Dynamics of Particles in Non Scaling Fixed Field Alternating Gradient Accelerators

Volume 1 PROGRESS IN PHYSICS January, 2010 SPECIAL REPORT Dynamics of Particles in Non Scaling Fixed Field Alternating Gradient Accelerators James K. Jones∗, Bruno D. Muratori†, Susan L. Smith‡, and Stephan I. Tzenov † STFC Daresbury Laboratory, Daresbury, Warrington, Cheshire, WA4 4AD, United Kingdom. E-mail: james.jones@stfc.ac.uk Daresbury Laboratory, Daresbury, Warrington, Cheshire, WA4 4AD, United Kingdom. E-mail: bruno.muratori@stfc.ac.uk ‡ STFC Daresbury Laboratory, Daresbury, Warrington, Cheshire, WA4 4AD, United Kingdom. E-mail: susan.smith@stfc.ac.uk STFC Daresbury Laboratory, Daresbury, Warrington, Cheshire, WA4 4AD, United Kingdom. E-mail: stephan.tzenov@stfc.ac.uk ∗ STFC 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. pared to the scaling one, among which are the relatively small transverse magnet aperture (tending to be much smaller than Fixed-Field Alternating Gradient (FFAG) accelerators were the one for scaling machines) and the lower field strength. proposed half century ago [1–4], when acceleration of elec- Unfortunately this lattice leads to a large betatron tune variatrons was first demonstrated. These machines, which were in- tion across the required energy range for acceleration as optensively studied in the 1950s and 1960s but never progressed posed to the scaling lattice. As a consequence several resbeyond the model stage, have in recent years become the fo- onances are crossed during the acceleration cycle, some of cus of renewed attention. Acceleration of protons has been them nonlinear created by the magnetic field imperfections, recently achieved at the KEK Proof-of-Principle (PoP) pro- as well as half-integer and integer ones. A possible bypass to ton FFAG [5]. this problem is the rapid acceleration (of utmost importance To avoid the slow crossing of betatron resonances associ- for muons), which allows betatron resonances no time to esated with a typical low energy-gain per turn, the first FFAGs sentially damage beam quality. designed and constructed so far have been based on the ”scalBecause non-scaling FFAG accelerators have otherwise ing” principle. The latter implies that the orbit shape and be- very desirable features, it is important to investigate analytitatron tunes must be kept fixed during the acceleration pro- cally and numerically some of the peculiarities of the beam cess. Thus, magnets must be built with constant field in- dynamics, the new type of fast acceleration regime (so-called dex, while in the case of spiral-sector designs the spiral an- serpentine acceleration) and the effects of crossing of linear gle must be constant as well. Machines of this type use con- as well as nonlinear resonances. Moreover, it is important to ventional magnets with the bending and focusing field be- examine the most favorable phase at which the cavities need ing kept constant during acceleration. The latter alternate in to be set for the optimal acceleration. Some of these problems sign, providing a more compact radial extension and conse- will be discussed in the present paper. quently smaller aperture as compared to the AVF cyclotrons. An example of the theory developed here is presented for The ring essentially consists of a sequence of short