Properties and excitation of solitary perturbations by electron beam in accelerator
The excitation of a solitary wave perturbation of electric potential hump type with a large amplitude of an electron beam at the accelerator has been considered. Its properties and dependencies of properties on the amplitude have been investigated. This perturbation propagates in the rest frame of t...
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| Veröffentlicht in: | Вопросы атомной науки и техники |
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| Datum: | 2001 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Zitieren: | Properties and excitation of solitary perturbations by electron beam in accelerator / V.I. Lapshin, V.I. Maslov, I.N. Onishchenko, V.L. Stomin // Вопросы атомной науки и техники. — 2001. — № 3. — С. 150-151. — Бібліогр.: 2 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859557079924080640 |
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| author | Lapshin, V.I. Maslov, V.I. Onishchenko, I.N. Stomin, V.L. |
| author_facet | Lapshin, V.I. Maslov, V.I. Onishchenko, I.N. Stomin, V.L. |
| citation_txt | Properties and excitation of solitary perturbations by electron beam in accelerator / V.I. Lapshin, V.I. Maslov, I.N. Onishchenko, V.L. Stomin // Вопросы атомной науки и техники. — 2001. — № 3. — С. 150-151. — Бібліогр.: 2 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The excitation of a solitary wave perturbation of electric potential hump type with a large amplitude of an electron beam at the accelerator has been considered. Its properties and dependencies of properties on the amplitude have been investigated. This perturbation propagates in the rest frame of the beam with a velocity, approximately equal to the thermal velocity of beam electrons. The perturbation forms hole and vortex in the electron phase space. The solitary perturbation is excited due to nonlocal interaction of the beam with a metallic wall of final conductivity. This hump of electric potential is the BGK perturbation.
|
| first_indexed | 2025-11-26T14:04:44Z |
| format | Article |
| fulltext |
PROPERTIES AND EXCITATION OF SOLITARY PERTURBATIONS BY
ELECTRON BEAM IN ACCELERATOR
V.I. Lapshin, V.I. Maslov, I.N. Onishchenko, V.L. Stomin
NSC Kharkov Institute of Physics & Technology, 61108 Kharkov, Ukraine
E-mail: vmaslov@kipt.kharkov.ua
The excitation of a solitary wave perturbation of electric potential hump type with a large amplitude of an electron
beam at the accelerator has been considered. Its properties and dependencies of properties on the amplitude have
been investigated. This perturbation propagates in the rest frame of the beam with a velocity, approximately equal
to the thermal velocity of beam electrons. The perturbation forms hole and vortex in the electron phase space. The
solitary perturbation is excited due to nonlocal interaction of the beam with a metallic wall of final conductivity.
This hump of electric potential is the BGK perturbation.
PACS numbers: 29.17.+w
1 INTRODUCTION
If the wall of the accelerator has a finite conductivi-
ty, then the penetration of electron beam field, propagat-
ing along the wall, into the wall is possible and the exci-
tation of volume charge perturbation, δq, is also possi-
ble in the wall. It means that the interaction of the elec-
tron beam with the wall is possible. We consider the
possibility of non-linear perturbation excitation in the
electron beam due to this interaction with the wall.
2 PROPERTIES OF A SOLITARY HUMP-
FIELD
The perturbation of volume charge of the wall is de-
scribed by the following equation
∂tδq = σ∂zE. (1)
Here σ is the conductivity of the wall. The Poisson
equation for the longitudinal component of electric field
in a long- wave approximation is as follows
∂zE = -4π(eδnb + Rδq). (2)
Here δnb is the perturbation of the beam density, R is
the small parameter or geometrical factor, taking into
account the skin depth and radial spatial distance be-
tween the beam and conducting wall. From equations of
a continuity and motion for beam electrons (1), (2) one
can obtain a dispersion relation for high-frequency per-
turbations with the frequency ω and wave vector k:
1 - ωb
2/(ω-kVb)2 - i4πσR/ω = 0. (3)
Here ωb is the plasma frequency of the electron beam,
and Vb is its velocity. From (3) it follows that the maxi-
mum growth rate of amplitude γL ≈ (πσRωb)1/2 has per-
turbations with the wave vector k ≈ ωb/Vb.
