Solid State

74%

Step 1: In FCC, atoms touch along the face diagonal: AC = 4r = a√2, so r = a√2/4. Step 2: Volume of one sphere = (4/3)π r³ = (4/3)π(a√2/4)³ = √2πa³/24. Step 3: Total volume of 4 spheres in FCC = 4 × (√2πa³/24) = √2πa³/6. Step 4: Packing efficiency = (√2πa³/6 / a³) × 100 = (1.414 × 3.14 / 6) × 100 ≈ 74%. Step 5: 52.4% is for Simple Cubic, 68% is for BCC. FCC (74%) has the highest packing efficiency among cubic systems, which is why it is called cubic close-packed (CCP).

The interplanar distance between two successive planes of atoms

Step 1: Bragg's equation is used in X-ray crystallography to determine crystal structure. Step 2: In 2d sinθ = nλ, each symbol has a specific meaning: 'd' is the interplanar spacing (distance between parallel planes of atoms), 'θ' is the angle of diffraction, 'n' is the order of diffraction, and 'λ' is the wavelength of X-rays used. Step 3: When X-rays hit crystal planes, they get reflected. If the path difference (2d sinθ) equals a whole number (n) of wavelengths (nλ), constructive interference occurs — this is diffraction. Step 4: By measuring θ and knowing λ and n, we can calculate d. Step

Schottky defect

Step 1: Schottky defect occurs when equal numbers of cations AND anions are missing from the crystal lattice, leaving behind vacancies (holes). Step 2: Since equal numbers of oppositely charged ions are missing, the electrical neutrality of the crystal is maintained. Step 3: This defect is common in ionic crystals where cations and anions are of similar size — like NaCl where Na⁺ and Cl⁻ have comparable sizes. Step 4: Schottky defect lowers the density of the crystal because mass decreases while volume stays nearly the same. Step 5: Frenkel defect is when an ion leaves its lattice site and goe

Cubic

Step 1: Each crystal system is defined by its edge lengths (a, b, c) and angles (α, β, γ). Step 2: Cubic system: a = b = c and α = β = γ = 90° — all edges equal, all right angles. This is the most symmetric system. Step 3: Tetragonal: a = b ≠ c and α = β = γ = 90° — two equal edges but the third is different. Step 4: Rhombohedral: a = b = c but α = β = γ ≠ 90° — all edges equal but angles are NOT 90°. This is a common trap! Step 5: Hexagonal: a = b ≠ c and α = β = 90°, γ = 120°. So cubic is the ONLY system where ALL edges are equal AND ALL angles are 90°.