Basic Structural Chemistry Crystalline state Structure types Degree of Crystallinity Single Crystalline Polycrystalline Amorphous Ťhe building blocks of these two are identical, but different crystal faces are developed Crystal Structure * Cleaving a crystal of rocksalt Crystals * Crystal consist of a periodic arrangement of structural motifs = building blocks * Building block is called a basis: an atom, a molecule, or a group of atoms or molecules * Such a periodic arrangement must have translational symmetry such that if you move a building block by a distance: then it falls on another identical building block with the same orientation. * If we remove the building blocks and replace them with points, then we have a point lattice or Bravais lattice. vectors.are,,andintegers,areand,,where 321 321 cbannn cnbnanT ++= Planar Lattice 2D Five Planar Lattices Unit Cell: An ,,imaginary" parallel sided region of a structure from which the entire crystal can be constructed by purely translational displacements Contents of unit cell represents chemical composition Space Lattice: A pattern that is formed by the lattice points that have identical environment. Coordination Number (CN): Number of direct neighbours of a given atom (first coordination sphere) Crystal = Periodic Arrays of Atoms Lattice point (Atom, molecule, group of molecules,...) Translation Vectors Primitive Cell: * Smallest building block for the crystal lattice. * Repetition of the primitive cell gives a crystal lattice a c ba, b , c Bravais Lattice (Lattice point = Basis of Spherical Symmetry) Crystal Structure (Structural motif = Basis of Arbitrary Symmetry) Number of point groups: 7 (7 crystal systems) 32 (32 crystallographic point groups) Number of space groups: 14 (14 Bravais lattices) 230 (230 space groups) Seven Crystal Systems Fourteen Bravais Lattices Add one atom at the center of the cube Body-Centered Cubic (BCC) a c b a = b = c a b c Simple Cubic (SC) Add one atom at the center of each face Face-Centered Cubic (FCC) Conventional Cell = Primitive Cell Conventional Unit Cell Primitive Cell Primitive Cell a a a Body-Centered Cubic (I) Unit Cell Primitive Cell A volume of space translated through all the vectors in a lattice just fills all of space without overlapping or leaving voids is called a primitive cell of the lattice. A primitive cell contains just one Bravais lattice point. The primitive cell is the smallest cell that can be translated throughout space to completely recreate the entire lattice. There is not one unique shape to a primitive cell and many possible shapes fulfill the definition. The primitive cell for the simple cubic lattice is equal to the simple cubic unit cell (they are identical). A common choice for the primitive cell of the body-centered cubic lattice is shown below. (magenta) Nonprimitive Unit Cell vs. Primitive Cell a a a Face-Centered Cubic (F) a Rotated 90 Primitive Cell Unit Cell The primitive cell is smaller or equal in size to the unit cell. The unit cells help to remind us of the symmetry (ie. Cubic). Primitive Cell of BCC ˇPrimitive Translation Vectors: Řhombohedron primitive cell 0.53a 109o28' Primitive Cell of FCC ˇAngle between a1, a2, a3: 60o 1) Find the intercepts