Longitudinal beam dynamics with phase slip in race-track microtrons

ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 596 (2008) 147–156

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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

Longitudinal beam dynamics with phase slip in race-track microtrons Yu.A. Kubyshin a,,1, A.P. Poseryaev b, V.I. Shvedunov b a b

Institute of Energy Technologies, Technical University of Catalonia, Campus Sud, Edif. ETSEIB, av. Diagonal 647, 08028 Barcelona, Spain Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia

a r t i c l e in fo

abstract

Article history: Received 14 April 2008 Received in revised form 26 July 2008 Accepted 30 July 2008 Available online 15 August 2008

Implementation of low-energy injection schemes in race-track microtron (RTM) designs requires a better understanding of the longitudinal beam dynamics. Unlike the high-energy case a low-energy beam slips in phase with respect to the accelerating field phase so that the standard notion of synchronous particle is not applicable. In the article, we generalize the concept of synchronous particle for the case of non-relativistic energies. An analytic approach for the description of the synchronous phase slip is developed and explicit, though approximate, formulas which allow to determine the equilibrium injection phase and to fix the parameters of the accelerator are derived. The approximation can be improved in a systematic way by calculating higher-order corrections. The precision of the analytic approach is checked by direct numerical computations and is shown to be quite satisfactory. Explicit examples of injection schemes and fixing of RTM global parameters are presented. We also address the issue of stability of synchrotron oscillations around the generalized synchronous trajectory and introduce the notion of critical energy. & 2008 Elsevier B.V. All rights reserved.

Keywords: Race-track microtron Longitudinal dynamics Phase-slip effect Critical energy

1. Introduction Race-track microtron (RTM) is a specific type of electron accelerator combining properties of linear accelerator (LINAC) and circular machine [1,2]. For applications in which a modest beam power at a relatively high beam energy is required this type of particle accelerators allows to get pulsed and continuous beams in the most cost and energy effective way with the most optimal dimensions of the machine. Main features of the longitudinal beam dynamics in RTMs, in particular small width of the region of stable phase oscillations and nonlinear resonances, are essentially the same as in case of the classical microtron [3–6]. However, the existence of a drift space between the bending magnets and effect of the fringe field in these magnets lead to an energy-dependent phase slip. This phenomenon affects the width of the machine acceptance and complicates the choice of initial beam parameters such as the injection energy and the injection phase. Till now the main approach to fixing these parameters is by means of numerical simulations of the phase motion using special codes [7,8]. One of the main difficulties in developing effective analytic techniques of RTM beam dynamics resides in non-applicability of standard perturbation methods, successfully used in beam

 Corresponding author. Tel.: +34 934 012 543; fax: +34 934 017 149.

E-mail address: [email protected] (Y.A. Kubyshin). On leave of absence from the Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia. 1

0168-9002/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2008.07.158

dynamics calculations in synchrotrons and storage rings, due to a large energy gain per turn and a low number of turns of the beam in RTMs. Recent intensive developments of accelerator technologies open new possibilities of applications of RTMs for cargo inspection of containers with detection of elemental composition of their content, production of short-life isotopes, some types of radiation therapy, etc. In designing of an RTM for each particular application one has to optimize machine parameters, in particular the RF wavelength, synchronous phase, energy gain per turn, harmonic number increase, etc. in order to achieve the most effective operation of the accelerator. For this, one needs better understanding of beam dynamics in such machines, therefore the development of analytic methods of RTM beam dynamics becomes an actual and important problem of the physics of this type of accelerators. Such methods would allow to analyze and to control analytically the dependence of machine characteristics on the choice of its main parameters, and also to give an initial approximation of the values of these parameters for their further more precise determination by numerical simulations. In the RTM design certain parameters of the machine (magnetic field in the end-magnets, energy gain in the accelerating structure (AS), length of the drift space, etc.) are adjusted in such a way that the condition of resonance acceleration is fulfilled for a reference synchronous particle with the relativistic factor b ¼ 1, and as a result this particle enters the AS at the fixed synchronous phase js. However, in many designs the electrons after the injection are not ultra-relativistic and have bo1, so that

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at the first orbits the resonance condition cannot be fulfilled and the beam phase at the AS slips, i.e. changes from turn to turn. As it is shown below the phase slip over an orbit is essentially   2l 1 ð2pÞ  1 l b¯ where l is the distance between the end magnets, l is the free space wavelength of the accelerating field and b¯ is an average value of the relativistic factor over the orbit. Thus, the effect is not negligible and should be taken into account in RTMs with (a) lowenergy injection; (b) long distance between the end magnets, and (c) short wavelength. There are a few factors in RTM designs which may require long distance between the end magnets, one of them is the use of an AS with a low gradient, a well-known example is the normal conducting continuous wave LINAC with an accelerating gradient 1 MeV/m of the MAMI facility [9]. As an example of design with non-negligible phase-slip effect due to a short wavelength, we can mention a compact 12 MeV RTM with lE5 cm proposed in Ref. [10]. An approach to reduce the phaseslip effect, described in Ref. [11], is to reflect the beam in the end magnet with the fringe field of special profile in the longitudinal direction [12] and accelerate it in the AS in the opposite direction, thus increasing the energy at the beginning of the beam recirculation. This method works well in case of pulsed RTMs with lE10 cm, solving at the same time the problem of the LINAC bypass, but not in the above-mentioned examples. Till now all analytical estimates of the main RTM parameters are obtained using the standard approach based on a simple formalism (see, for example, Ref. [2] and also Section 2.1 of the present article) which assumes that the particles of the beam move in the drift space at the speed of light and which does not take into account the fringe-field effect in the end magnets. As we argued above, for many RTMs such approximation does not give satisfactory description of their beam physics. In the present paper, we develop an analytic method of description of the phase motion in RTMs taking into account the phase slip in the drift space between the end magnets (but in the absence of the fringe field). This method allows to determine analytically, though approximately, a synchronous phase trajectory and main parameters of the beam. The obtained results are compared with results of numerical simulations. In our analysis, we will assume a zero-length acceleration gap and consider the stationary regime with constant amplitude of the accelerating field in the AS resonant cavities. The article is organized as follows. In Section 2, we describe the RTM model, introduce the notion of non-relativistic synchronous particle, develop the formalism and present a solution of the equations of phase motion taking into account the phase-slip effect. In Section 3, we introduce the notion of critical energy and discuss the stability of synchrotron oscillations around the synchronous particle trajectory. In Section 4, the analytic description of the synchronous particle is applied to the calculation of the injection phase in RTMs and tuning in resonance (adjusting the drift space length in this case). Also a method for numerical determination of the injection phase is described and a comparison between the analytic and numerical results is given. Section 5 contains some concluding remarks.