cells with the parameters of the Electron Machine with Many Applicavery large periodicity. tions (EMMA) [6], a prototype electron non-scaling FFAG Non scaling FFAG machines have until recently been con- to be hosted at Daresbury Laboratory. The Accelerators and sidered as an alternative. The bending and the focusing is pro- Lasers In Combined Experiments (ALICE) accelerator [7] is vided simultaneously by focusing and defocusing quadrupole used as an injector to the EMMA ring. The energy delivered magnets repeating in an alternating sequence. There is a num- by this injector can vary from a 10 to 20 MeV single bunch ber of advantages of the non-scaling FFAG lattice as com- train with a bunch charge of 16 to 32 pC at a rate of 1 to 20 72 James K. Jones, et al. Dynamics of Particles in Non Scaling Fixed Field Alternating Gradient Accelerators 1 Introduction January, 2010 PROGRESS IN PHYSICS Volume 1 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 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 [8–10] suitably modified to cover the case of a nonscaling FFAG lattice. Firstly, a sequence of canonical transformations within 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. 2 Generalities and first principles Using the median symmetry of the machine, it is straightforward to show that ψ can be written in the form ψ = a0 + a1 x + − b0 + b1 x + a2 x2 + ... z − 2! b2 x2 z3 z5 + ... + (c0 + c1 x + . . .) + . . . . (4) 2! 3! 5! 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 = a0 + Ka1 + a2 , b1 = −2Ka0 − K a0 + a1 − K 2 a1 + Ka2 + a3 , b2 = 6K 2 a0 + 6KK a0 − 4Ka1 − 2K a1 + +a2 + 2K 3 a1 − 2K 2 a2 + Ka3 + a4 , c0 = b0 + Kb1 + b2 . (5) (6) (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 , a2 = a1 = ∂2 Bz ∂x2 ∂Bz ∂x . x,z=0 , x,z=0 (9) (10) 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 + K x) × × (H − qϕ)2 − m2 0 c2 − (P x − qA x )2 − (Pz − qAz )2 − p c2 (1) 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 A x = − z F (x, z; s), Az = x F (x, z; s), A s = G (x, z; s), (11) where the Poincar´ gauge condition e xA x + zAz = 0 , (12) written in the natural coordinate system has been used. From Maxwell’s equation B= we obtain × A, (13) (14) (15) − q (1 + K x) A s , where m p0 is the rest mass of the particle. The guiding magnetic field can be represented as a gradient of a certain function ψ(x, z; s) B = ψ, (2) where the latter satisfies the Laplace equation 2 2F + (x ∂ x + z ∂z ) F = Bs , Kx G + (x ∂ x + z ∂z ) G = z Bx − x Bz . 1 + Kx Applying Euler’s theorem for homogeneous functions, we can write F= 1 (0) 1 (1) 1 (2) B + Bs + Bs + . . . , 2 s 3 4 (16) 73 ψ = 0. (3) James K. Jones, et al. Dynamics of Particles in Non Scaling Fixed Field Alternating Gradient Accelerators Volume 1 PROGRESS IN PHYSICS January, 2010 Gu = 1 + K x (0) 1 K x (1) Bu + + Bu + 2 2 3 1 K x (2) + + Bu + . . . , 3 4 z G x − x Gz G= . 1 + Kx (k) Bα × (17) E2 − m2 0 c2 − P x − qE x − qA x p c2 2 − Pz − qEz − qAz − (25) 2 − q (1 + K x) A s + E s , where (18) Es = dT E s (x, z, T ; s) = =− 1 1 + Kx dT ∂ϕ (x, z, T ; s) . ∂s (26) Here u = (x, z) and 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 A2 x 2 B2 x2 z2 ϕ = A0 + A1 x + + . . . − B0 + B1 x + + ... + 2! 2! 2! + (C0 + C1 x + . . .) z + .... 4! 