Technical Note H.-B Nguyen, H.-L Trinh, V.-T Chau and V.-T Nguyen Kerntechnik downloaded from www.hanser-elibrary.com by Purdue University Library TSS on February 1, 2016 For personal use only 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 energy increase of electron in the guide wave pipe following the linacs resonance cavity with circulated electromagnetic distribution and repeat-cycle of given number of resonance cavities has been developed The energy gain of electron following the electron way in Oz axle direction of the accelerator with the different 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 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 asymptotic electric field in Iris-loaded waveguide In this paper, the Hamiltonian theory using the full Bessel functions, i e without 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 injection phase of RF and cell-to-cell phase shift of periodic accelerator structure The energy oscillation of accelerated electron strongly 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 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]: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ c2 ỵ qV: H ẳ m2 c4 ỵ ~ p qA 258 1ị ~ ẳ Ar ; Ah ; Az Þ are [4]: where the components of vector A k a E ðzÞ P > n n z > J1 ðan rÞ cosðkn z xtÞ > Ar ¼ > > an x n¼ > > < n P0 Ah;j Ah ¼ > j¼1 > > > a E ðzÞ > P n z > > J0 an rị sinkn z xtị : Az ẳ x nẳ 2ị with an is the coefficient of development of Fourier series; Ah,j is the vector of statics magnetic; Ez(z) is the function of variable z ordination describing the electromagnetic towards Oz axial; J0(anr) and J1(anr) are Bessel equation; an2 = k2 – kn2 with k is the total wave number; k = x/c, c is the light speed in vacuum; x is the angle frequency of RF wave; kn is the wave number in Oz axial, kn = kf + 2pn/d with kf is the phase change following the length of resonance cavity; kf = U0/ d = xs/d = x/vg (vg is the velocity of RF in the wave pipe), U0 = x.s is the phase change of RF when passes to each resonance 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 Hẳ E20 ỵ pr eAr ị2 c2 ỵ ph r eAh c2 ỵ pz eAz ị2 c2 1=2 3ị where e is the element electric charge, E0 = m0c2 is the rest energy of particle; and pr, ph and pz are the momentum components 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 ¼ dz qKf pr;kin ¼ q& 00 pz;kin X n¼ 1 X en 2pn kn kJ1 ðan rịsin z ỵ k& 00 a d nẳ n en kJ0 an rịcos 2pn z ỵ k& 00 d ! ð4Þ where h = H/Hi, H is the energy of particle, Hi is the initial ean Ez ðzÞ ; Ar* = cAr/Hi, Hi k energy of particle; e0 = E0/Hi; en ¼ Carl Hanser Verlag, München 79 (2014) Technical Note 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/Hi are the extending momentum component of the particle in cylindrical coordinates, respectively; B = –ct + kfz/k and Kf is the Hamilton function after variable change Kf ¼ kf h k h2 e20 eAr pr ph r eAh 1=2 eAz ð5Þ Kerntechnik downloaded from www.hanser-elibrary.com by Purdue University Library TSS on February 1, 2016 For personal use only Therefore, the change of energy of particle moving on electromagnetic 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) qKf ¼ q& 00 n¼ en J0 ðan rịk cos 2pn z ỵ k& 00 d d an ẳ d& ẳ dz 6ị 7ị where v is the speed of electron along the axial of wave pipe; vg is the speed of RF wave in wave pipe In case of the electrons have speed which are approxid& 00 ¼ ) & 00 ¼ &0 ¼ mately with RF waves (v & vg), we have dz const This lets to kB@ : kB0 in Eq (6) also is constant for 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 ẳ hzị ẳ e0 kzJ0 a0 rịcosk& 00 ị X nẳ 1;n6ẳ0 en J0 an rị kd 2pn sin 2pn dz ỵ k& 00 Zd 8ị nẳ 1;n6ẳ0 en J0 an rị kd 2pn sin z ỵ k&0 2pn d 10ị 11ị an E0z ejxt kn zị 12ị nẳ 1 X Ez zị ¼ an E0z cos n¼ 2pn z d ð13Þ Applying to the delivery of electromagnetic in the wave pipe with cell-to-cell phase shift U0, the coefficient an might be counted as an ẳ Zd coskn zị p dz zðd zÞ J0 ðan Þ U0 U0 pJ0 þ np cos þ np d 2 ¼ J0 ðan Þ ð14Þ From Eq (14), it shows that the coeffients an depend on the phase shift U0, cell length d and aperture radius Using Eq (14) with the parameters of linear accelerator