Electrodynamics of helical slowing structure with high-current electron beam
In most theoretical and experimental research, devoted to studying the interactions of beams with hybrid structures, the beam density is assumed small, therefore it does not change a structure of eigenwaves of electrodynamic system the beam interacts with. It is interesting to consider another limit...
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| Дата: | 1999 |
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| Формат: | Стаття |
| Мова: | English |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
1999
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| Назва видання: | Вопросы атомной науки и техники |
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| Цитувати: | Electrodynamics of helical slowing structure with high-current electron beam / V.A. Buts, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 1999. — № 4. — С. 88-90. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-815032015-05-18T03:02:27Z Electrodynamics of helical slowing structure with high-current electron beam Buts, V.A. Tolstoluzhsky, A.P. In most theoretical and experimental research, devoted to studying the interactions of beams with hybrid structures, the beam density is assumed small, therefore it does not change a structure of eigenwaves of electrodynamic system the beam interacts with. It is interesting to consider another limiting case, when the beam density is already so large, that natural oscillations of beam partiсles ( wb ) are more than frequency of electromagnetic waves exited by beam, and when the presence of beam significantly changes the slowing structure electrodynamics. Thus, we have carried out analytical investigation and numerical analysis for dispersion chracteristics and have found increments of instability of nonequilibrium system – annular electron beam in helical slowing structure for a beam of large density, when the frequency of beam natural oscillations is more than frequency of oscillations excited by him. 1999 Article Electrodynamics of helical slowing structure with high-current electron beam / V.A. Buts, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 1999. — № 4. — С. 88-90. — Бібліогр.: 4 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/81503 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
In most theoretical and experimental research, devoted to studying the interactions of beams with hybrid structures, the beam density is assumed small, therefore it does not change a structure of eigenwaves of electrodynamic system the beam interacts with. It is interesting to consider another limiting case, when the beam density is already so large, that natural oscillations of beam partiсles ( wb ) are more than frequency of electromagnetic waves exited by beam, and when the presence of beam significantly changes the slowing structure electrodynamics. Thus, we have carried out analytical
investigation and numerical analysis for dispersion chracteristics and have found increments of instability of nonequilibrium system – annular electron beam in
helical slowing structure for a beam of large density,
when the frequency of beam natural oscillations is more
than frequency of oscillations excited by him. |
| format |
Article |
| author |
Buts, V.A. Tolstoluzhsky, A.P. |
| spellingShingle |
Buts, V.A. Tolstoluzhsky, A.P. Electrodynamics of helical slowing structure with high-current electron beam Вопросы атомной науки и техники |
| author_facet |
Buts, V.A. Tolstoluzhsky, A.P. |
| author_sort |
Buts, V.A. |
| title |
Electrodynamics of helical slowing structure with high-current electron beam |
| title_short |
Electrodynamics of helical slowing structure with high-current electron beam |
| title_full |
Electrodynamics of helical slowing structure with high-current electron beam |
| title_fullStr |
Electrodynamics of helical slowing structure with high-current electron beam |
| title_full_unstemmed |
Electrodynamics of helical slowing structure with high-current electron beam |
| title_sort |
electrodynamics of helical slowing structure with high-current electron beam |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/81503 |
| citation_txt |
Electrodynamics of helical slowing structure with high-current electron beam / V.A. Buts, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 1999. — № 4. — С. 88-90. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT butsva electrodynamicsofhelicalslowingstructurewithhighcurrentelectronbeam AT tolstoluzhskyap electrodynamicsofhelicalslowingstructurewithhighcurrentelectronbeam |
| first_indexed |
2025-07-06T06:28:53Z |
| last_indexed |
2025-07-06T06:28:53Z |
| _version_ |
1836877966679736320 |
| fulltext |
ELECTRODYNAMICS OF HELICAL SLOWING STRUCTURE WITH
HIGH-CURRENT ELECTRON BEAM
V.A. Buts, A.P. Tolstoluzhsky
NSC KIPT, Kharkiv, Ukraine
In most theoretical and experimental research,
devoted to studying the interactions of beams with
hybrid structures, the beam density is assumed small,
therefore it does not change a structure of eigenwaves of
electrodynamic system the beam interacts with. It is
interesting to consider another limiting case, when the
beam density is already so large, that natural oscillations
of beam partiсles (ω b ) are more than frequency of
electromagnetic waves exited by beam, and when the
presence of beam significantly changes the slowing
structure electrodynamics.
