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The figure-8-shaped ion collider ring of Jefferson Lab Electron-Ion Collider (JLEIC) is transparent to the spin. It allows one to preserve proton and deuteron polarizations using weak stabilizing solenoids when accelerating the beam up to 100 GeV/c. When the stabilizing solenoids are introduced into the collider’s lattice, the particle spins precess about a spin field, which consists of the field induced by the stabilizing solenoids and the zero-integer spin resonance strength. During acceleration of the beam, the induced spin field is maintained constant while the resonance strength experiences significant changes in the regions of “interference peaks”. The beam polarization depends on the field ramp rate of the arc magnets. Its component along the spin field is preserved if acceleration is adiabatic. We present the results of our theoretical analysis and numerical modeling of the spin dynamics during acceleration of protons and deuterons in the JLEIC ion collider ring. We demonstrate high stability of the deuteron polarization in figure-8 accelerators. We analyze a change in the beam polarization when crossing the transition energy. PRESERVATION OF ION POLARIZATION IN FIGURE-8 ACCELERATORS A characteristic feature of JLEIC [1] is its figure-8shaped rings [2]. Such a ring topology is transparent to the spin: the combined effect of arc fields on the spin in an ideal collider lattice reduces to zero after one particle turn on the design orbit, i.e. any orientation of the particle spin at any orbital location repeats from turn to turn. To preserve the polarizations of the proton and deuteron beams during acceleration from 8 GeV/c to 100 GeV/c in the ion collider ring, it is sufficient to use a weak solenoid with a field integral of 1.2 Tm, which does not perturb the design orbit and has practically no effect on the beam’s orbital parameters [3-10]. The solenoid then stabilizes longitudinal spin polarization at its location. A solenoid with the indicated field integral allows one to induce a spin tune  of 10-2 for protons and 310-3 for deuterons, i.e., when a particle with a vertical spin makes one turn on the design orbit, its spin tilts by an angle of  from its initial orientation. For polarization stability, one must ensure that the spin tune  induced by the solenoid significantly exceeds [3-6] the strength of the zero-integer spin resonance :  ≫ . The resonance strength is the average spin field ⃗ (the zero-integer Fourier harmonic of the spin perturbation without a stabilizing solenoid) determined by deviation of the trajectory from the design orbit due to machine element errors and beam emittances. In the absence of a solenoid, the spin precesses by an angle of  about the ⃗ direction in one particle turn. The resonance strength consists of two parts: a coherent part arising due to additional transverse and longitudinal fields on a trajectory deviating from the design orbit and an incoherent part associated with the particles’ betatron and synchrotron oscillations (beam emittances) [8, 9] ⃗ = ⃗ h + ⃗  , h ≫  In practice, the coherent part h significantly exceeds the incoherent one  . The coherent part does not cause beam depolarization and only results in a simultaneous rotation of the polarization about the field determined by the strength and alignment errors of collider elements. In principle, the direction and size of the coherent part of the resonance strength can be measured and taken into account for polarization control. To preserve the polarization, it is then sufficient to satisfy a weaker condition:  ≫  . CALCULATION OF ZERO-INTEGER SPIN RESONANCE STRENGTH IN JLEIC Figure 1 shows  functions of the JLEIC collider lattice in the acceleration mode [11] used in our spin dynamics calculations. The origin of the coordinate frame is located at the collider's IP. Figure 1 also indicates the location of the solenoid stabilizing the spin during acceleration. The difference from the collision mode [12] where functions in the IP region reach 2.5 km is that, in the acceleration mode, the functions in the detector section do not exceed 150 m. Figure 1: functions of the ion collider ring.  * Authored by Jefferson Science Associates, LLC under U.S. DOE Contracts No. DE-AC05-06OR23177 and DE-AC02-06CH11357. The U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce this manuscript for U.S. Government purposes. WEPIK038 Proceedings of IPAC2017, Copenhagen, Denmark ISBN 978-3-95450-182-3 3014 Co py rig ht © 20 17 CC -B Y3. 