RHIC optics and spin dynamics with snakes and rotators

The operation of RHIC collider rings in polarized proton runs includes helical snakes, which allow for preserving polarization during acceleration to store energies. The RHIC lattice also includes spin rotators, operated when nonvertical polarization or corrections to the orientation of polarization at the interaction points are required. Utilization of OPERA field maps of snakes and rotators has been systematized in the past decade, in order to assess in detail the effects of these spin devices on beam polarization, and their perturbative effects on beam optics. The method is also used in ongoing studies regarding the future Electron Ion Collider, to permit increasing average store polarization to at least 70% at 275 GeV and the acceleration of polarized helion with low polarization losses. This paper reviews various applications and outcomes of these field map methods. It is thereby also a review of studies undertaken as part of beam polarization research activities at RHIC in recent years.

Figure 1

Schematic layout of RHIC polarized beam complex. RHIC consists of two rings, respectively, known as Blue and Yellow. Beams go clockwise (CW) in the former, and counterclockwise (CCW) in the latter. Snakes are located at 9 o’clock and 3 o’clock, rotator pairs flank IP6 and IP8.

Figure 2

$\mathrm{R}+\mathrm{R}-\mathrm{R}+\mathrm{R}-$ and $\mathrm{R}-\mathrm{R}+\mathrm{R}-\mathrm{R}+$ four-helix snakes in RHIC Blue and Yellow rings, at 9 o’clock and 3 o’clock. The beam goes clockwise in Blue, counterclockwise in Yellow.

Figure 3

A sketch of RHIC snake and rotator layouts, each rectangle represents a 2.4 m long 360°-twist helix. Field maps from the OPERA model extend an extra 0.764 m at both ends, to encompass field falloffs.

Figure 4

OPERA model of RHIC four-helix snakes and rotators.

Figure 5

Numerical data used for the normalization of OPERA currents to real-life currents. Blue triangles: from field measurements in a prototype 360°-twist helical dipole. Red circle: low-field helix map, case of 100 A OPERA current; Blue circle: high field helix map, case of 322 A OPERA current. Black squares: a case of a 12 m extent OPERA model of a complete snake computed with $(I_{\mathrm{out}},I_{\mathrm{in}})=(100,322\mathrm{A})$ [12].

Figure 6

Field along the helical trajectory arc across a $\mathrm{R}+$ snake helix (using its OPERA field map, current $I_{\mathrm{in}}=322\mathrm{A}$, $B_0=4.07\mathrm{T}$ positive, $k>0$). Two cases are displayed: $x_0=23\mathrm{mm}$ or $y_0=23\mathrm{mm}$, all other coordinates are zero: the transverse field components are verified to be essentially independent of $x(s)$ and $y(s)$: $B_x\approx -B_0\sin (ks)$, $B_y\approx B_0\cos (ks)$ (Eq. (3), with yet a noticeable effect of field falloffs at helix ends.

Figure 7

Transverse projection of the helical motion across an $\mathrm{R}+\mathrm{R}-\mathrm{R}+\mathrm{R}-$ snake series, using OPERA field maps. A smaller excursion is in the outer, low-field helices. Solid lines are from motion integration through field maps, at 23 and 100 GeV and, very close to each other, 250 and 300 GeV; markers at 23 and 250 GeV are from theory [Eq. (5)]. RHIC closed orbit goes counterclockwise from the top of a smaller helix, “A” for instance at 23 GeV.

Figure 8

Evolution of the spin components along the helical trajectories of Fig. 7 (from vertical initial spin), at 23 and 250 GeV. It can be observed that spin motion across the snake is essentially independent of energy, as expected: except for the 3–7 m region where the 23- and 250-GeV $S_y$ components do not fully coincide, the $\overrightarrow{S}(s)$ components at both energies essentially superimpose at this scale. The small back-forth segments at helix ends are field map manipulation artifacts.

Figure 9

Energy dependence of $\mathrm{R}-\mathrm{R}+\mathrm{R}-\mathrm{R}+$ (index 1) and $\mathrm{R}+\mathrm{R}-\mathrm{R}+\mathrm{R}-$ (index 2) snake series angles, over [23.8, 300] GeV (the solid lines are to guide the eye). The OPERA model currents for precisely $\mu =180°$ and $\varphi =45°$ at 23.8 GeV ($G\gamma =45.5$) are $I_{\mathrm{out}}=93.35\mathrm{A}$, $I_{\mathrm{in}}=321.87\mathrm{A}$.

Figure 10

For reference, optical functions in RHIC Run 22 Yellow lattice injection (top) and 255 GeV (bottom) nominal models, prior to any insertion of snake or rotator OPERA field maps and their local centering bumps. The small vertical dispersion at 255 GeV results from vertical orbit separation bumps at the six interaction points (Sec. 3b). Origin of $s$ is at IP6 (STAR detector), $s$ increases going CCW.

