Current-carrying and electrostatic plasma-electron lenses controlled by the external programmed magnetic field
In this work the studies of long uniform, and non-uniform modified adiabatic plasma lenses of different types are performed that intended for middle and high energy ion beam focusing.
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irk-123456789-824952015-06-02T03:01:45Z Current-carrying and electrostatic plasma-electron lenses controlled by the external programmed magnetic field Ivanov, B.I. Butenko, V.I. Beams in plasma In this work the studies of long uniform, and non-uniform modified adiabatic plasma lenses of different types are performed that intended for middle and high energy ion beam focusing. 2000 Article Current-carrying and electrostatic plasma-electron lenses controlled by the external programmed magnetic field / B.I. Ivanov, V.I. Butenko // Вопросы атомной науки и техники. — 2000. — № 3. — С. 109-111. — Бібліогр.: 16 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/82495 533.9 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Beams in plasma Beams in plasma |
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Beams in plasma Beams in plasma Ivanov, B.I. Butenko, V.I. Current-carrying and electrostatic plasma-electron lenses controlled by the external programmed magnetic field Вопросы атомной науки и техники |
| description |
In this work the studies of long uniform, and non-uniform modified adiabatic plasma lenses of different types are performed that intended for middle and high energy ion beam focusing. |
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Article |
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Ivanov, B.I. Butenko, V.I. |
| author_facet |
Ivanov, B.I. Butenko, V.I. |
| author_sort |
Ivanov, B.I. |
| title |
Current-carrying and electrostatic plasma-electron lenses controlled by the external programmed magnetic field |
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Current-carrying and electrostatic plasma-electron lenses controlled by the external programmed magnetic field |
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Current-carrying and electrostatic plasma-electron lenses controlled by the external programmed magnetic field |
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Current-carrying and electrostatic plasma-electron lenses controlled by the external programmed magnetic field |
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Current-carrying and electrostatic plasma-electron lenses controlled by the external programmed magnetic field |
| title_sort |
current-carrying and electrostatic plasma-electron lenses controlled by the external programmed magnetic field |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2000 |
| topic_facet |
Beams in plasma |
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http://dspace.nbuv.gov.ua/handle/123456789/82495 |
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Current-carrying and electrostatic plasma-electron lenses controlled by the external programmed magnetic field / B.I. Ivanov, V.I. Butenko // Вопросы атомной науки и техники. — 2000. — № 3. — С. 109-111. — Бібліогр.: 16 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT ivanovbi currentcarryingandelectrostaticplasmaelectronlensescontrolledbytheexternalprogrammedmagneticfield AT butenkovi currentcarryingandelectrostaticplasmaelectronlensescontrolledbytheexternalprogrammedmagneticfield |
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2025-07-06T09:04:20Z |
| last_indexed |
2025-07-06T09:04:20Z |
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1836887742796005376 |
| fulltext |
Problems of Atomic Science and Technology. 2000. N 3. Series: Plasma Physics (5). p. 109-111 109
UDC 533.9
CURRENT-CARRYING AND ELECTROSTATIC PLASMA-ELECTRON
LENSES CONTROLLED BY THE EXTERNAL PROGRAMMED
MAGNETIC FIELD
B.I. Ivanov, V.I. Butenko
National Science Center “Kharkov Institute of Physics and Technology”,
Kharkov, 61108, Ukraine (E-mail: ivanovbi@kipt.kharkov.ua)
1. Introduction
At present there is a significant requirement for the
development of advanced devices for focusing of
intense ion beams of middle and high energies for actual
scientific and technological problems (inertial
thermonuclear fusion on heavy and light ions,
radiotherapy, high energy investigations, research of
radiation resistance of materials, implantation
metallurgy, etc.). Last time the attention was paid to
elaboration of plasma lenses with large focusing force
and charge compensation of the focused beams (e.g.,
see the review [1]). Under conditions of energy and
current growth of accelerated beams, plasma lenses
should replace the conventional ones. Several types of
these lenses are known [1]: Gabor electron lens,
Morozov electrostatic plasma lens, current-carrying
(magnetic) plasma lens. Theoretical and experimental
studies of the electrostatic plasma-electron lenses of
Gabor-Morozov type were published in many works
(e.g., see [2-6]. For high-energy ion beam focusing the
magnetic plasma lenses were investigated (e.g., [7-11]).
For focusing of ultra-high energy electron beams it was
proposed and studied «passive» plasma lenses based on
the magnetic self-focusing and wake-field focusing
(e.g., [12-14]). Earlier, the mechanism of charge
particles focusing by the fields that they radiate in a
plasma was studied theoretically by V. B. Krasovitskii
e.a. and experimentally by I.A. Soloshenko e.a.
In this work the studies of long uniform, and non-
uniform modified adiabatic plasma lenses of different
types are performed that intended for middle and high
energy ion beam focusing. The advantages of these
lenses are as follows. 1) One can minimize the spherical
aberrations by the condition that the focusing fields
forces are proportional to the focused particle deviation
from the axis through the most part of the lens. 2) The
force of the long plasma lens is greater on 1-2 orders
then of the short one with the same parameters. 3) For
the further enhancement of focusing efficiency of the
non-uniform lenses, one can use the external magnetic
field that changes along the device length so that the
focusing channel radius was close to the focused beam
radius, and it is decreased with beam radius decreasing.
