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 effects. In the present paper, we study these high aberrations by the power series technique and propose the method of the dynamic aperture recover. Results of analytic study are compared with those obtained by tracking.
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 …
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.
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…
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.
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…
After the successful completion of the SIDDHARTA experiment run with crab waist collisions, the electron- positron collider DAΦNE has started routine operations for the KLOE-2 detector. The new interaction region also exploits the crab waist collision scheme, but features certain complications including the experimental detector solenoid, compensating anti-solenoids, and tilted quadrupole magnets. We have performed simulations of the beam-beam collisions in the collider taking into account the real DAΦNE nonlinear lattice. In particular, we have evaluated the effect of crab waist sextupoles an…
Ultrafast electron diffraction (UED) is a technique in which short-pulse electron beams can probe the femtosecond-scale evolution of atomic structure in matter driven far from equilibrium. As an accelerator physics challenge, UED imposes stringent constraints on the brightness of the probe electron beam. The low normalized emittance employed in UED, often at the 10 nm scale and below, is particularly sensitive to both applied field aberrations and space charge effects. The role of aberrations is increasingly important in small probe systems that often feature multiple orders of magnitude varia…
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…
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…
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…
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 …
Artificial neural networks can be used for creating surrogate models that can replace computationally expensive simulations. In this paper, a surrogate model was created for a subset of the Compact Linear Collider (CLIC) final-focus system. By training on simulation data, we created a model that maps sextupole offsets to luminosity and beam sizes, thus replacing computationally intensive tracking and beam-beam simulations. This model was then used for optimizing the parameters of a random walk procedure for sextupole alignment.
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…
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.
The beam aperture of a particle accelerator defines the clearance available for the circulating beams and is a parameter of paramount importance for the accelerator performance. At the CERN Large Hadron Collider (LHC), the knowledge and control of the available aperture is crucial because the nominal proton beams carry an energy of 362 MJ stored in a superconducting environment. Even a tiny fraction of beam losses could quench the superconducting magnets or cause severe material damage. Furthermore, in a circular collider, the performance in terms of peak luminosity depends to a large extent o…
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…
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$^…
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…
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 …
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…
Low-energy medium-luminosity Muon Collider (MC) is being studied as a possible Higgs Factory (HF). Electrons from muon decays will deposit more than 300 kW in superconducting magnets of the HF collider ring. This imposes significant challenges to superconducting (SC) magnets used in the MC storage ring (SR) and interaction regions (IR). The magnet designs are proposed which provide high operating gradient and magnetic field in a large aperture to accommodate the large size of muon beams due to low \b{eta}* as well as the cooling system to intercept the large heat deposition from the showers in…
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.
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 …
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 low-energy medium-luminosity Muon Collider (MC) is being studied as a possible Higgs Factory (HF). Electrons from muon decays will deposit more than 300 kW in superconducting magnets of the HF collider ring. This imposes significant challenges to superconducting (SC) magnets used in the MC storage ring (SR) and interaction regions (IR). Magnet designs are proposed which provide high operating gradient and magnetic field in a large aperture to accommodate the large size of muon beams (due to low beta*), as well as a cooling system to intercept the large heat deposition from the showers induce…
Conceptual designs of superconducting magnets for the storage ring of a Muon Collider with a 1.5 TeV c.o.m. energy and an average luminosity of 10 34 cm-2s-1 are presented. All magnets are based on Nb3Sn superconductor and designed to provide an adequate operating field/field gradient in the aperture with the critical current margin required for reliable magnet operation in the machine. Magnet cross-sections were optimized to achieve the accelerator field quality in the magnet aperture occupied with beams. The magnets and corresponding protective measures are designed to handle about 0.5 kW/m …
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…
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.
A method is presented for characterizing the emittance dilution and dynamic aperture for an arbitrary closed lattice that includes guide field magnet errors, multipole errors and misalignments. This method, developed and tested at the Cornell Electron Storage Ring Test Accelerator (CesrTA), has been applied to the damping ring lattice for the International Linear Collider (ILC). The effectiveness of beam based emittance tuning is limited by beam position monitor (BPM) measurement errors, number of corrector magnets and their placement, and correction algorithm. The specifications for damping r…
A Super Proton-Proton Collider (SPPC) that aims to explore new physics beyond the standard model will be built in China. Here we focus on the impact of beam-beam interactions in the SPPC. Simulations show that with the current optics and nominal tunes, the dynamic aperture (DA) with all the beam-beam interactions is less than 6σ, the dominant cause being the long-range interactions. First, we show the results of a tune scan done to maximize the DA. Next, we discuss the compensation of the long-range interactions by increasing the crossing angle and also by using current carrying wires.
We report the design status and beam dynamics challenges for the electron-positron branch of the Future Circular Collider (FCC) study, as of August 2015. After recalling motivation and physics requirements for the FCC-ee, we briefly discuss configurations and parameters, collider layout, the superconducting RF system, possible staging scenarios, final-focus optics, interaction-region (IR) issues, machine detector interface and IR synchrotron radiation, dynamic aperture, beam-beam effects, top-up injection, mono-chromatization option, impedances, instabilities, energy calibration and polarizati…
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…
Beam-beam forces and collision optics can strongly affect beam lifetime, dynamic aperture, and halo formation in particle colliders. Extensive analytical and numerical simulations are carried out in the design and operational stage of a machine to quantify these effects, but experimental data is scarce. The technique of small-step collimator scans was applied to the Fermilab Tevatron collider and to the CERN Large Hadron Collider to study the effect of collisions on transverse beam halo dynamics. We describe the technique and present a summary of the first results on the dependence of the halo…
Halo dynamics influences global accelerator performance: beam lifetimes, emittance growth, dynamic aperture, and collimation efficiency. Halo monitoring and control are also critical for the operation of high-power machines. For instance, in the high-luminosity upgrade of the LHC, the energy stored in the beam tails may reach several megajoules. Fast losses can result in superconducting magnet quenches, magnet damage, or even collimator deformation. The need arises to measure the beam halo and to remove it at controllable rates. In the Tevatron and in the LHC, halo population densities and dif…
Several transverse noise sources, such as power supply ripples, can potentially act as an important limiting mechanism for the luminosity production of the Large Hadron Collider (LHC) and its future High-Luminosity upgrade (HL-LHC). In the presence of non-linearities, depending on the spectral components of the power supply noise and the nature of the source, such a mechanism can increase the diffusion of the particles in the distribution through the excitation of sideband resonances in the vicinity of the ones driven by the lattice non-linearities. For the HL-LHC, due to the reduction of the …
Square matrix-based six-dimensional convergence map for nonlinear beam dynamics analysis
Institute of High Energy Physics, Chinese Academy of Sciences（中国科学院高能物理研究所）；University of the Chinese Academy of Sciences（中国科学院大学）；Facility for Rare Isotope Beams, Michigan State University（密歇根州立大学稀有同位素束流设施）
