Particle beam dynamics in a magnetically insulated coaxial diode
The dynamics of charged particle beams emitted from a cathode into a smooth coaxial diode with magnetic insulation is studied with the aid of 3-D PIC simulation. The processes controlling space charge formation and its evolution in the diode are modeled for geometries typical of high-voltage millime...
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irk-123456789-1121882017-01-18T03:04:27Z Particle beam dynamics in a magnetically insulated coaxial diode Korenev, V.G. Magda, I.I. Sinitsin, V.G. Релятивистская электроника The dynamics of charged particle beams emitted from a cathode into a smooth coaxial diode with magnetic insulation is studied with the aid of 3-D PIC simulation. The processes controlling space charge formation and its evolution in the diode are modeled for geometries typical of high-voltage millimeter wave magnetrons that are characterized by very high values of emission currents, hence high space charge densities. Шляхом тривимірного чисельного моделювання досліджено динаміку пучка заряджених частинок, які емітовані з катоду в коаксіальний діод з магнітною ізоляцією. Процеси, які визначають формування та еволюцію просторового заряду, моделюються для таких геометричних параметрів діодної структури, що є характерними для високовольтних магнетронів міліметрового діапазону. Такі пристрої відрізняються дуже високим емісійним струмом, отже, високими значеннями щільності некомпенсованої плазми. Розглянуто розвиток плазмових нестійкостей, що ними визначається розподіл зарядів та електромагнітного поля в системі. Методом трехмерного численного моделирования исследуется динамика пучка заряженных частиц, эмиттируемых с катода в коаксиальный диод с магнитной изоляцией. Процессы, определяющие формирование и эволюцию пространственного заряда, моделируются для такой геометрии диодной структуры, которая характерна для высоковольтных магнетронов миллиметрового диапазона. Эти приборы отличаются очень высокими эмиссионными токами и, соответственно, высокими значениями плотности нескомпенсированной плазмы. Рассмотрены процессы развития плазменных неустойчивостей, которыми определяется распределение плотности заряда и электромагнитного поля в системе. 2015 Article Particle beam dynamics in a magnetically insulated coaxial diode / V.G. Korenev, I.I. Magda, V.G. Sinitsin // Вопросы атомной науки и техники. — 2015. — № 4. — С. 76-80. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 52.75.Pv; 52.80.Pi http://dspace.nbuv.gov.ua/handle/123456789/112188 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Релятивистская электроника Релятивистская электроника |
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Релятивистская электроника Релятивистская электроника Korenev, V.G. Magda, I.I. Sinitsin, V.G. Particle beam dynamics in a magnetically insulated coaxial diode Вопросы атомной науки и техники |
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The dynamics of charged particle beams emitted from a cathode into a smooth coaxial diode with magnetic insulation is studied with the aid of 3-D PIC simulation. The processes controlling space charge formation and its evolution in the diode are modeled for geometries typical of high-voltage millimeter wave magnetrons that are characterized by very high values of emission currents, hence high space charge densities. |
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Korenev, V.G. Magda, I.I. Sinitsin, V.G. |
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Korenev, V.G. Magda, I.I. Sinitsin, V.G. |
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Korenev, V.G. |
| title |
Particle beam dynamics in a magnetically insulated coaxial diode |
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Particle beam dynamics in a magnetically insulated coaxial diode |
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Particle beam dynamics in a magnetically insulated coaxial diode |
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Particle beam dynamics in a magnetically insulated coaxial diode |
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Particle beam dynamics in a magnetically insulated coaxial diode |
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particle beam dynamics in a magnetically insulated coaxial diode |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Релятивистская электроника |
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Particle beam dynamics in a magnetically insulated coaxial diode / V.G. Korenev, I.I. Magda, V.G. Sinitsin // Вопросы атомной науки и техники. — 2015. — № 4. — С. 76-80. — Бібліогр.: 11 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT korenevvg particlebeamdynamicsinamagneticallyinsulatedcoaxialdiode AT magdaii particlebeamdynamicsinamagneticallyinsulatedcoaxialdiode AT sinitsinvg particlebeamdynamicsinamagneticallyinsulatedcoaxialdiode |
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2025-07-08T03:30:57Z |
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ISSN 1562-6016. ВАНТ. 2015. №4(98) 76
PARTICLE BEAM DYNAMICS IN A MAGNETICALLY INSULATED
COAXIAL DIODE
V.G. Korenev, I.I. Magda, V.G. Sinitsin
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: magda@kipt.kharkov.ua
The dynamics of charged particle beams emitted from a cathode into a smooth coaxial diode with magnetic
insulation is studied with the aid of 3-D PIC simulation. The processes controlling space charge formation and its
evolution in the diode are modeled for geometries typical of high-voltage millimeter wave magnetrons that are
characterized by very high values of emission currents, hence high space charge densities.
