Dynamics of Particles in Non Scaling FFAG Accelerators

Dynamics of Particles in Non Scaling FFAG Accelerators James K. Jones,∗ Bruno D. Muratori,† Susan L. Smith,‡ and Stephan I. Tzenov§ STFC Daresbury Laboratory, Daresbury, Warrington, Cheshire, WA4 4AD, United Kingdom (Dated: April 14, 2009) Non scaling Fixed-Field Alternating Gradient (FFAG) accelerators have an unprecedented potential for muon acceleration, as well as for medical purposes based on carbon and proton hadron therapy. They also represent a possible active element for an Accelerator Driven Subcritical Reactor (ADSR). Starting from first principle the Hamiltonian formalism for the description of the dynamics of particles in non scaling FFAG machines has been developed. The stationary reference (closed) orbit has been found within the Hamiltonian framework. The dependence of the path length on the energy deviation has been described in terms of higher order dispersion functions. The latter have been used subsequently to specify the longitudinal part of the Hamiltonian. It has been shown that higher order phase slip coefficients should be taken into account to adequately describe the acceleration in non scaling FFAG accelerators. A complete theory of the fast (serpentine) acceleration in non scaling FFAGs has been developed. An example of the theory is presented for the parameters of the Electron Machine with Many Applications (EMMA), a prototype electron non scaling FFAG to be hosted at Daresbury Laboratory. PACS numbers: 29.20.-c, 29.20.D-, 41.85.-p arXiv:0904.2088v1 [physics.acc-ph] 14 Apr 2009 I. INTRODUCTION Fixed-Field Alternating Gradient (FFAG) accelerators were proposed half century ago [1, 2, 3, 4], when acceleration of electrons was first demonstrated. These machines, which were intensively studied in the 1950s and 1960s but never progressed beyond the model stage, have in recent years become the focus of renewed attention. Acceleration of protons has been recently achieved at the KEK Proof-of-Principle (PoP) proton FFAG [5]. To avoid the slow crossing of betatron resonances associated with a typical low energy-gain per turn, the first FFAGs designed and constructed so far have been based on the ”scaling” principle. The latter implies that the orbit shape and betatron tunes must be kept fixed during the acceleration process. Thus, magnets must be built with constant field index, while in the case of spiral-sector designs the spiral angle must be constant as well. Machines of this type use conventional magnets with the bending and focusing field being kept constant during acceleration. The latter alternate in sign, providing a more compact radial extension and consequently smaller aperture as compared to the AVF cyclotrons. The ring essentially consists of a sequence of short cells with very large periodicity. Non scaling FFAG machines have until recently been considered as an alternative. The bending and the focusing is provided simultaneously by focusing and defocusing quadrupole magnets repeating in an alternating sequence. There is a number of advantages of the non scaling FFAG lattice as compared to the scaling one, among which are the relatively small transverse magnet aperture (tending to be much smaller than the one for scaling machines) and the lower field strength. Unfortunately this lattice leads to a large betatron tune variation across the required energy range for acceleration as opposed to the scaling lattice. As a consequence several resonances are crossed during the acceleration cycle, some of them nonlinear created by the magnetic field imperfections, as well as half-integer and integer ones. A possible bypass to this problem is the rapid acceleration (of utmost importance for muons), which allows betatron resonances no time to essentially damage beam quality. Because non scaling FFAG accelerators have otherwise very desirable features, it is important to investigate analytically and numerically some of the peculiarities of the beam dynamics, the new type of fast acceleration regime (so-called serpentine