g., “perovskites” – BaTiO3 (ferroelectric) – YBa2Cu3O7 (superconductor)
• Lattices of + and - ions must interpenetrate since like ions cannot be neighbors
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
FCC Octahedral Interstitial: NaCl
• FCC parent – Stoichiometric formula AB – A-atoms on fcc sites – B-atoms in octahedral voids – Either can be “A”
• Found in: – Ionic compounds:
• NaCl, MgO (RB/RA ~ 0.5) • “Perovskites” (substitutional
ordering on both sites) – Metallic compounds
• Carbonitrides (TiC, TiN, HfC)
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
FCC Tetrahedral Interstitial: ß-ZnS
• Binary analogue of DC – Stoichiometric formula AB – A-atoms on fcc sites – B-atoms in 1/2 of tetrahedral voids
• AaBbCc – Either element can be “A”
• Found in: – Covalent compounds:
• GaAs, InSb, ß-ZnS, BN – Ionic compounds:
• AgCl • Large size difference (RB/RA < .414)
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Hcp Tetrahedral Interstitial: α-ZnS
• Hexagonal analogue of ß-ZnS – Stoichiometric formula AB – A-atoms on hcp sites – B-atoms in 1/2 of tetrahedral voids
• AaBbAaBb – Either element can be “A”
• Found in: – Covalent compounds:
• ZnO, CdS, α-ZnS, “Lonsdalite” C – Ionic compounds:
• Silver halides • Large size difference (RB/RA < .414)
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Interstitial Sites: “Octahedral” Voids in Bcc Crystals
• Octahedral voids in face center and edge center – Located at {1/2,1/2,0} and {1/2,0,0}
• Octahedral voids in bcc are asymmetric – Each has a short axis parallel to cube edge (Ox, Oy, Oz) – Total of six octahedral voids, three of each orientation
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Interstitial Sites: “Tetrahedral” Voids in Bcc Crystals
• Tetrahedral voids in each quadrant of each face – Located at {1/2,1/4,0} – 12/cell => 6/atom
• Tetrahedral voids in bcc are asymmetric
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Bcc Tetrahedral Interstitial: Α15
• Complex BCC derivative – Stoichiometric formula A3B – B-atoms on bcc sites – A-atoms in 1/2 of tetrahedral voids
• Form “chains” in x, y, and z
• Found in: – A15 compounds:
• Nb3Sn, Nb3Al, Nb3Ge, V3Ga – These are the “type-II”
superconductors used for wire in high-field magnets, etc.