4 We introduce the new scaled variables Pu = Pu Pu = , p0 m p0 c Θ = cT , γ= E E . = E p m p0 c2 (27) The new scaled Hamiltonian can be expressed as H= × H = − (1 + K x) × p0 γ2 − 1 − P x − q E x − q A x 2 − Pz − q E z − q A z 2 − (28) (19) − q (1 + K x) A s + E s , where q= q . p0 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 S 2 x, z, T , P x , Pz , E; s = x P x + z Pz + T E + +q where d T ϕ(x, z, T ; s) , T = −t (20) (21) (29) The quantities E x and Ez can be neglected as compared to the components of the vector potential A, so that H = βγ (1 + K x) ×     × − 1 − P x − qA x   − q (1 + K x) E s , where now q= q q = , p βγp0 Pu = Pu Pu = , p βγp0 u = (x, z) . (31)     − qA s  −   (30) 2 − Pz − qAz 2 is a canonical variable canonically conjugate to H. The relations between the new and the old variables are u= ∂S 2 ∂Pu = u, u = (x, z), ∂S 2 T = =T, ∂E (22) ∂S 2 Pu = = Pu − q ∂u dT Eu (x, z, T ; s) = Eu = − ∂ϕ , ∂u (23) Since Pu and u are small deviations, we can expand the square root in power series in the canonical variables x, P x and z, Pz . Tedious algebra yields H = H0 + H1 + H2 + H3 + H4 + . . . , H0 = −βγ − q(1 + K x) E s , H1 = βγ (qa0 − K) x , (24) H2 = H3 = q βγ 2 2 P x + Pz + (Ka0 + a1 )x2 − a1 z2 , 2 2 qa z βγ 2 2 K x P x + Pz + 0 z P x − x Pz + 2 3 (32) (33) (34) (35) = Pu − qEu (x, z, T ; s), ∂S 2 = E + qϕ (x, z, T ; s) = H= ∂T = m p0 γc2 + q ϕ(x, z, T ; s) . The new Hamiltonian acquires now the form H = − (1 + K x) × 74 James K. Jones, et al. Dynamics of Particles in Non Scaling Fixed Field Alternating Gradient Accelerators January, 2010 PROGRESS IN PHYSICS Volume 1 + q 3 Ka1 + a2 3 b0 x − Ka1 + a2 + x z2 , 2 2 2 (36) H4 = + q 2 βγ 2 2 P x + Pz 8 βγ a02 z2 18 + qxz Ka0 + 3a1 zP x − xPz + 12 Ka2 a3 4 + x − 2 6 (37) 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 (41) q x 2 + z2 + + 4 − Ka2 + a3 Kb0 b1 2 2 b1 4 + + z . x z + 3 2 2 6 The Hamiltonian decomposition (32) 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. 3 The synchro-betatron formalism and the reference orbit 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 3/2 q Xe = 1 + Xe2 Bz (Xe ; s) , (42) p0 βe γe where the prime implies differentiation with respect to s. To proceed further, we notice that equation (42) parameterizing the local curvature can be derived from an equivalent Hamiltonian 2 He (Xe , Pe ; s) = − β2 γe − P2 − q e e dXe Bz (Xe ; s) . (43) Taking into account Hamilton’s equations of motion Xe = Pe 2 β2 γ e e 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 Θ p = 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 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) , (38) (39) − P2 e , Pe = qBz (Xe ; s) , (44) and using the relation Pe = βe γe Xe 1 + Xe2 , (45) we readily obtain equation (42). Note also that the Hamiltonian (43) follows directly from the scaled Hamiltonian (28) with x = 0, P x = Pe , Pz = 0, A x = Az = 0 and the accelerating cavities being switched off respectively. Hamilton’s equations of motion (44) can be linearized and subsequently solved approximately by assuming that Pe βe γ e . (46) Thus, assuming electrons (q = −e), we have Pe = βe γe Xe , Xe = − ea1 (s) Xe − Xc − d(s) . (47) p0 βe γe 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 Lp . Xc = (40) 2 tan(Θ p /2) The three types of solutions to equations (47) are as follows: Drift Space Here L p is the length of the polygon side