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## Zumdahl’s Chapter 10 and Crystal Symmetries

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**Zumdahl’s Chapter 10 and Crystal Symmetries**Liquids Solids**Intermolecular Forces**The Liquid State Types of Solids X-Ray analysis Metal Bonding Network Atomic Solids Semiconductors Molecular Solids Ionic Solids Change of State Vapor Pressure Heat of Vaporization Phase Diagrams Triple Point Critical Point Contents**Intermolecular Forces**• Every gas liquifies. • Long-range attractive forces overcome thermal dispersion at low temperature. ( Tboil ) • At lower T still, intermolecular potentials are lowered further by solidification. ( Tfusion ) • Since pressure influences gas density, it also influences the T at which these condensations occur. • What are the natures of the attractive forces?**London Dispersion Forces**• AKA: induced-dipole-induced-dipole forces • Electrons in atoms and molecules can be polarized by electric fields to varying extents. • Natural electronic motion in neighboring atoms or molecules set up instantaneous dipole fields. • Target molecule’s electrons anticorrelate with those in neighbors, giving an opposite dipole. • Those quickly-reversing dipoles still attract.**Induced Dipolar Attraction**+••••••– –••••••+ • Strengths of dipolar interaction proportional to charge and distance separated. • So weakly-held electrons are vulnerable to induced dipoles. He tight but Kr loose. • Also l o n g molecules permit charge to separate larger distances, which promotes stronger dipoles. Size matters.**Permanent Dipoles**• Non-polar molecules bind exclusively by London potential R–6 (short-range) • True dipolar molecules have permanently shifted electron distributions which attract one another strongly R–4 (longer range). • Gaseous ions have strongest, longest range attraction (and repulsion) potentials R–2. • Size being equal, boiling Tpolar > Tnon-polar**Strongest Dipoles**• “Hydrogen bonding” potential occurs when H is bound to the very electronegative atoms of N, O, or F. • So H2O ought to boil at about – 50°C save for the hydrogen bonds between neighbor water molecules. • It’s normal boiling point is 150° higher!**The Liquid State (Hawaii?)**• The most complex of all phases. • Characterized by • Fluidity (flow, viscosity, turbulence) • Only short-range ordering (solvation shells) • Surface tension (beading, meniscus, bubbles) • Bulk molecules bind in all directions but unfortunate surface ones bind only hemispherically. • Missing attractions makes surface creation costly.**Type of Solids**• While solids are often highly ordered structures, glass is more of a frozen fluid. • Glass is an amorphous solid. “without shape” • In crystalline solids, atoms occupy regular array positions save for occasional defects. • Array composed by stacking of the smallest unit cell capable of reproducing full lattice.**Types of Lattices**• While there are quite a few Point Groups and hundreds of 2D wallpaper arrangments, there are only SEVEN 3D lattice types. • Isometric (cubic), Tetragonal, Orthorhombic, Monoclinic, Triclinic, Hexagonal, and Rhombohedral. • They differ in the size and angles of the axes of the unit cell. Only these 7 will fill in 3D space.**Isometric (cubic)**• Cubic unit cell axes are all • THE SAME LENGTH • MUTUALLY PERPENDICULAR • E.g.,“Fools Gold” is iron pyrite, FeS2, an unusual +4 valence.**Tetragonal**• Tetragonal cell axes: • MUTUALLY PERPENDICULAR • 2 SAME LENGTH • E.g., Zircon, ZrSiO4. This white zircon is a Matura Diamond, but only 7.5 hardness. • Real diamond is 10. Diamonds are not tetragonal but rather face-centered cubic.**Orthorhombic**• Orthorhombic axes: • MUTUALLY PERPENDICULAR • NO 2 THE SAME LENGTH • E.g., Aragonite, whose gem form comes from the secretion of oysters; it’s CaCO3.**Monoclinic**• Monoclinic cell axes: • UNEQUAL LENGTH • 2 SKEWED but PERPENDICULAR TO THE THIRD • E.g., Selenite (trans. “the Moon”) a fully transparent form of gypsum, CaSO4•2H2O**Triclinic**• Triclinic cell axes: • ALL UNEQUAL • ALL OBLIQUE • E.g., Albite, colorless, glassy component of this feldspar, has a formula NaAlSi3O8. • Silicates are the most common minerals.**Hexagonal**• Hexagonal cell axes: • 3 EQUAL C2 • PERPENDICULAR TO A C6 • E.g., Beryl, with gem form Emerald and formula Be3Al2(SiO3)6 • Diamonds are cheaper than perfect emeralds.