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DOE OSTI2019

Validation of transfer map calculation for electrostatic deflectors in the code COSY INFINITY

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). In our work, 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 ESP 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 ECL 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.

43 PARTICLE ACCELERATORS↗

DOE OSTI2019

Computation and consequences of high order amplitude- and parameter-dependent tune shifts in storage rings for high precision measurements

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 $δp$. 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 in this paper 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 g-2 Experiment at Fermilab (E989) is investigated.

43 PARTICLE ACCELERATORS↗

DOE OSTI2019

Chapter Six - Derivation, cross-validation, and comparison of analytic formulas for electrostatic deflector aberrations

Here, 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.

43 PARTICLE ACCELERATORS↗

DOE OSTI2024

Beam Dynamics of the Muon $g\textrm{-}2$ Experiment

The Muon $g\textrm{-}2$ Experiment (E989) at Fermilab aims to measure the muon anomalous magnetic moment $a_{\mu}$ with unprecedented precision, potentially uncovering physics beyond the Standard Model of particle physics. The result based on Runs 1-3, released in 2023, achieved a precision of 0.20 ppm. The experiment circulates muons in a storage ring, measuring $a_{\mu}$ from decay positron time and energy measurements collected with calorimeters. To achieve the required accuracy, it is crucial to measure and control the magnetic field in the ring with high precision. Beam dynamics corrections are necessary for muons not orbiting exactly in the midplane, for their oscillations, and for electric field effects. Highly accurate beam dynamics simulations are instrumental for quantifying and validating the beam dynamics corrections, ultimately improving the precision of the $a_{\mu}$ measurement and facilitating the achievement of the ambitious $70\:\mathrm{ppb}$ systematic uncertainty goal. The measured field data was incorporated into models for simulations using three codes: \texttt{gm2ringsim} (an internal Geant4-based code), \textit{COSY INFINITY}, and \textit{BMAD}. The advantages of \texttt{gm2ringsim} include using CAD-based geometry and modelling the detector effects. \textit{COSY INFINITY} is a highly accurate and efficient code that uses high-order differential-algebraic transfer maps, precise fringe field calculations, and advanced symplectification methods. Symplectification is important for maintaining the physical correctness of the muon beam behaviour with high precision over the storage time, ensuring conservation of phase space volume and preventing artificial damping or excitation of particle motion. The experiment completed its final Run 6 in July 2023, collecting 21 times more data than the previous BNL experiment. Analyses of data from Runs 4-6 are ongoing, with results planned for release in 2025, potentially resolving the current tension between experiment and theory.

43 PARTICLE ACCELERATORS↗

DOE OSTI2019

Advanced Simulation Tools for Muon-Based Accelerators

The project developed new software tools incorporating the most accurate theoretical calculations and experimental data available for crucial and not-yet-considered physics processes specific to muon accelerators. These new tools were applied to modeling muon ionization cooling channels for a muon collider or a neutrino factory, the facilities that have the potential to discover and explore new exciting fundamental physics but require the development of demanding technologies and innovative concepts. G4beamline, COSY Infinity, and WARP computer codes are particularly well-suited for addressing muon accelerator modeling problems. G4beamline was chosen for its flexibility and modularity, and for being one of the de facto standard codes used by the muon accelerator community. COSY was selected for speed, ability to take into account high-order nonlinear effects essential for the large beams under consideration, and availability of built-in optimization tools. WARP was chosen for its space charge calculation capabilities, speed and flexibility of implementing new lattice elements such as RF cavities. This final technical report summarizes the efforts on implementing advanced new tools in these computer codes addressing the shortcomings in modeling muon physics processes and using these tools to model muon ionization cooling channels.

