Energy spectra of accelerated electrons of chain relativistic electron bunches at excitation of wakefield in dielectric resonator
The energy spectra of a long sequence of relativistic electron bunches at the output of the dielectric resonator taking into account initial energy spread of the relativistic electron bunches are investigated. It is considered also the finite value of the Q-factor of the dielectric cavity. Досліджен...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Cite this: | Energy spectra of accelerated electrons of chain relativistic electron bunches at excitation of wakefield in dielectric resonator / V.A. Balakirev, I.N. Onishchenko, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2015. — № 4. — С. 94-96. — Бібліогр.: 1 назв. — англ. |
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| author | Balakirev, V.A. Onishchenko, I.N. Tolstoluzhsky, A.P. |
| author_facet | Balakirev, V.A. Onishchenko, I.N. Tolstoluzhsky, A.P. |
| citation_txt | Energy spectra of accelerated electrons of chain relativistic electron bunches at excitation of wakefield in dielectric resonator / V.A. Balakirev, I.N. Onishchenko, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2015. — № 4. — С. 94-96. — Бібліогр.: 1 назв. — англ. |
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| description | The energy spectra of a long sequence of relativistic electron bunches at the output of the dielectric resonator taking into account initial energy spread of the relativistic electron bunches are investigated. It is considered also the finite value of the Q-factor of the dielectric cavity.
Досліджено енергетичні спектри довгої послідовності релятивістських електронних згустків на виході діелектричного резонатора з урахуванням енергетичного розкиду релятивістських електронних згустків на вхідному торці резонатора. Враховано також скінченне значення добротності діелектричного резонатора.
Исследованы энергетические спектры длинной последовательности релятивистских электронных сгустков на выходе диэлектрического резонатора с учетом энергетического разброса релятивистских электронных сгустков на входном торце резонатора. Учтено также конечное значение добротности диэлектрического резонатора.
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ISSN 1562-6016. ВАНТ. 2015. №4(98)
94
NEW METHODS OF CHARGED PARTICLE ACCELERATION
ENERGY SPECTRA OF ACCELERATED ELECTRONS
OF CHAIN RELATIVISTIC ELECTRON BUNCHES AT EXCITATION
OF WAKEFIELD IN DIELECTRIC RESONATOR
V.A. Balakirev, I.N. Onishchenko, A.P. Tolstoluzhsky
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: tolstoluzhsky@kipt.khakov.ua
The energy spectra of a long sequence of relativistic electron bunches at the output of the dielectric resonator
taking into account initial energy spread of the relativistic electron bunches are investigated. It is considered also the
finite value of the Q -factor of the dielectric cavity.
PACS: 41.75.Jv, 41.75.Lx, 41.75.Ht, 96.50.Pw
INTRODUCTION
A characteristic feature of the experimental setup
"Almaz 2" is the presence of energy spread in a relativ-
istic electron bunches. Energy spread will influence on
the process of wakefield excitation in the dielectric cavi-
ty and thus on the acceleration of electrons by these
fields. Not less important is the account of the finite
value of the Q -factor of the dielectric cavity, as it limits
the wakefield excited by a long sequence of relativistic
electron bunches and the maximum energy of the accel-
erated electrons [1].
Let’s consider the process of self-acceleration of rel-
ativistic electron bunches which is realized under condi-
tions when there is a detuning between the repetition
frequency of the relativistic electron bunches and the
frequency of the excited oscillations, which is in che-
renkov synchronism with electron bunches. In this case,
electric field beat is excited. Wherein the first half of the
sequence of electron bunches is in the decelerating
phase and thus losses energy in the excitation of the
electric field. The second half of the sequence of bunch-
es is in the accelerating phase and therefore gaining
energy.
1. STATEMENT OF THE PROBLEM.
BASIC RELATIONS
The dielectric cavity is formed by a metallic cylin-
drical cavity which volume is completely filled a homo-
geneous dielectric with permittivity ε. The length of the
cavity is L, its radius is a. The cavity has a finite value
of the Q -factor. On the left end of the dielectric cavity
in the axial region periodic sequence of N relativistic
electron bunches with finite longitudinal and transverse
dimensions is injected. In the radial direction of the
charge density is distributed according to a Gaussian
law and in longitudinal direction charge density has a
rectangular profile. In addition, each relativistic electron
bunch has an initial energy spread, which is described
by the model distribution function
2
0
2
( )
0
1f e
γ γ
γ
π γ
−
−
∆=
∆
, (1)
where γ∆ is characteristic spread of relativistic factor,
0γ is the average value of the relativistic factor.