(At this consideration it is used the fact that focusing
channel radius is inversely proportional to the square
root of magnetic field strength. It should be noted that in
the works [10, 13] the criterion of adiabatic property of
plasma lenses was small changing of its parameters over
focusing length (slight non-uniformity). In this work the
another case is studied when that criterion is small
changing of the adiabatic lens parameters on much
smaller «cyclotron length» which is equal to the product
of electron cyclotron frequency by longitudinal velocity
of plasma electrons (the condition of the drift
approximation). For this case plasma parameters change
essentially on the focusing length (strong non-
uniformity).
The common approach for the problem studying was
developed, and it is described in the work for several
kinds of such lenses. Under the term «lens», the long
focusing channel will be understanded in this paper (i.e.,
«plasma focuser», as this term was used in [13]). In the
Part 2 the long non-uniform magnetic (current-carrying)
lenses are studied. The combined electrostatic-magnetic
(charge-current) lens is described in the Part 3. The
uniform and non-uniform long plasmaoptic focusing
devices of Morozov type were investigated in the
separate work [15].
2. Long non-uniform magnetic lens
Let us consider the problem of ion beam focusing by
the azimuthal magnetic field of the longitudinal current
in a plasma. We investigate the case that the current
radius is determined by the external non-uniform
longitudinal magnetic field. The problem is being
solved at the paraxial approximation. In this case the
equation of the magnetic surfaces is as follows:
( ) ( ) ( )
( )a z
a B
B z
z
z
2
2 0 0
= , (1)
where a(z) is the variable radius of the magnetic surface,
Bz(z) is the longitudinal magnetic field on the axis, Bz(0)
and a(0) are determined by the boundary conditions at z
= 0. We assume that in the case of the strong magnetic
field the electrons which transport the current in plasma
are moving along the cylindrical magnetic surfaces
enclosed one into another. The boundary conditions are
defined as it follows: at z=0, a(0) = b, where b is the
radius of an electrode that supply the current in the
plasma (e.g., it is the inner electrode of the plasma gun).
From Eq.(1) it is follows: if the equidistantness of the
magnetic surfaces is set in some cross-section, so it
conserves in any other one. As a result, if the current
density is homogeneous in the electron emitter region,
so it will be homogeneous in any other current channel
cross-section. It is necessary for focusing without
spherical aberration, because Lorentz force Fm focusing
an ion toward the axis is proportional to distance of the
ion from the axis:
( )F
e
c
vB
ev
c
I
cr
evI
c a z
rm = − = − = −ϕ
2 2
2 2
(2)
110
As a result, the equation for the focused ion trajec-
tories will take the form:
( )
( )′′ + =r k
B z
B
rz
z
2
0
0 , k
I e
Mc vb
2
2 2
2
= (3)
In Eqs.(2), (3), I is the current in plasma, e and M
are the charge and mass of the ion (i.e., the proton), c is
the light velocity, v is the ion velocity, Bz(0) is the
magnetic field intensity in the region of the plasma gun
output, Bϕ is the azimuthal magnetic field of the current.
Under condition Bz(z) = const (or 0) from Eq.(3) we
have: r r kz= 0 cos , and the focusing distance in the
plasma: L kf = π / 2 . For a lens of length l< L f :
)(klctgklL f
1−+= ,
whence at kl <<1 it is easy to receive the expression for
a thin lens: Lf=(k2l)-1.
In general case, the trajectories of focused particles
are calculated with help of a computer . For some cases
the Eq.(3) have analytic solution, e.g., for the «bell-
liked» distribution of the magnetic field :
( ) ( ) ( )[ ]B z B z dz z= +
−
0 1 2 2
/ (4)
In this case the Eq.(3) takes the form:
( )[ ]′′ + + =
−
r k r z d2 2 2
1 0/ (5)
The solution of the Eq.(5), that is known from the
electron optics, can be written as it follows:
( )
( )dz
dzdk
dk
r
r
arcctgsin
arcctgsin 22
22
0 1
1
+
+
= (6)
The coordinate of the ion beam focus corresponds to
the condition r=0, and is defined by the expression:
221 dk
dz
f
+
π= ctg (7)
The calculations based on the above formulae show:
due to compression of the current channel by the
external magnetic field, the one order decrease of the
focusing current can be reached. In this case, the
focusing of intense proton beams (of MeV range
energy) can be realized on the distance about 100 cm, in
the steady state regime.
During the ion focusing and compression of the
current channel by the magnetic field of a solenoid,
some ions (with large injection radius) can move partly
out of the current channel. They also deflected to the
axis but not get to the common focus. The moving
equation for them has the form:
′′ + =r
r
κ 0 , where
Mvc
eI
2
2=κ (8)
To put together all ions in the focus, it is needed an
optimization of the external magnetic field distribution.
