EH-undulative system for “effective cooling” of electron beams
The system for forming charged particles beams with small energy spread is proposed and studied. It is constructed on the basis of the Undulative Induction Accelerator (EH-accelerator). The obtained results of numerical modeling show that the proposed system allows to reduce the beam energy spread m...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Cite this: | EH-undulative system for “effective cooling” of electron beams / V.V. Kulish, I.V. Gubanov, O.A. Orlova // Вопросы атомной науки и техники. — 2001. — № 3. — С. 80-82. — Бібліогр.: 3 назв. — англ. |
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irk-123456789-792462015-03-31T03:02:37Z EH-undulative system for “effective cooling” of electron beams Kulish, V.V. Gubanov, I.V. Orlova, O.A. The system for forming charged particles beams with small energy spread is proposed and studied. It is constructed on the basis of the Undulative Induction Accelerator (EH-accelerator). The obtained results of numerical modeling show that the proposed system allows to reduce the beam energy spread more than ~40 times. The mechanism of energy “equalization” of electrons during the accelerating beam is investigated. This mechanism is called the "effective cooling". 2001 Article EH-undulative system for “effective cooling” of electron beams / V.V. Kulish, I.V. Gubanov, O.A. Orlova // Вопросы атомной науки и техники. — 2001. — № 3. — С. 80-82. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS numbers: 29.27.Ag http://dspace.nbuv.gov.ua/handle/123456789/79246 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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The system for forming charged particles beams with small energy spread is proposed and studied. It is constructed on the basis of the Undulative Induction Accelerator (EH-accelerator). The obtained results of numerical modeling show that the proposed system allows to reduce the beam energy spread more than ~40 times. The mechanism of energy “equalization” of electrons during the accelerating beam is investigated. This mechanism is called the "effective cooling". |
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Kulish, V.V. Gubanov, I.V. Orlova, O.A. |
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Kulish, V.V. Gubanov, I.V. Orlova, O.A. EH-undulative system for “effective cooling” of electron beams Вопросы атомной науки и техники |
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Kulish, V.V. Gubanov, I.V. Orlova, O.A. |
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EH-undulative system for “effective cooling” of electron beams |
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EH-undulative system for “effective cooling” of electron beams |
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EH-undulative system for “effective cooling” of electron beams |
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eh-undulative system for “effective cooling” of electron beams |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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EH-undulative system for “effective cooling” of electron beams / V.V. Kulish, I.V. Gubanov, O.A. Orlova // Вопросы атомной науки и техники. — 2001. — № 3. — С. 80-82. — Бібліогр.: 3 назв. — англ. |
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Вопросы атомной науки и техники |
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AT kulishvv ehundulativesystemforeffectivecoolingofelectronbeams AT gubanoviv ehundulativesystemforeffectivecoolingofelectronbeams AT orlovaoa ehundulativesystemforeffectivecoolingofelectronbeams |
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EH-UNDULATIVE SYSTEM FOR “EFFECTIVE COOLING”
OF ELECTRON BEAMS
V.V. Kulish, I.V. Gubanov1, O.A. Orlova1
Dep. of Physics I, National Aviation University,
1 Komarova Prospect, Kiev, 03058, Ukraine
1Dep. of Theoretical Physics, Sumy State University,
2 Rymski-Korsakov Str., 44007, Ukraine
The system for forming charged particles beams with small energy spread is proposed and studied. It is constructed
on the basis of the Undulative Induction Accelerator (EH-accelerator). The obtained results of numerical modeling
show that the proposed system allows to reduce the beam energy spread more than ~40 times. The mechanism of en-
ergy “equalization” of electrons during the accelerating beam is investigated. This mechanism is called the "effec-
tive cooling".
PACS numbers: 29.27.Ag
1 INTRODUCTION
The problem of forming beams with small particle
energy spread is rather "popular" and topical in various
areas of vacuum electronics and acceleration technolo-
gies. The main goal of this paper is to substantiate a
possibility of constructing the quasi-stationary EH-cool-
ers on the basis of "effective cooling" effect. The linear-
ly polarized stationary Undulative Induction Accelera-
tors (EH-accelerators) is proposed as a technological ba-
sis for such system.
Unlike to the earlier studied analogues EH-coolers
[1-3] the proposed system has a number of advantages.
Firstly, it is quasi-stationary. This means that ampli-
tudes and phases of the EH-fields are constant during
the time of beam passing. Hence, the particle energy
turns out to be the same as well for the first and the last
bunch particles. Secondly, the proposed system is more
compact and more acceptable technologically.
The project analysis shown that practical realization
of the proposed system should not meet any essential
difficulties because all its basic elements are well
known in technologies of Linear Induction Accelerators
(LINAC).
2 OPERATION PRINCIPLES
The proposed scheme of system for “effective cool-
ing” is shown on Fig. 1 (where only one period of undu-
lation is shown only, for simplicity). As it is mentioned
above, design of this system is based on the section of a
stationary linearly polarized EH-accelerator [1] (see
Fig.1).
