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The Muppet’s Guide to: The Structure and Dynamics of Solids 2. Simple Crystal Structures

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Bonding or E A is bonding dependent Figure adapted from Callister, Materials science and engineering, 7 th Ed. Already looked at vdW and ionic

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Covalent Bond Short range interaction between pairs of atoms Highly directional in space Number of bonds proportional to number of valence electrons Conduction band Valence band (semi-conductors or insulators) Figure adapted from Callister, Materials science and engineering, 7 th Ed.

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Covalent Bond Relies on orbital overlap (hybridisation) Total wavefunction must be anti-symmetric Figure adapted from hyperphysics Bonding orbital formed from overlap of symmetric wavefunctions, Electrons must be anti-symmetric - bond - bond

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Hybridisation

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Covalent Structures Methane – sp3 Diamond, Si, Ge – sp3 Graphite and Graphene– sp2

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Si, Ge, Diamond, Organic molecules and Polymers, SiH 4, CH 4, H 2 O, HNO 3, HF.. Strong angular preference of bonds due to overlap Low density materials Open structures, polymorphs Range of bond energies Diamond – >3550°C Bismuth – 270°C Covalent Materials sp 2 hybridisation – trigonal planar structure sp 3 hybridisation – tetragonal tetrahedra Large lattice parameters

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Metallic Bonds Complex bonding mechanism between the degenerate electrons and the ion cores but also between electrons. Range of bond energies Tungsten: 3410°C Mercury: -39°C Not all electrons involved in bonding – good electrical and thermal conductors Figure adapted from Callister, Materials science and engineering, 7 th Ed.

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Crystal Structures How do atoms pack given their bonding? Figures adapted from Callister, Materials science and engineering, 7 th Ed.

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Packing Fraction Nature 453, 629-632 (29 May 2008), Physics World The secrets of random packing May 29, 2008

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In 2 D, each atom has 6 nearest neighbours Dense Packed Structures Atoms modelled as incompressible spheres

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Unit Cell, Lattice and Basis A crystal is a parallelepiped that is made up of a regular repeat of some representative unit, called the unit cell. Unit Cell: A volume of space bounded by lattice points which describe the symmetry. It is defined in terms of their axial lengths (a,b,c) and the inter- axial angles ( , , ). TRANSLATIONAL SYMMETRY maps the unit cells across the entire volume of the crystal

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Crystal Structure Convolution of Basis and lattice

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A lattice is an infinite periodic set of points defined by the three basis vectors, a,b and c. Lattice vector: T 2D Bravais Lattices In 2D total of 5 distinct lattices

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P P P P I I F F I C C T – trigonal R- rhombohedral T P Bravais Lattices – 14 possible in 3D R All lattices have translational symmetry

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Simple Metals BASIS BCC LATTICE W W

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Molecular crystals BASIS FCC LATTICE

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Lattice and Basis The basis can be convolved with the lattice in different ways due to the symmetry of the basis and lattice

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SiF 4 BASIS LATTICE CRYSTAL NB: The point symmetries of the basis and lattice MUST be compatible!

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In 2 D, each atom has 6 nearest neighbours Dense Packed Structures Atoms modelled as incompressible spheres Extend to three dimensions by layering sheets on top of each other Repeat Patterns: ABABAB…. Hexagonal close packed ABCABCABC… Face centred cubic

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Simple Centred Cubic Packing Fraction=52.4% No. of Neighbours=6 AAAAAAAAAA Stacked symmetry is cubic Polonium Figure adapted from Callister, Materials science and engineering, 7 th Ed. P Centre of 4 unit cells is an octahedral site

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Hexagonal Close Packed The second layer (B) is translated with respect to the first (A) such that the atoms in layer B sit in the dimples between the atoms in layer A Packing Fraction=74%No. of Neighbours=12c/a=1.663 ABABABABAB Cd, Mg, Zn Co Figure adapted from Callister, Materials science and engineering, 7 th Ed. P

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Face Centred Cubic Initial stacking is the same as hcp. Then the third layer (C) is translated with respect to both the first and second such that the atoms in layer C sit in the dimples between the atoms in layer B. Packing Fraction=74% No. of Neighbours=12 ABCABCABCABC Noble Gases Cu, Ag, Au, Ni, Al, Pb [111] Figure adapted from Callister, Materials science and engineering, 7 th Ed. F

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FCC (111) [111]

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Body Centred Cubic Packing Fraction=68% No. of Neighbours=8 ABABABABAB Stacked symmetry is cubic not hexagonal Cr Fe W I

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Tetragonal Distortions In such cases the structure is usually written as bct or fct Figure adapted from Callister, Materials science and engineering, 7 th Ed.

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Covalent Elements - Diamond Group VI elements such as C, Si and Ge sp 3 hybridisation - tetrahedra http://cwx.prenhall.com http://www.ipap.jp/jpsj/news/jpsj-nc_17-fig1.gif Packing Fraction=37% Number of neighbours=12 2 FCC lattices

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Diatomic, AX type structures The three most common AX type structures are cubic and named after the representative examples: Rocksalt – NaCl Caesium Chloride – CsCl Zinc blende or sphalerite - ZnS Ionic Covalent

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Diatomic, AX type structures The three most common AX type structures are cubic and named after the representative examples: Rocksalt – NaCl Caesium Chloride – CsCl Zinc blende or sphalerite - ZnS Ionic Covalent

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The Rocksalt Structure Structure adopted for materials with strong ionic bonds MgO, MnS, LiF, FeO, Alkali halides and hydrides and II-VI compounds fcc lattice Each cation/anion is surrounded by 6 neighbours of the opposite kind in a perfect octahedral arrangement. Figure adapted from Callister, Materials science and engineering, 7 th Ed.

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Caesium Chloride Primitive lattice Each cation/anion is surrounded by 8 neighbours of the opposite kind. Resembles bcc lattice but it is not because the atom at the centre is different and so it is not a lattice point. Figure adapted from Callister, Materials science and engineering, 7 th Ed.

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Zinc Blende A structure that resembles the Diamond structure. Common in materials which exhibit low ionic character and thus favour sp 3 hybridised bonds and tetragonal bond angles III-V and I-VII as well as ZnS ( =18%), SiC ( =12%), CdTe, ZnTe, MnTe No. of Neighbours=12 Figure adapted from Callister, Materials science and engineering, 7 th Ed.

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