For this aim we can determine the form of the magnetic
surface that limit the current channel. Then we can
calculate the parameters of the solenoid (for producing
such magnetic surface) and determine the focusing ions
trajectories. The calculation can be carried out for
paraxial ion trajectories and paraxial magnetic surfaces
where particles and magnetic force lines go through
input and output butt-endes (faces) of a long cylindrical
focusing device (“lens”).
The limiting magnetic surface is determined from
the condition that its radius (R) coincides with the
current channel radius (a) and the radius of the focused
beam. The functions ( )R z and B zz ( ) are determined
from the equation similar to (8):
′′ + =R
R
κ 0 ,
Mvc
eI
2
2=κ (9)
The solution of the Eq.(9) (with initial conditions:
R R R R= ′ = ′0 0, at z=0) has the form:
z
dr
R R RR
R
= ±
′ −∫
0
2
02
0
κ ln /
(10)
Using the substitution:
t
R R
R
2 0
2
0
2=
′
−
κ
ln (11)
and the definition of the tabulated function (the
probability integral):
( )Φ R e dt
t
R
=
−∫2 1
2
0
2
π , (12)
we reduce the solution (10) to the form:
( )
( ) ( )
κ′Φ−
−κ′Φ×
×κ′κπ=
00
2
0
2
00
2
22
RRRR
RRz
ln
exp
(13)
In the case of the parallel ion beam injection, at z=0
we have ′ =R0 0 , besides, in the focusing region
z > 0 . As a result, the Eq.(13) takes the form:
z R
R
R
=
π
κ2
20 0
0Φ ln (14)
In the real experiment the current channel compression
leads to the certain value Rg (not equal to zero) that
corresponds to the coordinate z g . At this place the
current channel is finished (by a wire mesh or metallic
foil). Later on the inertial focusing of ions (which are
charge compensated) in the focal spot takes the place.
This point’s coordinate is defined as follows:
( )g
g
g
f
RR
R
R
R
Rz
0
0
00
2
2
2 ln
ln
κ
+
Φ
κ
π
= (15)
The numerical calculations of the Eq.(8) with help of
the PC give the result for z f coinciding with the (15).
The results of calculations are being compared with
the experimental ones [16] in the case of the 5 MeV
proton beam focusing by the current-carrying plasma
lens placed in the external non-uniform magnetic field.
In the Fig. 1 the calculated proton trajectories are
presented for the typical experimental conditions. In the
Fig. 2 the calculated distribution of the proton current
density is presented for the z-coordinate where the
fluorescent screen had been placed. The calculated and
measured sizes of the proton beam cross-section show
the satisfactory coincidence.
111
9080706050403020100
1
0
-1
Fig. 1 The calculated proton trajectories for the typical
experimental conditions: the proton beam energy is
5 MeV, the proton beam current is 10 mA, the initial
proton beam radius rb=1 cm, the beam divergence is
0.015 rad· r/rb , the focusing current is 1 kA, the initial
current channel radius is 1.5 cm, the final current
channel radius is 1.0 cm, the wire mesh cathode
coordinate is 60 cm, the solenoid face coordinate is 25
cm, the solenoid length is 19 cm, the solenoid inner
radius is 7.5 cm, the solenoid outer radius is 12.5 cm.
The parabola type curve presents the current channel
boundary. The horizontal and vertical scales are given
in cm.
21 . 510 . 50
8
6
4
2
0
Fig. 2. The calculated proton current density (mA/cm2)
versus the radius (cm) for the fluorescent screen
coordinate.
In conclusion of this part, we add the following
remark. For vacuum magnetic lenses the focusing length
2−κ∝fL , but for short magnetic plasma lenses Lf
1−κ∝ , i.e., it is much less. In this work it is shown that
for long plasma magnetic lenses Lf
211)( −κ∝ , i.e., it is
some more less. The suitable compression of the
focusing channel gives additional gaining of several
times over.
3. Combined electrostatic-magnetic lens
In this Part we are studying briefly the case of ion
beam focusing by the counter-stream intense electron
beam. As it is known, the electron concentration in such
beams can reach to 1012 – 1013 cm -3, therefore its using
for ion beam focusing can have good prospects.
The expression for the focusing force have the form:
rejvcrneFFF mer
22 22 −π−π−=+= , (16)
where n is the electron concentration, j is the current
density. Let us to express n and j by the current I, radius
a and velocity ve of the electron beam:
( ) ( )n z
I
ev a ze
=
π 2 , ( ) ( )j z
I
a z
=
π 2 (17)
The expression for the focusing force takes the form:
( )F
eI
a z v
vv
cr
e
e= − +
2
12 2 (18)
The equation for focused ion trajectories (with
account of Eq.(1)):
0
0
2 =+′′ rk
B
zB
r e
z
z
)(
)(
,
( )
2
0
2
2
2 12
avMv
cvveI
k
e
e
e
−+
= (19)
In the uniform or non-uniform cases it can be used
similar methods and formulae (with the own ke ) as it
have considered above in the case of the magnetic lens.
The uniform and non-uniform long plasmaoptic
focusing devices of Morozov type were investigated in
the separate work [15].
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