DSpace at VNU: A study of the energy enhancement of electron in radio frequency (RE) linear accelerator of iris loaded waveguards

The energy gain of electron following the elec-tron way in Oz axle direction of the accelerator with the differ-ent injection phase and phase shift of RF has been obtained.. The results

H.-B Nguyen, H.-L Trinh, V.-T Chau and V.-T Nguyen

A study of the energy enhancement of electron

in radio frequency (RF) linear accelerator

of iris loaded waveguards

In this paper, the Hamiltonian theory of particle motion has

been applied for developing the motion equations of electrons

in linear accelerator of Iris-loaded waveguides Using J C Slater

asumption for determining electric field in Oz direction, the

en-ergy increase of electron in the guide wave pipe following the

li-nacs resonance cavity with circulated electromagnetic

distribu-tion and repeat-cycle of given number of resonance cavities has

been developed The energy gain of electron following the

elec-tron way in Oz axle direction of the accelerator with the

differ-ent injection phase and phase shift of RF has been obtained

The results indicate that the energy increase of electron depends

on the injection phase of RF and cell-to-cell phase shift

1 Introduction

In theoretical scenario, to understand how charged particle is

gained energy by RF waves in Linac and how phase of RF

and geometrical structure of accelerator affect to accelerated

particle energy, orbit dynamics and particle energy in an RF

linear accelerator have been described using Hamiltonian

theory [1] The Bessel functions in the vector potential may

be expanded, yielding linearized equations and the electric

field can be estimated by J C Slater [2] asumption of

asymp-totic electric field in Iris-loaded waveguide In this paper, the

Hamiltonian theory using the full Bessel functions, i e

with-out linearization is used to calculate the equations of motion

and gain energy of particle travelling in this waveguide in the

term of z component electric field and its Fourier expansion

coefficients It showed that this energy depends on the

injec-tion phase of RF and cell-to-cell phase shift of periodic

accel-erator structure The energy oscillation of accelerated

elec-tron selec-trongly depends on RF phase shift from cell to cell For

certain accelerator structure there is only one phase shift of

RF which makes the smallest oscillation of particle energy

gain from cell to cell This phase shift is corresponding to

one which gives the best quality of particle energy gain

2 Hamilton equation of charged particle in electromagnetic

When an electric charge q moves relatively in the variable

electromagnetic, the Hamilton equation describes its motion

is as below [1, 3]:

H ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

m 2 c 4 þ ~ p q~ A 2c 2

r

A r ¼ P1 n¼ 1

k n x

a n E zð Þz

a n J 1ða n rÞ cos kð n z xtÞ

A h ¼P

n 0

j¼1 A h;j

A z ¼ P1 n¼ 1

a n E zð Þz

x J 0ða n rÞ sin kð n z xtÞ

8

>

>

>

>

>

>

ð2Þ

vari-able z ordination describing the electromagnetic towards Oz

with k is the total wave number; k = x/c, c is the light speed

reso-nance cavity or cell-to-cell phase shift; s is the time for RF pass over one cavity in wave pipe; and d is the length of cavity

in wave pipe of the accelerator

Basing on some definitions as above, the relative motion of

a charged particle e might be expressed by the Hamilton equation as

H ¼ E 2 þ p ð r eA r Þ2c 2 þ prh eA h

2

c 2 þ p ð z eA z Þ2c 2 1=2 ð3Þ

com-ponents of particle in cylindrical coordinates

In order to study the change of energy of charged particles following the Oz axial, the physics variables which depend

on time could be transferred to depend on the independent variable z Therefore, the energy of particle and phase can

be presented as the function of z

Using some variable change and Hamilton function change methods [4 – 7], the Eq (3) which expresses the energy vary to the phase and particle trajectory in Oz direction might be shown as:

dh

dz ¼

qK f q& 00 ¼ppr;kin z;kin

X 1 n¼ 1

e n

a n k n kJ 1 ða n rÞsin 2pnd z þ k& 00

!

