Muratori Bruno

Smith Susan Louise

Tzenov Stephan Ivanov

English

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HAMILTONIAN APPROACH TO THE DYNAMICS OF PARTICLES INNON SCALING FFAG ACCELERATORSB.D. Muratori, S.L. Smith and S.I. Tzenov∗, STFC/DL/ASTeC, Daresbury, Warrington, U.K.AbstractStarting from first principle the Hamiltonian formalismfor the description of the dynamics of particles in non scal-ing FFAG machines has been developed. The stationaryreference (closed) orbit has been found within the Hamil-tonian framework. The dependence of the path length onthe energy deviation has been described in terms of higherorder dispersion functions. The latter have been used sub-sequently to specify the longitudinal part of the Hamilto-nian. It has been shown that higher order phase slip coeffi-cients should be taken into account to adequately describethe acceleration in non scaling FFAG accelerators.INTRODUCTIONFixed field alternating gradient (FFAG) acceleratorswere proposed half century ago [1, 2], when accelerationof electrons was first demonstrated. In addition to that,acceleration of protons has been recently achieved at theKEK Proof-of-Principle (PoP) proton FFAG [3]. Machinesof this type use conventional magnets with the bending andfocusing field being kept constant during acceleration. Thelatter alternate in sign, thus providing a more compact ra-dial extension and consequently smaller aperture as com-pared to the AVF cyclotrons. The ring essentially consistsof a sequence of short periods with very large periodicity.Non scaling FFAG machines have until recently beenconsidered as an alternative. The bending and the focus-ing is provided simultaneously by focusing and defocus-ing quadrupole magnets repeating in alternating sequence.There is a number of advantages of the non scaling FFAGlattice as compared to the scaling one, among which arethe relatively small magnet aperture and the lower fieldstrength. Unfortunately this lattice leads to a large betatrontune variation across the required energy range for acceler-ation as opposed to the scaling lattice, where the betatrontune stays constant. As a consequence several resonancesare crossed during the acceleration cycle, some of themnonlinear created by the magnetic field imperfections, aswell as half-integer and integer ones.Because non scaling FFAG accelerators have otherwisevery desirable features, it is important to investigate ana-lytically and numerically some of the peculiarities of thebeam dynamics, the new type of fast acceleration regime(so-called serpentine acceleration) and the effects of cross-ing of linear as well as nonlinear resonances. Some of theseproblems will be discussed in the present paper.∗E-mail: s.i.tzenov@dl.ac.ukHAMILTONIAN FORMALISMGeneral DescriptionThe Hamiltonian describing the motion of a particle ina natural coordinate system associated with a planar refer-ence curve with curvature K is [4]H˜ = −(1 + Kx˜)×[√γ2 − 1−(P˜x − q˜Ax)2−(P˜z − q˜Az)2+ q˜As].(1)Here A = (Ax, Az , As) is the electromagnetic vector po-tential, while x˜ and z˜ are the horizontal and vertical de-viations from the reference orbit with canonical conjugatemomenta P˜x,z = Px,z/(m0c) = Px,z/p0. Furthermore,the distance s along the circumference of the machine ischosen as an independent variable, m0 is the particle restmass and q is the charge, where q˜ = q/(m0c) = q/p0. Thelongitudinal canonical coordinate Θ and its canonical con-jugate variable γ comprising the third degree of freedomare given byΘ = −ct, γ = EE0=Em0c2. (2)Since the longitudinal quantities are dominant, one canexpand the square root in power series in the transversecanonical coordinates. Tedious but straightforward algebrayields [4]H˜ = H˜0 + H˜1 + H˜2 + H˜3 + H˜4 + . . . , (3)H˜0 = −βγ + 1E0(dΔEds) ∫dΘ sinφ(Θ), (4)H˜1 = −(βγ − βeγe)Kx˜, (5)H˜2 =P˜ 2x + P˜2z2βγ+12[(g + βeγeK2)x˜2 − gz˜2], (6)H˜3 =Kx˜2βγ(P˜ 2x + P˜2z)+Kg6(2x˜3 − 3x˜z˜2), (7)H˜4 =(P˜ 2x + P˜2z)28β3γ3− K2g24z˜4 = q˜(∂Bz∂x)x=z=0, (8)where dΔE/ds is the energy gain per unit longitudinaldistance s, which in thin lens approximation scales asΔE/Δs, where Δs is the length of the cavity. In addition,γe is the energy corresponding to the reference orbit.g,THPP008 Proceedings of EPAC08, Genoa, Italy05 Beam Dynamics and Electromagnetic Fields3392D02 Non-linear Dynamics - Resonances, Tracking, Higher OrderReference OrbitThe FFAG lattice with polygonal structure in the hori-zontal plane will be considered here. To define and subse-quently determine the stationary reference orbit, it is con-venient to use a global Cartesian coordinate system whoseorigin is located in the centre of the polygon. To describestep by step the fraction of the reference orbit related to aparticular side of the polygon, we rotate each time the axesof the coordinate system by an angle Θp = 2π/Np, whereNp is the number of sides of the polygon.Let Xe and Pe denote the horizontal position along thereference orbit and the reference momentum, respectively.The vertical component of the magnetic field in the medianplane of a perfectly linear machine can be written asBz(Xe, s) = a1(s)[Xe −Xc − d(s)], a1 = gq˜, (9)where s is the distance along the polygon side. The latteris at a distanceXc =Lp2 tan(Θp/2), (10)from the centre of the polygon, where the length of thepolygon side Lp is actually representing the periodicity ofthe lattice. The quantity d(s) in Eq. (9) is the relative offsetof the magnetic centre from the polygon centre line.A design (reference) orbit corresponding to a local cur-vature K(Xe, s) can be defined according to the relationK(Xe, s) =qp0βeγeBz(Xe, s). (11)In terms of the reference orbit position Xe(s) the equationfor the curvature can be written as [5]X ′′e =qp0βeγe(1 + X ′2e)3/2Bz(Xe, s), (12)where the prime implies differentiation with respect to s.Note that Eq. (12) parameterizing the local curvature canbe derived from a HamiltonianHe(Xe, Pe; s) = −√β2eγ2e − P 2e − q˜∫dXeBz(Xe, s),(13)which is nothing but the stationary part of the Hamiltonian(1) evaluated on the reference trajectory (x˜ = 0 and theaccelerating cavities being switched off, respectively).DISPERSION AND BETATRON MOTIONIt is convenient to pass to new scaled variables as followsp˜x,z =P˜x,zβeγe, h =γβ2eγe, τ = βeΘ, Γe =βγβeγe.(14)Thus, expressions (4) – (8) becomeH˜0 = −Γe + 1β2eEe(dΔEds) ∫dτ sinφ(τ), (15)H˜1 = −(Γe − 1)Kx˜, (16)H˜2 =p˜2x + p˜2z2Γe+12[(ge + K2)x˜2 − gez˜2], (17)H˜3 =Kx˜2Γe(p˜2x + p˜2z)+Kge6(2x˜3 − 3x˜z˜2), (18)H˜4 =(p˜2x + p˜2z)28Γ3e− K2ge24z˜4, ge =gβeγe. (19)The longitudinal part (τe, γe) of the reference orbit can beextracted by means of the canonical transformationF2(x˜, ˜˜px, z˜, ˜˜pz , τ, η; s) = x˜˜˜px + z˜˜˜pz +(τ +s)(η + 1β2e),(20)σ = τ + s, η = h− 1β2e. (21)The linear and higher order dispersion can be introducedvia a canonical transformation aimed to cancel the first or-der Hamiltonian H˜1 in all orders of η. The explicit form ofthe generating function isG2(x˜, p̂x, z˜, p̂z, σ, η̂; s) = ση̂ + z˜p̂z + x˜p̂x+∞∑k=1η̂k[x˜Xk(s)− p̂xPk(s) + Sk(s)], (22)x˜ = x̂ +∞∑k=1η̂kPk, p˜x = p̂x +∞∑k=1η̂kXk, (23)σ = σ̂ +∞∑k=1kη̂k−1(Pkp̂x −Xkx̂)−∞∑k=1kη̂k−1(Sk + Xk∞∑m=1η̂mPm). (24)Equating terms of the form x̂η̂n and p̂xη̂n in the new trans-formed Hamiltonian, we determine order by order the con-ventional (first order) and higher order dispersions. Thefirst order in η̂ (terms proportional to x̂η̂ and p̂xη̂) yieldsthe well-known resultP ′1 = X1, X ′1 +(ge + K2)P1 = K. (25)From the requirement that the second sum in equation (24)is identically zero, we readily obtain S1 = 0, andS2 = −X1P12 . (26)In second order the dispersion equations take the formP ′2 = X2 −X1 + KX1P1, (27)X ′2 +(ge + K2)P2 = −KgeP21 − KX 212 − K2γ2e . (28)Proceedings of EPAC08, Genoa, Italy THPP00805 Beam Dynamics and Electromagnetic Fields D02 Non-linear Dynamics - Resonances, Tracking, Higher Order3393In addition, we haveS3 = −13(X1P2 + 2X2P1). (29)Up to third order in η̂ the new Hamiltonian describing thelongitudinal motion and the linear transverse motion ac-quires the formĤ0 = −K˜1η̂22+K˜2η̂33+1β2eEe(dΔEds) ∫dτ sinφ(τ),(30)Ĥ2 =p̂2x + p̂2z2+12[(ge + K2)x̂2 − geẑ2], (31)whereK˜1 = KP1 − 1γ2e, K˜2 = KP1γ2e−KP2 − X212− 32γ2e.