**_**3 Rhombohedral • Rhombohedral axis: • CUBE stretched (or squashed) along its diagonal. (a=b=c) • DIAGONAL is bar 3 • “rotary inversion” • E.g., Quartz, SiO2, the base for amethyst with it purple color due to an Fe impurity.**Lattice Type**Isometric Tetragonal Orthorhombic Monoclinic Triclinic Hexagonal Rhombohedral Essential Symmetry Four C3 C4 Three perpendicular C2 C2 None (or rather “i” all share) C6 C3 Identification (Point Symmetry Symbols)**Classes**• Although there’s only 7 crystal systems, there are 14 lattices, 32 classes which can span 3D space, and 230 crystal symmetries. • Only 12 are routinely observed. • Classes within a system differ in the symmetrical arrangement of points inside the unit cube. • Since it is the atoms that scatter X-rays, not the unit cells, classes yield different X-ray patterns.**Simple cubic**“Primitive” P Body-centered cubic “Interior” I Face-centered cubic “Faces” F “Capped” C if only on 2 opposing faces. BCC FCC Common Cubic Classes**Materials Density**• Density of materials is mass per unit volume. • Unit cells have dimensions and volumes. • Their contents, atoms, have mass. • So density of a lattice packing is easily obtained from just those dimensions and the masses of THE PORTIONS OF atoms actually WITHIN the unit cell.**INTERIOR atoms count in their entirety.**FACE atoms count for only the ½ inside. EDGE atoms count for only the ¼ inside. CORNER atoms are only 1/8 inside. Counting Atoms in Unit Cells**Gold is FCC.**a = b = c = 4.07 Å # Au atoms in cell: 1/8 (8) + ½ (6) = 4 M = 4(197 g) = 788 g Volume NAv cells: (4.0710–10 m)3 Nav 3.9010–5 m3 = 39.0 cc = M / V = 20.2 g/cc 4 Å Gold’s Density from Unit Cell**capped**Bravais Lattices • 7 lattice systems + P, I, F, C options • P: atoms only at the corners. • I: additional atom in center. • C: pair of atoms “capping” opposite faces. • F: atoms centered in all faces. • Totals 14 types of unit cells from which to “tile” a crystal in 3d, the Bravais Lattices. • Adding point symmetries yields 230 space groups.**–**– New Names for Symmetry Elements • What we learned as Cn (rotation by 360°/n), is now called merely n. 3’s a 3-fold axis. • Reflections used to be but now they’re m (for mirror). So mmm means 3 mirrors. • In point symmetry, Snwas 360°/n and then but now it is just n, still a 360°/n but now followed by an inversion (which is now 1).**Triclinic: **All 7 lattice systems have centrosymmetry, e.g., corner, edge, face, & center inversion pts! Designation: 1 These are inversion points only because the crystal is infinite! While all 7 have these, triclinic hasn’t other symmetry operations. It’s 1 meansinversion. Triclinic Lattice Designation – –**The principal rotation axes are “4”, but it is the**four3axes that are identifying for cubes. The 4–fold axes have an m to each. Each 3–fold axis has a trio of m in which it lies. All 3 to be shown. The cube is m3m All its other symmetries are implied by these. Cubic (isometric) Designation 3 m m**The Three Cubic Lattices**• Where before we called them simple, body-centered, and face-centered cubics, the are now P m3m, I m3m, and F m3m, resp. • The cubic has the highest and the triclinic the lowest symmetry. The rest of the Bravais Lattices fall in between. • We will designate only their primitive cells. • It will help when we get to a real crystal.**Orthorhombicall 90° but a b c. Trivial.**It’s mmm because: Rhombohedral all s= but 90°; a=b=c It’s 3m because: – 3 Ortho vs. Merely Rhombic – m**Tetragonal all 90° and a=b c**Principle axis is 4 which is m But it is also || to mm So it is designated as 4/mmm Abbreviated 4/mmm Last of the Great Rectangles 4 m m m**Monoclinic**a b c == 90° < Then b is a 2-fold axis and to m So it is 2/m b is a 2 because the crystal is infinite. Nature’s Favorite for Organics m 2**Hexagonal refers to the outlined rhomboid ( =120° ) of**which there are six around the hexagon! So a 6 That 6 has a m and two ||mm. m is a mirror because the crystal’s infinite. (finally) Hexagonal So it is 6/mmm 6 m m m**Lattice Type**Isometric “Cubic” Tetragonal Orthorhombic Monoclinic Triclinic Hexagonal Rhombohedral Crystal Symmetries m 3 m ( m4 + 3+||+||+|| ) 4 / m mm (4 m + ||+|| ) mmm (m m m) 2 / m ( 2 m) 1 (invert only) 6 / m mm (6 m + ||+|| ) 3 m ( 3 + ||+||+|| ) _ _ _ Lattice Notation Summary**X-ray Crystal Determination**• Since crystals are so regular, planes with atoms (electrons) to scatter radiation can be found at many angles and many separations. • Those separations, d, comparable to , the wavelength of incident radiation, diffract it most effectively. • The patterns of diffraction are characteristic of the crystal under investigation!