Paper  Title  Page 

MPPE002  Beam Propagation in Misaligned Magnetic Elements: A MatLab Based Code  826 


Funding: Fundacao de Amparo a Pesquisa do Estado de Sao Paulo  FAPESP Conselho Nacional de Desenvolvimento Cientifico e Tecnologico  CNPq. We present a method to calculate kinematical parameters of a beam subject to a misaligned magnetic element. The procedure consists in transforming the kinematical parameters of the beam to the reference frame in which the magnetic element is aligned, propagating the beam through the element, and transforming back to the original frame. This is done using rotation matrices around the X, Y, and Zaxes. These matrices are not Lorentz invariant, so the rotations must be performed in a reference frame where the beam is at rest. We describe the transformation matrices, present a MatLab based code that uses this method to propagate up to 100 particles trough a misaligned quadrupole, and show some graphical outputs of the code. 

MPPE003  Monte Carlo Simulations of Thin Internal Target Scattering In CELSIUS  


In the practical operation of the storage ring CELSIUS with the hydrogen pellet target, we simetimes observe a cooling phenomenon in the longitudinal phase space, that is, the circulating beam's phase space gets shrunk instead of blown up. This phenomenon occurs independently on the electron cooling. In this paper, we aim to investigate and interpret this phenomenon as well as the beam lifetime in the presence of hydrogen pellet target with and without rf and with and without electron cooling in CELSIUS using Monte Carlo simulations.  
MPPE004  Evaluation of Nonlinear Effects in the 3GeV Rapid Cycling Synchrotron of JPARC  916 


In order to accelerate an ultrahigh intense beam with small particle losses, the 3GeV Rapid Cycling Synchrotron (RCS) of JPARC, which is being constructed at JAERI, has a large acceptance. In this case the nonlinearity associated with the motion of particles at large amplitude and large momentum deviation plays a significant role. The sources of nonlinear magnetic fields in RCS are mainly connected with the fringe of the main dipole and quadrupole magnets and with the sextupole magnets used for the linear chromaticity correction. In this paper, we will present simulation results including such nonlinear effects. In addition, the possible correction scheme for the induced transverse resonances will be discussed.  
MPPE005  Dynamic Aperture and Resonance Correction for JPARCRCS  979 


Main intrinsic field nonlinearities, which are common for synchrotrons with large aperture, are the nonlinear field of the bending magnets, the fringing field of the magnets and the sextupole field nonlinearity, used for the chromaticity correction. The particle motion in the ring bending magnets has been analyzed by two methods: (1) by direct integration of the particle motion equations in the 3D magnetic field (Tosca output), based on the 4th order RungeKutta integrator and (2) by determination the transfer 8th order map of the bending magnet by using the Gaussian wavelet in the 3D space. The second technique allows us to use powerful tools such as the normal form analysis, to define the resonance driving terms, which can be used for the resonance correction. As the result of this study it was shown that the main limitation of the RCS dynamic aperture can be caused by the structure normal sextupoleorder resonance and the normal octupoleorder resonance. Other highorder resonances have smaller effects on the particles motion than the resonances mentioned above. The correction scheme to improve the dynamic aperture near the normal sextupoleorder resonance has been analyzed.  
MPPE006  Particle Distribution Function Forming in Nonlinear Systems  985 


Modern ionoptical systems are used in different fields of beam physics both independent facilities as consisting of largemachines. One of these destination is to create beams with a desired distribution of beams particles. Often there is a need to ensure a homogeneous distribution for a terminal beam phase portrait in a transverse configuration space. This is one of problems of nonlinear aberrations management. It is known that nonlinearity properties inhere to any beam lines. Such these nonlinearities have unremovable character, and their influence can be remove using only special nonlinear lattice elements, which are introduced artificially into the beam line. In this paper we suggest a procedure to find necessary nonlinear correcting control elements for purposive forming of beam particle distribution functions.  
MPPE007  Problems of Conservative Integration in Beam Physics  1087 


In this paper an approach to conservative integration methods development is discussed. This problem is very important for beam physics: from beam line synthesis up to long time evolution simulation. This approach is based on a Lie algebra technique. On the first step we find a special form of decomposition for a Lie map, describing the system under study. On the second step a researcher finds exact solutions for some classes of hamiltonians in symbolic forms. These steps allows forming an integration scheme, which have a desired symplectic property. The additional invariant and symmetry properties can be included using dynamical invariants conception.  
MPPE008  Synthesis of Beam Lines with Necessary Properties  1096 


