Calculation of ion focusing with a plasma electrostatic lens in a magnetic field formed by counter ring currents
Gespeichert in:
| Datum: | 2001 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
|
| Schriftenreihe: | Вопросы атомной науки и техники |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/79242 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Calculation of ion focusing with a plasma electrostatic lens in a magnetic field formed by counter ring currents / V.I. Butenko // Вопросы атомной науки и техники. — 2001. — № 3. — С. 74-76. — Бібліогр.: 6 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-79242 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-792422015-03-31T03:02:10Z Calculation of ion focusing with a plasma electrostatic lens in a magnetic field formed by counter ring currents Butenko, V.I. 2001 Article Calculation of ion focusing with a plasma electrostatic lens in a magnetic field formed by counter ring currents / V.I. Butenko // Вопросы атомной науки и техники. — 2001. — № 3. — С. 74-76. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS numbers: 41.85.– http://dspace.nbuv.gov.ua/handle/123456789/79242 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| format |
Article |
| author |
Butenko, V.I. |
| spellingShingle |
Butenko, V.I. Calculation of ion focusing with a plasma electrostatic lens in a magnetic field formed by counter ring currents Вопросы атомной науки и техники |
| author_facet |
Butenko, V.I. |
| author_sort |
Butenko, V.I. |
| title |
Calculation of ion focusing with a plasma electrostatic lens in a magnetic field formed by counter ring currents |
| title_short |
Calculation of ion focusing with a plasma electrostatic lens in a magnetic field formed by counter ring currents |
| title_full |
Calculation of ion focusing with a plasma electrostatic lens in a magnetic field formed by counter ring currents |
| title_fullStr |
Calculation of ion focusing with a plasma electrostatic lens in a magnetic field formed by counter ring currents |
| title_full_unstemmed |
Calculation of ion focusing with a plasma electrostatic lens in a magnetic field formed by counter ring currents |
| title_sort |
calculation of ion focusing with a plasma electrostatic lens in a magnetic field formed by counter ring currents |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/79242 |
| citation_txt |
Calculation of ion focusing with a plasma electrostatic lens in a magnetic field formed by counter ring currents / V.I. Butenko // Вопросы атомной науки и техники. — 2001. — № 3. — С. 74-76. — Бібліогр.: 6 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT butenkovi calculationofionfocusingwithaplasmaelectrostaticlensinamagneticfieldformedbycounterringcurrents |
| first_indexed |
2025-07-06T03:17:22Z |
| last_indexed |
2025-07-06T03:17:22Z |
| _version_ |
1836865912040325120 |
| fulltext |
CALCULATION OF ION FOCUSING WITH A PLASMA
ELECTROSTATIC LENS IN A MAGNETIC FIELD FORMED BY
COUNTER RING CURRENTS
V.I. Butenko
National Science Center "Kharkov Institute of Physics and Technology",
Academicheskaya St., 1, Kharkov, 61108, Ukraine (E-mail: butenko@kipt.kharkov.ua)
PACS numbers: 41.85.–
1 INTRODUCTION
The problems of intense ion beam focusing are im-
portant for the nuclear physics, physics of high energies,
physics and engineering of accelerators, beam technolo-
gies. 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 plasma-optic focusing systems which development is
initiated by A.I. Morozov and co-workers [1], and re-
cently successfully developed by A.A. Goncharov group
[2–4] is expedient.
In the plasma electrostatic lens of Morozov type the
magnetic surfaces are the equipotentials of the electrical
field. 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 the 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 "basic" ring electrodes. The experimental
researches [2–4] basically confirm the theoretical model
[1], but some problems remain, in particular, the reasons
of rather significant aberrations and methods of their
elimination. In this work computer modeling of plasma-
optic focusing devices is considered. Such simulations
require the development of special-purpose computer
codes aimed at modeling particular plasma-optic con-
figurations. The numerical codes will make it possible
to optimize efficiently the existing and planned experi-
mental devices.
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №3.
Серия: Ядерно-физические исследования (38), с. 74-76.
74
2 THE PROBLEM DEFINITION
In the previous work [5] a single-turn Morozov lens
is considered. Its disadvantage is far-reaching magnetic
surfaces. In experimental works [2–4] the configuration
of a magnetic field with counter inclusion of solenoids
is offered, that allows to locate basic electrodes near to
the central plane of a lens. In the given work the lens is
simulated by three coils with opposite currents.
The magnetic field of a ring current J (at radius of a
coil aс and coordinate l on an axis z) is described by
azimuth component of vector potential:
⎥⎦
⎤
⎢⎣
⎡ −−=ϕ )E()K()( kkk
r
a
ck
JA c
214 2
,
22
2 4
)()( lzra
rak
c
c
−++= .
where c is the light velocity, K and E are the complete
elliptic integrals of the 1-st and 2-nd kind.
