Moment aberrations in magneto-electrostatic plasma lenses (computer simulation)
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irk-123456789-789982015-03-25T03:02:34Z Moment aberrations in magneto-electrostatic plasma lenses (computer simulation) Butenko, V.I. Ivanov, B.I. 2001 Article Moment aberrations in magneto-electrostatic plasma lenses (computer simulation) / V.I. Butenko, B.I. Ivanov // Вопросы атомной науки и техники. — 2001. — № 5. — С. 95-98. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS number: 41.85.–p http://dspace.nbuv.gov.ua/handle/123456789/78998 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Butenko, V.I. Ivanov, B.I. Moment aberrations in magneto-electrostatic plasma lenses (computer simulation) Вопросы атомной науки и техники |
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Butenko, V.I. Ivanov, B.I. |
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Butenko, V.I. |
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Moment aberrations in magneto-electrostatic plasma lenses (computer simulation) |
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Moment aberrations in magneto-electrostatic plasma lenses (computer simulation) |
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Moment aberrations in magneto-electrostatic plasma lenses (computer simulation) |
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Moment aberrations in magneto-electrostatic plasma lenses (computer simulation) |
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Moment aberrations in magneto-electrostatic plasma lenses (computer simulation) |
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moment aberrations in magneto-electrostatic plasma lenses (computer simulation) |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2001 |
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Moment aberrations in magneto-electrostatic plasma lenses (computer simulation) / V.I. Butenko, B.I. Ivanov // Вопросы атомной науки и техники. — 2001. — № 5. — С. 95-98. — Бібліогр.: 13 назв. — англ. |
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Вопросы атомной науки и техники |
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ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №5.
Серия: Ядерно-физические исследования (39), с. 95-98.
95
MOMENT ABERRATIONS IN MAGNETO-ELECTROSTATIC
PLASMA LENSES (COMPUTER SIMULATION)
V.I. Butenko, B.I. Ivanov
NSC "Kharkov Institute of Physics and Technology "
Akademicheskaya st., 1, Kharkov, 61108, Ukraine
E-mail: butenko@kipt.kharkov.ua
PACS number: 41.85.–p
1 INTRODUCTION
As it is known, the questions of intense ion beam
focusing are important both for a problem of controlled
thermonuclear synthesis, and for other areas of science
and engineering: physics of high energies, accelerating
engineering, radiating technologies, implantation met-
allurgy and so on (see, e.g., [1, 2]). Essential feature of
intense ion beams is that for the blow-up avoidance,
their charge should be compensated by electrons during
transportation and focusing. In this case an application
of plasmaoptical focusing systems is expedient, which
development was begun by A.I. Morozov with the em-
ployees [3, 4], and recently basically is developed by
the group of A.A. Goncharov [5–8]. Additional benefits
of such systems are large focusing force, basic opportu-
nity of geometrical aberration elimination, and possibil-
ity of wide-aperture (non-paraxial) beam focusing. Cur-
rently the question consists in improvement of these
devices, reduction of aberrations, increase of efficiency
and focusing force.
The quality of the charged particle focusing is de-
fined by various aberrations: geometrical, depending
from radius and angle of the particle injection (in [4]
and in the subsequent works on plasmaoptics, concern-
ing focusing of parallel beams, they refer to as spherical
aberrations, or aberrations of wide beam); chromatic
aberrations, connected with a longitudinal momentum
of particles; at presence of magnetic fields to them aber-
rations connected with azimuthal movement of particles
are added (in the plasmaoptics [4] they refer to as mo-
ment aberrations, as they are connected to the angular
moment of a particle concerning an axis). Here it is per-
tinently to note, that in traditional electron optics, basi-
cally in connection with electronic microscopy, a little
another terminology is accepted [9-11]: chromatic aber-
rations have the same sense, the term "moment aberra-
tions" is not used, and geometrical aberrations are sub-
divided on eight kinds (one of which is the spherical
aberration of a dot source), and three of them depend on
the value and direction of a magnetic field and conse-
quently refer to as anisotropic aberrations.
Theoretically the moment aberrations were studied
in [4] by an example of compensated ion beam (CIB)
focusing in so-called "A-system", and also under CIB
recuperation (both was the plasmaoptical systems). In
the case of plasma electrostatic lenses (or rather mag-
neto-electrostatic ones, see below), moment aberrations
were considered in the experimental works [5, 6]. In
these works the conclusion was drawn that in the ab-
sence of spherical aberrations the relatively low coeffi-
cient of compression can be explained by the influence
of unremovable in principle moment aberrations associ-
ated with a finite azimuthal swirl of fast beam particles
in the magnetic field of a plasma lens.
