High-frequency properties of the electron flow slowing down in a planar diode
The instability of the electron flow injected into a planar diode and slowed down in it under the action of the external electric field is investigated through the exact solution of the set of inhomogeneous hydrodynamic first-approximation equations of the stability theory. A dispersion equation has...
Saved in:
| Published in: | Вопросы атомной науки и техники |
|---|---|
| Date: | 2011 |
| Main Authors: | , , , , , , , , |
| Format: | Article |
| Language: | English |
| Published: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2011
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/111467 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | High-frequency properties of the electron flow slowing down in a planar diode / A.V. Pashchenko, O.G. Melezhik, S.S. Romanov,D.A. Sitnikov, I.K. Tarasov, M.I. Tarasov,I.M. Shapoval, V.E. Novikov, V.A. Yatsyshin // Вопросы атомной науки и техники. — 2011. — № 5. — С. 86-92. — Бібліогр.: 4 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-111467 |
|---|---|
| record_format |
dspace |
| spelling |
Pashchenko, A.V. Melezhik, O.G. Romanov, S.S. Sitnikov, D.A. Tarasov, I.K. Tarasov, M.I. Shapoval, I.M. Novikov, V.E Yatsyshin, V.A. 2017-01-10T11:47:19Z 2017-01-10T11:47:19Z 2011 High-frequency properties of the electron flow slowing down in a planar diode / A.V. Pashchenko, O.G. Melezhik, S.S. Romanov,D.A. Sitnikov, I.K. Tarasov, M.I. Tarasov,I.M. Shapoval, V.E. Novikov, V.A. Yatsyshin // Вопросы атомной науки и техники. — 2011. — № 5. — С. 86-92. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk https://nasplib.isofts.kiev.ua/handle/123456789/111467 The instability of the electron flow injected into a planar diode and slowed down in it under the action of the external electric field is investigated through the exact solution of the set of inhomogeneous hydrodynamic first-approximation equations of the stability theory. A dispersion equation has been derived, which relates the frequencies and increments (decrements) of arising electromagnetic oscillations to the parameters of the electron beam and the diode. The solution of the dispersion equation shows that in the diode, through which the slowing down electron beam is propagating, there arises the microwave oscillatory instability not described before. This instability occurs in the case when in the steady-state condition there is no potential minimum in the diode. The experiments have confirmed the occurrence of oscillatory instability with theoretically predicted frequencies and decrements in the corresponding beam and diode parameter regions. На основі точного рішення системи неоднорідних гідродинамічних рівнянь першого наближення теорії стійкості вивчена нестійкість електронного потоку, інжектованного у плоский діод і сповільненого в ньому під дією зовнішнього електричного поля. Отримано дисперсійне рівняння, що пов'язує частоти і інкременти (декременти) виникаючих електромагнітних коливань з параметрами електронного потоку і діода. Рішення дисперсійного рівняння показує, що в діоді, крізь який розповсюджується сповільнений електронний потік, виникає не описана раніше нестійкість коливань у СВЧ діапазоні. Ця нестійкість має місце, коли в стаціонарному стані мінімум потенціалу в діоді не утворюється. Експериментально підтверджено виникнення нестійкості коливань з передбаченими теоретично частотами і декрементами у відповідних областях параметрів пучка і діода. На основе точного решения системы неоднородных гидродинамических уравнений первого приближения теории устойчивости изучена неустойчивость электронного потока, который инжектирован в плоский диод и замедляется в нём под действием внешнего электрического поля. Получено дисперсионное уравнение, связывающее частоты и инкременты (декременты) возникающих электромагнитных колебаний с параметрами электронного потока и диода. Решение дисперсионного уравнения показывает, что в диоде, через который распространяется замедляющийся электронный поток, возникает не описанная ранее неустойчивость колебаний в СВЧ диапазоне. Эта неустойчивость имеет место, когда в стационарном состоянии минимум потенциала в диоде не образуется. Экспериментально подтверждено возникновение неустойчивости колебаний с предсказанными теоретически частотами и декрементами в соответствующих областях параметров пучка и диода. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Электродинамика High-frequency properties of the electron flow slowing down in a planar diode Високочастотнi властивостi електронного потоку, сповiльненого у плоскому дiодi Высокочастотные свойства электронного потока, замедляющегося в плоском диоде Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
