Environmental Engineering Reference
In-Depth Information
Rutherford formula (2.35) we obtain
R Δ ε
J ln J
d
σ
R d
Δ ε D
σ D
(2.64)
in the limit that the energy of an incident electron is large compared with the atom-
ic ionization potential
ε
J .
2.2.3
Three Body Recombination of Electrons and Ions
If collision of two free atomic particles results in the formation of a bound state of
these particles, it is necessary to transfer excess energy to other degrees of freedom.
For example, for photorecombination of electrons and positive ions with the forma-
tion of atoms, it is necessary to transfer the excess energy to a photon. If a third
atomic particle takes away energy, the process proceeds according to the scheme
A
C
B
C
C
!
AB
C
C .
(2.65)
In this three body process, particles A and B form a bound system, and particle C
carries away the energy released thereby. The balance equation for the number
densities of particles in the collision process (2.65) has the form
d [ AB ]
dt D
K [ A ][ B ][ C ] ,
(2.66)
where [ X ] is the number density of particles X and K is the rate constant for the
three body process, with dimensionality centimeters to the sixth power per second.
Equation (2.66) may be considered as the definition of the three body rate constant.
We now estimate the rate constant K for the three body process (2.65) following
the Thomson theory [112], which allows us to determine the dependence of the
rate constant on the interaction parameters of colliding atomic particles. We make
the assumptions of the Thomson theory [112] that the binding energy of the end-
product molecule AB is much larger than thermal energies, and the motion of the
particles is governed by classical laws. The formation of a bound state of particles A
and B occurs in the following way. As particles A and B approach each other, their
energy increases as the potential energy of interaction is converted into kinetic
energy. If a third particle C interacts strongly with particle A or particle B when
these particles are close to each other, then the third particle may take excess energy
from the initial kinetic energy of particles A or B . The bound state of particles A
and B is thus formed as a result of a collision with the third particle.
This physical picture can be used as a basis for estimation of the rate constant
for a three body process. Typical kinetic energies of the particles are of the order
of the thermal energy T . We assume the mass of the third particle C to be com-
parable to the mass of either particle A or particle B . Since the energy exchange
must exceed the initial kinetic energy of particles A and B , the interaction potential
 
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