STATEMENT OF A PROBLEM
In the present report we shall investigate
theoretically and numerically the dispersion characteris-
tics and find increments of instability of nonequilibrium
systems with beams of large density that consist of
annular electron beam moving along axes of helical
slowing structure, which is immersed in strong external
magnetic field directed parallel to the system axis.
Let us consider a helical waveguide with radius
Rh , with helix period λ h and pitch angle of helixψ
immersed in the strong external magnetic field H Z0 || .
Along axis (axisZ ) the annular electron beam with
current Ibo is moved being localized near the mean
radius R Rb h< in the rather narrow region with
thickness ∆ < < Rh , so the electrons density is constant
in transverse direction and can be described by the
function:
n r I
eV R
R r r Rb
bo
o b
b b( ) ( ) ( )= − + − +
2 2 2π
θ θ
∆
∆ ∆
,
where θ ( )x is the unit Heaviside function, Vo is the
equilibrium beam velocity along the system axis, Ibo is
the beam current, 2π R Sb b∆ = is its cross section.
The electromagnetic fields in the system is described
usually by Maxwell’s equations. Let assume, that the
perturbed quantities varies as f r i k z t( ) exp ( )|| − ω .
THE BOUNDARY CONDITIONS
When solving the electrodynamics problem
about long waves propagation ( λ λ> > h ) we consider a
helix as infinitesimal thickness and perfectly conducting
anisotropy cylinder with standard boundary conditions
for fields (see, for example, [1-3]):
Eh
( ) ;1 0= Eh
( ) ;2 0=
E Er r
( ) ( ) ;1 2= H Hh h
( ) ( )1 2= . (1)
The index (2) and (1) are for fields inside and outside of
helix, accordingly, h and τ are direction along and
perpendicular wires in a plane tangent to helix. The
boundary conditions at thin-wall annular beam consist
of the condition for continuity of tangential components
of fields and presence of jump in the azimuthal
magnetic field due to the beam current:
E Ez zϕ ϕ,
( )
,
( ) ;1 2= H Hz z
( ) ( ) ;1 2= (2)
H H iR
c k V
E Rb
b
o
z bϕ ϕ
ω ω
ω
( ) ( )
||( )
( )3 2
2
2
− =
−
, (3)
where: ω πb boe n m2 24= / is the plasma frequency
of beam, n Vbo o, is the equilibrium density and beam
velocity, respectively, mo is the mass of electron, c is
the velocity of light. Since we assume, that the beam is
rather thin, and in the system the oscillations with a
wavelength λ > > ∆ are propagated, it is possible do
not take into account a beam stratification in transverse
direction. The value of the field acting on electrons can
be taken in a point equal to the mean beam radius.
THE DISPERSION EQUATION
Producing matching of the fields according to
boundary conditions (1-3) we obtain the dispersion
equation of the thin annular beam-helix system [4]:
D D M R F Fh b o h h b= ( )κ , (4)
where:
D k Fh o h= −2 2κ , D M R Fb o b b= −1 ( ) ,κ
F
k v
R I Rb
b
o
b o b=
−
ω
ω
κ κ
2
2
2 2
( )
( ) ( ),
||
F tg I R K R
I R K Rh
o h o h
h h
= 2
1 1
Ψ ( ) ( )
( ) ( )
,κ κ
κ κ
and κ = −k ko||
2 2 , M x K x
I xo
o
o
( )
( )
( )
= , k
co = ω
.
In this case the equation k Fo h
2 2 0− =κ is the
dispersion equation of the «cold» vacuum helix (without
electron beam), and the equation
1 0− =M R Fo b b( )κ is the dispersion equation of
thin annular electron beam in vacuum. When right-hand
side of Eq. (4) is small the dispersion equation is
decomposed, naturally, to two independent equations
for eigenwaves of helix and beam:
D Dh b= =0 0, . (5)
From Eq. (5) follow relations for phase velocities
of helical and beam modes:
β β1 2, ,= ± h β
β β
3 4 2
2
21
1
1
, ,=
+
± +
−o
b
o
bΩ Ω
(6)
where: Ω b b G2 2= ω is the reduced beam density,
G R c I R K Rb o b o b= ( / ) ( ) ( )2 2 κ κ is the beam de-
pression coefficient characterizing its space-charge,
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 1999. № 4.
