Simulations of wide-aperture ion beam focusing by the plasma lens formed by a magnetic coil and system of electrodes
This work is devoted to calculations of ion beam focusing by the lens of Morozov type formed by a current-carrying coil in a plasma, and a system of ring electrodes.
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2000 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Цитувати: | Simulations of wide-aperture ion beam focusing by the plasma lens formed by a magnetic coil and system of electrodes / V.I. Butenko, B.I. Ivanov // Вопросы атомной науки и техники. — 2000. — № 3. — С. 112-114. — Бібліогр.: 7 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860006763329224704 |
|---|---|
| author | Butenko, V.I. Ivanov, B.I. |
| author_facet | Butenko, V.I. Ivanov, B.I. |
| citation_txt | Simulations of wide-aperture ion beam focusing by the plasma lens formed by a magnetic coil and system of electrodes / V.I. Butenko, B.I. Ivanov // Вопросы атомной науки и техники. — 2000. — № 3. — С. 112-114. — Бібліогр.: 7 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | This work is devoted to calculations of ion beam focusing by the lens of Morozov type formed by a current-carrying coil in a plasma, and a system of ring electrodes.
|
| first_indexed | 2025-12-07T16:39:08Z |
| format | Article |
| fulltext |
Problems of Atomic Science and Technology. 2000. N 3. Series: Plasma Physics (5). p. 112-114 112
UDC 533.9
SIMULATIONS OF WIDE-APERTURE ION BEAM FOCUSING BY THE
PLASMA LENS FORMED BY A MAGNETIC COIL AND SYSTEM OF
ELECTRODES
V.I. Butenko, B.I. Ivanov
National Science Center “Kharkov Institute of Physics and Technology”,
Kharkov, 61108, Ukraine (E-mail: butenko@kipt.kharkov.ua)
In the plasma physics and problems of controlled
thermonuclear fusion a noticeable role are played the
trends connected with focusing of intense ion beams of
middle and high energies. In particular, such trends are
the inertial thermonuclear fusion on light and heavy
ions, researches of radiating resistance of the first wall
materials, generation of the high power neutral particle
beams by the charge exchange of intense ion beams, etc.
The problems of intense ion beam focusing are
important also for the nuclear physics, physics of high
energies, physics and engineering of accelerators, beam
technologies. The essential feature of intense ion beams
is that they should be charge compensated during the
focusing to prevent their destruction. In this case, the
application of plasmaoptic focusing systems is
expedient which development is initiated by A.I.
Morozov and co-workers [1, 2], and recently
successfully developed by A.A. Goncharov group (e.g.,
[3, 4]). Now the problem consists in optimization of
such lenses, mainly, in reduction of aberrations and
focusing force increasing.
So, this work is devoted to calculations of ion beam
focusing by the lens of Morozov type formed by a
current-carrying coil in a plasma, and a system of ring
electrodes.
In the plasma electrostatic lens of Morozov type the
magnetic surfaces are the equipotentials of the electrical
field [1]. It is supposed, that the current across a
magnetic field is absent, and intensity and spatial
distribution of electrical field in a plasma are completely
determined by magnetic field geometry and boundary
condition. The last one is given as a continuous function
( )zR,Φ , where Φ is the potential (that is set from the
outside), and R is the cylindrical surface radius. In
practice the electrical potentials are entered in plasma
by a discrete manner, using of ring electrodes, due to the
system of the "charged" magnetic surfaces can be
formed in the plasma. The experimental researches [2-4]
basically confirm the theoretical model [1], but some
problems remain, in particular, the reasons of rather
significant spherical aberrations and methods of their
elimination. On the basis of the experimental
experience, it is possible to consider that the probable
corrections to the theory can be taken into account as
additional aberrations.
In the large work of A.I. Morozov and S.V. Lebedev
[1] various problems of plasmaoptics are investigated
theoretically including consideration of the axial
electrostatic plasma lenses. In particular, the estimation
of the focal length for the elementary plasma lens
formed by the circular current is given. Meaning
importance of this problem for practical calculations of
electrostatic plasma lenses, we will consider it in more
details, with account of non-paraxial (wide-aperture)
focused beams and exact expression for a magnetic
field.
