Optimum plasma lens for focussing of high-current ion beams
The optimum plasma lens, intended for the focussing of high-current ion beams, has been investigated.
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| Zitieren: | Optimum plasma lens for focussing of high-current ion beams / A.A. Goncharov, V.I. Maslov, I.N. Onishchenko // Вопросы атомной науки и техники. — 2002. — № 4. — С. 152-154. — Бібліогр.: 1 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-803092025-02-10T01:41:10Z Optimum plasma lens for focussing of high-current ion beams Goncharov, A.A. Maslov, V.I. Onishchenko, I.N. Plasma electronics The optimum plasma lens, intended for the focussing of high-current ion beams, has been investigated. This work was supported by STCU grant 1596. 2002 Article Optimum plasma lens for focussing of high-current ion beams / A.A. Goncharov, V.I. Maslov, I.N. Onishchenko // Вопросы атомной науки и техники. — 2002. — № 4. — С. 152-154. — Бібліогр.: 1 назв. — англ. 1562-6016 PACS: 52.40.Mj; 52.59.-f https://nasplib.isofts.kiev.ua/handle/123456789/80309 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Plasma electronics Plasma electronics Goncharov, A.A. Maslov, V.I. Onishchenko, I.N. Optimum plasma lens for focussing of high-current ion beams Вопросы атомной науки и техники |
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The optimum plasma lens, intended for the focussing of high-current ion beams, has been investigated. |
| format |
Article |
| author |
Goncharov, A.A. Maslov, V.I. Onishchenko, I.N. |
| author_facet |
Goncharov, A.A. Maslov, V.I. Onishchenko, I.N. |
| author_sort |
Goncharov, A.A. |
| title |
Optimum plasma lens for focussing of high-current ion beams |
| title_short |
Optimum plasma lens for focussing of high-current ion beams |
| title_full |
Optimum plasma lens for focussing of high-current ion beams |
| title_fullStr |
Optimum plasma lens for focussing of high-current ion beams |
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Optimum plasma lens for focussing of high-current ion beams |
| title_sort |
optimum plasma lens for focussing of high-current ion beams |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2002 |
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Plasma electronics |
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https://nasplib.isofts.kiev.ua/handle/123456789/80309 |
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Optimum plasma lens for focussing of high-current ion beams / A.A. Goncharov, V.I. Maslov, I.N. Onishchenko // Вопросы атомной науки и техники. — 2002. — № 4. — С. 152-154. — Бібліогр.: 1 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT goncharovaa optimumplasmalensforfocussingofhighcurrentionbeams AT maslovvi optimumplasmalensforfocussingofhighcurrentionbeams AT onishchenkoin optimumplasmalensforfocussingofhighcurrentionbeams |
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2025-12-02T13:36:38Z |
| last_indexed |
2025-12-02T13:36:38Z |
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1850403820274712576 |
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OPTIMUM PLASMA LENS FOR FOCUSSING OF HIGH-CURRENT ION
BEAMS
A.A.Goncharov*, V.I.Maslov, I.N.Onishchenko
*Institute of Physics NASU, 252650 Kiev;
NSC Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine
E-mail: vmaslov@kipt.kharkov.ua
The optimum plasma lens, intended for the focussing of high-current ion beams, has been investigated.
PACS: 52.40.Mj; 52.59.-f
INTRODUCTION
Optimum plasma lens (PL) realized in experiments [1]
at weak magnetic fields, when the electron Larmor radius
is comparable with PL radius, has been theoretically
researched. The optimum PL is lens, in which the
perturbations are not excited and particle density is
uniform. Three possible reasons of perturbation damping
in PL have been researched: namely, finite time of ion
movement through PL, finite time of electron renovating
in it, proximity of PL parameters to optimum ones. It has
been shown, that the vortical perturbations are not excited
in PL, if the overbalance of ions by electrons is close to
limiting one, determined from a condition of balance
upset of radial forces confining rotating electrons in the
region of finite radius: magnetic confining, centrifugal and
electrical scattering forces. At the overbalance of ions by
electrons, close to limiting one, the vortical perturbations
are not excited in PL. Also it has been shown that the
spatial uniformity of electron density is easier supported
in the optimum PL.
The collective field instability suppression in PL due
to electron density increase on radius in near wall region
has been considered. The influence of such electron
distribution in PL on focussing of ion beam has been
estimated and simulated numerically. It has been shown,
that the aberrations, called by such electron density
distribution, are small.