on the axes in terms of the lattice constants a, b, c. The axes may be those of a primitive or nonprimitive unit cell. 2) Take the reciprocals of these numbers and then reduce to three integers having the same ratio, usually the smallest three integers. The result enclosed in parenthesis (hkl), is called the index of the plane. Index System for Crystal Planes (Miller Indices) Miller Indices Miller Indices Miller Indices Crystals * metallic (Cu, Fe, Au, Ba, alloys ) metallic bonding * ionic (NaCl, CsCl, CaF2, ... ) cations and anions, electrostatic interactions * covalent (diamond, graphite, SiO2, AlN,... ) atoms, covalent bonding * molecular (Ar, C60, HF, H2O, organics, proteins ) molecules, van der Waals and hydrogen bonding ˇ van der Waals bonds * Ionic bonds * Hydrogen bonds * Metallic bonds * Covalent bonds Crystal Bonding Three Cubic Cells SC or Primitive (P) BCC (I) FCC (F) a a a d D a = edge d = face diagonl (d2 = a2 + a2 = 2a2) D = body diagonal (D2 = d2 + a2 = 2a2 + a2 = 3a2) a2 =d a3 =D Cube CN 6 SC = Polonium Space filling 52% Space filling 68% CN 8 BCC = W, Tungsten a d D r Fe, Cr, V, Li-Cs, Ba Space filling 74% CN 12 FCC = Copper, Cu = CCP d r Close Packing in Plane 2D B and C holes cannot be occupied at the same time Close Packing in Space 3D Cubic CCP Hexagonal HCP hexagonal cubic cubichexagonal hexagonal cubic cubic hexagonal Mg, Be, Zn, Ni, Li, Be, Os, He, Sc, Ti, Co, Y, Ru Cu, Ca, Sr, Ag, Au, Ni, Rh, solid Ne-Xe, F2, C60, opal (300 nm) Structures with Larger Motifs Coordination Polyhedrons Coordination Polyhedrons Space Filling 74%42a/4FCC 34%83a/8Diamond 68%23a/4BCC 52%1a/2SC Space fillingNumber of Atoms (lattice points) Atom Radius CCP = FCC (ABC) Close packed layers of CCP are oriented perpendicularly to the body diagonal of the cubic cell of FCC CCP FCC Two Types of Voids (Holes) Tetrahedral Holes TN cp atoms in lattice cell N Octahedral Holes 2N Tetrahedral Holes Tetrahedral Holes T+ Octahedral Holes Two Types of Voids (Holes) Tetrahedral HolesOctahedral Holes Z = 4 number of atoms in the cell (N) N = 8 number of tetrahedral holes (2N) Tetrahedral Holes (2N) Octahedral Holes (N) Z = 4 number of atoms in the cell (N) N = 4 number of octahedral holes (N) Different Types of Radii Variation of atomic radii through the Periodic table Variation of ionic radii with coordination number The radius of one ion was fixed to a reasonable value (r(O2-) = 140 pm) (Linus Pauling) That value is then used to compile a set of self consistent values for all other ions. 1. Ionic radii increase down a group.(Lanthanide contraction restricts the increase of heavy ions) 2. Radii of equal charge ions decrease across a period 3. Ionic radii increase with increasing coordination number the higher the CN the bigger the ion 4. The ionic radius of a given atom decreases with increasing charge (r(Fe2+) > r(Fe3+)) 5. Cations are usually the smaller ions in a cation/anion combination (exceptions: r(Cs+) > r(F-)) 6. Frequently used for rationalization of structures: ,,radius ratio" r(cation)/r(anion) (< 1) General trends for ionic radii P ­ Pauling radius G ­ Goldschmidt radius S ­ Shannon radius. Variation of the electron density along the Li ­ F axis in LiF Cation/anion Radius