In this work moment aberrations in the plasma mag-
neto-electrostatic lenses are considered in more detail
with the use of the computer modeling. For solution of
the problem we have developed a special computer code
– the model of plasmaoptical focusing device, allowing
to display the main parameters and operations of ex-
perimental sample of a lens [5–8], to simulate the mo-
ment and geometrical aberrations and give recommen-
dations on their elimination.
2 STATEMENT OF THE PROBLEM
In the previous work [12] with the use of computer
simulation we have considered in detail Morozov
plasma lens, in which the magnetic field was created by
a single current ring. In order to locate the base elec-
trodes (which introduce electric potentials into the
plasma) near the central plane of the lens, in the ex-
perimental works [5–8] the magnetic field configuration
with oppositely wound three short solenoids was used.
In this work such a field configuration is simulated by
three turns with opposite currents. Besides, unlike [12],
the azimuth motion of focused particles, leading to mo-
ment aberrations, is taken into account. The case of ax-
ial symmetry is considered.
As it is known, in case of analytical investigation of
a Morozov lens it is assumed that the magnetic surfaces
coincide with equipotential lines of the electric field. So,
magnitude and spatial distribution of the electric field in
the plasma is determined completely by the magnetic
field geometry and boundary conditions, which are set
as a continuous distribution of the potential Φ(R, z) on a
cylindrical surface of radius R, bordering on plasma.
Thus it is supposed, that the ion beam propagation area
is filled with plasma, density and other characteristics of
which are sufficient for space charge neutralization of
the beam and creation of required focusing fields. In
practice [5–8], the electrical potentials are entered in the
plasma discretely, by means of n ring "basic" electrodes
of a radius R located in the central part of the lens, be-
tween the separatrices, where the longitudinal magnetic
field passes through a zero (in our case -
2.8 сm < z < 2.8 сm). The magnetic surfaces at the left
and right of the central area are considered as grounded.
The magnetic field of a current ring J with the radius
ac and the coordinate l along the z axis is described by
the azimuthal component of the vector potential (see,
96
[13], § 4):
22
2
2
4
214
)()(
)()()(
lzra
ra
k
kEkKk
r
a
ck
JA
c
c
c
−++
=
−−=ϕ
(1)
where c is the velocity of light, K и E are the complete
elliptic integrals of the first and second kind. The equa-
tion of a magnetic surface which is the exact integral of
the equation of the magnetic force line in the case of
axial symmetry is (see [13], § 3):
( ) ( ) constzrrAzr =≡Ψ ϕ ,, (2)
(Ψ is so called magnetic flux function [4]).
We have calculated the topography of magnetic sur-
faces for various current relations in a central and two
side rings (Jc and Js, respectively; they are the reverse
currents; the central ring is located at z = 0 and the side
rings are located at z = ±5 cm). Further the field line
topography at Jc = -1.5 Js was used, which gives the
satisfactory distribution over the lens volume.
3 BASIC EQUATIONS
In a Morozov lens the electric potential in the plas-
ma Φ changes from one magnetic surface to another that
is expressed mathematically as
( ) ( )Ψ=Φ Fzr , . (3)
The components of electric and magnetic fields can
be written as
zr
B r ∂
Ψ∂−= 1 ;
rr
B z ∂
Ψ∂= 1 , (4)
zr B
d
dFr
rd
dF
r
E
Ψ
−=
∂
Ψ∂
Ψ
−=
∂
Φ∂−= , (5)
rz B
d
dFr
zd
dF
z
E
Ψ
=
∂
Ψ∂
Ψ
−=
∂
Φ∂−= . (6)
Substituting these expressions in the equations of
motion for a cylindrical system of coordinates, we re-
ceive:
r
V
d
dFrV
c
B
M
q
dt
dV
z
r
2
1 ϕ
ϕ +
Ψ
−= , (7)
−
Ψ
= ϕV
cd
dFrB
M
q
dt
dV
r
z 1 , (8)
r
VV
r
V
z
V
Mcr
q
dt
dV r
rz
ϕϕ
−
∂
Ψ∂−
∂
Ψ∂−= , (9)
where c is the velocity of light, q, M and V are the
charge, mass and velocity of the ion respectively (in this
case the calculations were carried out for protons).
In case of injection into a lens of the homogeneous,
parallel to an axis, monoenergetic ion beam, the entry
conditions look as follow:
at t = 0 Vz = V0, Vr = Vϕ = 0, z = zi (zi < 0), r = ri, (10)
where zi is the coordinate of the injector output face, and
the ion injection radius ri is set from zero up to size so-
mewhat smaller than the radius of base electrodes R,
which in its turn is less than the radius of current rings
ac.