(32)The averaged over one lattice periodK1 = 1Lps+Lp∫sdλK˜1(λ), K2 = 1Lps+Lp∫sdλK˜2(λ),(33)represent the first and higher order phase slip coefficients,respectively.The Twiss parameters corresponding to a fixed refer-ence energy γe can be introduced in a conventional manner.For the sake of generality instead of (31) let us consider aHamiltonian of the typeĤb =∑u=(x,z)[Fup̂2u2+Ruûp̂u + Guû22]. (34)A generic Hamiltonian of the type (34) can be transformedto the normal formHb =∑u=(x,z)χ′u2(P2u + U2), (35)by means of a canonical transformation specified by thegenerating functionF2(x̂, P x, ẑ, P z ; s)=∑u=(x,z)(ûPu√βu− αuû22βu). (36)The old and the new canonical variables are related throughthe expressionsû = U√βu, p̂u =1√βu(Pu − αuU). (37)The phase advance χu(s) and the generalized Twiss param-eters αu(s), βu(s) and γu(s) are defined asχ′u =dχuds=Fuβu, (38)α′u =dαuds= Guβu −Fuγu, (39)β′u =dβuds= −2Fuαu + 2Ruβu. (40)The third Twiss parameter γu(s) is introduced according tothe well-known expressionβuγu − α2u = 1. (41)The corresponding betatron tunes are determined asνu =Np2πs+Lp∫sdλFu(λ)βu(λ). (42)It is worthwhile to note that if one takes into accounthigher order nonlinear dispersion function contributionsinto the longitudinal part of the Hamiltonian (30), the timeof flight variable τ becomes a polynomial function of theenergy γ. Moreover, the approach described here providesa systematic perturbative tool to uniquely determine thepolynomial coefficients up to arbitrary order. However, inpractice a parabolic approximation of the time of flight asa function of energy is sufficient [6].CONCLUDING REMARKSBased on the Hamiltonian formalism, the synchro-betatron approach for the description of the dynamics ofparticles in non scaling FFAG machines has been devel-oped. Its starting point is the specification of the static ref-erence (closed) orbit for a fixed energy as a solution of theequations of motion in the machine reference system. Theproblem of acceleration and dynamical stability can be se-quentially studied in the natural coordinate system associ-ated with the reference orbit thus determined.It has been further shown that the dependence of the pathlength on the energy deviation can be described in terms ofhigher order (nonlinear) dispersion functions. The methodprovides a systematic tool to determine the dispersion func-tions to every desirable order.REFERENCES[1] K. R. Symon et al., Phys. Rev. 103 (1956) 1837.[2] D. W. Kerst et al., Review of Science Instruments 31(1960)1076.[3] M. Aiba et al., ”Development of a FFAG proton synchrotron”,Proceedings of EPAC 2000, Vienna, 2000, p. 581.[4] S.I. Tzenov, ”Contemporary Accelerator Physics”, Singa-pore, World Scientific, 2004.[5] H.S.M. Coxeter, ”Introduction to Geometry”, 2nd ed., NewYork, Wiley, 1969.[6] B.D. Muratori, S.L. Smith and S.I. Tzenov, These Proceed-ings.THPP008 Proceedings of EPAC08, Genoa, Italy05 Beam Dynamics and Electromagnetic Fields3394D02 Non-linear Dynamics - Resonances, Tracking, Higher Order

Contemporary Accelerator Physics”,

Development of a FFAG proton synchrotron”,

Introduction to Geometry”, 2nd ed.,

These Proceedings.

Hamiltonian Approach to the Dynamics of Particles in Non-scaling FFAG Accelerators

Hamiltonian Approach to the Dynamics of Particles in Non-scaling FFAG Accelerators

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