Electrodynamics in e⁻ linacs
We report a new calculation of electron beam acceleration along linac using fundamental electrodynamics as basis. Following laws are considered: increasing of work equals multiplication of force to elementary interval; power equals work growth over time interval; force acting on charge equals charge...
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| Veröffentlicht in: | Вопросы атомной науки и техники |
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| Datum: | 2014 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2014
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Electrodynamics in e⁻ linacs / S.S. Proskin, V.A. Dvornikov, I.A. Kuzmin, I.S. Shchedrin // Вопросы атомной науки и техники. — 2014. — № 5. — С. 136-139. — Бібліогр.: 7 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860115848986886144 |
|---|---|
| author | Proskin, S.S. Dvornikov, V.A. Kuzmin, I.A. Shchedrin, I.S. |
| author_facet | Proskin, S.S. Dvornikov, V.A. Kuzmin, I.A. Shchedrin, I.S. |
| citation_txt | Electrodynamics in e⁻ linacs / S.S. Proskin, V.A. Dvornikov, I.A. Kuzmin, I.S. Shchedrin // Вопросы атомной науки и техники. — 2014. — № 5. — С. 136-139. — Бібліогр.: 7 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | We report a new calculation of electron beam acceleration along linac using fundamental electrodynamics as basis. Following laws are considered: increasing of work equals multiplication of force to elementary interval; power equals work growth over time interval; force acting on charge equals charge value multiplied to electric field value. Using electrodynamic characteristic, series impedance, equaled a square of electric eld value divided by power, and also taking into account laws mentioned above, an equation for the electric field radiated by the beam is calculated. A transient process of electric field radiated by beam is considered.
Рассмотрен расчет ускорения сгустков в линейных ускорителях электронов, основанный на постулатах электродинамики. Перечислены основные постулаты: увеличение работы равно произведению силы на элемент пройденного расстояния; мгновенная мощность равна отношению прироста работы на отрезок времени; сила, действующая на заряд, равна величине заряда, умноженного на напряженность электрического поля. Используя электродинамическую характеристику, последовательное сопротивление круглого диафрагмированного волновода, равное отношению квадрата напряженности электрического поля к мгновенной мощности, и выше названные постулаты электродинамики, получено выражение для электрического поля излучения. Рассмотрен переходный процесс поля излучения цуга сгустков.
Розглянуто розрахунок прискорення згусткiв у лiнiйних прискорювачах електронiв, оснований на постулатах електродинамiки. Перелiченi основнi постулати: збiльшення роботи дорiвнює здобутку сили на елемент пройденої вiдстанi; миттєва потужнiсть дорiвнює вiдношенню приросту роботи на вiдрiзок часу; сила, дiюча на заряд, рiвна величинi заряду, помноженого на напруженiсть электричного поля. Використовуючи електродинамiчну характеристику, послiдовний опiр круглого дiафрагмованого хвильопровода, який дорiвнює вiдношеннию квадрата напруженостi електричного поля на миттєву потужнiсть, та перелiченi вище постулати електродинамiки, отриманоно вираз для електричного поля випромiнювання. Розглянуто перехiдний процес поля випромiнювання цуга сгусткiв.
|
| first_indexed | 2025-12-07T17:36:24Z |
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| fulltext |
THEORY AND TECHNICS OF PARTICLE ACCELERATION
ELECTRODYNAMICS IN e− LINACS
S.S.Proskin, V.A.Dvornikov, I.A.Kuzmin, I. S. Shchedrin∗
National Research Nuclear University MEPhI, Moscow, Russia
(Received September 26, 2013)
We report a new calculation of electron beam acceleration along linac using fundamental electrodynamics as basis.
Following laws are considered: increasing of work equals multiplication of force to elementary interval; power equals
work growth over time interval; force acting on charge equals charge value multiplied to electric field value. Using
electrodynamic characteristic, series impedance, equaled a square of electric field value divided by power, and also
taking into account laws mentioned above, an equation for the electric field radiated by the beam is calculated. A
transient process of electric field radiated by beam is considered.
PACS: 29.27.-a
1. MAIN REMARKS ON TRAVELLING
WAVE ELECTRON LINAC THEORY
Considering electron bunch acceleration in travelling
wave acceleration structure few remarks on classical
theory describing bunch movement along acceleration
structure should be given [1].
