Muratori Bruno

Smith Susan Louise

Tzenov Stephan Ivanov

English

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INJECTION AND EXTRACTION ORBITS AND TWISS PARAMETERSFOR THE EMMA RINGB.D. Muratori, S.L. Smith and S.I. Tzenov∗, STFC/DL/ASTeC, Daresbury, Warrington, U.K.AbstractUsing the FFEMMAG code, the injection and extractionorbits for the EMMA ring at a variety of injection and ex-traction energies together with the Twiss parameters to beused for matching have been calculated. The orbits includetwo kickers together with a septum at both injection andextraction. The FFEMMAG code has been used in con-junction with several scripts so as to be able to scan theparameter space of the two kicker strengths for a sectionof the EMMA ring. The results confirm the choice of mag-net and vacuum pipe apertures as being adequate to operateEMMA from 10 to 20 MeV.INTRODUCTIONIn a FFAG accelerator the beam spirals from an orbit lo-cated to the inside of a reference curve (chosen arbitrarily)to an external orbit, so that every turn it occupies differ-ent intermediate orbits. Each orbit corresponds to a radialdisplacement and to a prescribed beam momentum.There are several disadvantages of the original scal-ing FFAG design among which are the large aperture re-quirements and the relatively complicated magnets. Therequired aperture in the non-scaling FFAG accelerator issignificantly smaller. The transverse magnetic field inthe alternating gradient magnets (focusing and defocusingquadrupoles) is linear so that particles with different mo-menta follow orbits oscillating around the ring.The non-scaling FFAG is a novel accelerator concept anda project of EMMA (Electron Machine for Many Applica-tions) to demonstrate its feasibility has been recently pro-posed [1]. It is also designed as the scale-down model of amuon accelerator for neutrino factory. The machine is to bebuilt in the ALICE (Accelerators and Lasers In CombinedExperiments) hall of Daresbury Laboratory in the UK andALICE itself is intended to be used as the injector.This paper addresses some issues of beam dynamics, in-jection and extraction details of the EMMA ring utilizingthe FFEMMAG computer code, which has been developedrecently at Daresbury Laboratory.REFERENCE TRAJECTORY ANDINJECTION REQUIREMENTSIt has been shown [2] that the equations for the referenceorbit can be obtained from the HamiltonianHe(Xe, Pe; s) = −√β2eγ2e − P 2e +ep0∫dXeBz(Xe; s).(1)∗E-mail: s.i.tzenov@dl.ac.ukThe Hamilton’s equations of motion can be linearized andsubsequently solved approximately by assumingPe  βeγe, (2)so thatPe = βeγeX ′e, X′′e = −ea1(s)p0βeγe[Xe −Xc − d(s)].(3)Here p0 = m0c, a1(s), and d(s) are the focusing strengthand the offset with respect to the machine polygon line ofthe (focusing or defocusing) quadrupole, respectively, andXc is the distance of the polygon line from the geometriccentre of the machine. Introducing the state vector Z e =(Xe, Pe)T, we can write the solution to Eqs. (3) as followsZf =MeZi + Ae, (4)where Zi is the initial value of the state vector, while Zfis its final value at the exit of the corresponding element.The transfer matrixMe and the shift vector Ae for variouslattice elements can be found in a straightforward mannerby solving the equations of motion (3).Since the reference trajectory must be periodic functionof the longitudinal position s, it clearly satisfies the con-dition Zf = Zi = Ze. In other words, the equation fordetermining the reference orbit becomesZe = (1−M)−1A. (5)HereM and A are the transfer matrix and the shift vectorfor one period, respectively.The basic requirements for the injection and extractionsystem are shown in Table 1. Taking into account the ma-chine requirements and beam characteristics, fast injectionand extraction schemes using two kickers and a septum areadopted. Provided the available space and aperture require-ments, injection and extraction for EMMA have proven tobe rather challenging.DESCRIPTION OF THE FFEMMAGPROGRAMMEThe computer code FFEMMAG implements the basicfeatures of the synchro-betatron formalism suitably mod-ified to study non scaling FFAG accelerators [2]. Themethod is developed starting from first principle. Its mile-stone is the full three-and-a-half-degree-of-freedomHamil-tonian describing the motion of a charged particle in theguiding and accelerating electromagnetic fields of the mostgeneral configuration. The Hamiltonian is written in theProceedings of EPAC08, Genoa, Italy THPP00903 Linear Colliders, Lepton Accelerators and New Acceleration Techniques A12 FFAG, Cyclotrons3395Table 1: Requirements for the injection and extraction sys-tem of the EMMA ringSpecification ValueInjection energy 10 – 20 MeV (variable)Extraction energy 10 – 20 MeV (variable)Acceptance 3π mm rad (normalized)with painting allowanceRange of tune variation 14.47 – 5.30 (Horizontal)Kicker rise time 30 ns (beam evolution periodof approximately 55.45 ns)natural coordinate system associated with a particle trajec-tory (reference orbit) for a fixed momentum/energy. There-fore, finding the reference orbit is an important task andcan be considered as the initial stage of the design of theEMMA ring. The formalism allows to further single outby orders of magnitude various linear and nonlinear con-tributions to the particle motion by a sequence of properlychosen canonical transformations.-8-6-4-2 0 2 4 0  0.2  0.4  0.6  0.8  1Offset [mm]Distance [m]10 MeV11 MeV12 MeV13 MeV14 MeV15 MeVFigure 1: Typical reference trajectory for energies in therange 10 – 15 MeV for two 0.394481 meter long EMMAcells.