Effects of vortices on ion beam focusing in a plasma lens
For high-current ion-beam focusing a vortical turbulence has been excited in plasma lens by the nonremovable gradient of an external magnetic field. The paper presents theoretical investigation, in the cylindrical approximation, of excitation of slow and fast vortices with taking into account the fi...
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| Cite this: | Effects of vortices on ion beam focusing in a plasma lens / A.A. Goncharov, S.N. Gubarev, V.I. Maslov, I.N. Onishchenko // Вопросы атомной науки и техники. — 2001. — № 3. — С. 152-154. — Бібліогр.: 1 назв. — англ. |
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| author | Goncharov, A.A. Gubarev, S.N. Maslov, V.I. Onishchenko, I.N. |
| author_facet | Goncharov, A.A. Gubarev, S.N. Maslov, V.I. Onishchenko, I.N. |
| citation_txt | Effects of vortices on ion beam focusing in a plasma lens / A.A. Goncharov, S.N. Gubarev, V.I. Maslov, I.N. Onishchenko // Вопросы атомной науки и техники. — 2001. — № 3. — С. 152-154. — Бібліогр.: 1 назв. — англ. |
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| description | For high-current ion-beam focusing a vortical turbulence has been excited in plasma lens by the nonremovable gradient of an external magnetic field. The paper presents theoretical investigation, in the cylindrical approximation, of excitation of slow and fast vortices with taking into account the finite length of plasma lens. It is shown that the growth rate of vortix excitation decreases with decreasing the length of plasma lens. The spatial structures of the vortices are constructed. The expression for the vortex amplitude of saturation is derived.
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EFFECTS OF VORTICES ON ION BEAM FOCUSING IN A PLASMA
LENS
A.A. Goncharov1, S.N. Gubarev1, V.I. Maslov, I.N. Onishchenko
1Institute of Physics NASU, 03650 Kiev
NSC Kharkov Institute of Physics & Technology, 61108 Kharkov, Ukraine
E-mail: vmaslov@kipt.kharkov.ua
For high-current ion-beam focusing a vortical turbulence has been excited in plasma lens by the nonremovable gra-
dient of an external magnetic field. The paper presents theoretical investigation, in the cylindrical approximation, of
excitation of slow and fast vortices with taking into account the finite length of plasma lens. It is shown that the
growth rate of vortix excitation decreases with decreasing the length of plasma lens. The spatial structures of the
vortices are constructed. The expression for the vortex amplitude of saturation is derived.
PACS numbers: 29.20.Bd
1 INTRODUCTION
Excitation of vortices in a plasma lens has been in-
vestigated analytically. The plasma lens is designed for
ion-beam focusing [1]. A focusing electric field is creat-
ed by the electron cloud. The cloud density exceeds the
ion beam density approximately by 10%. Electrons are
distributed on radius approximately homogeneously.
The lens represents the cylinder of a finite length,
placed in a short coil magnetic field.
As shown in [1], the plasma lens is unstable con-
cerning excitation of oscillating fields. The oscillation
excitation is realised owing to a positive radial gradient
of a short coil magnetic field.
As in the lens the crossed configuration of radial fo-
cusing electric, Eor, and longitudinal magnetic fields, Но,
is created, the electrons drift through an angle, θ, with a
velocity Vθo=-eEor/meωHe, ωHe=eНо/mec.
The density perturbation of primary homogeneous
electrons results in appearance of an electric field near
the perturbation. Therefore, near the perturbation the
crossed fields are realised. Thus, electron dynamics in a
field of perturbation is vortical.
In this paper a spatial structure and excitation of vor-
tical perturbations in a plasma lens are investigated the-
oretically.
2 INSTABILITY DEVELOPMENT IN PLAS-
MA LENS
We use the hydrodynamic and Poisson equations
∂tV+(V∇)V=(e/me)∇ϕ+[ωHe,V]-(V2
th/ne)∇ne (1)
∂tne+∇(neV)=0, ∇ϕ≡∇φ-Eor, ∆ϕ=4π(ene-qini) (2)
From (1) it is possible to derive equations
dt(α-ωHe)/ne=[(α-ωHe)/ne]∂zVz , dtVz=(e/m)∂zϕ (3)
dt=∂t+(V⊥∇⊥) , α≡ezrotV
From (1) one can obtain
V⊥=(e/mωHe)[ez,∇ϕ]-ω-1
He∂t[ez,V⊥]-ω-1
He[ez,(V∇)V⊥]≈
≈(e/mωHe)[ez,∇ϕ]+(e/mω2
He)∂t∇⊥ϕ, (4)
α≈2eEro/rmωHe+(eEro/m)∂r(1/ωHe)+(e/mωHe)∆⊥φ+
+(e/m)(∂rφ)∂r(1/ωHe)+(e/m)∂tez[∇,ω-2
He∇φ]. (5)
From (2), (5) it approximately follows, α≈(ω2
pe/ωHe)
δne/neo, that the vortical motion begins, as soon as there
appears a perturbation δne.