INFLUENCE OF PLASMA VORTICAL
PERTURBATIONS ON ION BEAM
FOCUSSING IN PLASMA LENS
In short cylindrical electrostatic PL for ion beam
focusing the vortical perturbations can be excited in wide
on radius, R-rt<r<R , with width, rt, near wall region. Two
kinds of vortical perturbations are excited. Namely, quick
vortical perturbations are excited due to development of
the resonant hydrodynamic instability of interaction of
electron vortical perturbations of PL with ions. Also slow
vortical perturbations are excited, which phase velocities
are much smaller then the electron drift velocity in PL
crossed fields. The excitation of the vortical perturbations
leads to anomalous electron radial transport, therefore, to
decrease of electron density and to PL focusing quality
deterioration. The numerical simulation shows (see Fig. 1)
that if the overbalance of ions by electrons decreases
monotonically due to vortical perturbation excitation in
wide near wall layer, which width equals 4/7 of PL radius,
then ion beam can be focused six times.
Fig. 1. Focusing of ion beam by plasma lens if ne-ni
decreases in its circumferential region, R-rt<r<R,
monotonically on radius to 0 due to excitation of vortical
perturbations
At increase of the PL magnetic field, Ho, electron
confining properties of the magnetic field increase and,
though vortical perturbations are excited, PL focusing
quality is improved. The latter dependence has been
observed in [1].
Let us consider damping of the vortical perturbation
excitation in PL. Namely, the optimum PL is considered
to be the lens, in which vortical perturbations are not
excited and particle density is homogeneous.
Vortical perturbations are not excited in PL, if the
overbalance of ions by electrons ∆n≡neo-nio is closed to
limit ∆nth. Let us determine ∆nth from the condition of
balance upset of the radial forces, confining rotating
electrons of the bunch in the region of finite radius:
magnetic confining, centrifugal and electrostatic scattering
forces ωHeVθme=meVθ
2/r-eEr. From this expression one can
obtain that two last forces exceed first one if overbalance
of ions by electrons, ∆n≡neo-nio, is not smaller then ∆nth,
determined by ∆nth=Ho
2/8πmec2. If ∆n could be formed
such that the following inequality is executed ∆n>∆nth,
then the electron cloud could propagate freely
transversally to magnetic field. Then the vortical
perturbations can not be excited in PL in the limiting case
∆n=∆nth. Let's explain it. The instability development of
the vortical perturbation excitation leads to the electron
bunch formation. But at ∆n, closed to ∆nth, the electron
bunches can not be formed due to the vortical perturbation
excitation, because any electron bunches formation leads
at once to their destruction by centrifugal and electric
scattering forces.
The following expression ∆nth=Ho
2/8πmec2 shows
observed in [1] quadratic relation of optimum electron
density neo on optimum Ho.
152 Problems of Atomic Science and Technology. 2002. № 4. Series: Plasma Physics (7). P. 152-154
The expression ∆nth=Ho
2/8πmec2 also shows fulfillment
of criterion that the radius, rHe=eEr/meω2
He, of the radial
oscillations of the electrons in crossed radial electric, Er,
and longitudinal magnetic, Ho, fields should be
approximately equal to the PL radius, R. This criterion is
used in [1] for determination of the optimum Ho.
Let us show that PL with ∆n, equal ∆n=∆nth, is the
lens, in which the electron density homogeneity is
supported more easy. Really, if in any part of PL the
electron density is such that the overbalance of ions by
electrons obeys ∆n>∆nth, then this electron density
overbalance is quickly disappeared by centrifugal and
electrostatic scattering forces. If in any part of PL the
electron density is such that the overbalance of ions by
electrons obeys ∆n<∆nth, then the vortical perturbations
are excited in this region, which provide anomalous
spatial diffusion of electrons, and non-uniformity of their
density is flattened.
SUPRESSION OF EXCITATION OF
VORTICAL PERTURBATIONS IN OPTIMUM
PLASMA LENS
Let us derive the dispersion relation and show that the
oscillation excitation in the cylindrical PL is damped at its
optimum parameters.
We take into account that the beam ions pass through
the plasma lens of length L during time, approximately
equal τi=L/Vbi. The electrons are renovated in PL also
during finite time, τe. Damping of perturbations of
densities and velocities of electrons and ions at recovery
of their unperturbed values we describe, using νi≡1/τi, νe≡
1/τe.
We use the electron, ion hydrodynamic eq.s
∂tV+νe(V-Vθo)+(V∇)V=(e/me)∇ϕ+[ωHe,V]-(V2
th/ne)∇ne
∂tne+(ne-neo)/τe+∇(neV)=0 (1)
∂tVi+νi(V-Vbi)+(Vi∇)Vi=-(qi/mi)∇ϕ ,
∂tni+(ni-nio)/τi+∇(niVi)=0 (2)
and Poisson eq. for the electrical potential, ϕ ,
∆ϕ=4π(ene-qini) (3)
Here V, ne are the velocity and density of electrons; Vth is
the electron thermal velocity; Vθo is the electron azimuth
drift velocity in crossed fields of PL; Vi, ni, qi, mi are the
velocity, density , charge and mass of ions.