Figure 11

Closed orbit (top) and optical functions (bottom) in RHIC Yellow ring lattice model at injection energy ($G\gamma =45.5$), with snake lattice elements represented by their OPERA field maps and including the local centering bumps.

Figure 12

Local horizontal (red) and vertical (blue) closed orbit bumps in the 3 o’clock snake region of the RHIC ring, using OPERA field maps. The 9 o’clock bump is similar, with reversed signs.

Figure 13

Spin ${\overrightarrow{n}}_0$ vector around Yellow ring and tilt angle. No snakes, here, $G\gamma =45.5$. Two trivial cases for reference, both yielding vertical spin closed orbit: on one hand bare, perfect, Yellow ring, flat orbit, and on the other hand, with 90 m extent vertical orbit bumps set in snake regions.

Figure 14

Spin ${\overrightarrow{n}}_0$ vector around Yellow ring and tilt angle. Case of snake field maps offset, thus no orbit bumps, with helix currents $I_{\mathrm{out}}=93.35\mathrm{A}$, $I_{\mathrm{in}}=321.87\mathrm{A}$. Spin closed orbit is vertical around the ring.

Figure 15

Spin ${\overrightarrow{n}}_0$ vector around Yellow ring and tilt angle. Regular polarized proton injection optics, snake field maps, and orbit bumps in place, helix currents as in Fig. 15. The spin tilt to vertical reaches 3.5° in the arcs.

Figure 16

Top: typical closed orbit with vertical separation bumps at IPs, in RHIC Yellow lattice nominal lattice, accounting for snake fields, at store (255 GeV); the vertical bar locates the p-C polarimeter, at $s=1846\mathrm{m}$ from IP6, CCW. Bottom: transverse projection of helical orbits in snakes.

Figure 17

Nominal RHIC Yellow ring settings, at store. The graph displays spin ${\overrightarrow{n}}_0$ components and tilt to vertical. The latter is about 2° around the ring, with up to 18° spikes at the fall or rise of the vertical orbit separation bumps. The vertical bar at $s=1850\mathrm{m}$ materializes the location of the p-C polarimeter, 1846 m from IP6, CCW, ${\overrightarrow{n}}_0$ tilt to vertical there is $\approx 3°$. Snake field maps are used here.

Figure 18

RHIC Blue ring with 9 o’clock (two-helix, 90%) and the 3 o’clock (four-helix, 91%) partial snakes. The graphs show the spin ${\overrightarrow{n}}_0$ components and tilt to vertical around the ring at injection (top) and store (bottom) energies. The 9 o’clock snake OPERA map current is $I_{2-\mathrm{hlx}}=322\mathrm{A}$, the 3 o’clock snake currents are $I_{\mathrm{out}}=130\mathrm{A}$, $I_{\mathrm{in}}=322\mathrm{A}$.

Figure 19

Energy dependence of ${\overrightarrow{n}}_0$ components (left axis) and ${\overrightarrow{n}}_0$ angle to the y axis (right axis) at STAR (top) and at p-C polarimeter (bottom); $n_{\mathrm{xy}}$ denotes the modulus of the projection of ${\overrightarrow{n}}_0$ in the ($x,y$) plane, which is the vector actually measured at the p-C polarimeter [14]. The right vertical bar at $G\gamma =487$ materializes the region of nominal operation with full snakes. The left vertical bar at $E=254.2\mathrm{GeV}$ was the optimal energy found from measurements, for operation with partial snake during RHIC Run 22, which this graph shows to be close to the expected energy value for a zero-tilt to vertical. Snake field maps are used for this energy scan.

Figure 20

Vertical local closed orbit bump in the two-helix snake region, at injection, using the helix field maps.

Figure 21

Transverse projection of the helical motion across a two-helix snake field map, at various energies.

Figure 22

Transmission of ${\overrightarrow{n}}_0$ from AGS H10 extraction septum to RHIC injection kickers, over the range $G\gamma \in [42,50]$, from numerical simulation. The graph displays ${\overrightarrow{n}}_{0,y}(G\gamma )$ at three different locations: H10 and the RHIC Blue and Yellow ring kickers. In this simulation, the periodic spin precession axis ${\overrightarrow{n}}_0(G\gamma )$ at the extraction septum in the AGS ring is computed using the AGS partial snakes OPERA field maps [15].

Figure 23

Turn-by-turn $⟨S_y⟩$ (left axis), $⟨S_x⟩$ and $⟨S_l⟩$ (right axis), an average over 1000 particles, during the crossing of the intrinsic resonance $G\gamma =393+{\nu }_y$, at a rate of $200\mathrm{kV}/\mathrm{turn}$. Observation point is IP6. All initial spins are vertical, whereas ${\overrightarrow{n}}_0$ is not. Top: a case of 90% 9 o’clock snake (two helices) and 91% 3 o’clock snake (four helices); nonzero $⟨⟨S_x⟩⟩$ and $⟨⟨S_l⟩⟩$ turn averages reflect ${\overrightarrow{n}}_0$ tilt. Bottom: both snakes are full snakes (four helices).

