Confinement theorem for high-current plasma lens
The theorem about a fraction of magnetized electrons, which could reach the cylindrical wall, is generalized for the case of collisionless plasma lens for focusing of ion beams. Long electron column is considered. The electrons are partly noncompensated by ions. It is shown from the conservation of...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Cite this: | Confinement theorem for high-current plasma lens / A.A. Goncharov, V.I. Maslov, I.N. Onishchenko // Вопросы атомной науки и техники. — 2000. — № 3. — С. 129-130. — Бібліогр.: 2 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-823992025-02-09T23:13:59Z Confinement theorem for high-current plasma lens Goncharov, A.A. Maslov, V.I. Onishchenko, I.N. Вeams in Plasma The theorem about a fraction of magnetized electrons, which could reach the cylindrical wall, is generalized for the case of collisionless plasma lens for focusing of ion beams. Long electron column is considered. The electrons are partly noncompensated by ions. It is shown from the conservation of angular momentum that if the radius of electron column is small in comparison with the distance from the column to the wall then the only small fraction of electrons from the column could reach the wall. This work was partly supported by the Science and Technology Center of Ukraine (project # 1596). 2000 Article Confinement theorem for high-current plasma lens / A.A. Goncharov, V.I. Maslov, I.N. Onishchenko // Вопросы атомной науки и техники. — 2000. — № 3. — С. 129-130. — Бібліогр.: 2 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/82399 533.9.01 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Вeams in Plasma Вeams in Plasma |
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Вeams in Plasma Вeams in Plasma Goncharov, A.A. Maslov, V.I. Onishchenko, I.N. Confinement theorem for high-current plasma lens Вопросы атомной науки и техники |
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The theorem about a fraction of magnetized electrons, which could reach the cylindrical wall, is generalized for the case of collisionless plasma lens for focusing of ion beams. Long electron column is considered. The electrons are partly noncompensated by ions. It is shown from the conservation of angular momentum that if the radius of electron column is small in comparison with the distance from the column to the wall then the only small fraction of electrons from the column could reach the wall. |
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Goncharov, A.A. Maslov, V.I. Onishchenko, I.N. |
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Goncharov, A.A. Maslov, V.I. Onishchenko, I.N. |
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Goncharov, A.A. |
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Confinement theorem for high-current plasma lens |
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Confinement theorem for high-current plasma lens |
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Confinement theorem for high-current plasma lens |
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Confinement theorem for high-current plasma lens |
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Confinement theorem for high-current plasma lens |
| title_sort |
confinement theorem for high-current plasma lens |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2000 |
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Вeams in Plasma |
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https://nasplib.isofts.kiev.ua/handle/123456789/82399 |
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Confinement theorem for high-current plasma lens / A.A. Goncharov, V.I. Maslov, I.N. Onishchenko // Вопросы атомной науки и техники. — 2000. — № 3. — С. 129-130. — Бібліогр.: 2 назв. — англ. |
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Вопросы атомной науки и техники |
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AT goncharovaa confinementtheoremforhighcurrentplasmalens AT maslovvi confinementtheoremforhighcurrentplasmalens AT onishchenkoin confinementtheoremforhighcurrentplasmalens |
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2025-12-01T15:45:38Z |
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1850321346105442304 |
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Problems of Atomic Science and Technology. 2000. N 3. Series: Plasma Physics (5). p. 129-130 129
UDC 533.9.01
CONFINEMENT THEOREM
FOR HIGH-CURRENT PLASMA LENS
A.A.Goncharov*, V.I.Maslov, I.N.Onishchenko
*Institute of Physics NASU, Kiev;
NSC Kharkov Institute of Physics & Technology, Kharkov, Ukraine
e-mail: vmaslov@kipt.kharkov.ua, fax: 38 0572 351688, tel: 38 0572 356611
The theorem about a fraction of magnetized electrons, which could reach the cylindrical wall, is generalized for
the case of collisionless plasma lens for focusing of ion beams. Long electron column is considered. The
electrons are partly noncompensated by ions. It is shown from the conservation of angular momentum that if the
radius of electron column is small in comparison with the distance from the column to the wall then the only small
fraction of electrons from the column could reach the wall.
Introduction
In [1] the theorem about a fraction of electrons,
which could reach the cylindrical wall in collisionless
case , is presented for the case of purely electron
magnetized plasma. In high-current plasma lens for
focusing of ion beams [2] the dynamics of electrons is
similar to their dynamics in purely electron plasma.
The latter is determined by the fact that the electrons
in high-current plasma lens are partly noncompensated
by ions. But the ions strongly influence on electron
behavior in plasma lens. Therefore we generalize this
theorem on the case of high-current plasma lens.