In this paper a new approach to the problem of synthesis of beam lines is discussed. Usually this problem can be overcome by the use of numerical simulation and optimal control theory methods. But this results in sufficiently great number of variable parameters and functions. Obviously, that this degrades quality of a modeling procedure. The suggested approach is demonstrated on a problem of a microprobe design problem. Essence of the problem is that necessary to design a high precision focusing system which satisfies some additional conditions. For solution of this problem we use an algebraic treatment based on Lie algebraic methods and computer algebra techniques. Using the symmetry ideology this approach allows rewriting beam properties to enough simple conditions for control parameters and functions. This leads a set of desired solutions and show results in some most suitable form. Moreover, this approach decreases the number of variable parameters.  
MPPE009  20032004 Nonlinear Optics Measurements and Modeling for the CERN SPS  1171 


In 2003 and 2004 nonlinear chromaticity, amplitude detuning, chromatic phase advance, resonance driving terms and offenergy orbits were measured in the CERN SPS at 14 GeV/c and 26 GeV/c, respectively. From the nonlinear chromaticity, the SPS optics model has been updated, by adjusting the strength of nonlinear field errors in dipoles and quadrupoles. Furthermore, we have added to the model the effect of the displacement of all main bends and the voluntary misalignments of all the other elements of the machine. We compare the field errors with those founded in 2002, 2001 and 2000. The tune shifts with transverse amplitude, driving terms, etc., predicted by this nonlinear optics model are compared with direct measurements.  
MPPE011  Expected Emittance Growth and Beam Tail Repopulation from Errors at Injection into the LHC  1266 


The preservation of the transverse emittance of the proton beam at injection into the LHC is crucial for luminosity performance. The population of the beam tails is also important for beam losses and collimation. The transfer and injection process is particularly critical in this respect, and several effects can contribute to the expected emittance increase and tail repopulation, like optical and geometrical mismatch, injection offsets and coupling, etc. The various effects are described, together with the tolerance limits on the parameters, and the expected contributions evaluated analytically where possible. The emittance growth and tail distributions are also simulated numerically using realistic errors. The implications for the tolerances on the matching of the transfer lines are discussed.  
MPPE012  MADX PTC Integration  1272 


MADX is CERN's successor for MAD8, a program for accelerator design with a long history. MADX is a modular, better maintainable rewrite of MAD8 with data structures written in C. Early on in the design of MADX we relied on the fact that older or doubtful modules could be replaced by new modules using the PTC code by E. Forest. Both codes remain independent entities but are linked via a converter to the MADX data structures. PTC is used for symplectic tracking of smaller machines and transfer line using better defined physical models of the elements and taking into account of how the elements are placed in the tunnel. The matching of the LHC will profit form the fact that the high order nonlinear parameters are provided by a PTC Normal Form analysis.  
MPPE013  High Precision Measurement of Muon Beam Emittance Reduction in MICE  1330 


Muon ionization cooling, an essential ingredient of a neutrino factory, will be demonstrated for the first time by the MICE experiment. The central part of MICE consists of a short section of a neutrino factory cooling channel and the emittance reduction achieved in this experiment is quite modest, 10% to 15%. In order to extrapolate the performance of a full cooling channel from these values, it is crucial for MICE to achieve an emittance measurement accuracy of 10^{}3 absolute. So far, beam emittance has never been measured with such a high level of precision and normailzed emittance in its present definition is not even conserved to 10^{}3 in the absence of dissipative forces. We present an improved definition of beam emittance and the requirements and constraints on MICE beam optics and spectrometers that are necessary to achieve the 10^{}3 level of accuracy.  
MPPE014  NonLinear Beam Dynamics Studies of the Diamond Storage Ring  1410 


The nonlinear beam dynamics have been investigated for the nonzero dispersion lattice of the Diamond storage ring. Effects in realistic lattice configurations such as the introduction of coupling errors, beta beating, closed orbit correction, quadrupole fringe field and invacuum and helical insertion devices have been studied in the presence of realistic physical aperture limitations. Frequency map analysis together with 6D tracking allows identification of the limiting resonances as well as the loss locations and calculation of the influence of nonlinear longitudinal motion on the Touschek lifetime. The sensitivity of the lattice to some of these effects leads to the identification of a better working point for the machine.  
MPPE015  NonLinear Ring Model Calibration with Frequency Analysis of Betatron Oscillations  1452 


