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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
Atomic Resolution Image of Gum Metal
• “Gum metal”: Ti-23Nb-0.7Ta-2Zr-1.2O
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.
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
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
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Hierarchical Description of Complex Crystal Structures
• Construct planar layers – Network (fcc or hcp) – Interstitial planes that contain atoms
• Identify ordered pattern – Primary and interstitial planes – Pattern is the same on all planes – Including vacancies, if necessary, as species
• Order layers – 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
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
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
= Cu = Pt
^
^
^
^
^
^ = W = C
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Octahedral Interstital X-Compounds
• Undifferentiated – 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
• Differentiated – Alternate lattice or interstitial planes differ – CdI2: like NiAs but octahedral Cd planes alternate with vacant planes
= Na
= Cl = As
= Ni
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Tetrahedral(I) X-compounds
• Lattice planes plus alternate planes of tetrahedral voids
• Examples: – Unary: diamond cubic, hexagonal diamond (Lonsdaleite) – Binary: α-ZnS, β-ZnS
= Zn
= S = Zn
= S
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Tetrahedral(II) X-Compounds
• Lattice planes plus planes on both tetrahedral sites
• Fcc-based: CaF2 (flourite) and Li2O
• Hcp-based: none known
= Ca
= F
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