Solid State: Unit Cell, Packing & Defects

Crystal Systems and Unit Cells: Crystalline solids have long-range order with repeating structural units called unit cells. Seven crystal systems exist: cubic, tetragonal, orthorhombic, hexagonal, monoclinic, triclinic, and rhombohedral (trigonal).
The cubic system (a = b = c, $\alpha$ = $\beta$ = $\gamma$ = 90 degrees) is most important for JEE. Three cubic unit cells: Simple Cubic (SC), Body-Centred Cubic (BCC), and Face-Centred Cubic (FCC/CCP).
Lattice points per unit cell: SC = 1 (8 corners x $\frac{1}{8}$), BCC = 2 (8 x $\frac{1}{8}$ + 1 body centre), FCC = 4 (8 x $\frac{1}{8}$ + 6 faces x $\frac{1}{2}$).

Relationship Between Edge Length and Atomic Radius: SC: a = 2r (atoms touch along edge).
BCC: a = 4r/$\sqrt{3}$ (atoms touch along body diagonal = a*$\sqrt{3}$).
FCC: a = 2*$\sqrt{2}$r (atoms touch along face diagonal = a$\sqrt{2}$). These relationships are essential for density calculations.

Coordination Number: SC: CN = 6 (edge neighbours). BCC: CN = 8 (body centre touches 8 corners). FCC: CN = 12 (face-centred atoms touch 12 nearest neighbours). Higher CN = more efficient packing.

Packing Efficiency: The fraction of total volume occupied by atoms.
SC: $\pi$/(6) = 52.36%.
BCC: $\pi$$\frac{\sqrt{3}}{8}$ = 68.02%.
FCC/HCP: $\pi$/(3$\sqrt{2}$) = 74.05%. FCC and HCP have the same packing efficiency — both are closest packed structures with different stacking sequences (ABC vs ABAB).

Close Packing: Hexagonal Close Packing (HCP): ABAB stacking of close-packed layers. Cubic Close Packing (CCP/FCC): ABCABC stacking. Both have 74.05% packing efficiency and CN = 12. In close-packed structures: tetrahedral voids = 2n, octahedral voids = n (where n = number of atoms). Tetrahedral void radius ratio: r/R = 0.225. Octahedral void radius ratio: r/R = 0.414.

Density Formula: d = ZM/($N_{A}$ * $a^{3}$), where Z = atoms per unit cell, M = molar mass, a = edge length, $N_{A}$ = Avogadro's number. This is the most commonly tested formula. Rearrange to find Z, M, or a.

Radius Ratio and Structures: Radius ratio = $\frac{r_{cation}}{r_{anion}}$ determines the coordination geometry. 0.155-0.225: trigonal planar (CN 3). 0.225-0.414: tetrahedral (CN 4, e.g., ZnS). 0.414-0.732: octahedral (CN 6, e.g., NaCl). 0.732-1.0: cubic (CN 8, e.g., CsCl). NaCl structure: Cl- in FCC, Na+ in all octahedral voids (4 NaCl per unit cell). CsCl structure: Cl- at corners, Cs+ at body centre (1 CsCl per unit cell, NOT BCC). ZnS (zinc blende): $S^{2}$- in FCC, $Zn^{2}$+ in alternate tetrahedral voids (4 ZnS per unit cell).

Crystal Defects: Point defects in ionic crystals: (1) Schottky defect: equal cation and anion vacancies. Observed in NaCl, KCl, CsCl (high CN, similar ion sizes). Density decreases. (2) Frenkel defect: ion displaced from normal site to interstitial site. Observed in ZnS, AgBr, AgCl (large size difference, low CN). Density unchanged. AgBr shows both Schottky and Frenkel defects. Non-stoichiometric defects: metal excess (F-centres, anion vacancies with trapped electrons — gives colour) and metal deficiency (cation vacancies compensated by higher oxidation state of neighbouring cation).

Band Theory: Metals: overlapping valence and conduction bands. Insulators: large band gap (> 3 eV). Semiconductors: small band gap (~ 1 eV). Intrinsic semiconductors: pure Si, Ge. n-type: doped with Group 15 (P, As) — extra electrons. p-type: doped with Group 13 (B, Al) — electron holes.

Electrical and Magnetic Properties: Conductors: $10^{4}$-$10^{7} \; ohm^{-1} \; cm^{-1}$. Semiconductors: $10^{-6}$-$10^{4}$. Insulators: $10^{-20}$-$10^{-10}$. Paramagnetism: unpaired electrons, attracted by magnetic field. Diamagnetism: all paired, weakly repelled. Ferromagnetism: permanent magnetisation (Fe, Co, Ni). Antiferromagnetism: equal antiparallel magnetic moments (MnO). Ferrimagnetism: unequal antiparallel moments (Fe₃O₄).