Lattice diffusion coefficient
An atom diffuses in the interstitial mechanism by passing from one interstitial site to one of its nearest neighboring interstitial sites. The movement of atoms can be described as jumps, and the interstitial diffusion coefficient depends on the jump frequency. The jump frequency, formula_1, is given by:
formula_2
where

formula_5 can be expressed as the sum of activation enthalpy term formula_7 and the activation entropy term formula_8, which gives the diffusion coefficient as:
formula_9
The diffusion coefficient can be simplified to an Arrhenius equation form:
formula_10
where

In the case of interstitial diffusion, the activation enthalpy formula_13 is only dependent on the activation energy barrier to the movement of interstitial atoms from one site to another. The diffusion coefficient increases exponentially with temperature at a rate determined by the activation enthalpy formula_13. The rate of self-diffusion can be measured experimentally by introducing radioactive A atoms (A
formula_17
where

An atom can make a successful jump when there are vacancies nearby and when it has enough thermal energy to overcome the energy barrier to migration. The number of successful jumps an atom will make in one second, or the jump frequency, can be expressed as:
formula_21
where


In thermodynamic equilibrium, 
formula_26 where formula_27 is the free energy of vacancy formation for a single vacancy. The diffusion coefficient in thermodynamic equilibrium can be expressed with formula_5 and formula_27, giving:
formula_30
Substituting ΔG = ΔH – TΔS gives:
formula_31
The diffusion coefficient can be simplified to an Arrhenius equation form:
formula_32
where

Compared to that of interstitial diffusion, the activation energy for self-diffusion has an extra term (ΔHv). Since self-diffusion requires the presence of vacancies whose concentration depends on ΔHv. Diffusion of a vacancy can be viewed as the jumping of a vacancy onto an atom site. It is the same process as the jumping of an atom into a vacant site but without the need to consider the probability of vacancy presence, since a vacancy is usually always surrounded by atom sites to which it can jump. A vacancy can have its own diffusion coefficient that is expressed as: 
formula_37
where formula_38 is the jump frequency of a vacancy. The diffusion coefficient can also be expressed in terms of enthalpy of migration (formula_7) and entropy of migration (formula_40) of a vacancy, which are the same as for the migration of a substitutional atom: 
formula_41
Comparing the diffusion coefficient between self-diffusion and vacancy diffusion gives: 
formula_42
where the equilibrium vacancy fraction formula_43 In a system with multiple components (e.g. a binary alloy), the solvent (A) and the solute atoms (B) will not move in an equal rate. Each atomic species can be given its own intrinsic diffusion coefficient formula_44 and formula_45, expressing the diffusion of a certain species in the whole system. The interdiffusion coefficient formula_46 is defined by the Darken's equation as:
formula_47
where formula_48 and formula_49 are the amount fractions of species A and B, respectively.
Diffusion Coefficient for Substitution Diffusion ; Vacancy Diffusion
In substitutional lattice diffusion (self-diffusion for example), the atom can only move by substituting place with another atom.