Structure and theory of virtual cathode in magnetic self-insulated transmission line
The possibility of formation of a virtual cathode in a self-insulated magnetic line is shown.
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Cite this: | Structure and theory of virtual cathode in magnetic self-insulated transmission line / A.V. Pashchenko // Вопросы атомной науки и техники. — 2000. — № 2. — С. 83-85. — Бібліогр.: 8 назв. — англ. |
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| author | Pashchenko, A.V. |
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| citation_txt | Structure and theory of virtual cathode in magnetic self-insulated transmission line / A.V. Pashchenko // Вопросы атомной науки и техники. — 2000. — № 2. — С. 83-85. — Бібліогр.: 8 назв. — англ. |
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| description | The possibility of formation of a virtual cathode in a self-insulated magnetic line is shown.
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STRUCTURE AND THEORY OF VIRTUAL CATHODE IN MAGNETIC
SELF-INSULATED TRANSMISSION LINE
A.V. Pashchenko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: paschenko@kipt.kharkov.ua, tel. (057) 335-25-64, fax: (057) 335-32-01
The possibility of formation of a virtual cathode in a self-insulated magnetic line is shown.
PACS: 52.25.Sw, 52.35.Py, 52.60+h
1. INTRODUCTION
The effect of formation of a virtual cathode (VC) is
known from the time of first studies of processes
happening in electron tubes [1].
It was revealed that at a large enough current that is
flowing between electrodes the potential barrier for
electrons is formed not far from cathode. This barrier
limits current of an electron beam. The potential barrier
together with an electron beam that is dissipated on
them also represents a virtual cathode.
Long time VC remained the extremely parasitic phe-
nomena limiting a current of electron devices. The first
sensible positive application of VC has found in the
generator of short electric (electromagnetic) impulses
[2].
One more application of VC is connected to
collective ion acceleration. It was established that
experiments [3-5], in which the accelerated ions were
detected in the presence of an electron beam, could be
explained by formation of VC in electron beam and its
movement.
VC, which is appeared in REB, was successfully
applied for generation of high power RF-radiation. It is
associated with the fact that the beam in VC makes
electric oscillations near electron plasma frequency that
converts VC into a generating element. This direction of
VC usage is most successfully realized as virtod [6] -
instrument, in which the self-organizing of generation
process is realized for reaching high arguments of
generated radiation.
All listed above applications of VC deal with VC as
quasi-one-dimensional spatial formation. The spatially
extended VC calls the more great interest, for example
as a generating element. The successful example of
creation of such VC is described in [7].
In the present paper the idea is stated and the
structure is described for the spatially extended VC that
is generated in electron beam, which is propagated in a
line with magnetic self-insulation. The feasibility of
vircator existence in a magnetic-insulated transmission
line (MITL) is demonstrated in this report. We describe
some structural peculiarities of vircator formation in
such lines and discuss the nature of electron motion in
such vircator. Usually, in the line only the potential
monotones radial profiles become realizable with zero
electric field on the cathode. The pre-supposed vircator
existence leaves a possibility of finding solutions,
involving the potential trough.
2. STRUCTURE AND PECULIARITIES OF
VIRCATOR FORMATION IN MITL
The necessity of ultra-powerful pulsed rf-generation,
using comparatively modest energies in drivers, calls for
development of such generation schemes that would
have high efficiency and reliability. There are several
similar schemes in existences that have both rf-seed
power generation and REB initial modulation. However,
more reliable would be usage of such master-oscillator
as the vircator modulator that naturally appears in REB
under certain conditions. Yet, feasibility of vircator
existence in a MTTL has not been demonstrated to date.
Below, we will try to fill the blank and describe some
structural peculiarities of vircator formation in such
lines.
First of all, let's discuss the nature of electron motion
in vircator. In the 1-D case vircator is formed by the
interpenetrating counter-streaming "incident" and
reflected electron flows. However, the single-stream
flows only are permissible in the MITL as constrained
by the formula υ=c(E/H) and the unambiquity of the
functions E and H. This purports to mean in this case
that electron bouncing from the potential barriers, if
realizable at all, must follow another path of trajectory.