2. Equilibrium phase Let us consider the longitudinal phase motion of electrons in an electron RTM with the magnetic field induction B in the end magnets, separation l between the magnets (straight section length), and the maximum energy gain DEmax in the AS or LINAC (see Fig. 1). In this article, we neglect effects of the magnet fringe

Fig. 1. Schematic view of our RTM model. The accelerating gap is represented by the vertical bar situated in the middle of the drift space between the end magnets.

field on the phase motion and model the LINAC by an infinitely thin accelerating gap, i.e. we suppose that the energy increase DE of a particle passing through the LINAC is instantaneous. As dynamical variables we choose the full particle energy E and its phase j with respect to the accelerating RF field so that DE ¼ DEmax cos j. Let (jn, En) be the values of these variables at the nth turn at some arbitrary but fixed point of the orbit. For the sake of convenience, we choose this point to be the entrance to the AS, so that (j0, E0) correspond to the particle phase and energy just before its first passage through the AS. We would like to note that in most of pulsed RTM designs the electrons after the injection and first passage through the AS reverse their trajectory in special reverse field magnets, in this case strictly speaking E0 is not the energy of injection but the energy before the second passage through the AS, i.e. after the first acceleration and reflection of the beam in the end magnet. Nevertheless, for the sake of simplicity we will use the term injection energy for E0 in this case as well. 2.1. Phase motion equations of the ultra-relativistic synchronous particle Let us consider first, the limiting case when the initial energy is already large enough and the electrons are ultra-relativistic so that their velocity is practically equal to c. As it is well known (see Refs. [2,6]), in this case the notion of synchronous particle can be introduced. In what follows, we will refer to such particle as ultra-relativistic synchronous particle. Its longitudinal dynamics is characterized by a synchronous phase js and the energy gain per turn equal to DEs ¼ DEmax cos js. The RTM is designed in such a way that its parameters satisfy the following condition of resonance motion of the ultra-relativistic synchronous particle: T ns ¼ T RF ðm þ nðn  1ÞÞ

(1)

where m and n are positive integers defining the mode of operation of the machine. According to Eq. (1) the time of the nth revolution Tns must be an integer multiple of the period of the RF field TRF. For our simple RTM model the time of the nth revolution is equal to T ns ¼

2l 2pr ns 2l Ens þ ¼ þ 2p 2 c c c ec B

(2)

where e is the elementary charge, rns is the radius of the trajectory of the synchronous particle in the end magnet, and E0,s and Ens ¼ E0+nDEs are its initial energy and energy at the nth turn, respectively. Here, we used the known relation between the orbit radius and the magnetic field induction in bending magnets with v/c ¼ 1. Since in accordance with Eq. (1) T 1;s ¼

2l E1;s 2l E0;s þ DEs þ 2p 2 ¼ þ 2p ¼ mT RF c c ec B ec2 B

(3)

the parameter m is equal to the number of RF field periods during the first turn and defines the synchronicity condition at the first orbit, whereas

DT s ¼ T nþ1;s  T n;s ¼ 2p

DEs ¼ nT RF ec2 B

(4)

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so that n is equal to the increase of the harmonic number due to the increase of the synchronous particle period of revolution in one turn. Relation (4) can equivalently be written in the form 2p DEs ¼n B eclRF

(5)

where lRF ¼ cTRF is the wavelength of the RF field in free space. For the ultra-relativistic synchronous particle the evolution of the phase and energy with turns is described by the following recursive relations [2]: Enþ1;s ¼ En;s þ DEmax cos jn;s

jnþ1;s ¼ jn;s þ 2p

(7)

Let us write them in a form which is more convenient for the further analysis. First of all, from the resonance condition, Eq. (1), it follows that the phase advance at the nth turn is equal to 2pTn,s/ TRF ¼ 2p/(m+(n1)n). Relation (7) can be easily resolved and gives

jn;s ¼ js þ 2pn½m þ nðn  1Þ=2

(8)

so that jns ¼ js(mod 2p). As a consequence DEmax cos jns ¼ DEmax cos js ¼ DEs, where DEs is the ultrarelativistic synchronous particle energy gain per turn introduced above. Combing this relation with resonance condition (1) map (6) and (7) can be written as Enþ1;s ¼ En;s þ DEs

(9)

jnþ1;s ¼ jn;s þ K 0 ðEnþ1;s Þ

(10)

where function K0(E) is given by K 0 ðEÞ ¼

4pl

l

þ 2pn

E

DEs

.

(11)

It gives the phase advance of an ultra-relativistic particle (with n ¼ c) of energy E in one turn. Phase solution (8) of system (9) and (10) of equations in finite differences should be complemented with the solution for the energy. As it was already mentioned and as it follows from Eq. (9) En,s is given by the formula En;s ¼ E0;s þ nDEs .

(12)

Relations (8) and (12) describe the trajectory of the ultrarelativistic synchronous particle. Using the condition of synchronous passage through the acceleration gap, Eqs. (3) and (4), we obtain the following formula for the injection energy of the ultra-relativistic synchronous particle:     l DEs 2l E0;s  E0;s mn ¼ (13)

l

n

For an arbitrary particle the recurrence relations between (jn+1, En+1) and (jn, En), analogous to Eqs. (9) and (10), are Enþ1 ¼ En þ DEmax cos jn

(16)

jnþ1 ¼ jn þ KðEnþ1 Þ

(17)

where the term K(En+1) is equal to the phase advance of a particle with energy En+1 in one turn, K(En+1) ¼ 2pTn+1/TRF. The function K(E) is given by the formula KðEÞ ¼

(6)

T nþ1;s . T RF

l

4pl 1 E þ 2pn . l bðEÞ DEs

It is obvious that the revolution time Tn given by Eq. (14) cannot satisfy resonance condition (1) with integer m and n for all n. Nevertheless, as we are going to show now, even in this case it is possible to introduce the notion of synchronous particle. For a given ultra-relativistic synchronous trajectory with synchronous phase js and energy En,s, we define synchronous particle as the particle whose phase variables (jn (mod 2p), En) tend to (js, En,s) as the orbit number n-N, i.e.

jn ðmod 2pÞ ! js ;

En ! Ens .

