Relativistic electron bunch excitation of wake fields in magnеtoactive plasma
We have been demonstrated that a multiple excess of the accelerated bunch energy εmax over the energy of the exciting. REB is possible in a magnetoactive plasma at a certain relationship between the parameters of the “plasmabunch-magnetic field” system owing to a hybrid volume-surface character of R...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Цитувати: | Relativistic electron bunch excitation of wake fields in magnеtoactive plasma / V.A. Balakirev, V.I.Karas`, I.V. Karas`, G.V. Sotnikov // Вопросы атомной науки и техники. — 2000. — № 1. — С. 113-116. — Бібліогр.: 10 назв. — англ. |
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irk-123456789-816202015-05-19T03:02:40Z Relativistic electron bunch excitation of wake fields in magnеtoactive plasma Balakirev, V.A. Karas, V.I. Karas, I.V. Sotnikov, G.V. Новые мeтoды ускорения заряженных частиц We have been demonstrated that a multiple excess of the accelerated bunch energy εmax over the energy of the exciting. REB is possible in a magnetoactive plasma at a certain relationship between the parameters of the “plasmabunch-magnetic field” system owing to a hybrid volume-surface character of REB-excited wake fields, even without using the REBs contoured in the longitudinal direction. 2000 Article Relativistic electron bunch excitation of wake fields in magnеtoactive plasma / V.A. Balakirev, V.I.Karas`, I.V. Karas`, G.V. Sotnikov // Вопросы атомной науки и техники. — 2000. — № 1. — С. 113-116. — Бібліогр.: 10 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/81620 533.9 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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English |
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Новые мeтoды ускорения заряженных частиц Новые мeтoды ускорения заряженных частиц |
| spellingShingle |
Новые мeтoды ускорения заряженных частиц Новые мeтoды ускорения заряженных частиц Balakirev, V.A. Karas, V.I. Karas, I.V. Sotnikov, G.V. Relativistic electron bunch excitation of wake fields in magnеtoactive plasma Вопросы атомной науки и техники |
| description |
We have been demonstrated that a multiple excess of the accelerated bunch energy εmax over the energy of the exciting. REB is possible in a magnetoactive plasma at a certain relationship between the parameters of the “plasmabunch-magnetic field” system owing to a hybrid volume-surface character of REB-excited wake fields, even without using the REBs contoured in the longitudinal direction. |
| format |
Article |
| author |
Balakirev, V.A. Karas, V.I. Karas, I.V. Sotnikov, G.V. |
| author_facet |
Balakirev, V.A. Karas, V.I. Karas, I.V. Sotnikov, G.V. |
| author_sort |
Balakirev, V.A. |
| title |
Relativistic electron bunch excitation of wake fields in magnеtoactive plasma |
| title_short |
Relativistic electron bunch excitation of wake fields in magnеtoactive plasma |
| title_full |
Relativistic electron bunch excitation of wake fields in magnеtoactive plasma |
| title_fullStr |
Relativistic electron bunch excitation of wake fields in magnеtoactive plasma |
| title_full_unstemmed |
Relativistic electron bunch excitation of wake fields in magnеtoactive plasma |
| title_sort |
relativistic electron bunch excitation of wake fields in magnеtoactive plasma |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2000 |
| topic_facet |
Новые мeтoды ускорения заряженных частиц |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/81620 |
| citation_txt |
Relativistic electron bunch excitation of wake fields in magnеtoactive plasma / V.A. Balakirev, V.I.Karas`, I.V. Karas`, G.V. Sotnikov // Вопросы атомной науки и техники. — 2000. — № 1. — С. 113-116. — Бібліогр.: 10 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT balakirevva relativisticelectronbunchexcitationofwakefieldsinmagnetoactiveplasma AT karasvi relativisticelectronbunchexcitationofwakefieldsinmagnetoactiveplasma AT karasiv relativisticelectronbunchexcitationofwakefieldsinmagnetoactiveplasma AT sotnikovgv relativisticelectronbunchexcitationofwakefieldsinmagnetoactiveplasma |
| first_indexed |
2025-07-06T06:49:25Z |
| last_indexed |
2025-07-06T06:49:25Z |
| _version_ |
1836879257969623040 |
| fulltext |
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ 2000. №1.