In this case we have set the boundary conditions as a
potential distribution on the radius in a plane of the
central coil Φ(r, 0), which, if necessary, can be repre-
sented as the potential distribution on a cylindrical sur-
face Φ(R, z).
On the basis of formulas (1)-(10), simulation of ion
trajectories in a plasma lens was made, to investigate the
dependence of moment and geometrical aberrations on
the lens parameters and position of the ion injector.
Thus, as it will be seen below, the connection of mo-
ment aberrations with the law of conservation of a gen-
eralized angular moment of an ion relative to the axis z
is established.
4 SIMULTANEOUS FOCUSING OF IONS
BY ELECTRIC AND MAGNETIC FIELDS IN
A MOROZOV LENS
As a rule, aberrations are calculated by a method of
perturbation of paraxial particle trajectories [4, 9–11].
With the use of computer modeling we have a possibili-
ty to consider in the nonparaxial approximation the case
of strong magnetic fields and large-aperture ion beams,
at which a significant moment and geometrical aberra-
tions take place. In this case the Morozov lens should be
considered as a magneto-electrostatic one, because ion
focusing in this case is caused by the action of both
electric, and (to a lesser degree) magnetic fields; be-
sides, moment aberrations occur through the action of a
magnetic field.
z, cm
14012010080604020
r,
c
m
0.5
0.4
0.3
0.2
0.1
0
z, cm
14012010080604020
r,
cm
0.5
0.4
0.3
0.2
0.1
0
a b
z, cm
12.111.911.711.511.311.1
r,
cm
0.12
0.08
0.04
z, cm
12.111.911.711.511.311.1
r,
cm
0.12
0.08
0.04
c d
Fig. 1.
As an example we shall consider the focusing by a
Morozov lens with the following parameters: radius of a
proton beam is 5.5 cm, radii of current rings are 6.5 cm,
current in the central ring is 30 kA (in the case of a short
coil it corresponds to 3⋅104 ampere-turns). In Fig. 1 the
proton trajectories are shown in the focus area for two
locations of an proton injector of: zi = -20 cm (a, c) and
zi = -50 cm (b, d) in two cases: (a, b) when the electric
field is switched off (it is the case of a "pure" magnetic
lens), and (c, d) at presence of the electric field opti-
mized on a minimum geometrical aberrations.
The influence of moment aberrations consists in that
the particles can not cross the lens axis; geometrical
aberration influence consists in that the particles cross
the axis in different points. As can be seen from Fig. 1,
97
the location of the injector practically does not influence
on geometrical aberrations, and moment aberrations de-
crease when the injector is moving back from the lens.
It will be shown below, that this behavior can be ex-
plained within the framework of the law of conservation
of generalized angular moment of a charged particle.
The electric field is picked up in view of action of a
magnetic field and has the following potential distribu-
tion in a plane of the central ring:
Φ = 1.5 r2 - 0.0232 r4 - 5·10-5·r6. (11)
When the magnetic field changes, the focusing is
appreciably worsened, therefore the Morozov lens in
this case should be considered as a magneto-
electrostatic one with moment and geometrical aberra-
tions inherent to it.
5 DEPENDENCE OF MOMENT
ABERRATIONS ON FOCUSING
CONDITIONS AND LENS PARAMETERS
It is possible to consider an action of different kinds
of aberrations as independent from each other, if they
are small enough (see. [11], chapter 5.7). Let us con-
sider the dependence of moment aberrations on focusing
conditions with the parameters accepted in the preced-
ing section.
In Fig. 2 the dependence of the minimum radius r of
a proton trajectory in the focus area versus the injector
coordinate zi is represented at the fixed current of the
central ring Jc = 30 kA (a) and versus the current in the
central ring Jc at the fixed location of the ion injector
zi = -15 cm (b) for various values of the injection radius
R. As can be seen from this figure, moment aberrations
grow when a radius of injection increases, an injector is
moved into the area of a stronger longitudinal magnetic
field, and also a magnetic field of a lens rises. When the
injector moves away from the current rings, moment
aberrations tend to zero.
-50
-45
-40
-35
-30
-25-20-15
0
1
2
3
4
5
0.05
0.10
0.15
0.20
0.25
R, cm
r,
cm
z, cm
1
5
9
13
17
21
2529
0
0.5
1.0
1.5
2.02.53.03.5
0.03
0.06
0.09
0.12
0.15
0.18
R, cm
r,
cm
I, k
A
a b
Fig. 2.
