Brillouin and kynetic flows in a magnetron diode
Two classes of approaches have received the most attention to describe the important space charge dynamics in a magnetron are Brillouin flow and double-stream kinetic model. Precise analysis of electron dynamics in fully selfconsistent kinetic equilibrium in a smooth-bore magnetron shows that for a...
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Agafonov, A.V. 2015-03-24T15:08:06Z 2015-03-24T15:08:06Z 2004 Brillouin and kynetic flows in a magnetron diode / A.V. Agafonov // Вопросы атомной науки и техники. — 2004. — № 1. — С. 131-133. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 52.35.Mw https://nasplib.isofts.kiev.ua/handle/123456789/78969 Two classes of approaches have received the most attention to describe the important space charge dynamics in a magnetron are Brillouin flow and double-stream kinetic model. Precise analysis of electron dynamics in fully selfconsistent kinetic equilibrium in a smooth-bore magnetron shows that for a given external magnetic field and a voltage there exists a multiplicity of natural equilibrium states, differing as to structure of electron trajectories and emission current density. The value of emission current density differs from one to other type of the equilibrium and can aspire to zero under the same condition of space charge limited flow due to a large number of revolutions of electrons around the cathode. The greater the number of revolutions the closer are the main parameters of the kinetic flow to Brillouin one. Work is supported by RFBR under grant 03-02-17301. У рамках аналітичного підходу показано, що в коаксіальній магнетронній гарматі можливе існування багатозначних стаціонарних станів пучка при заданих значеннях зовнішніх параметрів (геометрія діода, напруга на пушку і зовнішнє магнітне поле), що відрізняються числом обертів електронів навколо катода і струмом, що емітується з катода. Показано можливість граничного переходу від кінетичного потоку до бриллюенівського потоку. Робота виконана за підтримкою гранта РФФД 03-02-17301. В рамках аналитического подхода показано, что в коаксиальной магнетронной пушке возможно существование многозначных стационарных состояний пучка при заданных значениях внешних параметров (геометрия диода, напряжение на пушке и внешнее магнитное поле), отличающихся числом оборотов электронов вокруг катода и током, эмиттируемым с катода. Показана возможность предельного перехода от кинетического потока к потоку бриллюэновскому. Работа выполнена при поддержке гранта РФФИ 03-02-17301. Work is supported by RFBR under grant 03-02-17301. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Динамика пучков Brillouin and kynetic flows in a magnetron diode Бриллюенівський і кінетичний потоки у магнетронній гарматі Бриллюэновский и кинетический потоки в магнетронной пушке Article published earlier |
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| title |
Brillouin and kynetic flows in a magnetron diode |
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Brillouin and kynetic flows in a magnetron diode Agafonov, A.V. Динамика пучков |
| title_short |
Brillouin and kynetic flows in a magnetron diode |
| title_full |
Brillouin and kynetic flows in a magnetron diode |
| title_fullStr |
Brillouin and kynetic flows in a magnetron diode |
| title_full_unstemmed |
Brillouin and kynetic flows in a magnetron diode |
| title_sort |
brillouin and kynetic flows in a magnetron diode |
| author |
Agafonov, A.V. |
| author_facet |
Agafonov, A.V. |
| topic |
Динамика пучков |
| topic_facet |
Динамика пучков |
| publishDate |
2004 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Бриллюенівський і кінетичний потоки у магнетронній гарматі Бриллюэновский и кинетический потоки в магнетронной пушке |
| description |
Two classes of approaches have received the most attention to describe the important space charge dynamics in a
magnetron are Brillouin flow and double-stream kinetic model. Precise analysis of electron dynamics in fully selfconsistent kinetic equilibrium in a smooth-bore magnetron shows that for a given external magnetic field and a
voltage there exists a multiplicity of natural equilibrium states, differing as to structure of electron trajectories and
emission current density. The value of emission current density differs from one to other type of the equilibrium and
can aspire to zero under the same condition of space charge limited flow due to a large number of revolutions of
electrons around the cathode. The greater the number of revolutions the closer are the main parameters of the kinetic
flow to Brillouin one. Work is supported by RFBR under grant 03-02-17301.
