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)
•  Everything else
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
fcc and hcp from Stacking
Close-Packed Planes
A
A
B
A
A
A
B
B
C
A
A
A
B
C
A
A
BB
C
A
A
C
C
→
C
A
B
A
AB
A
A
A
B
→
C
C
A
A
B
A
B
C
A
A
ABA = hcp
A
B
C
C
A
A
A
B
B
C
A
A
A
•  There are two ways to stack spheres
•  The sequence ABA creates hcp
•  The sequence ABC creates fcc MSE 200A
Fall, 2008
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ABC = fcc
J.W. Morris, Jr.
University of California, Berkeley
Hexagonal Close-Packed
MSE 200A
Fall, 2008
• 
HCP does not have a primitive cell
• 
Common in
• 
Anisotropy limits engineering use of these elements
–  2 atoms in primitive cell of hexagonal lattice
–  6 atoms in cell drawn to show hexagonal symmetry
–  Divalent elements: Be, Mg, Zn, Cd
–  Transition metals and rare earths: Ti, Zr, Co, Gd, Hf, Rh, Os
J.W. Morris, Jr.
University of California, Berkeley
Face-Centered Cubic Structure
ABC stacking
Fcc cell
View along diagonal (<111>)
• 
FCC is cubic stacking of close-packed planes ({111})
• 
Common in
–  1 atom in primitive cell; 4 in cell with cubic symmetry
–  <110> is close-packed direction
–  Natural and noble metals: Cu, Ag, Au, Pt, Al, Pb
–  Transition metals: Ni, Co, Pd, Ir
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Interstitial Sites:
Octahedral Voids in fcc
MSE 200A
Fall, 2008
• 
Octahedral interstitial site is equidistant from six atoms
• 
There are 4 octahedral voids per fcc cell
–  “Octahedral void”
–  Located at {1/2,1/2,1/2} and {1/2,0,0}
–  One per atom
J.W. Morris, Jr.
University of California, Berkeley
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
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Interstitial Sites:
Voids between Close-packed Planes
A
B
C
C
A
A
A
B
B
C
A
A
A
A
B
C
C
A
A
A
B
B
C
A
A
• 
In both FCC and HCP packing:
• 
Stacking including voids:
A
–  Tetrahedral void above and below each atom (2 per atom)
–  Octahedral void in third site between planes
–  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
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
The Diamond Cubic Structure
• 
Fcc plus atoms in 1/2 of tetrahedral voids
• 
DC is the structure of the Group IV elements
–  Close-packed plane stacking is ...AaBbCc… or ... aAbBcC...
-  Each atom has four neighbors in tetrahedral coordination
-  Natural configuration for covalent bonding
–  C, Si, Ge, Sn (grey)
–  Are all semiconductors or insulators
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Binary Compounds: Examples
•  Substitutional:
–  Bcc: CsCl
–  Fcc: Cu3Au
•  Interstitial:
– 
– 
– 
– 
MSE 200A
Fall, 2008
Fcc octahedral: NaCl
Fcc tetrahedral: ß-ZnS
Hcp tetrahedral: α-ZnS
Bcc tetrahedral: Nb3Sn (A15)
J.W. Morris, Jr.
University of California, Berkeley
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
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
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)
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
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)
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
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)
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Description of
Complex Crystal Structures
•  Most crystals can be referred to a close-packed frame
–  Fcc or hcp network
–  Possibly plus small distortions along symmetry axes
•  Cubic → tetragonal (edge unique),
•  Cubic → rhombohedral (diagonal unique)
•  Atoms in ordered configurations in
–  Substitutional sites
–  Interstital sites: octahedral or tetrahedral
–  Vacancies are useful as “atoms” to complete the configuration
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Hierarchical Description of
Complex Crystal Structures
• 
Construct planar layers
• 
Identify ordered pattern
• 
Order layers
–  Network (fcc or hcp)
–  Interstitial planes that contain atoms
–  Primary and interstitial planes
–  Pattern is the same on all planes
–  Including vacancies, if necessary, as species
–  Physical pattern (fcc or hcp)
–  Chemical pattern
•  composition may change from layer to layer (differentiation)
–  Stacking pattern is the same for network and interstitial layers
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Representation of Crystal Structures
•  The basic pattern is L t o t L at 0, 1/4, 1/2, 3/4, 1)
FCC:
HCP:
AbcaBcabCabcA
AbcaBacbAbcaB
•  Levels of representation
–  1 - planar order: X, XY. X2Y, X3Y
–  2 - interstitial character: octahedral, tetrahedral(1,2)
–  3 - stacking pattern: fcc, hcp, polytypic, hexagonal
–  4 - differentiation: undifferentiated, differentiated
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Substitutional X-Compounds
•  Undifferentiated
–  All atoms are the same: fcc, hcp, polytypes (e.g., ABCBABCBA…)
•  Differentiated
–  Planes of atoms alternate: CuPt, WC
–  Note that cubic symmetry is broken in CuPt: rhombohedral
^
^
MSE 200A
Fall, 2008
^
^
^
^
= Cu
=W
= Pt
=C
J.W. Morris, Jr.