PACS: 52.75.Pv; 52.80.Pi
INTRODUCTION
The dynamics of charged particles in cross-field
devices has for decades been a subject of intense
investigations [1-5]. The theme has gained special
significance with the advent of high-voltage devices,
relativistic magnet-rons in particular, which are
characterized by extreme emission currents, high
densities of the space charge and great velocities of the
electron flows – both in the interaction space and out of
it. The importance of this problem stems from the
practical necessity to mitigate the tendency toward
reduced efficiency of relativistic devices, as compared
with the conventional magnet-rons [2], which is
particularly noticeable in the high-frequency part of the
microwave band [7]. To compre-hend the effects due to
space charge in a structure of ‘full cylindrical
geometry’, we have focused on a smo-oth-core model of
the magnetron with static E and B fields combined so as
to ensure absence of the anode current (magnetically
insulated operation mode). Earlier papers on the subject
(e.g., [3]) have provided a result of considerable
significance, namely that in the pre-ocillation (zero
anode current) regime of the magnetron the electron
flow may be very different from the classic Brillouin
flow. This finding followed from a 2-D PIC simulation
of a decimeter wave magnetron, involving a modest
number (about 3·104) of the macroparticles, all of which
retained their planar trajectories wihin respective cross-
section planes. In contrast to that, at millimeter
wavelengths the space charge-induced effects become
essentially three dimensional, because of the smaller
size of the device and greater emission cur-rents. The
electric field vector and the flow velocity acquire
components along the structure’s axis, in fact very soon
after start of the emission process (Fig. 1). The orbits of
individual particles are commensurate in size (Larmor
radius) with the cathode-anode gap.
1. PARTICLE FLOW DYNAMICS
1.1. MODEL DESCRIPTION
The dynamics of the non-neutral (negatively
charged) plasma existing in the coaxial diode has been
studied with the help of the 3-D PIC code CST Particle
Studio, taking full account of the self-fields produced by
the macroparticles. The diode was represented by two
coaxi-al circular electrodes extending 50 mm along the
z-axis of the cylindrical polar frame (ρ, φ, z) associated
with the diode. The perfectly conducting inner electrode
was 15 mm in diameter, while the size of the outer
electrode varied between dia=18 mm and dia=28 mm.
In some of the simulations the outer electrode (anode)
possessed a finite electric conductivity, σ=5.8·107 S/m.
Particles with a constant e/m ratio (that of the electron)
were emitted into the cathode-anode gap from a narrow
annular strip (Δz=6 mm) on the surface of the cathode,
in the form of mono-energetic packets, W=4.5 eV. The
number of point-sized centers of emission on the strip
was 1800, and the total number of (macro)particles in
the diode’s operating space was in excess of three
million (up to 5.9·106, see Fig. 1). The simulation code
permitted modeling the emission current only as gradu-
ally changing, from zero to a certain rated magnitude,
over a finite rise time (the values were 1 kA or 100 A
and 4 ns, respectively). The radial-oriented external
electric field was produced by applying a dc voltage
across the cathode-anode gap (-340 kV at the cathode
and 0 at the anode). The focusing magnetic field of axial
orientation was varied around B0=0.4 T, so as to pro-
vide, together with the dc electric field, for either full or
partial magnetic insulation.