Ratio 0.225 ­ 0.4144 ­ tetrahedral 0.414 ­ 0.7326 ­ octahedral 0.732 ­ 1.008 ­ cubic 1.00 (substitution)12 ­ hcp/ccp r/RCN optimal radius ratio for given CN ions are in touch Structure map: Dependence of the structure type (coordination number) on the electronegativity difference and the average principal quantum number (size and polarizability) AB compounds The lattice enthalpy change is the standard molar enthalpy change for the following process: M+ (gas) + X(gas) MX(solid) Because the formation of a solid from a ,,gas of ions" is always exothermic lattice enthalpies (defined in this way) are usually negative. If entropy considerations are neglected the most stable crystal structure of a given compound is the one with the highest lattice enthalpy. H L 0 H L 0 Lattice Enthalpy A Born-Haber cycle for KCl (all enthalpies: kJ mol-1 for normal conditions standard enthalpies) standard enthalpies of - formation: 438 - sublimation: +89 (K) - ionization: + 425 (K) - atomization: +244 (Cl2) - electron affinity: -355 (Cl) - lattice enthalpy: x Lattice enthalpies can be determined by a thermodynamic cycle Born-Haber cycle Hsluč o = - 411 kJ mol-1 Hsubl o = 108 kJ mol-1 D= 121 kJ mol-1 EA = - 354 kJ mol-1 IE = 502 kJ mol-1 L=?Na(s) + 1/2 Cl2 (g) Na(g) + 1/2 Cl2 (g) Na(g) + Cl (g) Na+ (g) + Cl (g) Na+ (g) + Cl- (g) NaCl (s) 0 = -Hsluč o + Hsubl o + 1/2 D + IE + EA + L 0 = 411 + 108 +121 + 502 + (-354) + L L = - 788 kJ mol-1 Born-Haber cycle all enthalpies: kJ mol-1 for normal conditions standard enthalpies Lattice Enthalpy L = Ecoul + Erep One ion pair Ecoul = (1/40) zA zB / d Erep = B / dn n = Born exponent (experimental measurement of compressibilty) Lattice Enthalpy 1 mol of ions Ecoul = NA (e2 / 4 0) (zA zB / d) A Erep = NA B / dn L = Ecoul + Erep Find minimum dL/d(d) = 0 nA BA A d B N d eZZ ANL += 0 2 4 N r ezz AV AB AB 0 2 4 -+-= Coulombic contributions to lattice enthalpies Calculation of lattice enthalpies Coulomb potential of an ion pair VAB: Coulomb potential (electrostatic potential) A: Madelung constant (depends on structure type) N: Avogadro constant z: charge number e: elementary charge o: dielectric constant (vacuum permittivity) rAB: shortest distance between cation and anion Madelung Constant Ecoul = (e2 / 4 0)*(zA zB / d)*[+2(1/1) - 2(1/2) + 2(1/3) - 2(1/4) + ....] Ecoul = (e2 / 4 0)*(zA zB / d)*(2 ln 2) Count all interactions in the crystal lattice Madelung constant A (for linear chain of ions) = sum of convergent series Calculation of the Madelung constant Na Cl ... 5 24 2 6 3 8 2 12 6 +-+-=A 3D ionic solids: Coulomb attraction and repulsion Madelung constants: CsCl: 1.763 NaCl: 1.748 ZnS: 1.641 (wurtzite) ZnS: 1.638 (sphalerite) ion pair: 1.0000 (!) = 1.748... (NaCl) (infinite summation) Madelung constant for NaCl Ecoul = (e2 / 4 0) * (zA zB / d) * [6(1/1) - 12(1/2) + 8(1/3) - 6(1/4) + 24(1/5) ....] Ecoul = (e2 / 4 0) * (zA zB / d) * A convergent series Madelung Constants for other Structural Types 1.64132ZnS Wurtzite 1.63805ZnS Sfalerite 2.519CaF2 1.76267CsCl 1.74756NaCl AStructural Type Born repulsion VBorn Because the electron density of atoms decreases exponentially towards zero at large distances from the nucleus the Born repulsion shows the same behavior approximation: r V nBorn B = B and n are constants for a given atom type; n can be derived from compressibility measurements (~8) r r0 Repulsion arising from overlap of electron clouds Total lattice enthalpy from Coulomb interaction and Born repulsion ).