In the first, the equation stated below is of great
importance in order to obtain electric field radiated
by moving along DLWG charge:
dW = −qEdz , (1)
dP
dz
= − ωP
vgQ
= −2αP , (2)
dE
dz
= −αE . (3)
Here dW is accelerated particle energy gain due
travelling dz distance along waveguide axis, q is
charge quantity, E is on-axis electric field amplitude,
α is attenuation factor, vg is group velocity, P is in-
stantaneous power and Q is disk loaded waveguide
(DLWG) Q-factor. It should be stated also that
equations (2)-(3) described by Perry B. Wilson are
correct for DLWG feeding by RF power source in
steady mode. It is wrong to believe that current en-
ergy speed equals group velocity. But actually group
velocity is responsible for transient mode and power
source current energy speed equals bunch velocity.
A relation mentioned by Wilson is series resis-
tance Rp which plays important part too (further Rsh
is shunt impedance):
Rp = αRsh =
ωRsh
2vgQ
, (4)
Rp =
E2
2P
. (5)
In the next equation minus sign should be put be-
fore multiplication of direct current and beam field:
dP
dz
= I0Eb − 2αP . (6)
It is considered that there is no power feed. En-
ergy balance in the whole accelerating structure gives
(6). Eb is beam field which resists short bunch with
I0 current. Energy gain generated by beam is power
loss.
The next equation means beam generates 2 waves.
The second component is in antiphase, i.e. in power
source phase and attenuating as well as power source
field:
Eb(z) = I0Rsh(1− e−αz) . (7)
It contradicts physics of the process. Moreover
boundary conditions are failed when z = 0 for bridged
waveguide.
It is cleared that fundamental theory of beam
loading which states that charge energy loss due radi-
ation equals half of induced voltage gives simple way
of beam field calculation.
2. BEAM FIELD CALCULATION BASED
ON WILSON AND RAMO THEOREMS
We consider a new approach to define relativistic elec-
tron beam field in DLWG. Relativistic electron bunch
with q charge is travelling along DLWG axis with con-
stant β = v/c = 1 velocity. It is assumed that charge
velocity is equaled to the speed of light. According
to Perry B. Wilson theorem, W , energy being lost by
travelling bunch with q charge, equals half of charge
and induced voltage U multiplication [1]:
W =
1
2
qU . (8)
Assume that U voltage induced on x length equals
current induced by q charge and multiplied by equiv-
alent impedance R on accelerating gap x length.
U = IR . (9)
∗Corresponding author E-mail address: ISShchedrin@mephi.ru, qstpss@gmail.com, +7(499)324-80-33
136 ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2014, N5 (93).
Series: Nuclear Physics Investigations (63), p.136-139.
According to Simon Ramo theorem induced cur-
rent equals multiplication of charge and velocity di-
vided by gap length [2]:
I =
qc
x
. (10)
Define R impedance on x gap length by means of
series resistance Rp:
R = Rpx
2 . (11)
According to definition:
Rp =
E2
2P
. (12)
In the last equation E is electric field strength on
DLWG axis, P is RF power in current DLWG cross-
section.
Using (8)-(12) for W , U , I, R and Rs, obtain
beam energy loss gain W on DLWG x length:
W =
1
2
qU =
1
2
q2cRpx . (13)
Dissipated power equals:
P =
W
t
=
1
2
q2cRp
x
t
. (14)
Within considered case beam velocity equals the
speed of light, i.e. x/t = c
P =
1
2
q2c2Rp =
E2
2Rp
. (15)
Which means that:
.E = qcRp (16)
Therefore, field E radiated by beam with q charge
is defined by charge, velocity and DLWG series resis-
tance. Current statement is also correct for velocities
under the speed of light v < c, i.e. it is correct for
any velocity of accelerating beam.
E = qvRp . (17)
3. ELECTRON BEAM FIELD
CALCULATION IN DLWG. CLASSICAL
THEORY
As it is expected simple calculation could raise
doubts. That is why consider classical approach by
means of electrodynamics fundamentals. Calculate q
charge beam field on DLWG axis. Beam is travelling
with v velocity. Point charge is considered. When
travelling beam interacts with decelerating system
like DLWG F force will act on charge. Force equals
multiplication of q charge on E beam field.
F = qE . (18)
Instantaneous power equals differential relation of
work dA divided by time interval dt, i.e.:
P =
dA
dt
. (19)
It should be noted that force equals relation dA
to dl, where dl is elementary distance on DLWG axis.
F =
dA
dl
. (20)
(19)-(20) conclude that instantaneous power
equals multiplication of force to velocity:
P = F
dl
dt
= Fv . (21)
Substitute F from (18) to (21) and get:
P = qEv . (22)
Define average power in DLWG P̄ by means of E
beam field and Rs series resistance:
P̄ =
E2
2Rp
. (23)
While reminding that electromagnetic field in-
stantaneous power P and average P̄ differ by factor
2, get the next relation:
P = 2P̄ , (24)
P =
E2
p
. (25)
(22) and (25) infer
E = qRpv . (26)
It is assumed indirectly that we have point charge
bunch with q charge, v velocity on TM010 mode in
DLWG.