X 1 n¼ 1

Ah *= cAh/Hi, Az*= cAz/Hi; pr,kin= pr– eAr*, ph,kin= ph– erAh *,

pz,kin= pz– eAz*; pr= cpr/Hi, ph= cph/Hi, pz= cpz/Hiare the

ex-tending momentum component of the particle in cylindrical

Hamil-ton function after variable change

K f ¼kkfh h 2 e 2 p r eAr 2 ph

r eAh

2 1=2

Therefore, the change of energy of particle moving on

elec-tromagnetic towards the Oz direction can be calculated by

integrating Eq (4) to find h(z) This means that if the initial

energy of particle Hi is known, the energy of particle in any

z position could be counted via h(z)

3 Energy gain of electron which have same trajectory

with Oz axial

Considering the electrons move along z direction of the wave

pipe which have trajectory the same with Oz axial It is

as-sumed that there is no motion of electron in r and h

direc-tions It means that the momentum components of the

Eq (4), the Hamilton can be expressed as

dh

dz ¼

qK f

q& 00 ¼ X

1 n¼ 1

e n J 0 ða n rÞk cos 2pnd z þ k& 00 ð6Þ The new variable B@ is

d& 00

dz ¼

c

v þ

c

where v is the speed of electron along the axial of wave pipe;

In case of the electrons have speed which are

the electron traveling beside the Oz direction

In Eq (6) at certain position r, due to h is the function of z

and when the particle starts to increase speed we have

h(z = 0) = –1, we might find out h by integrating Eq (6) with

h is from (–1) to (h) and z is from (0) to (z):

h ¼ hðzÞ ¼ e 0 kzJ 0 ða 0 rÞcosðk& 00 Þ

X 1 n¼ 1;n6¼0

e n J 0 ða n rÞ2pnkd sin 2pn

h zð Þ ¼ e0 kzJ 0ða 0 rÞ cos k&ð 0 Þ

X 1 n¼ 1;n6¼0

e n J 0 ð a n r Þ2pnkdsin 2pn

Eq (9) shows the relationship between electron energy and

its position towards the moving direction at the certain

con-stant position r In this case, the electrons move beside the

Oz axial and their speed are approximate with the velocity

delivery of electromagnetic at the z direction Therefore, it is

necessary to measure this parameter to find out the energy

of particle varies to the moving direction as Oz direction In

general case, the strength of electromagnetic in the wave pipe can be determined as [8]:

Ezðr; z; tÞ ¼ X1

n¼ 1

anE0zJ0ða n rÞej xt k ð n z Þ ð10Þ

when r is equal to aperture radius, and z is equal to zero and

a n ¼

2 d

Z d 0

e jk n z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z dð zÞ

The electromagnetic at Oz axial of wave pipe (or r = 0) could be as

E zðz; tÞ ¼ X1

n¼ 1

Due to the delivery of electromagnetic is symmetry, E(z,t) is

Ezð Þ ¼z X1 n¼ 1

anE0zcos 2pn

Applying to the delivery of electromagnetic in the wave pipe

counted as

a n ¼

2 d

Z d 0

cos k ð n z Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z dð zÞ

J 0 ð a n r a Þ ¼

2

dpJ0

U 0

2 þnp cos

U 0

2 þnp

Eq (14) with the parameters of linear accelerator RF as given

calculated and shown in Table 1 It is to noted that due to the

to 4, the equation of electron accelerating energy in electro-magnetic in the wave pipe with repeat after three speed en-hance cavities can be presented as:

h zð Þ ¼ e

H i a0z cos k&ð 0Þ þ X4

n¼ 3;n6¼0

an d 2pnsin

2pn

d z þ k& 0

!