The magnetic field of the circular current J (with the
radius of the coil añ and its coordinate l on the axis z) is
described by the azimuthal component of the vector-
potential (e.g., see [5]):
22
22 4
2
14
)()(
,)()()(
lzra
rakkEkKk
r
a
ck
JA
c
cc
−++
=
−−=ϕ (1)
where c is the light velocity, K and E are the complete
elliptic integrals of the 1-st and 2-nd kind.
Following [1], we enter the function of the magnetic
flow ϕ=ψ rA . The expression constzr =ψ ),( is the
equation of the magnetic surface and also the equation
of the magnetic force line on the plane (r, z). (the set of
such lines is calculated and given in the corresponding
figures, see below). In this lens the equipotential
property of magnetic surfaces is determined by relation
)(ψΦ=Φ , where Φ is the potential of the electrical
field.
Let's express the components of the electrical and
magnetic field through ψ and ϕA :
dr
d
d
dEr
ψ
ψ
Φ−= ;
dz
dA
rd
d
dz
d
d
dEz
ϕ
ψ
Φ−=
ψ
ψ
Φ−= ,
dz
dA
Hr
ϕ−= ; ϕ= rAdr
d
rH z
1 (2)
Hence it follows:
rHd
dE rz ψ
Φ= , rHd
dE zr ψ
Φ−= (3)
We will consider two cases of dependence Φ versus
ψ having practical meaning.
Case 1. In the work [1] the plasma lens formed by
the circular current is very shortly considered at the
electrical
potential distribution according to the condition
ϕ=ψ=Φ brAb , where constb = , (4)
and the estimation of its focusing length is given:
θΦ= 02qWaF c / , where W is the kinetic energy of
ions, Φ0 is the potential of the coil, θ ≈ 1 is the
dimensionless
parameter depended on the geometry of the system.
Let's consider this problem more in detail, with
application of computer modeling. For performance of
the relation (4) we will set the boundary condition as the
113
distribution of electrical potential on the cylindrical
surface with the radius Re2 (in practice it is set by
system of ring electrodes [2-4]):
),(),( zRAbRzR eee 222 ϕ=Φ (5)
The electrical and magnetic fields are connected by
the relations which follow from (3) and (4):
rz brHE = , zr brHE −= (6)
The constant b, on which the force of the lens
depends, is determined by setting (with help of the
electrodes) appropriate value of the electrical field
intensity Er2 in the point (r2, z0):
),(/ 0222 zrHrEb zr−= (7)
At focusing of ions with the mass Ì and charge q,
the equations of motion look like:
rqErM −=&& , zqEzM −=&& (8)
(Here the magnetic component of the Lorentz force
was neglected that in this case, for Hr ~ Hz, is allowable
at the energy ~ 10 keV/nuclon, and for paraxial ions is
allowable at energy ~ 1 MeV/nuclon).
The initial conditions will set as:
at t = 0 z = z0, vr = 0, vz = v0, r = r0, (9)
where the radius of ion injection r0 is set from 0 up to
size smaller the radius of electrodes, z0 = -10 cm, vr and
vz are the radial and longitudinal velocity of ions.
151050-5-10
5
4
3
2
1
0
Fig. 1. The results of the calculations of ion
trajectories for the case 1 (all values in cm).
The initial parameters of the beam are as follows:
the proton beam with energy 20 keV, current 1 A, radius
4 cm; the beam is uniform along the radius and charge
compensated by electrons. These conditions are close to
the experimental ones in Refs. [3, 4].
The results of calculations of ion trajectories for the
case 1 are given in the Fig. 1, whence it is visible, that
only the paraxial particles are well focused. On this base
it is found the distribution of the ion current density
versus the radius in the cross-section of the paraxial ion
focusing at the coordinate of zs=9.9 cm: the maximal
current density is j = 135 A/cì2, the half-width of the
focal spot is δr = 0.02 cm, and the relative amount of
ions within the limits of the half-width is about 10 %.