High-frequency properties of the electron flow slowing down in a planar diode |
| spellingShingle |
High-frequency properties of the electron flow slowing down in a planar diode Pashchenko, A.V. Melezhik, O.G. Romanov, S.S. Sitnikov, D.A. Tarasov, I.K. Tarasov, M.I. Shapoval, I.M. Novikov, V.E Yatsyshin, V.A. Электродинамика |
| title_short |
High-frequency properties of the electron flow slowing down in a planar diode |
| title_full |
High-frequency properties of the electron flow slowing down in a planar diode |
| title_fullStr |
High-frequency properties of the electron flow slowing down in a planar diode |
| title_full_unstemmed |
High-frequency properties of the electron flow slowing down in a planar diode |
| title_sort |
high-frequency properties of the electron flow slowing down in a planar diode |
| author |
Pashchenko, A.V. Melezhik, O.G. Romanov, S.S. Sitnikov, D.A. Tarasov, I.K. Tarasov, M.I. Shapoval, I.M. Novikov, V.E Yatsyshin, V.A. |
| author_facet |
Pashchenko, A.V. Melezhik, O.G. Romanov, S.S. Sitnikov, D.A. Tarasov, I.K. Tarasov, M.I. Shapoval, I.M. Novikov, V.E Yatsyshin, V.A. |
| topic |
Электродинамика |
| topic_facet |
Электродинамика |
| publishDate |
2011 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Високочастотнi властивостi електронного потоку, сповiльненого у плоскому дiодi Высокочастотные свойства электронного потока, замедляющегося в плоском диоде |
| description |
The instability of the electron flow injected into a planar diode and slowed down in it under the action of the external electric field is investigated through the exact solution of the set of inhomogeneous hydrodynamic first-approximation equations of the stability theory. A dispersion equation has been derived, which relates the frequencies and increments (decrements) of arising electromagnetic oscillations to the parameters of the electron beam and the diode. The solution of the dispersion equation shows that in the diode, through which the slowing down electron beam is propagating, there arises the microwave oscillatory instability not described before. This instability occurs in the case when in the steady-state condition there is no potential minimum in the diode. The experiments have confirmed the occurrence of oscillatory instability with theoretically predicted frequencies and decrements in the corresponding beam and diode parameter regions.
На основі точного рішення системи неоднорідних гідродинамічних рівнянь першого наближення теорії стійкості вивчена нестійкість електронного потоку, інжектованного у плоский діод і сповільненого в ньому під дією зовнішнього електричного поля. Отримано дисперсійне рівняння, що пов'язує частоти і інкременти (декременти) виникаючих електромагнітних коливань з параметрами електронного потоку і діода. Рішення дисперсійного рівняння показує, що в діоді, крізь який розповсюджується сповільнений електронний потік, виникає не описана раніше нестійкість коливань у СВЧ діапазоні. Ця нестійкість має місце, коли в стаціонарному стані мінімум потенціалу в діоді не утворюється. Експериментально підтверджено виникнення нестійкості коливань з передбаченими теоретично частотами і декрементами у відповідних областях параметрів пучка і діода.
На основе точного решения системы неоднородных гидродинамических уравнений первого приближения теории устойчивости изучена неустойчивость электронного потока, который инжектирован в плоский диод и замедляется в нём под действием внешнего электрического поля. Получено дисперсионное уравнение, связывающее частоты и инкременты (декременты) возникающих электромагнитных колебаний с параметрами электронного потока и диода. Решение дисперсионного уравнения показывает, что в диоде, через который распространяется замедляющийся электронный поток, возникает не описанная ранее неустойчивость колебаний в СВЧ диапазоне. Эта неустойчивость имеет место, когда в стационарном состоянии минимум потенциала в диоде не образуется. Экспериментально подтверждено возникновение неустойчивости колебаний с предсказанными теоретически частотами и декрементами в соответствующих областях параметров пучка и диода.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/111467 |
| citation_txt |
High-frequency properties of the electron flow slowing down in a planar diode / A.V. Pashchenko, O.G. Melezhik, S.S. Romanov,D.A. Sitnikov, I.K. Tarasov, M.I. Tarasov,I.M. Shapoval, V.E. Novikov, V.A. Yatsyshin // Вопросы атомной науки и техники. — 2011. — № 5. — С. 86-92. — Бібліогр.: 4 назв. — англ. |
