Geoscience Reference
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In a more accurate model one should take into account the angle included between
the electric force eE and the electron momentum.
The dynamical friction force is equal to the electron energy loss due to the
electron collisions per unit length, that is,
d"
d z :
F fr ."/ D
(3.18)
A main contribution to the energy losses from high-energy electrons is caused by the
ionization of air. A peculiarity of this ionization process is that the energy of the fast-
moving electron is much greater than the energies of atomic electrons. This means
that the high-energy electron interacts with atomic electrons and nuclei as with free
particles. In such a case the friction force can be estimated as F fr ."/ "=, where
the free length of the electrons .Zn m c / 1 depends on the number density
of molecules n m , the mean number of electrons in molecule Z, and the scattering
cross section c . In the non-relativistic energy range the interaction between charged
particles is governed by the Coulomb law through Rutherford scattering cross-
section c e 4 =" 2 (e.g., see Gurevich and Zybin 2001 ). Combining the above
relationships, we arrive at the following estimate F fr ."/ e 2 Zn m =". Notice that
this dependence is in good agreement with the equation derived by Bethe ( 1930 )in
a more accurate model:
2e 4 Zn m
"
ln "
J z
F fr ."/ D
:
(3.19)
Here J z " i , where " i is the energy of ionization.
A schematic plot of the dynamical friction force of electrons as a function of
their kinetic energy is displayed in Fig. 3.22 . Here we do not show a few resonance
peaks in the low-energy region although on average the friction force approximately
increases in this region as shown with dashed line. As is seen from this figure,
the friction force reaches a maximum value at the energy " . This maximum
corresponds to the so-called thermal runaway breakdown threshold, which occurs
at the electric field E th 260 kV/cm. So large electric field does not occur at
stratospheric and mesospheric altitudes.
Equation ( 3.19 ) can be applied to the 10 2 -10 6 eV energy range where the friction
force falls off with increasing the electron energy. At higher energies one should take
into account relativistic effects which were ignored in deriving the above estimates.
The contribution of the relativistic effects results in gradual changes in the above
tendency in such a way that the friction force reaches a minimum F min at the energy
" min 1:4 MeV, and then a logarithmically slow increase begins to prevail at higher
energies (Gurevich and Zybin 2001 ).
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