BNL, Upton, Long Island, New York, USA
With the aid of machine learning techniques, the genetic algorithm has been enhanced and applied to the multi-objective optimization problem presented by the dynamic aperture of the NSLS-II Ring. During the evolution employed by the genetic algorithm, the population is classified into different clusters. The clusters with top average fitness are given elite status. Intervention is implemented by repopulating some potentially competitive candidates based on the accumulated data. These candidates replace randomly selected candidates among the original data pool. The average fitness of the population is improved while diversity is not lost. The quality of the population increases and produces more competitive descendants accelerating the evolution process significantly. When identifying the distribution of optimal candidates, they appear to be located in isolated islands within the search space. Some of these optimal candidates have been experimentally confirmed at the NSLS-II storage ring. The machine learning techniques that exploit the genetic algorithm can also be used in other population-based optimization problems such as particle swarm algorithm.
UNM, Albuquerque, New Mexico, USA
University of Colorado at Boulder, Boulder, USA
DESY, Hamburg, Germany
In this talk we present some numerical and analytical results from our work on the spin polarization in high energy electron storage rings. Our work is based on the initial value problem of what we call the full Bloch equations (FBEs). The solution of the FBEs is the polarization density which is proportional to the spin angular momentum density per particle in phase space and which determines the polarization vector of the bunch. The FBEs take into account spin diffusion effects and spin-flip effects due to synchrotron radiation including the Sokolov-Ternov effect and its Baier-Katkov generalization. The FBEs were introduced by Derbenev and Kondratenko in 1975 as a generalization of the Baier-Katkov-Strakhovenko equations from a single orbit to the whole phase space. The FBEs are a system of three uncoupled Fokker-Planck equations plus two coupling terms, i.e., the T-BMT term and the Baier-Katkov term. Neglecting the spin flip terms in the FBEs one gets what we call the reduced Bloch equations (RBEs). The RBEs are sufficient for computing the depolarization time. The conventional approach of estimating and optimizing the polarization is not based on the FBEs but on the so-called Derbenev-Kondratenko formulas. However, we believe that the FBEs offer a more complete starting point for very high energy rings like the FCC-ee and CEPC. The issues for very high energy are: (i) Can one get polarization, (ii) are the Derbenev-Kondratenko formulas satisfactory at very high energy? If not, what are the theoretical limits of the polarization? Item (ii) will be addressed both numerically and analytically. Our numerical approach has three parts. Firstly we approximate the FBEs analytically using the method of averaging, resulting in FBEs which allow us to use large time steps (without the averaging the time dependent coefficients of the FBEs would necessitate small time steps). The minimum length of the time interval of interest is of the order of the orbital damping time. Seco
MSU, East Lansing, Michigan, USA
Fermilab, Batavia, Illinois, USA
Northern Illinois University, DeKalb, Illinois, USA
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.
BNL, Upton, Long Island, New York, USA
Electron polarization in a storage ring is subject to two very long term effects: Sokolov-Ternov polarization and depolarization by diffusion. This leads to an equilibrium state over a very long time scale, and, simulation-wise, is highly CPU-time and -memory consuming. Simulations aimed at determining optimal ring storage energy in an electron-ion collider in this context, are always based on tracking bunches with thousands of particles, and in addition for short time scales in comparison, due to HPC limitations. Based on considerations of ergodicity of electron bunch dynamics in the presence of synchrotron radiation, and on the very slow depolarization aimed at in a collider, tracking a single particle instead is investigated, here. This saves a factor of more than 2 orders of magnitudes in the parameter CPU-time*Memory-allocation, it allows much longer tracking and thus improved accuracy on the evaluation of polarization and time constants. The concept is illustrated with polarization lifetime and equilibrium polarization simulations at the eRHIC electron-ion collider.