Therefore, the synchronous particle phase trajectory asymptotically approaches that of the corresponding ultra-relativistic synchronous particle. The phase jn (mod 2p) of the synchronous particle defined here is not constant, as in the case of the ultra-relativistic particle (see Eq. (8)), but varies from turn to turn approaching the value js as n-N. This is precisely the phase-slip effect. Let us show that a solution for the synchronous particle exists and derive analytic formulas describing the evolution of jn and En. In what follows, we will use the notations jn,s and En,s for the phase and energy of the corresponding ultra-relativistic synchronous particle, respectively. Since, as the energy grows, the longitudinal phase coordinates asymptotically approach those of the ultra-relativistic synchronous particle it is reasonable to introduce the variables W n ¼ En  Ens

cn ¼ jn  jns

.

(19)

For the synchronous particle these new variables tend to zero as n-N. Combining Eqs. (9), (10) and (16), (17) one readily arrives at the following system of difference equations:   cosðjs þ cn Þ W nþ1 ¼ W n þ DEs 1 (20) cos js (21)

where the asymptotic energy Ens is given by Eq. (12). Since it grows infinitely with n it is natural to introduce the ratio

2.2. Phase equations of arbitrary particle with boc Now let us consider a particle which is not necessarily ultrarelativistic, i.e. with b ¼ n/co1. The time of nth revolution is equal to 2l 2pr ns 2l En þ ¼ þ 2p 2 bðEn Þc bðEn Þc bðEn Þc ec B

(18)

2.3. Synchronous particle

cnþ1 ¼ cn þ KðEnþ1;s þ W nþ1 Þ  K 0 ðEnþ1;s Þ

Tn ¼

149

(14)

where b(E) is the relativistic factor b ¼ n/c expressed in terms of the energy E: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2 mc bðEÞ ¼ 1  . (15) E

n ¼

DEs Ens

¼

1 n þ ðE0s =DEs Þ

(22)

which is a small parameter for sufficiently large n. We will search for a solution of system (19) as perturbation expansion in powers of en. For this, we introduce the dimensionless variable wn ¼ 2pnW n =DEs

(23)

and the dimensionless parameter

k ¼ mc2 =DEs .

(24)

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Then the system of recurrence relations (20) and (21) can be written as

oscillations is

wnþ1 ¼ wn þ Fðcn Þ

(25)

cnþ1 ¼ cn þ DKðnþ1 ; wnþ1 Þ

(26)

with C and w0 being constants of integration determined by initial conditions. The synchrotron oscillations are stable, i.e. solution (34) is oscillatory, if Q is real. This happens if the following well-known condition on the synchronous phase is fulfilled (see Ref. [2]):

where   cosðjs þ cn Þ 1 cos js 4pl DKðn ; wn Þ ¼ wn þ 0 l Fðcn Þ ¼ 2pn

0o tan js o 1

1 B C  @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1A. 1  k2 2n ð1 þ ðwn =2pnÞn Þ2

(27)

Assuming that the deviation from the ultra-relativistic synchronous trajectory described by cn and wn is small, we expand the function F(cn) in powers of cn and DK(en, wn) in a double series in powers of wn and en. We get

(28)

wnþ1 ¼ wn  2pn tan js cn

(29)

cnþ1 ¼ cn þ wnþ1 þ G0 ðnþ1 Þ

(30)

where the function G0(en) is the leading order term of the function G(en, wn)RDK(en, wn)wn and is equal to G0(en) ¼ (2pl/l)k2en2. Combining Eqs. (29) and (30) one arrives at the following linear inhomogeneous difference equation for cn:

(35)

The effect of the phase slip is described essentially by a solution of the inhomogeneous equation. To find it we represent the variable cn as

cn ¼ a3 an þ Oðnaþ1 Þ

(32)

the general solution of the homogeneous equation can be written as (33)

where t1,2 ¼ exp(7iQ) are solutions of the characteristic equation t22t cos Q+1 ¼ 0 and C1, C2 are arbitrary constants. A more common but equivalent form of this solution for synchrotron

k2

2l

ln tan js

3n þ Oð4n Þ.

(37)

The solution for wn is found immediately from Eq. (30): wn ¼ 

2pl

l

k2 2n þ Oð4n Þ.

(38)

We would like to note that higher order terms in powers of en in expansions (37) and (38) for cn and wn can be obtained in a systematic way. The corresponding procedure is described in Appendix A. Thus, the complete general solution of system (25) and (26) in the leading approximation is given by

cn ¼ C sinðQn þ w0 Þ  wn ¼ C

k2 3  þ Oð4n Þ ln tan js n 2l

(39)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pn tan js cosðQn þ w0  Q =2Þ

 (31)

Its homogenous part is the linearized equation of the RTM longitudinal dynamics for the phase deviation from the ultrarelativistic synchronous trajectory (see, for example, Ref. [2]). The terms on the r.h.s. of Eq. (30) are responsible for the deviation from this asymptotic regime and lead to the phase slip. In the limit n-N the difference G0(en+1)G0(en)pen3-0 and the phaseslip effect disappears. As it can be seen from Eq. (27) in the case of ultra-relativistic dynamics DK(en, wn) ¼ wn, the r.h.s. of Eq. (31) vanishes and this equation becomes the well-known linear equation for synchrotron oscillations in microtrons [2]. The general solution of Eq. (31) can be found as superposition of the general solution of its homogeneous part and a particular solution of the non-homogeneous one. Introducing the phase advance per turn Q by the relation

cn ¼ C 1 tn1 þ C 2 tn2

.

The general solution for wn is found from the second equation of the homogeneous system (Eq. (30) without the G0 term). One gets qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wn ¼ C 2pn tan js cosðQn þ w0  Q =2Þ. (36)

cn ¼ 

The dots in the r.h.s. of the expression for DK(en, wn) stand for higher powers in en and higher powers in wn multiplied by powers of en. As we will see shortly, in the leading order cnpen3, wnpen2. Therefore, in the leading approximation obtained by the consistent truncation of expansions (28) Eqs. (25) and (26) reduce to the following linear system of equations:

cnþ1  2ð1  pn tan js Þcn þ cnþ1 ¼ G0 ðnþ1 Þ  G0 ðn Þ.