Figure 24

Rotator pairs at IP6 in RHIC.

Figure 25

Field along RHIC closed orbit across a 360°-twist rotator helix. Top: outer “$\mathrm{L}+$” left-handed helix ($k<0$, $B_0>0$). Bottom: inner “$\mathrm{R}+$” right-handed helix ($k>0$, $B_0>0$).

Figure 26

Rotation of an initially vertical spin, along the closed orbit across a 360°-twist rotator helix. Top: $\mathrm{L}+$ left-handed helix. Bottom: $\mathrm{R}+$ right-handed helix.

Figure 27

RHIC Yellow ring, 255 GeV. This lattice includes the 16 OPERA field maps of the 9 o’clock and 3 o’clock snakes, and STAR rotators, the latter set for rotation of spin ${\overrightarrow{n}}_0$ to longitudinal at IP6. Origin of $s$ is at IP6, location of the STAR detector. Top: closed orbit excited when introducing the 16 field maps in the lattice, prior to correction. Bottom: after correction, based on local closed orbit bumps at both rotators and both snakes.

Figure 28

Local horizontal (red) and vertical (blue) closed orbit bumps in the IP6 (STAR) rotators in the present simulations, using OPERA field maps. IP6 (STAR) is at $s=0$, increasing $s$ is CCW. Top: $\mathrm{L}+\mathrm{R}+\mathrm{L}+\mathrm{R}+$ rotator, upstream of IP6, moves spin ${\overrightarrow{n}}_0$ from vertical in the arc to longitudinal at IP6. Bottom: $\mathrm{L}-\mathrm{R}-\mathrm{L}-\mathrm{R}-$ rotator downstream of IP6, moves spin ${\overrightarrow{n}}_0$ from longitudinal at IP6 to vertical in the arc.

Figure 29

Top: Polarization (${\overrightarrow{n}}_0$ vector components, left axis) around RHIC Yellow ring, and ${\overrightarrow{n}}_0$ tilt angle to vertical (right axis). Run 22 store optics (255 GeV), the lattice includes the 16 snake and rotator field maps. The ${\overrightarrow{n}}_0$ tilt spikes are located at the vertical orbit separation bumps at IPs. Bottom: details of ${\overrightarrow{n}}_0$ components in IR6 region, from vertical spin at YI6_B3 rotator entrance (at $s=0$ in this graph) to longitudinal at IP6 ($s\approx 77\mathrm{m}$).

Figure 30

Along RHIC Yellow 9 o’clock snake, case of $I_{\mathrm{out}}=185$ A, $I_{\mathrm{in}}=199$ A: $\varphi =-10°$ and $\mu =180°$. Left: transverse projection of the helical motion. Right: spin components, case of initially vertical spin (the small back-forth segments are field map artifacts).

Figure 31

Along RHIC Yellow 3 o’clock snake, case of $I_{\mathrm{out}}=164\mathrm{A}$, $I_{\mathrm{in}}=221\mathrm{A}$: $\varphi =0°$ and $\mu =180°$. Left: transverse projection of the helical motion. Right: spin components, case of initially vertical spin.

Figure 32

Required ramp of snake angles $\mu$ and $\varphi$, for rotation of RHIC Yellow ring polarization from vertical to radial; OPERA currents necessary for that, obtained by constraining the snake spin matrix; resulting closed orbit coordinate at snake entrance; and monitoring the convergence of this fit: fit penalty (right vertical axis). Top: 9 o’clock snake; bottom: 3 o’clock snake. The right axis in these graphs gives the penalty function values, other quantities use the left axis.

Figure 33

Components of the stable spin precession direction, and spin tune, over the snake power supply ramp. Top: at IP6; bottom: at the p-C polarimeter, location of the polarization measurement in this experiment. Data were obtained from the one-turn spin matrix.

Figure 34

Vertical closed orbits along 9 o’clock (top) and 3 o’clock (bottom) snakes, sampled over the helix current scan from vertical to radial polarization.

Figure 35

Sample particle motion in longitudinal phase space over an initial 5000 turns spin and longitudinal stabilization phase, followed by 10,000 turn snake supplies ramp. Bunch length is about RHIC nominal 25 ns, momentum excursion is $\pm {10}^{-3}$.

Figure 36

Polarization rotation from vertical to radial over 10,000 turns observed at p-C polarimeter (current ramp starts at turn 5000, materialized by the vertical bar). Dots: spin components of a few individual particles; superimposed solid lines: turn-by-turn average of 1000 particle spin components. Black solid line (right axis): departure from 1 of the bunch polarization (i.e., the average value of ${\overrightarrow{S}}_i·⟨\overrightarrow{S}⟩$, the projection of individual spins ${\overrightarrow{S}}_i$ on their average $⟨\overrightarrow{S}⟩$). Empty circle markers (left axis): spin tune ${\nu }_{\mathrm{sp}}$, times 10, from Fourier analysis.