Solid State
On this page
Direct answer
Fourteen Bravais lattices describe all possible crystal symmetries, but NEET lives on three cubic ones: simple cubic (Z = 1, 52.4 per cent packed), body-centred cubic (Z = 2, 68 per cent) and face-centred cubic (Z = 4, 74 per cent, matching hcp). Density follows d = Z·M/(a^3·NA), and in close packing every sphere touches 12 others, leaving two tetrahedral and one octahedral void per particle — the architecture that hosts NaCl (sodium in all octahedral holes) and CaF2. Schottky and Frenkel defects, F-centre colours, n- and p-type doping and magnetic ordering (Fe3O4 ferrimagnetic, MnO antiferromagnetic) complete the high-yield core.
What you must remember
- Edge-length relations: simple cubic a = 2r; bcc a = 4r/√3; fcc a = 2√2 r.
- hcp and ccp both give 74 per cent packing efficiency and coordination number 12 — hcp is ABAB, ccp is ABCABC.
- Radius rules in ionic lattices: octahedral void fits r+/r− ≥ 0.414; tetrahedral below that.
- Schottky defect (equal cation and anion vacancies, as in NaCl, CsCl, AgBr) lowers density; Frenkel defect (ion shifts to an interstitial site, as in AgCl, ZnS) leaves density unchanged.
- Heating NaCl in sodium vapour gives yellow colour from F-centres — electrons trapped in anion vacancies; KCl turns violet.
- n-type: group 15 dopant (As, Sb) in silicon; p-type: group 13 dopant (B, Al, In).
- MnO is antiferromagnetic; Fe3O4 and ZnFe2O4 are ferrimagnetic; ferromagnetism fades above the Curie temperature.
- Amorphous solids are isotropic; glass is a supercooled liquid rather than a true solid.
From unit cell to density: silver worked out
Silver (M = 108 g/mol) crystallises in fcc with edge 409 pm; find the density. Convert edge to centimetres first — 409 pm = 4.09 × 10^-8 cm — because d = Z·M/(a^3·NA) wants volume in cm^3. For fcc, Z = 4, so the numerator is 4 × 108 = 432 g per mol of cells. Cube the edge: (4.09 × 10^-8)^3 = 6.84 × 10^-23 cm^3, and multiply by NA = 6.022 × 10^23 to get 41.2 g per cell-mole. Density = 432/41.2 ≈ 10.5 g per cm^3 — the accepted value for silver. The workflow is always convert pm to cm, set Z by lattice type, then substitute; the two habitual errors are leaving the edge in picometres (introducing a factor of 10^30) and using Z = 1 for a "simple" metal that is actually fcc. When a question hands you density and asks for edge length, the same equation rearranged to a^3 = Z·M/(d·NA) gets you home in one line.
The packing-efficiency traps
The claim "fcc has the highest packing efficiency" needs care — hcp ties it at 74 per cent, and questions ask which two lattices share the maximum efficiency. Second, the defect pair gets muddled: Schottky needs similarly sized, highly ionic ions and drops the density; Frenkel needs a large size difference with small cations and keeps the density constant — AgBr uniquely shows both. Third, F-centre questions attach the colour to the metal: yellow for NaCl, violet for KCl, and the general rule that trapped electrons, not metal atoms, absorb light. Fourth, doping direction: adding a group 15 element donates extra electrons (n-type); group 13 creates holes (p-type) — a single swapped word flips the answer. Finally, do not call ferromagnetic and ferrimagnetic the same: in ferrimagnetic solids the domains align oppositely but unequally, so a net moment survives.
Frequently asked questions
What is the coordination number of an atom in hcp or ccp?
Twelve — six in its own layer plus three above and three below; bcc gives 8 and simple cubic 6.
Which point defect lowers the density of a crystal?
The Schottky defect, because equal numbers of cations and anions are missing from their lattice sites.
Why does NaCl turn yellow when heated in sodium vapour?
Anion vacancies trap electrons; these F-centres absorb visible light and impart colour — yellow for NaCl, violet for KCl.
How do n-type and p-type semiconductors differ?
n-type is doped with a group 15 element providing excess electrons; p-type with a group 13 element creating electron holes; both remain electrically neutral.
How many atoms belong to one fcc unit cell?
Four: eight corners contributing 1/8 each plus six faces contributing 1/2 each.