2

pn

(34)

and substitute it into Eq. (31). We obtain that a ¼ 3 and that in the leading approximation the solution for cn is

2

Fðcn Þ ¼  2pn tan js cn þ pncn þ     4pl 1 2 2 3 4 4 DKðn ; wn Þ ¼ wn þ k n þ k n 8 l 2  5 6 6 1 2 3 þ k n þ     k n wn þ    . 16 2pn

cos Q ¼ 1  pn tan js

cn ¼ C sinðQn þ w0 Þ

2pl

l

k2 2n þ Oð4n Þ.

(40)

In the general case a particle captured into acceleration, i.e. a particle whose phase variables (cn, wn) remain finite as n-N, experiences the slip in phase and energy, described by the second terms in Eqs. (39) and (40), and synchrotron oscillations, given by the sine and cosine terms. Among all possible initial conditions there exist the one for which the synchrotron oscillations are absent, i.e. C ¼ 0. The existence of such condition will be illustrated in Section 4. In this case, as the orbit number n grows, the phase trajectory monotonously approaches the phase trajectory of the ultra-relativistic synchronous trajectory, i.e. jn (mod 2p)-js, En-En,s. According to the definition this is precisely the synchronous particle. Its trajectory is described by Eqs. (37) and (38) or, taking into account relations (8), (12), (19) and (23) by the formulas   l jn ¼ js þ 2p½mn þ nnðn  1Þ þ cn n ; (41)

l

En ¼ E0;s þ nDEs þ

DEs 2pn

  l wn n ;

l

(42)

where the functions cn and wn in the leading approximation are given by Eqs. (37) and (38) respectively, and we indicated explicitly their dependence on the parameter en, defined by formula (22), and on l/l. Higher order corrections to these

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functions in powers of en are given in Appendix A. We would like to emphasize that cn and wn describe the approach to the ultrarelativistic synchronous trajectory (jn,s, En,s) given by Eqs. (8) and (12) respectively, with the orbit number n. A particle with arbitrary initial conditions makes phase oscillations around trajectory (41) and (42), they will be studied in the next section.

3. Critical energy and stability of oscillations In the previous section, we found the general synchronous trajectory as a deviation from the ultrarelativistic (asymptotic) synchronous one given by formulas (8) and (12). The longitudinal phase oscillations, Eqs. (34) and (36), were also described with respect to the same reference trajectory, the condition of their stability being given by inequalities (35). Such description is practical if the deviations cn and wn are small, i.e. the expansion parameter en is small enough, which means that the particle is close to the ultrarelativistic regime. In particular, in this case the function K(E) defined by Eq. (18), which gives the phase increase in one turn, grows monotonously with energy E. For low energies the parameter en and deviations cn, wn are relatively large and it may be appropriate to characterize the synchrotron oscillations using another reference trajectory, namely the synchronous trajectory given by Eqs. (41) and (42). In this case, one should take into account a new feature: the phase increase in one turn does not necessarily grow with energy. Indeed, as it can be seen from Fig. 2, the function K(E) is not monotonous in general and has a decreasing (for lower energies) and growing (for higher energies) branches. Using definition (18) we calculate 2 4

dKðEÞ 2pn 4pl m c 1 ¼  dE DEs l E3 b3 ðEÞ

(43)

and from equation dK(E)/dE ¼ 0 find that the function K(E) has a minimum and its derivative changes sign at the critical energy sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2=3 2l DEs 2 Ecr ¼ mc 1þ . (44) ln mc2

"



2l

2=3 #3=2

lnk

.

450 445 440 435 K (E)

430

dEnþ1 ¼ dEn þ

DEs ~ n þ djn Þ  cos j ~ n ½cosðj cos js

djnþ1 ¼ djn þ KðE~ nþ1 þ dEn Þ  KðE~ nþ1 Þ.

(45) (46)

In what follows, instead of dEn, we will use the dimensionless variable den ¼ 2pndEn/DEs. For small oscillations djn, den the system can be linearized and one gets

denþ1 ¼ den  2pn

~n sin j djn cos js

0

djnþ1 ¼ djn þ k~ nþ1 denþ1 where we introduced the notation  0 dKðEÞ DEs k~ n ¼ . dE E¼E~ n 2pn

(47)

(48)

(49)

Combining Eqs. (47) and (48) in a similar way as it was done in the derivation of Eq. (31), we arrive at the difference equation for djn: ! 0 0 ~ n ~0 k~ nþ1 k~ nþ1 sin j k nþ1 djn þ 0 djn1 ¼ 0. djnþ1  1 þ 0  2pn cos js k~ n k~ n (50)

It is known (see, for example, Ref. [2]) that the solution of this equation is oscillatory and does not have growing mode if the following condition is fulfilled:    ~  1  pn sin j k~ 0 o1.  cos js 

425 420 415 410

We write this condition as

405 400

We would like to note that the notion of critical energy does not appear in the case of classical microtron (case l ¼ 0). For RTMs Ecr may be not that low. For example, for an RTM with l/l ¼ 30 and DEs ¼ 1 MeV the critical energy EcrE5.04mc2E2.57 MeV. To characterize the longitudinal motion, let us denote in this ~ n ; E~ n ) the synchronous phase trajectory given by section by (j Eqs. (41) and (42) and by (jn, En) a general phase trajectory which ~ n þ djn , oscillates around the synchronous one, and write jn ¼ j En ¼ E~ n þ dEn . We are going to derive equations for the fluctuations djn, dEn and carry out a simplified analysis of their behavior. As it was already discussed, the evolution of the phase coordinates is governed by system of Eqs. (16) and (17). Subtracting from them the equations for the synchronous solution ~ n ; E~ n ) we obtain the following recurrence relations: (j

The equation is too complicated to be solved or analyzed analytically in the general case. To gain some intuition about the behavior of djn we make a few simplifying assumptions. We ~ n changes slowly with n, so that sin j ~ n  sin j ~, assume that sin j and that the derivative of the function K(E) changes slowly with 0 0 turns, so that k~ nþ1  k~ . This can be a good approximation if, for 0 example, the energy gain per turn DEs is small, so that E~ n and k~ n change slowly with n, and we consider the part of the trajectory near the minimum of the function K(E) where it changes slowly. With these assumptions Eq. (50) simplifies and takes the form   ~ ~0 sin j k djn þ djn1 ¼ 0. djnþ1  2 1  pn (51) cos js

The minimal value of the function is equal to KðEcr Þ ¼ 2pnk 1 þ

151

0o

1

2

3

4

5 E (MeV)

6

7

Fig. 2. Function K(E) for DEs ¼ 1 MeV and l/l ¼ 30.

