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GATE PH: Solid State Physics

GATE PH · Work through your exam, one topic at a time.

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Solid State Physics

Solid State Physics in GATE PH tests crystal structures, Miller indices, reciprocal lattices, Bragg diffraction, phonons, free-electron and band theory, semiconductors, magnetism, and superconductivity. Numerical questions frequently use a cubic crystal and a diffraction angle.

Core rules and concepts

  • For a cubic lattice, the interplanar spacing is d_hkl = a/√(h² + k² + l²). Bragg's law is 2d sinθ = nλ.
  • The reciprocal lattice vector is G = h b₁ + k b₂ + l b₃. Diffraction peaks occur when the scattering vector equals a reciprocal lattice vector.
  • The Debye model gives a phonon heat capacity proportional to T³ at low temperature and approaching the Dulong-Petit value at high temperature.
  • In band theory, the effective mass is m* = ħ²/(d²E/dk²). A negative curvature at a band maximum corresponds to a hole-like carrier.
  • Semiconductors: carrier concentration depends on the density of states and the Fermi-Dirac distribution; doping shifts the Fermi level.

Worked example

A cubic crystal has lattice constant a = 4 Å. For first-order Bragg reflection from (100) planes, d = a = 4 Å. With λ = 1.54 Å, Bragg's law gives sinθ = λ/(2d) = 1.54/8 = 0.1925. Thus θ ≈ 11.1°. The corresponding diffraction angle 2θ is about 22.2°.

Traps that cost marks

  • Using d = a for non-cubic or higher-index planes.
  • Forgetting the factor 2 in Bragg's law or reporting θ instead of 2θ.
  • Missing systematic absences caused by the basis, not the lattice.
  • Confusing the reciprocal lattice factor 2π with the crystallographic convention.