On the other hand, the vircator problem resolution
pre-supposes such possible evolution of the potential
relief within the MITL space that would make the
reflections quite conceivable. In according with the
classic magnetic insulation line theory [1-2] any suitable
potential relief in the case is unrealizable. Nevertheless,
analysis of the above theory indicates that this
impossibility steam from the condition imposed on the
cathode-emitted flow to be restricted by space charge.
This condition has the form: E|x=a = 0. Not being pre-
requisite for electrodynamics problem used to describe
the line, this condition, as an addition, constrains
severely any solution for the MITL. In particular, at any
Z-coordinate position in the line, only the potential
monotones radial profiles become realizable with zero
electric field on the cathode. Meanwhile, the pro-
supposed vircator existence leaves a possibility of
finding solutions, involving the potential well.
The condition E|x=a = 0 can be eliminated either in
the case of restricted cathode emission, or in the case of
transporting along the line of an electron beam with such
a large space charge that a sag of the potential appears
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, № 2.
Серия: Ядерно-физические исследования (36), с. 83-85.
83
in the cathode-anode space, causing the electric field
E>0 to appear near the cathode that would suppress the
emission. In the case, the line transports electron beam
from an outside source.
As it turns out, it is exactly in this case that vircator
becomes possible in the MILO line as vortex formation.
The structure of virtual cathode.
A visual impression of the structure and probable
scenario of such vircator can be made from Figure.
The electron beam, injected from an external source,
comes into a transmission line, in which the potential U
is applied between cathode and anode, propagating
along the coordinate 0Z until it reaches the region (z≈L)
with such impedance variations which preclude
propagation of the entire beam at z>L. A bit to the right,
only a portion of the injected beam can pass, while the
remainder must bounce back in the opposite direction.
Parameters for the beam and the line must be chosen
such that (he whole system should find itself in a near-
threshold state, i.e. when beam slowdown or stopping in
a certain region should cause the potential well with E|x≠
a = 0 to appear. Such macro perturbation, when
appearing mid-line, cannot but propagate to its
beginning, forming a trough-shaped potential well. The
electric field in its near-anode part ( rrr >>2 , where
r , 2r are the well minimum coordinates and beam
boundaries, respectively) retains its initial direction, i.e.
provides beam propagation to the right, while in its
near-cathode part ( 1rrr >> , 1r is the near-cathode
beam boundary) it assumes the opposite direction and
provides for the return of the reflected beam portion. In
the region z≥L there is only the passing flow, with the
appropriate electric field remaining in its initial
direction (E<0).
The structure of the forming zero-field formation is
such that the injected beam splits into two telescopic
parts (one inside the other). The outer, near-anode part
forms the passing beam, while the inner part does the
reflected beam returning to the near-cathode region.
Thus, the MITL-formed vircator is substantially a
multidimensional creation.
In the axis-symmetric case (Fig.), vircator is
described via the hydrodynamic electron motion
equations:
θυυυ H
c
eeE
z
P
r
P
t
P
zr
r
z
r
r
r −−=
∂
∂
+
∂
∂
+
∂
∂
, (1)
θυυυ H
c
eeE
z
P
r
P
t
P
rz
z
z
z
r
z −−=
∂
∂
+
∂
∂
+
∂
∂
, (2)
where rr mP υγ= , zz mP υγ= , ( ) 21221 −−= cυγ
, and the Maxwell equations
r
E
z
E
t
H
c
zr
∂
∂
−
∂
∂
=
∂
∂
− θ1
, (3)
z
H
t
E
cc
en rr
∂
∂
+
∂
∂
= θυπ 14 , (4)
( )
r
rH
rt
E
cc
en zz
∂
∂
−
∂
∂
= θυ
π 114 , (5)
( ) en
z
E
r
rE
r
zr π41 −=
∂
∂
+
∂
∂
. (6)
The potential and flow structure is one-dimensional
at z >> L and z << L being homogeneous along θ and z,
while it becomes intrinsically 2-D at z ≈ L. Next, we
derive solution far from the 2-D region.
