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Crystal defects and diffusion

A metal has a vacancy formation energy of 0.90 eV. Using n/N = exp(−Q/kT), which value is closest to the equilibrium vacancy concentration (the fraction of lattice sites that are vacant) at 1000 K? Take Boltzmann's constant as k = 8.617×10⁻⁵ eV/K.

  1. AAbout 5.4×10⁻³
  2. BAbout 2.9×10⁻⁵
  3. CAbout 3.6×10⁻¹¹
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Answer: B. About 2.9×10⁻⁵

kT = 8.617×10⁻⁵ × 1000 ≈ 0.0862 eV, so Q/kT = 0.90/0.0862 ≈ 10.44. Then n/N = exp(−10.44) ≈ 2.9×10⁻⁵, or one vacancy for roughly every 35,000 lattice sites. About 5.4×10⁻³ is the mistake of halving the exponent to exp(−Q/2kT), confusing it with the expression for the carrier density of an intrinsic semiconductor. About 3.6×10⁻¹¹ is the mistake of computing 10 to the power −10.44 with a common logarithm instead of exp. The vacancy concentration rises exponentially with temperature and reaches around 10⁻⁴ near the melting point.

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