A precise model of an accelerator ring is crucial to achieve ultimate performance both in synchrotron light sources and high energy synchrotrons. Algorithms have been developed to calibrate the linear model of the ring. They have been successfully applied experimentally to determine and correct the linear optics of the machine. More recently the Frequency Map Analysis has been used to model also the nonlinear optics. We propose here a technique based on the fit of nonlinear spectral lines to recover the nonlinear driving terms and to compensate the nonlinear field errors around the ring.  
MPPE016  Hamiltonian Analysis of Transverse Dynamics in Axisymmetric RF Photoinjector  1476 


A general Hamiltonian that governs the beam dynamics in an rf photoinjector is derived from first principles. With proper choice of coordinates, the resulting Hamiltonian has a simple and familiar form, while taking into account the rapid acceleration, rf focusing, magnetic focusing, and spacecharge forces. From the linear Hamiltonian, beamenvelope evolution is readily obtained, which better illuminates the theory of emittance compensation. Preliminary results on the thirdorder nonlinear Hamiltonian will be given as well.  
MPPE017  Longitudinal Acceptance in Linear NonScaling FFAGs  1532 


Funding: Work supported by U.S. Department of Energy contract DEAC0298CH10886. Linear NonScaling FFAGs have, particularly for muon acceleration, a unique type of longitudinal motion. This longitudinal motion can be approximated by a parabolic dependence timeofflight on energy. This motion can be described in dimensionless variables with two parameters. I describe the relationship between the parameters and the distortion of ellipses in longitudinal space. I discuss the relationship between the longitudinal acceptance and the time spent in the FFAG, the latter being especially relevant for decays in muon accelerators. I discuss what improvement one can expect to achieve by adding higherharmonic RF systems to the accelerator. 

MPPE020  Control of Dynamic Aperture for Synchrotron Light Sources  1670 


Funding: Under Contract with the U.S. Department of Energy Contract Number DEAC0298CH10886. Given the following frameworks: "A HamiltonianFree Description of Single Particle Dynamics for Hopelessly Complex Periodic Systems" (Forest, 1990), "Normal Form Methods for Complicated Periodic Systems" (Forest, Berz, Irwin, 1989), "The Correct Local Description for Tracking in Rings" (Forest, 1994), "The C++ Programming Language" (Stroustrup, 1985), we have designed a compact object oriented beam dynamics class by reusing existing FORTRAN libraries for: Truncated Power Series Algebra (Berz, SSC, 1988), and Map Normal Rorm (Forest, CBP, LBNL, 1990). In other words, implemented a numerical and analytical model for: 6dim phase space tracking, with classical radiation, and evaluation of equilibrium emittance, driving terms, amplitude dependent tune shifts, chromaticity, momentum compaction, etc., to arbitrary order, with selfconsistent treatment of magnet errors. The tool was developed for the lattice design of NSLSII. 

MPPE021  NonLinear Beam Transport System for the LENS 7 MeV Proton Beam  1704 


Funding: This work has been supported by the National Science Foundation under grants DMR0220560 and DMR–0320627, by the Indiana 21st Century Science and Technology Fund, and by the Department of Defense.
A beam transport system has been designed to carry a highintensity lowemittance proton beam from the exit of the RFQDTL acceleration system of the Indiana University Low Energy Neutron System (LENS)* to the neutron production target. The goal of the design was to provide a beam of uniform density over a 3cm by 3cm area at the target. Two octupole magnets** are employed in the beam line to provide the necessary beam phase space manipulations to achieve this goal. First order calculations were done using TRANSPORT and second order calculations have been performed using TURTLE. Second order simulations have been done using both a Gaussian beam distribution and a particle set generated by calculations of beam transport through the RFQDTL using PARMILA. Comparison of the design characteristics with initial measurements from the LENS commissioning process will be made.
*V.P. Derenchuk et al., "The LENS 7 MeV, 10mA proton Linac," these proceedings. **E. Kashy & B. Sherrill, Nuclear Instruments and Methods in Physics Research, B26 (1987) p. 610. 