In the following Chapter we will find solutions for
these equations that are now free from the condition E|
x=a = 0.
3. SPATIAL DISTRIBUTION OF THE
BRILLOUIN FLOW MAJOR
CHARACTERISTICS WITH THE MITL-
FORMED VIRCATOR
Difference of the Brillouin flow theory with vircator
in the transmission line from the classical transmission
line theory is formally manifested in the rejection of the
additional condition E|x=a = 0.
This signifies that the principal functional
dependencies, derived in [8] remain valid, although the
dependencies between the flow-determining parameters
are changed.
Thus, the set of equations for determining of ψ, s0
and s within this context corresponds to the set of
equations:
( ) ( )
( ) ( ) ( ) ( )DshssDshs
DchDch
ψψψ
ψγ
11
1
0
0
−++−+
+−++=
, (7)
( )Dchssic ψ0= , (8)
( ) ( )[ ]DchDchssib −+= ψψ0 , (9)
where
2
0 1 mceU+=γ ,
( ) ( )10 lnln rbabs = ,
( ) ( )121 lnln rrrbs = .
D is a new parameter which, similarly to ψ, is
determined by the set (7)-(9). D = 0, in the case with
additional condition E|x=a = 0.
Thus, equations (7)-(9) are determining the
depending parameters ψ, D, s0 (i.e. r1), if γ0, ib, ic are
initial independent parameters. The situation with D = 0
have described in [1]. Parameter s0 is determined by
principal of total energy minimum.
84
Let's write out for the beam, detached from
transmission line walls, the radial distributions of
potentials, fields, velocities and magnetic field.
In the region a ≤ r ≤ r1:
the electric field ( ) ρψε /01 Dshss−= ,
the potential ( )
+Φ=Φ
a
b
a
r
Dshssa
ln
ln
01 ψ ,
the magnetic field ( ) ρψθ /01 Dchssh = ,
where 2mc
eaE≡ε , 2mc
eϕ≡Φ ,
≡
a
b
mc
eaHh ln2
θ
θ ,
a
r≡ρ .
In the region r1 ≤ r ≤ r2:
( )
( )
+−= D
rr
rrshss
12
1
02 ln
ln1 ψ
ρ
ψε ,
( ) ( )
( )
( ) ( )
( )
+++−
+−Φ=Φ
D
rr
rr
chDch
rr
rb
Dshb
12ln
1ln
12ln
2ln
2
ψψ
ψψ
,
( )
( )
+= D
rr
rrchssh
12
1
02 ln
ln1 ψ
ρ
ψθ .
The potential minimum radius in vircator
( ) 01 ss
D
abrr ψ
−= .
Charge density
( )
( )
+
= D
rr
rrch
ss
n
12
1
2
0
ln
ln
* ψ
ρ
ψ
,
where
2
2
22
ln4*
≡
a
b
mc
naen π
,
and electron velocity
( )
( )
+= D
rr
rrthc
12
1
ln
lnψυ .
In the region r2 ≤ r ≤ b:
( )Dshss +−= ψ
ρ
ψε 1
03 ,
( ) ( )rbDshssb ln03 +−Φ=Φ ψψ ,
( )Dchssh += ψ
ρ
ψθ
1
03 .
The point rd in the vircator region separates the
passing beam part from the reflected and directed
backwards along the cathode. The radius rd is derived
from the condition of flow equality in the vircator region
for the case r1 < r < r and r < r < rd.
The flow separation radius proves to be
( )ξ+= 11rrd ,
where
( )
( )[ ]212
12
−+−
+=
ααα
αξ
k
k
, ( )Dchmc
eI c
3
2
=α ,
( ) ( )
( ) ( )DchDsh
DshDchk
α
α
+
+= , Ic – the cathode current.