Серия: Плазменная электроника и новые методы ускорения (2), с. 113-116.
113
UDK 533.9
RELATIVISTIC ELECTRON BUNCH EXCITATION OF WAKE FIELDS
IN A MAGNETOACTIVE PLASMA
V.A. Balakirev, V.I. Karas’, I.V. Karas’, G.V. Sotnikov
National Science Center “Kharkov Institute of Physics and Technology”,
Kharkov 61108, Ukraine,karas@kipt.kharkov.ua
We have been demonstrated that a multiple excess of the accelerated bunch energy εmax over the energy of the
exciting. REB is possible in a magnetoactive plasma at a certain relationship between the parameters of the “plasma-
bunch-magnetic field” system owing to a hybrid volume-surface character of REB-excited wake fields, even without
using the REBs contoured in the longitudinal direction.
The ideas of using collective fields for acceleration
in the plasma and noncom-pen-sated charged beams
were stated as early as in 1956 by V.I. Veksler, G.I.
Budker and Ya.B. Fainberg [1-3]. The appearance and
development of new powerful energy sources such as
lasers, high-current relativistic electron beams,
superhigh-power microwave generators, gave another
impetus to the development of the collective methods of
charged particle acceleration. As a result, in 1979 [4]
and 1985 [5], there appeared new modifications of the
method of charged particle acceleration in a plasma by
charge density waves (ref. [3]), where it was proposed
that the accelerating fields should be excited by laser
pulses and relativistic electron bunches.
In our opinion, the excitation of accelerating fields in
a plasma by an individual relativistic electron bunch
(REB) appears most preferable, because it is
nonresonant in character, and therefore, is little sensitive
to the longitudinal plasma density inhomogeneity
observed in experiment. Besides, to preclude the
electromagnetic filamentation slipping of instabilities,
etc. [6-9], it is reasonable to use the stabilizing external
longitudinal magnetic field [10]. Aside from
stabilization, the magnetic field also gives rise to a
multitude of new wave branches, and this, as will be
shown below, essentially extends the potentialities of the
wake-field method of charged particle acceleration.
The report presents the results from theoretical
studies into the processes of REB excitation of wake
fields in a magnetoactive plasma, both the cases of an
unbounded plasma and the waveguide with a partial
plasma filling.
1. Let us define the wake-field by an axially
symmetric REB moving in the magnetoactive plasma
along the axis z. We neglect the thermal motion of
electrons, and assume ions to be immobile. The REB
current density is written as
( )
0
02ext z
r zj I T t e
V
ψ
π
= − −
! !
,(1)
where I0 is the peak value of the total current of bunch,
ψ(r) is the function describing the current distribution in
the bunch cross section, the function T describes the
longitudinal profile of the bunch (maxT = 1), t is the
time, z and r are, respectively, the longitudinal and
radial coordinates, V0 is the bunch velocity, ez is the unit
vector in the direction of bunch motion and an external
magnetic field H0 = H0ez. Formula (1) describes the
REB current density in the approximation of assigned
motion, i.e., the “rigid” bunch model is used. The
function ψ(r) satisfies the condition
( )
0
1br r rdrψ =∫ ,
where rb is the maximum radius of REB.
