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The ever increasing demands placed upon machine performance have resulted in the need for more comprehensive particle accelerator modeling. Computer simulations are key to the success of particle accelerators. Many aspects of particle accelerators rely on computer modeling at some point, sometimes requiring complex simulation tools and massively parallel supercomputing. Examples include the modeling of beams at extreme intensities and densities (toward the quantum degeneracy limit), and with ultra-fine control (down to the level of individual particles). In the future, adaptively tuned models might also be relied upon to provide beam measurements beyond the resolution of existing diagnostics. Much time and effort has been put into creating accelerator software tools, some of which are highly successful. However, there are also shortcomings such as the general inability of existing software to be easily modified to meet changing simulation needs. In this paper possible mitigating strategies are discussed for issues faced by the accelerator community as it endeavors to produce better and more comprehensive modeling tools. This includes lack of coordination between code developers, lack of standards to make codes portable and/or reusable, lack of documentation, among others., 21 pages, 1 figure. To be published in JINST

The Muon g-2 Experiment at Fermi National Accelerator Laboratory (FNAL) has measured the muon anomalous precession frequency $\omega_a$ to an uncertainty of 434 parts per billion (ppb), statistical, and 56 ppb, systematic, with data collected in four storage ring configurations during its first physics run in 2018. When combined with a precision measurement of the magnetic field of the experiment's muon storage ring, the precession frequency measurement determines a muon magnetic anomaly of $a_{\mu}({\rm FNAL}) = 116\,592\,040(54) \times 10^{-11}$ (0.46 ppm). This article describes the multiple techniques employed in the reconstruction, analysis and fitting of the data to measure the precession frequency. It also presents the averaging of the results from the eleven separate determinations of \omega_a, and the systematic uncertainties on the result., Comment: 29 pages, 19 figures. Published in Physical Review D

The Fermi National Accelerator Laboratory has measured the anomalous precession frequency $a^{}_\mu = (g^{}_\mu-2)/2$ of the muon to a combined precision of 0.46 parts per million with data collected during its first physics run in 2018. This paper documents the measurement of the magnetic field in the muon storage ring. The magnetic field is monitored by nuclear magnetic resonance systems and calibrated in terms of the equivalent proton spin precession frequency in a spherical water sample at 34.7$^\circ$C. The measured field is weighted by the muon distribution resulting in $\tilde{\omega}'^{}_p$, the denominator in the ratio $\omega^{}_a$/$\tilde{\omega}'^{}_p$ that together with known fundamental constants yields $a^{}_\mu$. The reported uncertainty on $\tilde{\omega}'^{}_p$ for the Run-1 data set is 114 ppb consisting of uncertainty contributions from frequency extraction, calibration, mapping, tracking, and averaging of 56 ppb, and contributions from fast transient fields of 99 ppb., Comment: Added one citation and corrected missing normalization in Eqs (35) and (36)

This paper presents the beam dynamics systematic corrections and their uncertainties for the Run-1 data set of the Fermilab Muon g-2 Experiment. Two corrections to the measured muon precession frequency $\omega_a^m$ are associated with well-known effects owing to the use of electrostatic quadrupole (ESQ) vertical focusing in the storage ring. An average vertically oriented motional magnetic field is felt by relativistic muons passing transversely through the radial electric field components created by the ESQ system. The correction depends on the stored momentum distribution and the tunes of the ring, which has relatively weak vertical focusing. Vertical betatron motions imply that the muons do not orbit the ring in a plane exactly orthogonal to the vertical magnetic field direction. A correction is necessary to account for an average pitch angle associated with their trajectories. A third small correction is necessary because muons that escape the ring during the storage time are slightly biased in initial spin phase compared to the parent distribution. Finally, because two high-voltage resistors in the ESQ network had longer than designed RC time constants, the vertical and horizontal centroids and envelopes of the stored muon beam drifted slightly, but coherently, during each storage ring fill. This led to the discovery of an important phase-acceptance relationship that requires a correction. The sum of the corrections to $\omega_a^m$ is 0.50 $\pm$ 0.09 ppm; the uncertainty is small compared to the 0.43 ppm statistical precision of $\omega_a^m$., Comment: 35 pages, 29 figures. Accepted by Phys. Rev. Accel. Beams