From (3) one can derive
dtωHe/ne=(ωHe/ne)∂zVz. (6)
Taking into account the ion effect, from (2) one can
obtain
β∆ϕ/4πe=δne , β=1-ωpi
2/(ω-kzVib)2 , ne=noe+δne. (7)
At first let us consider instability development. We
search the following dependence δne∝exp(ikzz+ilθθ).
Then from (3) we derive
dt(ωHe/ne)=-(eωHe/meneo)ikz
2ϕ/(ω-lθωθo), ωθo=Vθo/r. (8)
From (4), (7), (8) we obtain the equation for φ
(ω2
pe/ω2
He)∇θφ∂rωHe+β(∂t∆φ+ωθo∂θ∆φ)=
=ikz
2φω2
pe/(ω-lθωθo). (9)
We obtain from (9) dispersion relation, describing
the instability development
1-ω2
pi/(ω-kzVbi)2-ω2
pe (lθ/r)∂r(1/ωHe)/k2(ω-lθωθo)-
-ω2
pekz
2/k2(ω-lθωθo)2=0. (10)
Let us take into account that the beam ions pass
through the plasma lens during τi=L/Vbi and electrons
are renovated during τe>τi. (10) can be presented as
1-ω2
pi/(ω-i/τi-kzVbi)2-ω2
pekz
2/k2(ω-i/τe-lθωθo)2-
-ω2
pe (lθ/r)∂r(1/ωHe)/k2(ω-i/τe-lθωθo)=0. (11)
Let us mean the quick perturbations for which the
phase velocity Vph≈Vθo. For them from (11) we derive in
approximation kz=0, ω=ω(o)+δω, δω<<ω(o) and ne-
glecting τe, τi
ω(o)= ωpi=lθωθo, ωθo=(ω2
pe/2ωHe)(∆n/noe) , δω=iγq
γq=(ωpe/k)[(ωpi/2)(lθ/r)∂r(1/ωHe)]1/2. (12)
From (12) it follows
lθ=(mi/me)1/2(ωHe/ωpe)(noe/∆n), (13)
that for typical parameters of experiments the perturba-
tions with lθ>1 are excited at a large magnetic field and
small electron density.
As γq grows with r, taking into account τe, τi, it can
lead to that the perturbations with r smaller than some
value can not be excited.
For slow perturbations fulfilled is Vph<<Vθo. We de-
rive for them from (11), in approximation kz=0 and ne-
glecting τe, τi, the following expressions
γs=(√3/24/3)[ω2
pilθ(ω2
pe/2ωHe)(∆n/noe)]1/3
k2=-(r/ωθo)ω2
pe∂r(1/ωHe), Reωs=γs/√3. (14)
Here γs is the growth rate of slow perturbation exci-
tation. As γs grows with r, because k grows with r, tak-
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №3.
Серия: Ядерно-физические исследования (38), с. 152-154.
152
ing into account τe, τi it can lead to that the perturbations
with r smaller than some value are not excited.
From (10) we obtain the growth rate with kz≠0
γs=(√3/24/3)ω2/3
pi(lθωθo-kzVbi)1/3×
×{1-kz
2/[2kz
2+(lθ/r)(lθωθo-kzVbi)∂r(1/ωHe))]}1/3. (15)
From (15) one can see that the particle longitudinal
dynamics results in reduction of γs. Perturbations with
least kz≈π/L have a maximum γs.