Energy i-th particle o f bunch with number α and
having the initial value of the relativistic factor lγ at the
input of the dielectric cavity is described by expression
,jl l jlα αγ γ γ= + ∆ (2)
where
2
20
0 2
1 0
4
( ) ,jl n m mn mn mn mn
n m
v
F r S G
Lα αγ ε α ω
∞ ∞
= =
∆ = − Λ
∑ ∑
sin
2
2
mn b
mn
mn b
t
G
t
ω
ω
= ,
mnω are frequencies of dielectric cavity oscillations, bt
is the bunch duration, 1/ 2mα = for 0m = and 1mα =
for 0m > ,
( ) ( )
( )
21
0 2
2
/ b nr
n a
n
n n
J r a
F r e
J
λλ
λ
−
= ⋅ ,
nλ are roots of Bessel’s function ( )( )0 0nJ λ = , br is
characteristic bunch radius,
( ) ( )
( )
( )
/ 2 /2
, ,
1Re
1
mn mn
mn j b mn j b
mn mn
T i T
t t i t t
m n T i T
ee e
e
α γ ω
γ ω
α γ ω
− −
− − −
−
−
Λ = ⋅
−
,
2
mn
mn Q
ω
γ = is damping decrement of the cavity oscilla-
tions, T is the repetition period of the bunches,
0/L jt T t z vα= + + is the Lagrangian time of j-th parti-
cle of the bunch with number α ;
( )2
0 02 2 2 2
0
4
1beQ L
a mc
ε β ε
ε β
= ⋅ − ,
bQ is charge of bunch, 0 0 /v cβ = , 0v is velocity of
bunches, ,e m are charge and mass of electron, c is
light velocity,
( )20
2 0
2 2 2
01
1 ( 1) cos
( )1
( )
m mn
mn
m mn
L
v
S d e
k v
γ γ
γ
ω
γ
γ
γ ωπ γ
−∞ −
∆
− −
=
−∆ ∫ ,
ISSN 1562-6016. ВАНТ. 2015. №4(98) 95
/mk m Lπ= .
Taking into account initial distribution function (1)
and the expression for the relativistic factor (2) the dis-
tribution function of sequence of N bunches at the exit
of the dielectric cavity can be written as
2
0
2
( )
1
1 l ljN
f e
αγ γ γ
γ
απ γ
+∆ −
−
∆
=
=
∆
∑ . (3)
2. NUMERICAL RESULTS
Numerical calculations of the energy distribution
function of sequence relativistic electron bunches at the
output dielectric cavity (3) were performed for the fol-
lowing parameters of the dielectric cavity and the se-
quence of relativistic electron bunches: 4.025a = cm is
the radius of the cavity, 31.925L = cm is the cavity
length, the dielectric constant is 2.045ε = , the initial
value of the relativistic factor is 0 9.8064γ = , the energy
spread is 0/ 3.6%γ γ∆ = , the charge of the electron
bunch is 0.32bQ = nC, longitudinal phase size of bunch
is 60°, / 1 / 8br a = is the radius of the bunch. The num-
ber of bunches in the sequence N = 1000, 1500 and the
quality factor of the cavity Q are varied. Simple esti-
mates show that Q -factor has no effect on the process
of self-acceleration of electron bunches sequence when
the condition is satisfied
Q Nωπ π
δω
>> = , (4)
where res мδω ω ω= − is the detuning between frequency
of resonant oscillation of the cavity resω and the repeti-
tion frequency of bunches мω , and was chosen from the
condition / 1 /res Nδω ω = . Note that for the chosen pa-
rameters of the dielectric cavity and the energy of
bunches the resonant oscillation have indexes
6, 1m n= = . Under condition (4) beat waves is formed
before stabilization of field growth due to the finite Q of
the resonator. In opposite case beat of wakefield is not
formed because in this case the field is stabilized at low
level.
