Polarized ${{}^3\mathrm{He}}^{+2}$ ions in the Alternate Gradient Synchrotron to RHIC transfer line

The proposed electron-hadron collider (eRHIC) to be built at Brookhaven National Laboratory (BNL) will allow the collisions of 20 GeV polarized electrons with 250 GeV polarized protons, or $100\mathrm{GeV}/\mathrm{n}$ polarized ${{}^3\mathrm{He}}^{+2}$ ions, or other unpolarized ion species. The large value of the anomalous magnetic moment of the ${}^3\mathrm{He}$ nucleus $G_{\mathrm{He}}=(g-2)/2=-4.184$ (where $g$ is the $g$-factor of the ${}^3\mathrm{He}$ nuclear spin) combined with the peculiar layout of the transfer line which transports the beam bunches from the Alternate Gradient Synchrotron (AGS) to the Relativistic Heavy Ion Collider (RHIC) makes the transfer and injection of polarized ${}^3\mathrm{He}$ ions from AGS to RHIC (AtR) a special case as we explain in the paper. Specifically in this paper we calculate the stable spin direction of a polarized ${}^3\mathrm{He}$ beam at the exit of the AtR line which is also the injection point of RHIC, and we discuss a simple modifications of the AtR beam-transfer-line, to perfectly match the stable spin direction of the injected polarized ${}^3\mathrm{He}$ beam to that of the circulating beam, at the injection point of RHIC.

Figure 1

The RHIC accelerator complex: The location of the AGS partial helices are shown at the A20 and E20 straight sections of the AGS. The AtR beam transfer line shown in the figure transports the beam from AGS to RHIC.

Figure 2

The first section of the AtR line where the beam is bent simultaneously along the vertical (blue rectangles) and horizontal (yellow rectangle) planes by 12.5 mrad and 15° respectively. The red or green rectangles represent proposed dipole magnets which can be placed at these locations along the AtR line to rotate the stable spin direction and match it to that at the RHIC injection point.

Figure 3

The second section of the AtR line where the beam is bent simultaneously along the vertical (blue rectangles) and horizontal (yellow rectangle) planes by 3.0 and 38.0 mrad respectively. The red rectangles represent proposed dipole magnets which can be placed at this location along the AtR line to rotate the stable spin direction and match it to that at the RHIC injection point.

Figure 4

The directional cosine $S_z$ of the stable spin direction at the extraction point of AGS (SSH10) as a function of $|G\gamma |$ in presence of two partial helices which are installed in the SSA20 and SSE20 straight sections and rotate the spin about the longitudinal axis by angles ${\delta }_{\text{ch}}$, and, ${\delta }_{\text{wh}}$ respectively.

Figure 5

The directional cosine $S_z$ of the stable spin direction at the AGS extraction point (blue trace) and at the RHIC injection point (red trace) as a function of $|G\gamma |$. There is considerable mismatch between the stable spin direction at the exit of the AtR line and that at the RHIC injection point for $|G\gamma |>51.5$.

Figure 6

The stable spin direction at the exit of the AtR line as a function of half integer $|G\gamma |$ values. The black circles and the red squares correspond to the $S_z$ values when the three dipoles at the location $♯1$ (green rectangles in Fig. 2) are not excited or excited respectively. The green diamonds is the integrated dipole field $\int B·d\ell /2.5$ at the location $♯1$.

Figure 7

The stable spin direction at the exit of the AtR line as a function of half integer $|G\gamma |$ values. The black circles correspond to the $S_z$ values when the three dipoles at the location $♯2$, shown in Fig. 2 as red rectangles, are set to zero and the red squares when the set of three dipoles are excited. The green diamonds is the integrated dipole field $\int B·d\ell /2.5$ at the location $♯2$.

Figure 8

The stable spin direction (red squares) at the exit of the AtR line as a function of half integer $|G\gamma |$ values when both sets of dipoles at the locations $♯1$ and $♯2$, (green and red rectangles shown in Fig. 2) are excited. The black circles correspond to the $S_z$ values when both sets of the three dipoles at the locations $♯1$ and $♯2$ shown in Fig. 2 are set to zero. The green diamonds and the blue triangles are the values of the integrated dipole field $\int B·d\ell /2.5$ for the dipoles at the locations $♯1$ and $♯2$ respectively.