As the sizes of vortical perturbations are much greater
than the electron Debye radius, rde≡Vth/ωpe, then the last
term in (1) can be neglected. Here ωpe≡(4πnoee2/me)1/2, noe
is the unperturbed electron density.
From eq.s (1) one can derive non-linear eq.s
dt[(α-ωHe)/ne]=[(α-ωHe)/ne]∂zVz-ανe/ne,
dtVz+νeVz=(e/me)∂zϕ (4)
describing both transversal and longitudinal electron
dynamics. Here
dt=∂t+(V⊥∇⊥) , ∂t≡∂/∂t , ∂z≡∂/∂z (5)
V⊥, Vz are the transversal and longitudinal electron
velocities, α is the vorticity, the characteristic of electron
vortical motion, α≡ezrotV.
Taking into account higher linear terms, from (1) one
can obtain
V⊥≈(e/mωHe)[ez,∇⊥ϕ]+(e/mω2
He)(∂t+νe)∇⊥ϕ (6)
From (6) we derive
α≈-2eEro/rmωHe-(eEro/m)∂r(1/ωHe)+
+(e/mωHe)∆⊥φ+(e/m)(∂rφ)∂r(1/ωHe)+
+(e/m)(∂t+νe)ez[∇⊥,ω-2
He∇⊥φ]≡µωHe , ∇ϕ≡∇φ-Eor (7)
Here Eor is the radial focusing electric field, φ is the
electric potential of the vortical perturbation; -2eEro/rmω
He=(ω2
pe/ωHe)(∆n/noe)≡ηωHe, ∆n≡noe-qinoi/e.
From (3), (7) α≈(ω2
pe/ωHe)δne/neo approximately
follows. Thus the vortical motion begins, as soon as the
electron density perturbation, δne, appears.
We use that, as it will be shown below, the
characteristic frequencies of perturbations approximately
equal to ion plasma frequency, ωpi.
As beam ions have large mass and propagate through
PL with fast velocity Vib, we will describe their dynamics
in linear approximation. We derive ion density
perturbation from eq.s (2)
δni=-nio(qi/mi)∆ϕ/(ω-kzVib+iνi)2 (8)
Here k, ω are wave number and frequency of perturbation,
Vib is the unperturbed longitudinal velocity of the ion
beam. Substituting (8) in Poisson eq. (3), one can obtain
β∆ϕ/4πe=δne, β=1-ωpi
2/(ω-kzVib+iνi)2, ne=noe+δne (9)
Let us consider instability development in linear
approximation. Then we search the dependence of the
perturbation on z, θ in the form δne∝exp(ikzz+ilθθ). Then
from (4) we derive
dt(ωHe/ne)(1-µ)=ανe/ne- (10)
-(eωHe/meneo)ikz
2ϕ(1-µ)/(ω-lθωθo+iνe), ωθo≡Vθo/r.
From (5), (6), (9), (10) we obtain, using the radial
gradient of the short coil magnetic field, the following
linear dispersion relation, describing the instability
development
1-(1-η)ω2
pe(lθ/r)∂r(1/ωHe)/k2(ω+i/τe-lθωθo)- (11)
-ω2
pi/(ω+i/τi-kzVbi)2-(1-η)ω2
pekz
2/k2(ω+i/τe-lθωθo)2=0.
It is necessary to note, that at optimum PL parameters, i.e.
at η=1, last two members in (11) equal zero.
Let us mean the quick vortical perturbations those,
which phase velocities, Vph, are closed to azimuth velocity
of the electron drift, Vph≈Vθo. For them from (11) we
derive in approximation kz=0, ω=ω(o)+δω, δω<<ω(o) and
neglecting τe, τi
ω(o)=ωpi=lθωθo , ωθo=(ω2
pe/2ωHe)(∆n/noe),
∆n≡noe-qinoi/e (12)
δω=iγq, γq=(ωpe/k)[(1-η)(ωpi/2)(lθ/r)∂r(1/ωHe)]1/2
Here noi is the unperturbed ion density. At obtaining (12)
we used a validity of an inequality
(1-η)(∆n/noe)(r/ωHe)ω2
pe∂r(1/ωHe)<<me/mi (13)
It is fulfilled at a weak overbalance of beam ion space
charge by electrons of PL, ∆n/noe<<1, at small radial non-
uniformity of ωHe, small plasma density, ωpe/ωHe<<1 and
for PL, closed to optimum one. From (12) one can see that
in PL, which parameters are closed to optimum ones, the
excitation of the quick vortical perturbations is damped.