43 PARTICLE ACCELERATORS↗

DOE OSTI2016

Closeout Report ARRA supplement to DE-FG02-08ER41546, 03/15/2010 to 03/14/2011 - Advanced Transfer Map Methods for the Description of Particle Beam Dynamics

The ARRA funds were utilized to acquire a cluster of high performance computers, consisting of one Altus 2804 Server based on a Quad AMD Opteron 6174 12C with 4 2.2 GHz nodes of 12 cores each, resulting in 48 directly usable cores; as well as a Relion 1751 Server using an Intel Xeon X5677 consisting of 4 3.46 GHz cores supporting 8 threads. Both systems run the Unix flavor CentOS, which is designed for use without need of updates, which greatly enhances their reliability. The systems are used to operate our COSY INFINITY environment which supports MPI parallelization. The units arrived at MSU in September 2010, and were taken into operation shortly thereafter.

43 PARTICLE ACCELERATORS↗

DOE OSTI2019

Dynamical simulations of the Muon Campus at Fermilab

The Muon Campus at Fermilab is a system through which muons are delivered to the storage ring of the Muon g-2 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.

43 PARTICLE ACCELERATORS↗

DOE OSTI2015

Software package for modeling spin–orbit motion in storage rings

A software package providing a graphical user interface for computer experiments on the motion of charged particle beams in accelerators, as well as analysis of obtained data, is presented. The software package was tested in the framework of the international project on electric dipole moment measurement JEDI (Jülich Electric Dipole moment Investigations). The specific features of particle spin motion imply the requirement to use a cyclic accelerator (storage ring) consisting of electrostatic elements, which makes it possible to preserve horizontal polarization for a long time. Computer experiments study the dynamics of 10{sup 6}–10{sup 9} particles in a beam during 10{sup 9} turns in an accelerator (about 10{sup 12}–10{sup 15} integration steps for the equations of motion). For designing an optimal accelerator structure, a large number of computer experiments on polarized beam dynamics are required. The numerical core of the package is COSY Infinity, a program for modeling spin–orbit dynamics.

43 PARTICLE ACCELERATORS↗

DOE OSTI2015

Design of in-flight fragment separator using high-power primary beams in the energy of a few hundred MeV/u

An in-flight fragment separator usually requires large acceptance and high momentum resolution to minimize the loss of a rare isotope beam of interest produced at a thin target, which is especially important when {sup 238}U fission reactions in the energy of 200 MeV/u are used. The production target and beam dump are located in the pre-separator, where a beam power of up to 400 kW is dissipated. The area is surrounded by thick radiation shielding walls, which result in long drift spaces between adjacent magnetic components at various locations and an asymmetrical layout. Efforts have been made to minimize non-linear effects in the pre-separator beam optics with trials of different separator configurations and correction schemes using COSY Infinity and GICOSY. The main separator is configured to be mirror symmetric such that correction with hexapole and octupole coils can be more readily applied. The separator configuration was finalized to allow the facility design to proceed and the key components including superconducting magnets have been designed and prototyped. In addition, the separator design has been evaluated using LISE++ including a set of wedge degraders at dispersive focal planes to improve the yield and purity of selected isotope beam.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2019

Analysis and fringe field scaling of a legacy set of electrostatic deflector aberration formulas [Chapter Seven]

In this work we performed an analysis of the derivation of first and second order analytic aberration formulas by Wollnik (1965) for the case of electrostatic deflectors in the horizontal plane. We found that these aberration formulas are valid for the main field of the deflector; however, they do not account for any fringe field effects. To address this issue, we modified the aberration formulas from Wollnik (1965) for a hard edge fringe field by scaling the particle inclinations at the entrance and exit edges of the deflector. These modified aberration formulas fully agree with electrostatic aberration formulas derived by Valetov and Berz (2020), as well as with differential-algebraic (DA) calculations of electrostatic deflector aberrations using the code COSY INFINITY.