UNM, Albuquerque, New Mexico, USA
University of Colorado at Boulder, Boulder, USA
DESY, Hamburg, Germany
In this talk we present some numerical results from our work on the spin polarization in high energy electron storage rings. The motivation of our work is to understand spin polarization in very high energy rings like the proposed Future Circular Collider* (FCC-ee) and Circular Electron Positron Collider** (CEPC). This talk is a supplement to K. Heinemann’s talk and gives further numerical details and results. As discussed in Heinemann’s talk our work is based on the initial value problem of the full Bloch equations*** (FBEs) which in turn determines the polarization vector of the bunch. The FBEs take into account spin diffusion effects and spin-flip effects due to synchrotron radiation. The FBEs are a system of three uncoupled Fokker-Planck equations plus coupling terms. Neglecting the spin flip terms in the FBEs one gets the reduced Bloch equations (RBEs) which poses the main computational challenge. Our numerical approach has three parts. Firstly we approximate the FBEs analytically using the method of averaging, resulting in FBEs which allow us to use large time steps (without the averaging the time dependent coefficients of the FBEs would necessitate small time steps). The minimum length of the time interval of interest is of the order of the orbital damping time. Secondly we discretize the averaged FBEs in the phase space variables by applying the pseudospectral method, resulting in a system of linear first-order ODEs in time. The phase space variables come in d pairs of polar coordinates where d = 1, 2, 3 is the number of degrees of freedom allowing for a d-dimensional Fourier expansion. The pseudospectral method is applied by using a Chebychev grid for each radial variable and a uniform Fourier grid for each angle variable. Thirdly we discretize the ODE system by a time stepping scheme. The presence of parabolic terms in the FBEs necessitates implicit time stepping and thus solutions of linear systems of equations. Dealing with 2d + 1 independent variables p
* See http://tlep.web.cern.ch
** See http://cepc.ihep.ac.cn
*** See http://ipac2018.vrws.de/papers/thpak144.pdf
BNL, Upton, Long Island, New York, USA
We discuss the longitudinal beam dynamics in storage rings in the presence of a higher-harmonic cavity (HHC) system for bunch lengthening. We first review the general conditions for HHC operations, either in active or passive mode, assuming the stability of the system. For uniform filling patterns, a distinction is made between operations with a normal-conducting HHC, where optimal conditions for bunch lengthening can be satisfied, and operations with super-conducting HHC, where optimal conditions can be met only approximately. The option to operate the NSLS-II storage ring with a passive, super-conducting third harmonic cavity (3HC) system is discussed next. The stability and performance of the system in the presence of a gap in the uniform filling, which corresponds to the present mode of operation of the NSLS-II storage ring, is investigated with self-consistent Vlasov-Fokker-Planck simulations performed with the code SPACE*.
* G. Bassi, A. Blednykh and V. Smaluk, Phys Rev. Accel. Beams 19, 024401 (2016).
SAGA, Tosu, Japan
MSU, East Lansing, Michigan, USA
We consider semi-infinite electrostatic deflectors with plates of different thickness, including plates with rounded edges, and we calculate their electrostatic potential and field using conformal mappings. To validate the calculations, we compare the fringe fields of these electrostatic deflectors with fringe fields of finite electrostatic capacitors, and we extend the study to fringe fields of adjacent electrostatic deflectors with consideration of electrostatic induction, where field falloffs of semi-infinite electrostatic deflectors are slower than exponential and thus behave differently from most magnetic fringe fields. Building on the success with electrostatic deflectors, we develop a highly accurate and fully Maxwellian conformal mappings 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 use this conformal mappings method to calculate the multipole terms of the high voltage quadrupole used in the storage ring of the Muon g-2 Experiment (FNAL-E-0989). Completing the methodological framework, we present a method for extracting multipole strength falloffs of a particle optical element from a set of Fourier mode falloffs. We calculate the quadrupole strength falloff and its effective field boundary (EFB) for the Muon g-2 quadrupole, which has explained the experimentally measured tunes, while simple estimates based on a linear model exhibited discrepancies up to 2%.