We express the electric field E, the magnetic
field H and the current density of REB (1) in terms of
the Fourier integral, then the set of Maxwell equations
takes the form
0 rkE k Hωϕ ω= , ( )0 1 2rkH k E i Eωϕ ω ωϕε ε= + ,
0
1
z
d rE ik H
r dr ωϕ ω= ,
( )0 1 2
r
r r
dH ikH ik E i E
dr
ω
ω ωϕ ωε ε− = − , (2)
0
z
r
dE ikE ik H
dr
ω
ω ωϕ− = ,
( ) ( )0
0 3
21
z
Id rH ik E r T
r dr cωϕ ωε ψ ω= − ,
where 0k Vω= , 0k cω= ,
( )
( )
2
1 2 2
1 pe
He
i
i
ω ω ν
ε
ω ω ν ω
+
= −
+ −
,
( )
2
2 2 2
pe He
Hei
ω ω
ε
ω ω ν ω
=
+ −
,
( )
2
3 1 pe
i
ω
ε
ω ω ν
= −
+
,ωpe,ωHe are, respectively, the
Langmuir and Larmor frequencies of plasma electrons, v
is the effective collision frequency, ϕ is the azimuthal
coordinate.
The set of first-order differential equations (2) is
conveniently reduced to coupled second-order
114
differential equations for longitudinal components of
electric and magnetic fields
( ) ( )
2 2 3
0
1
02
1
1
2
z
H z z
dHd r p H ikk E
r dr dr
Ik r T
c
ω
ω ω
ε ε
ε
ε ψ ω
ε
+ = − −
−
,
( ) ( ) ( )
2 2
0
1
2 2
0 1 0
0 1
1
2
z
E z z
dEd r p E ikk H
r dr dr
k k Ii r T
k c
ω
ω ω
ε
ε
ε
ψ ω
ε
+ = +
−
+
,(3)
where
2 2
2 2 21 2
0
1
Hp k kε ε
ε
−= − ,
2 2 2 3
0 3
1
Ep k k εε
ε
= − .
The solution to the nonuniform set of equations has
the form
( )
( )
( ) ( )
0 2
1
1 1 0 1
0 0 0
2 2 0 2
,
,
z
I kH i T
c
AG r r
r dr r
A G r r
ω
επ ω
ε
λ λ
ψ
λ λ
= ×
−
×
−
∫
,
( )
( ) ( ) ( )
0
0 3
1 1 0 1 2 2 0 2 0 0 0
1
, ,
z
IE T
c k
B G r r B G r r r dr r
ω π ω
ε
λ λ λ λ ψ
= − ×
× − ∫
,(4)
where
2
2 2
1 2
i
iA λ
λ λ
=
−
,
( )2 2 2
2 2
1 2
i H i
i
p
B
λ λ
λ λ
−
=
−
,
λ1,2 are the transverse wave numbers of ordinary and
inordinary waves, respectively, and are the roots of the
biquadratic equation
( )
2
4 2 2 2 2 2 2
0 3
1
2 2 0
E H
E H
p p k k
p p
ελ λ ε
ε
− + − +
+ =
,(5)
They can be written as:
( )2 22 22 2
2 2 2
1,2 0 3
12 4
E HE H
p pp p k k ελ ε
ε
− += ± +
,
( )
( ) ( ) ( )
( ) ( ) ( )
1
0 0 0 0
0 1
0 0 0 0
,
,
,
i i
i i
i i
H r J r r r
G r r
H r J r r r
λ λ
λ λ
λ λ
>=
<
where J0(λ ir), H0
(1) (λ ir) are the Bessel functions and
Hankel functions, respectively.
In the ultrarelativistic case, we can put V0 = c. Then,
instead of eq.(5) we have
( )
2
2 0
1,2 3 1 2 3
1
1kλ ε ε ε ε
ε
= − ± . (6)
In the frequency range ω2 < ω2
pe the transverse wave
numbers are complex.
In the limiting case of a strong magnetic field, ω2
He
>> ω2
pe, the expression for transverse wave numbers
takes the form
2 2 2
1 0k kλ = − , ( )2 2 2
2 0 3k kλ ε= − .