У рамках аналітичного підходу показано, що в коаксіальній магнетронній гарматі можливе існування
багатозначних стаціонарних станів пучка при заданих значеннях зовнішніх параметрів (геометрія діода,
напруга на пушку і зовнішнє магнітне поле), що відрізняються числом обертів електронів навколо катода і
струмом, що емітується з катода. Показано можливість граничного переходу від кінетичного потоку до
бриллюенівського потоку. Робота виконана за підтримкою гранта РФФД 03-02-17301.
В рамках аналитического подхода показано, что в коаксиальной магнетронной пушке возможно
существование многозначных стационарных состояний пучка при заданных значениях внешних параметров
(геометрия диода, напряжение на пушке и внешнее магнитное поле), отличающихся числом оборотов
электронов вокруг катода и током, эмиттируемым с катода. Показана возможность предельного перехода от
кинетического потока к потоку бриллюэновскому. Работа выполнена при поддержке гранта РФФИ 03-02-17301.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/78969 |
| citation_txt |
Brillouin and kynetic flows in a magnetron diode / A.V. Agafonov // Вопросы атомной науки и техники. — 2004. — № 1. — С. 131-133. — Бібліогр.: 9 назв. — англ. |
| work_keys_str_mv |
AT agafonovav brillouinandkyneticflowsinamagnetrondiode AT agafonovav brillûenívsʹkiiíkínetičniipotokiumagnetronníigarmatí AT agafonovav brillûénovskiiikinetičeskiipotokivmagnetronnoipuške |
| first_indexed |
2025-11-26T00:09:48Z |
| last_indexed |
2025-11-26T00:09:48Z |
| _version_ |
1850594212804820992 |
| fulltext |
BEAM DYNAMICS
BRILLOUIN AND KYNETIC FLOWS IN A MAGNETRON DIODE
A.V. Agafonov
Lebedev Physical Institute, 53, Leninsky pr., Moscow, V-333, GSP-1, 119991, Russia;
E-mail: agafonov@sci.lebedev.ru
Two classes of approaches have received the most attention to describe the important space charge dynamics in a
magnetron are Brillouin flow and double-stream kinetic model. Precise analysis of electron dynamics in fully self-
consistent kinetic equilibrium in a smooth-bore magnetron shows that for a given external magnetic field and a
voltage there exists a multiplicity of natural equilibrium states, differing as to structure of electron trajectories and
emission current density. The value of emission current density differs from one to other type of the equilibrium and
can aspire to zero under the same condition of space charge limited flow due to a large number of revolutions of
electrons around the cathode. The greater the number of revolutions the closer are the main parameters of the kinetic
flow to Brillouin one. Work is supported by RFBR under grant 03-02-17301.
PACS: 52.35.Mw
1. INTRODUCTION
Despite the great number of works about magnetron
operation, a detailed description of electron dynamics
under strong space charge influence is complicated by
the non-linear nature of a field-particles system. In par-
ticular, there is no satisfactory solution even to the prob-
lem of electron flow formation in a magnetron with a
smooth anode (a coaxial magnetron diode). In the work
presented here it is shown that within the framework of
accepted kinetic descriptions of a coaxial magnetron di-
ode, multiple steady states of electron flow are possible
for a given diode geometry and the same set of external
parameters (applied voltage and external magnetic
field). These states are distinguished by the number of
electron revolutions around the cathode and the current
emitted from the cathode. Direct transition from kinetic
flows to Brillouin flow as a limit is shown. Numerical
simulation helps to investigate the dependence of steady
state properties of electron flow on its history of forma-
tion. Comparison of analytical data and results of nu-
merical simulation is made with the purpose to analyse
the conditions of applicability for existing analytical
models.
