Crystal defects and diffusion
Learn the Arrhenius expression for vacancy concentration, dislocations and the Burgers vector, grain boundaries, Fick's laws, diffusion coefficients and the error-function solution with multiple-choice questions, including practice calculations for carburizing and dopant diffusion.
Diffusion coefficient and length →
Crystal defects and diffusion: key points
Real crystals contain point defects such as vacancies and interstitials, line defects called dislocations, and planar defects such as grain boundaries. The vacancy concentration rises steeply with temperature following exp(−Q/kT), and vacancies carry diffusion. Dislocations are characterized by the Burgers vector, which is perpendicular to the dislocation line for an edge dislocation and parallel to it for a screw dislocation.
Diffusion is described by Fick's laws; at steady state the flux is proportional to the concentration gradient. The diffusion coefficient depends strongly on temperature through D = D₀exp(−Q/RT), and √(Dt) gives a rough diffusion distance. When the surface concentration is held constant, as in carburizing, the error-function solution relates depth and time.
All Crystal defects and diffusion questions (12)
- Among point defects in ionic crystals, which correctly describes a Frenkel defect?
- Which correctly describes the relation between the Burgers vector and the dislocation line of an edge dislocation?
- 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.
- Carbon diffuses at steady state through a 5.0 mm thick steel plate whose two faces are held at carbon concentrations of 1.2 kg/m³ and 0.8 kg/m³. If the diffusion coefficient is 3.0×10⁻¹¹ m²/s, which value is closest to the diffusion flux of carbon? Assume the concentration varies linearly through the thickness.
- The diffusion coefficient of carbon in γ-iron follows the Arrhenius form D = D₀exp(−Q/RT) with D₀ = 2.3×10⁻⁵ m²/s and activation energy Q = 148 kJ/mol. Which value is closest to the diffusion coefficient at 927 °C? Take the gas constant as R = 8.314 J/(mol·K).
- The diffusion coefficient of an element was 1.0×10⁻¹⁴ m²/s at 800 K and 4.0×10⁻¹³ m²/s at 1000 K. Assuming the Arrhenius form D = D₀exp(−Q/RT), which value is closest to the activation energy Q? Take R = 8.314 J/(mol·K).
- A steel containing 0.20 wt% carbon is carburized with its surface carbon concentration held at 1.00 wt%. Which value is closest to the time needed for the carbon concentration to reach 0.60 wt% at a depth of 0.50 mm? Take the diffusion coefficient as 1.6×10⁻¹¹ m²/s, (C − C₀)/(Cₛ − C₀) = 1 − erf(x/2√(Dt)), erf(0.371) = 0.40 and erf(0.477) = 0.50.
- In a diffusion treatment at a fixed temperature, by what factor must the treatment time be increased to double the depth at which a given concentration is reached?
- In α-iron at the same temperature, the diffusion coefficient of carbon is many orders of magnitude larger than that of nickel. Which is the main reason?
- Which statement about diffusion paths in a polycrystalline metal is correct?
- When copper and brass (a Cu-Zn alloy) are bonded and heated, inert markers placed at the original interface move toward the brass side (the Kirkendall effect). What does this phenomenon show?
- The Burgers vector of a perfect dislocation in an FCC metal is (a/2)<110>. Which value is closest to its magnitude in aluminum (lattice parameter 0.405 nm)?