-4-2 0 2 4 6 8 10 12 0  0.2  0.4  0.6  0.8  1Offset [mm]Distance [m]15 MeV16 MeV17 MeV18 MeV19 MeV20 MeVFigure 2: Typical reference trajectory for energies in therange 15 – 20 MeV for two 0.394481 meter long EMMAcells.The first step of the method comprises in the calculationof the reference orbit. For a given energy, it must be a pe-riodic function of the particle location along the circumfer-ence of the ring with periodicity equal to the length of oneDOFO (or respectively FODO, depending on which onehas been chosen as a basic lattice configuration) cell. Tofind the reference trajectory, the exact relativistic equationsof motion following from (1) are integrated numerically byimposing periodic boundary conditions, which assures theperiodicity of the solution thus obtained. Typical referencetrajectories in the 10 – 20 MeV energy range are shown inFigures 1 and 2.Having determined the reference trajectory, one can ad-just the optimal strengths of the injection and extractionkickers for a variety of impact parameters (orbit locationand angle with respect to the reference coordinate system)after/before the septum magnet. The FFEMMAG code in-cludes a block which performs these calculations in de-tail. Some good and practically feasible cases of reason-able physical apertures and kicker strengths are shown inFigures 3 and 4.-0.04-0.02 0 0.02 0.04 0  0.2  0.4  0.6  0.8  1Offset [m]Distance from drift before second kicker (reverse order) [m]Reverse trajectory for 10 to 15 Mev, B1 = -0.0360 T, B2 = 0.0240 T10 MeV, initial angle  -0.1641035511 MeV, initial angle  -0.1495617612 MeV initial angle  -0.1319707013 MeV initial angle  -0.1161024014 MeV initial angle  -0.1032141715 MeV initial angle  -0.09331496Figure 3: Typical injection trajectory for energies in therange 10 – 15 MeV at suitable kicker strengths.-0.04-0.02 0 0.02 0.04 0  0.2  0.4  0.6  0.8  1Offset [m]Distance from drift before second kicker (reverse order) [m]Reverse trajectory for 15 to 20 Mev, B1 = 0.0320 T, B2 = -0.0340 T15 MeV, initial angle   0.1096338216 MeV, initial angle   0.0861691317 MeV initial angle   0.0662588518 MeV initial angle   0.0491822119 MeV initial angle   0.0343682320 MeV initial angle   0.02137073Figure 4: Typical injection trajectory for energies in therange 15 – 20 MeV at suitable kicker strengths.The next step includes the definition of dispersion to ar-bitrary order in terms of powers of the energy deviation.For a particle with an accurately specified design momen-tum, the position vector is precisely given by the refer-ence trajectory plus betatron oscillations (linear and non-linear) superimposed. The latter comprise the transverseTHPP009 Proceedings of EPAC08, Genoa, Italy03 Linear Colliders, Lepton Accelerators and New Acceleration Techniques3396A12 FFAG, Cyclotronsphase space. For off-momentum particles additional termsin the horizontal component of the position vector appear,which are proportional to the successive powers of the de-viation of the actual energy with respect to the design en-ergy. These terms specify the dispersion functions to all or-ders. The FFEMMAG code calculates the zero-order (con-ventional) dispersion function and the first-order dispersionfunction. These are used later to compute the correspond-ing phase slip coefficients characterizing the acceleration.Figure 5: Horizontal betatron tune as a function of energy.Figure 6: Vertical betatron tune as a function of energy.Further, the dynamics of the Twiss parameters describingthe betatron motion have been worked out in detail. Thesecrucially depend on the energy and on the characteristicshape of the reference trajectory. The FFEMMAG codecontains routines, which calculate the horizontal and verti-cal Twiss parameters (alpha and beta function) for all inter-mediate energies from injection to extraction as well as thecorresponding horizontal and vertical betatron tunes. Thedependence of the horizontal and vertical betatron tunes onthe energy is shown in Figures 5 and 6.The path length in an FFAG arc is often well approxi-mated as a quadratic function of energy (see Figure 7 foran example obtained using the FFEMMAG programme).A novel issue which has no analogue in other cyclic accel-erators is the acceleration process. Since an FFAG lattice isin principle highly dispersive, higher orders of phase slip-page must be taken into account. The acceleration processFigure 7: Path length as a function of energy for a single0.394481 meter EMMA cell.Figure 8: Time of flight as a function of energy for a single0.394481 meter EMMA cell.is described by a longitudinal Hamiltonian, which containsterms proportional to the zero-order (conventional phaseslip factor) and first-order phase slip factor. It usually suf-fices to take into account only terms to second order in theenergy deviation as suggested by Figure 8.CONCLUDING REMARKSThe FFAG accelerator EMMA aims to demonstrate thefeasibility of the non scaling FFAG concept. Due to thediversity of tasks to be fulfilled during its future commis-sioning, the injection system must be flexible enough andable to handle a broad variety of injection conditions. Toinvestigate the practicability of the injection and extrac-tion system, a tracking study using FFEMMAG has beencarried out. After a brief description of the programmesome results concerning the beam dynamics and injec-tion/extraction matching have been shown. These resultsindicate that the latter is feasible within reasonable apertureand strength specifications of septa and kicker magnets.REFERENCES[1] CONFORM website, http//www.conform.ac.uk[2] B.D. Muratori, S.L. Smith and S.I. Tzenov, These Proceed-ings.Proceedings of EPAC08, Genoa, Italy THPP00903 Linear Colliders, Lepton Accelerators and New Acceleration Techniques A12 FFAG, Cyclotrons3397

Injection and Extraction Orbits and Twiss Parameters for the EMMA Ring

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