Lecture Slides - 4

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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 (Cu 3 Au,NaCl, ß-ZnS) Hcp-based (α-ZnS) Bcc-based (CsCl, Nb 3 Sn)
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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, -ZnS) Hcp-based (-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: 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

Common in

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

MSE 200A Fall, 2008

J.W. Morris, Jr.University of California, Berkeley

fcc and hcp from Stacking Close-Packed PlanesA B C A B C A A B A C A A

B C A A B B C AA

C A B AA

A

A A B C C A A B B C A A

A

AB

ABA = hcp A B C C A A A B B C A AAThe image cannot be displayed. Your computer may not have enough memory to open the image, or the image may

There are two ways to stack spheres The sequence ABA creates hcp The sequence ABC creates fcc MSE 200A Fall, 2008

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

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

Anisotropy limits engineering use of these elements

J.W. Morris, Jr.University of California, Berkeley

Face-Centered Cubic Structure

ABC stacking

Fcc cell

View along diagonal ()

FCC is cubic stacking of close-packed planes ({111}) 1 atom in primitive cell; 4 in cell with cubic symmetry is close-packed direction Natural and noble metals: Cu, Ag, Au, Pt, Al, Pb Transition metals: Ni, Co, Pd, Ir

J.W. Morris, Jr.University of California, Berkeley

Common in

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} One per atom

There are 4 octahedral voids per fcc cell

MSE 200A Fall, 2008

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 atomMSE 200A Fall, 2008

J.W. Morris, Jr.University of California, Berkeley

Interstitial Sites: Voids between Close-packed PlanesA B C C A A A B B C A A A

A B C C A A A B B C A AA

In both FCC and HCP packing: Stacking including voids:

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

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

DC is the structure of the Group IV elements

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 moleculesMSE 200A Fall, 2008

J.W. Morris, Jr.University of California, Berkeley

Atomic Resolution Image of Gum Metal

Gum metal: Ti-23Nb-0.7Ta-2Zr-1.2OMSE 200A Fall, 2008

J.W. Morris, Jr.University of California, Berkeley

Binary Compounds: Examples Substitutional: Bcc: CsCl Fcc: Cu3Au

Interstitial: Fcc octahedral: NaCl Fcc tetrahedral: -ZnS Hcp tetrahedral: -ZnS Bcc tetrahedral: Nb3Sn (A15)

MSE 200A Fall, 2008

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 (-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: -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, -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: -ZnS Hexagonal analogue of -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, -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 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

MSE 200A Fall, 2008

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

Tetrahedral voids in bcc are asymmetric

MSE 200A Fall, 2008

J.W. Morris, Jr.University of California, Berkeley

Bcc Tetrahedral Interstitial: 15 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: These are the type-II superconductors used for wire in high-field magnets, etc.MSE 200A Fall, 2008

Nb3Sn, Nb3Al, Nb3Ge, V3Ga

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 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

Identify ordered pattern

Order layers

Stacking pattern is the same for network and interstitial layers

composition may change from layer to layer (differentiation)

MSE 200A Fall, 2008

J.W. Morris, Jr.University of California, Berkeley

Substitutional X-Compounds Undifferentiated Differentiated

All atoms are the same: fcc, hcp, polytypes (e.g., ABCBABCBA) Planes of atoms alternate: CuPt, WC Note that cubic symmetry is broken in CuPt: rhombohedral^ ^ ^

^ ^ = Cu = Pt MSE 200A Fall, 2008 =W =C ^

J.W. Morris, Jr.University of California, Berkeley

Octahedral Interstital X-Compounds

= Na = Cl

= As = Ni

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 Alternate lattice or interstitial planes differ CdI2: like NiAs but octahedral Cd planes alternate with vacant planes

Differentiated

MSE 200A Fall, 2008

J.W. Morris, Jr.University of California, Berkeley

Tetrahedral(I) X-compounds

= Zn =S

= Zn =S

Lattice planes plus alternate planes of tetrahedral voids Examples: Unary: diamond cubic, hexagonal diamond (Lonsdaleite) Binary: -ZnS, -ZnSMSE 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 knownMSE 200A Fall, 2008

J.W. Morris, Jr.University of California, Berkeley