With amplitude growth excited perturbations be-
come non-linear. We consider a limiting case, when per-
turbations are transformed in chain of short perturba-
tions. We investigate the excitation of such short soli-
tary perturbation at a non-adiabatic stage of its evolu-
tion. In other words, the approach is considered, when
during the time of perturbation excitation the beam elec-
trons are shifted relatively to perturbation on a distance,
not exceeding its width.
Fig. 1.
We consider the solitary electric potential hump,
propagating with velocity Vs along z in conductive
waveguide with a strong magnetic field and with high-
energy electron beam. At first we describe the solitary
perturbation of electric potential ϕ of a small amplitude,
ϕo. From Maxwell equations one can derive the equa-
tion for the electric field E
∆E-(Vs∇)2E/c2+4πe[∇n-(Vs∇)nv/c2]=0. (4)
Here E= -∂zϕ; n, v are the electron density and velocity.
For latter determining we use the kinetic equation for
the electron distribution function fe
∂fe/∂t+(v∇)fe-(e/me)(E+[v,B]/c)∂fe/∂v=0. (5)
As we consider a strong magnetic field Ho→∞, directed
along z-axis, then the electrons propagate along this ax-
is. Because, at first, we derive the solution in the form
of a stationary soliton, propagating with a velocity Vs,
we use dependence of fe on the coordinate and time ξ
=z-Vst. In this case from the Vlasov equation for high-
energy electrons (5) one can obtain expressions for a
high-energy electron density perturbation δn=n-no
δnh=noh[eϕ/Th-(4/3√π)(eϕ/Th)3/2+(eϕ/Th)2/2]. (6)
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №3.
Серия: Ядерно-физические исследования (38), с. 150-151.
150
Here Th, is the temperature of high-energy electrons.
Using (6), from (4) we derive the nonlinear equation
for small amplitudes
(ϕ’)2=ϕ2(k⊥
2-ωp
2/Vs
2+r-2
d)-(16/15)r-2
d(e/πTh)1/2ϕ5/2, (7)
r2
d=Th/4πe2noh.
Using ϕ’|ϕ=ϕo=0, one can derive the equation for the
velocity of solitary perturbation of electric potential
hump type
k⊥
2γ2
o-ωp
2/Vs
2γ3
o+ r-2
d=(16/15)r-2
d(e/πTh)1/2ϕ1/2
o. (8)
Using (8), one can solve the equation (7) as follows
ϕ=ϕo/ch4[ξ/∆ξ], ∆ξ=4rd(Th/eϕo)1/4 , (9)
where ξ is the coordinate along the direction of pertur-
bation propagation.
At large amplitudes the shape of the solitary electric
potential hump is determined by
(ϕ’)2=k⊥
2γo
2ϕ2+ (10)
+8πeno{-ϕ+(mVsc/e)[[(γo+eϕ/mc2)2-1]1/2-(γo
2-1)1/2]}+
+8πenoh {-T/e+2(Tϕ/eπ)1/2(1+Ttr/T)+
+exp(eϕ/T)[1-(2/√π)∫ o√(eϕ/T) dx exp(-x2)]T/e-
-(2Ttr/e√π)exp(-eϕ/Ttr)∫ o√(eϕ/T) dx exp(x2T/Ttr)}.
3 EXCITATION OF NONLINEAR PERTUR-
BATIONS IN AN ELECTRON BEAM DUE
TO DISSIPATIVE INSTABILITY DEVELOP-
MENT AT INTERACTION OF BEAM AND
WALL WITH A FINITE CONDUCTIVITY
Taking into account the nonstationary terms in a ki-
netic equation for beam electrons one can obtain, for
correction to density of beam electrons, δnbt, proportion-
al to ∂tφo, the expression
∂zδnbt ≈ ∂tφ[1/z - z + (2z2 - 1)φo/6z]Vth√2. (11)
Here φ=eϕ/Th, Th is the electron beam temperature, φo is
the amplitude of perturbation; Vth is the thermal velocity
of beam electrons, z=(Vs-Vb)/Vth√2. Similarly [1] one
can found that the velocity of the solitary perturbation is
approximately equal to Vs ≈ Vb - 1.32Vth.