X 4 n¼ 3

a n E 0z cos 2pn

Eq (15) can be used to count H energy of electron at any z ordinates

4 Results and discussion

In the study, the energy gain and oscillation of electron when

it moves on axial of wave pipe is considered Assumption that

a linear accelerator RF is designed by the parameters [4] pre-sented in Table 2

The parameters in Table 2 are set up for a linear accelerator

RF which is used to study the increase and decrease of the

electron energy at phase shift of 2p/3, p/3, p/2 and p

the delivery of electromagnetic in the wave pipe would be in

circulation and repeat after three speed enhance resonance

cavities Especially, if wave phase of RF is p/6, the delivery

of electromagnetic might be not in symmetry

The variation of electron energy has been investigated and

the results indicate that the electron energy might increase

up to 20 times when it moves in the wave pipe length of one

meter The average of energy enhance depends on the

param-eters of linear accelerator RF such as the maximum electric

field of RF wave, the initial energy of electron, the form and

size of speed enhance resonance cavity, injection phase and

phase shift of RF The initial energy of electron is higher, the

electric field of RF wave needs to be more strong in order to

the speed enhance is more effectiveness

Figure 1 shows the gradually enhancement of particle

en-ergy gain following z position when the electron moves This

indicates that electron energy increases gradually However,

the enhancement of electron energy has not been

continu-ously, it has decrease energy stage between the increase

en-ergy stages

It points out that the increase of speed of electron is not

lin-early It enhances with the combine of linear and sin function

The reason is the increase of speed of electron in wave pipe is not continuously as mentioned above This agrees with the real case due to there is a haft time for increase speed of elec-tron and other haft time for reducing speed in a cycle of RF wave with a position This effect make the accelerated elec-tron energy oscillate from cell to cell When the elecelec-tron moves to the RF wave area with suitable phase, the time for increase its speed would be higher than one for decrease speed Therefore, finally the electron energy would enhance gradually

injec-tion phase of RF wave as shown in Fig 2 As meninjec-tioned on

electron moves towards the Oz axial The motion of electron

in wave pipe with different phase of RF wave as p/6, p/3, and p/2 has been investigated The electron energy increases quickly when phase of RF wave is p/6 and the effectively of speed enhancement reduces rapidly for phase of RF wave is p/3 Furthermore, the increase of electron speed does not oc-cur with phase of RF wave is p/2 and the electron energy var-ies a little comparing to the initial value In opposite, the elec-tron energy reduces gradually with phase of RF wave is 2p/3

In this case, RF wave plays as a barrier to the motion of elec-tron In order to study the other effects on the speed enhance-ment of electron with different phase of RF wave, the change

of electron energy when it moves passing the first three speed

Table 1 The value of an

Table 2 Parameters of linear accelerator RF

enhancement cavities has also been investigated and shown in

Fig 3 The results are well agreement to the independent

study which were presented by the Eindhoven Technology

University, Netherland on 1996 [5]

The oscillation of accelerated electron energy which affects

to quality of accelerated beam at output of wave guide is

Fig 1 The enhancement of energy of electron during it moves in wave pipe (initial energy of electron is 1,5 MeV, accelerator works at U0= 2p/3, injection phase of RF wave is kB 0 = p/6)

Fig 2 The effect of kB 0 phase of RF wave on the speed enhancement of electron

Fig 3 The effects of phase of RF wave on the speed enhancement of electron when it moves passing the first three speed enhancement cav-ities

Fig 4 The greatest oscillation of energy gain occurs at the

the electron energy rapidly increases at the end of cell and

gets maximal value at position z = nd (n is n-th cavity) and

then this energy rapidly decreases to nearly the initial energy

As shown in the Fig 4, the more cell electron passes, the

greater peak of electron energy is In this phase shift, the

ob-tained energy of electron therefore is very not stable It

can-not be used to accelerate electron

The oscillation of electron energy is smaller in the other

phase shifts as shown in the Fig 5 for electron passing the first

three cells The electron energies in these phase shift are

stea-dily increased when the electron passes from cell to cell The

3 In this phase shift the increment energy of electron is

al-most linear in z This is the best phase shift for accelerating

electron in linac structure with parameters given in Table 2

and this phase shift is also operating phase shift of this

accel-erator structure given in reference [4] This result pointed

out that this simple method can be used to find the suitable

phase shift and injection phase of RF for some given structure

of linear accelerator RF

5 Conclusions

By using theoretical analysis, the accelerated particle energy which is altered by the effects of phase shift and injection phase of RF depending on geometrical accelerator structure and RF frequency are studied The quality of linear accelera-tor can be determined by the enhancement and oscillation of the accelerated particle energy This study showed that for given accelerator structure there is only the best couple of phase shift and injection phase of RF which makes the accel-erated particle energy steadily increase The other phases will make less energy enhancement or more energy oscillation This can be confirmed by applying this theory to the accelera-tor with parameters given in Table 2 [4] The results indicate

