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 Hydrogen Bonds Atomic Radii Analysis of Elastic Strains Elastic Compliance and Stiffness Constants Elastic Waves in Cubic Crystals

Introduction Cohesive 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 = eV/moleculeKJ/mol = eV/molecule

Crystals of Inert Gases Atoms: high ionization energy outermost shell filled charge distribution spherical Crystal: transparent insulators weakly bonded low melting point closed packed (fcc, except He 3 & He 4 ).

Van der Waals – London Interaction Van 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 R i and charge –Q at R i + x i. ( center of charge distributions ) 

H 0 = sum of atomic hamiltonians  0 = antisymmetrized product of ground state atomic functions 1 st order term vanishes if overlap of atomic functions negligible. 2 nd order term is negative &  R  6 (van der Waals binding).

Repulsive Interaction Pauli exclusion principle  (non-electrostatic) effective repulsion Lennard-Jones potential: ,  determined from gas phase data Alternative repulsive term:

Equilibrium Lattice Constants Neglecting K.E.  For a fcc lattice: For a hcp lattice: R  n.n. dist At equilibrium:  Experiment (Table 4): Error due to zero point motion

Cohesive Energy for fcc lattices For 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 Crystals ions: closed outermost shells ~ spherical charge distribution Cohesive/Binding energy =  5.14 = 6.4 eV

Electrostatic (Madelung) Energy Interactions involving ith ion: For N pairs of ions: z ﹦ number of n.n. ρ ~.1 R 0 ﹦ Madelung constant At equilibrium: →

Evaluation of Madelung Constant App. B: Ewald’s method KCl i fixed

Kcal/mol = 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 interaction H2H2

Ar : Filled outermost shell → van der Waal interaction (3.76A) Cl 2 : Unfilled outermost shell → covalent bond (2A) s 2 p 2 → s p 3 → tetrahedral bonds

Metals Metallic 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 H 2 O. Important in ferroelectric crystals & DNA.

Atomic Radii Na + = 0.97A F  = 1.36A NaF = 2.33A obs = 2.32A Standard ionic radii ~ cubic (N=6) Bond lengths: F 2 = 1.417A Na –Na = 3.716A  NaF = 2.57A Tetrahedral: C = 0.77A Si = 1.17A SiC = 1.94A Obs: 1.89A Ref: CRC Handbook of Chemistry & Physics

Ionic Crystal Radii E.g. BaTiO 3 : a = 4.004A Ba ++ – O – – : D 12 = = 2.94A → a = 4.16A Ti ++++ – O – – : D 6 = = 2.08A → a = 4.16A Bonding has some covalent character.

Analysis of Elastic Strains Letbe the Cartesian axes of the unstrained state be the the axes of the stained state Using Einstein’s summation notation, we have Position of atom in unstrained lattice: Its position in the strained lattice is defined as Displacement due to deformation: Define ( Einstein notation suspended ):

Dilation where

Stress Components X y = f x on plane normal to y-axis = σ 12. (Static equilibrium → Torqueless) 

Elastic Compliance & Stiffness Constants S = elastic compliance tensor Contracted indices C = elastic stiffness tensor

Elastic Energy Density Let then  Landau’s notations:

Elastic Stiffness Constants for Cubic Crystals Invariance under reflections x i → –x i  C with odd numbers of like indices vanishes Invariance under C 3, i.e.,  All C i j k l = 0 except for (summation notation suspended):

where 

Bulk Modulus & Compressibility Uniform dilation: δ = Tr e ik = fractional volume change B = Bulk modulus = 1/κ κ = compressibility See table 3 for values of B & κ.

Elastic Waves in Cubic Crystals Newton’s 2 nd law:don’t confuse u i with u α → Similarly 

Dispersion Equation → dispersion equation

Waves in the [100] direction → Longitudinal Transverse, degenerate  

Waves in the [110] direction → Lonitudinal Transverse

Prob 3.10