acceleration) and the effects of crossing of linear as well as nonlinear resonances. Moreover, it is important to examine the most favorable phase at which the cavities need to be set for the optimal acceleration. Some of these problems will be discussed in the present paper. An example of the theory developed here is presented for the parameters of the Electron Machine with Many Applications (EMMA) [6], a prototype electron non scaling FFAG to be hosted at Daresbury Laboratory. The Accelerators and Lasers In Combined Experiments (ALICE) accelerator [7] is used as an injector to the EMMA ring. The energy delivered by this injector can vary from a 10 to 20 MeV single bunch train with a bunch charge of 16 to 32 pC at a rate of 1 to 20 Hz. ALICE is presently designed to deliver bunches which are around 4 ps and 8.35 MeV from the exit of the booster of its injector line. These are then accelerated to 10 or 20 MeV in the main ALICE linac after which they are sent to the EMMA injection line. The EMMA injection line ends with a septum for injection into the EMMA ring itself followed ∗ Electronic † Electronic address: address: ‡ Electronic address: § Electronic address: james.jones@stfc.ac.uk bruno.muratori@stfc.ac.uk susan.smith@stfc.ac.uk stephan.tzenov@stfc.ac.uk 2 by two kickers so as to direct the beam onto the correct, energy dependent, trajectory. After circulation in the EMMA ring, the electron bunches are extracted using what is almost a mirror image of the injection setup with two kickers followed by an extraction septum. The beam is then transported to a diagnostic line whose purpose it is to analyze in as much detail as possible the effect the non scaling FFAG has had on the bunch. The paper is organized as follows. Firstly, we review some generalities and first principles of the Hamiltonian formalism [8] suitably modified to cover the case of a non scaling FFAG lattice. Subsequently the synchrobetatron framework is applied to determine the energy dependent reference orbit. Stability of motion about the stationary reference orbit is described in terms of betatron oscillations with energy dependent Twiss parameters and betatron tunes. Dispersion, measuring the effect of energy variation on the path length along the reference orbit is an essential feature of non scaling FFAGs. Within the developed synchrobetatron formalism higher order dispersion functions have been introduced and their contribution to the longitudinal dynamics has been further analyzed. Finally, a complete description of the so-called serpentine acceleration in non scaling lepton FFAGs is given together with conclusions. The calculations of the reference orbit and phase stability are detailed in the appendices. II. GENERALITIES AND FIRST PRINCIPLES Let the ideal (design) trajectory of a particle in an accelerator be a planar curve with curvature K. The Hamiltonian describing the motion of a particle in a natural coordinate system attached to the orbit thus defined is [8]: H = −(1 + Kx) (H − qϕ) − m2 c2 − (Px − qAx )2 − (Pz − qAz )2 − q(1 + Kx)As , p0 c2 2 (1) where mp0 is the rest mass of the particle. The guiding magnetic field can be represented as a gradient of a function ψ(x, z; s) B = ∇ψ, where the latter satisfies the Laplace equation ∇2 ψ = 0. (3) (2) + a′′ + 2K 3 a1 − 2K 2 a2 + Ka3 + a4 , 2 c0 = b′′ + Kb1 + b2 . 0 (7) (8) Prime in the above expressions implies differentiation with respect to the longitudinal coordinate s. The coefficients ak have a very simple meaning: a0 = (Bz )x,z=0 , a1 = ∂ 2 Bz ∂x2 ∂Bz ∂x , x,z=0 Using the median symmetry of the machine, it is straightforward to show that ψ can be written in the form ψ= a0 + a1 x + a2 x + ... z 2! 2 a2 = . x,z=0 (9) z5 z3 +(c0 + c1 x + . . . ) +. . . . 3! 5! (4) Inserting the above expression into the Laplace equation (3), one readily finds relations between the coefficients bk and ck on one hand and ak on the other: − b0 + b1 x + b0 = a′′ + Ka1 + a2 , 0 b1 = −2Ka′′ − K ′ a′ + a′′ − K 2 a1 + Ka2 + a3 , 0 0 1 b2 = 6K 2 a′′ + 6KK ′ a′ − 4Ka′′ − 2K ′ a′ 0 0 1 1 (5) (6) b2 x2 + ... 