| work_keys_str_mv |
AT pashchenkoav highfrequencypropertiesoftheelectronflowslowingdowninaplanardiode AT melezhikog highfrequencypropertiesoftheelectronflowslowingdowninaplanardiode AT romanovss highfrequencypropertiesoftheelectronflowslowingdowninaplanardiode AT sitnikovda highfrequencypropertiesoftheelectronflowslowingdowninaplanardiode AT tarasovik highfrequencypropertiesoftheelectronflowslowingdowninaplanardiode AT tarasovmi highfrequencypropertiesoftheelectronflowslowingdowninaplanardiode AT shapovalim highfrequencypropertiesoftheelectronflowslowingdowninaplanardiode AT novikovve highfrequencypropertiesoftheelectronflowslowingdowninaplanardiode AT yatsyshinva highfrequencypropertiesoftheelectronflowslowingdowninaplanardiode AT pashchenkoav visokočastotnivlastivostielektronnogopotokuspovilʹnenogouploskomudiodi AT melezhikog visokočastotnivlastivostielektronnogopotokuspovilʹnenogouploskomudiodi AT romanovss visokočastotnivlastivostielektronnogopotokuspovilʹnenogouploskomudiodi AT sitnikovda visokočastotnivlastivostielektronnogopotokuspovilʹnenogouploskomudiodi AT tarasovik visokočastotnivlastivostielektronnogopotokuspovilʹnenogouploskomudiodi AT tarasovmi visokočastotnivlastivostielektronnogopotokuspovilʹnenogouploskomudiodi AT shapovalim visokočastotnivlastivostielektronnogopotokuspovilʹnenogouploskomudiodi AT novikovve visokočastotnivlastivostielektronnogopotokuspovilʹnenogouploskomudiodi AT yatsyshinva visokočastotnivlastivostielektronnogopotokuspovilʹnenogouploskomudiodi AT pashchenkoav vysokočastotnyesvoistvaélektronnogopotokazamedlâûŝegosâvploskomdiode AT melezhikog vysokočastotnyesvoistvaélektronnogopotokazamedlâûŝegosâvploskomdiode AT romanovss vysokočastotnyesvoistvaélektronnogopotokazamedlâûŝegosâvploskomdiode AT sitnikovda vysokočastotnyesvoistvaélektronnogopotokazamedlâûŝegosâvploskomdiode AT tarasovik vysokočastotnyesvoistvaélektronnogopotokazamedlâûŝegosâvploskomdiode AT tarasovmi vysokočastotnyesvoistvaélektronnogopotokazamedlâûŝegosâvploskomdiode AT shapovalim vysokočastotnyesvoistvaélektronnogopotokazamedlâûŝegosâvploskomdiode AT novikovve vysokočastotnyesvoistvaélektronnogopotokazamedlâûŝegosâvploskomdiode AT yatsyshinva vysokočastotnyesvoistvaélektronnogopotokazamedlâûŝegosâvploskomdiode |
| first_indexed |
2025-11-24T02:19:04Z |
| last_indexed |
2025-11-24T02:19:04Z |
| _version_ |
1850837969674436608 |
| fulltext |
HIGH-FREQUENCY PROPERTIES OF THE ELECTRON
FLOW SLOWING DOWN IN A PLANAR DIODE
A.V. Pashchenko2, O.G. Melezhik1, S.S. Romanov1,
D.A. Sitnikov1, I.K. Tarasov1, M.I. Tarasov1,
I.M. Shapoval1∗, V.E. Novikov2, V.A. Yatsyshin3
1National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine;
2Institute for Electrophysics and Radiation Technologies, National Academy of Sciences of Ukraine,
61002, Kharkov, Ukraine;
3Science-and-Production Center ”Engineering-Technology Systems”, 04210, Kiev, Ukraine
(Received August 16, 2011)
The instability of the electron flow injected into a planar diode and slowed down in it under the action of the external
electric field is investigated through the exact solution of the set of inhomogeneous hydrodynamic first-approximation
equations of the stability theory. A dispersion equation has been derived, which relates the frequencies and increments
(decrements) of arising electromagnetic oscillations to the parameters of the electron beam and the diode. The solution
of the dispersion equation shows that in the diode, through which the slowing down electron beam is propagating,
there arises the microwave oscillatory instability not described before. This instability occurs in the case when in the
steady-state condition there is no potential minimum in the diode. The experiments have confirmed the occurrence of
oscillatory instability with theoretically predicted frequencies and decrements in the corresponding beam and diode
parameter regions.
PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk
1. INTRODUCTION
It is commonly believed that all electromagnetic
phenomena occurring in one of the simplest
radiophysical devices, namely, the planar diode,
have long been explored [1-4], and are the topics
of education courses rather than the subjects of
scientific research. However, it appears that a
sequential development of theory (in particular, the
application of Lagrangian description methods for
electrodynamic processes) reveals a broad spectrum
of new oscillation modes. The paper describes the
new oscillation modes and instability, the existence
of which has been corroborated by experiments.
