The possibility of using undulators for focusing and acceleration of charged particles in RF field is discussed. There are suggested that the RF field does not have harmonics in synchronism with the beam. The accelerating force is produced by a combination of RF and undulator fields. Examples illustrating the efficiency of the proposed method acceleration are given for low energy ion beams. In the undulator accelerator (UNDULAC) an electrostatic, a magnetic and radio frequency undulators can be used. The focusing conditions of the beam are studied. Methods for increasing of the ion beam intensity are discussed

Masunov, E.S.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

USING OF UNDULATORS FOR LOW ENERGY ION LINACE.S. MasunovMoscow Engineering Physics Institute,115409, Kashirskoe shosse, 31, Moscow, Russia, masunov@dinus.mephi.ruThe possibility of using undulators for focusing and acceleration of charged particles in RF field is discussed. There are suggested that the RF field does not have harmonics in synchronism with the beam. The accelerating force is produced by a combination of RF and undulator fields. Examples illustrating the efficiency of the proposed method acceleration are given for low energy ion beams. In the undulator accelerator (UNDULAC) an electrostatic, a mag-netic and radio frequency undulators can be used. The focusing conditions of the beam are studied. Methods for in-creasing of the ion beam intensity are discussed.PACS numbers: 41.75.L, 41.85.E, 29.27.F1  INTRODUCTIONIn a conventional RF linac the beam is accelerated by a synchronous wave. Another method to accelerate ions – in the fields without a synchronous wave – was suggested  in  Ref.  [1]  in  which  case  the  accelerating force is to be driven by a combination of two non-syn-chronous waves (two undulators). In undulator linac in question, one of the undulators must be of the RF type (it drives non-synchronous RF wave field), the second one being, optionally, of magnetic, electrostatic, or ra-dio frequency types. The 3D dynamics of the ion beam in undulator linear accelerator (UNDULAC) will be de-termined by the type of undulator and transverse struc-ture of its field. In this paper a theoretical aspects of the method of  ion acceleration and focusing in  the fields without synchronous wave and possible realization un-dulator accelerator for ion are investigated. The results are compared with a conventional ion RF linac.2  PARTICLE MOTION EQUATIONThe motion equation of a particle in the field of two waves can be written using Lagrangian function, as)-Av(P ΣΣ Φ⋅∇= edtd, (1)where  ApP e+=  is  the  generalized  momentum, )(rΣΦ  is  the  electrostatic  potential, ])r(A)r(A[A lnRe)( ϕ⊥ϕ⊥Σ +=ilin eer  is  the  total vector potential of the two fields with amplitudes l,nA , phases  ln,ln,ln,ln, tzk αωd +−=ϕ ∫  and wave numbers l,nk . When the phase velocities  ln,ln,ln,ph, k/v ω=  dif-fer significantly from the average velocity of the parti-cles bv , the trajectories of the individual particles of the beam in general have a complicated shape but can al-ways be represented as the sum of a slow variation  rand rapid oscillation r~ . Accordingly, the beam momen-tum p  can be represented as the sum of a slowly vary-ing and a rapidly oscillating component, ppp ~+= . Af-ter an average is taken over the rapid oscillations from (1) we can obtain the time-averaged equation of nonrel-ativistic  ion  motion  if  the  beam  velocity  ))/(( lnlncb vv kkωω ±±≡≈ :effUddtddrr−=22, (2)where 3D effective potential)r()r( ψ,UUU 21eff⊥⊥ += , (3)( )22224ln1meU AA += , (4)( )ψ⋅= i*ln2 emeU AA22Re2, (5)⊥r  and  )α(α)ω(ω)(ψ lnlnln tzkk ±+±−±=  are slowly varying transverse coordinate and phase.We see from (3) that the longitudinal bunching and accelerations of the beam are provided by a combined wave with phase velocity cv  which is close to the aver-age  particle  velocity. The  choice  of  the  functions )( ⊥rnA  and )(⊥rlA  is not arbitrary because simulta-neously to  acceleration it  is  necessary to keep up the transverse focusing of the beam. Equilibrium trajecto-ries can exist for all particle phases if the following con-ditions hold in the injection plane:01 =∂∂⊥ Ur, .U 2 0=∂∂⊥r(6)Conditions (6) are necessary but not sufficient con-ditions for focusing of the beam. In the absence of reso-nance there can be transverse stability only if the «effec-tive  potential»  Ueff has  a  minimum  in  the  transverse plane (X,Y) This imposes a limitation on the amplitudes )( ⊥rnA ,  )(⊥rlA  and  confine  configuration  of  the fields. The necessary condition of  simultaneous trans-verse and longitudinal  focusing is  the existence of  an absolute minimum for effU . For low energy ion accel-erator it is difficult to create resonator system with two generators, when ln ωω ≠ . It is interesting to consider some versions of the linear undulator accelerator:1. UNDULAC-E(M) that  employs  a  combination  of RF  field  ( /λ2πω cn = ,  λβ= nph,n ck /2π )  and static periodical  electric (or  magnetic) field of an undulator  ( 0ω =l ,  /λ2π/λ lk 00l +µ= ),  where l=0,1,2...; 0λ  is slowly varying period of structure, 0μ  is a phase advance of field per period;2. UNDULAC-RF that employs a combination of two ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №3.