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)
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
The Common Crystal Structures:
Body-Centered Cubic (BCC)
• 
Atoms at the corners of a cube plus one atom in the center
• 
Common in
MSE 200A
Fall, 2008
–  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}
–  Alkali metals (K, Na, Cs)
–  Transition metals (Fe, Cr, V, Mo, Nb, Ta)
J.W. Morris, Jr.
University of California, Berkeley
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
• 
Hcp: close-packed hexagonal planes stacked to fit one another
–  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
–  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
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
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
MSE 200A
Fall, 2008
J.W. Morris, Jr.
University of California, Berkeley
Atomic Resolution Image of Gum Metal
•  “Gum metal”: Ti-23Nb-0.7Ta-2Zr-1.2O
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
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)
MSE 200A
Fall, 2008
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
Interstitial Sites:
“Octahedral” Voids in Bcc Crystals
• 
Octahedral voids in face center and edge center
• 
Octahedral voids in bcc are asymmetric
MSE 200A
Fall, 2008
–  Located at {1/2,1/2,0} and {1/2,0,0}
–  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
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
• 
MSE 200A
Fall, 2008
Tetrahedral voids in bcc are asymmetric
J.W. Morris, Jr.
University of California, Berkeley
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.
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
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 alternate planes 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
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Crystal Structures of Interest •  Elemental solids: •  Binary compounds