3 STRUCTURE OF VORTEX
Let us describe structure of a quick vortex. Neglect-
ing nonstationary terms, we have from (4)
V⊥=-(e/meωHe)[ez,Ero]+(e/mωHe)[ez,∇φ], (16)
Vr=-(e/meωHe)∇θφ , Vθ=Vθo+(e/meωHe)∇rφ,
Vθo=-(e/meωHe)Ero=(ω2
pe/2ωHe)(∆n/noe)r. (17)
Vθ can be presented Vθ=Vph+δVθ. As Vθ equals Vθ=rdθ
/dt, we present dθ/dt=dα/dt+ωph, where ωph=(∆n/noe)(ω
2
pe/2ωHe)r=rv, rv is a vortex localisation. Then, decompos-
ing ωHe(r) on δr=r- rv near rv, from (17) we derive
dα/dt=-(ω2
pe/2ω2
He)(∆n/noe)δr(∂rωHe)r=rv+
+(e/meωHe)∂rφ, dr/dt=-(e/meωHer)∂θφ. (18)
Integrating (18), we obtain the equation, describing
oscillatory dynamics of electrons in a vortex field
(δr)2-(e/me)(no/∆no)(ωHe/ω2
pe)8φ/rv(∂rωHe)r=rv=const (19)
Fig. 1.
Let us consider the following dependence φ(θ)=φ
ocos[lθ(θ-ωpht)] and determine the boundary of a vortex
from the condition δrφ=-φo=0
δr=±2[2(e/m)(φ+φo)(n/∆n)(ωH/ω2
p)/rv(∂rωH)r=rv]1/2. (20)
The radial size of a vortex follows from (20)
δrv=2[2(e/me)φo(noe/∆n)(ωHe/ω2
pe)/rv(∂rωHe)r=rv]1/2. (21)
From (18), (20) it follows that the frequency of electron
oscillations Ωtr on closed trajectories is equal to
Ωtr=(lθ/2)(ωpe/ωHe)×
×[2(e/me)φo(∆n/2noe)(∂rωHe)r=rv/ωHerv]1/2. (22)
For a simplicity we consider structure of electron
trajectories in a slow vortex, Vph<<Vθo, with lθ=1 (see.
Fig. 2).
From (17) we approximately derive, similarly to
(18), equations describing electron oscillations in the
vortex field
dθ/dt≈(ω2
pe/ωHe)(∆n/2noe)+(e/meωHer)∂rφ,
dr/dt≈-(e/meωHer)∂θφ. (23)
Integrating (23), we obtain
r2+8(e/meω2
pe)(noe/∆n)=const. (24)
Let us consider the following dependence φ(θ)=-φ
ocos[lθ(θ-ωpht)] and determine the boundary of a vortex
by putting r=rmin at φ=φo. Here rmin is a minimum radius
of electrons oscillating in a field of a vortex at its
boundary. From (24) we derive
r=[r2
min+8(φo-φ)(e/meω2
pe)(noe/∆n)]1/2. (25)
From (25) the maximum radius of a vortex is as follows
rmax=[r2
min+16φo(e/meω2
pe)(noe/∆n)]1/2. (26)
From (26) we obtain the radial size, ∆rs, of a vortex
∆rs=[r2
min+16φo(e/meω2
pe)(noe/∆n)]1/2-rmin. (27)
In case rmin=0
∆rsrmin=0=[16φo(e/meω2
pe)(noe/∆n)]1/2. (28)
Fig. 2.
Instability is developed in a homogeneous plasma so
long as bunching of homogeneously distributed elec-
trons happens. Bunching ceases, when a slow (adiabat-
ic) stage of electron dynamics comes,
Ωtr≥γ. (29)
From (12), (22), (29) it follows that it happens at the
amplitude of an electrical potential of a fast vortex
φslh≈(ωHeωpi/2k2)(noe/∆n). (30)
However the amplitudes of a set of separate vortices
with a large distance between them can grow further.
The maximum amplitude of this set of vortices, φsm,
is determined by a condition, that the magnetic force
does not keep any more electrons of a vortex, rotated
around its axis on the closed trajectories. In other words
the electron bunch of a vortex can extend across a mag-
netic field. Thus bunching of electrons ceases. Thus
from a violation of forces balance,
meV2
θ/r-eEr≥meωHeVθ. (31)
one can obtain the amplitude of a vortex saturation, φsm.