8

9

~ ~0 sin j 2 ko . cos js pn

(52)

This is the condition of phase stability first formulated by Veksler [13] for microtrons and then independently by McMillan [14] for the case of synchrotrons.

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The existence of the critical energy and the necessity to adjust the phase in accordance with bounds (52) should be taken into account in the design of RTMs. If at first turns the accelerator 0 operates at energies EoEcr then the factor k~ o0. Since the synchronous asymptotic phase for n ¼ 1 is chosen around js ¼ 161 then, in order to avoid the presence of the growing ~ n at mode in synchrotron oscillations, the synchronous phase j the first turns, in accordance with inequalities (52), must satisfy the condition

j~ min oj~ n ðmod 2pÞo0

(53)

where ! ! 2 cos js p . ; pn jk~ 0 j 2

j~ min ¼ max arcsin

After the particle energy has crossed the critical energy, so that dK/dE40, the phase condition should be changed to     2 cos js p ~ max ¼ max arcsin ~ n ðmod 2pÞoj . (54) ; 0oj 0 pn k~ 2 It is easy to see that at higher orbits, when the energy is large enough and the synchronous phase is close to its asymptotic ~ n ðmod 2pÞ ¼ ðjns þ cn Þ ðmod 2pÞ  js , stability condition value, j (52) and its particular case, Eq. (54), reduce to the well-known condition for the asymptotic synchronous phase, Eq. (35). The change of the phase-stability condition from (53) to (54) is, of course, analogous to the change of the RF phase in synchrotrons when the transition energy is being crossed and the slippage factor changes sign (see, for example, Ref. [15]). We would like to note that since usually electrons in RTMs make only a few orbits relations (53) and (54) should not be interpreted as long-term stability conditions. However, it is important to take them into account in order to avoid growing modes in the longitudinal beam dynamics at the first orbits.

4. Calculation of the synchronous particle injection phase and RTM parameters Let us describe an analytic, though approximate, method for determining the initial conditions for the synchronous trajectory corresponding to an asymptotic synchronous phase js and asymptotic energy En,s. Suppose that the geometrical and physical RTM parameters l, m, n, l, DEmax, js are fixed. One has to check that for a given asymptotic energy En,s at the nth orbit they are consistent with relation   DEs 2l En;s ¼ m þ ðn  1Þn  (55)

n

l

following from Eqs. (12) and (13). In this case, the particle phase dynamics is completely defined. In particular, using definition (22) and the above expression it is easy to see that the expansion parameter is equal to

n ¼

n ðn  1Þn þ m  ð2l=lÞ

.

(56)

Taking into account that en depends on l/l it will be convenient to consider cn and wn also as functions of this ratio. We use for them the same notation and write             l l l l l l cn  cn n ; ; wn  w n n ;

l

l

l

l

l

l

in accordance with Eqs. (37) and (38) and relations (A.3) and (A.4) (see Appendix A). For us it is important that the functions cn and wn exist and that their dependence on the distance between the

end magnets l and the RF wavelength l enters only in combination l/l. This is a manifestation of the scale invariance of RTMs with respect to these parameters. Though in concrete examples considered further the wavelength is supposed to be fixed the existence of this scaling property allows to obtain results for other values of l by a simple rescaling of l. 4.1. Analytic approach For a given set of RTM parameters l, m, n, l, DEmax, js the phase trajectory of the synchronous particle is fully determined, and, in principle, the initial phase j0 and injection energy E0 can be calculated directly from solution (41) and (42) with n ¼ 0. However, one should keep in mind that the functions cn and wn are known only approximately, as expansions in parameter en. If e051, then even the leading order expressions, Eqs. (37) and (38) may give a sufficiently precise result. In the opposite case, one may include higher-order terms (see Appendix A for higher corrections) to get a better precision. If the series convergence is rather poor one should apply Eqs. (41) and (42) for some higher orbit number n ¼ n0 to get jn0 and En0 and then, using exact recurrence relations (16) and (17), master all the relations into equations for the initial values j0, E0. This is essentially the procedure we are going to describe in what follows. First of all let us note that if the initial energy of the electrons is large enough then the initial phase, i.e. the optimal phase of injection, is practically equal to the asymptotic synchronous phase js and the phase slip of the synchronous particle is negligible. This is the case of e051 considered above, the functions cn and wn only give small corrections to the ultra-relativistic synchronous trajectory. In practice, in the most of RTM designs the injection energy is not that high and the effect of phase slip is important. As it can be seen from Eqs. (39) and (40), to inject the beam into the synchronous trajectory one has to satisfy two conditions fixing the constants C and w0 in order to exclude the excitation of synchrotron oscillations. This can be done by tuning properly two free parameters of the machine. It is natural to consider l, n and js as fixed parameters and the injection phase j0 as a free one. As the second free parameter one can choose either the injection energy or the distance between the end magnets. In RTMs with a high-energy injection scheme from a pre-accelerator with variable beam energy it is natural to take the injection energy as the second free parameter [16]. If initially the beam is non-relativistic and is generated by an electron gun, then, as it was explained before, the injection energy is understood as the beam energy after the first passage through the AS. It is defined by the accelerating field amplitude and in practice cannot be changed [17]. In this case, it is more appropriate to consider the distance between the end magnets as the second free tuning parameter. Having in mind new areas of RTM applications which require accelerators with maximally simple design we are going to consider the second case. As an illustration of the method let us study two cases with e1 being small enough so that Eqs. (41) and (42) can be applied at the first orbit, i.e. n0 ¼ 1. Using the results of Section 2 and formulas in Appendix A it is not difficult to estimate the optimal number of terms in the e1-expansions of the functions c1(l/l) and w1(l/l) which give a good enough approximation of phase j1 and energy E1 at the first orbit. One should keep in mind that the convergence of the expansions depends also on the value of the parameter k defined by Eq. (24). For example, for the RTM proposed in Ref. [10] e1E0.4, kE0.25, the injection energy is practically equal to the energy gain per turn, i.e. the series convergence is rather slow, it is enough to include terms up to e41.