In experimental physics, it is a currently accepted
practice to determine high-frequency properties of
volumes under vacuum in accelerators, magnetic
traps, process devices, etc. In particular,
experimental and theoretical efforts are aimed at
determining the resonance properties of intrinsic
cavities of experimental devices, the influence of
cavity openings, branch pipes, projections, etc. on
the excited spectra.
The knowledge of electromagnetic oscillations
that may be excited in the setups is essential for
understanding the behavior of accelerated/confined
particles or plasma in the setups.
In this case, it is also of importance to determine
self-consistently the high-frequency properties of
experimental setups with due regard for both
the properties of accelerated/confined streams of
particles or plasma and the geometry of vacuum
chambers.
There are calculations made to date, which
have resulted in the development of a great variety
of electrodynamic devices for electromagnetic wave
generation in different frequency ranges of particle
accelerators and storage rings of various types,
plasma confinement devices. However, a similar
problem for the classical electrodynamics object, i.e.,
planar diode, still remained unsolved.
The authors of the present work have found the
spectrum of electromagnetic oscillations, which can
be excited in a planar diode on injection of the
electron beam into it. It has been shown that in
the case of electron beam deceleration in the diode
there occurs a new (not described in the literature)
instability, which is accompanied by excitation of
the above-mentioned spectrum. oscillations, which
can be excited in a planar diode on injection of the
electron beam into it. It has been shown that in
the case of electron beam deceleration in the diode
there occurs a new (not described in the literature)
instability, which is accompanied by excitation of the
above-mentioned spectrum.
∗Corresponding author E-mail address: shapoval@kipt.kharkov.ua
86 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2011, N5.
Series: Nuclear Physics Investigations (56), p.86-92.
2. CHOICE OF THE ELECTRON-FLOW
MODEL
In our studies the electron flow in the diode is
considered in the one-dimensional hydrodynamic
approximation. Fig.1 shows the electrode geometry
and the qualitative potential distribution pattern in
the diode.
0 l
j
j
2
j1
e
m
is
s
io
n
Fig.1. The qualitative distribution pattern in the
triode
The qualitative conclusions of the one-dimensional
model used here hold true in two-dimensional models,
too, though a detailed behavior of the flow changes
when passing from one- to two-dimensional problems.
The one-dimensional hydrodynamic equations of
motion and continuity as well as the Poisson equation
for the electric field are the initial equations:
∂v
∂t
+ v
∂v
∂z
=
∂Φ
∂z
,
∂n
∂t
+
∂
∂z
(nv) = 0 , (1)
∂2Φ
∂z2
= qn .
The equation of motion is, in effect, the second
Newton’s law for a macroscopically small volume of
the electron flow. The hydrodynamic equation of
continuity expresses the particle conservation law.
In the Coulomb gauge the scalar potential Φ̂ is
determined by the charge distribution.
The input flow parameters of the diode, namely,
n0, v0; the diode gap width l and the time of
flow transit of the distance l at a velocity v0, are
the units of measurement of density n, velocity v,
coordinate z and time t. The dimensionless electric-
field potential [Φ] = mv2
0
e is the ratio of the double
kinetic energy of the flow to the electric field energy,
q = 4πe2n0l
2/mv2
0 .
3. STEADY-STATE CONDITIONS OF THE
ELECTRON FLOW. LAGRANGIAN
FORMALISM
We write the time- and coordinate-dependent
quantities as follows
v(z, t) = v0(z) + ṽ(z, t) ,
n(z, t) = n0(z) + ñ(z, t) , (2)
Φ(z, t) = Φ0(z) + Φ̃(z, t) ,
where v0(z) , n0(z) , Φ0(z) are the stationary
quantities.
The relations describing the current density
stability and the flow energy conservation law are the
stationary solutions of the system (1):
n0(z)n0(z) = 1 ,
ϕ(z) =
1
2
− v0(z)2
2
. (3)
Let us consider the slowing down flow, i.e., the
potential of the second electrode is negative, Φl < 0
(Φ0 = −ϕ,ϕ > 0). With the use of integrals (3),
the solution of the problem for the steady-state
conditions leads to the following relations:
1
3
(v0 + c)
3
2 − c(v0 + c)
1
2 = −
√
q
2
(z − z̄) . (4)
The constants c and z are determined from the
boundary conditions
v0(z = 0) = 1, v0(z = 1) = vl , vl =
√
1− ϕl .