Серия: Ядерно-физические исследования (35), с. 88-90.
88
β h h hF F= +( / ( )) /1 1 2 is the phase velocity of the
helix without beam, β o oV c= / .
From Eq.(6) follows, that the beam density
influence becomes significant at Ω b o> β . This
relation allows to find the value of beam density, in
dependence on the geometry and its velocity, since
which the beam influence becomes dominant. In
dimensionless variables β ω β β= / , ,||k c o h the
equation (4) will be:
( )[( ) ( )]β β β β ω β µ ω2 2 2 2 2 21− − − − =h o b bG (7)
Here µ β β κ κ= −G M R M Rh o h o b( ) ( ) / ( )1 2 2 2 is
the coefficient of beam-wave coupling. If the beam
interacts with a forward wave of helix (propagated
along the beam), then the coupling coefficient in a right-
hand side of (7) is positive, and is negative in the case
of interaction with a backward wave.
In general case the analysis of dispersion
equation (7) can be carried out numerically, First of all,
we will be interested in cases of slow waves
propagation ( β 2 1< < , so k ko|| > > ; k || ≈ κ ). Thus,
the equation (7) becomes much easier – from
transcendental it turns into algebraic with respect to β :
( )[( ) ]β β β β α β2 2 2 2 2 2− − − =h o b h bΩ Ω (8)
where:α = M k R M k Ro h o b( ) / ( )|| || (magnitude α < 1 ).
For further analysis it will be convenient to use
the frame connected with beam β β δ= +o ,
β βh o= + ∆ and equation (8) can be written as:
β α
β δ δ
δh
o b
2 2
22
1
( )( )+ + −
= −
∆ ∆ Ω
(9)
Right-hand side of this equation corresponding to fast
and slow beam waves is the usual parabola. The left-
hand side has poles of first order in points δ = ∆ and
δ β= − −∆ 2 o are corresponding to forward and
backward wave of helix, and maximum in the point
δ β= − o which value is equal − α .
In order to understand in what conditions the
instability disappears, it is useful to plot the left-hand
and right-hand sides of equation (9) on the graph
(Fig. 1). It is easy to see that the solution of this
equation always contains two real roots in the region of
positive and negative values δ :
min ( ( ), )− + − >β β δs o bΩ
max ( ( ), )β β δs o b− <Ω
Whereas abs( )α < 1 , at values of the beam density
larger then some critical value ( n nb b> ∗ )
Ω b h
2 2 1> −β α/ ( ) and all roots of the equation (8)
become real.
Fig. 1. Plot of the right-hand side (dashed line) and left-
hand side (solid line) of the beam-loaded dispersion
equation in the frame of beam.
Under small beam densities β 2 2> > Ω b we
search a solution of the equation (8) near to intersection
of beam and helical modes β β δ= +0 , β βh= 0
and thus obtain:
δ δ α β( ) ( )2 2 21 2− =Ω Ωb h b (10)
When δ 2 2> > Ω b we find the ordinary cubic
increment of beam instability:
Im ( )( )/ / /δ α β= i h b3 24 3 1 3 2 3Ω . (11)
If the beam density grows ( Ω b
2 2> δ ) so, that
the beam influence on waves propagation in the system
becomes significant, the Cherenkov’s instability
disappears: δ α β= − 1 2/ h .
From the view point of physics one can explain it
in such a way. When the beam density increases the
splitting of its dispersion curve to fast and slow beam
modes becomes so large, that the dispersion curves of
helix and beam do not intersect.
Now the instability is possible in the case, when
the beam velocity Vo is more than wave phase velocity
in the system β βo> . Really, supposing that
conditions β β δ= +h , β βh o b= − Ω are fulfilled,
we can note, that the second condition is the condition
of anomalous Doppler radiation. Then from (8) we find
a quadratic increment (typical for instability on
anomalous Doppler): Im ( ) /δ α β= i h b2
1 2Ω .
