This differs from 03 Crystal Binding And Elastic

This differs from 03. _Crystal. Binding. And. Elastic. Constants. 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 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 e. V/molecule KJ/mol = 0. 0104 e. V/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 Ref: A. Haug, “Theoretical Solid State Physics”, § 30, Vol I, Pergamon Press (1972). Van der Waals forces ＝ induced dipole – dipole interaction between neutral atoms/molecules. Atom i charge +Q at Ri and charge –Q at Ri + xi. ( 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: At equilibrium: Experiment (Table 4): Error due to zero point motion R n. n. dist

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 = 7. 9+3. 61 5. 14 = 6. 4 e. V

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 = 0. 0434 e. V/molecule Prob 3. 6

Covalent Crystals H 2 • 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

Ar : Filled outermost shell → van der Waal interaction (3. 76 A) Cl 2 : Unfilled outermost shell → covalent bond (2 A) s 2 p 2 → s p 3 → tetrahedral bonds

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

Hydrogen Bonds • • Energy ~ 0. 1 e. V 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 Standard ionic radii ~ cubic (N=6) Na+ = 0. 97 A F = 1. 36 A Na. F = 2. 33 A obs = 2. 32 A Bond lengths: F 2 = 1. 417 A Na –Na = 3. 716 A Na. F = 2. 57 A Tetrahedral: C = 0. 77 A Si = 1. 17 A Si. C = 1. 94 A Obs: 1. 89 A Ref: CRC Handbook of Chemistry & Physics

Ionic Crystal Radii E. g. Ba. Ti. O 3 : a = 4. 004 A Ba++ – O– – : D 12 = 1. 35 + 1. 40 + 0. 19 = 2. 94 A → a = 4. 16 A Ti++++ – O – – : D 6 = 0. 68 + 1. 40 = 2. 08 A → a = 4. 16 A Bonding has some covalent character.

Analysis of Elastic Strains Let be the Cartesian axes of the unstrained state be 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 Xy = fx on plane normal to y-axis = σ12. (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 xi → –xi 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 eik = fractional volume change B = Bulk modulus = 1/κ See table 3 for values of B & κ. κ = compressibility

Elastic Waves in Cubic Crystals Newton’s 2 nd law: don’t confuse ui with uα → Similarly

Dispersion Equation → dispersion equation

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

Waves in the [110] direction → Lonitudinal Transverse

Prob 3. 10