J. Corbett, Y. Nosochkov, J. Safranek, A. Garren
The SSRL is investigating an accelerator upgrade project to replace the present 130 nm rad FODO lattice with an 18 nm rad double bend achromat lattice: SPEAR 3. In this paper, we review the methods used to maximize the SPEAR 3 dynamic aperture including optimization of linear optics, betatron tune, chromaticity and coupling correction, and effects of machine errors and insertion devices.
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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 incl…
The SPEAR 3 upgrade lattice will provide much reduced beam emittance to increase the brightness of synchrotron radiation beams from wigglers and undulators. Seven existing insertion devices will be used in the lattice. In this paper we review the wiggler parameters, outline the wiggler compensation scheme, and evaluate wiggler effect on the optics and dynamic aperture.
We propose to optimize the nonlinear beam dynamics of existing and future storage rings with direct online optimization techniques. This approach may have crucial importance for the implementation of diffraction limited storage rings. In this paper considerations and algorithms for the online optimization approach are discussed. We have applied this approach to experimentally improve the dynamic aperture of the SPEAR3 storage ring with the robust conjugate direction search method and the particle swarm optimization method. The dynamic aperture was improved by more than 5 mm within a short peri…
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…
Recently proposed novel idea of "crabbed waist" beam-beam collisions will be tested at DAFNE during the collider run for the Siddharta experiment. In order to achieve the goal luminosity, large dynamic aperture is a matter of primary importance. A new method of a dynamic aperture optimization based on step-by-step chromaticity compensation with choosing the "best" sextupole pair at each step was applied to the DAFNE upgrade lattice. Several tune points were considered taking into account both high luminosity and large dynamic aperture. Algorithm and results of optimization will be presented.
The DAFNE-II project is considered now as a possible candidate for upgrading the DAFNE electron-positron collider and improving its luminosity. 6D simulation of the DAFNE-II dynamic aperture under various assumptions (weak and strong RF focusing) is presented. Dynamic aperture is reduced as compared to the week focusing case or constant energy deviation due synchrobetatron resonances.
Compact low emittance lattice cell providing large dynamic aperture is essentual for development of extremely low (pm range) emittance storage rings. As it is well known, a pair of identical sextupoles connected by a minus-identy matrix transformer in ideal case of kick-like magnets provides infinite dynamic aperture. Though the finite sextupole length degrades the aperture, it is still large enough, and in this report we discuss development of the low emittance lattice cell providing the -I condition for both horizontal and vertical chromatic sextupoles. Such cell can be used as a module for …
A simplified Recycler lattice was created to fine tune injection straight, ring tune, and phase trombone. In this paper, we will present detailed modifications for further optimization of Recycler lattice which requires the investigation of tune footprint and dynamic aperture based on higher order momentum components of the magnetic fields, together with the space charge effects.
By considering delta function sextupole, octupole, and decapole perturbations and using difference action-angle variable equations, we find some useful analytical formulae for the estimation of the dynamic apertures of circular accelerators due to single sextupole, single octupole, single decapole (single 2$m$ pole in general). Their combined effects are derived based on the Chirikov criterion of the onset of stochastic motions. Comparisons with numerical simulations are made, and the agreement is quite satisfactory. These formulae have been applied to determine the beam-beam limited dynamic a…
Learning Dynamic Aperture from One-turn Maps
提出将动态孔径预测转化为图像分割问题，利用深度代理模型从编码的单圈映射直接学习粗粒度动态孔径，并在电子-离子对撞机电子储存环中验证了其泛化能力。
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…
Tracking studies have indicated that for a lattice whose elements all have a single field multipole present, all having the same order k, the dynamic aperture approaches a non zero limit when k becomes very large. The dynamic aperture and other properties of the lattice, as k becomes large, will be called the high multipole limit. It will be shown that the high multipole limit provides a reasonable estimate of the dynamic aperture of an accelerator, and the other properties of the high multipole limit found below are useful for understanding the stability of the accelerator. The high multipole…
Fast chaos indicator from auto-differentiation for dynamic aperture optimization
基于自动微分的快速混沌指标用于动态开口优化，通过切线映射范数高效检测混沌行为，降低计算成本。
Efficient algorithms for dynamic aperture and momentum acceptance calculation
针对环形加速器动态孔径与动量接受度计算问题，受光栅洪水填充启发提出高效动态孔径算法（提速数倍），还开发Fast Touschek Tracking技术，可将局部动量接受度计算提速1-2个数量级且精度损失有限。
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.