RESULTS OF NUMERICAL ANALYSIS
For numerical analysis of this equation the
following parameters were selected: β h = 01. ,
k Rb|| = 3 , and ratio R Rh b/ .= 11 . The numerical
solution’s results of the equation (7) one can see in
fig.2-3 as a dependence of normalized phase velocity (
V Vph h h/ (Re ) /= β β ) and increment (
γ β β= (Im ) / h ) on the beam density ν b (
ν ω βb b b hR c≡ 2 2 2 2/ ) for various values of
detuning between the beam velocity and wave phase
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 1999. № 4.
Серия: Ядерно-физические исследования (35), с. 88-90.
88
velocity in the system ξ ( ξ β β= o h/ ) for ξ = 10. ,
ξ = 11. , ξ = 12. .
Fig. 2. Normalized phase velocity versus beam density
for the varies value of detuning ξ : (a) – ξ =1.0, (b) – ξ
=1.1, (c) – ξ =1.2.
Fig. 3. Normalized increment versus beam density for
various values of detuning ξ : (a) – ξ =1.0, (b) – ξ =1.1,
(c) – ξ =1.2.
From these plots one can see the following most
important regularities for influence of the parameter ξ
and the beam density on Vph and γ change:
For the given detuning ξ there is some beam
density bounded range space at which the beam
instability develops and wave excitation by beam takes
place.
With growth of the beam density at some beam
density values there is a maximum increment γ γ= max
. At the further growth of beam density the value γ
decreases and at ν νb b
2 2= ∗ the instability disappears,
i.e. at these values ν νb b
2 2> ∗ there are no more dege-
neration in the system – phase velocities of two waves
which were equal earlier, become various now.
With growth of the detuning parameter ξ the
maximum values of increment and the values of beam
density at which this maximum can be reached increase.
Significant changing of phase velocities of waves
propagated along the beam takes the place. Two waves
which propagate along the beam are essentially slowing
(in comparison with a velocity of wave in the helical
waveguide without a beam), third is the fast.
The numerical analysis also shows that with
growth of the velocity detuning ξ the beam influence
on the phase velocity of backward wave is decreased.
But even at ξ ~ 1 the beam influence on the backward
wave phase velocity is relatively small and is distinct
only at rather large values of beam density. This result
is in a qualitative agreement with analytical
investigations carried out above.
CONCLUSION
Thus, we have carried out analytical
investigation and numerical analysis for dispersion
chracteristics and have found increments of instability
of nonequilibrium system – annular electron beam in
helical slowing structure for a beam of large density,
when the frequency of beam natural oscillations is more
than frequency of oscillations excited by him. Values of
beam density in dependence on geometry and of beam
velocity at which the beam influence on dispersion is
dominant, were determined analytically and
numerically. It is shown, that the density growth leads
not only to significant changes in dispersion properties
of the system, but also to modification of the
mechanism for generation of oscillations in the system -
from Cherenkov’s instability to instability on anomalous
Doppler effect; the further beam density growth leads to
the instability failure.
REFERENCES
1. A.I. Akhiezer, Ya.B. Fainberg, Slow electromagnetic
waves. // Usp.Fiz.Nauk.-1951.-V.44.-N3.-pp.321-368.
2. R.A.Silin, V.P.Sazonov, Slowing systems. //
M.:Sov.radio.-1966. 523p.
3. B.M.Bulgakov, V.P.Shestopalov, L.A.Shyshkin et al.
Slow waves in helical waveguide with plasma. //
Zh.Tekh.Fiz.-1960. -V.30. N7.-pp.840-850.
4. A.K.Berezin, V.A.Buts, I.K.Koval’chuk, V.I.Kurilko,
I.N.Onishchenko, Ya.B.Fainberg, A.P.Tolstoluzhsky
Electrodynamics of Helical-Plasma Structure. – Preprint
Kharkov: KIPT 1991 31p.
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 1999. № 4.
Серия: Ядерно-физические исследования (35), с. 88-90.
88
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