斯坦福大学同步辐射实验室正在研究一个加速器升级项目，该项目将用18 nm rad双弯曲消色差（DBA）晶格：SPF3取代目前的130 nm rad FODO晶格。低发射度设计产生高亮度光束，但DBA晶格中的较强聚焦增加了色度和对机器误差的光束灵敏度。为了确保有效的注入和长的Touschek寿命，已经进行了设计晶格和动态孔径的优化。在本文中，我们审查的方法，以最大限度地提高SPATIAL 3的动态孔径，包括必要的光学修改，选择的调谐和相位提前，优化的六极和耦合校正，和建模的机器误差，摇摆器和晶格周期性的影响。
英文摘要：
The Stanford Synchrotron Radiation Laboratory is investigating an accelerator upgrade project that would replace the present 130 nm rad FODO lattice with an 18 nm rad double bend achromat (DBA) lattice: SPEAR 3. The low emittance design yields a high brightness beam, but the stronger focusing in the DBA lattice increases chromaticity and beam sensitivity to machine errors. To ensure efficient injection and long Touschek lifetime, an optimization of the design lattice and dynamic aperture has been performed. In this paper, we review the methods used to maximize the SPEAR 3 dynamic aperture including necessary optics modifications, choice of tune and phase advance, optimization of sextupole and coupling correction, and modeling effects of machine errors, wigglers and lattice periodicity.
A novel approach to expedite design optimization of nonlinear beam dynamics in storage rings is proposed and demonstrated in this study. At each iteration, a neural network surrogate model is used to suggest new trial solutions in a multi-objective optimization task. The surrogate model is then updated with the new solutions, and this process is repeated until the final optimized solution is obtained. We apply this approach to optimize the nonlinear beam dynamics of the SPEAR3 storage ring, where sextupole knobs are adjusted to simultaneously improve the dynamic aperture and the momentum apert…
We briefly present recent progress with our algorithm and its implementation called SIMPA described in a previous paper. The algorithm has a new and unique approach to long-term 4D tracking of charged particles in arbitrary static electromagnetic fields. Using the improvements described in this paper, we made frequency analysis and dynamic aperture studies in ELENA. The effect of the end fields and the perturbation introduced by the magnetic system of the electron cooler on dynamic aperture is shown. A special feature of this study is that we have not introduced any multipole errors into the m…
Fast chaos indicator from auto-differentiation for dynamic aperture optimization
Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Facility for Rare Isotope Beams, Michigan State University（密歇根州立大学稀有同位素束设施）；Stanford University（斯坦福大学）
We developed a low emittance electron storage ring with large 6D dynamic aperture. Contrary to the traditional approach using strong focusing magnetic cells with optimized (and large) horizontal phase advance, which yields huge natural chromaticity, we employed a relatively weak focusing lattice with low chromaticity per cell and, consequently, wide on- and off-momentum dynamic aperture. Inevitable for weak focusing emittance growth, we compensated by slicing the lattice into many short, with small bending angle, elementary periodic cells. To reduce the size, we superimposed focusing gradient …
In the Hadron machine option, proposed in the context of the Future Circular Colliders (FCC) study, the dipole field quality is expected to play an important role, as in the LHC. A preliminary evaluation of the field quality of dipoles, based on the Nb$_{3}$Sn technology, has been provided by the magnet group. The effect of these field imperfections on the dynamic aperture, using the present lattice design, is presented and first tolerances on the b$_3$ and b$_5$ multipole components are evaluated.