43 PARTICLE ACCELERATORS↗

DOE OSTI2024

A compact and portable gamma-ray spectrometer (GRASP) for inertial confinement fusion and basic science experiments

A compact and portable gamma-ray spectrometer has been designed to diagnose different components of the inertial confinement fusion-relevant γ-ray spectrum with energies between ∼3.7–17.9 MeV. The system is designed to be as compact as possible for convenient transportation and fielding in diagnostic ports on the OMEGA laser, the National Ignition Facility, and other photon-source facilities. The system consists of a conversion foil for Compton scattering in front of four magnetic spectrometer “arms,” each covering a different energy range and constructed out of cylindrical permanent magnet Halbach arrays. Monte Carlo simulations have been used to optimize and assess the performance of the conversion foil, and COSY INFINITY ion-optical simulations have been used to optimize the spectrometer magnets. The performance of the design is assessed for a simulated direct-drive γ-ray spectrum. Spanning its total γ-ray energy bandwidth and using a 1.7 mm thick boron conversion foil, the system’s total energy resolution and efficiency are ∼15.8%–4.5% and 5.4 × 10−7–3.7 × 10−7e−/γ, respectively, with room for improvement. Spectral γ-ray measurements will provide guidance to the inertial confinement fusion program toward achieving high-energy gain relevant to inertial fusion energy and enable new measurement capabilities for basic discovery science.

Instruments & Instrumentation↗

DOE OSTI2018

Hybrid methods for muon accelerator simulations with ionization cooling

Muon ionization cooling involves passing particles through solid or liquid absorbers. Careful simulations are required to design muon cooling channels. New features have been developed for inclusion in the transfer map code COSY Infinity to follow the distribution of charged particles through matter. To study the passage of muons through material, the transfer map approach alone is not sufficient. The interplay of beam optics and atomic processes must be studied by a hybrid transfer map-Monte-Carlo approach in which transfer map methods describe the deterministic behavior of the particles, and Monte-Carlo methods are used to provide corrections accounting for the stochastic nature of scattering and straggling of particles. The advantage of the new approach is that the vast majority of the dynamics are represented by fast application of the high-order transfer map of an entire element and accumulated stochastic effects. The gains in speed are expected to simplify the optimization of cooling channels which is usually computationally demanding. In conclusion, progress on the development of the required algorithms and their application to modeling muon ionization cooling channels is reported.

43 PARTICLE ACCELERATORS↗

DOE OSTI2026

Analytic Chromaticity Formulas for the Muon g -2 experiment at fermilab

The Muon g-2 Experiment (E989) at Fermilab measured the muon anomalous magnetic moment aμ with unprecedented precision of 127 ppb using a storage ring. The experiment pushed the limits in terms of accuracy and systematic errors. Achieving the 78-ppb total systematic uncertainty required precise beam dynamics modeling and corrections for effects on the measured muon precession frequency. Here, we derived the first analytic aberration formulas up to the second order for the muon g-2 storage ring’s combined-function electrostatic quadrupoles (superimposed magnetic dipole and electric quadrupole fields) using an order-by-order perturbation method. From these, we obtained the exact chromaticity formulas for three ring models of different granularity and validated them against numerical calculations using COSY INFINITY, achieving analytic-numerical agreement to ⁠$\mathscr{O}(10^{-10})$. This work resolved discrepancies between previous approximate derivations and provided essential beam dynamics results for Runs 4-6 analyses. We also calculated nonlinear chromaticities up to ninth order. The experiment completed its final Run 6 in July 2023, collecting 21 times more data than the previous muon g-2 experiment at Brookhaven National Laboratory, with the final result announced in June 2025.

aberrations↗

DOE OSTI2019

Realistic Modeling of the Muon g-2 Experiment Beamlines at Fermilab

The main goal of the Muon g-2 Experiment at Fermilab (E989) is to measure the muon anomalous magnetic moment (a, also dubbed as the "anomaly’’) to unprecedented precision. This new measurement will allow to test the completeness of the Standard Model (SM) and to validate other theoretical models beyond the SM. Simulations of the beamlines from the pion production target to the entrance of the g-2 Storage Ring using COSY INFINITY contribute to the understanding and characterization of the muon beam production in relation to the statistical and systematics uncertainties of the E989 measurement. The effect of nonlinearites from fringe fields and high-order contributions on the beam delivery system performance are considered, as well as interactions with the beamline elements apertures, particle decay channels, spin dynamics, and beamline misalignments.