The system works in the following manner. The
magnetic component of joint EH-undulative field (see
item 1 in Fig.1) is generated by permanent magnetic un-
dulator 2 in the work bulk of the EH-cooler. The vortex
electrical component 6 is generated by special induction
coils 4, which are situated between the magnetic poles
of the magnetic undulator 2. Each these coils consist of
the ferrite core and the winding. The magnetic screens
are used in the system input and output for reducing in-
fluence of border magnetic fields on dynamics of the
beam motion.
The basic work principles of the acceleration process
had been described earlier in papers [1-3]. Therefore, let
us pay our attention to the “effective cooling” mecha-
nism only.
Fig. 1. Design of the stationary linearly polarized
EH-cooler (only one period). There: 1 is the vector
of induction of the undulative magnetic field B
,
2 are the poles of permanent undulator, 3 are di-
rections of magnetic fluxes in the ferrite cores,
4 are the inductors, which consist of the ferrite
cores and winding, correspondingly, 5 is the accel-
erating channel, 6 are vectors of intensity of the
vortex electrical field E
.
The electron bunch moves under influence of the un-
dulation magnetic field 1 on undulation-like trajectory.
Acceleration of the electrons occurs under influence of
the undulation vortex electrical field 6. Therein, it is im-
portant that different electrons with different initial en-
ergy are characterized by the different amplitudes of os-
cillations. Namely, the higher is electron energy the
smaller is its oscillation amplitudes. This means that the
electrons with higher energy move on shorter trajecto-
ries than the electrons with lower energy. Hence, the
particles with lower initial energy obtain some more en-
ergy during the acceleration process than the particles
with higher energy. As a result, the energy equalization
process realizes during the electron bunch acceleration.
As it is mentioned above, we referred this physical
mechanism of energy equalization to as the “effective
cooling” of electron beam in the EH-system. The sys-
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №3.
Серия: Ядерно-физические исследования (38), с. 80-82.
80
tem constructed on this basis is called the EH-coolers.
We choose for the project analysis the model, which
is close considerably to the real EH-accelerator. The
asymptotic hierarchical method (see [3]) is used as a ba-
sic mathematical tool.
The method of secondary sources and the method of
magnetic streams are used for calculation of undulative
magnetic and electric fields for the given geometry of
the magnetic poles and the ferrite cores. The calculated
parameters are shown in Table 1. The parameters of the
initial (i.e., non-cooled) electron bunch are also given in
this table.
As the analysis shows the two following typical pe-
culiarities of the considered real EH-accelerative struc-
ture could be noted. The first is presence of remarkable
expressed high harmonics of the EH-field. The second
is generation of the longitudinal component of the EH-
field. Both these features are not taken into account in
the earlier studying simplified models [1-3]. As it will
be shown below that the accounting these features could
essentially change the project characteristics the pro-
posed EH-cooler.
3 SINGLE-PARTICLES THEORY
Let us to expand the magnetic and electric compo-
nents of the EH-field in a Fourier series. Representing
the field harmonics in form icxB )( and ibxB )( iaxE )(
we get the looked for series:
( ) { } ( )ikzikychazxEeE
i
ix sin,
1
0 ∑
∞
=
=
, (1)
( ) { } ( ) −
= ∑
∞
=
ikzikychcezxBB
i
iy sin,
1
0
{ } ( )
− ∑
∞
=
ikzikyshbe
i
iz cos
1
, (2)
where ( ) ( ) ( )xBzQzxB =,0 ; ( ) ( ) ( )xEzQzxE =,0 are
amplitudes of corresponding harmonics. The multipliers
)(xB and )(xE take into account the possibility of
changing the magnetic and electrical fields along x-axis.
The multiplier
( ) { } ( ){ }( )LzthzthzQ −χ−χ=
2
1
, (3)
takes into account that the fields in the input and in the
output of the system decrease smoothly. Here χ is the
parameter of screening.
The Lorentz equation is chosen for single- particle
description of the electron dynamics in fields (1), (2).
Application of the hierarchic asymptotic method allows
to get the averaged system of equations and correspond-
ing set of total solutions. The latter are expressed via
these averaged values. Let’s accomplish the analysis of
the “effective cooling” process dynamics, using the
method of large particles and the obtained solutions.
We divide the cooled beam into ten large particles.
In what follows, their energy dynamics (on the normal-
ized longitudinal coordinate LzT /= , where L the sys-
tem length) is studied. The results of relevant calcula-
tions are shown on Fig. 2. It should be noted that be-
cause the parameters of the system don not change in
time the effect of “capture” in the system is absent. In
contrast, this effect essentially determines the dynamics
of non-stationary versions of the EH-coolers [1-3].
Some of their specific features are mentioned above in
this paper. Dynamics of the considered effect of equal-
ization of electron energies in such stationary EH-cooler
is obviously illustrated Fig. 2.
Fig. 2. Dependencies of averaged kinetic electron
energy iE , and the corresponding energy spread
( ) EEE minmax −=δ of ten large particles (
10,...,2,1=i ) on the normalized longitudinal co-
ordinate LzT = . All particles differ from each
other by initial energies only. Curves 1, 2 show
maximal and minimal electron energies, accord-
ingly, curve 3 shows the dependence of corre-
sponding energy spread ( ) EEE minmax −=δ
on the undimensional longitudinal coordinate. All
parameters are given in Table 1.