2. THEORETICAL MODELS
Usually it is supposed for a magnetron that at the
initial stage a magnetically insulated axially symmetric
rotating electronic flow is formed. As a rule, the de-
scription of the electron flow is based on two models: a
hydrodynamic parapotential model, or a Brillouin flow
[1,2], in which electrons rotate along the circular traject-
ories around the cathode [3,4], in which electrons move
along cycloid trajectories, beginning and coming to the
end on the cathode surface. It is necessary to emphasise
that in the latter case it was always supposed that an
electron makes a single revolution along a cycloid inde-
pendent of the geometry of the diode (plane or cyl-
indrical).
It is easy to show for a plane diode that at the top of
the cycloid trajectory the radial velocity and electro-
magnetic force both are equal to zero. Thus it is possible
to "connect" another descending (that usually is done),
or ascending trajectory, then continuing them symmet-
rically up to the cathode, i.e. the top of trajectories in the
plane diode is a point of solution branching.
But for a coaxial cylindrical diode it was shown
[5,6] that artificial connecting of trajectories is im-
possible: the cylindrical metrics removes degeneration.
And the structure of a kinetic flow differs in that the an-
gular movement of electrons around an axis can signi-
ficantly exceed 2π. The more the number of electron re-
volutions, the greater the time the electron stays in the
diode gap, and there should be less emission current
from the cathode surface. Thus, the steady state of an
electron flow depends on the value of the emission cur-
rent chosen (and an electric field on the cathode surface
equal to zero corresponding to a space charge limited
current; but the value of the emitted current is much less
than the limiting current and, basically, can approach to
zero).
3. BRILLOUIN (PARAPOTENTIAL) MOD-
EL [1]
A system of units in which c =e = me = 1 is hereafter
used. The electronic flow consists of circular trajectories
with radii r filling completely or partially the gap
between the cathode with radius rk and anode with radi-
us ra: rk < r < re, re ≤ ra.
Outer (boundary) parameters - anode voltage (γa –1)
and (kept in a short pulse regime) average value of mag-
netic induction in the diode gap B0 - are connected to
parameters of an electronic flow by the relations
2 2 2
0
2 2 2 2
0 0
ln ,
( )(1 )
ln ,
2 2
eA
e
r
a
a e e e
e
a k a k
a a e e
r dx
r C x x
rr E
r
r r r rr A B r A B
γ γ
=
+ +
= +
− −
= = +
∫
from which in that specific case re = ra (electronic flow
occupies the whole anode-cathode space) is essentially
only the last relation between an average magnetic in-
duction and diode voltage:
2 2 2
0( ) 2 1.a k a ar r B r γ− = −
4. KINETIC (EMISSIONING) MODEL [4]
___________________________________________________________
PROBLEMS OF ATOMIC SIENCE AND TECHNOLOGY. 2004. № 1.
Series: Nuclear Physics Investigations (42), p.131-133.
131
This analytical model, supposing presence of an
emission current from the cathode coming back on the
cathode, is described by a system of equations for γ(r)
and A(r):
0
2 2
0
2 2
1 ( ) ,
1
1( ) ,
1
1, ,
fd dr
r dr dr r A
fd d ArA
dr r dr r A
d dE rA B
dr r dr
γ γ
γ
γ
γ
=
− −
=
− −
= =
where the constant f0 is proportional to density of cath-
ode emission current. On the cathode γ(rk) =1, A(rk) =0,
E(rk) =0, B(rk)= Bk, and the outer boundary of electron
flow is at radius re, at which electrons turn back to the
cathode and the radial pulse of electrons
2 21rp Aγ= − − is equal to zero. In this model anode
voltage and average value of magnetic induction are
connected to parameters of an electronic flow by the re-
lations:
2 2 2 2
0
ln ,
.
