Dopants react with point defects via the reaction (3.2-35);
point defects and dopant point-defect pairs may react with electrons 
and holes 
(3.2-36) - (3.2-39). Reactions according
to the Frank-Turnbull mechanism (
and
) are neglected. The superscript 
represents the charge state of the particle.
Assuming that the reactions with electrons and holes is fast compared to
diffusion phenomena, the reactions (3.2-36) - (3.2-39)
have reached the equilibrium state, and we can use the law of mass action to
set up a relation between the concentrations of particles in various charge
states. Denoting 
the concentration of particle 
in charge
state , the total point defect concentration 
and the total
dopant point-defect pair concentration 
can be written
as (3.2-40) and (3.2-41), respectively. The concentrations of
particles in the different charges states and 
are related
to the concentrations of the neutral particles 
and 
by (3.2-42) and (3.2-43), respectively. There, 
is
the electron concentration, and 
is the intrinsic carrier concentration.
The basic idea of pair diffusion is that a substitutional particle cannot
diffuse just by itself. Particles residing on substitutional sites
get paired with point defects and move as dopant point-defect pairs. With
the charge state of the substitutional dopant we get the equations of
the particle fluxes (3.2-44) - (3.2-46).
Applying relations (3.2-40) - (3.2-43) we end up
at (3.2-47) and (3.2-48) for the fluxes of the total
concentrations, where 
, 
are mean diffusion coefficients
and 
, 
are mean electric charges.
The local charge neutrality approximation (3.2-51) together with
the relations from Boltzmann statistics (
,
) provide the expression for the electrostatic
potential (3.2-51). Neglecting the charges caused by point defects
and dopant point-defect pairs, which is only justified as soon as dopants
are primarily at substitutional sites, we obtain (3.2-52) for the
net active concentration and (3.2-53) for the electric field.
The reaction of dopants with point defects via (
)
accounts for the generation and recombination of dopant point-defect pairs
. A generation-recombination term 
has to be included in the
conservation laws for the dopant point-defect pairs (3.2-54) and
dopants (3.2-55), as well as point defects (3.2-56). The
Frenkel-pair mechanism (
) accounts for the
recombination of interstitials and vacancies and is included as 
in the
conservation law for point defects.
is modeled according to the law of mass action as (3.2-57).
The rate constants for generation 
and annihilation 
are
chosen to describe the kinetics between the neutral concentrations
, 
and 
. The Frenkel-pair constant 
is
the reaction rate with which the interstitial concentration 
and vacancy
concentration 
tend to approach their equilibrium values 
and
, respectively.