Due to the system is stationarity the electron beam
velocity don not depend on the time of electron entering
in the system. Therein, the maximum of “effective cool-
ing” efficiency can be determined as
min)/()( 2
1
2 →−><><−
=
=
∑
Lz
N
i
i
mcHHH , (4)
i.e., the maximum of cooling we determine as the sys-
tem state, when the relative energy spread is minimal.
Therein, we assume that the spatial distribution of the
initial bunch in the system input has a П-type form.
Analysis of the definition (4) gives an opportunity to get
the expression for the optimal amplitude of intensity of
the undulative electrical field:
LBe
pkc+kcp
E
0
2
2
0
2
00
0 4
2H
= . (5)
It should be pointed out that expression (5) has been ob-
tained in the zeroth Bogolyubov approximation. This
means that fast spatial electron oscillations here are not
accounted. More exact analogous results can be ob-
tained in the case, when the motion equations are inte-
grated in following higher approximations. However,
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №3.
Серия: Ядерно-физические исследования (34), с. 81-82.
81
the main features, which are described by expression
(5), conserve in these cases, too. For instance, the analo-
gous calculations accomplished for the first Bogolyubov
approximation give the result for 0E which differ at 5-7
% from the (5).
Table 1. Parameters of the EH-cooler
Parameters Magni-
tudes
The general parameters of the system
Induction of the magnetic field, B 180 Gs
Intensity of the electrical field E 1 MV/m
Length of the system L 1 m
Period of undulation Λ 20 cm
Width of the system H 40 cm
Distance between the magnets ( 1Y ) 3,5 cm
Sizes of the magnet poles ( 1L ) 3,5 cm
Height of the induction block 1H 5 cm
Parameters of electron beam in the system input
Average energy of the beam 0
E 150 keV
Energy spread E∆ 45 %
Diameter of the beam d 2 mm
Angle of the beam divergence α± 1,5 0
Duration of electron bunch τΔ ~3·10-8 с
Parameters of electron beam in the system output
Average energy of the beam 0
E 500 keV
Energy spread E∆ 1.07 %
Diameter of the beam d 6 mm
Angle of the beam divergence α± 0,017 0
4 INHOMOGENEOUS MODEL
OF THE EH-COOLER
As it is known [2] the inhomogeneous (with respect
to the field amplitudes) nonstationary EH-cooling sys-
tems could be rather affective for practice. At the same
time, these systems have one essential defect. Namely,
the capture effect here limits the rate of electron beam
acceleration and, consequently, the level of its cooling.
As a result, this version of the effective cooling effect
can be used practically for the cooling short bunches on-
ly. In contrast, the capture effect does not realize in the
stationary EH-coolers. Hence, we might expect that the
stationary system could be suitable for the cooling
longer bunches. So, let us discuss the inhomogeneous
stationary EH-coolers in more details.
We choose the longitudinal inhomogeneity in the
following form:
( )
β
+
= z
pc
eEsh
eE
pc
z
pc
eEchBzBy
0
0
00
0
2
0
0
0 2
2
2
,
where 0B is the induction of the magnetic field for the
equivalent homogeneous system, 0E is the intensity of
the vortex electrical field of the EH-system (that we, as
before, consider as homogeneous one).
Fig. 3. Dependencies of kinetic energy for ten elec-
trons (which differ by initial energies only - curve
1), and relative energy spread (curve 2) on the
non-dimensional longitudinal coordinate LzT /=
for the longitudinally inhomogeneous model. Here:
the intensity of the electrical field is =190 kV/m,
the period of undulation is 20 cm, the system
length is 1 m.
The dynamics of the “effective cooling” process in
such model is illustrated on Fig. 3. Curve 1 demon-
strates the dependence of non-averaged energy (first
Bogolyubov approximation) for ten electrons on the
longitudinal coordinate T. Curve 2 illustrates analogous
dynamics for the relative energy spread . As it is readily
seen the decreasing the initial energy spread at 75 times
(from ~12% to ~0.16%) can be attained in the discussed
inhomogeneous EH-cooler.
5 CONCLUSION
Thus, the performed analysis gives a hope that the
proposed linearly polarized stationary EH-cooler will to
solve a number of problems of forming electron beams
with low energy spread. The authors consider that pro-
posed EH-coolers could be useful for wide experimental
practice.
REFERENCES
1. V.V.Kulish, P.B.Kosel. A new principle of acceler-
ation of high power pulses of quasi-neutral plasmas
and charged particles // Proceeding of “11th IEEE
International Pulsed Power Conference. – Balti-
more, Maryland. 1998. v. 1, p. 667 – 672.
2. V.V.Kulish, P.B.Kosel, A.G.Kailyuk, I.V.Gubanov.
New acceleration principle of charged particles for
electronic applications. Examples // The Interna-
tional Journal of Infrared and Millimeter Waves.
1998, v. 19, №2, p. 251-328.
3. V.V.Kulish. Hierarchic oscillations and averaging
methods in nonlinear problems of relativistic elec-
tronics // The Journal of Infrared and Millimeter
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