pigments_lect4(1)

pigments_lect4(1) - Structures of Solids & X-ray...

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Unformatted text preview: Structures of Solids & X-ray Diffraction Chemistry 123 Dr. Patrick Woodward Supplemental Lecture 4 Crystalline Solids Crystalline CaF2 Unit Cell • Crystal Lattice – A 3D array of points where each point has an identical environment. • Unit Cell – The repeating unit (a unit cell is to a crystal, like a “brick” is in a house). In a given crystal all unit brick” cells are identical. 1 Crystal Systems Cubic Tetragonal Hexagonal Rhombohedral Orthorhombic Monoclinic Triclinic The crystal systems each have distinctive symmetry and unit cell dimensions Close Packed Array of Spheres The gray spheres represent a 2D Close Packed Array. In 3D the next layer of spheres could sit on the depressions marked in red (B) or those marked in blue (C). (C). AB Stacking AC Stacking 2 Cubic and Hexagonal Close Packing Hexagonal Close Packing (ABAB…) (ABAB… ABAB Stacking Cubic Close Packing (ABCABC…) (ABCABC… ABCABC Stacking Close Packed Spheres Hexagonal Close Packing (ABAB…) (ABAB… Cubic Close Packing (ABCABC…) (ABCABC… 3 Hexagonal Close Packing Body Centered Cubic Packing Cubic Close Packing CsCl Structure Cubic or Hexagonal Close Packing Coordination Number = 12 Packing Efficiency = 74% Body Centered Cubic Packing Coordination Number = 8 Packing Efficiency = 68% 4 Eutactic Structures Many ionic structure types can be described as a close packing of anions with cations filling voids or holes in the structure. Generally we will consider two types of holes (for the cations) • Octahedral holes - Voids are surrounded by 6 anions and lead to octahedral coordination of the cation • Tetrahedral holes - Voids are surrounded by 4 anions and lead to tetrahedral coordination of the cation Octahedral Holes Start with a close packed layer of anions (A) Insert cations in the triangular depressions (c) The resulting cation coordination is an octahedron Add another anion layer (B) 5 Tetrahedral Holes Start with a close packed layer of anions (A) Insert cations in the triangular depressions (b) The resulting cation coordination is a tetrahedron Add another anion layer (B) Eutactic Structures Structures obtained by filling Octahedral Holes Structure Fraction Packing Holes Filled Structures obtained by filling Tetrahedral Holes Structure Fraction Packing Holes Filled NaCl ccp Fluorite‡ 1 ccp NiAs 1 hcp Sphalerite 1/2 ccp CdCl2 1/2 ccp Wurtzite 1/2 hcp CdI2 1/2 hcp TiO2† 1/2 hcp Al2O3 † The 1 2/3 hcp ‡In fluorite (i.e. CaF2) the cations are close packed and the anions fill the tetrahedral holes. The opposite is true of the antifluorite structure (Na2O) hcp anion layers are buckled in rutile. rutile. 