MPPE022  Modification to the Lattice of the Fermilab Debuncher Ring To Improve the Performance of the Stochastic Cooling Systems  1799 


Funding: Supported by the Department of Energy and the National Science Foundation. The Fermilab Debuncher is used to collect antiprotons from the production target, reduce the momentum spread of the beam by an RF bunch rotation, and stochastically cool the transverse and longitudinal emittances of the beam prior to transfer to the Accumulator. A large value of the slip factor of the ring lattice is favored to provide a larger momentum acceptance for the bunch rotation process, while a small value of the slip factor is desirable for stochastic cooling. A dynamic change in the lattice from a large slip factor at injection to a smaller slip factor at extraction would optimize both processes and could lead to an improvement in antiproton stacking rate. This paper discusses the details of lattice modifications to the Debuncher, achievable with the existing hardware, which would result in a 60% increase in the slip factor, while maintaining the tunes and chromaticities fixed, and keeping the betatron functions within an acceptable range. 

MPPE023  Improvement of the Longitudinal Beam Dynamics Tuning Procedure for the MSU RIA Driver Linac  1826 


The Rare Isotope Accelerator (RIA) driver linac will use a superconducting, cw linac with independently phased superconducting radio frequency cavities for acceleration and, for the heavier ions, utilize beams of multiplechargestates (multiq). Given the acceleration of multiq beams and a stringent beam loss requirement in the RIA driver linac, a new beam envelope code capable of simulating nonlinearities of the multiq beam envelopes in the longitudinal phase space was developed. Using optimization routines, the code is able to maximize the linearity of the longitudinal phase space motion and thereby minimizing beam loss by finding values for the amplitude and phase of the cavities for a given accelerating lattice. Relative motion of the multiq beams is also taken into account so that superposition of the beam centroids and matching of their Twiss parameters are automatically controlled. As a result, the linac tuning procedure has been simplified and the longitudinal lattice performance has been improved. In this paper, the general architecture of the code and the results of using it to determine tuning parameters for the RIA driver linac are presented.  
MPPE024  Failure Modes Analysis for the MSURIA Driver Linac  1868 


Previous endtoend beam dynamics simulation studies* using experimentallybased input beams including alignment and rf errors and variation in chargestripping foil thickness have indicated that the Rare Isotope Accelerator (RIA) driver linac proposed by MSU has adequate transverse and longitudinal acceptances to accelerate light and heavy ions to final energies of at least 400 MeV/u with beam powers of 100 to 400 kW. During linac operation, equipment loss due to, for example, cavity contamination, availability of cryogens, or failure of rf or power supply systems, will lead to at least a temporary loss of some of the cavities and focusing elements. To achieve high facility availability, each segment of the linac should be capable of adequate performance even with failed elements. Beam dynamics studies were performed to evaluate the linac performance under various scenarios of failed cavities and focusing elements with proper correction schemes, in order to prove the flexibility and robustness of the driver linac lattice design. The result of these beam dynamics studies will be presented.
*X. Wu, "EndtoEnd Beam Simulations for the MSU RIA Driver Linac," Proceedings of the XXII Linac Conference, Lubeck, Germany, August 2004. 

MPPE025  Dynamical Effects Due to Fringe Field of the Magnets in Circular Accelerators  1907 


Funding: Work supported by the Department of Energy under Contract No. DEAC0276SF00515. The leading Lie generators, including the chromatic effects, due to hard edge fringe field of single multipole and solenoid are derived from the vector potentials within a Halmitonian system. These nonlinear generators are applied to the interaction region of PEPII to analyze the linear errors due to the feeddown from the offcentered quadrupoles and solenoid. The nonlinear effects of tune shifts at large amplitude, the synchrobetatron sidebands near half integer and their impacts on the dynamic aperture are studied in the paper. 

MPPE027  Modeling of Nonlinear Effects in RF Cavities  


Funding: DOE: DEFG0203ER83796.
Emerging accelerator applications require beam radii comparable to RF cavity apertures, placing new demands on the modeling of particle motion in RF cavities using mapbased techniques. A new method has been developed for computing nonlinear maps for arbitrary RF cavities.* We describe this new approach and present comparisons with more traditional approaches.
*D.T. Abell, in preparation. 

MPPE028  Non Linear Error Analysis from Orbit Measurements in SPS and RHIC  2012 


Funding: U.S. Department of Energy and Colciencias. Recently, an "action and phase" analysis of SPS orbits measurements proved to be sensitive to sextupole components intentionally activated at specific locations in the ring. In this paper we attempt to determine the strenght of such sextupoles from the measured orbits and compare them with the set values. Action and phase analysis of orbit trayectories generated by RHIC models with non linearities will also be presented and compare with RHIC experiments. 

