Frenkel & Schottky Point Defects

Defect Type

Conditions

Temperature T 900 K

Formation energy E_f 1.20 eV

Live State

n/N = exp(−E_f / k_BT)

Equilibrium n/N

0.0%

Defect count

0 / 512

k_BT

0.078 eV

E_f / k_BT

15.4

How it works

A crystal is never a perfect lattice at T > 0 — thermal agitation constantly creates and destroys point defects. The equilibrium number present at a given temperature is set purely by minimising free energy F = E − TS: making n defects costs energy nE_f but gains configurational entropy from the huge number of ways to arrange them, and the balance gives an Arrhenius law.

Schottky vacancy:
  n/N = exp(−E_f / k_BT)

Frenkel pair (vacancy + interstitial):
  n_F = √(N·N_i) · exp(−E_f / 2k_BT)

k_B = 8.617×10⁻⁵ eV/K

Schottky — an atom leaves its lattice site entirely (migrates to the surface), leaving one empty site (grey wire outline) per defect.
Frenkel — an atom is displaced into a nearby interstitial gap instead of leaving the crystal, so a vacancy (wire outline) and an interstitial atom (orange, off-lattice) always appear together. The extra freedom of "which interstitial site" halves the effective exponent, so at equal E_f Frenkel pairs are always more numerous than Schottky vacancies.
The scene relaxes the live defect count toward whichever equilibrium value the sliders currently predict — exactly the kinetic process (vacancy diffusion driven by the same Boltzmann factor) that lets a real crystal re-equilibrate after a temperature change.

This is why quenching a metal from high temperature "freezes in" excess vacancies, and why ionic crystals (which favour Frenkel/Schottky disorder depending on the ion size ratio) become significantly more conductive as they approach their melting point.

Lattice atom Vacancy Interstitial atom

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