which actually P0 represents the periodicity parameter of the lattice. Usually Xc (s − s0 ) , Pe = P0 , Xe = X0 + (48) βe γ e 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 where X0 and P0 are the initial position and reference moMeV orbit. The quantity d(s) in equation (39) is the relative mentum and s is the distance in longitudinal direction. offset of the magnetic center in the quadrupoles with respect Focusing Quadrupole to the corresponding side of the polygon. In what follows Xe = Xc + dF + (X0 − Xc − dF ) cos ωF (s − s0 ) + [see equations (49) and (52)] dF corresponds to the offset in the focusing quadrupoles and dD corresponds to the one in P0 sin ωF (s − s0 ) , (49) + the defocusing quadrupoles. Similarly, aF and aD stand for βe γ e ω F James K. Jones, et al. Dynamics of Particles in Non Scaling Fixed Field Alternating Gradient Accelerators 75 Volume 1 PROGRESS IN PHYSICS January, 2010 Pe = − βe γe ωF (X0 − Xc − dF ) sin ωF (s − s0 ) + + P0 cos ωF (s − s0 ) , where ω2 = F eaF . p0 βe γe (50) (51) of the cavity. It is convenient to pass to new scaled variables as follows γ Pu , h= 2 , (63) pu = ˜ βe γ e βe γ e τ = βe Θ , Γe = βγ = βe γ e β 2 h2 − e 1 . 2 β2 γ e e (64) Defocusing Quadrupole Xe = Xc + dD + (X0 − Xc − dD ) cosh ωD (s − s0 ) + P0 + sinh ωD (s − s0 ) , βe γ e ω D Pe = βe γe ωD (X0 − Xc − dD ) sinh ωD (s − s0 ) + + P0 cosh ωD (s − s0 ) , where ω2 = D eaD . p0 βe γe (53) (54) Thus, expressions (57)–(61) become (52) H0 = − Γe + Z d∆E Aβ2 Ee ds e dτ sin φ(τ) , (65) (66) (67) (68) H1 = −(Γe − 1) K x , ˜ 1 1 H2 = p2 + p2 + ˜ ˜z ge + K 2 x2 − ge z2 , ˜ ˜ 2Γe x 2 Kx 2 ˜ Kge ˜z H3 = p x + p2 + ˜ 2 x 3 − 3 x z2 , ˜ ˜˜ 2Γe 6 H4 = p2 + p2 ˜ x ˜z 8Γ3 e 2 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 X0 − Xc Xe = Xc + , cos Θ p − P0 sin Θ p /βe γe Pe = βe γe tan Θ p + arctan P0 βe γ e . (55) (56) E p = m p0 c2 , K 2 ge 4 z , ˜ 24 g . ge = βe γ e − (69) (70) The longitudinal part of the reference orbit can be isolated via a canonical transformation F2 x, p x , z, pz , τ, η; s = x p x + z pz + (τ + s) η + ˜ ˜ ˜ ˜ ˜˜ ˜˜ σ = τ + s, η=h− 1 , β2 e 1 , (71) β2 e (72) Once the reference trajectory has been found the corresponding contributions to the total Hamiltonian (32) can be written as follows H0 = − βγ + Z d∆E AE p ds dΘ sin φ(Θ) , (57) (58) (59) (60) where σ is the new longitudinal variable and η is the energy deviation with respect to the energy γe of the reference particle. 4 Dispersion and betatron motion 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, p x , z, pz , σ, η; s = ση + z pz + x p x + ˜ ˜ ˜ ˜ H1 = − (βγ − βe γe )K x , ˜ H2 = 1 1 P 2 + Pz2 + g + βe γ e K 2 x 2 − g z 2 , ˜ ˜ 2βγ x 2 H3 = Kx 2 ˜ Kg 3 P + Pz2 + 2 x − 3 x z2 , ˜ ˜˜ 2β γ x 6 H4 = 2 P x + Pz2 2 8β3 γ3 K2g 4 − z . ˜ 24 (61) ∞ + k=1 ∞ ηk xXk (s) − p x Pk (s) + Sk (s) , (73) ˜ ∞ Here, we have introduced the following notation g= q a1 . p0 (62) x= x+ ˜ k=1 ∞ ηk Pk , px = px + ˜ k=1 ηk Xk , (74) 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 76 σ=σ+ k=1 k η k−1 Pk p x − Xk x − ∞ − k=1 kη  k−1     Sk + Xk  ∞ m=1     η Pm  .   