In some limiting cases, behavior of moment aberra-
tions can be predicted on the basis of the input equa-
tions. To this effect, the equation of azimuthal motion of
an ion (9) after some transformations can be presented
as:
dt
d
c
e
dt
rVd
m Ψ−=
ϕ )(
, (12)
wherefrom after integration we obtain the expression
known in electronics as the theorem of Bush:
constC
mc
erV ==Ψ+ϕ 0 (13)
If the particles pass through an axis, then C0 = 0.
From (13) it follows, that a sufficient condition of
absence of moment aberrations at particle beam conver-
gence into a focus is the absence of initial azimuthal ve-
locities and the equality to zero of a magnetic field in
the injection area. For lenses used in experiments [5–8],
these conditions are fulfilled, hence it is necessary to
search for other reasons of not quite good focusing in
these experiments. One of such reasons can be not opti-
mum distribution of focusing fields on volume of a lens.
6 PLASMA LENS AND APPLICATION
OF ION BEAMS
As follows from above mentioned theses, in magne-
to-electrostatic plasma lenses the moment aberrations
are easily eliminated by placing an ion injector in a zero
magnetic field. However, they can be used, e.g., for a
uniform ion exposition of samples in the focus area
(about of part of square centimeter).
Fig. 3.
Placing an output face of an ion injector at a certain
distance from a lens (or placing its in the special sole-
noid with an adjustable longitudinal magnetic field), it
is possible to achieve in a focal plane the step-like dis-
tribution of a current density on radius. In particular,
such distribution of j(r) is shown in Fig. 3 for the fol-
lowing parameters: radius of a beam is 3.5 cm, coordi-
nate of an injector of ions is zi = -15 cm, current in the
central ring is Jc = 30 kA, potential distribution along
the radius in the central ring plane (in the GS units)
Φ = 1.5 r2 - 0.023 r4. In this case the ion current density
in the focal plane is j ≅ 9 A/см2, it is distributed rather
regularly on the radius from 0 up to 0.17 cm and 340
times as much as initial one. In such a mode the focused
beam remains laminar, i.e., the trajectories of ions are
not crossed, and the ion radius in the focal plane is pro-
portional to the ion radius in the plane of injection.
However, it should be noted that within the focal plane
the longitudinal ion velocity is rather non-uniform,
mainly because of a radial movement and to a lesser
degree because of azimuthal movement of ions. To re-
duce the ion velocity spread in the location of the sam-
ple, it is necessary to increase a focal length by reducing
polynomial factors that represent Φ(r), e.g., see (11).
98
REFERENCES
1. M.D.Gabovich, N.V.Pleshivtsev, N.N.Semashko.
Beams of Ions and Atoms for Controlled Thermo-
nuclear Fusion and Technology. Moscow: Ener-
goatomizdat, 1986, 249 p.
2. G.Hairapetian // AIP Conference Proceedings / Ad-
vanced Accelerator Concepts, Ed. P. Schoessow /
v. 335, p. 174, AIP Press, New York, 1995.
3. A.I.Morozov // Dokl. Akad. Nauk SSSR. 1965,
v. 163, p. 1363 [Sov. Phys. Dokl. 1966, v. 10,
p. 775].
4. A.I.Morozov and S.V.Lebedev, in Reviews of
Plasma Physics, Ed. By M.A. Leontovich (Ato-
mizdat, Moscow, 1974; Consultant Bureau, New
York, 1980), v. 8.
5. A.Goncharov, A.Dobrovolskii, I.Litovko et al. //
IEEE Trans. Plasma Sci. 1997. v. 25. p. 709.
6. A.Goncharov, S.Gubarev, A.Dobrovolskii et al. //
Proc. of the 18th ISDEIV, Eindhoven, 1998, p. 625.
7. A.A.Goncharov, A.N.Dobrovolskii, A.N.Kotsaren-
ko et al. // Plasma Physics Reports. 1994, v. 20,
p. 499.
8. A.A.Goncharov, I.M.Protsenko, G.Yu.Yushkov,
I.G.Brown // Appl. Phys. Lett. 1999, v. 75, p. 911.
9. W.Glaser. Grundlagen der Elektronen Optik, Wien,
Springer-Verlag, 1952.
10. V.M.Kel'man, S.Ya.Yavor. Electron Optics, Lenin-
grad: Nauka, 1988, 487 p.
11. M.Szilagui. Electron and Ion Optics, Plenum Press,
New York and London, 1988.
12. V.I.Butenko, B.I.Ivanov // Plasma Physics Reports.
2001, v. 27, p. 540.
13. A.I.Morozov and L.S.Solov'ev, in Reviews of
Plasma Physics, Ed. By M.A. Leontovich (Gosa-
tomizdat, Moscow, 1963; Consultant Bureau, New
York, 1966), v. 2.
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