2! In other words, this implies that, provided the vertical component Bz of the magnetic field and its derivatives with respect to the horizontal coordinate x are known in the median plane, one can in principle reconstruct the entire field chart. The vector potential A can be represented as Ax = −zF (x, z; s), Az = xF (x, z; s), As = G(x, z; s), (10) (11) where the Poincar` gauge condition e xAx + zAz = 0, written in the natural coordinate system has been used. From Maxwell’s equation B = ∇ × A, (12) 3 we obtain 2F + (x∂x + z∂z )F = Bs , Kx G + (x∂x + z∂z )G = zBx − xBz . 1 + Kx (13) + (C0 + C1 x + . . . ) z4 + .... 4! (18) (14) Applying Euler’s theorem for homogeneous functions, we can write F = 1 (0) 1 (1) 1 (2) B + Bs + Bs + . . . , 2 s 3 4 Kx (0) Bu + 2 1 Kx (1) Bu + 2 3 (15) Inserting the above expansion into the Laplace equation for ϕ, we obtain similar relations between Bk and Ck on one hand and Ak on the other, which are analogous to those relating bk , ck and ak . We consider the canonical transformation, specified by the generating function S2 x, z, T , Px , Pz , E; s = xPx + z Pz + T E Gu = 1+ +q where dT ϕ(x, z, T ; s), T = −t, (19) + 1 Kx (2) Bu + . . . , + 3 4 zGx − xGz . G= 1 + Kx (k) (16) (20) (17) is a canonical variable canonically conjugate to H. The relations between the new and the old variables are u= ∂S2 ∂ Pu = u, u = (x, z), T = ∂S2 = T , (21) ∂E Here u = (x, z) and Bα denotes homogeneous polynomials in x and z of order k, representing the corresponding parts of the components of the magnetic field B = (Bx , Bz , Bs ). Thus, having found the magnetic field represented by equation (4), it is straightforward to calculate the vector potential A. The accelerating field in AVF cyclotrons and FFAG machines can be represented by a scalar potential ϕ (the corresponding vector potential A = 0). Due to the median symmetry, we have ϕ = A0 + A1 x + A2 x2 + ... 2! z2 2! Pu = ∂S2 = Pu − q ∂u dT Eu (x, z, T ; s) = Pu − q Eu (x, z, T ; s), Eu = − ∂ϕ , ∂u (22) H= − B0 + B1 x + B2 x2 + ... 2! ∂S2 = E + qϕ(x, z, T ; s) = mp0 γc2 + qϕ(x, z, T ; s). ∂T (23) The new Hamiltonian acquires now the form H = −(1 + Kx) E2 − m2 c2 − Px − q Ex − qAx p0 c2 2 − Pz − q Ez − qAz 2 − q(1 + Kx) As + Es , (24) where Es = dT Es (x, z, T ; s) ∂ϕ(x, z, T ; s) . ∂s We introduce the new scaled variables Pu = (25) Pu Pu = , p0 mp0 c Θ = cT , γ= E E = . (26) Ep mp0 c2 =− 1 1 + Kx dT The new scaled Hamiltonian can be expressed as 4 H= H = −(1 + Kx) γ 2 − 1 − Px − q Ex − qAx p0 2 − Pz − q Ez − qAz 2 − q(1 + Kx) As + Es , (27) where q q= . p0 (28) The quantities Ex and Ez can be neglected as compared to the components of the vector potential A, so that H = βγ(1 + Kx) − 1 − P x − qAx 2 − P z − qAz 2 − qAs − q(1 + Kx)Es , (29) where now q= q q , = p βγp0 Pu = Pu Pu , = p βγp0 u = (x, z). H0 = −βγ − q(1 + Kx)Es , H1 = βγ(qa0 − K)x, (32) (30) Since P u and u are small deviations, we can expand the square root in power series in the canonical variables x, P x and z, P z . Tedious algebra yields H = H0 + H 1 + H 2 + H 3 + H 4 + . . . , (31) (33) H2 = q βγ 2 2 P x + P z + (Ka0 + a1 )x2 − a1 z 2 , (34) 2 2 H3 = βγ qa′ z q 2 2 Kx P x + P z + 0 zP x − xP z + 2 3 3 βγ 2 2 Px + Pz 8 q 4 2 Ka1 + a2 3 b0 x − Ka1 + a2 + xz 2 , 2 2 (35) H4 = + q 2 βγa′2 z 2 2 qxz 0 x + z2 (Ka′ + 3a′ ) zP x − xP z + 0 1 12 18 (36) + a3 Ka2 a3 4 Kb0 b1 2 2 b1 4 x − Ka2 + x z + z . + + + 2 6 3 2 2 6 The Hamiltonian decomposition (31) represents the milestone of the synchrobetatron formalism. For instance, H0 governs the longitudinal motion, H1 describes linear coupling between longitudinal and transverse degrees of freedom and is the basic source of dispersion. The part H2 is responsible for linear betatron motion and chromaticity, while the remainder describes higher order contributions. III. THE SYNCHRO-BETATRON FORMALISM AND THE REFERENCE ORBIT In the present paper we consider a FFAG lattice with polygonal structure. To define and subsequently calculate the stationary reference orbit, it is convenient to use a global Cartesian