m (75) James K. Jones, et al. Dynamics of Particles in Non Scaling Fixed Field Alternating Gradient Accelerators January, 2010 PROGRESS IN PHYSICS Volume 1 Equating terms of the form xηn and p x ηn 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 p x η) yields the wellknown result P1 = X1 , X1 + ge + K 2 P1 = K . (76) Since in the case, where betatron motion x = 0, p x = 0 can be neglected the new longitudinal coordinate σ should not depend on the new longitudinal canonical conjugate variable η, the second sum in equation (75) must be identically zero. We readily obtain S1 = 0, and S2 = − In second order we have P2 = X2 − X1 + KX1 P1 , X2 + ge + K 2 P2 = −Kge P2 − 1 KX2 K 1 − 2, 2 2γe (78) (79) X1 P1 . 2 (77) Fig. 1: Horizontal betatron tune for the EMMA ring as a function of energy. and in addition the function S3 (s) is expressed as 1 S3 = − (X1 P2 + 2X2 P1 ) . 3 (80) by means of a canonical transformation specified by the generating function    uPu αu u2    √ − .   (86) F2 x, P x , z, Pz ; s =   2βu βu 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 Close inspection of equations (76), (78) and (79) shows 1 that P1 is the well-known linear dispersion function, while P2 u = U βu , pu = √ Pu − αu U . (87) βu stands for a second order dispersion and so on. Up to third order in η the new Hamiltonian describing the longitudinal The phase advance χu (s) and the generalized Twiss pamotion and the linear transverse motion acquires the form rameters αu (s), βu (s) and γu (s) are defined as H0 = − K1 η2 K2 η3 Z d∆E + + 2 2 3 Aβe Ee ds dτ sin φ(τ) , (81) (82) αu = βu = χu = dχu Fu = , ds βu (88) (89) H2 = where K1 = KP1 − 1 2 1 p + p2 + ge + K 2 x2 − ge z2 , z 2 x 2 dαu = Gu βu − Fu γu , ds 1 , 2 γe K2 = X2 KP1 3 − KP2 − 1 − 2 . (83) 2 2 γe 2γe dβu = −2 Fu αu + 2Ru βu . (90) ds The third Twiss parameter γu (s) is introduced via the well-known expression βu γu − α2 = 1 . u (91) For the sake of generality, let us consider a Hamiltonian of the type Hb = u=(x,z) Fu 2 Gu 2 p + Ru u pu + u . 2 u 2 (84) The corresponding betatron tunes are determined according to the expression Np νu = 2π s+L p A generic Hamiltonian of the type (84) can be transformed to the normal form χu 2 2 Hb = Pu + U , 2 u=(x,z) (85) s dθ Fu (θ) . βu (θ) (92) Typical dependence of the horizontal and vertical betatron tunes on energy in the EMMA non-scaling FFAG is shown in Figures 1 and 2. 77 James K. Jones, et al. Dynamics of Particles in Non Scaling Fixed Field Alternating Gradient Accelerators Volume 1 PROGRESS IN PHYSICS January, 2010 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. It is worthwhile noting that the canonical transformation specified by the generating function (73) allowed us to cancel terms linear in the transverse canonical coordinates x and p x . In order to take a due account of the dependence of the longitudinal dynamics on the transverse one it is necessary to retain terms in the resulting Hamiltonian that are proportional to higher powers in η, x, p x and pz . Up to first order in η, this gives rise to additional terms in the longitudinal Hamiltonian of the form H0ad = − η 2 Kηx 2 p x + p2 − p x + p2 + . . . . z z 2 2 (93) 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 − Aγ2 , (97) as suggested by Figure 3. Here γm corresponds to the reference energy with a mini1 mum time of flight. Provided the time of flight Θi at injection 2 2 ∆Θ = − dθ 1 + K(θ) x(θ) p x (θ) + pz (θ) . (94) energy γi and the time of flight Θm at reference energy γm are 2βe s known, the constants entering equation (97) can