This differs from 03._CrystalBindingAndElasticConstants
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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 ):
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):
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  directionLongitudinalTransverse, degenerate
Waves in the  directionLonitudinalTransverseTransverse