A point defect in which an atom is missing from a lattice site. Its equilibrium concentration rises with temperature as exp(−Q/kT), and vacancies carry the diffusion of substitutional atoms.
Japanese name: 空孔
Vacancy appears in 7 practice questions across 3 topics, including Crystal defects and diffusion, Computational materials science, Ceramics and glasses.
Forming a vacancy breaks bonds and raises the energy, but it also increases the configurational entropy because the number of ways to arrange the vacancies grows. The Gibbs energy is therefore not at a minimum with zero vacancies, and the equilibrium vacancy concentration is given by n/N ≈ exp(−Qv/kT). With Qv = 1.0 eV, the fraction of vacant lattice sites is about 1.6×10⁻¹⁷ at 300 K and about 9×10⁻⁶ at 1000 K, a change of more than 10 orders of magnitude with temperature. The vacancy formation energy tends to be larger for metals with higher melting points.
If the change in length ΔL/L of a heated specimen and the change in lattice parameter Δa/a measured by XRD are recorded together, their difference gives the vacancy concentration as Δn/N ≈ 3(ΔL/L − Δa/a) (the method of Simmons and Balluffi). This works because atoms move to the surface when vacancies form, so the whole specimen expands more than the lattice parameter alone. Other methods measure the increase in electrical resistance from vacancies retained by quenching from high temperature, or use positron annihilation, in which positrons trapped at vacancies have longer lifetimes.
Ionic crystals must stay electrically neutral, so vacancies form as pairs of cation and anion vacancies (Schottky defects) or as pairs of a vacancy and an interstitial ion (Frenkel defects). When impurities of different valence are added, charge-compensating vacancies form in amounts fixed by the composition, the oxygen vacancies in stabilized zirconia being an example. Self-interstitials in metals have formation energies several times larger than vacancies, so they are far fewer in thermal equilibrium, and most of them are created by irradiation or plastic working. In reactor materials, for example, such vacancy–interstitial pairs are produced in large numbers.
Substitutional atoms diffuse by exchanging places with vacancies, so the diffusion rate is set by the concentration and mobility of vacancies. When an aluminum alloy is quenched from the solution-treatment temperature, excess high-temperature vacancies are retained and speed up solute diffusion even at room temperature, advancing natural aging. When the vacancy flow is biased in one direction, Kirkendall voids form. In nuclear reactor materials, vacancies created by irradiation cluster into voids, causing swelling, an increase in volume. Part of high-temperature creep is also carried by vacancy diffusion.
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.
Answer: A. About 1.15 eV
E_v = −400.10 − (107/108)(−405.00) = −400.10 + 401.25 = 1.15 eV. The factor (N−1)/N is applied to compare with a state in which the removed atom is returned with the energy of one atom in the crystal (−405.00/108 = −3.75 eV), so that the numbers of atoms match. About 4.90 eV takes only the difference of the two total energies and leaves out the contribution of returning the removed atom to the crystal. About 3.75 eV is the magnitude of the energy per atom of the perfect crystal and is not the vacancy formation energy. The correct value can also be written as 4.90 − 3.75 = 1.15 eV. In real calculations, a small cell leaves interactions between the periodically repeated vacancies, so convergence with respect to cell size is checked.
Creating vacancies raises the energy, but the number of ways to place them is enormous, so the configurational entropy rises and the Gibbs energy is lower with a small number of vacancies present. Vacancies always exist at any temperature above 0 K.
In metals, equilibrium vacancies are far more numerous. A self-interstitial distorts its surroundings strongly, so its formation energy is several times that of a vacancy, and its equilibrium concentration is negligibly small.
It follows exp(−Qv/kT), so it changes enormously. In many metals, about one lattice site in ten thousand (10⁻⁴) is vacant just below the melting point, while the equilibrium concentration at room temperature is many orders of magnitude smaller. Quenching retains some of the high-temperature vacancies at room temperature.
Then n/N = exp(−10.44) ≈ 2.9×10⁻⁵, or one vacancy for roughly every 35,000 lattice sites.
A metal has a vacancy formation energy of 0.90 eV. Using n/…About 3.75 eV is the magnitude of the energy per atom of the perfect crystal and is not the vacancy formation energy.
A 108-atom supercell (with periodic boundary conditions) of…A Frenkel defect forms when an ion (usually the smaller cation) moves from its lattice site to an interstitial position, creating a vacancy and an interstitial ion as a pair.
Among point defects in ionic crystals, which correctly desc…Substitutional atoms such as nickel diffuse by the vacancy mechanism and can move only when a vacancy arrives next to them; the vacancy formation energy is added to the activation energy, making diffusion far slower.
In α-iron at the same temperature, the diffusion coefficien…This historic experiment showed that diffusion proceeds by the vacancy mechanism.
When copper and brass (a Cu-Zn alloy) are bonded and heated…Density functional theory Diffusion coefficient Brass Solid electrolyte Monte Carlo method Molecular dynamics Burgers vector Dislocation Fick's laws