A simple method for taking into account the multiple Coulomb scattering in construction of a separatrix (the line separating the regions of runaway and decelerating electrons in an electric field) is described. The desired line is obtained by solving a simple transcendental equation.
1
INTRODUCTION
The so-called separatrix—a line separating the run-
away and deceleration regions is reported in almost
each study devoted to runaway electrons. In the sim-
plest case, the kinetic electron energy T is plotted on the
horizontal axis and the energy losses or the correspond-
ing strength of the accelerating field (Floss = dT/dx or Fel =
dU/dx) is plotted on the vertical axis. In this case, the
separatrix is a dependence of the loss on energy; the
point corresponding to a given electron energy T and a
given field Eel lies either above or below the curve. In
the first case, the electron is accelerated (runs away),
and, in the second case, it is decelerated. Separatrices
are plotted in such a form in [1–3]. In construction of
these curves, multiple Coulomb scattering was disre-
garded. Nevertheless, it is obvious that a scattered elec-
tron changes the motion direction with respect to the
field and, when the angle between the motion direction
and field increases, a particle is weaker accelerated and
stronger (due to the increase in the path length) decel-
erated. It seems that the curve should shift upwards;
however, the situation is complicated by the fact that
scattering is a random process, and, therefore, it is not
obvious if the particle is accelerated or decelerated. We
can only speak about the probability of being acceler-
ated or decelerated. Let us consider the separatrix to be
a curve connecting the points characterized by the same
escape probability. In this statement of the problem, the
“old” separatrix connects the points with zero escape
probability. The purpose of the calculations reported
below is to construct a separatrix for the case where
most beam electrons are accelerated (the escape proba-
bility is close to unity).
STATEMENT OF THE PROBLEM
Let us consider a beam of N electrons propagating
downward, along the accelerating field directed parallel
to the z axis (also directed downward). The beam can be
described by the distribution function n(p) in the space
of momenta p = (z) or p = (t); here, n is the number
of particles per volume unit dp, and
( )d = N (one
particle with the probability density function f(p) can
be also considered; in this case, f(p) = n(p)/N and
( )d = 1).
We can easily determine the fraction of runaway
electrons if the function n(p) is known. To this end, we
have to solve the kinetic equation with appropriate ini-
tial conditions. However, according to [4], it is impos-
sible to determine n(p) analytically, and an adequate
approximation should be used.
Numerical calculations were reported in [1–3],
where the solution is sought for particular cases of elec-
tron motion. These calculations are fairly complicated
and, apparently, not very reliable since they yield dif-
ferent results. In this study, we also perform numerical
calculations, bearing in mind their maximum simplifi-
cation with conservation of reasonable accuracy.
CONSIDERATION OF SCATTERING
Analysis shows that a significant (if not the main)
source of uncertainty is the incorrect consideration of
electron scattering in air. For example, the ionization
loss function was used in [2, 5] to calculate the scatter-
ing; this approach is justified for electron–electron scat-
tering but has a poor accuracy for electron scattering by
nuclei (the case where energy losses are almost absent).
The difference is especially high at low energies (<100
keV).
A formula for the root-mean-square scattering angle
after passage of X radiation units is given in [6, 7]:
(1)
Here, p is the momentum of a particle, β is its velocity,
and ES = 21 MeV is a characteristic energy of multiple
p
p
n∫
p
p
f∫
p
p
θ
2
〈
〉
ES
X
pβ
-------.
=
Effect of Multiple Scattering on the Critical Electric Field
for Runaway Electrons in the Atmosphere
Ya. S. Elensky
Institute for Nuclear Research, Russian Academy of Sciences, pr. Shestidesyatiletiya Oktyabrya 7a, Moscow, 117312 Russia
e-mail: yansa@gcnet.ru
A
simple method for taking into account the multiple Coulomb scattering in construction of a sep-
aratrix (the line separating the regions of runaway and decelerating electrons in an electric field) is described.
The desired line is obtained by solving a simple transcendental equation.
DOI: 10.3103/S1062873807070428
2
scattering. This formula is exact only for fast particles
(E >100 MeV); for slow particles, it gives a very large
maximum angle (several rad) for each scattering and,
thus, the result is overestimated.
Somewhat other formula was reported in [8]:
,
(2)
where s is the passed distance, r0 is the classical elec-
tron radius, and υ is the electron velocity. The expres-
sion under the logarithm is transformed in [8] as fol-
lows:
130.60p/
mc;
(2a)
however, the numerical coefficient may range from
65.3 to 384. The above examples show that the accu-
racy of solution of the kinetic equation is low because
changes in angles are directly related to the coefficients
of this equation. Hence, it is reasonable to simplify it
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