The nonlinear theory of the electron and ion beames interaction in a supercritical mode
The problem of REB and ion beam interaction at currents above space charge limiting values has been mathematically formulated. The method of Lagrangian variables has been used for the description of selfconsistent nonlinear dynamics of electrons and ions in collective fields of the beams. The simula...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2002
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| Zitieren: | The nonlinear theory of the electron and ion beames interaction in a supercritical mode / V.A. Balakirev, N.I. Onishchenko // Вопросы атомной науки и техники. — 2002. — № 4. — С. 149-151. — Бібліогр.: 4 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859606933052325888 |
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| author | Balakirev, V.A. Onishchenko, N.I. |
| author_facet | Balakirev, V.A. Onishchenko, N.I. |
| citation_txt | The nonlinear theory of the electron and ion beames interaction in a supercritical mode / V.A. Balakirev, N.I. Onishchenko // Вопросы атомной науки и техники. — 2002. — № 4. — С. 149-151. — Бібліогр.: 4 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | The problem of REB and ion beam interaction at currents above space charge limiting values has been mathematically formulated. The method of Lagrangian variables has been used for the description of selfconsistent nonlinear dynamics of electrons and ions in collective fields of the beams. The simulation has been performed for the parameters of the experimental installation.
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| first_indexed | 2025-11-28T04:54:13Z |
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THE NONLINEAR THEORY OF THE ELECTRON AND ION BEAMES
INTERACTION IN A SUPERCRITICAL MODE
V.A.Balakirev, N.I.Onishchenko
National Science Center " Kharkov Institute of Physics and Technology"
Academicheskaya Str. 1, Kharkov, 61108, Ukraine
E-mail: onish@kipt.kharkov.ua
The problem of REB and ion beam interaction at currents above space charge limiting values has been mathematically
formulated. The method of Lagrangian variables has been used for the description of selfconsistent nonlinear dynamics
of electrons and ions in collective fields of the beams. The simulation has been performed for the parameters of the
experimental installation.
PACS: 52.59.-f; 52.65.-y
1. INTRODUCTION
For implementation of collective methods of
acceleration of ions by quasistatic electrical fields of high-
current relativistic electron beam (REB), in particular in a
mode of formation of a virtual cathode (VC) [1-4], the
research low frequency (LF) processes of interaction of
ions flows with REB has a great important. The similar
situation arises at injection of plasma in area of VC for
destruction of VC potential well and time modulation of
REB [1]. The self-consistent influence of ions on VC can
result in to synchronic motion of a potential well of VC
and essential increase of final energy of accelerated ions
[2,3]. The transversal oscillations of ions in two-
dimensional quasistatic electrical field of VC under
certain conditions cause the low frequency (LF) relaxation
oscillations of VC field and by that provide deep LF
modulation of REB density. This phenomenon can be laid
in the fundamentals of the operation of the first stage of a
two-stage collective ion accelerator working on a
principle of simultaneous spatial and temporal LF
modulation of REB density [1,4].
In the present paper some results of non-linear
dynamics investigation of the interaction of ions flow with
VC field produced by REB is represented. The system of
the non-linear integral-differential equations describing
indicated process is formulated and is obtained and its
solution and analysis is carried out by numerical methods.
2. PROBLEM FORMULATION
Problem will be decided in the following statement. In
the semi-infinite metallic drift chamber, closed at z=0 by
the conductive diaphragm, it is injected along axis in
direction z > 0 thin-wall tubular electron and ion beams.
These beams have different radii, energy of particles and
currents. The drift chamber is under zero electric
potential. Electron and ion beams are propagating in a
homogeneous leading magnetic field. The beam is
considered as magnetized ones (motion strictly along of
magnetic field lines), and influence of the magnetic field
on motion of ions is neglected.