(0 VVMin BornABL += ) 1 1( 4 00 2 0 n N r ezz A L --= -+ (set first derivative of the sum to zero) Measured (calculated) lattice enthalpies (kJ mol-1): NaCl: ­772 (-757); CsCl: -652 (-623) (measured from Born Haber cycle) The Kapustinskii equation Kapustinskii found that if the Madelung constant for a given structure is divided by the number of ions in one formula unit () the resulting values are almost constant: 6:40.834.172-Al2O3 8:40.842.519CaF2 6:60.871.748NaCl 8:80.881.763CsCl CoordinationA/Madel. const.(A)Structure general lattice energy equation that can be applied to any crystal regardless of the crystal structure -+ -+ -= rr zz L 10 5 0 079.1 Most important advantage of the Kapustinski equation it is possible to apply the equation for lattice calculations of crystals with polyatomic ions (e.g. KNO3, (NH4)2SO4 ...). a set of ,,thermochemical radii" was derived for further calculations of lattice enthalpies Lattice Enthalpy += nd eZZ MNL BA A 1 1 4 0 2 nEl. config. 10Kr 12Xe 9Ar 7Ne 5He Born ­ Mayer d* = 0.345 Born ­ Lande -= d d d eZZ MNL BA A * 0 2 1 4 Lattice Enthalpy Kapustinski M/v je přibližně konstantní pro všechny typy struktur v = počet iontů ve vzorcové jednotce M nahrazeno 0.87 v, není nutno znát strukturu -= dd ZZ vL BA 345,0 11210 structure M CN stoichm M / v CsCl 1.763 (8,8) AB 0.882 NaCl 1.748 (6,6) AB 0.874 ZnS sfalerite 1.638 (4,4) AB 0.819 ZnS wurtzite 1.641 (4,4) AB 0.821 CaF2 fluorite 2.519 (8,4) AB2 0.840 TiO2 rutile 2.408 (6,3) AB2 0.803 CdI2 2.355 (6,3) AB2 0.785 Al2O3 4.172 (6,4) A2B3 0.834 v = the number of ions in one formula unit Kapustinski Lattice Enthalpy of NaCl Born ­ Lande calculation L = - 765 kJ mol-1 Only ionic contribution Experimental Born ­ Haber cycle L = - 788 kJ mol-1 Lattice Enthalpy consists of ionic and covalent contribution Applications of lattice enthalpy calculations: thermal stabilities of ionic solids stabilities of oxidation states of cations Solubility of salts in water calculations of electron affinity data lattice enthalpies and stabilities of ,,non existent" compounds Cation/anion Radius Ratio 0.225 ­ 0.4144 ­ tetrahedral 0.414 ­ 0.7326 ­ octahedral 0.732 ­ 1.008 ­ cubic 1.00 (substitution)12 ­ hcp/ccp r/RCN Structure Types Derived from CCP = FCC ˇ Cation-Anion distance is determined by sums of ionic radii. Cation coordination environment is determined by radius ratio. * The bond valence sum of each ion should equal oxidation state. * Avoid shared polyhedral edges and/or faces. (particularly for cations with high oxidation state & low coordination number) * In a crystal containing different cations those with large valence and small coord. number tend not to share anions. * The number of chemically different coordination environments for a given ion tends to be small. Pauling Rules Structure Types Derived from CCP = FCC Characteristic Structures of Solids = Structure Types Rock salt NaCl LiCl, KBr, AgCl, MgO, TiO, FeO, SnAs, UC, TiN, ... Fluorite CaF2 BaCl2, K2O, PbO2 ... Lithium bismutide Li3Bi Sphalerite (zinc blende) ZnS CuCl, HgS, GaAs ... Nickel arsenide NiAs FeS, PtSn, CoS ... Wurtzite ZnS ZnO, MnS, SiC Rhenium diboride ReB2 Structure Types Derived from CCP = FCC Anions/cell (= 4) Oct. (Max 4) Tet. (Max 8) Stoichiometry Compound 4 100% = 4 0 M4X4 = MX NaCl (6:6 coord.) 