be expressed as Θm − Θi 2 5 Acceleration in a non-scaling FFAG accelerator A= , Θ0 = Θm − Aγm . (98) (γm − γi )2 The process of acceleration in a non-scaling FFAG accelerNext, we pass to a new variable ator can be studied by solving Hamilton’s equations of motion for the longitudinal degree of freedom. The latter are γ = γ − γm , Θ = Θm − Aγ2 , (99) obtained from the Hamiltonian (43) supplemented by an additional term [similar to that in equation (57)], which takes similar to the variable η introduced in the previous section. into account the electric field of the RF cavities. They read as Then, Hamilton’s equation of motion (95) can be rewritten in an equivalent form γ dΘ =− , (95) dΘ Θm Aγ2 ds β2 γ2 − P2 = − . (100) ds Lp Lp Nc ωc Θ ZeUc dγ δ p (s − sk ) sin =− − ϕk . (96) In what follows, it is convenient to introduce a new phase ds 2AE p k=1 c ϕ and the azimuthal angle θ along the machine circumference Here Uc is the cavity voltage, ωc is the RF frequency, Nc as an independent variable according to the relations is the number of cavities and ϕk is the corresponding cavity NL L p ωc Θ (101) , R= . ds = Rdθ , ϕ= phase. c 2π s+L p The lengthening of the time of flight for one period of the machine due to betatron oscillations can be expressed as 78 James K. Jones, et al. Dynamics of Particles in Non Scaling Fixed Field Alternating Gradient Accelerators January, 2010 PROGRESS IN PHYSICS Volume 1 It is straightforward to verify (see the averaging procedure below) that the necessary condition to have acceleration is ωc NL |Θm | = h, 2πc (102) where h is an integer (a harmonic number). Averaging Hamilton’s equations of motion dϕ = −h − haγ2 , dθ ZeUc dγ =− dθ 2AE p Nc a= A , |Θm | (103) δ p (θ − θk ) sin (ϕ − ϕk ) , k=1 (104) 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. (103) is taken to be 2.686310−5 . we rewrite them in a simpler form as dϕ = haγ2 , dθ where ϕ = − ϕ − hθ + ψ0 , D= Nc dγ = λ sin ϕ , dθ λ= ZeUc D , 4πAE p (105) (106) (107) In the general case where H0 θ= where 0, we have (114) A2 + A2 , c s cos (hθk + ϕk ) , As ψ0 = arctan , Ac Nc J 1 1 1 4 J3 J3 C1 F1 ; , ; ; , − − , √ 3 2 2 3 a1 c b b a1 c Ac = k=1 As = k=1 sin (hθk + ϕk ) . (108) The effective longitudinal Hamiltonian, which governs the equations of motion (105) can be written as H0 = ha 3 γ + λ cos ϕ . 3 (109) H0 H0 a1 = 1 + , c=1− , λ λ   J3   1 1 1 4 J3 Ji     C1 = √ F1  ; , ; ; i , − i  .   3 2 2 3 a1 c a1 c (115) (116) 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 Since the Hamiltonian (109) is a constant of motion, the of the longitudinal Hamiltonian H0 is illustrated in Figure 5. second Hamilton equation (105) can be written as The important question on whether the serpentine acceleration along the separatrix H0 = 0 is stable is addressed in Ap2 dγ 1 ha 3 pendix B. = ±λ 1 − 2 H0 − γ . (110) A qualitative analysis of the fast serpentine acceleration dθ 3 λ has been presented earlier [11, 12]. However, to the best of Let us first consider the case of the central trajectory, for our knowledge the results presented here comprise the first which H0 = 0. It is of utmost importance for the so called gut- attempt to describe the process quantitatively. Although the ter (or serpentine) acceleration. Equation (110) can be solved exact solution is expressed in the form of standard and generin a straightforward manner to give alized hypergeometric functions, it can be easily incorporated in modern computational environments like Mathematica. C J 1 1 7 θ = 2 F1 , ; ; J 6 − , (111) b 6 2 6 b 6 Concluding remarks where J=γ 3 ha , 3λ b=λ 3 ha , 3λ (112) (113) C = 2 F1 1 1 7 6 , ; ; J Ji . 