(vl +c)
1
2 ·(−vl +2c)−(1+c)
1
2 (−1+2c) =
√
9
2
q , (5)
z̄ =
1
2
− 1
3
√
2q
[(vl + c)
1
2 · (−vl + 2c)−
−(1 + c)
1
2 (−1 + 2c)] . (6)
Equation (6) defines the coordinates of the plane,
where the flow rate is equal to c (5). We introduce
the Lagrange time τ . Then by definition we have
v0(z) = dz
dτ , and both the rate of the stationary
flow and its coordinate are given by the following
relations, depending on τ and the parameters of flow
and diode:
v0(τ) =
q
2
(τ − 1√
qγ
)2 + 1− 1
2γ
,
z =
q
6
τ3 − 1
2
√
q
γ
τ2 + τ , (7)
c =
1
2γ
− 1 .
Since |c| ≤ 1, then we have ∞ > γ ≥ 1/4 . Equation
(5) that defines c can be rewritten as
q(γ, vl) =
1
γ
[1− 1
3γ
+
87
+
1
3
(−vl +
1
γ
− 2)
√
1− 2γ(1− vl)]2 , (8)
and the coordinate of the plane, where the flow rate
is equal to c, is written as
z̄ =
1
2
− 1
2
√
qγ
[−1 +
1
3γ
+
+
1
3
(−vl +
1
γ
− 2)
√
1− 2γ(1− vl)]2 . (9)
For example, for the drift space at γ=1 and vl=0.5
we obtain q=4/9 and z =1. Note that the gap transit
time (z = zl =1) for the stationary flow is equal to
τl =
1√
qγ
(1 +
√
1− 2γ(1− vl)) . (10)
4. DEVIATIONS FROM THE
STATIONARY STATE
We assume the deviations from stationary values
to be small and consider a linearized set of equations
(1). According to expression (2) we have
−iωṽ(z) +
d
dz
[v0(z)ṽ(z)] = −dΦ0
dz
,
−iωñ(z) +
d
dz
[n0(z)ṽ(z) + ñ(z)v0(z)] = 0 , (11)
d2Φ0(z)
dz2
= −qñ(z) .
The time dependence of deviation variables is
proportional to e−iωt and from here on the tilde over
the deviation terms will be omitted. From expression
(11) we obtain the following equation:
[v0(z)]3
d2u
dz2
+ qu =
= C1[v0(z)]2 exp(−iω
∫ z
0
dx
v0(x)
) , (12)
where u(z) = v0v(z)exp(−iω
∫ z
0
dx
v0(x) ), and C1 is
the integration constant proportional to the Fourier
amplitude being the total current component. Eq.
(11) is solved in terms of Lagrange variables. Using
the relations derived, we rewrite eq. (12) as
v0(τ)
d2u
dτ2
− dv0(τ)
dτ
du
dτ
+qu = C1[v0(τ)]2e−iωτ . (13)
With the known solution of the homogeneous
equation, i.e., eq. (13), where the right-hand member
equals zero, and using the method of variation of
arbitrary constants we find the following solution
u(θ) = (θ − 1)D1 + D2[θ(θ − 1) + 1− 2γ]+
+Ce−σθ[1− 2γ − 2
σ
(θ − 1)− (θ − 1)2] . (14)
In relation (14), the Lagrange variable is denoted
as θ = τ
√
qγ, the parameter σ is given by iω/
√
γq,
C = −C1/2σ2qγ2, D1 and D2 are the integration
constants. Going in eq.(11) from z to θ we obtain
the following formula
ϕ(θ) = ϕ(θ) +
∫ θ
0
dx
du
dx
eσx . (15)
The substitution of the velocity disturbance (14) into
formula (15) results in
ϕ(θ) = ϕ(0)− D1
σ
(eσθ − 1)−
−D2
σ
[eσθ(2θ − 2
σ
− 1) + 1 +
1
σ
]+
+Cσ(1− 2γ +
2
σ2
θ − (θ − 1)3 + 1
3
) . (16)
The density deviation from the stationary value can
now be found by eq. (15) as
n(θ) =
γ
[v0(τ)]2
(D1e
σθ[σ − θ − 1
γv0
]+
+D2e
σθ[2σθ + 2− σ − θ − 1
γv0
])+
+Cσ[(2θ−1)− θ − 1
γv0
(−1+2γ+(θ−1)2− 2
σ2
)] . (17)
5. FREQUENCY SPECTRUM OF
SPACE-CHARGE WAVES
The integration constants in eqs. (15)-(17) are
found with the help of the boundary conditions (18)
v(z = 0) = 0, n(z = 0) = 0 ,
ϕ(z = 0) = 0, ϕ(z = 1) = 0 . (18)
The conditions imply that at the diode input
there are no deviations of hydrodynamic quantities
from their stationary values. Similarly, at the
diode output there is no deviation of the potential
from its stationary value. That is to say, to a