When v ̸= c (26) could be rewritten as:
E = qRpβc . (27)
In the current equation β = v/c and when β = 1
get:
E = qRpc . (28)
(28) could be rewritten also as follows. Since the
speed of light equals wave length λ divided by oscil-
lation period it is:
c =
λ
T
, (29)
q
T
= I . (30)
I is average q charge bunch current. Time be-
tween bunches is equaled to oscillation period of RF
power. Substitute (29), (30) to (28). As a result
beam field in relativistic case (β = 1) equals:
E = IRpλ . (31)
For v ̸= c:
E = IRpβλ . (32)
(32) might be used for calculation of electron linac
bunchers. (31) is usually used for calculation of elec-
tron linac acceleration sections.
Therefore, obviously equation (28) for obtain-
ing single bunch field in travelling wave accelerat-
ing structure calculated by means of classical theory
identical with (17) calculated by means of Wilson and
Ramo theorems.
137
4. MULTIPLE BUNCHES BEAM FIELD
TRANSIENT PROCESS IN DLWG
Consider radiation process for a chain of electrons
with q charge, interval λ which is equaled to wave
length in DLWG (Fig.1).
The first bunch radiates field:
E = IRpλ . (33)
It is taken into account that beam is relativistic
(v = c) and interval between point charge bunches
is equaled wave length λ. Using results of [3] and
considering DLWG as chain of feedthrough quarter-
wave cavities (θ = π/2 mode) or third-wave cavities
(θ = 2π/3 mode) beam field E transient process is
defined as:
E1 = lRpλe
− πt
Q1T . (34)
In the statement above t is current time, T
is oscillations period and Q1 is DLWG loaded Q-
factor. While t = 0: E1 = IRpλ.
Fig.1. The chain of bunches travelling in DLWG
When the second bunch enters considered DLWG
cross-section beam field is:
E2 = IRpλ+ IRpλe
− π
Ql . (35)
And after the third bunch:
E2 = IRpλ+ IRpλe
− π
Ql + IRpλe
− 2π
Ql . (36)
Net series is a geometric sequence. Summary field
in considered cross-section is equaled:
En =
IRpλ
(
1− e
−nπ
Ql
)
1− e
− π
Ql
. (37)
When infinite quantity of bunches is taken into
calculation following beam field is observed in DLWG
with finite Ql:
E0 =
IRpλ
1− e
− π
Ql
. (38)
Field envelope amplitude growth is obvi-
ously characterized by Ql size (Fig. 2).
Fig.2. Transient process for the chain of bunches
beam field
We take into account power losses in walls under
T oscillation period. As it is seen from (37) maxi-
mum field is generated after travelling of n bunches
in DLWG cross-section. After the bunch number n
beam field decreases according to law:
E =
IRpλ
(
1− e
−nπ
Ql
)
1− e
− π
Ql
. (39)
It should be mentioned that time is counted here
before the bunch number n enters considered cross-
section.
5. CONCLUSIONS
As a result few remarks on classical theory for calcu-
lation of electron linac electrodynamic parameters are
given within current research. Method of beam field
calculation based on electrodynamics fundamentals
and different from classical one is proposed. Derived
way of beam field calculation is used in consideration
of transient process for the chain of bunches beam
field.
Moreover, this technique could be used not for
DLWG only but for other electron linac accelerating
structures which have their own benefits and draw-
backs.
Calculation of electron beam field presented
within current work allows to make more accurate
calculations of some electrodynamic characteristics
used in design and optimization of electron linac ac-
celerating systems including improvement of acceler-
ating structures design software.
Presented research has been conducted in realiza-
tion of two State Contracts in Compact Accelerators
Laboratory of National Research Nuclear University
MEPhI which has been financed within mentioned
contracts. It includes development of electron linac
with 4MeV output energy (with usage of modern
klystrons and magnetrons of different wave lengths)
for generation of powerful bremmstrahlung in THz
wavelength range and also development of linear elec-
tron accelerator with 15MeV output energy, maxi-
mum of 1 kA beam current and pulse duration about
1ns.
138
This work has been accomplished under support
of The Ministry of Education and Science of The Rus-
sian Federation within the program ”Science and ed-
ucation of the innovative Russia” 2009− 2013, State
Contracts P433, P1222, the program ”Developing of
the High School science potential” 2009 − 2010, sci-
entific work number 1.49.09 and also by State Order
from The Ministry of Education and Science of The
Russian Federation number N00− Γ− 611− 4056.
