3 Crystal Binding Crystals of Inert Gases Ionic

3. Crystal Binding • • • Crystals of Inert Gases Ionic Crystals Covalent Crystals Metals Hydrogen Bonds Atomic Radii

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

Van der Waals – London Interaction

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.