In a collider final focus (FF) system chromaticity generated by FF quadrupoles is compensated by sextupole pairs with -I map in between to cancel geometrical aberrations. However, it works perfectly only for zero-length sextupole kicks; for realistic thick magnets only quadratic aberrations are cancelled exactly while higher order terms appear and may deteriorate the transverse dynamic aperture. The latter is particularly important for prospective projects of the future circular colliders based on the Crab-Waist collision approach where extremely low beta at IP generates very large chromatic e…
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
Beam Dynamics Framework Incorporating Acceleration to Define the Minimum Aperture in Two Focusing Schemes for Proton Radiotherapy Linac
本文提出一个包含加速效应的横向束流动力学框架，用于质子放疗直线加速器，通过解析求解类FODO方案和最小孔径方案，实现束流尺寸控制与最小孔径定义。
Optimizing Dynamic Aperture Studies with Active Learning
本研究提出一种结合深度神经网络与主动学习框架的新方法，利用CERN大型强子对撞机模拟数据快速准确估计动态孔径，在保证精度的同时显著提升参数扫描效率。
Achromatic Telescopic Squeezing for Dynamic Aperture Optimization in the Electron Storage Ring of the EIC
本文探讨了色差消散望远镜压缩在EIC电子存储环动态 aperture 优化中的应用，通过对比传统六极磁铁方案，发现ATS光学能降低六极磁铁强度并改善动量接受度，但会导致色度增长，需在存储环设计中权衡其优势。
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…
Physically speaking, the delta function like beam-beam nonlinear forces at interaction points (IPs) act as a sum of delta function nonlinear multipoles. By applying the general theory established in ref. 1, in this paper we investigate analytically the beam-beam interaction limited dynamic apertures and the corresponding beam lifetimes for both the round and the flat beams. Relations between the beam-beam limited beam lifetimes and the beam-beam tune shifts are established, which show clearly why experimentally one has always a maximum beam-beam tune shift, ξ_{y, max}, around 0.045 for e-^+e$^…
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…
Beam-beam simulations predict that PEP-II luminosity can be increased by operating the horizontal betatron tune near and above a half-integer resonance. However, effects of the resonance and its synchrotron sidebands significantly enhance betatron and chromatic perturbations which tend to reduce dynamic aperture. In the study, chromatic variation of horizontal tune near the resonance was minimized by optimizing local sextupoles in the Interaction Region. Dynamic aperture was calculated using tracking simulations in LEGO code. Dependence of dynamic aperture on the residual orbit, dispersion and…
Third order tracking calculations show that a wavelength shifter on the basis of a commercially available 12 T split-pair solenoid is compatible with the ANKA storage ring
Convexity of the longitudinal variation of third-order resonance driving terms and its application in dynamic aperture optimization
本文证明三阶共振驱动项纵向变化的RMS值在六极子强度空间中为凸函数，利用此特性开发了基于粒子跟踪的快速动态孔径优化方法。
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…
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…
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…
The step-by-step chromaticity compensation method for chromatic sextupole optimization and dynamic aperture increase was proposed by E. Levichev and P. Piminov (E. Levichev and P. Piminov, 2006 [1]). Although this method can be used to enlarge the dynamic aperture of storage ring, it has some drawbacks. In this paper, we combined this method with evolutionary computation algorithms, and proposed an improved version of this method. In the improved method, the drawbacks are avoided, and thus better optimization results can be obtained.
A further study on the high-precision optics for HIAF-BRing
本文基于实测磁场对HIAF-BRing的切片光学系统进行模拟研究，考察误差引起的闭合轨道畸变、轨道校正及动力学孔径，结果表明切片光学在COD控制和动力学孔径方面优于或相当于原始光学系统，为加速器调束优化提供参考。
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 …
Flexible Spectral-Normalized Neural Gaussian Process for Dynamic Aperture Prediction
针对大规模科学应用中先进机器学习方法计算不可行的问题，提出一种经验贝叶斯方法自动调整谱归一化神经高斯过程超参数，在大型强子对撞机动态孔径预测中实现低计算成本下的高精度与良好不确定性校准。
Efficient Dynamic and Momentum Aperture Optimization for Lattice Design Using Multipoint Bayesian Algorithm Execution
本文提出利用多点贝叶斯算法执行实现存储环设计的高效动态和动量孔径优化，通过减少计算成本提升设计质量。
Recently, the Hamiltonian Control Theory was used in [Boreux et al.] to increase the dynamic aperture of a ring particle accelerator having a localized thin sextupole magnet. In this letter, these results are extended by proving that a simplified version of the obtained general control term leads to significant improvements of the dynamic aperture of the uncontrolled model. In addition, the dynamics of flat beams based on the same accelerator model can be significantly improved by a reduced controlled term applied in only 1 degree of freedom.
Proton accumulator rings and other circular hadron accelerators are susceptible to intensity-driven parametric instabilities because the zero-current charged particle dynamics are characterized by a single tune. Landau damping can suppress these instabilities, which requires energy spread in the beam or introducing nonlinear magnets such as octupoles. However, this approach reduces dynamic aperture. Nonlinear integrable optics can suppress parametric instabilities independent of energy spread in the distribution, while preserving the dynamic aperture. This novel approach promises to reduce par…
Persistent homology of Hamiltonian orbits with applications to beam physics: Euler-Betti classification and topological chaos indicators
本文利用持续同调对哈密顿轨道进行分类，提出拓扑混沌指标，以短轨道段预测粒子加速器动力学孔径，减少长期跟踪模拟需求。
Convergence map with action-angle variables based on square matrix for nonlinear lattice optimization
本文提出基于方阵与作用量-角变量的收敛图，通过 Fourier 微扰迭代判断非线性运动稳定性；在 NSLS-II 上获得与跟踪相当或更优的动力学孔径，速度提升30–300倍。
Space charge effects can be very important for the dynamics of intense particle beams, as they repeatedly pass through nonlinear focusing elements, aiming to maximize the beam's luminosity properties in the storage rings of a high energy accelerator. In the case of hadron beams, whose charge distribution can be considered as "frozen" within a cylindrical core of small radius compared to the beam's dynamical aperture, analytical formulas have been recently derived \cite{BenTurc} for the contribution of space charges within first order Hamiltonian perturbation theory. These formulas involve dist…
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…
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