We describe a benchmark study of collective and nonlinear dynamics in an APS storage ring. A 1-mm long bunch was assumed in the calculation of wakefield and element by element particle tracking with distributed wakefield component along the ring was performed in Elegant simulation. The result of Elegant simulation differed by less than 5 % from experimental measurement
The Future Circular Collider (FCC) study investigates the feasibility of circular colliders in the post-LHC era. The sub-study FCC-ee is a 100 km electron positron collider in the energy range of 90-365 GeV. In order to achieve a design luminosity in the order of $10^{36}$cm$^{-2}$s$^{-1}$ continuous top-up injection is required. The injector chain therefore includes a 100 km booster synchrotron in the same tunnel as the collider rings. This paper presents the lattice design of this booster synchrotron and the first dynamic aperture studies based on the chromaticity correction sextupole scheme…
In recent years, modelling the evolution of beam losses in circular proton machines starting from the evolution of the dynamic aperture has been the focus of intense research. Results from single-particle, non-linear beam dynamics have been used to build simple models that proved to be in good agreement with beam measurements. These results have been generalised, thus opening the possibility to describe also the luminosity evolution in a circular hadron collider. In this paper, the focus is on the derivation of scaling laws for luminosity, which include both burn off and additional pseudo-diff…
A fast method for dynamic aperture (DA) optimization of storage rings has been developed through the use of reversal integration. While chaotic dynamical systems have exact time-reversal symmetry, numerical forward integration differs from its reversal due to scaled cumulative round-off errors. The difference, intrinsically associated with the Lyapunov exponent, is a generic indicator of chaos because it represents the sensitivity of chaotic motion to an initial condition. A chaos indicator of the charged particle motion is then obtained by comparing the forward integrations of particle trajec…
This report summarizes progress made towards a new approach for enlarging the dynamic aperture of particle accelerators. Unlike prior methods which attempted to move the location of select resonances outward in phase space, our approach aims to move all resonances concurrently. These resonances are in one-to-one correspondence with fixed points of symplectic maps, which in turn are in a one-to-one correspondence with the critical points of their generating function. Thus in this approach, the problem of enlarging dynamic aperture boils down to an approximation problem: given a generating funct…
In recent years, modelling the evolution of beam losses in circular proton machines starting from the concept of dynamic aperture its time evolution has been the focus of intense research. Results from single-particle non-linear beam dynamics have been used to build simple models that proved to be in good agreement with beam measurements. These results have been generalised, thus opening the possibility to describe also the luminosity evolution in a circular hadron collider. In a companion paper (Part I), the derivation of a scaling law for luminosity, which includes both burn off and pseudo-d…
Dynamic aperture (DA) is an important nonlinear property of a storage ring lattice, which has a dominant effect on beam injection efficiency and beam lifetime. Generally, minimizing both resonance driving terms (RDTs) and amplitude dependent tune shifts is an essential condition for enlarging the DA. In this paper, we study the correlation between the fluctuation of RDTs along the ring and the DA area with double- and multi-bend achromat lattices. It is found that minimizing the RDT fluctuations is more effective than minimizing RDTs themselves in enlarging the DA, and thus can serve as a very…
A multi-objective genetic algorithm is developed for optimizing nonlinearities in diffraction limited storage rings. This algorithm determines sextupole and octupole strengths for chromaticity correction that deliver optimized dynamic aperture and beam lifetime. The algorithm makes use of dominance constraints to breed desirable properties into the early generations. The momentum aperture is optimized indirectly by constraining the chromatic tune footprint and optimizing the off-energy dynamic aperture. The result is an effective and computationally efficient technique for correcting chromatic…
In a lepton storage ring of very high energy (e.g. in the $e^+e^-$ Higgs factory) synchrotron radiation from quadrupoles constrains transverse dynamic aperture even in the absence of any magnetic nonlinearities. This was observed in tracking for LEP and the Future Circular $e^+e^-$ Collider (FCC-ee). Here we describe a new mechanism of instability created by modulation of the particle energy at the double betatron frequency by synchrotron radiation in the quadrupoles. Energy modulation varies transverse focusing strength at the same frequency and creates a parametric resonance of the betatron …
The upgrade of the Swiss Light Source, called SLS 2.0, is scheduled for 2023-24. The current storage ring will be replaced by one based on multi-bend achromats, allowing for about 30 times higher brightness. Due to the stronger focusing and the required chromatic compensation, finding a reasonably large dynamic aperture (DA) for injection, as well as an energy acceptance for a sufficient beam lifetime, is challenging. In order to maximize the DA and prolong the beam lifetime, we combine the well-known tracking code tracy with a massively parallel implementation of a multi-objective genetic alg…