For the theoretical description of non-linear dynamics
of electron and ion beams interaction in the drift chamber
the following approach was used. At first we find the
electric potential created in the drift chamber by infinitely
thin driving charged ring with particle density
)(
2
)(
0 L
L
L zz
r
rrdNdn −−= δ
π
δ
(1)
where 0dN – number of particles in a ring, ),( 0ttrL – ring
radius, that varies during motion, t – current time, 0t –
time moment of an entry of a ring in the drift chamber,
00 ),( rttrL = – initial value of radius of a ring, ),( 0ttzL –
Lagrangian longitudinal coordinate of a ring, 0),( 00 =ttzL
, zr, – radial and longitudinal coordinate in the drift
chamber. The electric potential Gφ of a ring (1) is
governed by a Poisson equation
)(
)(
21
02
2
L
L
LGG zz
r
rr
edN
zr
r
rr
−
−
=
∂
∂
+
∂
∂
∂
∂ δ
δφφ
(2)
For determinacy we consider a potential (Green
function) for negatively charged ring (electron ring). The
electric potential becomes zero on a lateral area of the
drift chamber and also at the conductive butt-end of the
drift chamber.
For Green function we receive
∑
φ
∞
=
−
λ
−
−
+
λ
−
λλ
λλ
⋅
⋅=
1
)(
2
1
00
0
)(
)()(
2),(
n
zz
a
nzz
a
n
nn
n
L
n
G
LL ee
J
a
rJ
a
rJ
a
edNzr
(3)
The number of particles in each ring 0dN is
connected with a current of the electron beam )( 0tIe by a
simple relationship
edttIdN e /)( 000 = (4)
The electric potential of the electron beam is being
found by integrating of a right side of eq. (3) over time of
an entry of "macroparticles" formed by elementary rings
with number of particles (4). The contribution to a
potential of the ion beam is being similarly determined. In
result for the electric potential of a system consisting of
infinitely thin electron and ion beams moving in the
conductive drift chamber, we shall receive the following
expression
Problems of Atomic Science and Technology. 2002. № 4. Series: Plasma Physics (7). P. 149-151 149
∫
−−
−∫
−
⋅
⋅=
−−+−
−−+−
∞
=
∑
t zz
a
zz
aLi
ni
t zz
a
zz
a
en
nn
n
n
Li
n
Li
n
Le
n
Le
n
eedt
a
rJtI
eedttI
a
rJ
J
a
rJ
a
zr
0
)(
000
0
)(
00
0
0
2
1
0
1
)()(
)()(
)(
)(2),(
λλ
λλ
λ
λ
λλ
λφ
(5)
Let's comment briefly the expression (5). The
integrating over time of an entry in right side is carried out
from time of entry in the drift chamber of the first
particles 00 =t up to current time tt =0 . The radial
coordinates of electrons (in (5) indexes "e") remain
unchanged, i.e. -independent of an entry time. In an
integral over an entry time we has taken into consideration
this circumstance. As to ions (in (5) indexes "i"), under an
integral over an entry time of ions both their radial
),( 0ttrLi , and longitudinal ),( 0ttzLi coordinates are
contained.
Knowing electric potential, it is not difficult to
determine longitudinal and radial component of electrical
field. Inserting expressions for the components of
quasistatic electrical field in motion equations for
electrons and ions we obtain self-consistent nonlinear
equations set, which describes complex dynamics of
interaction of overlimiting REB with ion flow.
dz
d
e
dt
dPLe φ
= , Le
Le v
dt
dz
= , 222 cmP
cPv
Le
Le
Le
+
= (6)
dz
d
M
e
dt
zd Li φ
−=
2
2
,
dr
d
M
e
dt
rd Li φ
−=2
2
(7)
The dependencies of electron )( 0tIe and ion )( 0tI i
currents upon time at the entry into the drift chamber are
taken in the form )()( 0,,0, TItI ieieie ψ= , where the
functions )( 0, Tieψ describe the shape of pulses of the
corresponding currents. If in the system the direct currents
are continuously injected, then. 1)( 0, =Tieψ .
3. NUMERICAL RESULTS
The set of motion equations (5-7) was solved by
Runge-Kutta method of second order. Integrals in right
sides were approximated by Simpson method. Depending
on parameters the step in time was varied.