Серия: Ядерно-физические исследования (38), с. 62-64.62space nonsynchronous harmonics of RF field in the periodical  resonator  structure  ( /λ2πωω cln == , ln kk ≠ ),  where  00vn n2k /λπ/λμ += ,  vμ  is phase  advance  per  period  of  RF  structure,  l, n=0,1,2,... and n≠l.3  ION BEAM FOCUSING AND ACCELERA-TION IN UNDULACIn UNDULAC-E (M) the rate of acceleration is pro-portional to the amplitudes of the RF field Ev and undu-lator fields  Eo (Bo). Still,  increase of the beam energy occurs due to RF field only. As is shown in [2], the en-ergy gain is  found by  cvme,c ψcosEeT/dzWd =  where Te,m are the acceleration efficiency factors for UNDU-LAC-E and UNDULAC-M. The choice of RF field am-plitude  Ev and undulator field amplitude  Eo (Bo)  is not independent, since it is necessary to realize beam focus-ing and to provide a large transmission coefficient  K. For example, if the RF field and the undulator electric field have only single space harmonics ( 00 == l,n ); phase advance per  period of  structure  0=µ v  for  RF field and  pi=µ 0  for undulator, focusing and accelera-tion occur for all paraxial particles, when λλ=β /c 02  and 2/3/χ ≥= v0 EE . In this case the acceleration effi-ciency factors  sve mceET βpiλχ= 24 .  For  a  plane magnetic undulator the function  effU  has minimum, and the fo-cusing takes place for all phase ψ  if 0cv BB β= . This condition connects values vB , 0B  and cβ and permits to express the factor mT  through the amplitude of RF field, c2vm mceETβπ4λ= . This value is close by eT .The combined acceleration field can be driven with-out use of a magnetic or an electrostatic undulator. In-deed, consider ion beam dynamics in the periodical RF structure without synchronous space harmonics. Interac-tion of the beam with each harmonic can be treated as ion interaction with a radio frequency undulator (UN-DULAC-RF).  The  combined  field  of  two  harmonics would accelerate the beam if ccb ck/ωββ =≈  where kc is a wave number of a combined wave field, 0,1,2,... ,(   2)/( ,nkkkkkk pncpnc =≠≠±=)0,1,2,...  =p .  The  rate  of  energy  gain 2ψsin  d ⋅= v,1rfc ΕeΤ/dzW . The acceleration and phase stability of the beam are possible when the phase of a synchronous particles  ψc in the combined field is in an intervals  [ ]π/2π/4,  and  [ ]/23  /4,5 pipi .  In  this  case  the frequency of ion beam bunching is double ( 2ωω =b ).For undulator, where RF field has a phase advanceµ=0 per a period, and only two harmonics n=0, p=1 are taken into acount, the transverse focusing of the beam is possible if the amplitude of the first harmonic is larger than  zero  one  (E1 > E0)  and  the  beam  velocity /λ2λββ 0cb =≈ .  For  this  case  the  acceleration  effi-ciency  factor  is  s2v,0rf βmc4λeETpi= .  The  rate  of  energy gain is the same as in UNDULAC-E(M).For the undulator, where the RF field has a phase ad-vance  per  period  piµ =v  and  beam  velocity /λλββ 0cb =≈ ,  value  of  the  acceleration  efficiency factor s2v,0rf βmc2λeETpi= . Besides, the choice of harmonic amplitudes 0E  and 1E  are independent because the con-dition of focusing in UNDULAC-RF can be fulfilled for any value of  E0/ 1E .  The maximum values of  0E  and 1E  can  be  found  from RF  characteristic  of  the  res-onator, the transverse acceptance and beam current.4  UNDULAC AND RF FOCUSING IN LINACIt is interesting to compare the methods of accelera-tion  in  UNDULAC  and  in  linac  under  consideration with axisymmetric RF focusing (ARF). The main prin-ciples of APF can be described by means of two wave approximation method: in a periodical RF structure the beam is accelerated by a synchronous space harmonic of the wave while  another,  nonsynchronous,  harmonic is only responsible for focusing the particles. By means of Hamiltonian analysis in the 4-dimensional phase space it is possible to find the relationship between the defined longitudinal acceptance and the limit value of transverse beam emittance, which provides the maximal transmis-sion coefficient  K≈1 [3].  For two- wave approach the rate of energy gain  cnv,sc ψcosEeT/dzWd = . The beam focusing takes place and  K≈1 if the amplitude of non-synchronous harmonic  nE >> sE . For two fundamental harmonics 10 == n,s  and a phase advance per period piµ =v , the efficiency factor for axisymmetric RF fo-cusing  s2ns βmc4λeETpi≈ .  