Here Er is the electrical field of a vortex, the perturba-
tion of electron density in which is δnev. From (31) it
follows that electrons of a vortex at inequality fulfilment
ω2
pe(δnev+∆n)/noe≥ω2
He/2. (32)
can freely move across a magnetic field. Thus, using
(2), we have found, that the amplitude of a vortex is sta-
bilised at [1]
φsm≈(me/ek2)[ω2
He/2-(∆n/noe)ω2
pe]. (33)
From (33) one can see that if ∆n is close to
∆n≈Ho
2/8πmec2, (34)
the vortical perturbations are not excited.
Let us compare φslh with φsm
φslh/φsm≈2(noe/∆n)(ωpi/ωHe). (35)
One can see that in case of heavy ions and large
magnetic field the inequality φslh<<φsm is fulfilled.
153
4 EXCITATION OF NON-LINEAR VOR-
TICES
Let us describe excitation of non-linear vortices us-
ing equations
β∆ϕ=4πeδne, β=1-ω2
pi/ω2 , δne=ne-neo, (36)
dt(α-ωHe)/ne=0 , dt=∂t+(V∇), (37)
V=-(e/meωHe)[ez,Ero]+(e/mωHe)[ez,∇φ]-
-ω-1
He∂t[ez,V]-ω-1
He[ez,(V∇)V]. (38)
Selecting a time derivative ∂τ and a vortex motion in the
crossed fields with a velocity Vsθ, we have
dt=∂τ+(V∇)-Vsθ∇θ. (39)
From (36)-(38) we obtain
α≈-2eEro/rmeωHe-(eEro/m)∂r(1/ωHe)+(e/mωHe)∆φ+
+(e/m)(∂rφ)∂r(1/ωHe)+
+(e/me)∂t[r-1∂r (1/ω2
He)∂θφ-r-1ω-2
He∂2
θrφ], (40)
dt(ωHe/ne)=0 , ne=neo+(qi/e)δni+∆ϕ/4πe, (41)
Vθ≈Vθo+(e/meωHe)[ez,∇φ], (42)
Vθo=-(e/meωHe)Ero=(ω2
pe/2ωHe)(∆n/noe)r. (43)
From (36), (41), (42) we derive the non-linear evolu-
tion equation, describing an excitation of non-linear vor-
tices
[∂τ+(Vθo-Vsθ)∇θ+ (44)
+(e/meωHe)([ez,∇φ]∇)]ωHe/(neo+(qi/e)δni+∆φ/4πe)≈0.
In stationary approximation the slow vortex is de-
scribed according to (44) by equation
[(Vθo-Vsθ)∇θ+ (45)
+(e/meωHe)([ez,∇φ]∇)](ω2
pe+∆φe/me)/ωHe≈0.
The equation (45) can be presented in the form
(1-Vsθ/Vθo)∇θ∆φ-2(noe/∆n)(∇θφ)r-1∂r(1/ωHe)+
+2(e/meωHeω2
pe)r-1(noe/∆n){φ,∆φ}r,θ=0, (46)
{φ,∆φ}r,θ≡(∇rφ)∇θ∆φ-(∇θφ)∇r∆φ=
=r-1[(∂rφ)(∂θ∆φ)-(∂θφ)(∂r∆φ)].
Taking into account in (44) terms with ∂t and effect
of ions we derive
∂t∆φ≈-4πqi(Vθo∇θ))δni. (47)
One can obtain from the hydrodynamic equations the
equation for the ion density perturbation, δni,
∂2
tδni≈noi(qi/mi)∆φ. (48)
From (47), (48) we obtain the equation, describing the
vortex excitation
∂t
3∆φ≈-ω2
pi(Vθo∇θ)∆φ. (49)
The solution (49) we search as
φ=φoη[θ-∫dt δωθs]. (50)
Here η is the quasi-stationary shape of the vortex, deter-
mined by (46); δωθs is the shift of the angle frequency of
the vortex, determined by its interaction with ions.
From (46), (49), (50) one can show that the non-lin-
ear growth rate of the vortex excitation is proportional
to γNLs∝γs∝(me/mi)1/3.
From (44) and hydrodynamic equations for ions in
stationary approximation we derive for a quick vortex
[(Vθo-Vsθ)∇θ+(e/meωHe){φ, }r,θ](∆φ+qiδni4π)≈0. (51)
{φ, }r,θ=r-1[(∂rφ)∂θ-(∂θφ)∂r] , (Vsθ∇θ)2δni≈noi(qi/mi)∆φ.