first approximation, nothing from the outside is
brought into the diode, and at the output the
system responses in a self-consistent way. The third
condition in (18) is satisfied by itself. The first,
second and fourth conditions form a homogeneous
system for the unknown quantities D1, D2, C:
D1 −D2(1− 2γ) + 2C(γ − 1
σ
) = 0,
D1(1 +
1
γσ
) + D2(
2
σ
− 1
γσ
− 1)− C · 2
γσ2
= 0,
D1(eσθl − 1) + D2[eσθl(2θl − 2
σ
− 1) + 1 +
2
σ
]+
88
+Cσ2[(−1 + 2γ − 2
σ2
)θl +
(θl − 1)3 + 1
3
] = 0 . (19)
This system has non-zero solutions when the
determinant composed of the coefficients of the
unknowns is equal to zero.
The calculations lead to the following equation for
the frequency spectrum of space-charge waves, which
can exist in the diode at conditions of no particle
reflection, when the potential minimum is absent:
e2α(1− α) + 4Gα3 − α− 1 = 0 . (20)
Here we introduced the notation
2α = σθl,
G =
−1 + 2γ(vl + 2)−
√
1− 2γ(1− vl)
6[1 +
√
1− 2γ(1− vl)]2
. (21)
To find the complex roots (20), we put
σ = −Q + iP. (22)
So, in accordance with the notation introduced, we
have
ω =
√
qγ(P + iQ) . (23)
The time dependence is given as e−iωt, therefore at
Q > 0 the instability takes place, and at Q < 0 we
have the stability. From eq. (20), after substitution
of (22) and separation of imaginary and real parts,
we obtain the following equations
4G =
e−2Q[(1 + Q)cos2P + Psin2P ] + Q− 1
Q(Q2 − 3P 2)
,
(24)
4G =
e−2Q[(1 + Q)sin2P + Pcos2P ]− P
P (P 2 − 3Q2)
. (25)
Equations (24), (25) enable us to find P(q,vl) and
Q(q,vl). Their plots are presented in Figs.1 and 2.
The first five solutions of eqs. (24), (25) indicate
that with an increase in q stationary states set
in. Their frequency is proportional in the order of
magnitude to the electronic Langmuir frequency (see
Fig.2).
-2.0
-1.5
-1.0
-0.5
0
0.5
1.0
0.008 0.009 0.010 0.011 0.012 0.013 0.014 0.015
Q
q
Fig.2. Oscillation increments at Vl = 0.8
The range of q values, where Q > 0 is rather narrow.
For example, in Fig.1, at q=0.0115 we have Q=0.5.
Q > 0 case corresponds to the onset of the instability,
and the Q < 0 case corresponds to the damping of
arising oscillations.
The oscillation increments for Q > 0 (see Fig.2)
have their maximum values for a wide frequency
range in the narrow parameter q region in the
neighborhood of zero. The oscillation spectrum,
which can be excited in the diode, presents a set of
discrete frequencies. This permits the excitation of
both the oscillations with dedicated frequency and
a set of oscillations. The oscillation frequencies and
oscillation increments can be found by multiplying
P and Q by the dimension factor v0
l
√
qγ. As it can
be seen in Fig.3, the increment of instability Q takes
on small values (Q ≈ 0.5...0.7), which prevent the
instability from causing the beam breakup during
the transit time.
4
6
8
10
12
14
16
0.00 0.05 0.10 0.15 0.20 0.25 0.30
P
q
Fig.3. Frequencies at Vl = 0.8
The analysis of the dispersion equation has enabled
us to determine the parameter regions (q and Vl),
where the oscillations get excited at the expense of
the instability under study (Fig.4).
89
0.2 0.4 0.6 0.8 1.0
0.1
0.2
0.3
0.4
q
V
l
0
Fig.4. Excitatory regions of the first and second
frequency bands
Curves 1 in Fig.4 define the excitatory region
of the first frequency band (lower curve in Fig.2).