4 0 100% = 8 M8X4 = M2X Li2O (4:8 coord.) 4 0 50% = 4 M4X4 = MX ZnS, sfalerite (4:4 coord.) 4 50% = 2 0 M2X4 = MX2 CdCl2 4 100% = 4 100% = 8 M12X4 = M3X Li3Bi 4 50% = 2 12.5% = 1 M3X4 MgAl2O4, spinel o/t fcc(ccp) hcp all oct. NaCl NiAs all tetr. CaF2 ReB2 o/t (all) Li3Bi (Na3As) (!) problem t sphalerite (ZnS) wurtzite (ZnS) o CdCl2 CdI2 Comparison between structures with filled octahedral and tetrahedral holes Fluorite (CaF2, antifluorite Li2O) Fluorite structure = a face-centered cubic array (FCC) of cations = cubic close packing (CCP) of cations with all tetrahedral holes filled by anions = a simple cubic (SC) array of anions. Antifluorite structure = a face-centred cubic (FCC) array of anions = cubic close packing (CCP) of anions, with cations in all of the tetrahedral holes (the reverse of the fluorite structure). K2[PtCl6], Cs2[SiF6], [Fe(NH3)6][TaF6]2 Fluorite (CaF2, antifluorite Li2O) F / Li Fluorite structures (CaF2, antifluorite Li2O) Oxides: Na2O, K2O, UO2, ZrO2, ThO2 alkali metal sulfides, selenides and tellurides K2[PtCl6], (NH4)2[PtCl6], Cs2[SiF6], [Fe(NH3)6][TaF6]2. CaF2, SrF2, SrCl2, BaF2, BaCl2, CdF2, HgF2, EuF2, -PbF2, PbO2 Li2O, Li2S, Li2Se, Li2Te, Na2O, Na2S, Na2Se, Na2Te, K2O, K2S Sphalerite (zincblende, ZnS) Cubic close packing of anions with 1/2 tetrahedral holes filled by cations Sphalerite (zincblende, ZnS) Sphalerite (zincblende, ZnS) 13-15 compounds: BP, BAs, AlP, AlAs, GaAs, GaP, GaSb, AlSb, InP, InAs, InSb 12-16 compounds: BeS, BeSe, BeTe, -MnS (red), -MnSe, -CdS, CdSe, CdTe, HgS, HgSe, HgTe, ZnSe, ZnTe Halogenides: AgI, CuF, CuCl, CuBr, CuI, NH4F Borides: PB, AsB Carbides: -SiC Nitrides: BN Diamond 6,16 2,50 4,10 cubic hexagonal SiO2 cristobalite SiO2 tridymite ice Diamond Cubic Diamond Diamond Structure C, Si, Ge, -Sn * Add 4 atoms to a FCC * Tetrahedral bond arrangement * Each atom has 4 nearest neighbors and 12 next nearest neighbors Elements of the 14th Group Diamond Lattice (100) Diamond Lattice (110) Diamond Lattice (111) Diamond Lattice (111) Hard Sphere Model Diamond Lattice (111) Hard Sphere Model Face Centered Cubic Lattice (111) Hard Sphere Model Wurzite, ZnS Hexagonal close packing of anions with 1/2 tetrahedral holes filled by cations Wurzite, ZnS ZnO, ZnS, ZnSe, ZnTe, BeO, CdS, CdSe, MnS, AgI, AlN Semiconductors of 13-15 and 12-16 type Rock Salt, NaCl Cubic close packing of anions with all octahedral holes filled by cations Rock Salt, NaCl Anion and cation sublattices Rock Salt, NaCl Rock salt structures (NaCl) Hydrides: LiH, NaH, KH Borides: ZrB, HfB Carbides: TiC, ZrC, VC, UC Nitrides: ScN, TiN, UN, CrN, VN, ZrN Oxides: MgO, CaO, SrO, BaO, TiO, VO, MnO, FeO, CoO, NiO Chalcogenides: MgS, CaS, SrS, BaS, -MnS, MgSe, CaSe, SrSe, BaSe, CaTe Halides: LiF, LiCl, LiBr, LiI, NaF, NaBr, NaI, KF, KCl, KBr, KI, RbF, RbCl, RbBr, AgCl, AgF, AgBr Intermetallics: SnAs Other FeS2 (pyrite), CaC2 NiAs - type Hexagonal close