Paper  Title  Page 

MOPLG03  Spin Dynamics in Modern Electron Storage Rings: Computational and Theoretical Aspects  127 


Funding: U.S. Department of Energy, Office of Science, Office of High Energy Physics, Award Number DESC0018008 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 spinflip effects due to synchrotron radiation including the SokolovTernov effect and its BaierKatkov generalization. The FBEs were introduced by Derbenev and Kondratenko in 1975 as a generalization of the BaierKatkovStrakhovenko equations from a single orbit to the whole phase space. The FBEs are a system of three uncoupled FokkerPlanck equations plus two coupling terms, i.e., the TBMT term and the BaierKatkov 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 socalled DerbenevKondratenko formulas. However, we believe that the FBEs offer a more complete starting point for very high energy rings like the FCCee and CEPC. The issues for very high energy are: (i) Can one get polarization, (ii) are the DerbenevKondratenko 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 

Slides MOPLG03 [0.465 MB]  
DOI •  reference for this paper ※ https://doi.org/10.18429/JACoWICAP2018MOPLG03  
About •  paper received ※ 20 October 2018 paper accepted ※ 24 October 2018 issue date ※ 26 January 2019  
Export •  reference for this paper using ※ BibTeX, ※ LaTeX, ※ Text/Word, ※ RIS, ※ EndNote (xml)  
MOPAF04  Spin Dynamics in Modern Electron Storage Rings: Computational Aspects  146 


Funding: This material is based on work supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DESC0018008. 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* (FCCee) 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 spinflip effects due to synchrotron radiation. The FBEs are a system of three uncoupled FokkerPlanck 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 firstorder 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 ddimensional 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 

Slides MOPAF04 [0.993 MB]  
DOI •  reference for this paper ※ https://doi.org/10.18429/JACoWICAP2018MOPAF04  
About •  paper received ※ 20 October 2018 paper accepted ※ 24 October 2018 issue date ※ 26 January 2019  
Export •  reference for this paper using ※ BibTeX, ※ LaTeX, ※ Text/Word, ※ RIS, ※ EndNote (xml)  