FZJ, Jülich, Germany
HHUD, Dusseldorf, Germany
University of Crete, Heraklion, Crete, Greece
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
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 present our experiences in optimizing the proposed Rapid Cycling Synchrotron (RCS) injector for the eRHIC Storage ring and the RHIC 2017 lattice. We have develop python code to drive lattice calculations in MADX which are then used to calculate spin resonances using the DEPOL algorithm. This approach has been used to minimize intrinsic spin resonances during the RCS acceleration cycle while controlling lattice parameters such as dispersion and beta functions. This approach has also been used to construct localized imperfection bumps using a spin response matrix and SVD. This approach has also been used to reduce interfering intrinsic spin resonances during the RHIC acceleration ramp.