This differs from 03._CrystalBindingAndElasticConstants

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This differs from 03._CrystalBindingAndElasticConstants.ppt only in the section “Analysis of Elastic Strain” in which a modified version of the Kittel narrative is used. 3. Crystal Binding and Elastic Constants. Crystals of Inert Gases Ionic Crystals Covalent Crystals Metals - PowerPoint PPT Presentation

Transcript of This differs from 03._CrystalBindingAndElasticConstants

  • This differs from 03._CrystalBindingAndElasticConstants.pptonly in the section Analysis of Elastic Strain in which a modified version of the Kittel narrative is used.

  • 3. Crystal Binding and Elastic ConstantsCrystals of Inert GasesIonic CrystalsCovalent CrystalsMetalsHydrogen BondsAtomic RadiiAnalysis of Elastic StrainsElastic Compliance and Stiffness ConstantsElastic Waves in Cubic Crystals

  • IntroductionCohesive energy energy required to break up crystal into neutral free atoms.Lattice energy (ionic crystals) energy required to break up crystal into free ions.

  • Kcal/mol = 0.0434 eV/moleculeKJ/mol = 0.0104 eV/molecule

  • Crystals of Inert GasesAtoms: high ionization energyoutermost shell filledcharge distribution sphericalCrystal: transparent insulatorsweakly bondedlow melting pointclosed packed (fcc, except He3 & He4).

  • Van der Waals London InteractionVan der Waals forces induced dipole dipole interaction between neutral atoms/molecules.Ref: A.Haug, Theoretical Solid State Physics, 30, Vol I, Pergamon Press (1972).Atom i charge +Q at Ri and charge Q at Ri + xi.( center of charge distributions )

  • H0 = sum of atomic hamiltonians 0 = antisymmetrized product of ground state atomic functions1st order term vanishes if overlap of atomic functions negligible.2nd order term is negative & R6 (van der Waals binding).

  • Repulsive InteractionPauli exclusion principle (non-electrostatic) effective repulsion Lennard-Jones potential: , determined from gas phase dataAlternative repulsive term:

  • Equilibrium Lattice ConstantsNeglecting K.E. For a fcc lattice:For a hcp lattice:R n.n. distAt equilibrium:Experiment (Table 4):Error due to zero point motion

  • Cohesive Energyfor fcc latticesFor low T, K.E. zero point motion. For a particle bounded within length , quantum correction is inversely proportional to the atomic mass:~ 28, 10, 6, & 4% for Ne, Ar, Kr, Xe.

  • Ionic Crystalsions: closed outermost shells ~ spherical charge distributionCohesive/Binding energy = 7.9+3.615.14 = 6.4 eV

  • Electrostatic (Madelung) EnergyInteractions involving ith ion:For N pairs of ions:z number of n.n. ~ .1 R0Madelung constantAt equilibrium:

  • Evaluation of Madelung ConstantApp. B: Ewalds methodKCli fixed

  • Kcal/mol = 0.0434 eV/moleculeProb 3.6

  • Covalent Crystals Electron pair localized midway of bond. Tetrahedral: diamond, zinc-blende structures. Low filling: 0.34 vs 0.74 for closed-packed.Pauli exclusion exchange interactionH2

  • Ar : Filled outermost shell van der Waal interaction (3.76A)Cl2 : Unfilled outermost shell covalent bond (2A)s2 p2 s p3 tetrahedral bonds

  • MetalsMetallic bonding: Non-directional, long-ranged. Strength: vdW < metallic < ionic < covalent Structure: closed packed (fcc, hcp, bcc) Transition metals: extra binding of d-electrons.

  • Hydrogen Bonds Energy ~ 0.1 eV Largely ionic ( between most electronegative atoms like O & N ). Responsible (together with the dipoles) for characteristics of H2O. Important in ferroelectric crystals & DNA.

  • Atomic RadiiNa+ = 0.97AF = 1.36ANaF = 2.33Aobs = 2.32AStandard ionic radii ~ cubic (N=6)Bond lengths:F2 = 1.417ANa Na = 3.716A NaF = 2.57ATetrahedral:C = 0.77ASi = 1.17ASiC = 1.94AObs: 1.89ARef: CRC Handbook of Chemistry & Physics

  • Ionic Crystal RadiiE.g. BaTiO3 : a = 4.004ABa++ O : D12 = 1.35 + 1.40 + 0.19 = 2.94A a = 4.16ATi++++ O : D6 = 0.68 + 1.40 = 2.08A a = 4.16ABonding has some covalent character.

  • Analysis of Elastic StrainsLetbe the Cartesian axes of the unstrained state be the the axes of the stained stateUsing Einsteins summation notation, we havePosition of atom in unstrained lattice:Its position in the strained lattice is defined asDisplacement due to deformation:Define ( Einstein notation suspended ):

  • Dilationwhere

  • Stress ComponentsXy = fx on plane normal to y-axis = 12 . (Static equilibrium Torqueless)

  • Elastic Compliance & Stiffness ConstantsS = elastic compliance tensorContracted indicesC = elastic stiffness tensor

  • Elastic Energy DensityLetthen Landaus notations:

  • Elastic Stiffness Constants for Cubic CrystalsInvariance under reflections xi xi C with odd numbers of like indices vanishes Invariance under C3 , i.e., All C i j k l = 0 except for (summation notation suspended):

  • where

  • Bulk Modulus & CompressibilityUniform dilation: = Tr eik = fractional volume changeB = Bulk modulus= 1/ = compressibilitySee table 3 for values of B & .

  • Elastic Waves in Cubic CrystalsNewtons 2nd law:dont confuse ui with u Similarly

  • Dispersion Equationdispersion equation

  • Waves in the [100] directionLongitudinalTransverse, degenerate

  • Waves in the [110] directionLonitudinalTransverseTransverse

  • Prob 3.10