The equation of magnetic surfaces has the form:
constrA =ϕ .
The topography of magnetic surfaces for a various
ratio of currents in central and lateral coils (Jc and Js,
respectively) was calculated. Further we used topogra-
phy of force lines Jc = 1.5 Js. The central coil is located
at z = 0, lateral at z = 5.7 cm.
In the central part of a lens (-2.8 cm < z < 2.8 cm)
the potentials of basic ring electrodes are applied to
magnetic surfaces, and the magnetic surfaces at the left
and to the right of the central area are considered as
grounded.
The topography of equipotential surfaces at
Jc = 1.5 Js. are presented in Figs. 2, 3, 6 and others.
3 BASIC EQUATIONS
Following [1], we shall enter the function of a mag-
netic flow:
( ) ( zrrAzr ,, ϕ )=Ψ ,
where Aϕ is the azimuth component of the vector poten-
tial. In a Morozov lens the equipotentiality of magnetic
surfaces is determined by the relation [1]:
( ) (Ψ )=Φ Fzr , ,
where Φ is the potential of an electric field.
Let's express components of an electric and mag-
netic fields through Ψ:
zr
B r ∂
Ψ∂
−=
1 ,
zr B
d
dFr
rd
dF
r
E
Ψ
−=
∂
Ψ∂
Ψ
−=
∂
Φ∂
−= ,
rz B
d
dFr
zd
dF
z
E
Ψ
=
∂
Ψ∂
Ψ
−=
∂
Φ∂
−= .
The equations of motion in cylindrical system of
coordinates:
r
V
BV
mc
eE
m
e
dt
dV
zr
r
2
ϕ
ϕ ++= ,
( )
r
VV
BVBV
mc
e
dt
dV r
zrrz
ϕϕ −−= ,
rz
z BV
mc
eE
m
e
dt
dV
ϕ−= .
Substituting here the expressions for a component of
electric and magnetic fields, we have:
r
V
d
dFrV
c
B
m
e
dt
dV
z
r
2
1 ϕ
ϕ +⎟
⎠
⎞
⎜
⎝
⎛
Ψ
−= ,
( )
r
VV
BVBV
mc
e
dt
dV r
zrrz
ϕϕ −−= ,
⎟
⎠
⎞
⎜
⎝
⎛ −
Ψ
= ϕV
cd
dFrB
m
e
dt
dV
r
z 1 .
If the initial azimuthal velocity of the beam Vϕ0 = 0,
then 0Ψ=Ψ+ϕ mc
e
mc
erV , (*)
where Ψ0 is a function of a magnetic flow in the injec-
tion region.
4 RESULTS OF CALCULATIONS
Let's carry out calculations of the ion trajectories at
parameters comparable to the Kiev lens [2–4]: energy of
protons W = 20 keV, radius of an injected beam
r0 = 3.5 cm, beam is parallel, radii of current rings
ra = 6.5 cm, coordinate of protons injector z0 = -30 cm,
proton current of is 1 A. In the regions of a beam injec-
tion and its focus the magnetic field is practically equal
to zero, thus in the focus , and moment aberra-
tions are absent, see (*).
0=ϕrV
For focusing of ions the distribution 2r∝Φ is usu-
ally considered. But in this case it does not carry to sat-
isfactory results (see calculation of ion trajectories in
Fig. 1 at Φ = 0.65 r 2 and current density in the focus
region in Fig. 2).
z, cm
242220181614121086420-2-4
r,
cm
6
4
2
0
z, cm
24222018
r,
cm
0.04
0.03
0.02
0.01
Fig. 1.
In a plane of the central coil the optimized distribu-
tion of potential on radius (in GS) is a polynomial:
Φ = 0.75 r 2 – 0.0121 r 4,
where the factors are adjusted such that the focusing is
best. In Fig. 3 the proton trajectories for this case are
submitted. In Fig. 4 the current density distribution of
protons on radius is submitted in the focus region. From
distribution of potential on radius in the plane of the
central coil we shall pass to distribution of potential on
length of a cylindrical surface with radius R, see Fig. 5.
In experimental work [4] optimum focusing was re-
ceived at distribution of the potential on a cylindrical
surface proportional to distribution of a longitudinal
magnetic field on an axis of lens. For such distribution
we have found proton trajectories of (Fig. 6) and distri-
bution of current density on radius (Fig. 7). Thus den-
sity of the current in focus Jmax = 1.2 A/cm2 is received
at average beam radius 0.5 cm, that will be matched to
the experimentally found results, but considerably con-
cedes to the above-mentioned optimized calculated re-
sults.
