summary-b - Summary 1 Chemical binding 1.1 Covalent binding...

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Unformatted text preview: Summary 1 Chemical binding 1.1 Covalent binding Overlap of partially filled atomic wave functions ⇒ delocalized orbitals 2 atoms: ψ = c Aψ A + cBψ B N atoms in solids quasi continuous energy bands Splitting of energy levels increases with overlap. • Small coordination numbers ≤ 4 ⇒ open structions, low density • Tight binding typ. 4 - 8 eV/atoms ⇒ high Tm • Strongly directed bonds • Saturated bonds } very hard, little plasticity ⇒ fully occupied valence band (semiconductors, insulators) S-4 Summary 1.2 Ionic binding Smoth transition to covalent binding depending on ∆EN! U rep ∝ e − r / ρ or 1 rn n > 10 (Pauli principle) very short-ranged! U Coulomb ∝ 1 n • High binding energy: 6-11eV ⇒ High Tm, very hard, brittle (repulsion of equally charged ions) • Undirected bounds: ⇒ close packing • Electrons bound tightly: ⇒ No electronic conductivity (but ionic conductivity via defects at high temperature) S-5 Summary 1.3 Van der Waals binding Binding between fluctuating dipoles: U dipole ∝ 1 r6 • Contributes always to bonding • Very small binding energy 0.02-0.2 eV ⇒ low Tm, large r0 • Undirected bonds ⇒ close packing E.g., noble gas crystals, unpolar molecules, polymers (covalent bonds along chains) Because of filled orbitals only van der Waals binding possible! 1.4 Hydrogen bonds Bonds via static dipole moment of molecules of F, N, C covalently bound to H Binding energy typ. 0.1 eV 1.5 Metallic binding Ideal metallic binding: delocalized valence electrons around lattice of positive ion cores ⇒ high conductivity Binding energy typ. 1 eV E.g., alkali metals Undirected bonds ⇒ close packing, high plasticity Often strong: covalent bonds between d-orbitals ⇒ brittleness, high Tm. S-6 Summary 2 Crystal structure Crystal structure = lattice + basis Lattice: r1a1 + n2a2 + n3a3, ni = 0, ±1, ±2, ... Primitive unit cell: cell with smallest volume Close packing: Z = 12 1) fcc: stacking sequence of {111}: ABCABC ... 2) hcp: stacking sequence ABAB ... hexagonal lattice + basis (0 0 0), (2/3 1/3 1/2) Diamond structure: Z = 4 fcc latice + basis (0 0 0), (1/4 1/4 1/4) Different atoms in basis ⇒ Cubic ZnS (zinc blende): III-V semiconductors GaAs, GaP, ... Hexagonal ZnS (wurzite, hexagonal zinc blende) hcp + basis: II - VI compounds ZnS, ZnO, CdS, CdSe, ... NaCl Structure: fcc + basis (0 0 0) (1/2 0 0) Like bcc but different ions on sc sublattices. S-7 Summary Noncrystallline solids Short-range but no long-range order Pair correction function: g ( r ) = ρ ( r ) ρ 0 Metallic glasses: dense random packing of (soft) spheres Amorphous semiconductors: continuous random network • Very pronounced short-range order! • Substantial bond-angle variations • Dangling bonds ⇒ electronic properites Oxide glasses: Tg variable via Na+ ions ⇒ non-bridging O-. S-8 Summary 3 Diffraction by solids a 2 × a3 g 1 = 2π a ⋅ g = 2πδ ij Reciprocal lattice: a1 ⋅ ( a 2 × a3 ) and cyclic, i j Scattering condition: k - k0 = G ⇒ Ewald construction λ sin Θ = Bragg condition: 2 d hkl Miller indices hkl: Ghkl ⊥ (h k l) Brillouin zones: (important fcc, bcc, hcp) Enclosed by planes that perpendicularly bisect Ghkl Bragg reflection at Brillouin zone boundaries! Structure factor: S hkl = ∑ α ∈EZ fα e iG hkl ′ r α ⇒ fα: atomic structure factor selection rules unless fα different Experimental methods: Requirement: λ of order of magnitude of a Photos: 1 - 100 keV f ∝ Z2 ⇒ difficulties in detecting light elements Electrons: 10 - 1000 eV f very large ⇒ surface sensitive Neutrons: 0.01 - 1 eV f varies non-systematically ⇒ discrimination of neighboring elements f very small ⇒ large sample volume Techniques: Laue (λ continuum), rotating crystal, Debye Scherrer (powder) S-9 ...
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