4. CONCLUSIONS
This report demonstrates the idea, structure and sta-
tionary theory of MITL vircator, which is a substantially
multi-dimensional through-shaped vortex creation.
The structure of vircator is such that the injected
beam splits into two telescopic parts. The outer, near-
anode part forms the passing beam, while the inner part
does the reflected beam returning to the near-cathode
region. In this case, vircator becomes possible in the
MITL as vortex formation.
ACKNOWLEDGEMENT
I wish to express my gratitude to Novikov V.E. for
fruitful discussions on the results.
REFERENCES
1. D.S. Gvozdover. The theory of electron HPM
devices. M.: SPHТТL, 1956.
2. A.V. Pashchenko, B.N. Rutkevich, V.D. Fedor-
chenko, Yu.P. Mazalov. State hysteresis and charge
reset in electron flow // JTP, 1983, v. 53, No 1, p. 75-
80.
3. А.А. Plutto, P.E. Belensov, Е.D. Korop at al. Ion
acceleration in electron beams // Letters to JETP.
1967, v. 6, p. 540-542.
4. J.S. Luse, H.L. Sahlin, T.R. Crites. Collective
acceleration of intense ion beams in vacuum // IEEE
Trans. on N.S. NS-20. 1973, p. 336-342.
5. I.А. Stepanenko, Yu.V. Tkach, А.V. Pashchenko at
al. Collective ion acceleration in diode with anode
plasma // Kharkov: KIPT AS UkSSR, 1988.
6. I.I. Magda and Yu.V. Prokopenko Cooperative high-
power radiation of two beams – two vircator
assembly // Proc. Beams'96, Prague, Czech Republic,
June 1996, v. 1, p. 422-425.
7. М.G. Nikulin, S.D. Stolbetsov, V.G. Тarakanov,
А.V. Fedotov, А.G. Shkvarunets. Transision dyna-
mics between stationary states in diode // Radiotech-
nica & Electronica (in Russian), 1992, No 9,
p. 1665-1670.
8. E.L. Baranchikov, A.V. Gordeev, V.D. Koroljov,
V.P. Smirnov. Magnetic self-insulation of electron
beams in a vacuum transmission lines // JETP. 1978,
p. 2102-2121.
85
ACKNOWLEDGEMENT
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-82283 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:31:23Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Pashchenko, A.V. 2015-05-27T13:24:52Z 2015-05-27T13:24:52Z 2000 Structure and theory of virtual cathode in magnetic self-insulated transmission line / A.V. Pashchenko // Вопросы атомной науки и техники. — 2000. — № 2. — С. 83-85. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 52.25.Sw, 52.35.Py, 52.60+h https://nasplib.isofts.kiev.ua/handle/123456789/82283 The possibility of formation of a virtual cathode in a self-insulated magnetic line is shown. I wish to express my gratitude to Novikov V.E. for
 fruitful discussions on the results. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Тheory and technics of particle acceleration Structure and theory of virtual cathode in magnetic self-insulated transmission line Структура и теория образующегося в линии с магнитной самоизоляцией виртуального катода Article published earlier |
| spellingShingle | Structure and theory of virtual cathode in magnetic self-insulated transmission line Pashchenko, A.V. Тheory and technics of particle acceleration |
| title | Structure and theory of virtual cathode in magnetic self-insulated transmission line |
| title_alt | Структура и теория образующегося в линии с магнитной самоизоляцией виртуального катода |
| title_full | Structure and theory of virtual cathode in magnetic self-insulated transmission line |
| title_fullStr | Structure and theory of virtual cathode in magnetic self-insulated transmission line |
| title_full_unstemmed | Structure and theory of virtual cathode in magnetic self-insulated transmission line |
| title_short | Structure and theory of virtual cathode in magnetic self-insulated transmission line |
| title_sort | structure and theory of virtual cathode in magnetic self-insulated transmission line |
| topic | Тheory and technics of particle acceleration |
| topic_facet | Тheory and technics of particle acceleration |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82283 |
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