Abstract
We develop a symplectic charged particle tracking method for phase space coordinates and polarization in all electric storage rings. Near the magic energy, the spin precession tune is proportional to the fractional momentum deviation ${\ensuremath{\delta}}_{m}$ from the magic energy, and the amplitude of the radial and longitudinal spin precession is proportional to $\ensuremath{\eta}/{\ensuremath{\delta}}_{m}$, where $\ensuremath{\eta}$ is the electric dipole moment for an initially vertically polarized beam. The method can be used to extract the electron electric dipole moment of a charged particle by employing narrow band frequency analysis of polarization around the magic energy.
Highlights
Most particle accelerators are built out of the magnetic elements, because the magnetic fields guide high energy particles more effectively compared to the electric fields
This paper presents an explicit symplectic orbit and spin tracking method under static electric fields using the Lorentz covariant Hamiltonian
This work extends that of Ref. [16] of tracking methods for s-dependent magnetic field system to element-byelement tracking for electric elements
Summary
Most particle accelerators are built out of the magnetic elements, because the magnetic fields guide high energy particles more effectively compared to the electric fields. It is important to build and benchmark precise and explicit symplectic numerical particle tracking codes for electrostatic fields. This paper presents an explicit symplectic orbit and spin tracking method under static electric fields using the Lorentz covariant Hamiltonian. We present self-adjoint spin tracking method such that higher order integrator can be built using composition method [15] We find that this method can be beneficial to reduce numerical error on closed orbit for very compact magnetic rings.
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