coordinate system whose origin is located in the center of the polygon. To describe step by step the fraction of the reference orbit related to a particular side of the polygon, we rotate each time the axes of the coordinate system by the polygon angle Θp = 2π/NL , where NL is the number of sides of the polygon. Let Xe and Pe denote the reference orbit and the reference momentum, respectively. The vertical component 5 of the magnetic field in the median plane of a perfectly linear machine can be written as Bz (Xe ; s) = a1 (s)[Xe − Xc − d(s)], a0 (Xe ; s) = Bz (Xe ; s), (37) and the accelerating cavities being switched off respectively. Hamilton’s equations of motion (42) can be linearized and subsequently solved approximately by assuming that Pe ≪ βe γe . Thus, assuming electrons (q = −e), we have ea1 (s) (Xe − Xc − d(s)). p0 βe γe (45) The three types of solutions to equations (45) are as follows: Drift Space ′ Pe = βe γe Xe , ′′ Xe = − (44) where s is the distance along the polygon side, and Xc is the distance of the side of the polygon from the center of the machine Xc = Lp . 2 tan(Θp /2) (38) Here Lp is the length of the polygon side which actually represents the periodicity parameter of the lattice. Usually Xc is related to an arbitrary energy in the range from injection to extraction energy. In the case of EMMA it is related to the 15 MeV orbit. The quantity d(s) in equation (37) is the relative offset of the magnetic center in the quadrupoles with respect to the corresponding side of the polygon. In what follows [see equations (47) and (50)] dF corresponds to the offset in the focusing quadrupoles and dD corresponds to the one in the defocusing quadrupoles. Similarly, aF and aD stand for the particular value of a1 in the focusing and the defocusing quadrupoles, respectively. A design (reference) orbit corresponding to a local curvature K(Xe ; s) can be defined according to the relation K(Xe ; s) = q Bz (Xe ; s), p0 βe γe (39) Xe = X0 + P0 (s − s0 ), βe γe Pe = P0 , (46) where X0 and P0 are the initial position and reference momentum and s is the distance in longitudinal direction. Focusing Quadrupole Xe = Xc + dF + (X0 − Xc − dF ) cos ωF (s − s0 ) + P0 sin ωF (s − s0 ), βe γe ωF (47) Pe = −βe γe ωF (X0 − Xc − dF ) sin ωF (s − s0 ) + P0 cos ωF (s − s0 ), where 2 ωF = (48) where γe is the energy of the reference particle. In terms of the reference orbit position Xe (s) the equation for the curvature can be written as q ′′ ′2 3/2 Xe = 1 + Xe Bz (Xe ; s), (40) p0 βe γe where the prime implies differentiation with respect to s. To proceed further, we notice that equation (40) parameterizing the local curvature can be derived from an equivalent Hamiltonian He (Xe , Pe ; s) = − 2 2 2 βe γe − Pe − q eaF . p0 βe γe (49) Defocusing Quadrupole Xe = Xc + dD + (X0 − Xc − dD ) cosh ωD (s − s0 ) + P0 sinh ωD (s − s0 ), βe γe ωD (50) dXe Bz (Xe ; s). (41) Taking into account Hamilton’s equations of motion ′ Xe = Pe = βe γe ωD (X0 − Xc − dD ) sinh ωD (s − s0 ) + P0 cosh ωD (s − s0 ), where 2 ωD = (51) Pe , 2γ2 − P 2 βe e e ′ Pe = qBz (Xe ; s), (42) and using the relation Pe = ′ βe γe Xe , ′2 1 + Xe eaD . p0 βe γe (52) (43) we readily obtain equation (40). Note also that the Hamiltonian (41) follows directly from the scaled Hamiltonian (27) with x = 0, Px = Pe , Pz = 0, Ax = Az = 0 In addition to the above, the coordinate transformation at the polygon bend when passing to the new rotated coordinate system needs to be specified. The latter can be written as Xe = Xc + X0 − Xc , cos Θp − P0 sin Θp /βe γe 6 Pe = βe γe tan Θp + arctan P0 βe γe . (53) H4 = 2 p2 + pz x 8Γ3 e 2 − K 2 ge 4 z , 24 g . βe γe (66) Once the reference trajectory has been found the corresponding contributions to the total Hamiltonian (31) can be written as follows H0 = −βγ + Z AEp d∆E ds dΘ sin φ(Θ), (54) Ep = mp0 c2 , ge = (67) The longitudinal part of the reference orbit can be isolated via a canonical transformation F2 x, px , z, pz , τ, η; s = xpx + z pz + (τ + s) η + 1 , 2 βe (68) (69) H1 = −(βγ − βe γe )K x, H2 = (55) 1 1 2 P 2 + Pz + g + βe γe K 2 x2 − gz 2 , (56) 2βγ x 2 Kg Kx 2 2 Px + Pz + 2x3 − 3xz 2 , 2βγ 6 2 2 Px + Pz 2 σ = τ + s, η =h− 1 , 2 βe H3 = (57) where σ is the new longitudinal variable and η is the energy deviation with respect to the energy γe of the reference particle. H4 = 8β 3 γ 3 − K 2g 4 z . 