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Having chosen the number of terms in the expansions of the functions c1(l/l) and w1(l/l), on one hand, we write the phase and the energy of the synchronous particle at the end of the first orbit as   l (57) j1 ¼ js þ c1

l

E1 ¼ E1s þ

DEs 2pn

w1

  l

l

.

(58)

Note that in Eq. (57) we write js instead of j1s. Since the difference between them is a multiple of 2p and the energy gain is defined by cos j such substitution does affect the results. On the other hand, for a given injection energy E0 from Eqs. (16) and (17) one gets   l j1 ¼ j0 þ K E1 ; (59)

l

E1 ¼ E0 þ DEs

cos j0 . cos js

(60)

As before, the function K is defined by Eq. (18), in relation (59) we indicate explicitly its dependence on l/l. Taking into account the scaling property with respect to the change of the distance between the end magnets l and the accelerating field wavelength l we will derive an equation for l/l as independent variable. For this, we express j0 from Eq. (59) and use Eq. (57). We obtain       l l l j0 ¼ js þ c1  K E1 ; . (61)

l

l

l

Here, we understand that the energy E1 depends on l/l both through the function w1(l/l) and the term E1,s ¼ E0,s+DEs (see Eq. (55)):       l DEs l DE l m  2 þ s w1 ¼ . (62) E1 l n l 2pn l The function K in Eq. (61) depends on l/l both explicitly (first term in definition (18)) and through the function E1. Using relations (56) and (62) and formula (A.4) for w1(l/l) it is easy to show that in fact the function K on the synchronous trajectory up to order e41 is equal to     l l 6k2 l 4 K E1  þ Oð51 Þ. (63) ; ¼ 2pm þ l l n tan js l 1 Finally, substituting expressions (61) and (62) into relation (60) we arrive at the equation for l/l we have been looking for     1 l 1 l E0 ¼ w1 m2 þ 2pn n l l DEs        1 l l l þ cos js þ c1  K E1 ; . (64) cos js l l l It can be solved by means of standard numerical methods. Having found the ratio l/l we calculate the injection phase of the synchronous particle j0 using formula (61). As a result, for a given injection energy and RF wavelength we obtain the value of the distance between the RTM end magnets l and the initial (injection) phase of the synchronous trajectory. Let us recall that the solution depends also on the asymptotic synchronous phase js and the asymptotic beam energy En,s, i.e. energy at orbits with high n. This latter dependence enters through the choice of the integer parameter m. As illustrations we consider here the following two examples. In both of them we consider the mode with n ¼ 1 and take the energy gain per turn DEs ¼ 2 MeV and the asymptotic synchronous phase js ¼ 161.

153

Example 1. Low-energy injection. E0 ¼ 2.536 MeV, m ¼ 12. The solution of Eq. (64) is (l/l)th ¼ 4.8344, e1E0.4 and from Eq. (61) we obtain the initial phase of the synchronous trajectory to be j0,th ¼ 0.78871. Example 2. High-energy injection. E0 ¼ 12.536 MeV, m ¼ 17. In this case the solution is (l/l)th ¼ 4.8622, e1E0.14 and the initial phase of the synchronous trajectory is j0,th ¼ +15.59941. In the examples above the critical energy, defined by Eq. (44), is equal to EcrE1.8 MeV, so that the energies En4Ecr for all n. In the case E04Ecr the situation can be more complicated, in particular it is not clear whether solutions for l/l and j0 exist. We carried out a preliminary numerical study of this problem for l/l ¼ 5 as an example. Our results show that the solution can always be found and provides a non-zero acceptance for large enough synchronous energy gain DEs. However, we did not find any solution for values DEso0.6 MeV. More complete study of this issue, which is of interest for better understanding of RTM beam dynamics, is still missing. The solution of Eq. (64) can also be found with rather good accuracy analytically as expansion in a small parameter. Indeed, taking into account expression (63) for the function K and Eqs. (A.3) and (A.4) for the functions c1 and w1 one can readily check that to the leading order Eq. (64) reduces to   1 l E0 m2 þ1 (65) ¼ n l DEs so that the leading order solution for l/l is equal to      l 1 E ¼ mn 0 þ1 . l 0 2 DEs

(66)

This is similar to the relation between the resonance value of l/l and the injection energy of the ultra-relativistic synchronous particle (see Eq. (55)). Thus, to the leading order the solution for l/l is as if the synchronous particle was already ultrarelativistic. The true solution has a correction which can be written as an expansion in

1;0 ¼

n 1 ¼ m  2ðl=lÞ0 ðE0 =DEs Þ þ 1

(67)

i.e. the value of e1(l/l) taken at the leading order solution. Re-expanding the parameter e1 and all the functions in Eq. (64) in powers of e1,0 one can easily find the approximate solution      l l k2 2 ¼ 1   k2 31;0 l ap l 0 2 1;0    k2 12 k2 4 3 1;0 þ Oð51;0 Þ . (68) þ 2 n tanðjs Þ 4 Using Eqs. (61) and (63) we get the solution for the injection phase      2k2 l 3 j0;ap ¼ js   31;0 þ 41;0 þ Oð51;0 Þ ðmod 2pÞ. 2 n tanðjs Þ l 0 (69) Of course, this perturbation approach works if the expansion parameter e1,0 given by Eq. (67) is small enough. For the above examples we obtain the following results. Example 1. (Low-energy injection). The leading order solution is (l/l)0 ¼ 4.866, the expansion parameter e1,0 ¼ 0.43 and the solution for l/l and the injection phase are (l/l)ap ¼ 4.8056, j0,ap ¼ 2.081. Example 2. (High-energy injection). In this case the leading order solution is (l/l)0 ¼ 4.866, the expansion parameter e1,0 ¼ 0.14 and

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Fig. 3. Phase jn as function of n for E0 ¼ 2.536 MeV (a) and E0 ¼ 12.536 MeV (b). Curves (I) and (II) are obtained by applying mapping (16) and (17) with l/l, j0 fixed by the following methods: (I) numerically from minimization of functional (70); (II) by solving Eq. (64). Straight line (III) is the ultra-relativistic synchronous trajectory.