Nonlinear problems in accelerator physics (Mais) Moment methods for nonlinear maps (Pusch) Differential algebraic formulation of normal form theory (Berz) Analytical determination of 5th-order transfer matrices of magnetic quadrupole fringing fields (Hartmann, et al ) Nonlinear beam transport effects in highly charged positive ion beams extracted from ECR ion sources (Antaya) Status of MAD (version 8.5) and future plans (Iselin) COSY INFINITY version 6 (Berz) The arbitrary order design code Tlie 1.0 (van Zeijts and Neri) The Chalk River differential algebra code "DACYC" and the role of differential and lie algebras in understanding the orbit dynamics of cyclotrons (Davies, et al) Optics programs at TRIUMF (Servranckx) A comparison of methods for long-term tracking using symplectic maps (Gjaja, et al) A generalization of the Henon map: stability of the orbits, symmetries and connections to accelerator physics (Todesco) Chaotic path at a nonlinear resonance (Lee) Third-order achromats based on mirror symmetries (Wan, et al) Design of modern high resolution magnetic spectrometers (Zeller) Alternating-phase focusing: a model to study nonlinear dynamics (Sagalovsky and Delayen) Review of the dynamic aperture experiment at the CERN SPS (Gareyte, Scandale and Schmidt) Review of nonlinear beam dynamics experiments (Lee).

International audience; Establishing long-term relative bounded motion between orbits in perturbed dynamics is a key challenge in astrodynamics to enable cluster flight with minimum propellant expenditure. In this work, we present an approach that allows for the design of long-term relative bounded motion considering a zonal gravitational model. Entire sets of orbits are obtained via high-order Taylor expansions of Poincarè return maps about reference fixed points. The high-order normal form algorithm is used to determine a change in expansion variables of the map into normal form space, in which the phase space behavior is circular and can be easily parameterized by action–angle coordinates. The action–angle representation of the normal form coordinates is then used to parameterize the original Poincarè return map and average it over a full phase space revolution by a path integral along the angle parameterization. As a result, the averaged nodal period and drift in the ascending node are obtained, for which the bounded motion conditions are straightforwardly imposed. Sets of highly accurate bounded orbits are obtained, extending over several thousand kilometers, and valid for decades.

We derived first and second order analytic aberration formulas in the horizontal transverse plane for an electrostatic deflector specified by the reference orbit radius, the central angle spanning the deflector, and its inhomogeneity coefficients. The derivation was performed using an iterative order-by-order perturbation method in a Frenet–Serret beamline coordinate system. We produced a C program edabrt for calculation of the first and second order aberrations using the formulas that we derived. The electrostatic deflector aberration formulas from Wollnik (1965) disagree with COSY INFINITY and the aberration formulas that we derived here, and they also deviate from the first and second order symplecticity conditions. The reason for this discrepancy is that the aberration formulas from Wollnik (1965) consider only motion in the main field of the deflector and do not properly account for fringe field effects, in particular the necessarily occurring change of kinetic energy due to the change in potential. The code GIOS uses electrostatic aberration formulas from Wollnik (1965) and exhibits the same discrepancy. We performed a comparison of first and second order electrostatic deflector aberrations for (1) the analytic formulas derived in this work, (2) differential-algebraic (DA) numerical integration of the equations of motion using the code COSY INFINITY, (3) the aberration formulas from Wollnik (1965) , (4) aberrations computed using the code GIOS, and (5) the aberration formulas from Wollnik (1965) adjusted to account for the occurring change in potential. An electrostatic spherical deflector and an electrostatic cylindrical deflector were used as test cases for this comparison. There is excellent agreement between methods (1), (2), and (5), and these three methods satisfy the first and second order symplecticity conditions.