University of California, Berkeley
Octahedral Interstital X-Compounds
• 
Undifferentiated
• 
Differentiated
= Na
= As
= Cl
= Ni
–  Fcc or hcp planes alternate with filled octahedral planes: NaCl, NiAs
–  Note that o-sites in NiAs are ccc, can tell which atom is in octahedral hole
–  Alternate lattice or interstitial planes differ
–  CdI2: like NiAs but octahedral Cd planes alternate with vacant planes
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Tetrahedral(I) X-compounds
= Zn
= Zn
=S
=S
•  Lattice planes plus one plane of tetrahedral voids
•  Examples:
–  Unary: diamond cubic, hexagonal diamond (Lonsdaleite)
–  Binary: α-ZnS, β-ZnS
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Tetrahedral(II) X-Compounds
= Ca
=F
•  Lattice planes plus planes on both tetrahedral sites
•  Fcc-based: CaF2 (flourite) and Li2O
•  Hcp-based: none known
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Distributions of XY in a plane
•  XY:
–  3 basic patterns
–  Label I, II, III
•  X2Y:
–  1 basic pattern
•  X3Y:
–  2 basic patterns
–  Label I, II
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Binary (XY) Patterns in a Plane
XY(I)
(common)
Note: all planes in the
stacking have the
same type of order.
XY(II)
XY(III)
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Substitutional XY undifferentiated
= Cu
= Au
•  Only known example is fcc-based CuAu(I)
–  Has the XY(I) pattern in every plane
–  Creates structure in which Cu, Au fill alternate (100) planes
–  Cubic symmetry lost: tetragonal
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Differentiated Substitutional XY
C
A
B
= Cu
= Pt
•  Fcc example: CuPt3
–  CuPt planes in XY(I) order alternate with Pt planes
•  Note Pt plane has XY(I) pattern with X=Y
–  Stacking: A(CuPt)B(Pt)C(CuPt)A(Pt)B(CuPt)C(Pt)
–  Cubic symmetry broken: rhombohedral
•  No hcp-based examples
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
FCC-based Octahedral XY
= Fe
=N
•  (Fe,Ni)2N
–  Fcc solution of Fe and Ni
–  XY(I) pattern of N and ∅ on octahedral layers
–  Cubic symmetry broken: tetragonal
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
HCP-based Octahedral XY
•  Many MO2 oxides are hcp
– 
– 
– 
– 
TiO 2
å - PbO 2
= O at A-sites 0,1
= O at B-sites (1/2)
= M at c1 sites (1/4)
= M at c2 sites (3/4)
FeO(OH)
MSE 200A
Fall, 2008
O on hcp sites
M∅ on octahedral planes
M alternates to fill all sites
Pattern Ac1Bc2A
•  Examples:
–  TiO2 (rutile) = Ti in XY(I), O
on HCP sites
–  α-PbO2 = Pb in XY(II)
–  FeO(OH) (geothite) = Fe in
XY(III), O and (OH) planes
alternate
J.W. Morris, Jr.
University of California, Berkeley
X2Y Pattern
•  Note that 2d unit cell contains three atoms
–  Cell outlined in red
•  Requires three planes for symmetric coverage of sites
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Octahedral X2Y: Corundum
•  Analogue of NiAs
–  X2Y order in the plane
•  Examples:
–  Al2O3
–  FeTiO3
MSE 200A
Fall, 2008
…c1Ac2Bc3Ac1B…
Note 6-layer repeat pattern
J.W. Morris, Jr.
University of California, Berkeley
Most Common X3Y Pattern
•  Note that 2d unit cell contains four atoms
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Octahedral X3Y Perovskite
•  Perovskite: CaTiO3
•  Also Fe4N, Fe8N, Ni4N
•  Also YBCO and other oxide superconductors
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Defects in Crystals
•  Imperfections are present in all real crystals
•  Often, they are added to control properties
–  Materials engineering is largely “defect”
engineering
•  Classify defects by dimension
–  Point defects: solute atoms (strength, conductivity)
–  Line defects: dislocations (plastic deformation)
–  Surface defects: external surface (crystal shape)
–  Volume defects: voids, inclusions (fracture)
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
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Crystal Structures of Interest •  Elemental solids: •  Binary compounds