0 an d by th er es pe ct iv ea ut ho rs 01 Circular and Linear Colliders A19 Electron-Hadron Colliders Incoherent Part of the Spin Resonance Strength Figure 2 shows the incoherent part of the proton resonance strength for normalized emittance values of 1 mmmrad in both radial and vertical directions. As we can see, the value of the incoherent part does not exceed 210-4 practically in the whole momentum range of the collider with the exception of narrow “interference” peaks where spin perturbations add up coherently in the arc magnets. The presented calculation confirms that the spin tune value of 10-2 induced by the solenoid field is sufficient to stabilize the spin with normalized emittances of the betatron motion equal to 1 mmmrad. Figure 2: Incoherent part of the proton resonance strength in the JLEIC ion collider ring. Figure 3 shows a similar graph of the dependence of the incoherent part of the deuteron resonance strength on momentum. Our calculations assumed that the transverse size of the deuteron beam was equal to the proton beam size, i.e. the transverse beam emittances were 0.5 m. Figure 3: Incoherent part of the deuteron resonance strength in the JLEIC ion collider ring. In contrast to protons, due to the small value of the deuteron anomalous magnetic moment, deuterons have only one interference peak at the momentum of 93 GeV/c whose value does not exceed 210-8. Thus, a deuteron spin tune of 310-3 induced by a solenoid significantly exceeds the incoherent part of the resonance strength. Coherent Part of the Spin Resonance Strength Let us calculate the resonance strength for a proton beam using a model with random shifts of all quadrupoles in the transverse directions. Figure 4 shows the coherent part of the proton resonance strength in the ion collider ring with random quadrupole misalignments resulting in a transverse closed orbit distortion of about 100 m rms. The statistical model calculates the most probable magnitude of the coherent part of the resonance strength not specifying its direction, which lies in the collider’s plane. As in the case of the incoherent part, the coherent part of the resonance strength has interference peaks whose maximum values do not exceed 1.510-2, which has an order of magnitude comparable to the field induced by the stabilizing solenoid. Figure 5 shows a graph of the coherent part of the deuteron resonance strength calculated using the statistical model of random quadrupole misalignments. Figure 4: Coherent part of the proton resonance strength in the ion collider ring. Figure 5: Coherent part of the deuteron resonance strength in the ion collider ring. ACCELERATION OF POLARIZED IONS IN THE JLEIC COLLIDER The spin dynamics during acceleration in the ion collider ring is a precession about the spin field h⃗ , which consists of the field h⃗ sol induced by the stabilizing solenoid and the resonance strength ⃗ : h⃗ = h⃗ sol + ⃗ [8]. During acceleration the field h⃗ sol is maintained constant while the resonance strength ⃗  experiences significant changes in the regions of interference peaks. The beam polarization substantially depends on the field ramp rate in the arc magnets. When using superconducting magnets with a field ramp rate of ~3 T/min, acceleration happens adiabatically, which means that, in a characteristic time of change in the spin field, the spin makes a large number of turns. During adiabatic acceleration, the spin follows the h⃗ field direction, which can significantly deviate from the longitudinal direction at the locations of the interference peaks of the coherent part of the resonance strength. However, this does not signify polarization loss, the beam polarization restores its longitudinal direction in places where hsol ≫ h. Let us present calculations of the spin dynamics during acceleration of protons and deuterons in the JLEIC ion collider ring made using a spin tracking code Zgoubi [13]. Figure 6 shows the longitudinal spin components in the ion collider ring during acceleration of 3 protons with Δ/ = (green line), Δ/ = −3 (red line) and Δ/ = − −3 (blue line). As we can see, the graphs of the longitudinal spin components practically do not differ from each other (the red line covers up the blue and green lines), i.e. synchrotron energy modulation does not give a noticeable contribution to the ion spin motion when stabilizing the polarization by a weak solenoid in the JLEIC Proceedings of IPAC2017, Copenhagen, Denmark WEPIK038 01 Circular and Linear Colliders A19 Electron-Hadron Colliders ISBN 978-3-95450-182-3 3015 Co py rig ht © 20 17 CC -B Y3. 0 an d by th er es pe ct iv ea ut ho rs ion collider ring. All particles were launched with the same initial conditions: 0 = , 0 = . mm, 0′ = rad, 0 = . mm, 0′ = rad. The field ramp rate was ~3 T/min (the particles were accelerated in 8.3 million turns). During acceleration, the spin preserves its component along the spin field, which lies in the orbit plane and noticeably deviates from the longitudinal direction in the regions of the “interference” peaks of the coherent part of the resonance strength at momenta of about 60 GeV/c and 75 GeV/c, where the resonance strength becomes approximately equal to the size of the solenoid spin field. The spin tune induced by the solenoid during acceleration is 10-2. The simulation in Fig. 6 is do

Acceleration of polarized protons and deuterons in the ion collider ring of JLEIC