The Supper Proton-Proton Collider is a next-generation hadron collider that is now being designed. A baseline design aims for a peak luminosity of about 1*10^35 cm^-2 s^-1. The focus of this article is the effect of beam-beam interactions which are expected to strongly influence stability in the beams. We start with a discussion of a scheme to generate the crossing angles at the interaction points while also correcting the dispersion thus created. The optics constraints on the achievable \b{eta}* were studied. Weak-strong simulations were performed to study single particle dynamics via tune fo…
A numerical method to design nonlinear double- and multi-bend achromat (DBA and MBA) lattices with approximate invariants of motion is investigated. The search for such nonlinear lattices is motivated by Fermilab's Integrable Optics Test Accelerator (IOTA), whose design is based on an integrable Hamiltonian system with two invariants of motion. While it may not be possible to design an achromatic lattice for a dedicated synchrotron light source storage ring with one or more exact invariants of motion, it is possible to tune the sextupoles and octupoles in existing DBA and MBA lattices to produ…
Determining a model for the time scaling of the dynamic aperture of a circular accelerator is a topic of strong interest and intense research efforts in accelerator physics. The motivation arises in the possibility of finding a method to reliably extrapolate the results of numerical simulations well beyond what is currently possible in terms of CPU time. In earlier work, a proposal for a model based on Nekhoroshev theorem and Kolmogorov--Arnold--Moser theory was made. This model has been studied in detail and proved successful in describing the evolution of the dynamic aperture in numerical si…
We discuss the choice of the magnetic lattice and parameter optimization of the fourth generation light source SKIF (Russian acronym for Siberian Circular Photon Source) under construction in Novosibirsk. The study compares several basic lattice cells to procure one with low emittance and large dynamic aperture. The result is a developed lattice of SKIF with 72~pm natural emittance (at zero beam current and absent betatron coupling) at 3 GeV beam energy and 476 m circumference. Only two families of sextupoles provide horizontal and vertical dynamic apertures of 12 mm and 3.5 mm respectively an…
For any storage ring-based large-scale scientific facility, one of the most important performance parameters is the dynamic aperture (DA), which measures the motion stability of charged particles in a global manner. To date, long-term tracking-based simulation is regarded as the most reliable method to calculate DA. However, numerical tracking may become a significant issue, especially when lots of candidate designs of a storage ring need to be evaluated. In this paper, we present a novel machine learning-based method, which can reduce the computation cost of DA tracking by approximately one o…
Fermilab is constructing the Integrable Optics Test Accelerator (IOTA) as the centerpiece of the Accelerator R&D Program towards high-intensity circular machines. One of the factors limiting the beam intensity in present circular accelerators is collective instabilities, which can be suppressed by a spread of betatron frequencies (tunes) through the Landau damping mechanism or by an external damper, if the instability is slow enough. The spread is usually created by octupole magnets, which introduce the tune dependence on the amplitude and, in some cases, by a chromatic spread (tune dependence…
Modern synchrotron light source storage rings, such as the Swiss Light Source upgrade (SLS 2.0), use multi-bend achromats in their arc segments to achieve unprecedented brilliance. This performance comes at the cost of increased focusing requirements, which in turn require stronger sextupole and higher-order multipole fields for compensation and lead to a considerable decrease in the dynamic aperture and/or energy acceptance. In this paper, to increase these two quantities, a multi-objective genetic algorithm (MOGA) is combined with a modified version of the well-known tracking code tracy. As …
The non-linear effects caused by the intrinsic field transverse roll-off in an Elliptically Polarizing Undulator with 72-mm period and maximum peak field of 0.72 T (EPU72) on the dynamics aperture of the Pohang Light Source II (PLS-II) were investigated. A kick map and Frequency map analysis both demonstrated that EPU72 will not reduce the lifetime or cause injection problems in PLS-II.
In this paper the experimental results of the recent dynamic aperture at top energy for the CERN Large Hadron Collider are analysed by means of a diffusion model whose novelty consists of deriving the functional form of the diffusion coefficient from Nekhoroshev theorem. This theorem provides an optimal estimate of the remainder of perturbative series for Hamiltonian systems. As a consequence, a three-parameter diffusion model is built that reproduces the experimental results with a high level of accuracy. A detailed discussion of the physical interpretation of the proposed model is also prese…