In Fig. 1 (curve 1) phase portrait of electron beam in
absence of ions is pictured. Parameters of electron beam
are the followings: energy keVEe 280= , current
kAIe 6.5= , radius cmR e 6.10 = . For indicated values of
energies, radii of the beam and drift chamber the limiting
current is equal kAI 8.3lim = that is less than the current
of injected REB.
From Fig. 1 it is seen, that in the drift chamber the VC is
formed. It is possible to see passed through and reflected
particles. The value of voltage drop (see Fig. 1, curve 2)
coincides with initial energy of the beam.
-1 0 1 2 3 4 5 6 7 8 9
-1,0
-0,8
-0,6
-0,4
-0,2
0,0
0,2
0,4
0,6
0,8
400
320
240
160
80
0
80
160
240
320
4
3
2
1
po
ten
tia
l, k
V
v/c
z, cm
Fig.1 Phase plane of electron beam without ion beam (1)
and with ion beam (3). Electrical potentials for these
cases, accordingly (2 and 4). Ie=Ii=5.6kA, Ee=280keV,
Ei=25keV, m/M=40, Re=R0i=1.6cm, Ni/Ne=1.5
Let's consider now the case, interesting for us, of
simultaneous injection REB and ion flux. We begin from
the analysis of the simplest situation, when the initial radii
of beams coincide cmRR ie 6.100 == . The initial energy
of ions was selected equal keVEi 25= , and linear density
(number of particles per unit length of beam) 5.1/ =ei NN
, where ieieie evIN ,,, /= , where iev , - initial velocities of
particles. The model mass ratio was selected equal
40/ =Mm . These conditions correspond to ion current
AIi 540= . In Fig. 1 (curve 3) the phase portrait of
electrons for this case is pictured. In the same Fig. 1 the
distribution of the electric potential in the region of VC is
shown (curve 4). It is seen, that the influence of ions has
resulted in displacement of VC inside drift chamber. The
distance from the injection plane up to VC was increased
more than twice comparatively to the case of ion flow
absence. By that the depth of a voltage drop has not
changed. On the curve of the potential additional
minimum has appeared. Accordingly, this minimum has
found its reflection on a phase portrait of electrons in a
view of slowing and gathering of electrons in this area.
In Fig. 2 the phase portrait of ions is shown. It is well
seen that starting from injection plane the ions are
experienced accelerating by VC field. Then, having
reached the bottom of the potential well and having
received gain of energy, approximately equal to REB
energy, the ions start slowing. Their density increases, that
results in distorting of the potential and formation with the
help of electrons of the next minimum of the potential.
Energy of ions again slightly grows up. The level of these
oscillations of energy is rather low.
150
0 1 2 3 4 5 6
0,04
0,06
0,08
0,10
0,12
0,14
0,16
0,18
v/
c
z, cm
Fig.2 Phase plane of ion beam with electron beam.
Ie=Ii=5.6kA, Ee=280keV, Ei=25keV m/M=40,
Re=R0i=1.6cm, Ni/Ne=1.5
Let's consider now the dynamics of VC when ion
current is increasing. In Fig. 3 the phase portrait of
electrons and potential distribution along the system is
represented for 5.4/ =ei NN .
-1 0 1 2 3 4 5 6 7 8 9
-1,0
-0,8
-0,6
-0,4
-0,2
0,0
0,2
0,4
0,6
0,8
-400
-320
-240
-160
-80
0
80
160
240
320
po
ten
tia
l,
kV
2
1
v/
c
z, cm
Fig.3 Phase plane of electron beam with ion beam (1)
and electrical potential (2). Ie=Ii=5.6kA, Ee=280keV,
Ei=25keV m/M=40, Re=R0i=1.6cm, Ni/Ne=4.5
It is seen, that the displacement of VC into chambers
increases. The distribution of potential, as well as in the
previous case, is not monotonic.
In Fig. 4 (curve 1) the phase portrait of ions is shown.
It is well seen, that near the plane of injection the ion
virtual anode (VA) is formed. The part of ions was
reflected, and the others are involved in process of
acceleration. On the curve of the electric potential (curve
2) in proximity of the injection plane it is visible the
positive maximum formed by ion VA.