Curves 2 define the region for the second frequency
band. As it can be seen from the figure, the
instability is observed in a rather narrow range of
small q values.
6. EXPERIMENTAL STUDIES OF
INSTABILITY OF ELECTRON-BEAM
STATIONARY STATES IN THE DIODE
The electron beam instability was found and
investigated experimentally at the devices, which
simulated the conditions of the instability onset
defined by theoretical calculations. The electron
flow was investigated in a planar triode electrode
geometry. The experimental arrangement is
presented in Fig.5.
The electron flow produced by the indirectly
heated cathode 1 propagated towards anode 3 and
grid 2. The linear dimensions of the electrodes (grid,
anode) substantially exceeded the interelectrode
spacing. The cathode-grid and grid-anode separation
distances are in the ratio of 1:5.
U
HEAT
U
1
U
2
OSCILLOSCOPE
SPECTRAL ANALYSER
1 2 3
Fig.5. Schematic of the experimental arrangement.
1 - indirectly heated cathode, 2 - grid, 3 - anode
In the experiments, we have measured such
parameters as the anode/grid voltages (U1and U2),
and also the emission current density and the
oscillation spectrum. The amplitude of oscillations
was also measured, and the increments were
estimated.
In the first case,the vacuum chamber, in which
the experiments were performed, was pumped down
to a residual pressure of ∼ 2× 10−6Torr.
In the second case, when the vacuum chamber
was operated, the residual pressure was estimated to
be about ∼ 10−7 Torr. In both cases, the potentials
were applied to the grid and the anode relative to
the grounded cathode.
Based on the measurements of the emission
current for different potential values at the grid
and the anode, we have plotted the voltage-current
characteristics of the grid-anode distance (Fig.6).
0 50 100 150 200 250 300
0
5
10
15
20
25
30
35
I A
(m
A
/c
m
2
)
U
1
(V)
3V
6V
9V
12V
Fig.6. Voltage-current characteristics of the
grid-anode distance for different anode potential
values
The instability was detected and observed only in
the case when the potential applied to the grid was
in excess of the anode potential. This corresponds to
the mode of beam slowing-down in the diode.
2 4 6 8 10 12 14 16
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
lower threshold
upper threshold
U
1
(V
)
U
2
(V)
Fig.7. Upper and lower thresholds of oscillation
excitation
90
The process of excitation had a threshold
character in relation to the grid/anode potential
values. The threshold values of anode/grid potentials
are presented in Fig.7.
The excitation of oscillations takes place in the
region bounded by the curves shown in Fig.7.
The treatment of these and similar curves
provides the comparison between the experimental
data and the theoretical predictions.
The measurements and the analysis of the
experimental data have demonstrated that the
excitation of oscillations takes place in the q and
vl parameter region predicted by theory, this being
confirmed by the data presented in Fig.8.
0.2 0.4 0.6 0.8 1.0
0.1
0.2
0.3
0.4
q
V
l
0
Fig.8. Regions of oscillation excitation and the
points, at which the instability was observed
experimentally
In Fig.8, the points corresponding to the
experimental measurements are plotted against the
background of the theoretical curves bounding the
regions of oscillation excitation. Each point presents
a set of parameters, at which the excitation of
oscillations was observed.
The frequencies were measured with the help
of the spectrum analyzer. In the process of
measurements the frequencies of the first and second
frequency bands were recorded. In some modes of
operation, the excitation of the third frequency band
could also be observed.
For comparison between experimental and
theoretical results, Fig.9 shows the points obtained
from the oscillation frequency measurements at
different grid/anode potentials against the theoretical
boundaries of oscillation P-excitation bands. It
can be seen that with an increase in the anode
potential the ratio of the points falling in the band
indicated by theory to those lying beyond the band
changes in favor of the first ones. At the anode
voltage U2 = 3 V their number makes 50% of the
total number of points, 71.4% and 100% at U2 = 6 V
and U2 = 9 V , respectively. So, it can be stated that
with a further going into the mode of slowing down
(from the reflection mode) the agreement between the
experimental and theoretical results is considerably
improved.
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
0.0 0.2 0.4 0.6 0.8 1.0
upper q
lower q
3V
6V
9V
P
V
l
Fig.9. P excitation band boundaries obtained
theoretically (solid lines) and frequencies observed
experimentally at different anode potentials (points)
The mode shape was investigated with the help
of a high-speed oscillograph. The amplitude of
oscillations was registered at different moments of
time as the instability under study was developing.