the solution for l/l and the injection phase are (l/l)ap ¼ 4.8621, j0,ap ¼ 15.6011. We see that the agreement between the exact numerical solution of Eq. (64) and the approximate solution (68) and (69) is much better for Example 4. This is because the expansion parameter e1,0 is smaller in this case and the expansions up 4 to e1,0 give more accurate result. 4.2. Numerical solution Approximate values of the distance between end magnets and the injection phase found by solving equations (64) and using relation (61) can be compared with an exact solution found numerically, for example, by minimization of the functional f ½jn ðl=l; j0 Þ ¼

n1 X

synchronous phase j ¼ js. This is the synchronous trajectory. The curve with l/l and j0 obtained by the analytic method from Eq. (64) oscillates around the synchronous trajectory. The origin of these oscillations is, of course, the error in the calculation of the injection phase and distance between the end magnets. In the case of injection with higher energy (Fig. 3b) the residual phase oscillations due to the inaccuracy in the determination of l/l, j0 are small, whereas in the low-energy injection case (Fig. 3a) the oscillation amplitude is comparable with the width of the RTM phase stability region. Nevertheless, even in this case the values of the distance l and injection phase found analytically are useful, because they can be used as initial approximation for RTM beam dynamics simulation with dedicated codes [7,8].

5. Summary and discussion ðjn ðl=l; j0 Þ ðmod 2pÞ  js Þ2

(70)

n¼n0

constructed as a sum of quadratic deviations of the phase jn of a test particle from the asymptotic synchronous phase js calculated for N ¼ n1n0+1 turns. Here n0b1 is the number of an orbit at which the summation begins and n1 is the orbit number at which it ends. To start the procedure of minimization some initial values of l/l and j0 are used. The values of the phase jn in Eq. (70) are then calculated from the recurrence relations (16) and (17). For the two examples considered above we obtain the following results. Example 1. (Low-energy injection). E0 ¼ 2.536 MeV, m ¼ 12. The solution is (l/l)num ¼ 4.8278 and j0,num ¼ 4.941471. Therefore, the accuracies of the analytic procedure are |lnumlth|/lnum103 and |j0,numjth|61. Such low accuracy is due to a relatively large value of the parameter e1E0.4. Example 2. (High-energy injection). E0 ¼ 12.536 MeV, m ¼ 17. In this case the solution is (l/l)num ¼ 4.86225 and j0,num ¼ 15.630611. The accuracies are |lnumlth|/lnum105 and |j0,numj0,th|0.031. Good accuracy of the analytic procedure is due to the smallness of the expansion parameter e1E0.14. In this case, even the expansions of cn and wn to the leading order in Eqs. (37) and (38) already give rather good approximation. In Fig. 3 the phase trajectories for Examples 1 and 2 calculated by applying mapping (16) and (17) are shown. The curves corresponding to the values of l/l and j0 determined by the minimization of functional (70) approach monotonously the

We developed an analytic approach to the analysis of the beam phase motion in RTMs which takes into account the phase-slip effect in the drift space. We introduced the notion of nonrelativistic synchronous particle and found its phase trajectory analytically though approximately (see Eqs. (41) and (42)). As the energy grows, this synchronous trajectory approaches monotonously the ultra-relativistic synchronous trajectory known in the literature. The synchronous trajectory corresponds to the optimal acceleration of electrons in the AS and, therefore, should be taken as the reference trajectory in RTM designs, especially in cases with low-energy injection. The synchronous trajectory can be found numerically, for example by the method explained in Section 4.2. It can also be found analytically, and we developed the corresponding formalism. Its analytic description is obtained as an expansion in terms of the parameter en ¼ DEs/Ens, where DEs is the energy gain per turn and Ens is the energy of the ultra-relativistic synchronous particle. Using this result we proposed an approach for fixing the injection phase j0 and the ratio l/l (case with fixed injection energy) and derived a corresponding equation. We showed that the solution for non-relativistic synchronous particle exists, considered two examples, with high- and low-energy injection, and discussed the accuracy of the method. The accuracy of the solutions can be improved by including higher order terms in en. Also synchrotron oscillations around the synchronous trajectory were studied. We derived conditions of absence of growing modes, showed that these conditions are different for high and

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low energies and calculated the critical energy value separating these two regimes. As it was shown in Section 2.3 the phase slip is given by the term Gðn ; wn Þ  DKðn ; wn Þ  wn  KðEn;s þ W n Þ  K 0 ðEn;s Þ  wn

(71)

which enters into the r.h.s. of Eqs. (26), (30) and (31). Using formula (27) one can easily see that to the leading order l Gðn ; wn Þ  2p k2 2n .

l

(72)

The phase slip can be quite considerable, thus for the example of low-energy injection with l/l ¼ 5, considered in Section 4, at the first orbit G(e1, w1)E191. Assuming for simplicity that the particle energy is close to the energy of the corresponding ultrarelativistic synchronous particle, EnEEn,s, we obtain from Eq. (72) that the phase slip is roughly  2 l DEs . (73) GjE / l E In our calculations the end magnets were considered in the hard edge approximation. However, the fringe-field effects may be essential, in particular for RTM designs with low-energy injection. A preliminary study of the phase slip due to the fringe field is given in Ref. [18]. Using formulas derived there one can evaluate the increase of the electron path due to the fringe field and obtain that the corresponding phase slip to the leading order in inverse energy is   3  d DEs 2 Gfringe jE / (74) l E where d is the length of the fringe-field region in the longitudinal direction and the proportionality factor depends on the specific magnetic field profile. In fact, the same relation can be obtained by calculating the deviation of an electron trajectory in a region of constant magnetic field B1 of length d along the direction of incidence. To relate this setting to the case considered previously we can interpret B1 as an average value of the field in the fringe zone. Considering this simple system one can easily get that the increase of the particle path is

DL ¼ R1 arcsin

d d R1

(75)

where R1 is the radius of the orbit curvature in the field region. Taking into account that R1p1/p ¼ c/(Eb) and expanding function (75) in powers of d/R1 one obtains that the additional phase slip Dj ¼ 4pDL/(lb) due to the fringe field in the leading order is, indeed, given by relation (74). We see that the phase slip G is smaller than Gfringe by a factor of  3 Gfringe d l / . (76) G l l In general this ratio is small, so that in the first approximation the phase slip due to the fringe field can be neglected. Thus, for the examples studied in Section 4 with l/lE5 assuming that the fringe-field zone length dE2 cm and l ¼ 5 cm, ratio (76) is about 0.013. More detailed analysis of the fringe-field effect on the phase slip which takes into account the specific profile of the magnetic field will be published in another article which is now in preparation. A way to reduce the phase slip is to use a highenergy injection scheme, an alternative approach, as it was mentioned in the Introduction, is to increase the initial particle energy by first orbit beam reflection by the end magnet [11]. In the article, we supposed that the LINAC has zero length acceleration gap, i.e. particles get the energy increase at one point.

