P/2292
USA
Theory of Runaway Electrons
By H, Dreicer*
This paper treats the problem of electrons moving
through an infinite gas of positive ions under the
influence of a static uniform electric field of arbitrary
strength. In evaluating the electrical conductivity
of such a gas the conventional treatment involves a
perturbation solution of the time-independent Boltzmann equation, and results in the well-known (temperature )3/2 law.1 Two assumptions are basic to these
treatments: (1) that a steady state electron velocity
distribution is attained several mean-free collision
times after the electric field is applied, and (2) that
the terminal electron drift velocity is small compared
to the average random electron speed. Both assumptions are avoided in this paper. In the next
section the problem is formulated starting with
the Boltzmann equation and a review of approximate
analytic solutions appropriate to the weak and strong
electric field cases is presented. We then describe
a time-dependent numerical solution to the Boltzmann equation and compare these results with the
approximate solutions.
All of these treatments lead to the conclusion that
this problem does not admit a time-independent
solution. Because of the strong energy dependence
of the Rutherford scattering law, the electron drift
velocity is not bounded by a terminal value, rather it
grows monotonically with time. This is the so-called
runaway effect predicted by Giovanelli.2
Collective effects, or plasma oscillations, are ignored
in this work, although these undoubtedly play an
important role in the conduction of electricity through
a plasma.
distribution function F therefore possesses cylindrical
symmetry and we may consider it to be a function
of the radial velocity variable с and cz. We use the
Fokker-Planck equation to describe the effect of
Coulomb encounters on F. However, in the interest
of clarity we defer the details of this collision term
to the next section.
Strong Field Regime
In the limit of strong fields we may neglect the
interaction between particles compared to their interaction with the applied field. The solution to Eq. (1)
which satisfies the initial condition
F {с, cz, 0) =
{т/2якТ0)Ч*
exp-[(w/2*T 0 ) (<* + <**)] (2)
is then simply the displaced Maxwellian distribution
F(c, cZy t) = (т/ШТ0)шЬ
exp - {(m/2kT0) (c« + [cz - »(*)]*)},
(3)
where v, the electron drift velocity, is determined by
the equation of motion
dv/dt = eE/m.
(4)
If we assume that this solution has approximately
the correct form for all values of applied field, then
we may use it to satisfy the Boltzmann equation on
LI M * ( z ) LEADS TO
LAW CONDUCTIVITY
REVIEW OF APPROXIMATE ANALYTIC
TREATMENTS
Following standard procedure, the statistical behavior of the electrons is described by a velocity
distribution function F which satisfies the Boltzmann
equation
dF/dt + {eE/m) 3F/dcz = dFjdt^.
1)
The electric field is applied along the negative z axis
of a stationary cartesian coordinate system. The
Figure 1. The velocity dependence, ^ ( Z ) , of the dynamical
friction force as a function of Z
* Los Alamos Scientific Laboratory, University of California,
Los Alamos, New Mexico.
57
58
SESSION A-5
P/2292
H. DREICER
1.0
I
I
I I I I I [I
5 -
10'
kT
10
Ю12
Ю13
I014
Ю15
I016
Ю17
Ю18
DENSITY IN CM"3
nm
Figure 2. The critical electric field as a function of electron
density with average electron energy in volts as parameter
10""
10
10"
20
88
64
: /|=4.0
7 7/
/
;
/
/
- 1 /
. I /
40
32
24
8
100
120
140
160
180
I /
.
.
.
.
10"
/
/
zo
/h
Z 56
48
16
80
.
80 1
72
60
Normalized temperature, T/To, as a function of т
with £/Ec as a parameter
Figure 3.
96
40
/
I* 10
/
/
E
c
'
;
10
/
/
-|=02
'
-
•
•
\
\
Figure 4.
/
I
i
IO¿
i
i
i i i I 1I
5
I0 1
\
I
i
i i I I l
5
1.0
/
/
/
4=0.1
—
•
—
*
*
С
__————•"" s
Normalized drift velocity, Z, as a function of т with
£/£ c as parameter
Figure 5.
Runaway rate X as a function of £/£ c
THEORY OF RUNAWAY ELECTRONS
The Maxwellian velocity distribution shown at the
initial instant of time with V as a parameter
Figure 8. The velocity distribution shown at т = 2 with V as
a parameter
59
Figure 7. The velocity distribution shown at т =
a parameter
Figure 9. The velocity distribution shown at т = 3 with V as
a parameter
60
SESSION A-5
.o1
i
1
1
1
1
P/2292
H. DREICER
Table 1
г"
T¡T0
-2
0
1
2
3
3.75
4
ILOO
]1.04
]L22
]1.48
]L71
1L.79
5
2.14
6
7
8
9
10
2.51
2.89
3.27
3.64
4.00
25
10
.ó3
5
ю4
5
ю5
and furnishes a useful criterion for separating the
weak and strong field cases. Figure 2 gives values
of Ec for a range of temperatures and densities.