2 2
a
a e e e
e
a k a k
a a e e e
rr E
r
r r r rr A B r A B
γ γ= +
− −
= = +
The results of calculations on these models are shown in
Fig. for the same external conditions (voltage and ex-
ternal magnetic flux) and with zero electric field on the
cathode. The Brillouin smooth solution and three kinetic
ones from a set of possible number of layers (n = 1,2,4)
are shown. With increasing number of layers, emission
current from the cathode tends to zero, electrons make
more and more revolutions before coming back to the
cathode, and a kinetic solution gradually approaches the
Brillouin one. In the situation when magnetic field is
much stronger than electric field, the Brillouin solution
Kinetic (n=1,2,4) and Brillouin models of a rotating
beam (E(r) upper, B(r) lower) for the same external
conditions (voltage and magnetic flux)
appears to be the only possible one. Thus the kinetic
solution occurs only if the electric field on the cathode
is not equal to zero. We note that in the inverted mag-
netron diode (with the anode on inner surface) only a
single-layer kinetic solution exists, which points to the
essential influence of the cylindricity on the electron
flow within the diode. In the "plane" approximation for
the kinetic model the criterion for restriction of number
of layers by any value is not presented, and the usual
choice of n=1 is in essence arbitrary.
Defining 2 21p Aγ= − − and introducing the new
variable t the system can be written in the following
form for numerical integration
0
2
0
, , ( ) ,
( ) , ,
d d dr p pE rE f
dt dt dt
fd d d ArA rpB B A p E AB
dt dt r dt r
γ γ
γ
= = =
= = = − +
with initial conditions for t = 0:
, 1, 0, 0, , 0.k kr r E A B B pγ= = = = = =
Using the notations
0 1 2 3 4 5, 1, , , ,x r x x rE x rA x B x pγ= = − = = = =
the final form of the system for numerical integration
can be written as
/
0 5
/ 2 5
1
0
/
2 1 0
/
3 0 4 5
/ 3
4 0 2
0
2
/ 1 2 2 3 0 3 4
5
0
,
,
(1 ) ,
,
,
( / ) .
x x
x xx
x
x x f
x x x x
xx f
x
x x x x x x xx
x
=
=
= +
=
=
+ + −=
Initial step for numerical integration is calculated using
the following expansion in series of a necessary high or-
der obtained by means of MATHEMATICA 2.2:
3 2 2 3
0 0
0 2
2 4 2 2
0
1 2
2 4
0
2 0 2
3 2 2 3
0
3 2
2 4 2 2 6
0 0
4 2 2
2 2 2
0
5
(1 ),
6 20 30
(1 ),
188
(1 ),
40
(1 ),
6 20 20
(1 ),
24 720
(1 ).
2 12
k
k
k k
k
k
k
k k k
k
k
k
k k
k
k
f t B t f tx r
r r
f t B tx
r
f tx f t
r
f B t B t B tx
r
f t f B tx B
r r
f t B tx
r
= + − −
= −
= +
= − +
= + −
= −
5. TEMPORAL EFFECTS AND THEORETI-
CAL MODELS
Discussion of effects observed in a magnetron di-
ode had, as its basic purpose, to show that theoretical
models describing one and the same situation, actually
correspond to various physical conditions. We shall il-
132
lustrate this by an example, comparing analytical results
and numeric simulation realised with a PIC-code KAR-
AT [7].
Modelling of particle emission in the KARAT code
can be realised in two ways: (1) by setting the law for
temporal change of the emitted current, the value of
which can be less than, or more than the limiting cur-
rent, the applied voltage being fixed (this situation cor-
responds to emission from photocathodes or external in-
jection of a beam through the surface of the emitter);
and (2) by setting the law for temporal rising of diode
voltage to some constant value, the emission current be-
ing fixed at a value greatly exceeding the value limited
by the space charge (this situation corresponds to the
thermionic cathode).
Typical distribution functions of particles in these
two cases are different. In the first case a single electron
revolution in a cycloid is realised (symmetric two-peak
distribution of electrons on a pulse pr) without accumu-
lation of charge in an accelerating gap; in the second
case - symmetric distribution of particles on a pulse pr
with electron capture during voltage increase, growth of
number of particles in the diode gap and multiturn dy-
namics of electrons.
