HEXAGONAL CLOSE-PACKED STRUCTURE An HCP crystal is a close-packed structure with the stacking sequence ...ABABAB... To construct: 1st layer: 2D HCP array (layer A) 2nd layer: HCP layer with each sphere placed in alternate interstices in 1st layer (B) 3rd layer: HCP layer positioned directly above 1st layer (repeat of layer A) …ABABABAB… A B A B A A HCP is two interpenetrating simple hexagonal lattices displaced by a /3 + a /3 + a /2 1 2 3 70 HCP STRUCTURE • not a Bravais lattice Orientation alternates with each layer • each sphere touches 12 equidistant nearest neighbors (CN = 12) Six in plane, six out-of-plane • structure has maximum packing fraction possible for single-sized spheres (0.74) 71 HCP STRUCTURE • unit cell is a simple hexagonal lattice with a two-point basis a (0,0,0) • ideal ratio c/a of 8/ 3 1.633 a (2/3,1/3,1/2) • {0002} planes are close packed • ranks in importance with FCC and BCC Bravais lattices Plan view 72 HCP STRUCTURE • about 30 elements crystallize in the HCP form 73 CUBIC CLOSE-PACKED STRUCTURE A CCP crystal is a close-packed structure with the stacking sequence ...ABCABC... To construct: 1st layer: 2D HCP array (layer A) 2nd layer: HCP layer with each sphere placed in alternate interstices in 1st layer (B) 3rd layer: HCP layer placed in the other set of interstitial depressions (squares, C) 4th layer: repeats the 1st layer (A) …ABCABCABC… stacking of HCP layers along body diagonals A B C It turns out that the CCP structure is just the FCC Bravais lattice! 74 CCP STRUCTURE • CN = 12, packing fraction 0.74 • {111} planes are close packed • 4 atoms in unit cell Plan view 75 CLOSE-PACKED STRUCTURES • most common are HCP and CCP • an infinite # of alternative stacking sequences exist Example: silicon carbide has over 250 polytypes e.g., 6H-SiC stacking sequence …ABCACB… 76 STACKING FAULTS Stacking faults are one or two layer interruptions in the stacking sequence that destroy lattice periodicity e.g., an <110> projection of an FCC lattice: - A A [111] B B C C missing A B [110] B plane C C of atoms A - A [001] B - B [110] C C perfect FCC faulted FCC ABCABCABC ABCBCABC The stacking fault is an example of a planar defect • stacking fault energy γ ~100 mJ m-2 • also results in a linear defect called a dislocation 77 EXAMPLE InAs nanowires - <110> projection 78 Caroff, P. et al. Nature Nanotechnology 4, 50 - 55 (2009). • CCP and HCP have very similar lattice energies • no clear cut trends 79 Nature 353, 147 - 149 (12 Sep 1991) Rare Gases: Ne, He, Ar, Kr, Xe (CCP) 80 gold nanocrystals X. M. Lin 81 ANOTHER VIEW OF CLOSE PACKING 82 . It was reviewed by a panel of 12 referees; the panel reported in 2003, after 4 years of work, that it was “99% certain” of the correctness of the proof, but couldn’t verify all of the computer calculations. Hales and Ferguson (his student) received the Fulkerson Prize for outstanding papers in the area of discrete mathematics in 2009. In 2003, Hales announced that he would pursue a formal proof of the Conjecture that could be verified by computer. He estimated that the proof would be finished by 2023…but it was announced complete on Aug 10, 2014!!!! In January 2015 Hales and 21 collaborators published "A formal proof of the Kepler conjecture". The proof was accepted in 2017. http://en.wikipedia.org/wiki/Kepler_conjecture PACKING FRACTIONS