Spin resonances for stored deuteron beams in COSY. Vector polarization. Tracking with Spink

Results of measureemnts of vector and tensor ploarizatin of a deuteron beam in the storage ring COSY have been published by the SPIN@COSY collaboration[l]. In this experiment a RF Dipole was used that produced spin flip. The strength of the RFD-induced depolarizing resonance was calculated from the amount of spin flipping and the results shown in the figures of the cited paper. In this note we present the simulation of the experimental data (vector polarization) with the spin tracking code Spink 1 Spink Spink is a spin tracking code through the lattice of a synchrotron[2],[3]. In this code each machine element is treated as “thin” for spin motion and the spin angle rotation p is then calculated by a “flattened” version of the BMT spin equation[4] = dS ‘ S x F dt y m containing the magnetic field components transverse and longitudinal with respect to the velocity v, where G is the particle gyromagnetic ratio (=-0.142987 for deuterons) and y the Lorentz factor for the energy.. The spin rotation is calculated along the particle orbit, while the magnetic field that appear in the equations are calculated in the laboratory reference frame. The vector spin is treated as a 3-dimension real vector. spink uses L L ~ y n ~ h r ~ t r ~ n ’ ’ coordinates, defined as: x, radial, y, vertical, and z , longitudinal. 2 RF Dipole simulation In the COSY experiment that we wanted to simulate, an RF Dipole was used, that produced an horizontal oscillating magnetic field at a location of the beam trajectory. The RF field gives a vertical periodic kick to the orbit and produces a proportional kick to the deuteron spin. The orbit kick modulates the vertical betatron motion of the beam. When the frequency of modulation of the orbit and the frequency of spin precession (spin tune) match through the condition where fRF is the frequency of the RF, fc the frequency of circulation of the beam in the ring, Gy the spin tune, or the relative frequency of spin precession in the machine, and k is an integer, the spin flips from its direction (originally vertical) to a direction closer to the horizontal. To obtain this flip, the RF of the Dipole was linearly varied in time, while the momentum of the deuteron beam was kept constant, until the condition (3) was met, with k = 1.