For simplicity sake we consider an infinitely thin
annular bunch of radius rb
( ) 1T ω = . ( ) ( )b
b
r r
r
r
δ
ψ
−
= (7)
Then, for the electric field on the axis of the system r
= 0 we obtain the following integral representation
( ) ( ) ( )10
02 2
0 0
1 exp
2z b
QE i t H k r d
V
ω ω ω
γ
∞
⊥
−∞
=− −∫ , (8)
where Q0 is the total charge of bunch, γ0 is the
relativistic factor, k⊥ 2 = (k0
2 - k2)ε3. Note that in the
frequency range ω < ωpe the bunch emits an
electromagnetic field in the radial direction, because
k⊥ 2 > 0.
The argument of the Hankel function has two branch
points ω = ±ωpe - iv/2 lying in the lower half-plane of the
complex variable ω. Let us draw a cut along the line of
segment connecting the branch points, and close the
integration contour in the lower half-plane over the
semi-circle of infinite radius. As a result, the initial
integral (8) reduces to the integral over the cut shores.
Then, using the path-tracing rule for the Hankel
function, we obtain the following expression for the
longitudinal component of the electric wake field
( )
( )
2 1
0
2 2 2
0 0 1 0
exp
1
pe
z
isQ
E i
V J s sds
τω
γ µ−
− ×
= −
× −∫ , (9)
where
0
pe
zt
V
τ ω
= −
,
0 0
pe br
V
ω
µ
γ
= .
After calculation of the integral we obtain from (9)
115
.
2
0
2 2 2 2
0 0
2 2
2 2
2 2
2
sin
cos
pe
z
Q
E
V
ω τ
γ τ µ
τ µ
τ µ
τ µ
= ×
+
+
−
× +
− +
(10)
At large distances behind the bunch, the wake-wave
field decreases as 1/τ. This is due to the fact that the
oscillations in the plasma placed in a strong magnetic
field have the finite group velocity. The radiation of
plasma waves from the near-axis region causes the wake
field to decrease in the longitudinal direction.
2. Let us consider the plasma waveguide with a
partial plasma filling in the external magnetic field, i.e.,
the waveguide, where between the plasma boundary r =
a and the conducting housing r = b there is a vacuum
gap.
To derive a dispersion equation of eigenwaves of this
plasma wave guide, it is necessary to find the
electromagnetic fields in the vacuum gap b<r<a and
join them with the plasma fields through the use of
boundary conditions on the plasma surface. The
boundary conditions are standard, i.e., the continuity of
electromagnetic-field tangential components. As a
result, we obtain the dispersion equation, which can be
conveniently written as a determinant
Det 0A = , (11)
where the matrix A has the following components
11 1A = , 12 1A = , 13 1A = − , 14 0A = ,
( )
( )
1 1
21 1
1 0 1
J a
A
c J a
λω
λ λ
= Γ ,
( )
( )
1 2
22 2
2 0 2
J a
A
c J a
λω
λ λ
= Γ ,
23 0A = , 24A
cw
ω= ,
31 1A = Γ , 32 2A = Γ , 33 0A = , ( )34A Q wa= ,
( )
( )
1 1
41 3
1 0 1
J a
A
c J a
λωε
λ λ
= ,
( )
( )
1 2
42 3
2 0 2
J a
A
c J a
λωε
λ λ
= ,
( )43 1A F wa
cw
ω= − , 44 0A = ,
where
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0 1 0 1
1 1 1 1
I wa K wb K wa I wb
Q wa
I wa K wb I wb K wa
+
=
−
,
( )
( )
1 3
2
2
1 22
0
1 222
1,2 1 3 1 2
1 0
2
2
2
2
2 3 2
0
1
2
4
k
k
k
k
k
k
ε ε
ε ε
ε ε εε
ε
ε ε
− ×
− − ±
Γ = − − − × + ± − +
( )
( )
2
1 3 1 2
0
1 222
0 1 3 1 2
1,2 0
1 2 2
2 2
2
2
2 3 2
0
2
4
k
k
kk
k
k
k
ε ε ε
ε ε ελ
ε
ε ε
ε ε
+ − −
− − − = + − ± −