packing of anions with all octahedral holes filled by cations NiS, NiAs, NiSb, NiSe, NiSn, NiTe, FeS, FeSe, FeTe, FeSb, PtSn, CoS, CoSe, CoTe, CoSb, CrSe, CrTe, CoSb, PtB (anti-NiAs structure) ReB2 - type Hexagonal close packing of anions with all tetrahedral holes filled by cations [Cr(NH3)6]Cl3, K3[Fe(CN)6] bcc Li3Bi - type (anti BiF3) Li3Bi - type (anti BiF3) Fe3Al [Cr(NH3)6]Cl3 K3[Fe(CN)6] Cubic close packing of anions with all tetrahedral and octahedral holes filled by cations CsCl Primitive cubic packing of anions with all cubic holes filled by cations CsCl CsCl is not BCC CsBr, CsI, CsCN, NH4Cl, NH4Br, TlCl, TlBr, TlI, CuZn, CuPd, LiHg SC of ReO6 octahedra ReO3 NaCl structure with 3/4 of cations removed and 1/4 of anions removed UO3, MoF3, NbF3, TaF3, Cu3N Perovskite, CaTiO3 Two equvivalent views of the unit cell of perovskite Ti CaO Ti O Ca Cubic "close packing" of Ca and O with 1/4 octahedral holes filled by Ti cations TiO6 ­ octahedra CaO12 ­ cuboctahedra (Ca2+ and O2- form a cubic close packing) preferred structure of piezoelectric, ferroelectric and superconducting materials Perovskite structure CaTiO3 Similarity to CsCl Perovskite, CaTiO3 Cubic "close packing" of A and X with 1/4 octahedral holes filled by B cations Perovskite, CaTiO3 KNbO3, KTaO3, KIO3, NaNbO3, NaWO3, LaCoO3, LaCrO3, LaFeO3, LaGaO3, LaVO3, SrTiO3, SrZrO3, SrFeO3. Rutile, TiO2 CN ­ stoichiometry Rule AxBy CN(A) / CN(B) = y / x Distorted hexagonal close packing of anions with 1/2 octahedral holes filled by cations (giving a tetragonal lattice) Rutile, TiO2 GeO2, CrO2, IrO2, MoO2, NbO2, -MnO2, OsO2, VO2 (>340K), RuO2, CoF2, FeF2, MgF2, MnF2 TiO6 ­ octahedra OTi3 ­ trigonal planar (alternative to CaF2 for highly charged smaller cations) The rutile structure: TiO2 fcc array of O2- ions, A2+ occupies 1/8 of the tetrahedral and B3+ 1/2 of the octahedral holes normal spinel: AB2O4 inverse spinel: B[AB]O4 (Fe3O4): Fe3+[Fe2+Fe3+]O4 basis structure for several magnetic materials The spinel structure: MgAl2O4 Spinel AB2X4 Spinel normal: Cubic close packing of anions with 1/2 octahedral holes filled by B cations and 1/8 tetrahedral holes by A cations MgAl2O4, CoAl2O4, MgTi2O4, Fe2GeO4, NiAl2O4, MnCr2O4 AB2X4 Spinel inverse: As for spinel but A cations and 1/2 of B cations interchanged MgFe2O4, NiFe2O4, MgIn2O4, MgIn2S4, Mg2TiO4, Zn2TiO4, Zn2SnO4, FeCo2O4. Layered Structures CdI2 Hexagonal close packing of anions with 1/2 octahedral holes filled by cations CoI2, FeI2, MgI2, MnI2, PbI2, ThI2, TiI2, TmI2, VI2, YbI2, ZnI2, VBr2, TiBr2, MnBr2, FeBr2, CoBr2, TiCl2, TiS2., TaS2. CdCl2 Cubic close packing of anions with 1/2 octahedral holes filled by cations CdCl2, CdBr2, CoCl2, FeCl2, MgCl2, MnCl2, NiCl2, NiI2, ZnBr2, ZnI2, Cs2O* (anti-CdCl2 structure) CdI2 Hexagonal close packing CdCl2 Cubic close packing CdCl2 Cubic close packing Fázové přeměny za zvýšeného tlaku Zvýšení koordinačního čísla Zvýšení hustoty Prodloužení vazebných délek Přechod ke kovovým modifikacím Sfalerit Chlorid sodný Důsledky zvýšení tlaku X-ray structure analysis with single crystals Principle of a four circle X-ray diffractometer for single crystal structure analysis CAD4 (Kappa Axis Diffractometer) IPDS (Imaging Plate Diffraction System)