24 (58) IV. DISPERSION AND BETATRON MOTION Here, we have introduced the following notation g= qa1 . p0 (59) The (linear and higher order) dispersion can be introduced via a canonical transformation aimed at canceling the first order Hamiltonian H1 in all orders of η. The explicit form of the generating function is G2 (x, px , z, pz , σ, η; s) = ση + z pz + xpx ∞ Moreover, Z is the charge state of the accelerated particle, A is the mass ratio with respect to the proton mass in the case of ions, and φ(Θ) is the phase of the RF. For a lepton accelerator like EMMA, A = Z = 1. In addition, (d∆E/ds) is the energy gain per unit longitudinal distance s, which in thin lens approximation scales as ∆E/∆s, where ∆s is the length of the cavity. It is convenient to pass to new scaled variables as follows pu = Pu , βe γe h= γ , 2 βe γe 1 . 2 2 βe γe (60) + k=1 η k [xXk (s) − px Pk (s) + Sk (s)], ∞ (70) ∞ x=x+ k=1 η Pk , ∞ k px = px + k=1 η k Xk , (71) τ = βe Θ, Γe = βγ = βe γe 2 βe h 2 − (61) σ=σ+ k=1 kη k−1 (Pk px − Xk x) Thus, expressions (54) – (58) become Z H0 = −Γe + 2 Aβe Ee d∆E ds dτ sin φ(τ ), (62) − ∞ ∞ k=1 kη k−1 Sk + Xk m=1 η m Pm . (72) H1 = −(Γe − 1)K x, H2 = 1 1 ge + K 2 x2 − ge z 2 , p2 + p2 + z 2Γe x 2 Kge Kx 2 2x3 − 3xz 2 , px + p2 + z 2Γe 6 (63) (64) Equating terms of the form xη n and px η n in the new transformed Hamiltonian, we determine order by order the conventional (first order) and higher order dispersions. The first order in η (terms proportional to xη and px η) yields the well-known result ′ P1 = X1 , ′ X1 + ge + K 2 P1 = K. (73) H3 = (65) Since in the case of vanishing betatron motion (x = 0, px = 0) the new longitudinal coordinate σ 7 should not depend on the new longitudinal canonical conjugate variable η, the second sum in equation (72) must be identically zero. We readily obtain S1 = 0, and S2 = − In second order we have ′ P2 = X2 − X1 + KX1 P1 , 2 ′ X2 + ge + K 2 P2 = −Kge P1 − 2 KX1 K − 2, 2 2γe 16 14 Horizontal Betatron Tune X1 P1 . 2 (74) 12 10 (75) 8 (76) 6 4 and in addition the function S3 (s) is expressed as 1 S3 = − (X1 P2 + 2X2 P1 ). 3 (77) 10 12 14 16 18 20 Energy [MeV] Close inspection of equations (73), (75) and (76) shows that P1 is the well-known linear dispersion function, P2 stands for a second order dispersion and so on. Up to third order in η the new Hamiltonian describing the longitudinal motion and the linear transverse motion acquires the form K1 η 2 K2 η 3 Z H0 = − + + 2 2 3 Aβe Ee d∆E ds dτ sin φ(τ ), (78) (79) FIG. 1: Horizontal betatron tune for the EMMA ring as a function of energy. The phase advance χu (s) and the generalized Twiss parameters αu (s), βu (s) and γu (s) are defined as χ′ = u Fu dχu , = ds βu (85) 1 1 ge + K 2 x2 − ge z 2 , H2 = p 2 + p 2 + x z 2 2 where 1 K1 = KP1 − 2 γe 2 X1 α′ = u dαu = Gu βu − Fu γu , ds (86) KP1 3 K2 = 2 −KP2 − − 2 . (80) γe 2 2γe ′ βu = dβu = −2Fu αu + 2Ru βu . ds (87) For the sake of generality, let us consider a Hamiltonian of the type Hb = u=(x,z) The third Twiss parameter γu (s) is introduced via the well-known expression βu γu − α2 = 1. u (88) Gu 2 Fu 2 p + Ru upu + u . 2 u 2 (81) The corresponding betatron tunes are determined according to the expression s+Lp A generic Hamiltonian of the type (81) can be transformed to the normal form Hb = χ′ 2 2 u Pu + U , 2 (82) Np νu = 2π s dθFu (θ) . βu (θ) (89) u=(x,z) by means of a canonical transformation specified by the generating function F2 x, P x , z, P z ; s = uP α u2 √ u − u . 