From (12) it follows lθ=(me/mi)1/2(ωHe/ωpe)(noe/∆n), that
for typical parameters of experiments, ∆n/noe≈0.1, the
perturbations with lθ>1 are excited at a large magnetic
field and at small electron density.
153
As γq grows with r, taking into account τe, τi can lead
to those perturbations with r smaller then critical value
cannot be excited.
Let us introduce slow vortical perturbations as those
ones, whose phase velocity is much less than azimuth
velocity of the electron drift, Vph<<Vθo. We derive for
them from (11) in approximation kz=0 and neglecting τe, τi
the following expressions
γs=(√3/24/3)[ω2
pilθ(ω2
pe/2ωHe)(∆n/noe)]1/3
k2=-(1-η)(1/Vθo)ω2
pe∂r(1/ωHe) , Reωs=γs/√3 (14)
Here γs is the growth rate of excitation of slow vortical
perturbations of small amplitudes, Reωs is the real part of
the frequency. As γs grows with r, then taking into account
τe, τi should lead to that perturbations with r smaller than
some value are not excited. In other words, the vortical
perturbations are excited far from the PL axis.
(14) is obtained in approximation of a validity of a
following inequality lθ>>(noe/∆n)2ωpiωHe/ω2
pe. From (14)
one can show that really the following inequality Vph<<Vθo
is fulfilled, and not large lθ are excited. From (21) it also
follows that in optimum PL the slow vortical perturbations
are not excited. One can also show that the radius of
localization of the slow vortex rs is less then the radius of
localization of the quick vortex, rs<rq.
So, one can see that the quick vortical perturbations
are excited at relatively small plasma density, and at
parameters of optimum PL the quick vortical
perturbations are not excited. The slow vortical
perturbations are excited at large plasma density, however
growth rate of their excitation decreases for parameters,
closed to optimum PL parameters, and the growth rate
equals zero for optimum PL parameters.
SIMULATION OF HIGH-CURRENT ION
BEAM FOCUSSING IN PLASMA LENS
Let us numerically simulate the ion beam focusing in
high-current PL of different structures. In particular, let us
compare the results with ion beam focusing in the ideal
high-current PL. The ideal PL is considered to be the
cylindrical homogeneous electron cloud of finite length,
L, and of radius, R, which overbalances and focuses ion
beam. The results of beam focusing in the ideal PL are
shown in Fig. 2. On a horizontal axis the longitudinal
coordinate (cm) is postponed. On a vertical axis the
current radius (cm) of the focused beam ion is postponed.
At first ions are focused inside PL. After escaping a lens
the ions are focused on inertia. One can see apparent
result, that beam ions are focused in a point at some
longitudinal coordinate.
Fig. 2. Focusing of high-current ion beam by ideal
plasma lens.
Let us compare now the focusing in the ideal PL with
focusing in the real lens. Namely, the spatial structure of
magnetic field lines of short cylindrical permanent magnet
or short coil such that the electron cloud of PL can be
solid cylinder of length L and radius R with hollow cones
of radius R and with cones lengths Z1 and Z2 on the ends.
In Fig. 3 the simulation results of ion beam focusing in
such PL are presented.
Fig. 3. Focusing of ion beam by plasma lens if electron
column has hole cones on its ends.
One can see that all trajectories do not converge in one
point, however if the altitudes of hollow cones are small,
the aberrations are small.
In Fig. 1 the results of the ion beam focusing in PL, in
which in wide near wall region, R-rt<r<R, the strong
vortical perturbations are excited, are shown. These
perturbations lead in the region of their excitation to
anomalous radial transport of electrons, and certainly to
strong decrease of electron density in this region.
The excitation of the oscillated fields in the near wall
region of PL can be damped by providing of the positive
radial gradient of the electron density. Certainly, this
gradient of density could lead to aberrations of focused
beam. The numerical simulation, presented in Fig. 4,
shows that aberrations are not large.
Fig. 4. Focusing of ion beam by plasma lens if electron
density grows in region, R-rt<r<R, on 1/3.
This work was supported by STCU grant 1596.
REFERENCES
1. A.A.Goncharov, S.M.Gubarev, I.M.Protsenko,
I.Brown. Influence of magnetic field strength on the
focusing properties of a high-current plasma lens //
Problems of Atomic Science and Technology.
Kharkov. 2001. N3.
INTRODUCTION
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