Structure of Solids

Recap of Primary and Secondary Bonds

  • Atomic orbitals can hybridize into molecular orbitals (bonding or antibonding).

  • Different electronegativities lead to electron transfer (ionic bond) vs. hybridization (covalent bond).

  • Primary bonds: Covalent, ionic, metallic.

  • Secondary bonds: Van der Waals interactions, hydrogen bonding.

Crystalline vs. Amorphous Materials

  • Crystalline: Solid-state materials with long-range order (periodicity).

  • Amorphous: Lack long-range order.

Crystal Structure Fundamentals

  • Crystal Structure = Lattice x Motif.

  • Lattice: Periodic array of points in space.

  • Motif: Atom or group of atoms positioned on each lattice point.

  • Translation vector (t): t = ua + vb + wc where a,b,c are lattice vectors and u,v,w are integers.

  • Unit cell: Most basic repeating volume of the lattice, characterized by lattice parameters [a, b, c, \alpha, \beta, \gamma].

  • Fractional coordinates: Point positions expressed as fractions of unit-cell distances (e.g., 0,0,0).

Lattice Directions and Planes

  • Lattice direction [uvw]: Determined by subtracting tail coordinates from head coordinates; integers, with negative values indicated by a bar (e.g., [1\bar{1}0]).

  • **Lattice plane (hkl): Determined by reciprocals of axial intercepts, converted to integers; negative values indicated by a bar (e.g., (11\bar{1})).

  • Families of directions and planes {hkl}: Group equivalent directions and planes.

Lattice Types and Systems

  • Primitive (P): One lattice point per unit cell at 0,0,0.

  • Non-primitive:

    • Body-centered (I): Two lattice points (0,0,0 and 0.5,0.5,0.5).

    • Face-centered (F): Four lattice points (0,0,0; 0.5,0.5,0; 0.5,0,0.5; 0,0.5,0.5).

    • Base-centered (C): Two lattice points (0,0,0 and 0.5,0.5,0 or similar).

  • 7 Crystal Systems: Cubic, Hexagonal, Tetragonal, Orthorhombic, Rhombohedral, Monoclinic, Triclinic.

  • 14 Bravais Lattices: Combinations of crystal systems and centering types.

Metallic Crystal Structures

  • Often adopt close-packed structures: Face-Centered Cubic (FCC) or Hexagonal Close-Packed (HCP).

    • Examples FCC: Al, Cu, Ni.

    • Examples HCP: Ti, Zr, Mg.

Covalent Crystal Structures

  • Example: Diamond (Cubic system, covalent bonding).

  • Example: GaAs (Cubic system, ~1% ionic character calculated by \text{% Ionic Character} = 100 * {1 - \text{exp}[-0.25(cA - cB)^2] } ).

Ionic Crystal Structures and Pauling's Rules

  • Examples: MgO, CsCl (determined by high electronegativity difference).

  • Pauling's Rules:

    1. Cation coordination: Determined by the radius ratio ( R{cation} / R{anion} ).

    2. Local electroneutrality: Anion coordination governed by electrostatic bond strength ( Z{cation} / N{coordination} ).

    3. Polyhedral sharing: Ionic crystals favor corner sharing (rather than edge or face) to reduce cation repulsion.

Polymer Crystal Structures

  • Example: Polyethylene (Orthorhombic system, covalent bonds within chains).

  • Monomer: CH2=CH2 . Polymer: (-CH2-CH2-)_n .

  • Molecular Weight = n \times m $$ (n=degree of polymerization, m=molecular weight of repeat unit).