43 PARTICLE ACCELERATORS↗

DOE OSTI2014

Space Charge Simulation in COSY using The Fast Multipole Method

A method is implemented in COSY Infinity that allows the computation of space charge effects of arbitrary and large distributions of particles in an efficient and accurate way based on a variant of the Fast Multipole Method (FMM). It relies on an automatic multigrid-based decomposition of charges in near and far regions and the use of high-order differential algebra methods to obtain decompositions of far fields that lead to an error that scales with a high power of the order. Given an ensemble of N particles, the method allows the computation of the self-fields of all particles on each other with a computational expense that scales as O(N). Further- more, the method allows the computation of all high-order multipoles of the space charge fields that are necessary for the computation of high-order transfer maps and all resulting aberrations. Space charge effects are crucial in modeling the latter stages of the six-dimensional (6D) cooling channel for the Muon Collider. Results of simulating the 6D cooling channel for the Muon Collider using the FMM method and other tools and improvements implemented for ionization cooling lattices are presented.

Loseth, B.↗

DOE OSTI2022

Beam-Based Diagnostics of Electric Guide Fields and Lattice Parameters for Run-1 of the Muon g-2 Storage Ring at Fermilab

A portion of the Muon g-2 Storage Ring electric system, which provides vertical beam focusing, exhibited an unexpected time dependence that produced a characteristic evolution of the stored beam during Run-1 of the Muon g-2 Experiment at Fermilab (E989). A method to reconstruct the Run-1 electric guide fields has been developed, which is based on a numerical model of the muon storage ring and optimization algorithms supported by COSY INFINITY. This method takes beam profile measurements from the Muon g-2 straw tracking detectors as input, and it produces a full reconstruction of the time-dependent fields. The fields can then be used for the reproduction of detailed beam tracking simulations and the calculation of ring lattice parameters for acceptance studies and systematic error evaluations.

43 PARTICLE ACCELERATORS↗

DOE OSTI2025

On Final Results From the Muon $g\mathrm{-}2$ Experiment at Fermilab

The Muon g-2 Experiment at Fermilab has measured the muon anomalous magnetic moment, a_mu, with unprecedented precision, leveraging a dataset from Runs 1 6 that is 21 times larger than its Brookhaven predecessor. This talk will present the experiment s final result, which serves as a benchmark for testing the Standard Model with high precision. The measurement was performed in a storage ring with a highly uniform magnetic field, where precise beam dynamics understanding is vital to determine the muon anomalous precession frequency. Key to this effort were advanced simulation tools including COSY INFINITY, precise fringe field modeling, and calculations of beam dynamics characteristics like tunes and chromaticity. We will conclude by exploring how the experiment's findings reshape our understanding of particle physics.

Valetov, Eremey [Michigan State U.]↗

NASA NTRS2022

High-energy Photon Opacity in the Twisted Magnetospheres of Magnetars

Magnetars are neutron stars characterized by strong surface magnetic fields generally exceeding the quantum critical value of 44.1 TG. High-energy photons propagating in their magnetospheres can be attenuated by QED processes like photon splitting and magnetic pair creation. In this paper, we compute the opacities due to photon splitting and pair creation by photons emitted anywhere in the magnetosphere of a magnetar. Axisymmetric, twisted dipole field configurations embedded in the Schwarzschild metric are treated. The paper computes the maximum energies for photon transparency that permit propagation to infinity in curved spacetime. Special emphasis is given to cases where photons are generated along magnetic field loops and/or in polar regions; these cases directly relate to resonant inverse Compton scattering models for the hard X-ray emission from magnetars and Comptonized soft gamma-ray emission from giant flares. We find that increases in magnetospheric twists raise or lower photon opacities, depending on both the emission locale and the competition between field-line straightening and field strength enhancement. Consequently, given the implicit spectral transparency of hard X-ray bursts and persistent "tail" emission of magnetars, photon splitting considerations constrain their emission region locales and the twist angle of the magnetosphere; these constraints can be probed by future soft gamma-ray telescopes such as COSI and AMEGO. The inclusion of twists generally increases the opaque volume of pair creation by photons above its threshold, except when photons are emitted in polar regions and approximately parallel to the field.

radiation mechanisms: non-thermal↗