This  magnitude  is  similar  to. me,T , but it is twice as low than rfT .The using of undulators for acceleration and focus-ing of ion beams is extremely promising. First, the prob-lem of design of UNDULAC RF system is simplified considerably,  since focusing and acceleration of parti-cles is possible for both transverse (TE or TEM) and longitudinal (TM) RF fields without any external focus-ing elements and dedicated slow-wave systems. No drift tubes are required for TEM wave. Second, an efficient bunching and a large transmission coefficient of parti-cles can be achieved solely by changing the amplitude and period of the static undulator field. This eliminates the serious problems involving adjustment and matching of the RF system since the latter can be made uniform. Third, an UNDULAC can be used for acceleration high intensity ion beams [4]. Indeed, the main factor limiting beam intensity in ion accelerator is space charge forces. There exist, at least, three way to increase ion beam in-ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №3.Серия: Ядерно-физические исследования (34), с. 63-64.63tensity in a linear undulator accelerator: (i)  to enlarge beam cross-section; (ii) to accelerate several beams in a channel  of  RF  structure;  (iii)  to  compensate  for  the space charge by accelerating ions with opposite charge signs within the same bunch.(i) In a UNDULAC where there are no drift tubes, a ribbon or  a  hollow beams having large cross-sections can be accelerated. Acceleration of a ribbon ion beam with the current J>1A in a plane electrostatic undulator was studied in [5]. It was shown that a large cross-sec-tion and electrostatic shielding of the space charge field decreases  Coulomb defocusing  of  the  particles  in  the narrow accelerating channel.(ii)  In  the  new accelerator  one  can  accelerate  the several beams in a single channel of RF structure since there  are  no  drift  tubes  involved.  The  problem is  to choose a dedicated symmetry of the transverse radio-frequency and periodic magnetic field. The RF system must be a small transverse size. It is preferable to use a shielded  multielectrode  line  where  transverse  electro-magnetic waves (TEM) can propagate. Configuration of the RF field and magnetic undulator field must be such as to maintain several equilibrium trajectories simulta-neously [4].(iii) Study of feasibility of simultaneous acceleration of both positive and negative ions with identical charge-to-mass ratio within the same bunch is of great interest. The current limit of the ion beam can be increased sig-nificantly  by using the space charge compensation of positively H+ (D+) and negatively H- (D-) charged ions accelerated in the same bunch. This conclusion can be drawn from Eq.(1). Indeed, the effective potential effU  depends  on  the  particle  charge  squared,  i.e.  averaged motions of positive and negative charged ions are iden-tical. It allows to increase the beam current limit.All possible methods of focusing and acceleration in undulator linear accelerator would be effective for low energy ion, when  2nb mc4λeEβpi≅ .  The acceleration effi-ciency factor rf,e,mT  decreases with growth of the beam velocity like in the RFQ accelerator. Therefore UNDU-LAC can be used as an initial part of the high intensity linear accelerator (buncher) or as aninjector for the neu-tron generators and nuclear fusion reactor.5  CONCLUSIONTheoretical studies of using undulators in the accel-eration system showed a possibility to create a new type of ion linear accelerator (UNDULAC). Three types of undulator for ion acceleration linac are suggested. The requirement of particle focusing imposes a limitation on the field amplitudes in undulators of all  types besides UNDULAC-RF with  pi=µ v , where the focusing con-dition holds for any relation between harmonic ampli-tudes. The rate of energy gain in UNDULAC is compa-rable with analogous value for RFQ and conventional linac, where RF focusing is realized. But the new accel-erator has a number of advantages. In this accelerators it is possible to use not only a longitudinal (TM) but also a transverse (TE or TEM) radio frequency field. For the TEM wave the drift tubes are absent. The beam intensi-ty can be increased in these RF structures by means of construction of  a  multibeam channel  or  enlarging the beam cross-section. The other important way to increase the beam intensity is using the space charge compensa-tion. In UNDULAC positive and negative ions would be within the same bunch and the current limit of the ion beam can be substantially increased.REFERENCES1. E.S.Masunov // Sov. Phys.-Tech. Phys. 1990, v. 35, No. 8, p. 962-965.2. E.S.Masunov //  Proc. of the XVIII LINAC Confer-ence,  Geneva,  Switzerland,  CERN  96-07,  1996. V. 2, pp. 487 – 4893. E.S.Masunov, N.E.Vinogradov // Phys. Rev. ST Ac-cel. Beams. 2001, No 7, 070101.4. E.S.Masunov // Proc of the 1995 Particle Accelera-tor  Conference  and  International  Conference  on  High-Energy  Accelerator,  Dallas,  1995.  v. 2,  p. 1149.5. E.S.Masunov, A.S.Roshal //  Proc of the 1997 Par-ticle Accelerator Conference, Vancouver, 1997. v. 4, p. 2835.64

Using of undulators for low energy ion linac

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Using of undulators for low energy ion linac

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)