Properties of solitary electric potential hump in electron plasma
The properties of collective electric trap for electrons of charged beam are investigated theoretically. This electron beam propagates along magnetic field in cylindrical tube. It is shown that part of beam electrons are trapped in the system.
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| Date: | 2000 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Cite this: | Properties of solitary electric potential hump in electron plasma / V.I. Maslov, V.L. Stomin // Вопросы атомной науки и техники. — 2000. — № 3. — С. 128. — Бібліогр.: 4 назв. — англ. |
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Maslov, V.I. Stomin, V.L. 2015-05-29T08:48:54Z 2015-05-29T08:48:54Z 2000 Properties of solitary electric potential hump in electron plasma / V.I. Maslov, V.L. Stomin // Вопросы атомной науки и техники. — 2000. — № 3. — С. 128. — Бібліогр.: 4 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/82398 533.9.01 The properties of collective electric trap for electrons of charged beam are investigated theoretically. This electron beam propagates along magnetic field in cylindrical tube. It is shown that part of beam electrons are trapped in the system. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Вeams in Plasma Properties of solitary electric potential hump in electron plasma Article published earlier |
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Properties of solitary electric potential hump in electron plasma |
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Properties of solitary electric potential hump in electron plasma Maslov, V.I. Stomin, V.L. Вeams in Plasma |
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Properties of solitary electric potential hump in electron plasma |
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Properties of solitary electric potential hump in electron plasma |
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Properties of solitary electric potential hump in electron plasma |
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Properties of solitary electric potential hump in electron plasma |
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properties of solitary electric potential hump in electron plasma |
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Maslov, V.I. Stomin, V.L. |
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Maslov, V.I. Stomin, V.L. |
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Вeams in Plasma |
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Вeams in Plasma |
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2000 |
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Вопросы атомной науки и техники |
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The properties of collective electric trap for electrons of charged beam are investigated theoretically. This electron beam propagates along magnetic field in cylindrical tube. It is shown that part of beam electrons are trapped in the system.
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1562-6016 |
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Properties of solitary electric potential hump in electron plasma / V.I. Maslov, V.L. Stomin // Вопросы атомной науки и техники. — 2000. — № 3. — С. 128. — Бібліогр.: 4 назв. — англ. |
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AT maslovvi propertiesofsolitaryelectricpotentialhumpinelectronplasma AT stominvl propertiesofsolitaryelectricpotentialhumpinelectronplasma |
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2025-11-26T06:07:18Z |
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2025-11-26T06:07:18Z |
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Problems of Atomic Science and Technology. 2000. N 3. Series: Plasma Physics (5). p. 128 128
UDC 533.9.01
PROPERTIES OF SOLITARY ELECTRIC POTENTIAL HUMP IN
ELECTRON PLASMA
V.I.Maslov, V.L.Stomin
NSC Kharkov Institute of Physics & Technology, Kharkov, Ukraine
e-mail: vmaslov@kipt.kharkov.ua, fax: 38 0572 351688, tel: 38 0572
The properties of collective electric trap for electrons of charged beam are investigated theoretically. This
electron beam propagates along magnetic field in cylindrical tube. It is shown that part of beam electrons are
trapped in the system.
In this paper the properties of collective electric
trap for electrons of charged beam are investigated
theoretically. This electron beam propagates with
velocity Vb along magnetic field in conducting
cylinder. It has been shown that part of the beam’s
electrons may be tarpped by selfconsistently formed
electric potential hump. This hump keeps these
electrons inside the conducting cylinder. The
dependence of width of electric trap on amplitude of its
electric field is investigated.
In magnetized, electron-plasma-filled conducting
tube slow solitary structure may exist [1, 2]. During
time interval of electron beam injection into a
conducting tube an electric potential hump, ϕ(z), is
formed due to the dissipative instability or Pierce
instability development with maximum growth rate
γ≈Vb/L . Here L is the length of the system. This
potential hump can be transformed into a solitary
perturbation on nonlinear stage of instability
development. This potential hump could trap fraction of
beam electrons during the time of potential hump
formation. This paper is concerned with the properties
of this kind potential solitary structure. We consider
the case of the strong external longitudinal magnetic
field, Ho→ ∞ . Then the electron dynamics is one-
dimensional. We choose the initial perturbation in the
form of electric potential hump of small amplitude and
of width ∆z smaller than the system length. In the case
of small soliton amplitude, ϕo (<< Te , here Te is the
electron temperature), from the Vlasov equation one
can obtain the expression for the velocity distribution
function of electrons. Integrating the latter over
velocities, one can derive the expression for the
electron density perturbation in the second order of
φo= eϕo/Te
∂zδn=∂tφ[y+(1-2y2)(1-R(y))/y]+ ∂zφR(y)+
+φ∂zφ[1-y2+(1.5-y2)(R(y)-1)], (1)
R(y) = 1 + (y/√π)∫-∞∞dt exp(-t2)/(t-y),
y=(Vb-Vo)/Vth√2
Here Vo, ϕ are the velocity and potential of the
soliton. Substituting (1) in the Poisson equation, one
can derive the KdV evolution equation
∂tφ[y+(1-2y2)(1-R(y))/y]+ ∂zφR(y)+
+φ∂zφ[1-y2+(1.5-y2)(R(y)-1)]- ∂zzzφ=0 (2)
From (2) one can obtain the equation describing the
space distribution of the potential:
(∂zφ)2 = φ2R(y) - [1 + (2y2 - 3)R(y)]φ3/6 (3)
From (3) and ∂zφ|φ=φo = 0 the expression for R(y)
and Vo follows (similarly to [3])
R(y) ≈ φo/6, Vb-Vo ≈1.32Vth (4)
From (4) one can see that, if beam velocity is close to
1.32Vth , the potential hump is approximately fixed.
We determine roughly the soliton width from (3),
(4):
∆z = (48Te/eϕo)1/2 (5)
The soliton is the "hole" in the electron phase space.
In the case of large amplitudes, eϕo/Te > 1 , from
the Vlasov equation one can have the expression for the
velocity distribution function of plasma electrons
(without electrons trapped by a soliton field) f =
fo[(u2-2eϕ/m)1/2+Vosign(u)] for |u|=|V-Vo|>(2eϕ/m)1/2
. Here fo is the Maxwellian distribution function. Thus
one can derive the equation for the soliton shape
(∂zφ)2 =- φ + (2/√π)1/2 ∫-∞∞dt (t-y)2exp(-t2){[1 + φ/(y-
t)2]1/2 - 1} (6)
From (6) the expression for the soliton width follows
∆z = [2eφo/Te(√2 - 1)]1/2 (7)
From (7) one can conclude that the soliton width
increases with φo . Hence, taking into account the
trapped electrons is important. We assume for their
density distribution the following expression nt r(z) =
ntroexp [eφ(z)/Tt r]. Here Tt r is the effective
temperature of trapped electrons. Using last expression
one can obtain similarly to (7) that the width of soliton
increases wi th amplitude growing.
These properties of soliton and its amplitude
dependences were observed in experiments and
numerical simulations [1].
Similar electron trap has been observed in [4].
References
1. I.P.Lynov et al. // Physica Scripta , (20). 1979, p.328.
2. H.Tanaka et al. // Proc. of 9th Int. Conf. On Plasma
Physics. Toki. 1998.
3. H.Schamel // Plasma Physics, (14). 1972, p.905.
4. I.K.Tarasov // VANT, 1998.
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