RF as given in Table 2, the value of an in the linear accelerator works on the different phas shifts of U0 = 2p/3, p/3, p/2, and p have been calculated and shown in Table It is to noted that due to the value of an decreases when subscript n increases, some value of an are tiny with n = £ and n ‡ Substituting coefficient an into Eq (9) with n runs from –3 to 4, the equation of electron accelerating energy in electromagnetic in the wave pipe with repeat after three speed enhance cavities can be presented as: hzị ẳ e Hi X a0 z cosk&0 ị ỵ an E0z cos 2pn z d X n¼ d 2pn an sin z ỵ k&0 2pn d 3;n6ẳ0 ! ð15Þ Eq (15) can be used to count H energy of electron at any z ordinates ð9Þ Eq (9) shows the relationship between electron energy and its position towards the moving direction at the certain constant position r In this case, the electrons move beside the Oz axial and their speed are approximate with the velocity of wave group Furthermore, en in Eq (9) depends on the 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 79 (2014) X nẳ e0 kzJ0 a0 rị cosk&0 Þ X kn zÞ e jkn z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dz zðd zÞ Substituting B@ = B0 into Eq (8), we have: hzị ẳ an E0z J0 an rịejxt J0 an ị d c c ỵ v vg n¼ Due to the delivery of electromagnetic is symmetry, E(z,t) is the even function and substitute xt –knz = xz/vg – knz = (kf – kn)z = –2pnz/d, the Eq (12) can be rewritten as The new variable B@ is 00 X Using J C Slater asumption [7] for asymtoptic of Ez(r,z) when r is equal to aperture radius, and z is equal to zero and d, an will be determined as Ez z; tị ẳ Considering the electrons move along z direction of the wave pipe which have trajectory the same with Oz axial It is assumed that there is no motion of electron in r and h directions It means that the momentum components of the particle in these direction are equal to zero pr = pr = ph = ph = and Ar* = Ar = Ah* = Ah = This could let to pr,kin = pr – eAr* = = ph,kin = ph – erAh* Substituting these variables into Eq (4), the Hamilton can be expressed as dh ¼ dz Ez r; z; tị ẳ where is aperture radius The electromagnetic at Oz axial of wave pipe (or r = 0) could be as Energy gain of electron which have same trajectory with Oz axial X general case, the strength of electromagnetic in the wave pipe can be determined as [8]: Hzị ẳ Hi hzị 16ị 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] presented in Table 259 Kerntechnik downloaded from www.hanser-elibrary.com by Purdue University Library TSS on February 1, 2016 For personal use only Technical Note The parameters in Table 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 When the linear accelerator works at regime of U0 = 2p/3, 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 parameters 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 shows the gradually enhancement of particle energy gain following z position when the electron moves This indicates that electron energy increases gradually However, the enhancement of electron energy has not been continuously, it has decrease energy stage between the increase energy stages It points out that the increase of speed of electron is not linearly 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 electron and other haft time for reducing speed in a cycle of RF wave with a position This effect make the accelerated electron energy oscillate from cell to cell When the electron 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 The effectively of speed enhancement depends on kB0 injection phase of RF wave as shown in Fig As mentioned on section II, kB0 phase of RF wave has not changed when the 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 occur with phase of RF wave is p/2 and the electron energy varies a little comparing to the initial value In opposite, the electron energy reduces gradually with phase of RF wave is 2p/3 In this case, RF wave plays as a barrier to the motion of electron In order to study the other effects on the speed enhancement of electron with different phase of RF wave, the change of electron energy when it moves passing the first three speed Table The value of an U0 