Fig. 1. Process of diode filling by charged (macro)
particles: nominal emission current Ie=1 kA
1.2. RESULTS AND DISCUSSION
As can be seen from Fig. 1, the particles leaving the
cathode take some time to fill the space right above the
annular emitting strip, though shorter than the rise time
mailto:magda@kipt.kharkov.ua
ISSN 1562-6016. ВАНТ. 2015. №4(98) 77
necessary for the emission current to reach an estab-
lished nominal value. The particle motion is governed
both by the external applied fields E0 and B0, and the
fields produced by the space charge. As long as the net
magnetic field strength B remains equal to or higher
than Hull’s cutoff magnitude, the particles cannot reach
the anode, and the diode remains magnetically insulated
(see the lower panel in Fig. 2: The dc magnetic field
B0=0.51 T is capable of maintaining the insulation re-
gime both during the current formation period and
through that of steady-state emission). A non-zero an-
ode current through an initially insulated diode may
appear owing to the space charge-produced electrostatic
field. The situ-ation is shown in the upper panel of
Fig. 2 where the ap-plied dc magnetic field of 0.46 T
had been sufficient for diverting the particles from the
anode until at t ≈1.5 ns the diode became conductive as
a result of further build-up in the space charge. Next, the
start of two opposite particle flows along the z-axis can
be seen at t ≈2.6 ns both in Figs.1, and 2. Till that mo-
ment the motion of individual particles was two-
dimensional, with each of them moving along a planar
trajectory within the diode’s cross-section plane (ρ, φ)
which it was emitted to. The sum of the anode current
and the axial currents, including such to auxiliary struc-
ture elements, shall be (and is) equal to the emitted cur-
rent – however, not the nominal but rather the space
charge-controlled value. The latter is formed under du-
al-stream conditions, with particles moving at each radi-
al distance from the cathode both toward the anode and
back. Note the ‘real’ emission current to settle at a low-
er level when the flow to the anode is absent.
Fig. 2. Balance of currents through the diode (electrode
diameters 15, and 28 mm), with a 1 kA nominal emis-
sion current, for two magnitudes of the focusing mag-
netic field: 1 – nominal emission current;
2 – space charge-limited emission current; 3 – current
flow to the anode; 4 – total current of axially directed
flows; 5 – current to structure supports, etc
The motion of the non-neutral plasma can be de-
scribed, in the cold fluid approximation, by the equation
mc
e
dt
d BpEp ×
+= , (1)
where p is momentum of the fluid element and m its
mass. By projecting (1) to the (ρ-φ) plane, it is easy to
derive the ‘radial force equilibrium’ equation [6],
c
BVBVe
eE
mV zz )(2
ϕϕ
ρ
ϕ
ρ
γ
−
+=− , (2)
with 2/1222 ]./)(1[ cVV z+−= ϕγ Here γ is the relativ-
istic factor, Eρ stands for the radial component of the
electric field vector, and ρρωϕ )(=V is the azimuthal
velocity, with )(ρω the respective angular velocity of
the rotating ‘hub’ of particles. The magnetic component
Bz includes both the applied field and the self-induced,
axially oriented contribution. The azimuthal component
Bφ is fully self-induced, owing to the space charge, and
is often neglected in analyses of plasma equilibria and
instabilities [3, 4]. Apparently, this may be plausible at
an early stage of the process when no axial flow has
formed yet, Vz ≈ 0 and Bφ is small, i.e. in the case of
Fig. 2 at times earlier than 2.2 or 2.3 ns. Space charge
density simulations for these early periods are shown in
Fig. 3 where the left-hand column presents results for a
nominal emission current of 100 A and the right-hand
one for 1 kA, all in the coaxial structure with a cathode
of ρc=7.5 mm and anode, ρa =14 mm (15 mm/28 mm
diode), placed in a 0.46 T magnetic field. The charge
density distributions refer to the central cross-section
plane of the diode, z=0. Estimating representative mag-
nitudes of the plasma frequency, 2/1)/4( mQep πω = , and
cyclotron frequency, cBmeH /)/(=ω (where Q is
charge density), from the data of Fig. 3 we find ωH ≈
8·1010 s-1 and ωp ranging from 0.7·1010 to 1.5·1010 s-1 in
the left column, and 2.4·1010 to 3.9·1010 s-1 in the right
one. The ‘self-field parameter’ 22 /2 Hpes ωω= [6] is less
than one throughout, which suggests a relatively weak
repulsion effect of the space charge against the focusing
action of the magnetic field, owing to the low charge
density at early stages of the process. Meanwhile, even
