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Page 1: Lecture Slides - 4

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)

Page 2: Lecture Slides - 4

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)

Page 3: Lecture Slides - 4

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

Page 4: Lecture Slides - 4

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

Page 5: Lecture Slides - 4

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

Page 6: Lecture Slides - 4

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>)

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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

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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

Page 9: Lecture Slides - 4

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

Page 10: Lecture Slides - 4

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

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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

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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

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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)

Page 14: Lecture Slides - 4

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)

Page 15: Lecture Slides - 4

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

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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)

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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)

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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)

Page 19: Lecture Slides - 4

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

Page 20: Lecture Slides - 4

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

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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.

Page 22: Lecture Slides - 4

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

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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

Page 24: Lecture Slides - 4

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

Page 25: Lecture Slides - 4

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

Page 26: Lecture Slides - 4

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

Page 27: Lecture Slides - 4

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