Taking into account in (44) the term with ∂τ and ra-
dial non-uniformity of ωHe we derive the following
equation for a quick vortex
∂τδne≈(e/me)(∇θφ)noe∂r(1/ωHe). (52)
Using (41), we rewrite (54) in the following form
∂τ∆φ+4πqi∂τδni=(∇θφ)ω2
pe∂r(1/ωHe). (53)
From the hydrodynamic equations for ions one can
obtain the equation
(Vsθ∇θ-∂τ)2δni≈noi(qi/mi)∆φ. (54)
From (53), (54) we have, with taking into account that
for main terms of (51) the following equality (Vsθ∇θ)2φ
≈-ω2
piφ is approximately true, the equation
∂2
τ∆φ=(ω2
pi/2Vsθ)φω2
pe∂r(1/ωHe). (55)
Let us introduce a non-linear wave number, κNL, fol-
lowing the expression ∆φ≡-κ2
NLφ. Then from (55) we
obtain the non-linear growth rate of the quick vortex ex-
citation
γ(q)
NL≈(ωpi/κNL)[(ω2
pe/2Vθo)∂r(1/ωHe)]1/2 ∝ γq. (56)
5 CONCLUSION
So, it is shown that the electron density perturbation
in the plasma lens leads to the vortices. Two kinds of
vortices are excited: quick perturbations with Vph close
to the drift velocity, Vph≈Vθo, and slow perturbations
with Vph<<Vθo. Owing to the fast pass of ion beam
through the plasma lens and finite time of electron
renovating the perturbations are not excited in the
neighbourhood of its axis. The radial width of vortices
is proportional to the radical square from the amplitude
of the vortex electric potential, √φo. The radial width of
a quick vortex depends also on the radial gradient of a
magnetic field. The instability of excitation of a homo-
geneous vortical turbulence is saturated at a low level,
when a frequency of electron oscillations in the vortex
on the closed trajectories begins to exceed the growth
rate of the instability development. At large amplitudes
the set of separated non-linear vortices is excited. The
non-linear growth rates are proportional to the linear
growth rates.
6 ACKNOWLEDGEMENTS
This work was supported by STCU under grant
1596.
REFERENCES
1. A.A.Goncharov et al. Static and dynamic properties
of a high-current plasma lens // Plasma Physics
Rep. 1994, v. 20, № 5, p. 499-505.
154
3 STRUCTURE OF VORTEX
|
| id | nasplib_isofts_kiev_ua-123456789-79264 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:35:14Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Goncharov, A.A. Gubarev, S.N. Maslov, V.I. Onishchenko, I.N. 2015-03-30T08:25:11Z 2015-03-30T08:25:11Z 2001 Effects of vortices on ion beam focusing in a plasma lens / A.A. Goncharov, S.N. Gubarev, V.I. Maslov, I.N. Onishchenko // Вопросы атомной науки и техники. — 2001. — № 3. — С. 152-154. — Бібліогр.: 1 назв. — англ. 1562-6016 PACS numbers: 29.20.Bd https://nasplib.isofts.kiev.ua/handle/123456789/79264 For high-current ion-beam focusing a vortical turbulence has been excited in plasma lens by the nonremovable gradient of an external magnetic field. The paper presents theoretical investigation, in the cylindrical approximation, of excitation of slow and fast vortices with taking into account the finite length of plasma lens. It is shown that the growth rate of vortix excitation decreases with decreasing the length of plasma lens. The spatial structures of the vortices are constructed. The expression for the vortex amplitude of saturation is derived. This work was supported by STCU under grant 1596. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Effects of vortices on ion beam focusing in a plasma lens Влияние вихревой турбулентности на фокусировку ионного пучка плазменной линзой Article published earlier |
| spellingShingle | Effects of vortices on ion beam focusing in a plasma lens Goncharov, A.A. Gubarev, S.N. Maslov, V.I. Onishchenko, I.N. |
| title | Effects of vortices on ion beam focusing in a plasma lens |
| title_alt | Влияние вихревой турбулентности на фокусировку ионного пучка плазменной линзой |
| title_full | Effects of vortices on ion beam focusing in a plasma lens |
| title_fullStr | Effects of vortices on ion beam focusing in a plasma lens |
| title_full_unstemmed | Effects of vortices on ion beam focusing in a plasma lens |
| title_short | Effects of vortices on ion beam focusing in a plasma lens |
| title_sort | effects of vortices on ion beam focusing in a plasma lens |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79264 |
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