at this early stage the laminar Brillouin flow condition,
)](
2
11[)( 4
44
22
b
cb
p H
ρ
ρρ
ωρω −
−= (3)
(ρb ≤ ρa is the beam radius), known for the transverse
(2-D) motion of the particles [6], is not met. The
squared plasma frequency 2
pω which is proportional to
charge density Q remains less than the magnitude in the
right-hand side of (3) because of the axial outflow of
particles. Within each cross-section, the radial profiles
of space charge density are, on the average, close to a
parabola, which they should be in the relativistic case
[4], and demonstrate several points of change in the sign
of ρ∂∂ /Q between the cathode and the anode. On the
one hand, these changes may be regarded as indicators
of a developing diocotron (slipping stream) instability
which gives rise to azimuthal waves ~ )exp( tiin ωϕ −
around the plasma column.
ISSN 1562-6016. ВАНТ. 2015. №4(98) 78
Fig. 3. Radial space-charge density distributions in the 15/28 mm diode, at several successive instants during the
early stage of hub formation. Left column: nominal emission current Ie =100 A; right column: Ie =1 kA.
The changes in the sign of ρ∂∂ /Q within the C-A gap, where Q is charge density, indicate conditions
for the development of the diocotron instability
On the other hand, a suspicion remains that the radi-
al non-uniformity of the early-stage charge density
might be an effect arising from the discrete model of the
emission. Assuming that the instability does evolve, we
expect the waves with n = 1, 2, 3, etc. to be overlaid on
the general rotational motion of the plasma hub.
Fig. 4. Cross-sectional view of charge density distribu-
tion in the 15/28 mm diode, featuring a n=3 azimuthal
wave. With B0 =0.46 T, the degree of magnetic insula-
tion is close to ‘critical’ and ρb ≈ ρa =14 mm
Figs. 4 and 5 show the faceted particle patterns that
have developed at later times (t>3 ns) from the unstable
density profiles like in Fig. 3. The number n of the pre-
ferred azimuthal mode, hence the quantity of facets, is
determined by the effective thickness of the plasma
beam along the radial, i.e. by the ratio bc ρρ / , where the
beam radius ρb may be less than or equal to the radius of
the anode, depending on the degree of magnetic insula-
tion. The ‘critical’ insulation in Fig. 4 where the plasma
barely touches to the anode gives rise to an almost equi-
librium state of the particle beam with three broad
‘rays’ of charge density, i.e. the preferred azimuthal
mode is n =3. The case of Fig. 5 is characterized by a
smaller width of the cathode-anode gap (the anode radi-
us is 12 mm rather than 14 mm), plus a noticeably high-
er magnetic field, B0 = 0.707 T.
Fig. 5. Cross-sectional view of charge density distribu-
tion in the 15/24 mm diode, featuring a n=4 azimuthal
wave. The full magnetic insulation with
B0=0.707 T results in a ρb < ρa =12 mm
ISSN 1562-6016. ВАНТ. 2015. №4(98) 79
Fig. 6. Axially accelerated particles and formation
of annular beams away from the emitter. The upper
panel represents a diode with a full-sized coaxial
cathode, and the lower one shows an asymmetric
pattern in a structure with a ‘shortened’ electrode
As a result, in the fully insulated beam the cross-
sectional density distribution takes the form of a round-
ed tetragon, n=4. By further reducing the gap width, we
could obtain charge density patterns (and the corre-
sponding distributions of the electric self-fields) charac-
teristic of higher-order azimuthal modes, for example
n=12 for ρc =11 mm. These peripheral waves are not
confined to the axial limits of the emitting strip on the
cathode but propagate along the z-axis, giving rise to a
(hollow) annular particle beam. The azimuthal patterns
are retained in the near-surface layers of the propagating
beam, taking the form of helical traces. Examples of
such annular beams are given in Fig. 6 which shows
simulation results for a coaxial diode with a ‘full sized’
cathode (upper panel) and for a diode with a cathode cut
off at one end. In the upper panel, it can be seen that the
beam particles gradually emerge from the area near the
emitter, where their motion was predominantly two-
dimensional, and acquire z-component of velocity under
the action of the space-charge field. This axially orient-
ed acceleration reveals itself particularly brightly in the
asymmetric pattern of particle positions presented in the
lower part of Fig. 6. The radial size of the beam reduces
non-uniformly along z away from the emitter because of
the now existing axial current and its associated azi-
muthal magnetic field.