2βu βu (83) Typical dependence of the horizontal and vertical betatron tunes on energy in the EMMA non scaling FFAG is shown in Figures 1 and 2. V. ACCELERATION IN A NON SCALING FFAG ACCELERATOR u=(x,z) Here the prime implies differentiation with respect to the longitudinal variable s. The old and the new canonical variables are related through the expressions u=U βu , 1 P u − αu U . pu = √ βu (84) The process of acceleration in a non scaling FFAG accelerator can be studied by solving Hamilton’s equations of motion for the longitudinal degree of freedom. The latter are obtained from the Hamiltonian (41) supplemented by an additional term [similar to that in equation (54)], 8 16 1.3205 1.3200 14 1.3195 Vertical Betatron Tune Time of Flight [ns] 12 1.3190 1.3185 10 1.3180 1.3175 8 1.3170 1.3165 6 10 12 14 16 18 20 10 12 14 16 18 20 Energy [MeV] Energy [MeV] FIG. 2: Vertical betatron tune for the EMMA ring as a function of energy. FIG. 3: Time of flight as a function of energy for a single 0.394481 meter EMMA cell. Next, we pass to a new variable which takes into account the electric field of the RF cavities. They read as dΘ =− ds γ β2γ2 − P2 , (90) γ = γ − γm , Θ = Θm − Aγ 2 , (94) similar to the variable η introduced in the previous section. Then, Hamilton’s equation of motion (90) can be rewritten in an equivalent form Aγ 2 Θm dΘ − , = ds Lp Lp (95) dγ ZeUc =− ds 2AEp Nc k=1 δp (s − sk ) sin ωc Θ − ϕk . c (91) Here Uc is the cavity voltage, ωc is the RF frequency, Nc is the number of cavities and ϕk is the corresponding cavity phase. One could use the results obtained in the previous section with the additional requirement that the phase slip coefficient K1 averaged over one period vanishes. Instead, we shall use an equivalent but more illustrative approach. The path length in a FFAG arc and therefore the time of flight Θ is often well approximated as a quadratic function of energy. The acceleration process is then described by a longitudinal Hamiltonian, which contains terms proportional to the zero-order (conventional phase slip) factor and first-order phase slip factor. It usually suffices to take into account only terms to second order in the energy deviation Θ = Θ0 + 2Aγm γ − Aγ 2 , (92) In what follows, it is convenient to introduce a new phase ϕ and the azimuthal angle θ along the machine circumference as an independent variable according to the relations ds = Rdθ, ϕ= ωc Θ , c R= NL L p . 2π (96) It is straightforward to verify (see the averaging procedure below) that the necessary condition to have acceleration is ωc NL |Θm | = h, 2πc (97) where h is an integer (a harmonic number). Averaging Hamilton’s equations of motion dϕ = −h − haγ 2 , dθ Nc a= as suggested by Figure 3. Here γm corresponds to the reference energy with a minimum time of flight. Provided the time of flight Θi at injection energy γi and the time of flight Θm at reference energy γm are known, the constants entering equation (92) can be expressed as A= (γm − γi )2 Θm − Θi , 2 Θ0 = Θm − Aγm . A , |Θm | (98) ZeUc dγ =− dθ 2AEp k=1 δp (θ − θk ) sin (ϕ − ϕk ), (99) we rewrite them in a simpler form as dϕ = haγ 2 , dθ dγ = λ sin ϕ, dθ (100) (93) 9 20 C = 2 F1 1 1 7 6 , ; ; J Ji . 6 2 6 i (108) 18 Energy MeV 16 14 In the above expressions 2 F1 (α, β; γ; x) denotes the Gauss hypergeometric function of the argument x. This case is illustrated in Figure 4. In the general case where H0 = 0, we have 1 1 1 4 J3 J3 J ,− F1 ; , ; ; θ= √ b a1 c 3 2 2 3 a1 c 80 85 90 Phase ° 95 100 12 − C1 , b (109) 10 where a1 = 1 + H0 , λ c=1− H0 , λ (110) FIG. 4: An example of the so-called serpentine acceleration for the EMMA ring for the central trajectory, where the longitudinal H0 = 0. The harmonic number is assumed to be 11, with the RF wavelength 0.405m. The parameter a from Eq. (98) is taken to be 2.686310−5 . 