References
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plications to Storage Ring RF Systems and Lin-
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2. S. Ramo. Currents Induced by Electron Motion
// Proceedings of the I/R/E. September 1939,
p. 584-585.
3. D.Altman. Microwave circuits / Translation
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2009, p. 48-56.
6. V.A.Dvornikov, I.A.Kuzmin, S.S. Proskin, et al.
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7. S.S. Proskin, V.A.Dvornikov, I.A.Kuzmin, et
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uide Sections // Proceedings of IPAC 2012, New
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ÝËÅÊÒÐÎÄÈÍÀÌÈÊÀ Â ËÈÍÅÉÍÛÕ ÓÑÊÎÐÈÒÅËßÕ ÝËÅÊÒÐÎÍÎÂ
Ñ.Ñ.Ïðîñêèí, Â.À.Äâîðíèêîâ, È.À.Êóçüìèí, È.Ñ.Ùåäðèí
Ðàññìîòðåí ðàñ÷åò óñêîðåíèÿ ñãóñòêîâ â ëèíåéíûõ óñêîðèòåëÿõ ýëåêòðîíîâ, îñíîâàííûé íà ïîñòóëàòàõ
ýëåêòðîäèíàìèêè. Ïåðå÷èñëåíû îñíîâíûå ïîñòóëàòû: óâåëè÷åíèå ðàáîòû ðàâíî ïðîèçâåäåíèþ ñèëû íà
ýëåìåíò ïðîéäåííîãî ðàññòîÿíèÿ; ìãíîâåííàÿ ìîùíîñòü ðàâíà îòíîøåíèþ ïðèðîñòà ðàáîòû íà îòðåçîê
âðåìåíè; ñèëà, äåéñòâóþùàÿ íà çàðÿä, ðàâíà âåëè÷èíå çàðÿäà, óìíîæåííîãî íà íàïðÿæåííîñòü ýëåê-
òðè÷åñêîãî ïîëÿ. Èñïîëüçóÿ ýëåêòðîäèíàìè÷åñêóþ õàðàêòåðèñòèêó, ïîñëåäîâàòåëüíîå ñîïðîòèâëåíèå
êðóãëîãî äèàôðàãìèðîâàííîãî âîëíîâîäà, ðàâíîå îòíîøåíèþ êâàäðàòà íàïðÿæåííîñòè ýëåêòðè÷åñêîãî
ïîëÿ ê ìãíîâåííîé ìîùíîñòè, è âûøå íàçâàííûå ïîñòóëàòû ýëåêòðîäèíàìèêè, ïîëó÷åíî âûðàæåíèå
äëÿ ýëåêòðè÷åñêîãî ïîëÿ èçëó÷åíèÿ. Ðàññìîòðåí ïåðåõîäíûé ïðîöåññ ïîëÿ èçëó÷åíèÿ öóãà ñãóñòêîâ.
ÅËÅÊÒÐÎÄÈÍÀÌIÊÀ  ËIÍIÉÍÈÕ ÏÐÈÑÊÎÐÞÂÀ×ÀÕ ÅËÅÊÒÐÎÍIÂ
Ñ.Ñ.Ïðîñêií, Â.À.Äâîðíiêîâ, I.À.Êóçüìií, I.Ñ.Ùåäðií
Ðîçãëÿíóòî ðîçðàõóíîê ïðèñêîðåííÿ çãóñòêiâ ó ëiíiéíèõ ïðèñêîðþâà÷àõ åëåêòðîíiâ, îñíîâàíèé íà ïî-
ñòóëàòàõ åëåêòðîäèíàìiêè. Ïåðåëi÷åíi îñíîâíi ïîñòóëàòè: çáiëüøåííÿ ðîáîòè äîðiâíþ¹ çäîáóòêó ñèëè
íà åëåìåíò ïðîéäåíî¨ âiäñòàíi; ìèòò¹âà ïîòóæíiñòü äîðiâíþ¹ âiäíîøåííþ ïðèðîñòó ðîáîòè íà âiäði-
çîê ÷àñó; ñèëà, äiþ÷à íà çàðÿä, ðiâíà âåëè÷èíi çàðÿäó, ïîìíîæåíîãî íà íàïðóæåíiñòü ýëåêòðè÷íîãî
ïîëÿ. Âèêîðèñòîâóþ÷è åëåêòðîäèíàìi÷íó õàðàêòåðèñòèêó, ïîñëiäîâíèé îïið êðóãëîãî äiàôðàãìîâàíî-
ãî õâèëüîïðîâîäà, ÿêèé äîðiâíþ¹ âiäíîøåííèþ êâàäðàòà íàïðóæåíîñòi åëåêòðè÷íîãî ïîëÿ íà ìèòò¹âó
ïîòóæíiñòü, òà ïåðåëi÷åíi âèùå ïîñòóëàòè åëåêòðîäèíàìiêè, îòðèìàíîíî âèðàç äëÿ åëåêòðè÷íîãî ïîëÿ
âèïðîìiíþâàííÿ. Ðîçãëÿíóòî ïåðåõiäíèé ïðîöåñ ïîëÿ âèïðîìiíþâàííÿ öóãà ñãóñòêiâ.