MPPE030  Comparison of OffLine IR Bump and ActionAngle Kick Minimization  2116 


Funding: Work supported by U.S. DOE under contract No. DEAC0298CH10886. The interaction region bump (IR bump) nonlinear correction method has been used for the sextupole and octupole field error online corrections in the Relativistic Heavy Ion Collider (RHIC). Some differences were found for the sextupole and octupole corrector strengths between the online IR bump correction and the predictions from the actionangle kick minimization. In this report we compare the corrector strengths from these two methods based on the RHIC Blue ring lattice with the IR nonlinear modeling. The comparison confirms the differences between resulting corrector strengths. And the reason for the differences is found and discussed. It is followed by a further discussion of the operational IR bump applications to the octupole, and skew sextupole and skew quadrupole field error corrections. 

MPPE031  Simulation of Resonance Streaming at the eRHIC Electron Storage Ring  2215 


Funding: Work performed under the auspices of the U.S. Department of Energy. To estimate electron beam lifetime and detector background at the future electronion collider eRHIC, knowledge of the electron beam halo region is essential. Simulations have been performed to determine the deviation of the transverse beam profile from a Gaussian distribution. 

MPPE034  Symmetries and Invariants of the Timedependent Oscillator Equation and the Envelope Equation  2315 


Funding: Research supported by the U.S. Department of Energy.
Singleparticle dynamics in a timedependent focusing field is examined. The existence of the CourantSnyder invariant* is fundamentally the result of the corresponding symmetry admitted by the oscillator equation with timedependent frequency.** A careful analysis of the admitted symmetries reveals a deeper connection between the nonlinear envelope equation and the oscillator equation. A general theorem regarding the symmetries and invariants of the envelope equation, which includes the existence of the CourantSnyder invariant as a special case, is demonstrated. The symmetries of the envelope equation enable a fast algorithm for finding matched solutions without using the conventional iterative shooting method.
*E.D. Courant and H.S. Snyder, Ann. Phys. 3, 1 (1958). **R.C. Davidson and H. Qin, Physics of Intense Charged Particle Beams in High Energy Accelerators (World Scientific, 2001). 

MPPE035  Transfers from High Power Hadron Linacs to Synchrotrons  2375 


The Fermilab Proton Driver is an example of a high power H^{} linear accelerator proposed as a new source of high brightness protons for the Main Injector synchrotron. Because of the elevated radioactive activation of accelerator components associated with beam losses during injection and acceleration, extra attention must be paid to RF manipulations wherein small losses were once deemed acceptable. Especially when injecting into existing synchrotrons from upgraded injectors, instabilities and beam loading make loss free manipulations especially problematic. This paper discusses some options for reducing the losses associated with common longitudinal beam manipulations.  
MPPE036  Characterization of the Chaotic or Regular Nature of Dynamical Orbits: A New, Fast Method  2449 


A new method of characterization of the regular or chaotic nature of dynamical orbits is introduced. It takes advantage of both morphological and dynamical properties of orbits, and can be applied to systems of all degrees of freedom. The new technique has been designed to analyze timeindependent, timedependent and Nbody systems. It can provide straightforward information about the transition of orbits from regular to chaotic and vice versa, which can be found in timedependent regimes. Equally important is the distinction it can make in timeindependent regimes between sticky and wildly chaotic epochs during the evolution of chaotic orbits. Its most important advantage over the existing methods is, that it characterizes an orbit using information from a very small number of orbital periods. For these reasons the new method is extremely promising to be useful and effective in a broad spectrum of disciplines.  
MPPE038  Synchrotron Sidebands of a Linear Differential Coupling Resonance  2538 


Sidebands of a linear differential coupling resonance are observed in the tune survey of the SPring8 storage ring. The vertical beam size and the Touschek beam lifetime blow up at a distance by synchrotron tune from the linear differential resonance. The synchrotron sidebands of a linear betatron coupling resonance are excited by the vertical dispersion at sextupole magnets. Although the vertical dispersion of the SPring8 storage ring is well reduced to be small, order of 1 mm, the linear betatron coupling resonance is further suppressed. In addition, the sextupole magnets are relatively strong as well as other third generation light source facilities. Hence at the SPring8 storage ring we can observe the synchrotron sidebands of the coupling resonance. By means of the tracking simulation based on the ring model obtained by the response matrix measurement we confirm the existence of the synchrotron sidebands of a linear differential coupling resonance. In order to incorporate synchrotron motion in the simulation, the 6dimensional tracking code developed at SPring8 is used.  
MPPE039  A C++ Framework for Conducting Highspeed, Longterm Particle Tracking Simulations  2565 