We developed a highly accurate and fully Maxwellian conformal mapping method for calculation of main fields of electrostatic particle optical elements. A remarkable advantage of this method is the possibility of rapid recalculations with geometric asymmetries and mispowered plates. We used this conformal mapping method to calculate the multipole terms of the high voltage quadrupoles in the storage ring of the Muon [Formula: see text] Experiment (FNAL-E-0989). Next, we demonstrate that an effect where the observed tunes correspond to a voltage that is about [Formula: see text] higher compared to the voltage to which the Muon [Formula: see text] quadrupoles are set is explained by the conceptual and quantitative differences between the beam optics quadrupole voltage and the quadrupole voltage at the plates. Completing the methodological framework for field computations, we present a method for extracting multipole strength falloffs of a particle optical element from a set of Fourier mode falloffs. We calculated the quadrupole strength falloff and its effective field boundary (EFB) for the Muon [Formula: see text] quadrupole, which has explained the experimentally measured tunes, while simple estimates based on a linear model exhibited discrepancies up to [Formula: see text].

Nuclear instruments & methods in physics research / A Accelerators, spectrometers, detectors and associated equipment Section A 935, 56 - 64 (2019). doi:10.1016/j.nima.2019.04.086, For high-resolution separators like the projected Super-FRS at FAIR, an adapted and accurate ion-optical model considering realistic -dependent magnet parameters is crucial in achieving the desired parameters (e.g. resolution) and to enable a fast optimization. Starting from the magnetic field measurements and simulations, rigidity-dependent Taylor transfer maps are generated for the Super-FRS preseparator dipole magnets. The effects of the magnetic saturation in the steel yoke on the image aberrations are analyzed., Published by North-Holland Publ. Co., Amsterdam

Nonlinear effects of the various electric field and magnetic field components of storage rings to confine the particles and bend their trajectory can cause substantial amplitude-dependent tune shifts within the beam. Furthermore, tune shifts are often sensitive to variations of system parameters, e.g. total particle momentum offsets [Formula: see text]. Such amplitude- and parameter-dependent tune shifts influence the dynamics and stability of a beam in particle storage rings. Thus, it is critical for high precision measurements to analyze and understand these influences. On this basis, we present normal form methods for the calculation of high order amplitude and system parameter dependencies of the horizontal and vertical tunes in storage rings using the differential algebra (DA) framework within COSY INFINITY. A storage ring is simulated using COSY INFINITY to generate a DA Poincaré return map describing the transverse phase space behavior after each revolution in the storage ring. The map is expanded around the parameter-dependent closed orbit of the system before transforming the resulting map into normal form coordinates to extract the high order tune dependencies on the phase space amplitude and variation in the system parameters. As a specific example, a storage ring similar to the Storage Ring of the Muon [Formula: see text]-2 Experiment at Fermilab (E989) is investigated.

The Muon Campus at Fermilab is a system through which muons are delivered to the storage ring of the Muon [Formula: see text] Experiment (E989). It consists of a set of 1 km beamlines that transport and prepare a highly polarized muon beam out of secondaries produced downstream a target station. Realistic simulations of this beam delivery system (BDS) using COSY INFINITY, and presented here, contribute to the understanding and characterization of the muon beam production in relation to the statistical and systematic uncertainties of the E989 measurement, intended to be smaller than 0.14 parts per million to achieve the goals of the experiment. The impact of nonlinearities from fringe fields and high-order contributions on the BDS performance are presented, as well as detailed studies of the interactions between secondaries and the beamline elements apertures, particle decay channels, spin dynamics and beamline misalignments.

The code COSY INFINITY uses a beamline coordinate system with a Frenet–Serret frame relative to the reference particle, and calculates differential algebra-valued transfer maps by integrating the ODEs of motion in the respective vector space over a differential algebra (DA). We described and performed computation of the DA transfer map of an electrostatic spherical deflector in a laboratory coordinate system using two conventional methods: (1) by integrating the ODEs of motion using a numerical integrator and (2) by computing analytically and in closed form the properties of the respective elliptical orbits from Kepler theory. We compared the resulting transfer maps with (3) the DA transfer map of COSY INFINITY’s built-in electrostatic spherical deflector element [Formula: see text] and (4) the transfer map of the electrostatic spherical deflector computed using the program GIOS, which uses analytic formulas from a paper1 by Hermann Wollnik regarding second-order aberrations. In addition to the electrostatic spherical deflector, we studied an electrostatic cylindrical deflector, where the Kepler theory is not applicable. We computed the DA transfer map by the ODE integration method (1), and we compared it with the transfer maps by (3) COSY INFINITY’s built-in electrostatic cylindrical deflector element [Formula: see text] and (4) GIOS. The transfer maps of electrostatic spherical and cylindrical deflectors obtained using the direct calculation methods (1) and (2) are in excellent agreement with those computed using (3) COSY INFINITY. On the other hand, we found a significant discrepancy with (4) the program GIOS.