As a result of multiple studies of the mode
shape during instability excitation, a number of
increments of growth were obtained for different
stages of instability development. Based on the
results obtained, average increments of growth of the
amplitude of oscillations were estimated at different
q values.
Figure 10 gives the values of increments obtained
experimentally against the curves resulting from the
theoretical study of the instability.
0.040 0.048 0.056 0.064 0.072 0.080
-1.5
-1.0
-0.5
0.0
0.5
1.0
Q
q
Fig.10. Comparison of experimental increments
of instability growth (crosses) with theoretical
predictions for different q at vl = 0.5
7. CONCLUSIONS
The instability of accelerated and slowed down beams
has been investigated in the mode without the
potential minimum. For slowed down beams, the
91
instability has been revealed, which corresponds to
small parameter q values. The instability can be
interpreted as a linear stage of transition from a
quasi-single-particle mode of beam propagation to
the hydrodynamic mode.
Experiment has been performed to detect and
investigate the instability. The experiment has
confirmed the occurrence of the instability in the
region of parameters predicted by theory.
References
1. A.V. Pashchenko, B.N. Rutkevich. The stability of
electron flow in diode // Plasma Physics. 1977, v.3,
p.774.
2. R.B.Miller. Intense charged particle beams.
”Plenum Press”, New York and London: 1982,
432 p.
3. I.I.Magda, V.E.Novikov, A.V. Pashchenko,
S.S.Romanov, I.M. Shapoval. To the theory of
beam feedback in the generators with a virtual
cathode // VANT. 2003, v.6, N46, p.167-170.
4. J. Pierce. Limiting stable current in electron
beams in the presence of ions //J. Appl. Phys.
1944, v.15, p.721.
ВЫСОКОЧАСТОТНЫЕ СВОЙСТВА ЭЛЕКТРОННОГО ПОТОКА,
ЗАМЕДЛЯЮЩЕГОСЯ В ПЛОСКОМ ДИОДЕ
А.В. Пащенко, О.Г. Мележик, С.С. Романов, Д.А. Ситников, И.К. Тарасов,
М.И. Тарасов, И.Н. Шаповал, В.Е. Новиков, В.А. Яцышин
На основе точного решения системы неоднородных гидродинамических уравнений первого прибли-
жения теории устойчивости изучена неустойчивость электронного потока, который инжектирован в
плоский диод и замедляется в нём под действием внешнего электрического поля. Получено дисперси-
онное уравнение, связывающее частоты и инкременты (декременты) возникающих электромагнитных
колебаний с параметрами электронного потока и диода. Решение дисперсионного уравнения показы-
вает, что в диоде, через который распространяется замедляющийся электронный поток, возникает не
описанная ранее неустойчивость колебаний в СВЧ-диапазоне. Эта неустойчивость имеет место, когда в
стационарном состоянии минимум потенциала в диоде не образуется. Экспериментально подтверждено
возникновение неустойчивости колебаний с предсказанными теоретически частотами и декрементами
в соответствующих областях параметров пучка и диода.
ВИСОКОЧАСТОТНI ВЛАСТИВОСТI ЕЛЕКТРОННОГО ПОТОКУ,
СПОВIЛЬНЕНОГО У ПЛОСКОМУ ДIОДI
А.В. Пащенко, О.Г. Мележик, С.С. Романов, Д.А. Ситников, И.К. Тарасов,
М.I. Тарасов, I.М. Шаповал, В.Е. Новiков, В.А. Яцишин
На основi точного рiшення системи неоднорiдних гiдродинамiчних рiвнянь першого наближення теорiї
стiйкостi вивчена нестiйкiсть електронного потоку, iнжектованного у плоский дiод i сповiльненого в
ньому пiд дiєю зовнiшнього електричного поля. Отримано дисперсiйне рiвняння, що пов’язує часто-
ти i iнкременти (декременти) виникаючих електромагнiтних коливань з параметрами електронного
потоку i дiода. Рiшення дисперсiйного рiвняння показує, що в дiодi, крiзь який розповсюджується спо-
вiльнений електронний потiк, виникає не описана ранiше нестiйкiсть коливань у СВЧ-дiапазонi. Ця
нестiйкiсть має мiсце, коли в стацiонарному станi мiнiмум потенцiалу в дiодi не утворюється. Експе-
риментально пiдтверджено виникнення нестiйкостi коливань з передбаченими теоретично частотами i
декрементами у вiдповiдних областях параметрiв пучка i дiода.
92
|