The non-paraxial ions (which is much more since their
amount in a layer is proportional to the radius of
injection) are «overfocused», moreover, the larger an
injection radius, the earlier an ion intersects the axial
line. The explanation is connected with the fact that rAϕ
and Φ increase too rapidly versus r near by the coil
surface. In the Fig. 2 it is presented the distribution of
the ion current density versus the radius in the cross-
section near the minimum beam radius that
demonstrates bad focusing of non-paraxial ions: at the
cross-section coordinate of zs=8.2 cm there is the
maximal current density j = 11 A/cì2, the half-width of
the focal spot is δr = 0.15 cm, and the relative amount
of ions within the limits of the half-width is about 50 %.
Optimization of the case 1. The condition of ideal
focusing is the requirement, that in any cross-section of
a lens the focusing force can be proportional to a
deviation of an ion from the axis, that is Er ∝ r. As it is
visible from (6), it is reduced to the condition
Hz(r) = const, that is realized in long solenoids (see
Ref.[6]).
0.80.70.60.50.40.30.20.10
1 0
8
6
4
2
0
Fig. 2. Distribution of ion current density (A/cm2) versus
radius (cm) near the minimum beam radius (case 1).
Case 2. Let's consider the variant, when in the plane
of the coil z = z0 the linear normalized distribution of
the radial electrical field is set:
1
1
1
010 r
rE
r
rzrEzrE rrr == ),(),( (9)
In practice such statement of the problem can be carried
out by setting potentials on the electrodes that insert into
the plasma, and measuring the distribution of the
electrical field intensity in the plasma. (The method of
local non-contact measurements of electrical field
intensity was proposed and experimentally grounded in
Ref.[7]).
In this work the electrical field in the plasma is
determined by the calculation way. Thus the functions
)(ψΦ=Φ and ψΦ dd / are set parametrically:
( ) ( )
=ψ
−=ψΦ
ϕ 00
2
1
1
0 2
1
zrrAzr
r
r
E
zr r
,,
)),((
( ) ( )
=ψ
−=
ψ
ψΦ
ϕ 00
01
10
zrrAzr
zrHr
E
d
zrd
z
r
,,
),(
)),((
(10)
On the cylindrical surface with radius Re1 we will set the
boundary condition as the distribution of electrical
potential:
)),(()),(( zRARzR eee 111 ϕΦ=ψΦ (11)
Using the equations (9-11) for determination of
electrical fields from the formulas (3), and then the
motion equations (7) and initial conditions (8), it is
possible to calculate the ion trajectories.
The results of calculations of ion trajectories for the
case 2 are given in the Fig. 3. On this base it is found
the distribution of the ion current density versus the
radius in the cross-section of the paraxial ion focusing at
114
20151050-5-10
4
3
2
1
0
Fig.3. The results of the calculations of ion
trajectories for the case 2 (all values in cm).
the coordinate zs=10.3 cm: the maximal current density
is j = 66 A/cì2, the half-width of the focal spot is
δr = 0.03 cm, and the relative amount of ions within the
limits of the half-width is about 13 %. From the
distribution of ion radii at zs=10.3 cm versus its
injection radii; it is evident that well focused paraxial
ions have initial radii 0–1.5 cm. The non-paraxial ions
this time are underfocused, moreover, the larger an
injection radius, the later an ion intersects the axial line.
Now the explanation is connected with the fact that, due
to magnetic surfaces curvature, the non-paraxial ions
have not sufficient time-of-flight in the region of the
high focusing fields. In the Fig. 4 it is presented for the
case 2 the distribution of the ion current density versus
the radius in the cross-section near the minimum beam
radius that demonstrates bad focusing of non-paraxial
ions: at the cross-section coordinate of zs=11.5 cm there
is the maximal current density j = 4.3 A/cì2, the half-
width of the focal spot is δr = 0.25 cm, and the relative
amount of ions within the half-width limits is about
50 %. (As we have studied, in the case of long
solenoids, and at the Φ ∝ r2 distribution, the good
focusing takes place because the focusing force is
proportional to the ion deflection from the axis [6]).