Fig. 1 shows the distribution of energy (relativistic
factor) chains of bunches at the input and output of the
dielectric cavity for without ohmic losses. Noticeable
broadening of distribution function takes place.
Fig. 1. Initial distribution function (magenta line),
distribution function 0( / 0.36%)γ γ∆ = (brown line),
1000N =
Thus its maximum value slightly decreased since the
total area (total number of particles) is always con-
served. At half height of distribution function we ob-
serve increasing of energy equal to 34 keV, on
0.1 height increasing of energy is 73keV.
For a sequence of 1000N = bunches inequality (4)
takes the form 1000Q π>> . Fig. 2 illustrates the case
9000Q = when Q -factor has little influence on the
process of self-acceleration of bunches chain.
Fig. 2. Initial distribution function (magenta line),
distribution function 0( / 0.36%)γ γ∆ = (brown line),
1000N = , 9000Q =
Fig. 3. Initial distribution function (magenta line),
distribution function 0( / 0.36%)γ γ∆ = (brown line),
1000N = , 3000Q =
Energy distribution has decreased, though not signif-
icantly. The picture radically changes when the ine-
quality (4) is not satisfied. Fig. 3 shows the distribution
function for 3000Q = . It can be seen that the accelerat-
ed particles are absent and there are only slow particles.
The maximum of the distribution function is shifted
somewhat to the left.
Let us now discuss effects due to increasing of the
number of bunches ( 1500N = ). Fig. 4 shows the distri-
bution function at the output of the dielectric cavity is
lossless Q = ∞ . Increasing the number of bunches leads
to a marked broadening of the distribution function. At
half height of distribution function we observe increas-
ing of energy equal to 84 keV, on 0.1 height increasing
of energy is 136 keV.
ISSN 1562-6016. ВАНТ. 2015. №4(98)
96
Fig. 4. Initial distribution function (magenta line),
distribution function 0( / 0.36%)γ γ∆ = (brown line),
1500N =
Thus, increasing of bunches number in the sequence
requirements on Q -factor ( Q Nπ>> ) of the dielectric
cavity are increased. If this condition is not satisfied, the
proportion of accelerated particles is decreased and the
number of deceleration particles is increases.
ACKNOWLEDGEMENTS
This work was supported by the US Department of
Energy/NNSA through the Global Initiatives for Prolif-
eration Prevention (GIPP) Program in Partnership with
the Science and Technology Center in Ukraine (Project
ANL-T2-247-UA and STCU Agreement P522).
REFERENCES
1. N.I. Onishchenko, G.V. Sotnikov, T.C. Marshal.
Amplitudes and Spectra of Wake Fields in a Planar
Dielectric Cavity with Finite Q-factor // 12th Ad-
vanced Accelerator Concepts Workshop, Lake Ge-
neva, Wisconsin (USA), 10-15 July 2006; ISBN
978-0-7354-0378-9; AIP Conf. Proceedings / Edi-
tors: Manoel Conde and Catherine Eyberger. 2006,
v. 877, p. 866-872.
Article received 30.06.2015
ЭНЕРГЕТИЧЕСКИЕ СПЕКТРЫ УСКОРЕННЫХ ЭЛЕКТРОНОВ ЦЕПОЧКИ РЕЛЯТИВИСТСКИХ
ЭЛЕКТРОННЫХ СГУСТКОВ ПРИ ВОЗБУЖДЕНИИ КИЛЬВАТЕРНОГО ПОЛЯ
В ДИЭЛЕКТРИЧЕСКОМ РЕЗОНАТОРЕ
В.А. Балакирев, И.Н. Онищенко, А.П. Толстолужский
Исследованы энергетические спектры длинной последовательности релятивистских электронных сгуст-
ков на выходе диэлектрического резонатора с учетом энергетического разброса релятивистских электрон-
ных сгустков на входном торце резонатора. Учтено также конечное значение добротности диэлектрического
резонатора.
