Solid State

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Solid State — Key Facts for CUET Crystal systems: 7 systems (cubic, tetragonal, orthorhombic, hexagonal, rhombohedral, monoclinic, triclinic); 14 Bravais lattices Cubic: simple (SC) — 1 atom/unit cell; bcc — 2 atoms/unit cell; fcc — 4 atoms/unit cell Packing efficiency: fcc = hcp = 74% (closest packing); bcc = 68%; simple cubic = 52% Coordination number: fcc/bcc/hcp = 12; bcc = 8; simple cubic = 6 Defects: Schottky (cation + anion vacancies), Frenkel (cation displaced to interstitial site), F-centre (electron trapped by anion vacancy) ⚡ Exam tip: Density of unit cell $\rho = \frac{n \times M}{N_A \times a^3}$; know how to calculate for any cubic lattice

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding of crystal structures and properties.

Solid State — CUET Chemistry Study Guide

Solids are classified into crystalline (ordered, definite geometry) and amorphous (disordered, no definite crystal structure). Crystalline solids have a repeating pattern called a lattice, with smallest repeating unit being the unit cell.

Seven Crystal Systems: Based on symmetry elements. The seven systems are: cubic ($a=b=c$, $\alpha=\beta=\gamma=90°$), tetragonal ($a=b≠c$, all 90°), orthorhombic ($a≠b≠c$, all 90°), hexagonal ($a=b≠c$, $\alpha=\beta=90°$, $\gamma=120°$), rhombohedral ($a=b=c$, $\alpha=\beta=\gamma≠90°$), monoclinic ($a≠b≠c$, $\alpha=\gamma=90°$, $\beta≠90°$), triclinic ($a≠b≠c$, all angles different).

The 14 Bravais Lattices: Combinations of lattice parameters with lattice types (primitive P, body-centred I, face-centred F, C-centred). In cubic alone: simple cubic (SC/P), body-centred cubic (bcc/I), face-centred cubic (fcc/F).

Atomic Packing in Cubic Systems:

• Simple Cubic: atoms touch along cube edge. Coordination number = 6. Packing efficiency = 52.4%. Body diagonal = $a\sqrt{3}$.
• Body-centred Cubic (bcc): atoms touch along body diagonal. Coordination number = 8. Packing efficiency = 68%. Body diagonal = $2R\sqrt{3} = a\sqrt{3}$.
• Face-centred Cubic (fcc): atoms touch along face diagonal. Coordination number = 12. Packing efficiency = 74%. Face diagonal = $2R\sqrt{2} = a\sqrt{2}$.

Relationship between edge length and atomic radius:

• SC: $a = 2R$
• bcc: $a = \frac{2R\sqrt{3}}{1}$ (since $a\sqrt{3} = 4R$)
• fcc: $a = 2R\sqrt{2}$ (since $a\sqrt{2} = 4R$)

Radius Ratio Rule: For ionic crystals, the ratio of radii $r_+/r_-$ determines coordination number and geometry. For CN = 4 (tetrahedral): $0.225 \leq r_+/r_- < 0.414$. For CN = 6 (octahedral): $0.414 \leq r_+/r_- < 0.732$. For CN = 8 (cubic): $0.732 \leq r_+/r_- < 1.00$.

Example: Calculate packing efficiency of fcc. In fcc, face diagonal = $a\sqrt{2} = 4R$ (4 atomic radii across face diagonal of 2 atoms touching). $a = \frac{4R}{\sqrt{2}} = 2\sqrt{2}R$. Volume of unit cell = $a^3 = (2\sqrt{2}R)^3 = 16\sqrt{2}R^3$. Number of atoms per unit cell = 4 (8 corners × 1/8 + 6 face centres × 1/2 = 1 + 3 = 4). Packing efficiency = $\frac{4 \times \frac{4}{3}\pi R^3}{16\sqrt{2}R^3} \times 100 = \frac{16\pi R^3}{3 \times 16\sqrt{2}R^3} \times 100 = \frac{\pi}{3\sqrt{2}} \times 100 \approx 74%$.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Solid State — Complete CUET Chemistry Notes