**X-rays have d.**X-rays mirror reflect from adjacent planes in the crystal. If the longer reflection exceeds the shorter by n, they reinforce. If by (n+½), cancel! 2d sin = n, Bragg d Diffraction’s Source reinforced d sin**Relating Cell Contents to **• Atomic positions replicate from cell to cell. • Reflection planes through them can be drawn once symmetries are known. • Directions of the planes are determined by replication distances in (inverse) cell units. • Interplane distance, d, is a function of the direction indices (Miller indices).**The index for a full cell move along axis b is 1. Its**inverse is 1. That for ½ a cell on b is ½. Its inverse is 2. Intersect on a parallel axis is ! Its inverse makes more sense, 0. Shown is (3,2,0) b/2 a/3 Inverse Distances c a b**Set of 320 planes at right (looking down c).**Their normal is yellow. (h,k,l) = (3,2,0) Shifts are a/h, b/k, c/l Inverses h/a, k/b, l/c Pythagoras in inverse! d–2hkl = (h/a)–2 + (k/b)–2 + (l/c)–2 for use in Bragg Interplane Spacings (cubic lattice)**c** Bragg Formula b a • 2 sin / = 1 / d (conveniently inverted) • Let the angles opposite a, b, and c be , , and . (All 90° if cubic, etc.) • Then Bragg for cubic, orthorhombic, monoclinic, andtriclinic becomes: • 2 sin / = [ (h/a)2 + (k/b)2 + (l/c)2 + 2hkcos/ab + 2hlcos/ac + 2klcos/bc ]½**Triclinic**Monoclinic Orthorhombic Tetragonal Rhombohedral Hexagonal Cubic a b c; a b c; = = 90° < a b c; = = = 90° a = b c; = = = 90° a = b = c; = = 90° a = b = c; = = 90°; = 120° a = b = c; = = = 90° Unit Cell Parameters from X-ray**Glide Plane**Simultaneous mirror with translation || to it. a, b, or c if glide is ½ along those axes. n if by ½ along a face. d if by ¼ along a face. Screw axis, nm Simultaneous rotation by 360°/n with a m/n translation along axis. New Space Symmetry Elements cell 2 32 screw cell 1 a glide**Systematic Extinctions**• Both space symmetries and Bravais lattice types kill offsome Miller Index triples! • Use missing triples to find P, F, C, I • E.g., if odd sums h+k+l are missing, the unit cell is body-centered and must be I. • Use them to find glide planes and screw axes. • E.g., if all oddh is missing from (h,k,0) reflections, then there is an a glide (by ½) c. • http://tetide.geo.uniroma1.it/ipercri/crix/struct.htm**Nature’s Choice Symmetries**• 36.0% P 21 / c monoclinic • 13.7% P 1 triclinic • 11.6% P 21 21 21orthorhombic • 6.7% P 21monoclinic • 6.6% C 2 / c monoclinic • 25.4% All (230 – 5 =) 225 others! • 75% these5; 90% only 16 total for organics. • Stout & Jensen, Table 5.1 _**Packing in Metals**ABA : hexagonal close pack ABC : cubic close pack**A**B C A Relationship to Unit Cells Is FCC ABC : cubic close pack**A**B A 90° 120° ABA (hcp) Hexagonal The white lines indicate an elongated hexagonal unit cell with atoms at its equator and an offset pair at ¼ & ¾. If we expand the cell to see it’s shape, we get a diamond at both ends…3 make a hexagon whose planes are 90° to the sides of the (expanded) cell.**Alloys (vary properties of metals)**• Substitutional • Heteroatoms swap originals, e.g., Cu/Sn (bronze) • Intersticial • Smaller interlopers fit in interstices (voids) of metal structure, e.g., Fe/C (steels) • Mixed • Substitutional and intersticial in same metal alloy, e.g., Fe/Cr/C (chrome steels)**Phase changes mean**Structure reorganization Enthalpy changes, H Volume changes, V Solid-to-Solid E.g., red to white P Solid-to-Liquid Hfusion significant Vfusion small Solid-to-Gas Hsublimation very large Vsublimation very large Liquid-to-Gas Hvaporization large Vvaporization very large All occur at sharplydefinedP,T, e.g., P 1 bar; Tfusionnormal FP Phase Changes**Heating Curve (1 mol H2O to scale)**Csteam T 60 steam heats water becomes steam heat (kJ) Hvaporization ice warms water warms Hfusion CiceT Cwater T ice becomes water 0 0°C T 100°C**Equilibrium Vapor Pressure, Peq**• At a given P,T, the partial pressure of vapor above a volatile condensed phase. • If two condensed phases present, e.g., solid and liquid, the one with the lower Peq will be the more thermodynamically stable. • The more volatile phase will lose matter by gas transfer to the less (more stable) one because such equilibrium are dynamic!