The fraction of the total crystal volume that is occupied by spheres CCP (and HCP) a 2 radius 4 42a 4 ( )3 Vatoms 34 fraction 3 2 0.7405 a 2 Vcell a 6 a 74% BCC SC a 3 radius 4 a 4 ()3 fraction 32 0.5236 43a a3 2 ( )3 34 fraction 3 0.6802 52% a 68% 84 85 DENSITY CALCULATION n: number of atoms/unit cell m nA 1 A: atomic mass atoms VC: volume of the unit cell VNVcell A C NA: Avogadro’s number (6.023×1023 atoms/mole) Calculate the density of copper. RCu = 0.128 nm, Crystal structure: FCC, ACu= 63.5 g/mole 3 3 3 n = 4 atoms/cell, VC a (2 2 R ) 16 2 R (4)(63.5) 8.89g / cm3 [16 2(1.28 108 ) 3 (6.023 10 23 )] 8.96 g/cm3 in the literature 86 INTERSTICIAL SITES IN CP STRUCTURES A large number of ionic structures can be regarded as built of CP layers of anions with the cations placed in interstitial sites for every anion, there is 1 Octahedral site and 2 Tetrahedral sites 87 Octahedral holes coordinates: ½00 0½0 00½ ½½½ = O site cavities have <100> orientation 88 Tetrahedral holes in CCP T+ sites: ¾¼¼ ¼¾¼ ¼¼¾ ¾¾¾ T- sites: ¼¼¼ ¾¾¼ ¼¾¾ ¾¼¾ cavities have <111> orientation 89 Holes in HCP O sites: 2/3,1/3,1/4 2/3,1/3,3/4 (1/3,2/3,1/2) T+ sites: 1/3,2/3,1/8 0,0,5/8 a (0,0,0) 1 T- sites: a 2 0,0,3/8 1/3,2/3,7/8 90 LOCATION OF OCTAHEDRAL HOLES 91 LOCATION OF TETRAHEDRAL HOLES (3/8 of a unit cell directly above/below each anion) 92 SIZE OF OCTAHEDRAL CAVITY Only cations smaller than the diameter of the cavity can fit without forcing the anion lattice to expand from cell edge a22 rMX r from face diagonal ar 22X M = cation X = anion . The cavity radius is 41% of the anion radius SIZE OF TETRAHEDRAL CAVITY The tetrahedral holes are twice as numerous but six times smaller in volume The cavity radius is 22.5% of the anion radius EUTACTIC STRUCTURES Structures in which the arrangement of ions is the same as in a close packed array but the ions are not necessarily touching Within certain loose limits (given by the radius ratio rules), cations too large to fit in the interstices can be accommodated by an expansion of the anion array • anions don’t like to touch anyway • modern techniques show that, in many cases, anions (cations) are not as large (small) as previously thought • we still describe eutactic structures as CCP or HCP lattices with ions in some fraction of the interstitial sites 95 CRYSTALS THAT CAN BE DESCRIBED IN TERMS OF INTERSTITIAL FILLING OF A CLOSE-PACKED STRUCTURE 96 SOME EUTACTIC CRYSTAL STRUCTURES Variables: 1) anion layer stacking sequence: CCP or HCP array? 2) occupancy of interstitial sites 97 98 NaCl (ROCK SALT, HALITE) STRUCTURE → → → 99 POLYHEDRAL REPRESENTATION • shows the topology and indicates interstitial sites • tetrahedra and octahedra are the most common shapes • • • 100 ROCK SALT - OCCURANCE Very common (inc. 'ionics', 'covalents' & 'intermetallics' ) • Most alkali halides (CsCl, CsBr, CsI excepted) • Most oxides / chalcogenides of alkaline earths • Many nitrides, carbides, hydrides (e.g. ZrN, TiC, NaH) 101 COMPLEX ION VARIANT OF ROCK SALT • 2 • S2 102 ZINC BLENDE (ZnS, SPHALERITE) → → → 103 ZINC BLENDE • bonding is less ionic than in rock salt • common for Be, Zn, Cd, Hg chalcogenides (i.e., ZnS, ZnSe, ZnTe) • common for III-V compounds (B, Al, Ga, In with N, P, As, Sb) 104 DIAMOND STRUCTURE Fd3m → 105 FLUORITE (CaF2) & ANTIFLUORITE (Na2O) → → → 106 ALTERNATIVE