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The transient phenomenon analysis for low-energy high-current
electron beam transportation in plasma
Evgeny Vagin1,a, Vladimir Grigoriev1,b
1
National Research Tomsk Polytechnic University, 634050, Russia, Tomsk, Lenin Avenue, 30
a
[email protected], [email protected]
Keywords: electron beam, virtual cathode, charge neutralization.
Abstract. The paper examines the role and influence of transient phenomenon on the transport
efficiency of high current (5-20 kA) low energy (tens of keV) electron beams in a pipe filled with
low-pressure plasma in an external magnetic field. The research is based on the self-consistent
mathematical model that takes into account beam and plasma particles dynamic, current and charge
neutralization of electron beam. Particle in cell (PIC) method was using in a numerical study.
Physical effects arising in the beam drift region were considered on the particular case.
Introduction
High efficiency of beam transportation with high current densities and low electron energy
(LHEB) is possible only in case of almost full charge neutralization [1]. LHEB should be
transported in the pipe with low-pressure plasma or neutral gas of (10–1 … 10–2 Pa) to provide this
condition. Besides, self-magnetic field can cause a beam pinching, leading to a low efficiency. But
sufficient current neutralization can reduce this effect and external magnetic field can be used to
reduce this effect.
Thus, LHEB transportation with high efficiency represents a sophisticated challenge.
This paper described mathematical model, equations and results of numerical research of
transient phenomenon in the case on LHEB transportation in the pipe filled with plasma in the
external magnetic field.
The basic equations of a physical model
There is an interaction of a beam to plasma at transportation of an intensive electron beams.
Injection of beam forms a potential in the area that forces plasma electrons to leave injection area.
Thus ions of plasma remain in area and provide charging neutralization of a transported beam
because of high relative mass (usually it is the one-charging ions of an argon), therefore on the basic
part of an impulse operates only focalizing forces from self and external magnetic fields. The
plasma channel on which the beam is transported is as a result shaped. At the expense of space
charge neutralization sagging of potential and the beam diffusion decreases. It allows to transport
the self-focalized beams with currents above, than in vacuum channels.
The mathematical model of self-consistent dynamics of the beam in the field of a space charge
and magnetic fields at its transportation in drift space (fig. 1), filled with plasma with the
homogeneous density n0, and is based on the description of electrons of the beam and plasma by
macroparticles. The model is constructed for the area coinciding with range of the tube, and has
dimensionality 2,5 (three-dimensional on dynamics, two-dimensional on fields).
Following assumptions are made at model build-up:
• Axial symmetry of processes;
• Prevalence of the longitudinal current of the beam: Jz >> Jr, Jθ;
• The plasma ions are considered as immobile and its density homogeneous and constant
ni=n0, because of its huge mass relatively electrons.
Fig. 1. The tube of drift of particles of the beam
Dynamics of electrons of the beam and plasma is featured by system of the relativistic equations
in a cylindrical coordinate system:
 d (γ z ) e

( r Bθ  Ez )

m0
 dt
 d (γ r ) e


(r θ Bz *  z Bθ  Er )  γ r θ 2 (1)

m0
 dt
 1 d (γ r 2θ)
e



(r Bz * )
dt
m0
 r
where m0 – an electron rest mass; e – an electron charge; Ez, Er, Вθ – components of self
electromagnetic field of the beam; Bz*=const – component of an external magnetic field; γα – the
relativistic factor of particles α; α – electron of the beam and plasma.
The beam internal field is featured by Poisson equations for the scalar potential Φ and
longitudinal component of the vector potential Az:
1   Ф   2Ф
1
r
  2   ρ , (2)
r r  r  z
ε0
1   Az   2 Az
 μ 0 J z , (3)
r