The Muon [Formula: see text] Experiment at Fermilab (E989) is directed toward measuring the muon magnetic anomaly, [Formula: see text], with total statistical and systematic errors of 0.14 ppm. This new measurement will serve as strong probe of effects of as yet undiscovered particles beyond the Standard Model (SM), and perhaps validate or disprove other theoretical models beyond the SM. Of special interest is the reduction of muon losses from the storage ring to achieve the precision needed at the Muon [Formula: see text] Experiment. For this purpose, we have developed a detailed and precise symplectic model of the Muon [Formula: see text] Storage Ring using COSY INFINITY that considers measured inhomogeneities of the magnetic field; high-order representation of the Electrostatic Quadrupole System (EQS) electrostatic field at different stages of the experiment including fringe fields; injection to the ring based on measurements; and beam collimation. Specifically, we have performed numerical analyses of the rate of muons that are lost before they have a chance to decay for several possible configurations of the EQS in order to find the best possible scenarios that minimize muon losses and understand the resonance mechanisms that contribute to betatron and possibly spin resonances. Additionally, comparisons with measurements have permitted the determination of whether observed resonances come from anticipated features of the [Formula: see text] storage ring or from unexpected sources of error whose effect could be detrimental to the precision of E989.

Physical review accelerators and beams 21(4), 042002 (2018). doi:10.1103/PhysRevAccelBeams.21.042002, Published by American Physical Society, College Park, MD

This letter reports the successful use of feedback from a spin polarization measurement to the revolution frequency of a 0.97 GeV/$c$ bunched and polarized deuteron beam in the Cooler Synchrotron (COSY) storage ring in order to control both the precession rate ($\approx 121$ kHz) and the phase of the horizontal polarization component. Real time synchronization with a radio frequency (rf) solenoid made possible the rotation of the polarization out of the horizontal plane, yielding a demonstration of the feedback method to manipulate the polarization. In particular, the rotation rate shows a sinusoidal function of the horizontal polarization phase (relative to the rf solenoid), which was controlled to within a one standard deviation range of $\sigma = 0.21$ rad. The minimum possible adjustment was 3.7 mHz out of a revolution frequency of 753 kHz, which changes the precession rate by 26 mrad/s. Such a capability meets a requirement for the use of storage rings to look for an intrinsic electric dipole moment of charged particles.

A frontier challenge in implementing femtosecond electron microscopy is to gain precise optical control of intense beams to mitigate collective space charge effects for significantly improving the throughput. Here, we explore the flexible uses of an RF cavity as a longitudinal lens in a high-intensity beam column for condensing the electron beams both temporally and spectrally, relevant to the design of ultrafast electron microscopy. Through the introduction of a novel atomic grating approach for characterization of electron bunch phase space and control optics, we elucidate the principles for predicting and controlling the phase space dynamics to reach optimal compressions at various electron densities and generating conditions. We provide strategies to identify high-brightness modes, achieving ∼100 fs and ∼1 eV resolutions with 106 electrons per bunch, and establish the scaling of performance for different bunch charges. These results benchmark the sensitivity and resolution from the fundamental beam brightness perspective and also validate the adaptive optics concept to enable delicate control of the density-dependent phase space structures to optimize the performance, including delivering ultrashort, monochromatic, high-dose, or coherent electron bunches.

Numerical methods assuring confidence involve the treatment of entire sets instead of mere point evaluations. We briefly review the method of interval arithmetic that is long known for rigorous, verified computations, and all operations are conducted on intervals instead of numbers. However, interval computations suffer from overestimation, the dependency problem, the dimensionality curse, and the wrapping effect, to name a few, and those difficulties often make conventional interval based verified computational methods useless for practical challenging problems.