6. DISCUSSION
Both theoretical models give about the same physic-
al results. Recall that the equations describing Brillouin
flow can be deduced on the basis of the same approach
used for the kinetic model [8, 9]. Under conditions of
conservation of full particle energy and canonical angu-
lar momentum
( ) ,P r p A constθ θ= + =
the next general set of equations can be derived:
/
2 / / /
2 /
2 / / /
( ) ( ) 0,
( ) ( ) 0,
z
z
v Pr v
v
v Pr v r
r r
θ θ
θ θ
θ
γ γ
γγ γ
+ =
− − =
where the prime denotes d/dr.
For the case of a cathode surface coinciding with a
magnetic flux surface we have Pθ = const and
2 / .zr v constγ =
The constant equals to zero (i.e., vz = const in this case
and we have the same equations as in [9]. Therefore, it
is not surprising that a direct transition exists from the
kinetic flow to a Brillouin one as a limit.
7. CONCLUSION
Precise analysis of electron dynamics in fully self-
consistent kinetic equilibrium in the smooth-bore mag-
netron shows that for a given external magnetic field
and a voltage there exists a multiplicity of natural equi-
librium states, differing as to structure of electron tra-
jectories and emission current density. The value of the
emission current density differs from one to other type
of the equilibrium and can aspire to zero under the same
condition of the space charge limited flow due to the
different number of revolutions of electrons around the
cathode.
REFERENCES
1. A.V.Agafonov, V.S.Voronin, A.N.Lebedev, K.N.
Pazin // Sov. JTF. 1974, v.44, p.1909.
2. J.M.Creedon // J. Appl. Phys. 1977, v.48, p.1070.
3. R.V.Lovelace, Ott E. // Phys. Fluids. 1974, v.17,
p.1263.
4. V.S.Voronin, A.N.Lebedev // Sov. JTP. 1973, v.43,
p.2591.
5. A.V.Agafonov, D.B.Orlov // Proceed. of the 1989
IEEE PAC, Chicago, USA, v.2, p.1397.
6. A.V.Agafonov, V.S.Voronin // Proceed. of the
1997 IEEE PAC, Vancouver, Canada, v.2, p.1302.
7. P.V.Kotetashwily, P.V.Rybak, V.P.Tarakanov. In-
stitute of General Physics: Preprint N 44, Moscow,
Russia, 1991.
8. M.Raiser // Phys. Fluids. 1974, v.20, p.477.
9. A.V.Agafonov // Proceed. of the 1995 IEEE PAC,
Dallas, USA, v.5, p.3272.
БРИЛЛЮЭНОВСКИЙ И КИНЕТИЧЕСКИЙ ПОТОКИ В МАГНЕТРОННОЙ ПУШКЕ
А.В. Агафонов
В рамках аналитического подхода показано, что в коаксиальной магнетронной пушке возможно
существование многозначных стационарных состояний пучка при заданных значениях внешних параметров
(геометрия диода, напряжение на пушке и внешнее магнитное поле), отличающихся числом оборотов
электронов вокруг катода и током, эмиттируемым с катода. Показана возможность предельного перехода от
кинетического потока к потоку бриллюэновскому. Работа выполнена при поддержке гранта РФФИ 03-02-
17301.
БРИЛЛЮЕНІВСЬКИЙ І КІНЕТИЧНИЙ ПОТОКИ У МАГНЕТРОННІЙ ГАРМАТІ
А.В. Агафонов
У рамках аналітичного підходу показано, що в коаксіальній магнетронній гарматі можливе існування
багатозначних стаціонарних станів пучка при заданих значеннях зовнішніх параметрів (геометрія діода,
напруга на пушку і зовнішнє магнітне поле), що відрізняються числом обертів електронів навколо катода і
струмом, що емітується з катода. Показано можливість граничного переходу від кінетичного потоку до
бриллюенівського потоку. Робота виконана за підтримкою гранта РФФД 03-02-17301.
___________________________________________________________
PROBLEMS OF ATOMIC SIENCE AND TECHNOLOGY. 2004. № 1.
Series: Nuclear Physics Investigations (42), p.131-133.
133
Бриллюэновский и кинетический потоки в магнетронной пушке
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