+
0 0
1
0 0
( ) ( , )
;
( ) ( , )
p b
p
I w r wr wb
F
I w a wa wb
∆
=
∆
1
2 2 2
0 3( ) ;pw k k ε= −
2 2 2
0 ;w k k= −
0 0 0 0 0( ) ( ) ( ) ( );I wr K wb I wb K wr∆ = −
Character of field distribution in the cross section for the
waveguide is determined by the transverse wave
numbers. If λ1,22 > 0, then the wave is three-dimensional
(volumetric). However, if λ1,22 < 0, then the wave is
pertai pertaining to surface. And finally, if λ1,22 are the
complex variables, then the wave is hybrid. The
boundaries of the region, where λ1,22 become complex,
are defined by the inequalities
1 2ω ω ω> > ,
where
( )
1,2
1 22 2 4 2 2 2 2
2 2 2
2
4
pe He pe He pe He
He
kc
k c
k c
ω
ω ω ω ω ω ω
ω
= ×
+ ± + −
×
+
(12)
To excite the REB of the hybrid wave, the relativistic
factor must satisfy the following condition
0 2
He
pe
ωγ
ω
> . (13)
The field pattern and the frequency of the hybrid
wave being in synchronism with the bunch were found
by the numerical methods.
116
We attained the characteristic radial distribution of
the longitudinal electric-field component at the
following plasma and waveguide
parameters: 6.3He
pe
ω
ω
= , 23.3pea
c
ω
= , 2.4b
a
= ,
0 4.6γ = . The wake hybrid wave frequency is here
equal to 0.35 ωpe. It is shown that for the radius
r
a
= 0.8
the magnitude of the longitudinal electric-field
component has a deep maximum corresponding to the
energy transformation coefficient
( )
max
0
37z
E
z r
ER
E =
= = .
Note that a great value of the transformation
coefficient corresponds to a significant ( ER times)
excess of the maximum energy obtained by the
accelerated bunch as compared to the energy of the
bunch exciting the wake field, because the energy
transformation coefficient RE is equal to the ratio of the
electric field amplitude accelerating the guided bunch to
the electric field amplitude decelerating the bunch that
excites the accelerating wake field.
So, it has been demonstrated that a multiple excess of
the accelerated bunch energy εmax over the energy of the
exciting. REB is possible in a magnetoactive plasma at a
certain relationship between the parameters of the
“plasma-bunch-magnetic field” system owing to a
hybrid volume-surface character of REB-excited wake
fields, even without using the REBs contoured in the
longitudinal direction, namely:
( )2
max 0 1Emc Rε γ= − . (14)
References
1. V.I.Veksler //Proc. Symp.CERN, 1956, vol. 1.p. 80.
2. G.I.Budker //Proc. Symp.CERN, 1956, vol. 1, p. 68.
3. Ya.B.Fainberg//Proc. Symp .CERN, 1956, vol. 1, p. 84.
4. T Tajima, J.M. Dawson // Phys. Rev. Lett., 1979,
vol. 43,p. 267.
5. Chen P., J.M. Dawson J.M., T Katsouleas., et al. //
Phys. Rev. Lett., 1985, vol. 54, p. 693
6. T.Katsouleas // Phys. Rev. A, 1986, vol. 33. p. 2066.
7. R.,Keinigs, M.E.Jones // Phys. Fluids, 1987, vol. 30,
# 1, p.252.
8. Ya.B. Fainberg Ya.B. // Plasma Physics Reports,
1997, vol. 23, # 4, p. 251.
9. Ya.B. Fainberg // Plasma Physics Reports, 2000, vol.
26, # 4, p.324.
10. A.Ts.Amatuni., E.V.Sekhposyan., A.G.Khachatryan,
and S.S.Elbakyan // Plasma Physics Reports, 1995,
vol. 21, # 11, p. 945.
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