REPRESENTATIONS 107 FLUORITE / ANTIFLUORITE • origin of the term “fluorescence” (George Stokes, 1852) • fluorite common for fluorides of large, divalent cations and oxides of large 2+ 4+ tetravalent cations (M F2 and M O2) • antifluorite common for oxides/chalcogenides of alkalis (M2O) 108 FLUORESCENT MINERALS = fluorite http://en.wikipedia.org/wiki/Fluorescence109 COMPARING NaCl, ZnS, Na2O 110 111 Li3Bi EXAMPLE 112 NiAs STRUCTURE 2 → → → 113 114 •Transition metals with chalcogens, As, Sb, Bi e.g. Ti(S,Se,Te); Cr(S,Se,Te,Sb); Ni(S,Se,Te,As,Sb,Sn) 115 WURTZITE (ZnS) STRUCTURE → → → 116 117 118 119 HCP VERSION OF CaF2? No structures are known with all Tetra. sites (T+ and T-) filled in HCP - i.e. there is no HCP analogue of the Fluorite /Anti-Fluorite structure Why? The T+ and T- interstitial sites above and below a layer of close-packed spheres in HCP are too close to each other (distance = 0.25c) to tolerate the coulombic repulsion generated by filling with like-charged ions. 120 Face-linking is unfavorable RUTILE STRUCTURE (TiO2) → → → 121 ANATASE STRUCTURE (TiO2) → → → 122 RUTILE AND ANATASE → 123 CdI2 STRUCTURE → → → 124 125 CdI2 - OCCURANCE • Iodides of moderately polarizing cations; bromides and chlorides of strongly polarizing cations; e.g. PbI2, FeBr2, VCl2 • Hydroxides of many divalent cations e.g. (Mg,Ni)(OH)2 • Di-chalcogenides of many quadrivalent cations e.g. TiS2, ZrSe2, CoTe2 126 CdCl2 STRUCTURE Chlorides of moderately polarizing cations e.g. MgCl2, MnCl2 Di-sulfides of quadrivalent cations e.g. TaS2, NbS2 (CdI2 form as well) 127 Type and fraction Formula of sites occupied CCP HCP NaCl NiAs AB All octahedral Rock Salt Nickel Arsenide Half ZnS ZnS tetrahedral Zinc Blende Wurtzite (T+ or T-) Na2O Anti-Fluorite A2B All tetrahedral not known CaF2 Fluorite All octahedral A B Li Bi 3 & tetrahedral 3 not known Half octahedral AB2 (Alternate layers CdCl2 (Cadmium Chloride) CdI2 (Cadmium Iodide) full/empty) Half octahedral (Ordered CaCl2 framework TiO2 (Anatase) TiO2 (Rutile) arrangement) Third octahedral AB3 Alternate layers YCl3 BiI3 2 128 /3 full/empty PEROVSKITE STRUCTURE ABX3 (CaTiO3) → → → 129 PEROVSKITE CONNECTIVITY 3D network of corner-sharing Network of face-sharing octahedra cuboctahedra 130 Perovskites: the most widely studied oxide structure • Wide range of chemistries possible - thousands of examples known • Cubic, tetragonal, and orthorhombic symmetries are common Unique properties of perovskites - high Tc cuprate superconductors - Colossal Magneto-Resistance (La,SrMnO3) + 2- - fast ion conduction (Li , O in SrSnO3), batteries, fuel cells - mixed electronic/ionic conduction, fuel cells - oxidation/reduction catalysts (CH3NH3)PbI3 organic-inorganic - ferroelectric / piezoelectric ceramics (BaTiO3, Pb(ZrTi)O3) - important mineral structure in lower mantle (MgSiO3, pyroxene) - frequency filters for wireless communications : Ba(Zn1/3Ta2/3)O3 131 X A Perovskite Structure: ABX3 Tolerance factor (t): rrAX B X t 2(rrBX ) t Effect Likely structure > 1 A cation too large to fit in Hexagonal interstices perovskite 0.9 - 1.0 ideal Cubic perovskite 0.71 - 0.9 A cation too small Orthorhombic perovskite < 0.71 A cation same size as B cation Possible close packed lattice 132 PEROVSKITES Most perovskites contain distorted octahedra and are not cubic, at least at lower temperatures.