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)$$: Determined by reciprocals of axial intercepts, converted to integers; negative values indicated by a bar (e.g.,$(11\bar{1})$).</p></li><li><p><strong>Families of directions$$and planes${hkl}$</strong>: Group equivalent directions and planes.</p></li></ul><h4 collapsed="false" seolevelmigrated="true">Lattice Types and Systems</h4><ul><li><p><strong>Primitive (P)</strong>: One lattice point per unit cell at$0,0,0$.</p></li><li><p><strong>Non-primitive</strong>:</p><ul><li><p><strong>Body-centered (I)</strong>: Two lattice points ($0,0,0$and$0.5,0.5,0.5$).</p></li><li><p><strong>Face-centered (F)</strong>: Four lattice points ($0,0,0$;$0.5,0.5,0$;$0.5,0,0.5$;$0,0.5,0.5$).</p></li><li><p><strong>Base-centered (C)</strong>: Two lattice points ($0,0,0$and$0.5,0.5,0$or similar).</p></li></ul></li><li><p><strong>7 Crystal Systems</strong>: Cubic, Hexagonal, Tetragonal, Orthorhombic, Rhombohedral, Monoclinic, Triclinic.</p></li><li><p><strong>14 Bravais Lattices</strong>: Combinations of crystal systems and centering types.</p></li></ul><h4 collapsed="false" seolevelmigrated="true">Metallic Crystal Structures</h4><ul><li><p>Often adopt close-packed structures: Face-Centered Cubic (FCC) or Hexagonal Close-Packed (HCP).</p><ul><li><p>Examples FCC: Al, Cu, Ni.</p></li><li><p>Examples HCP: Ti, Zr, Mg.</p></li></ul></li></ul><h4 collapsed="false" seolevelmigrated="true">Covalent Crystal Structures</h4><ul><li><p>Example: Diamond (Cubic system, covalent bonding).</p></li><li><p>Example: GaAs (Cubic system, ~1$ \text{% Ionic Character} = 100 * {1 - \text{exp}[-0.25(cA - cB)^2] } $).</p></li></ul><h4 collapsed="false" seolevelmigrated="true">Ionic Crystal Structures and Pauling's Rules</h4><ul><li><p>Examples: MgO, CsCl (determined by high electronegativity difference).</p></li><li><p><strong>Pauling's Rules</strong>:</p><ol><li><p><strong>Cation coordination</strong>: Determined by the radius ratio ($ R{cation} / R{anion} $).</p></li><li><p><strong>Local electroneutrality</strong>: Anion coordination governed by electrostatic bond strength ($ Z{cation} / N{coordination} $).</p></li><li><p><strong>Polyhedral sharing</strong>: Ionic crystals favor corner sharing (rather than edge or face) to reduce cation repulsion.</p></li></ol></li></ul><h4 collapsed="false" seolevelmigrated="true">Polymer Crystal Structures</h4><ul><li><p>Example: Polyethylene (Orthorhombic system, covalent bonds within chains).</p></li><li><p>Monomer:$ CH2=CH2 $. Polymer:$ (-CH2-CH2-)_n $.</p></li><li><p>Molecular Weight =$ n \times m $$ (n=degree of polymerization, m=molecular weight of repeat unit).