Since the positive ions are assumed to be stationary
Eq. (5) also yields the rate of Joule heating
5
ю6
5
= eEG
7
ю
ZW{Z).
The appropriate dimensionless variables for this
problem are
5 ;
,ô8
0
0.36
0.59
0.82
1.00
1.07
1.35
1.67
2.01
2.38
2.77
3.18
E/Ec, T/To, Zt and r = {eEc/m) (m/2kT0)H, (8)
1
1
1
-6
-2
2292.10
Figure 10. The velocity distribution shown at т = 4 with V as
a parameter
with Ec and Z defined in terms of the initial electron
temperature To, and r measured in terms of the meanfree time for collisions between electrons moving
with the thermal speed (2kT0/tn)b. Equations (5)
the average. The drift velocity in Eq. (3) is then the
solution of an equation of motion which includes
the effects of collisions and has the form t
dv/dt + eEc1¥(Z)/m = eE/m,
(5)
where
{m¡2kT0) In Л
eEclm = Ann
n = electron particle density,
Z-Xe = ionic charge,
Z = (ml2kT0)iv
dE
Г
£-* 2 í?ií.
(6)
Z is a dimensionless measure of the electron velocity v.
The additional term in Eq. (5) arises from the dynamical friction force exerted by the ions (assumed to be
stationary) upon the electrons. As illustrated in
Fig. 1, this friction force follows a Stokes law for small
Z, whereas for Z > 1 the strong velocity dependence
of the Rutherford scattering law leads to a Z~2 decay.
The friction force is a maximum at Z = 1. Examination of Eq. (5) shows that it admits no static solution
when E exceeds £ C Y(1). The parameter EG therefore plays the role of a critical field in the theory
t Details of the strong and weak field approximation will
be submitted for publication elswhere.
-6
Figure 11. The velocity distribution shown at т = 5 with V as
a parameter
THEORY OF RUNAWAY ELECTRONS
61
probability Q(r) that all electrons have crossed into
the runaway region in the time r. To a good approximation Q(r) is given by
The variation of X with E/Ec is shown in Fig. 5.
Again r and EG are defined in terms of the initial
temperature of the electrons.
WTe conclude, therefore, that runaway in the weak
field limit proceeds under the combined action of
Joule heating and diffusion into the high-energy tail
of the distribution.
NUMERICAL SOLUTION
A more detailed analysis of this problem was carried out with the aid of an IBM 704 digital computer.
A straightforward difference equation version of the
Boltzmann equation was solved subject to the initial
condition given in Eq. (2). In terms of dimensionless
variables, this equation has the form
dF{V, Vz, r)/8r + (E/Ec)8F/dVz = (dF/dr)C0Ïh
(10)
where the velocity components are defined by
V =
Vz =
Figure 12.
The velocity distribution shown at т = 6 with V as
a parameter
and (7) have been solved simultaneously, by numerical
means, for a variety of applied fields. From the
results shown in Figs. 3 and 4 we may conclude
that runaway occurs even when E < EGW(l).
This
is simply a statement of the fact that Joule heating
transforms any weak field into a strong field as time
proceeds. The variation of T/To and Z as a function
of time is tabulated in Table 1 for E/Ec = J.
Weak Field Regime
In the weak field limit, E < Ec, Eq. (5) leads to the
well-known T*l* conductivity law. This result is not
strictly valid, quite apart from the Joule heating
effects just discussed. By making use of the displaced Maxwellian distribution we ignore the fact
that certain fast electrons in the high-energy " tail "
of the distribution make collisions so infrequently
that for these almost any applied field may be
considered to be strong. When this effect is examined,
we find that velocity space can roughly be divided into
a runaway region where the applied field plays the
role of a strong field and into a non-runaway region
where the same field is weak. The time scale for
appreciable depletion of the original distribution is
therefore determined by the diffusion of electrons
into the runaway region. We have employed the
time-dependent Boltzmann equation, supplemented
by Fokker-Planck collision terms, to calculate the
(m/2kT0)lc
(m/2kT0)bcz.
In cylindrical coordinates the Fokker-Planck collision
term describing electron encounters with electrons
and stationary ions has the form
v
dF
Vz
(F2 +
dVe
2
dG
3 V2
2
xid G
+ F,2)3/,
F2
1 IdG ^
F
dF
2
(F + F/
FF,
3VdVz
(F 2
3VdVz
(11)
The function G(V, Vz) describes Coulomb encounters
between electrons and is defined by
G(F, F2) - J *" J ^ J " F(F', F',,
X (7/2
+
T)
+ 72 _ 2 F F ' cos ф
( 7 г __ 7 2 ')2)i 7 ' ^ F ^ F , ' ^ .