Highresolution tracking studies such as the ones presented in*,** require unprecented amounts of CPU power. Usually, flexibility of a simulation code compromises performance; we have developed a C++ framework for parallel simulation of circular accelerators which provides a high degree of flexibility and programmability (parsing of MAD beamline descriptions, manipulation of beamlines and interfaces, optimization and matching of beamlines, tracking of particles or differentialalgebraic objects) while achieving raw tracking speeds comparable to and surpassing handcoded Fortran code. We describe some of the techniques used, such as compiletime polymorphism, metaprogramming, and present benchmarking results.
*A. Kabel, Y. Cai, this conference. **A. Kabel, Y. Cai, T. Sen, V. Shiltsev, this conference. 

MPPE040  Efficient Modeling of Nonlinear Beam Optics Using Parametric Model Independent Analysis  


Funding: Research supported by DOE grant number: DEFG0204ER86225. Based on precision beam orbit measurements, Model Independent Analysis(MIA) has been used successfully to build a computer model that matches the linear optics of the real accelerator. We report a parametric extension of MIA that will allow efficient modeling of the nonlinear beam optics to account for energy dispersions. A simulation study is presented where the nonlinear dependency of lattice parameters on beam energy is captured by constrained training of a universal nonlinear approximator. These parametric nonlinear models of beam optics are easy to construct, diagnose, and modify. They can be very useful for more accurate model predicted beam operation and control. 

MPPE058  Virtual Accelerator for Accelerator Optics Improvement  3426 


Funding: Work supported by Department of Energy contract DEAC0276SF00515. Through determination of all quadrupole strengths and sextupole feeddowns by fitting quantities derivable from precision orbit measurement, one can establish a virtual accelerator that matches the real accelerator optics. These quantities (the phase advances, the Green's functions, and the coupling eigenplane ellipses tilt angles and axis ratios) are obtained by analyzing turnbyturn Beam Position Monitor (BPM) data with a modelindependent analysis (MIA). Instead of trying to identify magnet errors, a limited number of quadrupoles are chosen for optimized strength adjustment to improve the virtual accelerator optics and then applied to the real accelerator accordingly. These processes have been successfully applied to PEPII rings for beta beating fixes, phase and working tune adjustments, and linear coupling reduction to improve PEPII luminosity. 

MPPE064  Dynamic Aperture Study and Lifetime Improvement at the Advanced Photon Source  3632 


Funding: Work supported by U.S. Department of Energy, Office of Basic Energy Sciences, under Contract No. W31109ENG38. Over past few years, the optics of the Advanced Photon Source storage ring was optimized to provide lower natural emittance. Presently, APS operates at 2.5 nmrad emittance. The optimization was done at the expense of stronger sextupoles and shorter lifetime. Here we present our work on measurement and understanding the dynamic aperture of APS in lowemittance mode. We found good agreement between the dynamic aperture measurements and that of the model derived from the response matrix analysis. Based on the model, we were able to increase the lifetime significantly by optimizing sextupoles, correcting optics, moving working point, and adjusting rf voltage. The higher lifetime allowed us to decrease operating coupling from 2.5% to 1%. 

MPPE075  Simulation of the Effect of an InVacuum Undulator on the Beam Dynamics of the ALS  3949 


Funding: Work supported by the Director, Office of Energy Research, Office of Basic Energy Science, Material Sciences Division, U.S. Department of Energy, under Contract No. DEAC0376SF00098. The femtosecond slicing project at the Advanced Light Source (ALS) requires that a short period (3 cm) and narrow gap (5.5 mm) in vacuum undulator to be installed. The combination of the short period and the narrow gap raised concern of the impact on the beam dynamics. A 3D field model was established based on numerical data using 8 longitudinal and 4 transverse harmonics. At first fourthorder symplectic integrator was used. It was to our surprise that the dynamic aperture decreased by a fact of 3. To understand the cause of the drastic change in the dynamic aperture, the field model was implemented in a differential algebraic code and the Taylor map of the undulator was obtained. Tracking result using the Taylor map showed little change in the dynamic aperture, which was latter corroborated by that using the symplectic integrator with 150 slices per period (as opposed to 10 before). Yet it is simply too time consuming to use the symplectic integrator with such thin slices. For this case, Taylor proves to be a much faster alternative. 