6 2 6 i 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. Based on the Hamiltonian formalism, the synchro-betatron 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. 79 James K. Jones, et al. Dynamics of Particles in Non Scaling Fixed Field Alternating Gradient Accelerators Volume 1 PROGRESS IN PHYSICS January, 2010 1. Polygon Bend. Within the approximation (46) considered here we can linearize the second of equations (56) and write    1/ cos Θ p −Xc tan Θ p / βe γe cos Θ p         , (119) Mp =        2 0 1/ cos Θ p   Xc 1 − 1/ cos Θ p    Ap =     βe γe tan Θ p 2. Drift Space. 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.      .    (120)   1    MO =    0  LO /βe γe    ,    1 AO = 0 , (121) 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 and their derivatives 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 have been 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 (quantitative) theory describing the acceleration in nonscaling FFAGs has been developed. A Calculation of the reference orbit 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 )        , (122)   MF =      −βe γe ωF sin (ωF LF ) cos (ωF LF )   (Xc + dF )[1 − cos (ωF LF )]    AF =    βe γe ωF (Xc + dF ) sin (ωF LF )     ,    (123) 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 )        , (124)    MD =     βe γe ωD sinh (ωD LD ) cosh (ωD LD )      AD =    (Xc + dD )[1 − cosh (ωD LD )]     ,    (125) The explicit solutions of the linearized equations of motion −βe γe ωD (Xc + dD ) sinh (ωD LD ) (47) can be used to calculate approximately the reference orbit. To do so, we introduce a state vector where LD is the length of the defocusing quadrupole. Since the reference orbit must be a periodic function of s Xe Ze = . (117) with period L p , it clearly satisfies the condition P e The effect of each lattice element can be represented in a simple form as Zout = Mel Zin + Ael . (118) 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: 80 Zout = Zin = Ze . (126) Thus, the equation for determining the reference orbit becomes Ze = MZe + A, or Ze = 1 − M −1 A. (127) Here M and A are the transfer matrix and the shift vector for one period, respectively. The inverse of the matrix 1 − M James K. Jones, et al. Dynamics of Particles in Non Scaling Fixed Field Alternating Gradient Accelerators January, 2010 PROGRESS IN PHYSICS Volume 1 can be expressed as 1−M −1 = × 1 + 1 − SpM cos3 Θ p  M12  1 − M22   ×   M21 1 − M11 cos3 Θ p    .    (128) For the EMMA lattice in particular, the components of the one period transfer matrix and shift vector can be written explicitly as M11 = 1 ωD − L0 L1 ω F ω D s F s D + cF cD + cp ωF (129) 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. + (L0 + L1 ) ωD cF sD − L1 ωF sF cD , M12 = 1 βe γ e c p L0 + L1 − Xc t p cF cD + cp ωF L1 ωD Xc t p − sF sD + ωF ωD c p For the sake of brevity, the following notations c p = cos Θ p , cF = cos (ωF LF ), cD = cosh (ωD LD ) , (135) t p = tan Θ p , sF = sin (ωF LF ) , sD = sinh (ωD LD ), (136) (130) have been introduced in the final expressions for the components of the one period transfer matrix and shift vector. B Phase stability in FFAGs To study the stability of the serpentine acceleration in FFAG accelerators, we write the longitudinal Hamiltonian (109) in an equivalent form H0 = λ J 3 + cos ϕ . (132) Hamilton’s equations of motion can be written as dϕ = 3bJ 2 , dθ dJ = b sin ϕ . dθ (138) (137) + L0 L1 ω F ω D − + + 1 − (L0 + L1 ) ωD Xc t p cF sD + ωD c p L0 L1 ω F 1 + L1 ωF Xc t p − sF cD , ωF c p cp M21 = − M22 = βe γ e (ωF sF cD + L0 ωF ωD sF sD − ωD cF sD ) , (131) cp 1 cF cD ωF + L0 ωF ωD Xc t p − sF sD + cp cp ωD c p L0 s F c D − ω D Xc t p c F s D , + ω F Xc t p − cp A1 = Xc + dF + (dD − dF )(cF − L1 ωF sF ) + × L1 ωF sF cD − cF cD − (L0 + L1 ) ωD cF sD − − ωD sF sD + L0 L1 ω F ω D s F s D + ωF cF sD + ωD Xc + dD × cp Let ϕa (θ) and Ja (θ) be the exact solution of equations (138) 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 (133) dJ1 = bϕ1 cos ϕa . dθ (139) + t p (L0 + L1 )cF cD + + s F c D L1 ω F s F s D − − L0 L1 ω F s F c D , ωF ωD Xc + dD × cp The latter should be solved provided the constraint 2 3Ja J1 − ϕ1 sin ϕa = 0 , (140) A2 = − βe γe ωF (dD − dF )sF + βe γe × ω F s F c D + ω F ω D L0 s F s D − ω D c F s D + + βe γ e t p c F c D − ωF sF sD − ωF L0 sF cD . ωD (134) following from the Hamiltonian (137) holds. Differentiating the second of equations (139) with respect to θ and eliminating ϕ1 , we obtain d2 J1 6b2 H0 4 Ja J1 + 15b2 Ja J1 = 0 . − λ dθ2 (141) 81 James K. Jones, et al. Dynamics of Particles in Non Scaling Fixed Field Alternating Gradient Accelerators Volume 1 PROGRESS IN PHYSICS January, 2010 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 (141) can be written as d2 J1 4 + 15Ja J1 = 0 . 2 dJa (142) The latter possesses a simple solution of the form            C J  5 |J |3  + C Y  5 |J |3  , (143)          J1 = |Ja |  1 1/6  2 1/6      3 a   3 a  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 dJ1 (Ji ) = ϕ1 (Ji ) cos ϕi , dJa J1 (Ji ) = J1i . (144) Submitted on November 12, 2009 / November 16, 2009 References 1. Kolomensky A.A. and Lebedev A.N. Theory of cyclic accelerators. North-Holland Publishing Company, 1966. 2. Kolomensky A.A. et al. Some questions of the theory of cyclic accelerators. Edition AN SSSR, 1956, v.26, no. 2, 7. 3. Symon K.R. et al. Phys. Rev., 1956, v.103, 1837. 4. Kerst D.W. et al. Review of Science Instruments, 1960, v.31, 1076. 5. Aiba M. et al. Development of a FFAG proton synchrotron. Proceedings of EPAC, 2000, 581. 6. Edgecock R. et al. EMMA — the world’s first non-scaling FFAG. Proceedings of EPAC, 2008, 3380. 7. Smith S.L. The status of the Daresbury Energy Recovery Linac Prototype (ERLP). Proceedings of ERL, 2007, 6. 8. Tzenov S.I. Contemporary accelerator physics. World Scientific, 2004. 9. Symon K.R. Derivation ANL/APS/TB-28, 1997. of Hamiltonians for accelerators. 10. Suzuki T. Equation of motion and Hamiltonian for synchrotron oscillations and synchrotron-betatron coupling. KEK Report, 1996, 96–10. 11. Koscielniak S. and Johnstone C. Nuclear Instrum. and Methods A, 2004, v. 523, 25. 12. Scott Berg J. Physical Review Special Topics Accel. Beams, 2006, v.9, 034001. 82 James K. Jones, et al. Dynamics of Particles in Non Scaling Fixed Field Alternating Gradient Accelerators