The ensemble of particles rotating in the diode at
angular velocities ω= (2…3)·1010 1/s should be capable
of exciting some of the proper oscillations pertinent to
the diode as a wave supporting structure (a resonant
cavity or section of a transmission line). Indeed, oscilla-
tory regimes for currents and the cathode-anode voltage
happened to appear at times about 2 ns or 2.5 ns, i.e.
well before the diode space has been completely filled
with the particles. Analysis of dispersive proper-ties and
excitation characteristics would be more appropriate if it
were a realistic magnetron model with a slow wave
structure in one of the electrodes, however a few re-
marks concerning the present model are in order. The
oscillation frequencies excited are close, though not
equal, to estimated eigenfrequencies of an empty trans-
mission line of the same cross-section as the diode. Both
amplitudes and “exact” frequencies of the oscillations
are dependent on the amount of magnetic insulation or,
rather, on the effective size of the particle hub which
determines the transverse (radial) eigenvalues. Exam-
ples of the oscillations appearing under different condi-
tions in the 15/28 mm diode are given in Fig. 7 in a
spectral representation. The smaller effective sizes of
the plasma column (i.e., higher magnetic fields) corre-
spond to higher oscillation frequencies.
Fig. 7. Spectra of C-A voltage oscillations in the
15/28 mm diode with various magnitudes
of the applied magnetic field
CONCLUSIONS
The motion of charged particles in the high voltage,
smooth-core coaxial diode operated in the regime of
either full or partial magnetic insulation is essentially
three-dimensional, and largely different in the 2-D
cross-sectional area from the laminar Brillouin flow
dynamics.
The space-charge density distribution along the radi-
al coordinate is not monotonic, being modified by rela-
tivistic effects and axial outflow of particles. The equili-
brium distributions are established with account of axial
motion of the particles.
The particles moving in the axial direction are accel-
erated by the self-field of the space charge, and ulti-
mately give rise to an annular beam with a helical densi-
ty structure on the outer surface.
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1. O. Buneman. Crossed-field microwave devices / ed.
F. Okress, v. 1. New York: Academic Press, 1961,
p. 209.
2. F.J. Agee. Evolution of Pulse Shortening Research
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IEEE Trans. Plasma Sci. 1998, v. 26, p. 235.
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study of relativistic magnetrons // J. Appl. Phys.
1993, v. 73, № 11, p. 7053-7060.
4. I. Kotelnikov, M. Romé, R. Pozzoli. On the relativ-
istic cold fluid radial equilibrium of a nonneutral
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p. 1445-1450.
5. Y.Y. Lau, J.W. Luginsland, K.L. Cartwright,
D.H. Simon, W.Tang, B.W. Hoff, R.M. Gilgenbach.
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http://www.sciencedirect.com/science/article/pii/S0375960107013710
http://www.sciencedirect.com/science/article/pii/S0375960107013710
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A re-examination of the Buneman-Hartree condition
in a cylindrical smooth-bore relativistic magnetron //
Phys. Plasmas. 2010, v. 17, 033102; doi:
0.1063/1.3328804.
6. R.C. Davidson. Physics of non-neutral plasmas.
New York, Addison-Wesley. 1992, p. 290-310.