Electron cloud or long electron column is considered
here. It is trapped by longitudinal magnetic field, Bo, in
the system of finite radial dimension, R. Conservation
of angular momentum leads to estimation for fraction
of electrons which could reach the cylindrical wall in
radial direction. It is shown that if the radius of
electron column ro is small in comparison with the
distance from the column to the wall R then the only
small fraction of electrons ∆NR =ne(R)2πR∆R from
the column could reach the wall. Here ∆R is the
thickness of the hollow cylinder of electrons which
reached the wall. Also it is shown that the fraction of
electrons from the column which could reach the wall
in collisionless case is depended on the difference of
electron and ion densities ne-ni .
Influence of ions on confinement of electrons in
plasma lens
One kind of plasma lens for ion beam focusing
consists of a long electron column. For providing of
good quality focusing the electron column should be
homogeneous in radial direction. For supporting this
required homogeneous state it is important to control
radial electron transport. Note that charged magnetized
plasma of finite radial dimension has good confinement
properties. We do not consider the axial confinement
properties. But we worry about the radial confinement.
We use an approximation of an infinitely long electron
column. The initial number of electrons equals
No=noπro
2. We suppose that they are distributed
homogeneously in radial direction on dimension ro.
The electrons are partly neutralized by ions with
homogeneous density noi. In approximation of
homogeneous radial particle distribution electrons drift
on angle with velocity Vθ≈2πecr(ne-ni)/Bo . Here r is
the distance from the column axis, c is the light
velocity. One can use the conservation of angular
momentum Pθ of electrons and field for estimation of
electron fraction ∆NR/No which could reach the wall.
We consider conditions when ∆NR/No is small. We
neglect by electron collisions with atoms and ions.
Also we neglect by dissipation of electron column due
to electron radiation. We suppose that the hollow
electron cylinder with thickness ∆r , density ne(r∆)
and small number of electrons ∆N=n∆2πr∆∆r has
reached the radius r∆. But the radius of remaining
electron cylinder has been decreased from ro to ra and
their density became nae. Thus the radial electric field
equals
Er=-2πeδnoro
2/r , ro<r<R,
Er=-2πe(noro
2/r-noir , ra<r<ro, (1)
Er=-2πeδnar , r<ra
Here δno=noe-noi, δna=nae-noi.
According to the equation
Vθ
2/r=eEr/me+ Vθωce (2)
one can obtain that the electrons drift on angle with
velocity
Vθ=(ωcer/2)[1-(1-2A/ωce
2)1/2] (3)
Here ωce=eBo/mec is the cyclotron frequency of
electron,
A=ωpo
2ro
2/r2 , ro<r<R,
A=ωpa
2 , r<ra (4)
130
ωpo
2=4πe2δno/me , ωpa
2=4πe2δna/me .
The angular momentum for one electron equals
Pθe = mrVθ + (e/c)rAθ (5)
Here Aθ is the θ component of vector potential. Aθ
is determined by external magnetic field and by
selfconsistent magnetic field or electron current with
velocity Vθ . Initially Aθ equals
Aθt=o=(Bor/2)[1-(r2ωpoo
2/8c2)(1-(1-2ωpo
2/ωce
2)1/2)]
(6)
For final state with two electron cylinder: hollow one
with radius r∆ and solid one with radius ra Aθ equals
Aθ=(Bor/2){1-(r2ωpao
2/8c2)[1-(1-2ωpa
2/ωce
2)1/2]} (7)
Here ωpoo
2=4πe2no/me , ωpao
2=4πe2na/me .
For electrons with density ne , distributed in radial
direction from axis to r , the angular momentum equals
Pθ = 2π∫or dr rne(r)[mrVθ + (e/c)rAθ] (8)
The conservation of Pθ can be written as follow
2π∫oro dr r2noe[mVθ + (e/c)Aθ]=
=2π∫ora dr r2na[mVθ + (e/c)Aθ]+ (9)
+∆Nr∆(mVθ∆ + (e/c)Aθ∆)
From (9) one can derive at ra →0 and r∆→R for ∆NR
(is the number of electrons, which could reach the wall
of the system)
∆NR/No≈ (10)
≈(ro
2/2R2){1+(1-ro
2ωpoo
2/12c2)[1-(1-2ωpo
2/ωce
2)1/2]}
From (10) it follows that the small fraction of
electrons ∆NR/No could reach the wall of the system
for small ro in comparison with R and it depends on
noe-noi .
This work was partly supported by the Science
and Technology Center of Ukraine (project # 1596).
References
1. T.M.O’Neil. A confinement theorem for nonneutral
plasmas // Phys. Fluids (23). 1980, p.2216.
2. A.Goncharov, I.Litovko. Electron Vortexes in
High-Current Plasma Lens // IEEE Trans. Plasma
Sci. (27). 1999, p.1073.
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