Optimization of the case 2. In the formula (9) for the
distribution of the radial electrical field on the radius the
terms of a high degree on r were added. The coefficients
at them were selected by testing of variants. The
essential improvement of focusing is received at the
distribution Er=390r+0.285r7 (where Er in V/cm, and r
in cm). For this case the results of calculation of ion
trajectories are given in the Fig. 5. In the Fig. 6 it is
presented the distribution of the ion current density
versus the radius in the cross-section near the minimum
beam radius: at the cross-section coordinate zs=12.7 cm
there is the maximum current density j = 230 A/cì2, the
half-width of the focal spot is δr = 0.025 cm, and the
relative amount of ions within the half-width limits is
about 50 %.
In principle, the problem of the optimum electrical
field distribution can be solved by the special algorithm
developing. Furthermore, it is expedient to proceed
from this simplest lens to the lens with an arbitrary
solenoid, using the field superposition principle. Thus
for each experimental sample of the lens it is possible to
create its computational model intended for
determination of the optimum parameters and modes of
operation.
0.80.70.60.50.40.30.20.10
4
3
2
1
0
Fig.4. Distribution of ion current density (A/cm2 ) versus
radius (cm) near the minimum beam radius (case 2).
20151050-5-10
4
3
2
1
0
Fig. 5. The results of the calculations of ion trajectories
for the case 2, after optimization (all values in cm).
0.120.10 .080.060.040.020
200
160
120
8 0
4 0
0
Fig. 6. Distribution of ion current density (A/cm2)
versus radius (cm) in the cross-section near the
minimum beam radius (case 2, after optimization).
REFERENCES
1. A.I. Morozov, S.V. Lebedev, Plasma Theory
Problems, V.8, p.247, Moscow, Atomizdat, 1974.
2. V.V. Zhukov, A.I. Morozov, G.Ya. Shchepkin,
JETP Letters, 1969, V.9, p.14.
3. A.A. Goncharov e.a., Plasma Physics // 1994, V.20,
P.499.
4. A.A. Goncharov e.a. // Appl. Phys. Lett. 1999.
Vol.75. P.911.
5. A.I. Morozov, L.S. Solov'ev. Plasma Theory
Problems, V.2, P.3, Moscow, Atomizdat, 1963.
6. V.I. Butenko, B.I. Ivanov, Adiabatic Plasma Lenses
of Morozov Type... // see this issue.
7. B.I. Ivanov, V.P. Prishchepov, V.M. Kodyakov //
Proc. of the 13th Intern. Conf. on High Energy
Accelerators, V.2, P.229, Novosibirsk, 1987.
|
| id | nasplib_isofts_kiev_ua-123456789-82382 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:39:08Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Butenko, V.I. Ivanov, B.I. 2015-05-29T07:44:08Z 2015-05-29T07:44:08Z 2000 Simulations of wide-aperture ion beam focusing by the plasma lens formed by a magnetic coil and system of electrodes / V.I. Butenko, B.I. Ivanov // Вопросы атомной науки и техники. — 2000. — № 3. — С. 112-114. — Бібліогр.: 7 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/82382 533.9 This work is devoted to calculations of ion beam focusing by the lens of Morozov type formed by a current-carrying coil in a plasma, and a system of ring electrodes. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Вeams in Plasma Simulations of wide-aperture ion beam focusing by the plasma lens formed by a magnetic coil and system of electrodes Article published earlier |
| spellingShingle | Simulations of wide-aperture ion beam focusing by the plasma lens formed by a magnetic coil and system of electrodes Butenko, V.I. Ivanov, B.I. Вeams in Plasma |
| title | Simulations of wide-aperture ion beam focusing by the plasma lens formed by a magnetic coil and system of electrodes |
| title_full | Simulations of wide-aperture ion beam focusing by the plasma lens formed by a magnetic coil and system of electrodes |
| title_fullStr | Simulations of wide-aperture ion beam focusing by the plasma lens formed by a magnetic coil and system of electrodes |
| title_full_unstemmed | Simulations of wide-aperture ion beam focusing by the plasma lens formed by a magnetic coil and system of electrodes |
| title_short | Simulations of wide-aperture ion beam focusing by the plasma lens formed by a magnetic coil and system of electrodes |
| title_sort | simulations of wide-aperture ion beam focusing by the plasma lens formed by a magnetic coil and system of electrodes |
| topic | Вeams in Plasma |
| topic_facet | Вeams in Plasma |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82382 |
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