2p/3 a–3 –8.8 · a–2 10–3 –4.0 · 10–2 a–1 –1.9 · 10–1 a0 1.0 a1 2.3 · 10–1 a2 5.1 · a3 1.1 · 10–2 10–2 a4 2.5 · 10–3 U0 a–3 a–2 a–1 a0 a1 a2 a3 a4 p/3 4.5 · 10–3 2.0 · 10–2 7.8 · 10–2 1.0 2.2 · 10–1 5.0 · 10–2 1.0 · 10–2 2.4 · 10–3 U0 a–3 a–2 a–1 a0 a1 a2 a3 a4 p/2 –4.5 · 10–4 –2.7 · 10–3 –2.3 · 10–2 1.0 2.3 · 10–1 5.1 · 10–2 1.1 · 10–2 2.5 · 10–3 U0 a–3 a–2 a–1 a0 a1 a2 a3 a4 p 2.5 · 10–1 6.8 · 10–1 1.0 1.0 6.8 · 10–1 2.5 · 10–1 7.8 · 10–2 2.2 · 10–2 Table Parameters of linear accelerator RF 260 Parameters Symbol Value Unit length of wave pipe L 1.0 M length of speed enhance cavity d 0.033 M aperture radius 0.008 M maximun strength of electromagnetic on Oz direction E0z 3.0 · 107 V/m frequency of RF wave f 2.998 · 109 Hz angle frequency of RF wave x 1.884 · 1010 rad/s velocity group of RF wave vg 2.968 · 108 m/s phase change of RF wave per length kf 63.467 rad/m electron energy when it starts increase speed Hi 1.5 MeV injection phase of RF wave in wave pipe k&0 p/6 rad 79 (2014) Technical Note Kerntechnik downloaded from www.hanser-elibrary.com by Purdue University Library TSS on February 1, 2016 For personal use only enhancement cavities has also been investigated and shown in Fig 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 investigated at the different phase shifts U0 = 2p/3, p/3, p/2, and p with the injection phase kB0 = p/6 as shown in the Fig 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 kB0 = p/6) Fig The effect of kB0 phase of RF wave on the speed enhancement of electron Fig The effects of phase of RF wave on the speed enhancement of electron when it moves passing the first three speed enhancement cavities 79 (2014) 261 Kerntechnik downloaded from www.hanser-elibrary.com by Purdue University Library TSS on February 1, 2016 For personal use only Technical Note Fig The greatest oscillation of energy gain occurs at the phase shift U0 = p When the electron passes from cell to cell, 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 obtained energy of electron therefore is very not stable It cannot be used to accelerate electron The oscillation of electron energy is smaller in the other phase shifts as shown in the Fig for electron passing the first three cells The electron energies in these phase shift are steadily increased when the electron passes from cell to cell The smallest energy oscillation occurs at the phase shift U0 = 2p/ In this phase shift the increment energy of electron is almost linear in z This is the best phase shift for accelerating electron in linac structure with parameters given in Table and this phase shift is also operating phase shift of this accelerator 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 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 accelerator 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 accelerated 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 accelerator with parameters given in Table [4] The results indicate that the best phase shift and injection phase are U0 = 2p/3 and kB0 = p/6 respectively These phases are consistent with its operating phases given by reference [4] To confirm this legitimate approach, however, the further calculations in the other linear accelerator structures have to be carried out in our next work Fig The effects of phase shift of RF wave on the oscillation of energy gain of electron Fig The effects of phase shift of RF wave on the energy gain of electron when it moves passing the first three speed enhancement cavities 262 79 (2014) Technical Note Acknowledgements The authors appreciate the support received from the National Key Laboratory of Digital Control and System Engineering (DCSELAB), Vietnam National University Hochiminh City, Vietnam, under contract No 01TK/2012/HÐ/ KHCN-DCSELAB (Received on 17 November 2012) Kerntechnik downloaded from www.hanser-elibrary.com by Purdue University Library TSS on February 1, 2016 For personal use only References Reiser, M.: Theory and Design of Charged Particle Beams Wiley – VCH Slater, J C.: Electromagnetic Waves In Iris-Loaded Waveguides Technical report No 48 (1947) MIT Rosenzweig, J B.: Fundamentals of Beam