ЕНЕРГЕТИЧНІ СПЕКТРИ ПРИСКОРЕНИХ ЕЛЕКТРОНІВ ЛАНЦЮЖКА РЕЛЯТИВІСТСЬКИХ
ЕЛЕКТРОННИХ ЗГУСТКІВ ПРИ ЗБУДЖЕННІ КІЛЬВАТЕРНОГО ПОЛЯ
В ДІЕЛЕКТРИЧНОМУ РЕЗОНАТОРІ
В.А. Балакірев, І.М. Онищенко, О.П. Толстолужський
Досліджено енергетичні спектри довгої послідовності релятивістських електронних згустків на виході
діелектричного резонатора з урахуванням енергетичного розкиду релятивістських електронних згустків на
вхідному торці резонатора. Враховано також скінченне значення добротності діелектричного резонатора.
|
| id | nasplib_isofts_kiev_ua-123456789-112147 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-01T20:32:10Z |
| publishDate | 2015 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Balakirev, V.A. Onishchenko, I.N. Tolstoluzhsky, A.P. 2017-01-17T18:30:26Z 2017-01-17T18:30:26Z 2015 Energy spectra of accelerated electrons of chain relativistic electron bunches at excitation of wakefield in dielectric resonator / V.A. Balakirev, I.N. Onishchenko, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2015. — № 4. — С. 94-96. — Бібліогр.: 1 назв. — англ. 1562-6016 PACS: 41.75.Jv, 41.75.Lx, 41.75.Ht, 96.50.Pw https://nasplib.isofts.kiev.ua/handle/123456789/112147 The energy spectra of a long sequence of relativistic electron bunches at the output of the dielectric resonator taking into account initial energy spread of the relativistic electron bunches are investigated. It is considered also the finite value of the Q-factor of the dielectric cavity. Досліджено енергетичні спектри довгої послідовності релятивістських електронних згустків на виході діелектричного резонатора з урахуванням енергетичного розкиду релятивістських електронних згустків на вхідному торці резонатора. Враховано також скінченне значення добротності діелектричного резонатора. Исследованы энергетические спектры длинной последовательности релятивистских электронных сгустков на выходе диэлектрического резонатора с учетом энергетического разброса релятивистских электронных сгустков на входном торце резонатора. Учтено также конечное значение добротности диэлектрического резонатора. This work was supported by the US Department of Energy/NNSA through the Global Initiatives for Proliferation Prevention (GIPP) Program in Partnership with the Science and Technology Center in Ukraine (Project ANL-T2-247-UA and STCU Agreement P522). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Новые методы ускорения заряженных частиц Energy spectra of accelerated electrons of chain relativistic electron bunches at excitation of wakefield in dielectric resonator Енергетичні спектри прискорених електронів ланцюжка релятивістських електронних згустків при збудженні кільватерного поля в діелектричному резонаторі Энергетические спектры ускоренных электронов цепочки релятивистских электронных сгустков при возбуждении кильватерного поля в диэлектрическом резонаторе Article published earlier |
| spellingShingle | Energy spectra of accelerated electrons of chain relativistic electron bunches at excitation of wakefield in dielectric resonator Balakirev, V.A. Onishchenko, I.N. Tolstoluzhsky, A.P. Новые методы ускорения заряженных частиц |
| title | Energy spectra of accelerated electrons of chain relativistic electron bunches at excitation of wakefield in dielectric resonator |
| title_alt | Енергетичні спектри прискорених електронів ланцюжка релятивістських електронних згустків при збудженні кільватерного поля в діелектричному резонаторі Энергетические спектры ускоренных электронов цепочки релятивистских электронных сгустков при возбуждении кильватерного поля в диэлектрическом резонаторе |
| title_full | Energy spectra of accelerated electrons of chain relativistic electron bunches at excitation of wakefield in dielectric resonator |
| title_fullStr | Energy spectra of accelerated electrons of chain relativistic electron bunches at excitation of wakefield in dielectric resonator |
| title_full_unstemmed | Energy spectra of accelerated electrons of chain relativistic electron bunches at excitation of wakefield in dielectric resonator |
| title_short | Energy spectra of accelerated electrons of chain relativistic electron bunches at excitation of wakefield in dielectric resonator |
| title_sort | energy spectra of accelerated electrons of chain relativistic electron bunches at excitation of wakefield in dielectric resonator |
| topic | Новые методы ускорения заряженных частиц |
| topic_facet | Новые методы ускорения заряженных частиц |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/112147 |
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