its operating phases given by reference [4] To confirm this le-gitimate approach, however, the further calculations in the other linear accelerator structures have to be carried out in our next work

Fig 4 The effects of phase shift of RF wave on the oscillation of energy gain of electron

Fig 5 The effects of phase shift of RF wave on the energy gain of electron when it moves pas-sing the first three speed enhancement cavities

The authors appreciate the support received from the

Na-tional Key Laboratory of Digital Control and System

Engi-neering (DCSELAB), Vietnam National University

Hochi-minh City, Vietnam, under contract No 01TK/2012/HÐ/

KHCN-DCSELAB

(Received on 17 November 2012)

References

1 Reiser, M.: Theory and Design of Charged Particle Beams Wiley –

VCH

2 Slater, J C.: Electromagnetic Waves In Iris-Loaded Waveguides.

Technical report No 48 (1947) MIT

3 Rosenzweig, J B.: Fundamentals of Beam Physics Oxford 2002

4 de Leeuw, R W.: The accelerator injection chain of the electron

sto-rage ring EUTERPE, Eindhoven University of Technology,

Nether-lands, 1996

5 Hammen, A F J.; Corstens, J M.; Botman, J I M.; Hagedoorn, H.L.;

Theuws, W H C.: Hamiltonian calculation on particle Motion in

Lin-ear electron accelerators Proc of the fifth European Particle

Accel-erator Conference, Barcelona (1996) pp 716 – 718

6 Corstens, J M.; Hammen, A F J.; Botman, J I M.: Particle Dynamics

In Low-Energy Travelling – Waves Linacs Proceedings of the

1999 Particle Accelerator Conference, New York (1999) pp 866 –

868, DOI:10.1109/PAC.1999.792964

7 Terrall, J R.; Slater, J C.: Particle dynamics in the linear accelerator.

Massachusetts Institute of Technology, USA, 1951

8 Pruiksma, J P.; de Leeuw, R W.; Botman, J I M.; Hagedoorn, H L.;

Tijhuis, A G.: Electromagnetic Fields In Periodic Linear

Travelling-Wave Structures Proc XVIII International Linear Accelerator

Con-ference (1996) pp 89 – 91

The authors of this contribution

Huy-Bich Nguyen, Ph D.

Faculty of Engineering and Technology

Nong Lam University

Linh Trung Ward, Thu Duc District, Hochiminh City

Vietnam

E-mail: nhbich@hcmuaf.edu.vn or nguyenhuybich@gmail.com

and

National Key Laboratory of Digital Control and System Engineering (DCSELAB) National University

268 Ly Thuong Kiet St., District 10, Hochiminh City Vietnam

Hoa-Lang Trinh, MSc.

Faculty of Physics – Physical Engineering Natural Science University

227 Nguyen Van Cu St., District 5, Hochiminh City Vietnam

and National Key Laboratory of Digital Control and System Engineering (DCSELAB) National University

268 Ly Thuong Kiet St., District 10, Hochiminh City Vietnam

Van-Tao Chau, Assoc Prof Dr.

Faculty of Physics – Physical Engineering Natural Science University

227 Nguyen Van Cu St., District 5, Hochiminh City Vietnam

Van-Tuong Nguyen, MSc.

Faculty of Physics – Physical Engineering Natural Science University

227 Nguyen Van Cu St., District 5, Hochiminh City Vietnam

Bibliography DOI 10.3139/124.110325 KERNTECHNIK

79 (2014) 3; page 258 – 263

ª Carl Hanser Verlag GmbH & Co KG ISSN 0932-3902

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