Taking into account the electron beam interaction
with perturbation of the charge density in the wall from
(2) one can derive
e δnbt ≈ -Rδq. (12)
From (11), (12) the equation follows, describing the ex-
citation of perturbation. In case of not so small ampli-
tudes it has a form
∂τφo∂τφ ≈ 8√2∂3
yφ. (13)
Here y = z/rb , rb = (T/4πnbe2)1/2 , τ = tγL . The solution of
equation (13) we search as
φ(y, τ) = φo(τ)µ[y - ∫ τ
-∞dτoδVs(φo(τo))]. (14)
δVs is the change of a soliton velocity, appeared as a re-
sult of its interaction with environment, µ(y) is the Spa-
tial distribution of a soliton potential, described by
equation (7). From (7), (13), (14) one can obtain the fol-
lowing expression for the growth rate of excitation of
the solitary perturbation
γ ≈ (γL/3)φo
1/4 [2/(√3-1)]1/2. (15)
Thus, the possibility of non-linear perturbation for-
mation in an electron beam as a result of its interaction
with a wall of a final conductivity is under considera-
tion. The formation of perturbations in the electron
beam was observed in [2].
REFERENCES
1. H.Schamel. Electron holes, ion holes and double
layers // Phys. Rep. 1986. V. 140. N 3. P.163-191.
2. L.K.Spentzouris, P.L.Colestock, F.Ostiguy // Proc.
of the PAC’95. 1995.
151
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| id | nasplib_isofts_kiev_ua-123456789-79263 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-26T14:04:44Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Lapshin, V.I. Maslov, V.I. Onishchenko, I.N. Stomin, V.L. 2015-03-30T08:24:32Z 2015-03-30T08:24:32Z 2001 Properties and excitation of solitary perturbations by electron beam in accelerator / V.I. Lapshin, V.I. Maslov, I.N. Onishchenko, V.L. Stomin // Вопросы атомной науки и техники. — 2001. — № 3. — С. 150-151. — Бібліогр.: 2 назв. — англ. 1562-6016 PACS numbers: 29.17.+w https://nasplib.isofts.kiev.ua/handle/123456789/79263 The excitation of a solitary wave perturbation of electric potential hump type with a large amplitude of an electron beam at the accelerator has been considered. Its properties and dependencies of properties on the amplitude have been investigated. This perturbation propagates in the rest frame of the beam with a velocity, approximately equal to the thermal velocity of beam electrons. The perturbation forms hole and vortex in the electron phase space. The solitary perturbation is excited due to nonlocal interaction of the beam with a metallic wall of final conductivity. This hump of electric potential is the BGK perturbation. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Properties and excitation of solitary perturbations by electron beam in accelerator Свойства и возбуждение одиночных возмущений электронным пучком в ускорителе Article published earlier |
| spellingShingle | Properties and excitation of solitary perturbations by electron beam in accelerator Lapshin, V.I. Maslov, V.I. Onishchenko, I.N. Stomin, V.L. |
| title | Properties and excitation of solitary perturbations by electron beam in accelerator |
| title_alt | Свойства и возбуждение одиночных возмущений электронным пучком в ускорителе |
| title_full | Properties and excitation of solitary perturbations by electron beam in accelerator |
| title_fullStr | Properties and excitation of solitary perturbations by electron beam in accelerator |
| title_full_unstemmed | Properties and excitation of solitary perturbations by electron beam in accelerator |
| title_short | Properties and excitation of solitary perturbations by electron beam in accelerator |
| title_sort | properties and excitation of solitary perturbations by electron beam in accelerator |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79263 |
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