r r  r  z 2
where ε0, µ0 – the dielectric and magnetic stationary values; ρ, Jz – densities of the charge and
current in the drift space depending on level of fields.
Density of charge and a beam current in the equations (2), (3) are related by the continuity
equation:
ρ
div J  b  0. (4)
t
The net charge density in the equation (2) is featured by a relation:
ρ  ρb   ρi  ρe  , (5)
where ρb, ρe – charge densities of beam and plasma electrons; ρi=n0 qi=const - charge density of
plasma ions. The initial conditions for the charge density electrons of a bundle are set as, that
corresponds to lack of a beam in a drift tube.
Boundary conditions for potentials are set proceeding from requirements of ideal conductivity of
a surface of tube walls (r=R) and requirements of a continuity of potentials on the tube axis (r=0)
and at tube end faces (z=0 and z=L):
Az
Az
Ф
Ф r  R  Az r R  0 ,
Ф z 0  Ф z  L  0 ,

0,
r r 0 r r 0
z
Fields of the beam are calculated under formulas of potentials differentiation:
A
Ф Az
Ф
Ez  

, Er  
, B   z . (7)
z
t
r
r

z 0
Az
z
 0 (6)
zL
The transient phenomenon analysis
Transient analysis will performed on the numerical simulation case of the beam transportation
with the following parameters: energy of beam electrons W0=20 keV, beam current I0=15 kA,
plasma temperature 2 эВ, gas pressure p=10–1 Pa, gas ionization degree 10 %, plasma dencity
n0=2.5 1011 cm–3, magnetic field induction Bz*= 1.5 kGs. Pipe geometric parameters: L=20 cm,
R=10 cm; beam radius: Rb=4,3 cm. The injected beam current has rise time (τФ) with the linear
increase, and constant current (I0) after.
Few conditionally phases can be allocated during transient period of LHEB transportation. They
have not clear borders but each of them have physical processes that can characterize each phase.
Dedicated phases can be illustrated by figures 2-4. We will provide the characteristic of each stage:
1) Initial phase (0 - 5 ns)
Initial filling of the drift tube by beam electrons is fast enough, as the transit time of the beam
is about 2 ns. Beam current doesn’t reaches great values (~ 100 A) by this time. Further
current increase during 2-3 ns does not lead to significant field gradients.
2) The second phase (5 ns – 50 ns)
Beam current reaches a value of 2 kA. Increasing of beam current leads to significant
fluctuations in the electrons density of plasma and beam with Langmuir plasma frequency (~
5 GHz). Phase is characterized by the formation of periodic local virtual cathode, which may
slow down electron beam overflight, as well as the formation of large electric field gradient.
Under the action of fields plasma electrons begin to leave the beam drift region. The degree
and uniformity of charge neutralization is increases.
3) The third phase (50 ns – 150 ns)
4) The period is characterized by a massive output of plasma electrons and the achievement of
the charge neutralization of the beam, which corresponds to a total output of plasma electrons
from the beam drift area. There is a falling gradient fields.
5) Переход в стационарный режим (150-350 нс)
During the current phase gradually reaches its maximum value (15 kA). By this stage is
reached the charge neutralization of the beam, but with increasing current output process of
the plasma electrons continues, but without causing significant fluctuations in density and
fields. Increasing self-magnetic (up to 0.7 kGs) leads to increased influence gyromagnetic
frequency (~4.5 GHz).
Further transport of the beam is conserved paintings reached to 300 ns, which allows us to
conclude that the transition process is completed and the process of transportation is carried out
with minimal losses (fig 4).
Fig 2. Current density Jz(t)
Fig 3. Scalar potential Φ(t)
Fig 4. Electric field Ez(t)
Fig 5. Gained current of the beam
Summary
Transient phenomenon has significant role on the formation of steady state of LHEB
transportation in the low-pressure plasma. Time of formation steady state of beam transportation,
instabilities evolution and current of the transported beam depends on this phenomenon.
Local virtual cathode generated as a result of the fields perturbations generated. This leads to
beam inhibition and the partial reflection of electrons. Decisive for the transient process is the
charge neutralization of the beam.
High efficiency of LHEB transportation can be provided by selection of parameters of plasma
and beam that have influence on charge neutralization: beam current, plasma density and current
rise time.
References
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[4]
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