Physics Oxford 2002 de Leeuw, R W.: The accelerator injection chain of the electron storage ring EUTERPE, Eindhoven University of Technology, Netherlands, 1996 Hammen, A F J.; Corstens, J M.; Botman, J I M.; Hagedoorn, H.L.; Theuws, W H C.: Hamiltonian calculation on particle Motion in Linear electron accelerators Proc of the fifth European Particle Accelerator Conference, Barcelona (1996) pp 716 – 718 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 Terrall, J R.; Slater, J C.: Particle dynamics in the linear accelerator Massachusetts Institute of Technology, USA, 1951 Pruiksma, J P.; de Leeuw, R W.; Botman, J I M.; Hagedoorn, H L.; Tijhuis, A G.: Electromagnetic Fields In Periodic Linear TravellingWave Structures Proc XVIII International Linear Accelerator Conference (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 Books · Bücher Near Surface Disposal Facilities for Radioactive Waste IAEA Safety Standards Series No SSG-29, Published by the International Atomic Energy Agency, 2014, ISBN 978-92-0114313-6, English, 103 pp., 36.00 EUR The objective of this Safety Guide is to provide guidance and recommendations relating to the development, operation, closure and regulatory control of facilities for the near surface disposal of radioactive waste to meet the safety requirements established in SSR-5 It is primarily intended for use by those involved with policy development and with the regulatory control and use of near surface disposal The term \near surface disposal" is used in this Safety Guide to refer to a range of disposal methods, including the emplacement of solid radioactive waste in earthen trenches, above ground engineered structures, engineered structures just below the ground surface and rock caverns, silos and tunnels excavated at depths of up to a few tens of metres underground This Safety Guide provides general guidance for the development, operation and closure of facilities of this a near surface disposal facility is not provided in this Safety Guide IAEA safety standards for fuel cycle facilities and for the predisposal management of waste apply for this type of facility Nuclear security aspects of the disposal of radioactive waste 79 (2014) in near surface facilities are outside the scope of this publication However, this Safety Guide does identify where security measures are relevant for safety purposes Guidance on addressing nuclear security aspects can be found in the IAEA Nuclear Security Series publications Section provides an overview of near surface disposal and its implementation, and the step by step approach to developing a near surface disposal facility Section provides guidance on legal and organizational infrastructure Section discusses the safety approach and design principles, and Section provides guidance for the preparation of the safety case and safety assessment Section presents guidance for specific steps in the development, operation and closure of a near surface disposal facility Section provides guidance on assurance for safety, and Section deals with existing disposal facilities Appendices I and II provide additional information and guidance concerning the siting of near surface disposal facilities, specifically concerning data needs IAEA Safety Standards Series SSR-5 establishes 26 safety requirements that are applicable to the near surface disposal of radioactive waste For convenience and traceability, the text of each requirement in SSR-5 is reproduced in this Safety Guide and is followed by the related recommendations 263 ... parameters of linear accelerator RF as given in Table 2, the value of an in the linear accelerator works on the different phas shifts of U0 = 2p/3, p/3, p/2, and p have been calculated and shown in Table... stages It points out that the increase of speed of electron is not linearly It enhances with the combine of linear and sin function The reason is the increase of speed of electron in wave pipe is... effects of phase shift and injection phase of RF depending on geometrical accelerator structure and RF frequency are studied The quality of linear accelerator can be determined by the enhancement and