Till now in calculations we accepted continuous dis-
tribution of potential on coordinates. As against it in
experiments [2–4] the potentials in plasma are entered
with the help of finite number (5 or 9) cylindrical elec-
trodes.
Fig. 2.
z, cm
242220181614121086420-2-4
r,
cm
6
4
2
0
z, cm
222120
r,
cm
0.04
0.03
0.02
0.01
Fig. 3.
r, cm
0.040.030.020.01
j,
A/
cm
^2
400
300
200
100
0
Fig. 4.
z cm
32.521.510.50
Ф
, G
S
6
4
2
0
Fig. 5.
z, cm
26242220181614121086420
r,
cm
6
4
2
0
Fig. 6.
r, cm
1.510.50
j,
A/
cm
2̂
4
3
2
1
0
Fig. 7.
Let's consider a case of 9-electrode lens, that corre-
sponds to the preset of 6 discrete values of potential on
half of lens (6-th potential corresponds to zero potential
on an axis). If these 6 values be set on the optimized
curve submitted in Fig. 5, and be smooth by splines, the
satisfactory concurrence of these curves will turn out.
75
However in experiment the electrodes of a finite length
specifying the step distribution of potential were applied
which in plasma smoothed out. Characteristics of this
smoothing are not investigated yet experimentally.
Formally, in calculations this smoothing was simulated
by B-splines of the 3-rd order, and the degree of
smoothing was defined by a ratio of effective electrode
length that can be smaller than real one, and effective
gap lenght between electrodes (this length can be larger
than real one). For example, at a gap between electrodes
of 5 mm the smoothed distribution was observed
(Fig. 8). The proton trajectories, corresponding to this
case, and their distribution on radius in the focal plate
are presented in Fig. 9 and Fig. 10.
Fig. 8.
z, cm
26242220181614121086420
r,
cm
6
4
2
0
Fig. 9.
r, cm
1.2510.750.50.250
j, A
/cm
^2
0.8
0.6
0.4
0.2
0
Fig. 10.
z, cm
2.521.510.50
Φ
, G
S
8
7
6
5
4
3
2
1
Fig. 11.
z, cm
26242220181614121086420
r,
cm
6
4
2
0
Fig. 12.
At the effective gap between electrodes of 2 mm
(that is close to the experimental value) the smoothed
distribution with more distinct stairs is received
(Fig. 11). The proton trajectories which correspond to
this case are shown in Fig. 12 and the current density in
the focus region in Fig. 13. The current density (about
0.1 A/cm2) and halfwidth of a focal spot (about 1 cm)
under the order coincide with the experimental results
[2–4]. In this case because of the step distribution of
potential in plasma, bad beam focusing takes place. As
the line of magnetic force in the marginal electrode re-
gion passes very closely and near to the axis and they
settle down far apart, paraxial ions appear at the same
potential, and therefore are not focused. Thus a diver-
gence between experiment and calculations is signifi-
cant, hence, in real plasma the potential smoothing is
much stronger. The current density (∼0.3 A/cm2) and
the focal spot half-length (∼1 cm, see Fig. 13) corre-
sponded to the experimental results by the order of val-
ues. Besides, in the paper [6] the ion focusing improve-
ment was observed for the case while the magnetic field
strength of the lens of such type decreased. Except of
another causes, the smoothing of the potential step dis-
tribution can influence this effect in such case.
r, cm
21.510.50
j, A
/c
m
2̂
0.3
0.2
0.1
0
Fig. 13.
Taking into account these investigations, it is expe-
dient to move the recommendations as follows. 1. Using
the adequate computer model of the plasma lens for op-
timization of the electric field distribution. 2. Increasing
the number and decreasing the thickness of the basic
ring electrodes. 3. Testing the optimum electric field
distribution in the plasma by a suitable precision ex-
perimental method.
The author would like to thank B.I. Ivanov for the
proposition of the work subject, help in the work, and
fruitful discussions.
REFERENCES
1. A.I.Morozov, S.V.Lebedev. // Plasma Theory
Problems, V.8, P.247, Moscow, 1974.
2. A.A.Goncharov et.al. // Plasma Physics Reports,
1994, v. 20, p. 499.
3. A.A.Goncharov et.al. // IEEE Trans. Plasma Sci.
1997, v. 25, p. 709.
4. A.A.Goncharov et.al. // Appl. Phys. Lett. 1999,
v. 75, p. 911.
5. V.I.Butenko, B.I.Ivanov. // Problems of Atomic
Science and Technology. Issue: Plasma Electronics
and New Acceleration Methods (2). 2000, No. 1,
p. 164.
6. A.A.Goncharov et.al. // Problems of Atomic Sci-
ence and Technology. Issue: Plasma Physics (6).
2000, No. 6, p. 124.
76
|