1 1 1 4 Ji3 Ji J3 F1 C1 = √ ,− i . ; , ; ; a1 c 3 2 2 3 a1 c (111) where ϕ = −ϕ − hθ + ψ0 , λ= ZeUcD , 4πAEp As , Ac (101) Here now, F1 (α; β, γ; δ; x, y) denotes the Appell hypergeometric function of the arguments x and y. The phase portrait corresponding to the general case for a variety of values of the longitudinal Hamiltonian H0 is illustrated in Figure 5. D= A2 + A2 , c s Nc ψ0 = arctan (102) VI. CONCLUDING REMARKS Nc Ac = cos (hθk + ϕk ), k=1 As = sin (hθk + ϕk ). k=1 (103) The effective longitudinal Hamiltonian, which governs the equations of motion (100) can be written as H0 = ha 3 γ + λ cos ϕ. 3 (104) Since the Hamiltonian (104) is a constant of motion, the second Hamilton equation (100) can be written as 1 dγ ha 3 = ±λ 1 − 2 H0 − γ dθ λ 3 2 . (105) Let us first consider the case of the central trajectory, where H0 = 0. It is of utmost importance for the so called gutter acceleration. Equation (105) can be solved in a straightforward manner to give θ= where J =γ 3 1 1 7 6 J , ; ;J 2 F1 b 6 2 6 − C , b (106) ha , 3λ b=λ 3 ha , 3λ (107) Based on the Hamiltonian formalism, the synchrobetatron approach for the description of the dynamics of particles in non scaling FFAG machines has been developed. Its starting point is the specification of the static reference (closed) orbit for a fixed energy as a solution of the equations of motion in the machine reference frame. The problem of dynamical stability and acceleration is sequentially studied in the natural coordinate system associated with the reference orbit thus determined. It has been further shown that the dependence of the path length on the energy deviation can be described in terms of higher order (nonlinear) dispersion functions. The method provides a systematic tool to determine the dispersion functions to every desired order, and represents a natural definition through constitutive equations for the resulting Twiss parameters. The formulation thus developed has been applied to the electron FFAG machine EMMA. The transverse and longitudinal dynamics are explored and an initial attempt is made at understanding the limits of longitudinal stability of such a machine. Unlike the conventional synchronous acceleration, the acceleration process in FFAG accelerators is an asynchronous one in which the reference particle performs nonlinear oscillations around the crest of the RF waveform. To the best of our knowledge, it is the first time that such a fully analytic theory describing the acceleration in non scaling FFAGs has been developed. 10 20 18 Energy MeV 16 14 12 10 0 50 Phase ° 100 150 where LO is the length of the drift. Every cell of the EMMA lattice includes a short drift of length L0 and a long one of length L1 . 3. Focusing Quadrupole. The transfer matrix can be written in a straightforward manner as   cos (ωF LF ) sin (ωF LF )/(βe γe ωF ) , (A5) MF =  −βe γe ωF sin (ωF LF ) cos (ωF LF ) AF =   (Xc + dF )[1 − cos (ωF LF )] βe γe ωF (Xc + dF ) sin (ωF LF )  , (A6) FIG. 5: Examples of serpentine acceleration for the EMMA ring, with varying value of the longitudinal Hamiltonian. The limits of stability are given at values of the longitudinal Hamiltonian of ±0.31272, corresponding to either a 0 phase at 10MeV, or a π phase at 20MeV. APPENDIX A: CALCULATION OF THE REFERENCE ORBIT The effect of each lattice element can be represented in a simple form as Zout = Mel Zin + Ael . The explicit solutions of the linearized equations of motion (45) can be used to calculate approximately the reference orbit. To do so, we introduce a state vector   Xe . Ze =  (A1) Pe where LF is the length of the focusing quadrupole. 4. Defocusing Quadrupole. The transfer matrix in this case can be written in analogy to the above one as   cosh (ωD LD ) sinh (ωD LD )/(βe γe ωD ) , MD =  βe γe ωD sinh (ωD LD ) cosh (ωD LD ) (A7) AD =   (Xc + dD )[1 − cosh (ωD LD )] −βe γe ωD (Xc + dD ) sinh (ωD LD )  , (A8) where LD is the length of the defocusing quadrupole. Since the reference orbit must be a periodic function of s with period Lp , it clearly satisfies the condition Zout = Zin = Ze . (A9) (A2) Here Zin is the initial value of the state vector, while Zout is its final value at the exit of the corresponding element. The transfer matrix Mel and the shift vector Ael for various lattice elements are given as follows: 1. Polygon Bend. Within the approximation (44) considered here we can linearize the second of equations (53) and write   1/ cos Θp − Xc tan Θp /(βe γe cos Θp ) , Mp =  0 1/ cos2 Θp Ap =   Xc (1 − 1/ cos Θp ) βe γe tan Θp  . (A3) Thus, the equation for determining the reference orbit becomes −1 Ze = MZe + A, or Ze = 1 − M A. (A10) Here M and A are the transfer matrix and the shift vector for one period, respectively. The inverse of the matrix 1 − M can be expressed as −1 1−M  = cos3 Θp 1 + 1 − SpM cos3 Θp M12  2. Drift Space.  × (A4) 1 − M22 M21 1 − M11 . (A11) MO =  1 LO /βe γe 0 1  , AO = 0, For the EMMA lattice in particular, the components of the one period transfer matrix and shift vector can be written explicitly as 11 M11 = 1 cF cD + cp ωD − L0 L1 ωF ωD sF sD + (L0 + L1 )ωD cF sD − L1 ωF sF cD , ωF ωD ωF L1 X c tp − sF sD ωF ω D cp (A12) M12 = 1 βe γe cp L0 + L1 − X c tp c F c D + cp L0 L1 ωF ωD − + 1 − (L0 + L1 )ωD Xc tp cF sD + ω D cp 1 L0 L1 ωF + L 1 ω F X c tp − ω F cp cp sF cD , (A13) M21 = − βe γe (ωF sF cD + L0 ωF ωD sF sD − ωD cF sD ), cp (A14) M22 = L0 ωF 1 cF cD s F s D + ω F X c tp − s F c D − ω D X c tp c F s D , + L 0 ω F ω D X c tp − cp cp ω D cp cp Xc + dD cp (A15) A1 = Xc + dF + (dD − dF )(cF − L1 ωF sF ) + × L1 ωF sF cD − cF cD − (L0 + L1 )ωD cF sD − ω D sF sD + L 0 L 1 ω F ω D sF sD ωF + tp (L0 + L1 )cF cD + cF sD sF cD L 1 ω F sF sD + − − L 0 L 1 ω F sF cD , ωD ωF ωD Xc + dD (ωF sF cD + ωF ωD L0 sF sD − ωD cF sD ) cp ω F sF sD − ω F L 0 sF cD . ωD (A16) A2 = −βe γe ωF (dD − dF )sF + βe γe + βe γe tp cF cD − (A17) For the sake of brevity, the following notations cp = cos Θp , tp = tan Θp , cF = cos (ωF LF ), sF = sin (ωF LF ), cD = cosh (ωD LD ), (A18) nian (104) in an equivalent form H0 = λ J 3 + cos ϕ . Hamilton’s equations of motion can be written as dϕ = 3bJ 2 , dθ dJ = b sin ϕ. dθ (B2) (B1) sD = sinh (ωD LD ), (A19) have been introduced in the final expressions for the components of the one period transfer matrix and shift vector. APPENDIX B: PHASE STABILITY IN FFAGS Let ϕa (θ) and Ja (θ) be the exact solution of equations (B2) described already in Section V. Let us further denote by ϕ1 and J1 a small deviation about this solution such that ϕ = ϕa + ϕ1 and J = Ja + J1 . Then, the linearized equations of motion governing the evolution of ϕ1 and J1 are dϕ1 = 6bJa J1 , dθ dJ1 = bϕ1 cos ϕa . dθ (B3) To study the stability of the serpentine acceleration in FFAG accelerators, we write the longitudinal Hamilto- 12 20 18 Energy MeV 16 Next, we examine the case of separatrix acceleration with H0 = 0. In Section V we showed that to a good accuracy the energy gain [Ja (θ) = bθ + Ji ] is linear in the azimuthal variable θ. Therefore, equation (B5) can be written as d2 J1 4 + 15Ja J1 = 0. 2 dJa The latter possesses a simple solution of the form 80 85 90 Phase ° 95 100 14 (B6) 12 10 FIG. 6: Phase stability of the standard EMMA ring, for the central trajectory at H0 = 0. The errors are given as 0.1MeV in energy and 1.3o in phase. 5 3 , |Ja | 3 (B7) where Jα (z) and Yα (z) stand for the Bessel functions of the first and second kind, respectively. In addition the constants C1 and C2 should be determined taking into account the initial conditions J1 = |Ja | C1 J1/6 + C2 Y1/6 dJ1 (Ji ) = ϕ1 (Ji ) cos ϕi , dJa J1 (Ji ) = J1i . (B8) 5 3 |Ja | 3 The latter should be solved provided the constraint 2 3Ja J1 − ϕ1 sin ϕa = 0, (B4) following from the Hamiltonian (B1) holds. Differentiating the second of equations (B3) with respect to θ and eliminating ϕ1 , we obtain d2 J1 6b2 H0 4 − Ja J1 + 15b2 Ja J1 = 0. 2 dθ λ (B5) [1] A. A. Kolomensky and A. N. Lebedev 1966, “Theory of Cyclic Accelerators”, North-Holland Publishing Company. [2] A. A. Kolomensky et al. 1955, “Some questions of the theory of cyclic accelerators”, Edition AN SSSR, page 7, PTE, N0. 2, 26(1956). [3] K. R. Symon et al. 1956, Phys. Rev. 103 (1956) 1837. [4] D. W. Kerst et al. 1960 , Review of Science Instruments 31 1076. [5] M. Aiba et al. 2000, “Development of a FFAG proton syn- chrotron”, Proceedings of EPAC 2000, p. 581. [6] R. Edgecock et al., ”EMMA - the World’s First Nonscaling FFAG”, Proceedings of EPAC 2008, p. 3380. [7] S. L. Smith, ”The Status of the Daresbury Energy Recovery Linac Prototype (ERLP)”, Proceedings of ERL 2007, p. 6. [8] S. I. Tzenov 2004, “Contemporary Accelerator Physics”, World Scientific.