139
|
| id | nasplib_isofts_kiev_ua-123456789-80497 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:36:24Z |
| publishDate | 2014 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Proskin, S.S. Dvornikov, V.A. Kuzmin, I.A. Shchedrin, I.S. 2015-04-18T15:13:54Z 2015-04-18T15:13:54Z 2014 Electrodynamics in e⁻ linacs / S.S. Proskin, V.A. Dvornikov, I.A. Kuzmin, I.S. Shchedrin // Вопросы атомной науки и техники. — 2014. — № 5. — С. 136-139. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 29.27.-a https://nasplib.isofts.kiev.ua/handle/123456789/80497 We report a new calculation of electron beam acceleration along linac using fundamental electrodynamics as basis. Following laws are considered: increasing of work equals multiplication of force to elementary interval; power equals work growth over time interval; force acting on charge equals charge value multiplied to electric field value. Using electrodynamic characteristic, series impedance, equaled a square of electric eld value divided by power, and also taking into account laws mentioned above, an equation for the electric field radiated by the beam is calculated. A transient process of electric field radiated by beam is considered. Рассмотрен расчет ускорения сгустков в линейных ускорителях электронов, основанный на постулатах электродинамики. Перечислены основные постулаты: увеличение работы равно произведению силы на элемент пройденного расстояния; мгновенная мощность равна отношению прироста работы на отрезок времени; сила, действующая на заряд, равна величине заряда, умноженного на напряженность электрического поля. Используя электродинамическую характеристику, последовательное сопротивление круглого диафрагмированного волновода, равное отношению квадрата напряженности электрического поля к мгновенной мощности, и выше названные постулаты электродинамики, получено выражение для электрического поля излучения. Рассмотрен переходный процесс поля излучения цуга сгустков. Розглянуто розрахунок прискорення згусткiв у лiнiйних прискорювачах електронiв, оснований на постулатах електродинамiки. Перелiченi основнi постулати: збiльшення роботи дорiвнює здобутку сили на елемент пройденої вiдстанi; миттєва потужнiсть дорiвнює вiдношенню приросту роботи на вiдрiзок часу; сила, дiюча на заряд, рiвна величинi заряду, помноженого на напруженiсть электричного поля. Використовуючи електродинамiчну характеристику, послiдовний опiр круглого дiафрагмованого хвильопровода, який дорiвнює вiдношеннию квадрата напруженостi електричного поля на миттєву потужнiсть, та перелiченi вище постулати електродинамiки, отриманоно вираз для електричного поля випромiнювання. Розглянуто перехiдний процес поля випромiнювання цуга сгусткiв. This work has been accomplished under support
 of The Ministry of Education and Science of The Russian
 Federation within the program ”Science and education
 of the innovative Russia” 2009 − 2013, State
 Contracts P433, P1222, the program ”Developing of
 the High School science potential” 2009 − 2010, scientific
 work number 1.49.09 and also by State Order
 from The Ministry of Education and Science of The
 Russian Federation number N00 − Γ − 611 − 4056. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Теория и техника ускорения частиц Electrodynamics in e⁻ linacs Электродинамика в линейных ускорителях электронов Електродинамiка в лiнiйних прискорювачах електронiв Article published earlier |
| spellingShingle | Electrodynamics in e⁻ linacs Proskin, S.S. Dvornikov, V.A. Kuzmin, I.A. Shchedrin, I.S. Теория и техника ускорения частиц |
| title | Electrodynamics in e⁻ linacs |
| title_alt | Электродинамика в линейных ускорителях электронов Електродинамiка в лiнiйних прискорювачах електронiв |
| title_full | Electrodynamics in e⁻ linacs |
| title_fullStr | Electrodynamics in e⁻ linacs |
| title_full_unstemmed | Electrodynamics in e⁻ linacs |
| title_short | Electrodynamics in e⁻ linacs |
| title_sort | electrodynamics in e⁻ linacs |
| topic | Теория и техника ускорения частиц |
| topic_facet | Теория и техника ускорения частиц |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80497 |
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