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V.G. Korenev, A.N. Lebedenko, M.I. Marchenko,
I.I. Magda, O.G. Melezhik, V.G. Sinitsin,
V.A. Soshenko. Special traits of the millimeter wave
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Article received 17.06.2015
ДИНАМИКА ПУЧКОВ ЗАРЯЖЕННЫХ ЧАСТИЦ В КОАКСИАЛЬНОМ ДИОДЕ
С МАГНИТНОЙ ИЗОЛЯЦИЕЙ
В.Г. Коренев, И.И. Магда, В.Г. Синицын
Методом трехмерного численного моделирования исследуется динамика пучка заряженных частиц,
эмиттируемых с катода в коаксиальный диод с магнитной изоляцией. Процессы, определяющие
формирование и эволюцию пространственного заряда, моделируются для такой геометрии диодной
структуры, которая характерна для высоковольтных магнетронов миллиметрового диапазона. Эти приборы
отличаются очень высокими эмиссионными токами и, соответственно, высокими значениями плотности
нескомпенсированной плазмы. Рассмотрены процессы развития плазменных неустойчивостей, которыми
определяется распределение плотности заряда и электромагнитного поля в системе.
ДИНАМІКА ПУЧКІВ ЗАРЯДЖЕНИХ ЧАСТИНОК У КОАКСІАЛЬНОМУ ДІОДІ
З МАГНІТНОЮ ІЗОЛЯЦІЄЮ
В.Г. Коренєв, І.І. Магда, В.Г. Сініцин
Шляхом тривимірного чисельного моделювання досліджено динаміку пучка заряджених частинок, які
емітовані з катоду в коаксіальний діод з магнітною ізоляцією. Процеси, які визначають формування та
еволюцію просторового заряду, моделюються для таких геометричних параметрів діодної структури, що є
характерними для високовольтних магнетронів міліметрового діапазону. Такі пристрої відрізняються дуже
високим емісійним струмом, отже, високими значеннями щільності некомпенсованої плазми. Розглянуто
розвиток плазмових нестійкостей, що ними визначається розподіл зарядів та електромагнітного поля в
системі.
INTRODUCTION
1. PARTICLE FLOW DYNAMICS
1.1. MODEL DESCRIPTION
Fig. 3. Radial space-charge density distributions in the 15/28 mm diode, at several successive instants during the early stage of hub formation. Left column: nominal emission current Ie =100 A; right column: Ie =1 kA. The changes in the sign of with...
On the other hand, a suspicion remains that the radial non-uniformity of the early-stage charge density might be an effect arising from the discrete model of the emission. Assuming that the instability does evolve, we expect the waves with n = 1, 2, 3...
Figs. 4 and 5 show the faceted particle patterns that have developed at later times (t>3 ns) from the unstable density profiles like in Fig. 3. The number n of the preferred azimuthal mode, hence the quantity of facets, is determined by the effective ...
Fig. 6. Axially accelerated particles and formation of annular beams away from the emitter. The upper panel represents a diode with a full-sized coaxial cathode, and the lower one shows an asymmetric pattern in a structure with a ‘shortened’ electrode
As a result, in the fully insulated beam the cross-sectional density distribution takes the form of a rounded tetragon, n=4. By further reducing the gap width, we could obtain charge density patterns (and the corresponding distributions of the electri...
The ensemble of particles rotating in the diode at angular velocities ω= (2…3)∙1010 1/s should be capable of exciting some of the proper oscillations pertinent to the diode as a wave supporting structure (a resonant cavity or section of a transmission...
CONCLUSIONS
The motion of charged particles in the high voltage, smooth-core coaxial diode operated in the regime of either full or partial magnetic insulation is essentially three-dimensional, and largely different in the 2-D cross-sectional area from the lamina...
The space-charge density distribution along the radial coordinate is not monotonic, being modified by relativistic effects and axial outflow of particles. The equili-brium distributions are established with account of axial motion of the particles.
The particles moving in the axial direction are accelerated by the self-field of the space charge, and ultimately give rise to an annular beam with a helical density structure on the outer surface.
references
1. O. Buneman. Crossed-field microwave devices / ed. F. Okress, v. 1. New York: Academic Press, 1961, p. 209.
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