Defects in Crystals:

• Schottky defect: equal number of cations and anions missing from lattice (maintains charge neutrality). Common in ionic compounds with similar-sized ions (NaCl, KCl). Increases with temperature. More common in compounds with high coordination numbers.
• Frenkel defect: a cation displaced from its lattice site to an interstitial site. Common in compounds where cation is much smaller than anion (AgCl, ZnS). Does not change overall stoichiometry.
• F-centre: electron trapped in an anion vacancy. Gives colour to crystals (e.g., NaCl appears yellow, KCl appears violet). The trapped electron can absorb visible light.
• Non-stoichiometric defects: variable composition (e.g., FeO is actually Fe₀.₉₅O due to Fe³⁺ occupying some Fe²⁺ sites).

Magnetic Properties:

• Diamagnetic: all electrons paired, weakly repelled by magnetic field (e.g., NaCl, TiO₂)
• Paramagnetic: unpaired electrons, weakly attracted (e.g., O₂, Cu²⁺ complexes)
• Ferromagnetic: unpaired electrons aligned parallel, strongly attracted, retains magnetism (e.g., Fe, Co, Ni, Gd)
• Antiferromagnetic: unpaired electrons aligned antiparallel, cancelling magnetic moments (e.g., MnO)
• Ferrimagnetic: unequal antiparallel alignment, net magnetic moment (e.g., Fe₃O₄, MgFe₂O₄)

Superconductivity: Certain materials conduct electricity with zero resistance below critical temperature $T_c$. Type I: abrupt loss of resistivity at $T_c$ (e.g., Hg, Pb). Type II: gradual penetration of magnetic field (e.g., Nb₃Sn, YBCO). Meissner effect: superconductor expels magnetic field from its interior. Applications: MRI machines, particle accelerators, power transmission.

Band Theory: In solids, atomic orbitals combine to form bands. Valence band (highest occupied) and conduction band (lowest unoccupied) may overlap (metal), have a small gap (semiconductor), or have a large gap (insulator). Band gap $E_g$ for Si = 1.1 eV; Ge = 0.67 eV; C (diamond) = 5.5 eV.

Semiconductors:

• Intrinsic: pure Si, Ge — electrons thermally excited across band gap
• n-type: doping with P (group 15) → extra electron, Fermi level near conduction band
• p-type: doping with B (group 13) → hole (absence of electron), Fermi level near valence band
• p-n junction: forms basis of diodes, transistors, solar cells

X-ray Diffraction: Used to determine crystal structure. Bragg’s law: $n\lambda = 2d\sin\theta$. Different crystal planes (hkl) give different $\theta$. From $\theta$ values, $d$-spacings can be calculated and unit cell dimensions determined.

Liquid Crystals: States of matter between crystalline solid and isotropic liquid. Thermotropic LCs: phases appear at certain temperatures (nematic, smectic, cholesteric). Used in LCD displays — liquid crystal molecules align with applied electric field, controlling light passage.

CUET Exam Patterns (2022–2024):

• Packing efficiency and density calculations for cubic systems are most frequent (2–3 marks)
• Crystal systems and Bravais lattices are commonly tested
• Schottky vs Frenkel defects is a common MCQ
• Magnetic properties and semiconductor doping occasionally appear
• Common mistakes: forgetting that atoms at corners contribute 1/8 each (cube has 8 corners); not converting units (pm to m); confusing coordination numbers

⚡ Key insight: For any density calculation problem, first identify the lattice type to get the number of atoms per unit cell ($n$). Then calculate volume of unit cell ($a^3$). Use $\rho = \frac{n \times M}{N_A \times a^3}$. Always check units: if $a$ is given in pm, convert to cm ($1 \text{ pm} = 10^{-10} \text{ cm} = 10^{-12} \text{ m}$, and $a^3$ in cm³). If calculating from radius, remember $a = 2R$ for SC, $a = 4R/\sqrt{3}$ for bcc, $a = 2R\sqrt{2}$ for fcc.

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