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bound states. However, the contribution of these states is dependent on the particle
size. The reason for this lies in the fact that in the free molecule zone the incident
ion is assumed not to experience any collisions with the carrier gas molecules and
thus eventually returns back to the diffusion zone. Hence, according to this theory
only the trajectories intersecting the particle surface are able to contribute to the
recombination efficiency. Meanwhile, ion states lying deeper in the Coulomb energy
at the separation distance should contribute to recombination efficiency independent
of particle size. The chance for the ion to return to the state with higher energies is
negligibly small. Here we report on an attempt to apply this flux-matching theory
for determining the efficiency of ion-particle recombination. We assume that the
three-body trapping becomes efficient only inside the limiting sphere. We thus solve
the kinetic equation in the free molecule zone assuming that the ion can loose a
part of its energy and thus be trapped by the particle. The rate of the losses is
introduced and is considered to be known. In principle, its value can be found
from the solution of the Boltzmann equation for the ion moving in the Coulomb
field. Here we estimate this constant from the data on the ion-ion recombination.
Hence, our theory includes three steps. We first solve the diffusion equation for the
ion moving in the Coulomb field created by the particle. Instead of the boundary
condition to this diffusion equation, we introduce the value of the ion flux toward
the target particle. Then, we solve the Boltzmann equation that takes into account
only the losses of ions caused by the three-body trapping. Their returns to states
with higher energies are ignored. The next step assumes the calculations of the
ion profiles in the diffusion and free molecule zones and their first derivatives.
The conditions of continuity of the ion profile and its derivative at the surface of
the limiting sphere give a set of two equations for determining the radius of the
free molecule zone and ion flux onto the aerosol particle. The final expression for
the recombination efficiency thus consists of two parts. The first term contains the
geometric particle cross section times the Coulomb enhancement factor. This part
depends on the particle size as its first power. The second part is proportional to the
area of the limiting sphere times a factor linearly dependent on the rate of the ion
energy losses in the free molecule zone. This term is independent of the particle size
in the limit of very small particles. The ideology of our approach reminds us of that
adopted in the theory of nuclear reactions, where the optical potential is introduced
whose imaginary part is responsible for capturing the incident nucleons by atomic
nuclei. This imaginary part of the optical potential is also an empirical constant,
although it can be calculated once the rates of doorway inelastic channels can be
properly accounted.
However, under some reasonable approximation it is possible to derive a simple
expression for recombination efficiency. Namely, we assume that at the border of
the limiting sphere all ions disappear; that is, the recombination process inside the
limiting sphere goes so fast that diffusion transport in the diffusion zone is not able
to deliver enough ions to restore the non-zero ion concentration at r D R.a/.Then
the efficiency of recombination is given by the formula
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