(12)
Its second derivatives
(i.e., 32G/dV2, 32G/dVz2,
2
3 G/dVdVz) may be related to the rate at which
fluctuations about the dynamical friction force bring
about a diffusion of particles in velocity space.
Rosenbluth, MacDonald and Judd, 3 have shown
that G may be related to F through an auxiliary
H(V, Vz) function as follows
62
SESSION A-5
Figu re 13. Cu rves of constant F i n velocity
space shown at т = 0. These curves represent the Maxwellian distribution
P/2292
H. DREICER
Figure 14. Curves of constant F in
velocity space shown at
Figure 15. Curves of constant
F in velocity space shown
atx = 2
1
ICT = F
IO"5-F
V,
V,
г -
i
_L
Figure 16.
v
Curves of constant F in velocity
space shown at т = 3
i V
Figure 17. Curves of constant F in
velocity space shown at т = 4
Figure 18. Curves of constant F
in velocity space shown a t T = 5
63
THEORY OF RUNAWAY ELECTRONS
sary boundary values of H and G are obtained from
the asymptotic limits of Eqs. (12) and (14) in the
form
G(V,VZ)~[V* +
oo or
(Vz-
( F-> oo or
\Vz-
where
"oo
Го
— °° Jo
(15)
(16)
FVdVdVz.
The entire calculation is carried out over the rectangular region in velocity space bounded by the lines
7 = 0, V = 6,
Vz=±6
and the mesh points in this region are spaced apart
by the intervals
A single-cycle time-step of size Дт = 10~3 was used
and G was recalculated after every 10 cycles.
The following physical quantities were recalculated
periodically
iV = Г °° f°° F2nVdVdVz
J— °° Jo
(17)
(18)
(19)
Figure 19. Curves of constant F in velocity space shown at т = 6
<FZ> = m <czy/2kT0.
(21)
(13a)
, F z , T).
(13b)
The normalization N provides a check on the conservation of particles, and the energy integrals in
Eqs. (18), (19) and (21) can be related to check the
conservation of energy in the following way
(14)
(22)
The definition of H(F, F2) is
X
F(V, Vz\ r)
(F'> + F 2 - 2FF' cos ^ + (7, - F z ') 2 )i
and its gradient in velocity space gives the dynamical
friction force.
In our numerical scheme G is obtained from Eqs.
(13a) and (13b) by a relaxation method. The neces-
Random average energies are defined by
(23)
Table 2
c
N
T
0
1
2
3
4
5
6
0
0.43
0.77
1.07
1.39
1.72
2.06
1.00
1.096
1.196
1.296
1.296
1.296
1.296
(20)
*[<TV> + <T">Í
0.50 kT0
0.61
0.76
0.97
1.22
1.49
1.76
1.00 kT0
1.02
1.13
1.32
1.56
1.84
2.13
0
- 0.088
- 0.150
-0.100
- 0.071
- 0.052
- 0.034
<F 2 > d T
SESSION A-5
64
P/2292
(24)
In Table 2 we have recorded (17), (20), (23), (24)
and the degree to which (22) is satisfied over a duration covering 6 mean-free collision times. The applied
field in this calculation had the value E/Ec = J.
Figures 6 through 19 illustrate the evolution of
F(V, Vz, r) during the same interval of time.
If we choose as our runaway criterion the equality
of the drift velocity and the initial thermal speed; i.e.,
(Vzy = 1, then comparison of Tables 1, 2 and Fig.
5 yields reasonably good agreement among the
various runaway times, т/.
Numerical
treatment
Strong field
approximation
3.0
3.75
Weak field
approximation
2.0
The normalization tabulated in Table 2 indicates
that the number of particles in the rectangular mesh
at first increases with time and then reaches a steady
value. This is associated with the error in the
assumed boundary values of F. The algebraic sign
of the error in the energy balance indicates that
fictitious particles are probably entering the mesh
H. DREICER
across the negative Vz boundary. With the onset
of the runaway effect, particles also begin to leave
the mesh across the positive Vz boundary, and a
balance between these currents produces the steady
value for the normalization calculated after т = 3.
In spite of these inaccuracies, we believe that the
absence of a time-independent solution has also been
illustrated in this numerical calculation.
ACKNOWLEDGEMENTS
It is a pleasure for the author to express his gratitude to Mr. A. Feldstein, who carried through the
numerical solution of Eqs. (5) and (7). The author
is also greatly indebted to Mr. E. Kinney, who coded
Eqs. (10) and (11) for the computer.
REFERENCES
1. L. Spitzer and R. Harm, Phys. Rev., 89, 977 (1953).
2. R. G. Giovanelli, Phil. Mag., 40, 206 (1949).
3. M. N. Rosenbluth, Wm. M. MacDonald and D. L. Judd,
Phys. Rev., 107, 1 (1957).