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The EMMA Main Ring Lattice BNL-79961-2008-IR CONFORM-EMMA-ACC-RPT-0001-V1.0-JSBERG-LATTICE The EMMA Main Ring Lattice J. Scott Berg March 2008 Physics Department/Bldg. 901A Brookhaven National Laboratory P.O. Box 5000 Upton, NY 11973-5000 www.bnl.gov Notice: This manuscript has been authored by employees of Brookhaven Science Associates, LLC under Contract No. DE-AC02-98CH10886 with the U.S. Department of Energy. The publisher, by accepting the manuscript for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any le- gal liability or responsibility for the accuracy, completeness, or any third party’s use or the results of such use of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Ref- erence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof or its contractors or subcontractors. The views and opinions of au- thors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. Printed on recycled paper The EMMA Main Ring Lattice J. Scott Berg 1 Brookhaven National Laboratory; Building 901A; P.O. Box 5000; Upton, NY 11973-5000 Abstract I give a brief introduction to the purpose and goals of the EMMA experiment and describe how they will impact the design of the main EMMA ring. I then describe the mathematical model that is used to describe the EMMA lattice. Finally, I show how the different lattice configurations were obtained and list their parameters. Key words: linear non-scaling fixed field alternating gradient accelerator PACS: 29.20.D-, 29.27.Bd, 41.85.-p 1. Goals of the Experiment and Lattice (magnet excitations and positions, RF voltages and fre- quencies, etc.) are correct. The purpose of the EMMA experiment is to provide a There are several effects that appear in linear non-scaling platform for the study of a linear non-scaling FFAG. Such FFAGs, in particular for the configuration used for muon a machine has been contemplated for a number of applica- acceleration, that are known to be of significance. The first tions: muon acceleration for a neutrino factory or muon col- is the accelerating mode itself, often referred to as “ser- lider [1–4], high-power proton drivers [5,6], medical accel- pentine” acceleration, that appears in linear non-scaling erators [7–9,6,10], and other applications [11,12]. The most FFAGs that use high-frequency RF and are isochronous extensive studied of these applications is the muon acceler- within their energy range [13–17]. This mode of accelera- ation application, and was therefore used as a basis for the tion is has never been used in an accelerator before. When design of the EMMA main ring. This application requires it is used to accelerate rapidly, it is linac-like, but it is more that the acceleration be very rapid (under 20 turns in most interesting when it is used to accelerate as slowly as possi- cases), and uses RF with a constant frequency. The par- ble. The EMMA experiment will therefore try to probe the ticles are highly relativistic, and the lattice is adjusted to parameter space of this accelerating mode, to verify that it minimize the variation of the time of flight with respect to behaves as expected. In particular, this requires that one energy (which is what allows the use of constant-frequency be able to change the time of flight as a function of energy, RF). while making minimal changes to the tunes as a function The EMMA experiment will not simply demonstrate of energy. As a result, the machine must be able to vary that one can accelerate a beam in a linear non-scaling the dipole and quadrupole components of the lattice inde- FFAG. The machine will be used to study single-particle pendently. We chose to do this by constructing quadrupole beam dynamics in such a system. While we may not have magnets and putting them on sliders that allowed them to a precise match between the dynamics predicted by simu- be moved horizontally. In addition to varying the time of lation and the behavior of the machine, due to imperfect flight, we will vary the RF voltage in the lattice as well to modeling of magnets as well as errors in the machine, we further explore the parameter space for serpentine acceler- should still be able to verify that the dependency of mea- ation. surable quantities in the machine on machine parameters The second effect of interest is resonances (though “res- onance” may not be precisely the right term [18]). Because the machine tune varies with energy, one expects to cross a number of resonances. A linear non-scaling FFAG is de- Email address: [email protected] (J. Scott Berg). signed to minimize the effect of these resonances by mini- URL: http://pubweb.bnl.gov/people/jsberg/ (J. Scott Berg). 1 Work Supported by the United States Department of Energy, mizing the driving terms for the resonances and by cross- Contract No. DE-AC02-98CH10886. ing any resonances rapidly. The driving of the resonances 2 is minimized by constructing a lattice consisting entirely of where identical, compact cells, and by using linear magnets which 1 s > 0 should reduce the driving of nonlinear magnets. Any break- H(s) = 1/2 s = 0 (4) ing of the perfect symmetry can lead to emittance growth 0 s < 0 and orbit distortion [18]. In addition, some authors have found significant effects due to nonlinearities when realistic Note the distinction between the function H(s) and the magnets are used in a linear non-scaling FFAG [19,4,18]. Hamiltonian H: the function will always have an argument To study imperfections, we plan to vary the position and in parentheses. The vector potential components Ax and gradient in some individual magnets. For the study of non- Ay are 0. linear resonance effects, we will vary the tune of the lattice to change which nonlinear resonances we cross during the 2.3. Integration Algorithm acceleration cycle. The equations of motion (2) are integrated using the im- 2. Mathematical Description plicit midpoint method which is turned into a fourth or- der integrator using Yoshida’s algorithm [20]. This results The computations here are based on a code which sym- in a symplectic integrator. For the purposes of defining plectically integrates Hamiltonian equations of motion. The the EMMA lattice, this discussion is merely academic: the code also has a method for estimating the effects of mag- equations of motion are integrated to high accuracy, and net end fields by using a lumped approximation known as the results are thus independent of the integration method. a “hard-edge” approximation. Furthermore, the code care- The implicit midpoint performs a step of size ∆s for a fully treats coordinate system translations and rotations. vector field v(z,s) by solving the implicit equations This section gives a mathematical description of the algo- zf = zi + ∆s v (zi + zf)/2,s + ∆s/2 , (5) rithms used in the code. ¡ ¢ where zi are the initial phase space variables, and zf are the v 2.1. Hamiltonian phase space variables at the end of the step. If is derived from a Hamiltonian, for which case I will use a rectilinear coordinate system to describe the ∂H v = J , (6) motion. The coordinates are x, y, and s, the latter being i ij ∂z Xj j the independent variable in the equations of motion. x is in the horizontal direction, and y is in the vertical direction. where Jxpx = Jypy = JEt = 1, Jpxx = Jpy y = JtE = 1, − There is a third coordinate variable, the arrival time t. The and Jij = 0 otherwise, the algorithm is symplectic. The al- three coordinates x, y, and t have conjugate momenta px, gorithm is accurate to second order in ∆s. This algorithm py, and E respectively. is particularly convenient for cases that have a transverse The motion− is governed by the Hamiltonian vector potential. Its disadvantage is that it is iterative, typ- ically requiring around 6 evaluations of the vector field v 2 2 2 H = qAs (E/c) (px qAx) (py qAy) , (1) and its derivative. − − − − − − q The implicit equation is solved using Newton’s method. where A , A , and A are the components of the magnetic x y s Application of Newton’s method requires the derivative A vector potential in the x, y, and s directions respectively. of the vector field, which is Hamilton’s equations of motion are ∂2H dx ∂H dy ∂H dt ∂H Aij = Jik . (7) = = = ∂zk∂zj ds ∂px ds ∂py ds − ∂E Xk (2) dp ∂H dp ∂H dE ∂H x = y = = . Newton’s method then gives ds − ∂x ds − ∂y ds ∂t −1 zf,n+1 = zi,n + (I ∆sA/2) δn, (8) − 2.2. Quadrupole Field where δn = zi + ∆s v (zi + zf)/2,s + ∆s/2 zf. (9) The magnet will be described using a so-called “hard- − ¡ ¢ edge” model.
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