6 Cubic close packed (CCP) anion array Rock salt structure (NaCl) (NaCl) (Octahedral Holes) Space Group = Fm3m Atom Site x Anion 4a 0 Oct Hole 4b ½ y 0 ½ z 0 ½ Antifluorite structure (Na2O) (Tetrahedral Holes) Space Group = Fm3m Atom Site x Anion 4a 0 Tetr Hole 8c ¼ y 0 ¼ z 0 ¼ CCP Anion Array & Tetrahedral Holes Zinc Blende Structure (ZnS) (ZnS) (50% Tetrahedral Holes) Space Group = F43m Atom Site x Anion 4a 0 Tetr Hole 4b ¼ y 0 ¼ z 0 ¼ Antifluorite structure (Na2O) (100% Tetrahedral Holes) Space Group = Fm3m Atom Site x Anion 4a 0 Tetr Hole 8c ¼ y 0 ¼ z 0 ¼ 7 Sphalerite (ZnS) (ccp, 50% Tetr. Holes Filled) ccp, Tetr. Space Group = F-43m FAtom Site x Zn 4a 0 S 4c ¼ y 0 ¼ z 0 ¼ Cation Coord. → Tetrahedron Coord. Anion Coord. → Tetrahedron Coord. Connectivity → Corner sharing Oct. Wurtzite (ZnO) (hcp, 50% Tetr. Holes Filled) hcp, Tetr. Space Group = P63mc Atom Site x 1/3 Zn 2b 1/3 O 2b 1/3 y 2/3 2/3 z 0 0.38 Cation Coord. → Tetrahedron Coord. Anion Coord. → Tetrahedron Coord. Connectivity → Corner sharing Oct. Hexagonal Close Packed Anion Array Nickel Arsenide Structure (Octahedral Holes) Space Group = P63/mmc Atom Site x y 1/3 2/3 Anion 2c Anion Oct Hole 2a 0 0 z 1/4 0 8 HCP Anion Array - Tetrahedral Holes No such structure exists Space Group = P63/mmc Atom Site x y 1/3 2/3 Anion 2c Anion Tetr Hole 4f 1/3 2/3 1/3 z 1/4 z z ~ 0.632 X-ray Diffraction 9 Diffraction Demo Take home message • The diffraction pattern is related but not equal to the grid pattern • Diffraction is most effective for monochromatic light whose wavelength is similar to the spacing of “slits” slits” • For crystals X-rays have a Xwavelength comparable to spacings of atoms Powder Diffractometer Divergence Slit Horizontal Diffraction Circle Sample (Vertical Flat Plate) Antiscatter Slit Receiving Slit θ Divergent X-ray Source 2θ Horizontal Soller Slits Detector 10 Single Crystal Diffraction Powder Diffraction Diffracted Beam Diffracted Beam Incident Incident Beam Incident Beam In powder diffraction only a small fraction of the crystals (shown in blue) are correctly oriented to diffract. Bragg’s Law 11 X-ray Powder Pattern Bond Distance from XRD Pattern (Ex. PbS, Rock Salt Structure) PbS, 1. Determine the 2-theta value, 2θ, and hkl values for a diffraction peak Int. 2θ = 38.7° 20 h=2, k=2, l=0 00 80 60 40 20 0 20 25 30 Bragg’s Law: 35 40 2Theta 45 50 55 60 65 λ = 2dhkl sin θhkl 12 Bond Distance from XRD (Cont.) (Ex. PbS, Rock Salt Structure) PbS, 2. Use Bragg’s Law and the wavelength of radiation (typically λ = 1.541 Å) to calculate dhkl λ = 2dhkl sin θhkl dhkl = λ /(2 sin θhkl) dhkl = 1.541 Å /{2 sin (38.7°/2)} = 2.10 Å 3. The interplanar spacing, dhkl, is related to the unit cell size. For a cubic crystal: a = (h2 + k2 + l2)1/2 dhkl a = (22 + 22 + 02)1/2(2.10 Å) = 5.94 Å Bond Distance from XRD (Cont.) (Ex. PbS, Rock Salt Structure) PbS, 4. Now that we know the unit cell size, the Pb-S distance can be determined from the unit cell using simple geometry. dist (Pb-S) = a/2 dist (Pb-S) = 5.94 Å/2 dist (Pb-S) = 2.97 Å 13 Peak Positions Bragg’s Law: λ = 2dhkl sin θhkl The distance between different planes of atoms in a crystal, dhkl, where h, k and l are integers that correspond to different planes Cubic: 1/d2 = (h2 + k2 + l2)/a2 Tetragonal: 1/d2 = {(h2 + k2)/a2} + (l2/c2) Orthorhombic: 1/d2 = (h2/a2) + (k2/b2) + (l2/c2) Hexagonal: 1/d2 = (4/3){(h2 + hk + k2)/a2} + (l2/c2) 14 ...
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