This differs from 03._CrystalBindingAndElasticConstants

1. 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.

2. 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

3. 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.

4. Kcal/mol = 0.0434 eV/molecule KJ/mol = 0.0104 eV/molecule

7. Crystals of Inert Gases • Crystal: • transparent insulators • weakly bonded • low melting point • closed packed (fcc, except He3 & He4). • Atoms: • high ionization energy • outermost shell filled • charge distribution spherical

10. 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 ) 

11. H0 = sum of atomic hamiltonians 0= antisymmetrized product of ground state atomic functions 1st order term vanishes if overlap of atomic functions negligible. 2nd order term is negative &  R6 (van der Waals binding).

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

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

14. 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.

15. Ionic Crystals ions: closed outermost shells ~ spherical charge distribution Cohesive/Binding energy = 7.9+3.615.14 = 6.4 eV

17. Electrostatic (Madelung) Energy Interactions involving ith ion: For N pairs of ions: z ﹦number of n.n. ρ ~ .1 R0 ﹦Madelung constant At equilibrium: →

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

19. Kcal/mol = 0.0434 eV/molecule Prob 3.6

20. Covalent Crystals H2 • 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

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

22. 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.

23. 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.

24. Atomic Radii Standard ionic radii ~ cubic (N=6) Na+ = 0.97A F = 1.36A NaF = 2.33A obs = 2.32A Bond lengths: F2 = 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

25. Ionic Crystal Radii E.g. BaTiO3 : a = 4.004A Ba++– O– – : D12 = 1.35 + 1.40 + 0.19 = 2.94A → a = 4.16A Ti++++ – O – – : D6 = 0.68 + 1.40 = 2.08A → a = 4.16A Bonding has some covalent character.

26. Analysis of Elastic Strains Let be 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 ):

27. Dilation where

28. Stress Components Xy = fx on plane normal to y-axis = σ12 . (Static equilibrium → Torqueless) 

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

30. Elastic Energy Density Let then  Landau’s notations:

31. Elastic Stiffness Constants for Cubic Crystals Invariance 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):

32. where 

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

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

35. Dispersion Equation → dispersion equation

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

37. Waves in the [110] direction → Lonitudinal Transverse Transverse

38. Prob 3.10