155

A finite length of the resonant cavities and the accelerating field variation during the particle flight-through cause additional phase slip which deserves a special study. A preliminary analysis shows that for beam energies EX2 MeV it is at least one order of magnitude smaller than the main phase-slip effect due to the beam motion with bo1 in the drift space considered in the present article. We would like to note also that our analysis of the phase-slip effect was obtained for the stationary regime of the AS operation when the amplitude of the accelerating field is constant, i.e. we did not consider the non-stationary period with the field amplitude setting because of the beam loading and a complicated evolution of the particle phase. In RTMs the effects due to the beam loading usually can be neglected in cases when the beam power is o10–20% of the power dissipated in the LINAC walls. Acknowledgements The authors would like to thank B.S. Ishkhanov for his interest in the work and valuable comments and Juan Pablo Rigla for checking some calculations. The work was supported by Grant 08-02-00273-a of the Russian Fund for Basic Research and Grants PCI2005-A7-0284 and FIS2006-07016 of the Spanish Ministry of Science and Education.

Appendix A. Analytic calculation of the synchronous trajectory As it was explained in Section 2.3 parameters of the synchronous particle are calculated as expansions in powers of enRDEs/En,s introduced by Eq. (22). To find the coefficients of cn and wn, describing the deviation of the particle phase and energy at the nth orbit from those of the ultra-relativistic synchronous particle, we write

cn ¼

1 X

ai in

(A.1)

bi in

(A.2)

i¼1

wn ¼

1 X i¼1

where ai, bi are coefficients independent of en, substitute these series into system of Eqs. (25) and (26) and calculate the expansions of F(cn), DK(en+1, wn+1), defined by Eqs. (27) and (28), in powers of eni. In working out the expressions one should take into account that

nþ1 ¼ n  2n þ 3n þ Oð4n Þ, n1 ¼ n þ 2n þ 3n þ Oð4n Þ. Since en are independent variables by matching the coefficients of equal powers of en in both sides of formulas (25) and (26) one gets a set of recurrence relations from which ai, bi up to any given order can be obtained. The leading order terms of the expansions of cn and wn are given by formulas (37) and (38), expressions with terms up to en5 are the following:

cn ¼ 

2l

ln tan js

k2 3n

    3 3 6  1  n þ 2 1 þ k2  2n þ Oð3n Þ 2 2 pn tan js wn ¼ 

2pl

(A.3)

k2 2n

l    3 4 2l  1 þ  k2  2n þ k2 3n þ Oð4n Þ 4 pn tan js ln

(A.4)

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where k and en are defined by Eqs. (22) and (24). Recall that the phase and energy of the particle are calculated using relations jn ¼ jn,s+cn and En ¼ En,s+(DEs/2pn)wn correspondingly (see Eqs. (19) and (23)). We would like to note that as expansion parameter one can also use the product enk ¼ mc2/En,s, the coefficients of the expansion in powers of enk are obtained by simple calculation from the coefficients in Eqs. (A.3) and (A.4). The analysis of the convergence of expansions (A.3) and (A.4) shows that for each particular case there exists an optimal number of terms for which the error of the approximation is minimal. References [1] S.P. Kapitza, V.N. Melekhin, The Microtron, Harwood Academic Publishers, London, 1978. [2] R.E. Rand, Recirculating Electron Accelerators, Harwood Academic Publishers, New York, 1984. [3] A.A. Kolomensky, Theoretical study of the particles motion in modern cyclic accelerators, Dr. Sci. Thesis, Lebedev Physical Institute, Moscow, 1956 (in Russian).

[4] V.N. Melekhin, Zh. Eksp. Teor. Fiz. 61 (1971) 1319 (in Russian). [5] V.K. Grishin, M.A. Sotnikov, V.I. Shvedunov, Vestnik MGU, Ser. Phys. Astron. 27 (1986) 26 (in Russian). [6] P. Lidbjo¨rk, Microtrons, in: S. Turner (Ed.), CERN Accelerator School. Fifth General Accelerator Physics Course (CERN94-01), vol. II, pp. 971–981. [7] K.-H. Kaiser, PTRACE, private communication. [8] G. Gevorkyan, A.B. Savitsky, M.A. Sotnikov, V.I. Shvedunov, RTMTRACE, VINITI preprint 183-B89, 1989 (in Russian). [9] H. Euteneuer et al., Experience with the 855 MeV RTM-Cascade MAMI, in: Proceedings of the EPAC-92, Berlin, 1992, pp. 418–421. [10] B.S. Ishkhanov, N.I. Pakhomov, N.V. Shvedunov, V.I. Shvedunov, V.P. Gorbachev, Conceptual design of the miniature electron accelerator dedicated to IORT, in: Proceedings of the RuPAC XIX, Dubna, 2004, pp. 474–476. [11] R.A. Alvinsson, M. Eriksson, Report TRITA-EPP-76-07, Royal Inst. of Tech., Stockholm, 1976, p. 1. [12] H. Babic, M. Sedlacek, Nucl. Instr. and Meth. 56 (1967) 170. [13] V.I. Veksler, Dokl. Akad. Nauk (USSR) 44 (1944) 393. [14] E.M. McMillan, Phys. Rev. 75 (1945) 143. [15] H. Wiedemann, Particle Accelerator Physics, vol. 1, Springer, Berlin, 2003. [16] V.I. Shvedunov, et al., Nucl. Instr. and Meth. A 531 (2004) 346. [17] V.I. Shvedunov, et al., Nucl. Instr. and Meth. A 550 (2005) 39. [18] Yu.A. Kubyshin, J.P. Rigla, A.P. Poseryaev, V.I. Shvedunov, Analytic description of the phase slip effect in race-track microtrons, in: Proceedings of the PAC-2007, Albuquerque, 2007, p. 3369.