-1 0 1 2 3 4 5 6 7 8
-0,05
0,00
0,05
0,10
0,15
0,20
v/
c
z, cm
Fig.4 Phase plane of ion beam with electron beam.
Ie=Ii=5.6kA, Ee=280keV, Ei=25keV m/M=40,
Re=R0i=1.6cm, Ni/Ne=4.5
The ion VA can be considered as an emitter of ions with
space charge restriction. Therefore passed ion current will
be determined by a voltage drop in the region of electron
VC and spacing interval from an ion VA (plane of
injection) up to an electron VC (Child-Langmuir law).
4. CONCLUSIONS
In the paper the system of non-linear integral-
differential equations describing the non-linear interaction
of an ion flux with a VC, produced by REB, is
formulated, in two-dimensional approximation for ions
and one-dimensional for electrons (the electrons are
supposed to be magnetized). Using a numerical code
differing from both the method of “macroparticles” and
the full numerical simulation, but combining some of their
advantages, the results of dynamics of electron and ion
beams are obtained.
This work is partly supported by STCU project №1569.
REFERENCES
1. V.A.Balakirev, A.M.Gorban’, I.I.Magda et al.
Collective Ion Acceleration by a Modulated High-
Current REB // Plasma Physics Reports, 1997, Vol.
23, No.4, pp.350-354.
2. G.L.Olson. Theory ion acceleration by drifting
intense relativistic electron beams. // Phys. Fluids
1975, Vol.18, No.5, pp.585-597
3. W.Peter, R.Faehl, C.Snell, N.Yonas. Collective ion
acceleration by means of virtual cathode. // IEEE
Trans. of Nucl. Sci. 1985, Vol.NS-32, No.5, pp.3506.
4. V.V.Belikov, A.G.Lymar, and N.A.Khizhnyak. Ions
acceleration by modulated electron beam. // Pis’ma
Zh. Tekh. Fiz. 1975, Vol.1, No.13, p.615.
151
References
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| id | nasplib_isofts_kiev_ua-123456789-80308 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-28T04:54:13Z |
| publishDate | 2002 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Balakirev, V.A. Onishchenko, N.I. 2015-04-14T17:36:54Z 2015-04-14T17:36:54Z 2002 The nonlinear theory of the electron and ion beames interaction in a supercritical mode / V.A. Balakirev, N.I. Onishchenko // Вопросы атомной науки и техники. — 2002. — № 4. — С. 149-151. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 52.59.-f; 52.65.-y https://nasplib.isofts.kiev.ua/handle/123456789/80308 The problem of REB and ion beam interaction at currents above space charge limiting values has been mathematically formulated. The method of Lagrangian variables has been used for the description of selfconsistent nonlinear dynamics of electrons and ions in collective fields of the beams. The simulation has been performed for the parameters of the experimental installation. This work is partly supported by STCU project №1569. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Plasma electronics The nonlinear theory of the electron and ion beames interaction in a supercritical mode Article published earlier |
| spellingShingle | The nonlinear theory of the electron and ion beames interaction in a supercritical mode Balakirev, V.A. Onishchenko, N.I. Plasma electronics |
| title | The nonlinear theory of the electron and ion beames interaction in a supercritical mode |
| title_full | The nonlinear theory of the electron and ion beames interaction in a supercritical mode |
| title_fullStr | The nonlinear theory of the electron and ion beames interaction in a supercritical mode |
| title_full_unstemmed | The nonlinear theory of the electron and ion beames interaction in a supercritical mode |
| title_short | The nonlinear theory of the electron and ion beames interaction in a supercritical mode |
| title_sort | nonlinear theory of the electron and ion beames interaction in a supercritical mode |
| topic | Plasma electronics |
| topic_facet | Plasma electronics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80308 |
| work_keys_str_mv | AT balakirevva thenonlineartheoryoftheelectronandionbeamesinteractioninasupercriticalmode AT onishchenkoni thenonlineartheoryoftheelectronandionbeamesinteractioninasupercriticalmode AT balakirevva nonlineartheoryoftheelectronandionbeamesinteractioninasupercriticalmode AT onishchenkoni nonlineartheoryoftheelectronandionbeamesinteractioninasupercriticalmode |